Student Misconceptions About Newtonian Mechanics: Origins and Solutions Through Changes to Instruction Dissertation by Aaron

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Student Misconceptions about Newtonian Mechanics:

Origins and Solutions through Changes to Instruction

Dissertation

Presented in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Graduate School of The Ohio State University

Aaron Michael Adair, M.S.

Graduate Program in Physics

The Ohio State University

2013

Dissertation Committee:

Lei Bao, Advisor

Andrew Heckler

Gordon Aubrecht

Samir Mathur

Aaron Michael Adair

2013

ABSTRACT

In order for Physics Education Research (PER) to achieve its goals of significant learning

gains with efficient methods, it is necessary to figure out what are the sorts of pre-

existing issues that students have prior to instruction and then to create teaching methods

that are best able to overcome those problems. This makes it necessary to figure out what

is the nature of student physics misconceptions—prior beliefs that are both at variance to

Newtonian mechanics and also prevent a student from properly cognizing Newtonian

concepts. To understand the prior beliefs of students, it is necessary to uncover their

origins, which may allow instructors to take into account the sources for ideas of physics

that are contrary to Newtonian mechanics understanding. That form of instruction must

also induce the sorts of metacognitive processes that allow students to transition from

their previous conceptions to Newtonian ones, let alone towards those of modern physics.

In this paper, the notions of basic dynamics that are common among first-year college students are studied and compared with previous literature. In particular, an analysis of historical documents from antiquity up to the early modern period shows that these conceptions were rather widespread and consistent over thousands of years and in numerous cultural contexts. This is one of the only analyses in PER that considers the original languages of some of these texts, along with appropriate historical scholarship.

Based on the consistent appearance of these misconceptions, a test and interview module was devised to help elucidate the feelings students have that may relate to fictitious forces. The test looked at one-dimensional motion and forces. The first part of the

ii interview asked each student about their answers to the test questions, while the second part asked how students felt when undergoing three cases of constant acceleration in a car. We determined that students confabulated relative motion with the experience of

force; students claim to feel a force in the direction of relative motion even when the

actual force is in the opposite direction. The interview process also showed how students

had both their intuitive sense of physics as well as Newtonian concepts from instruction,

and how each model was activated could be influenced by questions from the

interviewer.

In order to investigate how changes to instructional method and pedagogy may

affect students’ ability to overcome their non-Newtonian intuitions, an experimental

lecturing series was devised that used individual voting machines (“clickers”) to increase

class participation and dialog in a fashion that was more student-centered. The

experimental section also had video recordings of the lectures as well as concept-based

video homework solutions. The initial availability of the videos hindered early use, and

overall students rarely used these additions. The clicker system also had technical issues

due to the volume of students and an interface that was not streamlined. Nonetheless, the

results showed the experimental section to have significantly greater learning gains (d >

0.5, p ~ 0.01), and we determined that this was most likely due to the clicker system.

iii

VITA

June 2003...... Bay City Western High School

2008...... B.S. Physics, Michigan State University

2008...... B.S. Astrophysics, Michigan State University

2008...... B.S. Mathematics, Michigan State University

2011...... M.S. Physics, The Ohio State University

2008 to present...... Graduate Teaching Associate, Department of

Physics, The Ohio State University

Publications

Adair, A. & Bao, L. (2012). Project-Based Learning: Theory, Impact, and Effective

Implementation. REAL: Research in Education, Assessment, and Learning 3(1),

6-21.

Adair, A. (2012). The Star of Christ in the Light of Astronomy. Zygon: Journal of

Science & Religion 47(1), 7-29.

Fields of Study

Major Field: Physics Education

TABLE OF CONTENTS

Abstract ...... ii Vita ...... iv List of Tables ...... viii List of Figures ...... ix Chapter 1. Introduction ...... 1 A. Physics Education Research—its Problems and Goals ...... 1 B. Lines of Investigation ...... 4 i. Learning Approaches ...... 4 ii. Student Misconceptions ...... 9 iii. Use of Technology in Curricula ...... 13 C. Purpose of this Thesis ...... 16 Chapter 2. Students Physics Misconceptions: Previous Research and Assessment Tools 18 A. Newtonian Misconceptions ...... 19 i. Forces and Linear Motion ...... 19 a. Modern Examples ...... 20 b. Historical Examples ...... 24 1. Aristotle’s Physics of Motion ...... 25 2. Ancient Thinkers on Motion after Aristotle...... 29 3. Medieval Islamicate Physicists ...... 32 4. Medieval European Physicists ...... 33 5. Physics of the Scientific Revolution ...... 36 6. Conclusions from History of Science ...... 39 ii. Circular Motion ...... 39 a. Modern Examples ...... 40 b. Historical Examples ...... 44 1. Circular Forces in Ancient and Medieval Commentary ...... 45 2. The Scientific Revolution and Circular Motion ...... 51

3. Conclusions from History ...... 57 iii. Sources of Misconceptions ...... 58 B. Assessment Tools ...... 60 i. Force Concept Inventory ...... 61 a. Creation and Assessment with Factor Analysis ...... 61 b. Model Analysis and FCI ...... 66 ii. Other Inventories ...... 73 Chapter 3. Student Physics Misconceptions: Tests and Interviews ...... 76 A. Previous Research and History of Science ...... 76 B. Research Questions ...... 79 C. Experimental Designs ...... 81 i. 1D Motion Test and Interview ...... 82 a. Multiple Choice Test ...... 82 b. 1D Force Question Interviews ...... 85 ii. Experience of Forces and Motion during Constant Acceleration ...... 87 a. Constant Acceleration in a car ...... 87 1. Car Undergoing Forward Acceleration ...... 88 2. Car Undergoing Violent Braking ...... 89 3. Car Undergoing Circular Motion at Constant Speed ...... 90 b. Data Collection of Feelings of Force ...... 90 1. Student Interpretation of Forces and Motion ...... 91 2. Deliberative and Quick-Thinking Students ...... 92 3. Methods of Analysis to Force and Motion Questions ...... 94 c. High School Experience ...... 98 D. Results ...... 99 i. 1D Multiple Choice Answers ...... 99 a. Pre- and Post-test Results ...... 99 ii. Experiences of Motion and Force under Constant Acceleration ...... 106 a. Forward Acceleration ...... 109 b. Backward Acceleration ...... 110 c. Circular Motion ...... 111 d. Correlations between Accelerated Motion Answers...... 112 e. Deliberative vs. Quick-Thinking Student Responses...... 116 iii. High School Physics Experience ...... 119

E. Discussion...... 120 i. Experiential Origins of Common Student Conceptions of Force and Motion .... 120 ii. Relationship between Some Force Misconceptions ...... 121 iii. Gut vs. Brain: Mental Models in Conflict ...... 127 a. Effects of Prompting by Instructor ...... 127 b. Mixed Models ...... 128 iv. Final Assessment ...... 129 v. Future Research ...... 130 Chapter 4. Technological Enhancements ...... 132 A. Previous Research ...... 132 B. In-Class Additions ...... 142 i. Demonstrations ...... 142 ii. Clickers ...... 144 C. Outside Additions ...... 147 i. Lecture Videos and Slides ...... 147 ii. Homework Solutions ...... 149 Chapter 5. Testing Methods and Results ...... 151 A. Standard Physics Class...... 151 B. Additional Lecture Materials ...... 151 C. Online Materials ...... 153 D. Assessment Methods ...... 154 E. Difficulties in Implementation ...... 157 i. Clickers ...... 157 ii. Homework Solutions ...... 160 F. Results...... 162 Chapter 6. Conclusions ...... 172 A. Summary of Findings ...... 172 i. Misconception Origins ...... 172 ii. Lecture Changes ...... 173 B. Future Goals ...... 178 Bibliography ...... 182 Appendix A: FCI Question categories ...... 189 Appendix B: The Case of Constant Acceleration, Gravity, and Abu’l-Barakat ...... 191

vii

LIST OF TABLES

Table 1. 1D Force Question Interview Process ...... 86 Table 2. Feelings of Force Interview Process ...... 92 Table 3. Student Answers to 1-Dimensional Force Questions. Correct answers have an asterisk (*)...... 100 Table 4. Forces Identified or Felt by Students under Constant Acceleration...... 108 Table 5. Contingency Matrix for Forward and Backward Linear Acceleration...... 114 Table 6. Contingency Matrix for Forces during Linear Acceleration and Circular Motion ...... 114 Table 7. Contingency Matrix for Forces during Forward Acceleration and Circular Motion ...... 114 Table 8. Contingency Matrix for Forces during Backward Acceleration and Circular Motion ...... 114 Table 9. Number of Deliberative and Quick-Thinking Student Answers to Acceleration Questions...... 117

viii

LIST OF FIGURES

Figure 1. Dropping a Ball with Constant Velocity. (A) Initial setup, (B) correct answer, (C) straight-down response, (D) backward-falling response. From McCloskey et al. (1983)...... 11 Figure 2. Thrown Objects with different Trajectories and Speeds. From Viennot (1979)...... 22 Figure 3. The Curved Tube Problem. The setup (A), the correct response (B), the incorrect response (C). From Kaiser et al. (1986)...... 43 Figure 4. Pseudo-Aristotle’s Idea of Objects Forced into the Center of Rotation. From Winter (2007)...... 47 Figure 5. Model Analysis Graph. From Bao & Redish (2006)...... 71 Figure 6. Flow Chart of Experimental Process ...... 93 Figure 7. Answer Transitions for 1D Force Questions with Pushed Sled ...... 103 Figure 8. Answer Transitions 1D Force Questions with Thrown Marble ...... 105 Figure 9. Normalized gains for each section and the control group average for physics taught in Fall 2012. Error bars represent the standard error...... 164 Figure 10. Model Analysis of Control Group...... 166 Figure 11. Model Analysis of Experimental Group...... 167 Figure 12. Correlation between Normalized Gains and Pretest Scores for the Experimental Group...... 168

CHAPTER 1. INTRODUCTION

A. Physics Education Research—its Problems and Goals

The field of physics has been successful in expanding our knowledge of the mechanics of

the universe at the most fundamental levels. From the workings of space and time to the

interactions of the smallest particles, the magnitude of detail that can be modeled using

current theories has been astounding. However, this effort has taken thousands of years of

interest in what was once called natural philosophy, the original name applied to the field of physics.

But with this advanced knowledge remains the baggage from the sorts of minds

that originally created the erroneous models of the world that lasted for 2000 years between Aristotle and Newton. The modern scientific understanding of the natural world has not permeated far beyond the papers and books of physicists. Even though the subject is taught at the college and the high school level in the United States, the Newtonian model of the everyday world has not become the dominant understanding by the majority of people whether they have had a physics course or not. This disturbing realization has led to treating the teaching of physics in a more scientific manner.

The beginnings of physics education research (PER) are found in the mid-20th

century, particularly in the work of Arnold Arons (1916-2001), but it came into its own

during the 1980s and 90s and was subsequently recognized by the American Physical

Society (APS) as a legitimate field of physics. The field has also made considerable

progress. Hake (1998) has shown that classrooms implementing PER methods had about

double the normalized gain in understanding the materials than “traditional” classes.

Examination of these sorts of methods fulfills one of the goals of PER: to increase

physics understanding. So far, this research among students has been focused at the college level, primarily because of the access to large student bodies by university researchers, but the goal applies to the K-12 curriculum as well. In addition to helping students understand the subject material, PER attempts to maintain this more accurate understanding of the physical world following the period of instruction. A decay of knowledge after classroom instruction has ended is expected, and this had effect has been measured (Heckler & Sayre 2010), but any utility of instruction is lost if the material has

been completely forgotten after several months.

To achieve the objectives of long-term understanding of physics, PER is led to two other major goals: to find ways to measure how much and how well the physics is

understood, and to identify and mitigate misconceptions that may interfere with student understanding and acceptance of the more accurate, scientific paradigm.

The term ‘misconception’ used here requires some explanation, as it has a controversial meaning in education literature. Different researchers disagree on the term’s meaning. As defined here, a physics misconception is a belief about the mechanics of the physical world that is 1) at variance with the scientific models (i.e., Newtonian mechanics) and 2) interferes with understanding the more accurate paradigm. While a misconception may be a reasonably good approximation of the world for day-to-day life, it will not work well when it requires deeper analysis and may instead lead to confusion and inconsistency. Rather than acting as a stepping stone or a scaffold to scientific

2 models, misconceptions (as defined above) are a distraction and hold the student back from comprehension. This should be sufficient reason for instructors to identify and overcome these prior, contra-Newtonian beliefs.

The task of identifying misconceptions has made significant progress. There have been several measuring tools used by PER investigators, some tailored for concepts in electromagnetism, others for thermodynamics, and many for Newtonian mechanics. For classical physics, the most commonly-used tool is the Force Concept Inventory (FCI), and several others are popular. These inventories are built to exploit the sorts of concepts that students misunderstand as distractors, and their details are discussed in Chapter 2.

This leads to the sorts of misconceptions that students bring with them before taking their first physics class. By the time most students have had their first day with a physics instructor, they have had decades of experience in the world, creating their own mental models of how the everyday world works as built up from interactions with objects as well as from social exchanges. Rarely do such interactions take place in the simplified and ideal settings of the laws of motion, leading to models of the world that do not simply conform to what is described by the forces laws of Newton or the concepts of energy and momentum. This means that instructors need to: categorize the common misconceptions, identify students who have a given misconception, and find an effective and efficient way to help the students overcome it and transcend to the more accurate understanding.

This last point leads into the goal of all education research to make instruction more effective: the creation of tools and techniques to better teach the materials. This comes in the form of new curricula and innovations in teaching.

B. Lines of Investigation

There are several avenues for exploration in PER. One is determining what general pedagogy to apply for instruction. Should traditional methods simply be modified with new curricula, or are there pedagogies that work better or have a better foundation in psychology and education? Another avenue is the identification of student misconceptions as defined above. Is there a systematic problem, such as a complete but contrary model that students apply, or is the situation more complex and context- dependent? There is also the question of where these misconceptions come from, especially because they can in many cases be clearly at variance to Newtonian explanations that describe the world of experience far better than the misconceptions that physics teachers hope to defeat. Another point of research concerns what tools to use in the classroom, especially when it comes to technological additions. What sorts of tools can new technology provide to help teachers teach and learners learn?

i. Learning Approaches

The most common way to instruct in academic settings is also considered the most traditional method. The instructor presents information to the students, usually in the form of lectures or book readings, and the students are expected to “absorb” the knowledge bequeathed to them. In conjunction with this sort of instruction are homework problems that apply the lectured-over materials, and later there is an examination with problems of similar design as found in the homework. In physics, homework problems

tend to be calculation exercises requiring the student to find the appropriate equation and use the correct mathematical manipulations to get the desired answer.

While this sort of method has been used for generations, it is not based on the best theoretical insights and phenomenological observations in educational theory. Instead, one should start creating a teaching method from what is known from investigations into the theoretical underpinnings of instructional pedagogies.

The predominant theory of knowledge explored in the field of education is that of constructivism. This theory treats the mind of the student not as a blank slate or a computer drive that can be wiped clean with new information. Instead, constructivism takes into account the fact that a student already has a collection of ideas and experiences that will interact with the new ideas and experiences the instructor provides. Knowledge in this theory is not simply information but something created by the learner; it is a construct that organizes the accepted ideas and experiences into something that can maintain coherence. “Intelligence organizes the world by organizing itself” (Piaget 1937).

The origins of constructivism are often said to have started with the philosopher and psychologist John Dewey in the early 20th century in the United States, but its most lasting form was born out of the work of the developmental psychologist Jean Piaget from Switzerland in the mid-20th century. In Piaget’s view, under the label of genetic

epistemology, the learner goes through stages, especially notable for PER being the move

from a concrete phase in which a particular situation is understood, to a formal

operational stage in which the knowledge can be transferred to new and different

scenarios. The movement to more abstract and more advanced understandings of a

concept is explained by the idea of the zone of proximal development (ZPD) as

established by Vygotsky (1935). The ZPD defines what learners can do on their own,

what the learners can understand with guidance (within the ZPD), and what the learners cannot understand at their current state of development (outside the ZPD).

The task of instructors is no longer simply to relay information, but to help the students in the creation of their own knowledge constructs. The principle of this support is called scaffolding, and it holds to the metaphor of building construction; the scaffolds

are there to help in the initial construction and are slowly taken away as the new construct

can support itself. This means a teacher needs to know what sorts of instruction to

provide to a student and how to move away to allow the students to build their knowledge

construct more and more on their own. It is the tailoring of scaffolds that allows the

students to expand their ZPD and move on to new concepts, continuing to integrate them

with the old. The expansion of the ZPD through scaffolding requires that a task not

simply have the students regurgitate some absorbed piece of factual knowledge, but

rather the students must think about how to apply their understanding to some challenge.

The process of thinking about what they know, how to apply it to a challenge, and how

their thinking may be mistaken, will fall under the heading of metacognition, that is,

thinking about thinking. Having the students reflect on their challenge and how they

approach it is at the core to helping the students build a new knowledge construct as well

as tear down a prior construct that proved inapplicable or incorrect. Under

constructivism, learning takes place because the instructor provides metacognitive

activities; contrast this with the idea of learning as recalling information.

Of course, in order to have metacognition there must already be knowledge upon

which to reflect, so the teacher must also consider how much the students already know

and how much they can absorb in a given sitting. As students will rely on their previous

experiences without having to think actively about them, similarly, they need to be able

to have recall of factual knowledge to form those constructs that teachers are trying to

help them form. There is the need to provide long-term memory of factual knowledge, freeing up the mental processes for reflection. The limit to how much information a student can process at once is known as working memory, and research indicates that students can only think about several ideas at once (Cowan 2005). Being able to hold only three or four new ideas or facts in our minds at once means that minds must rely on previously digested knowledge and experiences, as working memory is simply too limited. Previously established patterns free up the mind. This can be seen in the following example. Try to hold in your memory the following string of sixteen numbers:

385 947 84 90 482 910.

It is unlikely that looking at this string for ten seconds will make it memorable. Now try

to hold the following string of sixteen letters:

The cat is on the mat.

The latter string is very easy to remember because the mind is able to recall past

experiences with the words the letters create, and the rhyming scheme also simplifies the

process of remembering (cf. Hirsch 1987). Similar principles apply for physics

knowledge, so one already must have a sense of force, energy, coordinate systems,

7 vectors, and so on, before being able to form knowledge constructs and apply them with anything like expert ability. The teacher has a dual role: provide the necessary factual information for the students to remember, and induce student reflection on this information in order to create deep knowledge and be able to transfer said knowledge into new and different contexts.

When it comes to inducing metacognition, a process of questioning is desirable. It is through inquiry that one thinks about a problem and how to solve it. Part of that process is figuring out what you need to know in order to reach a resolution and figuring out how to apply that information to the problem. In effect, the inquiry-based approach to learning turns students into scientists as they research their problem and test various ways at resolving it.

There are several approaches to inquiry-based learning that have been described as confirmation inquiry, structured inquiry, guided inquiry, and open inquiry (Banchi &

Bell 2008). Confirmation inquiry has the students reach a conclusion they already expected in advance, reinforcing the idea or practicing another skill such as data collection and error analysis. Structured inquiry provides the question to answer as well as the method for reaching the conclusion. These first two forms may well describe most physics labs done at the undergraduate level. Guided inquiry, on the other hand, provides the research question, but the method is to be decided by the participants. Lastly, open inquiry has the student or students form their own questions and methods to answer them.

In creating a curriculum, an instructor has to decide what sorts of inquiry processes are needed. Some tasks may be best done as an act of confirmation, while in other cases an open project is called for. As for how to implement such tasks, an

instructor has several options. The most common and closely-related pedagogies are problem-based and project-based learning. In both cases, an inquiry-based approach is

used to apply previous and knowledge in order to solve a problem and to further students’

understanding of the concepts involved. The project-based approach differs primarily in

that there is a final, physical artifact that a student or group of students provides at the end, while the problem-based approach does not (Adair & Bao 2012). Though these two

pedagogies unfortunately have the same acronym, PBL is used here to describe project-

based learning.

For a PBL or any sort of inquiry assignment to provide the desired results, the task ought to be within the ZPD of the students, apply the knowledge that ought to be acquired in a given class, and provide sufficient challenge in areas that students may enter with misconceptions. This last point leads to figuring out what are the misconceptions.

ii. Student Misconceptions

As noted above, when a student enters into a physics class for the first time, he or she will

already have some idea of what will happen in a given scenario with macroscopic

objects. By the late teen years, a person will not need to be told that an object thrown up

and forward will come back down and have traveled forward some distance. However,

these ideas can be, in the important details, at variance with the models provided by the

instructor, and they can be persistent. Again, these ideas will have had many years to be

codified and demonstrated to students in their life experiences, so a few hours a week for

perhaps only a single week on a certain topic will likely be insufficient to change the

students’ conceptions. Moreover, once class is done, the student may find the new model

of the world alien to their experience, so a reversion to the old paradigm happens.

One important feature of the misconceptions that physics instructors encounter is

that they are not random. A population of students will likely have the same or similar

misunderstandings, which makes it possible to create diagnostic tests such as FCI to

examine what misconceptions are likely in play. The greatest amount of research has

taken place concerning misconceptions in mechanics, while topics in electromagnetism

have received about half as much attention in the PER literature (Duit 2005). On the one

hand, there is the strangeness of the number and stubbornness of misconceptions about

Newtonian mechanics. They are at the level of daily interaction and apparent to the eye of

the observer without special tools, unlike misconceptions in, say, electromagnetism or

special relativity. On the other hand, the great number of experiences that students have

that involved numerous sources of force makes it difficult to overcome misconceptions.

The student has already contextualized years of experience, unlike the lack of experience with electromagnetism. Thus, it is difficult to provide everyday scenarios that relate to the simple situations described by mechanics problems in a physics course.

When it comes to the sorts of misconceptions students have about the mechanical world, one collection of results suggested that students had a view of the world like that of Aristotle and the medieval natural philosophers prior to Newton (McCloskey et al.

1983). A prime example concerns the path of a ball dropped by a person moving at constant speed (see Figure 1). If you were carrying a ball in your hand while steadily walking forward, would you expect the ball land in front of your feet, behind, or at your feet after you let go without throwing the ball? About half of students believed the ball

Figure 1. Dropping a Ball with Constant Velocity. (A) Initial setup, (B) correct answer, (C)

straight-down response, (D) backward-falling response. From McCloskey et al. (1983). would fall behind, fitting with the Aristotelian idea that an object begins to stop moving forward when that object is no longer pushed. This feeds into the medieval idea of

impetus which later lost sway to Newton’s laws and the principle of inertia (see Section

2.A for more discussion).

However, it is probably inaccurate to believe that people will have a unified, complete and coherent theory of motion. Rather, it may be the case that people develop

heuristics that work in a given situation, and their application has so far not led to serious cognitive dissonance. diSessa (1993) proposed the idea of phenomenological primitives

or p-prims, a collection of physical ideas that someone will rely on, which may not

cohere well but generally will provide the needed heuristic to understand and predict the

motion of an object. These p-prims are not provided by formal instruction; they explain

an observed phenomenon, and when asked to explain them that person is likely to have

difficulty reducing them to some other, more primitive—that is, fundamental—idea. In other words, the p-prims are like axioms that are not reliant on other propositions, and

that axiomatic, self-evident nature of p-prims allows them to explain apparently so much of experience, and makes them persistently held even in the face of contradictory information.

Of these misconceptions, many have been identified with respect to forces.

Perhaps one of the most commonly held beliefs about force and motion is the necessity of a continuing force in order to maintain motion. This is seen in students’ belief that there is a force in the direction of motion; when there is a constant velocity, there is a tendency to believe that there is also a force in that direction, even after college physics instruction

(Viennot 1979; Clement 1982; Hestenes et al. 1992; Planinic et al. 2010). This notion is found in the physics of Aristotle, and it was used by the astronomer Claudius Ptolemy to argue against dynamic geocentrism and heliocentrism because of the belief in a large

force to keep the Earth rotating and there would be an incredible force acting on the

planet’s inhabitants (Toomer 1998; Taub 1993). As such, this fictitious force has had a

significant impact on the history and development of science, and it still affects students

today.

While force in the direction of motion is hardly the only common misconceptions

that students are likely to have, it receives the greatest amount of attention here. When

analyzing student misconceptions during interviews with students (see Sections 2.B, D), this was sought for in student responses. In particular, we investigated for reasons that students tend to have this force misconception.

iii. Use of Technology in Curricula

Knowing the existence and prevalence of the conceptual issue discussed above permits

educators to consider how best to overcome them and impart a more scientific view.

Along with the considerations of pedagogy, there are the tools that can help to fulfill

teaching goals that can be fitted to the method an instructor desires. In particular, various

technologies have been created to try to allow instructors to be more productive and

perhaps fit their instructions into the best supported educational pedagogies.

Some of the technological changes have taken place in the lecture room. For

example, films or projected images have been used to describe or demonstrate some

phenomenon or concept. A video clip may show what happens in a collision. A film

could become a lecture in itself with all the information provided in a visually and

audibly more impressive fashion than what a single professor can do in from of a class.

These changes do not provide for a revolution in pedagogy but rather a change in the

efficiency and (hopefully) effectiveness in teacher-centered instruction.

There also exist technologies to try and make the lecture period more student-

centered or provide more active engagement for the students. Audience response systems

or “clickers” have found wide usage in various university contexts. Clicker systems can

allow for students to think about some given problem or question in lecture, and their

answers can provide nearly instant feedback to the instructor about the state of the class.

Perhaps this system can change the dynamics of the 100+ student lecture experience.

Beyond the lecture hall, there have been attempts to make the homework system

take on an electronic form. Having students do homework in the traditional way has

proved to be ineffective in helping students overcome misconceptions. The solution is not

more problems; research shows that even doing 1000 traditional physics problems does

not help (Kim & Pak 2002). But an electronic version of homework has the potential to

change this, among other advantages. Grading now becomes less of a burden on

instructors and teaching assistants, while students can receive instant feedback about their

answers. More importantly, changes to the way homework is done have the potential to

make them into tutoring software rather than just answer recording devices. If a student

provides a wrong answer, appropriate responses can be printed to help the students

realize where they may be mistaken or what concepts they are failing to consider.

Homework can move from a series of “plug ’n’ chug” problems to an opportunity for

metacognition.

Technology also has the power to provide other individual-level assistance. Now

that computers have become portable, widespread, and relatively inexpensive, their use

has become commonplace in laboratory settings and supplemental materials. Simulations

also have the power to provide students an actual experience to give them the ability to

overcome their preconceived ideas about a variety of physical phenomena and concepts.

For example, Zhou et al. (2011) were able to use a virtual environment of a ball with a

force acting upon it with a joystick to create circular motion; a student would have to

apply the force with the joystick and try to move the ball (with some initial velocity) in a

circle. Circular motion and centripetal force are topics with several misconceptions

attached to them, so it is particularly difficult to change students’ minds with traditional

lectures and homework. Compared to a group of student with problem-solving activities

as a control, the subjects that used the simulation had learning gains with an effect size

about double that of the control group. This and similar results from other studies suggest

that a wider use of this sort of technology can become a great boon to instructors.

However, the largest component of modern physics instruction continues to be the

lecture. Thus, if technology can prove to be of help in other contexts of the learning

process, perhaps, when used with effective pedagogy, new technological additions to

lecture can increase the effectiveness of the instructor. The various new tools have their

particular uses, some of them able to fit into both traditional, teacher-centered instruction as well as student-centered instruction; lectures can be done in the traditional fashion, just with the material more efficiently presented, or they can be made more interactive with student participation with the use of, for example, clickers. The nature of these innovations is considered Chapter 4. Considerable work has gone into this subject, especially because it has been shown that just introducing technology into the classroom has not been sufficient in helping increase student learning (Polman 2000; Mayer 2009).

C. Purpose of this Thesis

Considering the prevalence of the student misconceptions that instructors wish to

overcome, this research thesis has several goals. One is to investigate the origins of some

of these misconceptions, primitive ideas that students have before instruction, which is

discussed in Chapter 2. This chapter also looks at the consistency of those physics

misconceptions, both culturally and historically, as seen in previous research and records.

Those misconceptions are reproduced when testing batches of students for the current experiments. In Section 3.C, the interview process with the tested students is described,

especially how individuals reported they feel under various circumstances of constant

acceleration. These interviews help to elucidate the possible sources of student belief in

fictitious forces. Chapter 3 looks into some of the reasons why these beliefs seem to be

persistent even with students who had prior instruction in physics. Furthermore, the sorts

of real-world experiences that students point to when describing their beliefs about the

world are analyzed.

In Chapter 4, technological enhancements to the classroom are described,

especially those that are used in the experimental section, along with other innovations.

Those innovations include not only additions to the classroom, but this paper also

considers changes to learning outside of the classroom and standard homework, either

paper-based or electronic.

In Chapter 5, an experimental section of physics is described that used innovative

learning tools. The lectures have additional classroom technologies not used in a student-

centered way in the control groups, and online additions that affect the students in the

16 experimental group—access to materials designed to help explain the physics concepts with respect to homework problems. The implementation of these tools is described, as well as difficulties encountered, and the effect of the experimental setup compared to the control group are all described.

Chapter 6 concludes with an analysis of how effective those tools proved to be compared to lecture sections that did not use them. In part, the investigation determines how much students used these additional learning tools and compares that condition to students who used little or no additional learning tools provided by the technological supplements. The possible future of such research is also described, based on the results from those technological enhancements.

CHAPTER 2. STUDENTS PHYSICS MISCONCEPTIONS: PREVIOUS RESEARCH

AND ASSESSMENT TOOLS

As has been seen across time and culture, there are ideas about the physical world that

people build up in their lifetimes that are at variance with the best understandings of

physics today or to the most useful approximations of those physical theories. While

Newtonian physics will significantly diverge from the predictions of special relativity and

quantum mechanics in certain cases, it applies so well to the world of normal human

experience that classical mechanics continues to be of high utility. Strangely, while this approximation breaks down only in realms outside of direct human experience, it is not the model that most adults have about the world. Instead, their model is contrary even to classical physics.

The ideas that people have are not random. Instead, it seems various heuristics are developed, which seem to work well enough to allow for functionality in the world with minimal difficulty. Moreover, there appears to be a continuity of these heuristics between different times in history, even for different cultures. If the goal is to bring students into at least the classical understanding of forces, motion, and energy, it is necessary to know what the misconceptions are and how pervasive they are. Part A of this section explores the research into the prevalence of some of these misconceptions, namely linear and circular impetus concepts. It provides historical evidence that these sorts of beliefs are more than a cultural phenomenon but constitute something innate to human experience.

The concern about student misconceptions also leads to the desire to figure out if

these beliefs are found among certain individuals or groups of students. The last few

decades has seen the development of several assessment tools that allow us to measure

this. Those tools tend to be inventories of the most common misconceptions that have

been discovered. Questions from these inventories are given to students to see if they

hold onto such beliefs or if they are in line with classical descriptions of the mechanics of

the world. Those tools, especially the Force Conception Inventory, are appraised in Part

B of this section, noting their strengths and weaknesses. The review includes some of the

ways to use tests similar to FCI for research purposes.

A. Newtonian Misconceptions

Under the rubric of classical or Newtonian mechanics, there are concepts such as forces, motion (both linear and circular), energy, and momentum, which all have their own characteristic issues. This paper concentrates on the issues of force and motion.

i. Forces and Linear Motion

The defining equation of a force given by Newton is fairly simple in form, the multiple of

an object’s mass with the instantaneous rate of change in its velocity (acceleration), and the special case of no force implying no change in speed or direction is Newton’s first law of motion. However, it appears that this is not the concept that students have. In their lives, mathematically defined objects such as vectors are not natural, and mass is a concept that is difficult to distinguish from weight, a physical concept tied more directly in experience. So perhaps it should come as little surprise that the modern, sophisticated

definition of force is not the same as the intuitive ideas that most people hold before

receiving instruction.

In this section, the misconceptions concerning force and linear motion are

examined in two general time frames. One part concerns recent evidence that students

have various conceptions that do not align with Newtonian dynamics. The literature and

methods that show the pervasiveness of these misconceptions are discussed. In the next

part, the history of these ideas is considered, demonstrating that the repeated occurrence

of these notions is not a modern or cultural anomaly but something more constant in

human experience and perception.

a. Modern Examples

There are several levels where confusion may come into play. One is that students tend to conflate velocity and acceleration. This can be seen in student interpretation of kinematic graphs, in which they tend to think graphs of acceleration, velocity, and even position ought to be the same (Beichner 1994). When looking at physics problems, commonly students believe it is not possible to have an acceleration without a velocity or that acceleration vectors point in the same direction as velocity vectors (Kim & Pak 2002;

Trowbridge & McDermott 1981; Whitaker 1983; Halloun & Hestenes 1985b).

Acceleration, velocity, and force all seem to be poorly differentiated in the minds of most people, and those concepts are deeply embedded such that even doing a thousand traditional problems fail to alter them (Kim & Pak 2002).

The conflation can be seen in the form of the misconception of the force in the direction of motion. Most adults tend to believe that when an object is in motion, then

there must be a force acting on it in the direction it travels. This belief entails that an

object moving at constant speed on a given bearing has a constant, non-zero force acting

on it in the direction it travels. This is in clear contradiction to Newton’s first and second

laws, which state that to have constant velocity requires there to be no net force acting on

a body. This erroneous idea is well embedded into the intuition of subjects that have been

tested (Viennot 1979; Trowbridge & McDermott 1981; Clement 1982; Whitaker 1983;

Halloun & Hestenes 1985b; Hestenes et al. 1992; Planinic et al. 2010), which suggests

that this is a notion built up by real-world experience and is formed when constructing a

mental model of the physical world. In interviews done by myself and colleagues, it is

seen that even for students in their second semester of physics in the highly abstract cases

of charged particles in a magnetic field continue to believe there is a force in the direction

of motion, at least until this fails to explain the motion of the particle (i.e., what is

pushing the particle in that direction).

Let us consider the sorts of tests that showed the existence of this misconception.

Viennot (1979) was the earliest to provide such evidence, and from multiple routes. He

examined classes of French Belgian students at the university level and those just ending

secondary school. For a test, Viennot used the case of a juggler throwing up an object.

Students were asked if the force acting on the object changed with height or when it was flying up or down. The question also had objects with different levels of horizontal velocity (see Figure 2). The key point was that in all of these situations the only force is the weight, so a student should answer that the force is the same in all cases. For French students in their last year of secondary school, 55% incorrectly stated that the forces are not equal in all of these cases; 42% of French university students also said the forces

were unequal; and 54% of Belgian university students concurred. In other words, it is not

uncommon to see half of tested subjects to believe in the incorrect notion that velocity is

somehow indicative of the force acting on an object at an instant. This test did not show

what other force was in play, but we could infer that the force was related to velocity rather than acceleration. This was also confirmed with questions about spring oscillators.

Figure 2. Thrown Objects with different Trajectories and Speeds. From Viennot (1979).

Another experiment, done by Clement (1982), was more direct in figuring out

what forces are in play. The examination was done on first-year engineering students

(presumably at the University of Massachusetts Amherst) who had not had college-level

physics yet. In a written test, students were asked about the forces acting on a coin that

was tossed up into the air. They were to draw force arrows and as many as they thought

were acting on the coin at a certain point on its way up. The typical wrong answer was

that there was a “force from your hand” pushing up on the coin while in upward flight.

The labeling of this upward force as the “force from your hand”, the “force up from

velocity”, or the “momentum force” all came from interviews with students after this

testing. 88% of students gave an incorrect answer, and almost all of those (90%) had an

upward-acting force. After taking a physics course, 72% still gave an incorrect response.

Engineers who had taken two semesters of physics were still wrong 70% of the time. This

study showed not only the existence of, but the persistence of, the idea of a force in the direction of motion. These results are otherwise typical in the research, with a small to significant majority of students holding onto this misconception even after instruction.

While it is common to find students with this idea about force among those who are examined in the literature, research is almost exclusively done on subjects who are in

Europe or North America and attend college, usually at universities that conduct research.

This means that there is a notable selection bias of the population, namely first-world, generally more well-off individuals who are probably not ethnically diverse. This is compounded if the subjects are students who are taking calculus-based physics at college, a group that will largely be engineering and science majors, a population that is primarily male and Caucasian. There is the possibility that the effect exists not so universally, and testing outside of the most common groups that Western university professors are likely to see in their physics classes would falsify this possibility. As an example of this, Bao et al. (2009) examined students in China in the first years of college and compared them to

United States university students in their freshman year. Tests included the Force

Concept Inventory (FCI), the Brief Electricity and Magnetism Assessment (BEMA), and the Lawson test for scientific reasoning. The first two examined the understanding that students had in physics with respect to Newtonian mechanics and the basics of

electromagnetism; the last test looked at abstract, scientific reasoning and is more

content-knowledge-independent. With respect to FCI and BEMA, the Chinese students

were significantly better. On FCI alone, the average score for Chinese students was

85.9% ± 13.9%, while American students averaged at 49.3% ± 19.3%. This is likely a

product of the very different way that Chinese K-12 education is conducted with respect to physics; rather than taking an elective physics class in high school as many US students may have, in China national standards students have physics courses from grade

8 through 12—that is, 5 years of training. This makes a comparison between US and

Chinese students about the commonality of physics misconceptions difficult unless it is

done before either group of students has had formal instruction. As such, it is not possible

to see directly if physics misconceptions among young adults are more or less prevalent

in a different, modern cultural context.

b. Historical Examples

As noted above, it cannot currently be concluded that the misconceptions about one-

dimensional force and motion are widespread rather than a selection bias from the

population of students normally studied in first-world, Western (especially Anglo-

American) nations. However, instead of comparing geographically different subjects, it is

possible to investigate temporally distant thinkers about physics. When looking for the

misconceptions that are so commonly found in the literature, it becomes obvious that they

are not new, as numerous researchers have compared these ideas of motion to those of the

ancient Greek philosopher and scientist Aristotle (4th century BCE) and the medieval

natural philosophers of Europe. In particular, there are comparisons to the theory of

impetus (cf. McCloskey et al. 1983; Hallouns & Hestenes 1985b; Espinoza 2005),

described by Jean Buridan in the 14th century. However, Buridan was formulating ideas based on centuries of discussion and debate from the ancient Greeks and Romans to the

Arab scientists and then European medieval scholars, so the idea of impetus had a fair amount of articulation in the writings of some of the most intelligent people in the ancient and medieval world. That may make for a less helpful comparison to the average student,

who probably has not read commentaries on Aristotle, such as those by John Philoponus

(6th century). It may be more helpful to look at the earliest articulations of the ideas of force, letting their natural language try to express what is happening rather than the

refined ideas of the late medieval period. Their struggles to explain their own concepts

can be revealing, especially if it is not refined by dialogue with other minds.

1. Aristotle’s Physics of Motion

Most everyone looks to the beginnings of physics with Aristotle, and he is justifiably one

of the earliest scientists in history. As opposed to the methods of non-empirical reflection

as is done in much of early Greek philosophy, Aristotle was more empirically-oriented

and studied numerous natural phenomena and creatures. In particular he is known for his

careful studies of animals, many of which he personally dissected. He also wrote about

physics and dynamics, including a treatise titled Physics. However, Aristotle did not

leave behind a single, comprehensive volume explaining his ideas about dynamics, and in

some ways he may not have produced a complete articulation of forces and motion.

Nonetheless, his works can provide enough insight into what he believed in simple cases

of force and motion. In particular, his book about astronomy, De Caelo (On the Heavens)

includes the majority of what he had to say on the subject, though readings from Physics,

Metaphysics, and even his works on animal motion provide clues.

Aristotle is also not easy to read, even in translation, and those without expertise of his works may indeed reach conclusions that are not believed by classical scholars. For example, Lythcott (1985) argued that many of Aristotle’s beliefs about force and dynamics were not the same as those of 20th century physics students (and she is correct),

but Lythcott makes mistakes interpreting the philosopher. For example, she believes

Aristotle said that a falling object did not have a force acting on it; however, there seems to be confusion about Aristotle’s idea of a natural motion and a self-mover. Indeed

Aristotle did say a falling object had a “power” (dunamis, cf. De Caelo 312a16-19) or

“strength” (ischus, cf. De Caelo 297a1-3) acting on it, giving it its downward inclination or “impulsion” (rhopê) (cf. De Caelo 297a26-30; Physics 216a12-21); this same

vocabulary is used with respect to throwing forces, so it seems clear Aristotle did believe

falling objects have a force acting on them. This misinterpretation is partially due to

focusing only on the book Physics and not as examining De Caelo, and also in part because of the reliance on English translations that may convey different meanings to modern physicists. As such, this review of Aristotle’s ideas will rely on his own words as well as modern interpretations by expert scholars of the philosopher (for example,

Hussey 1991).

First, the vocabulary Aristotle uses needs comment. The term that is often translated as force or impulse is rhopê, a noun that comes from the verb rhepô. The verb is used to mean to weigh, to be stronger/heavier than something else, to come down on one side or another, or to incline. The noun rhopê has similar meanings. Hussey (1991)

argues that for the philosopher it has a meaning like impulse or impetus, and so he

translates it as “impulsion.” Here the term will be similarly translated as ‘impulsion’ or as

‘force’, avoiding ‘impetus’ because of what it means in the larger history of science and

discussed later in this section. ‘Force’ will also capture the meaning, but it will not be the

same as in the Newtonian definition, which will be seen below.

Aristotle’s ideas of impulsion had two categories, natural (phusis) and violent

(bia). Natural forces included those such as weight (or gravitational force), and that force was natural because the object was trying to move to its natural location in the universe.

Materials made of earth were heavy and tried to move towards the center of the universe, while fire and air were light and moved upward to the heavens in the common geocentric view. In Aristotle’s system, fire provided an upward inclination, while earth provided a downward inclination. Objects are a mixture of these elements, so the balance of forces or “excess of weight or lightness” (Physics 215a28-29) determines if the natural motion will be down or up. Violent motions, on the other hand, are caused by forces that resist the natural positioning of bodies—for example, the act of throwing a rock. In Aristotle’s scheme, the velocity of a body came about because of a force on it, and the velocity and force were in direct proportion; pushing twice as hard would mean an object traveled twice as fast.

Because Aristotle was the first to articulate this idea of forces, he was also first to realize that his own characterization of forced motion lead to the conclusion that without a force an object would instantly stop moving. However, anyone throwing a rock or firing an arrow realizes that it continues to move forward until it hits the ground. Aristotle does not provide a clear explanation for this, but a reasonable hypothesis can be synthesized

27 from careful reading of his works. One interpretation provides a strained solution to objects moving by violent impulsion is this: the force also imparts the ability for locomotion into the air surrounding an object, so the air would be able to push the object for some time until that locomotive action had dissipated too much. This has some similarity to an idea by Aristotle’s teacher, Plato, in the dialogue Timaeus, though

Aristotle was critical of the particulars of Plato’s hypothesis (Lindsay 2009). However, in

De Caelo 301b17-30 Aristotle says that the object “uses the air as an instrument,” and the air “collaborates” with the object. While vague, it suggests the object is less passive in moving forward, not unlike what he thought of how animals used the ground to push for their movement (Hussey 1991). Other ancient interpreters, such as Philoponus, would not have this more nuanced reading of Aristotle; nonetheless, Aristotle appears to have something like the idea of impetus in mind. However, his ‘impetus’ differs from later inquiries because of his need for the medium (air, water, etc.) to help push the object forward.

The continued effects of the violent impulsion also have the property of dispersion. With time, an object must move through a medium, and this medium provides resistance. Thus, the object will lose its impulsion with travel, slow down, and eventually stop. How far an object travels before stopping is directly related to the density of the medium, where a doubling of density halves the distance traveled and halves the speed of travel. Fully understood, the medium acts as both resistance to the movement of a body and as the cause of its continued motion. It is difficult to know if Aristotle empirically verified these ideas, but it would have been difficult to know the density of another

28 medium of travel (namely, water), so any such tests would not have provided definite answers.

Also vague are Aristotle’s views on natural motion, and in fact he provides no confident answers. Aristotle did seem to believe that objects moved more quickly as they fell, and in De Caelo 277a29-33 he seems to suggest that the weight increases as the object falls. There Aristotle argues that if natural motion were without end, then the weight (baros) would go up to infinity, as would the speed. Hussey (1991) disagrees with this reading, but his reasons are not strong. He stated that if weight increased then

Aristotle needed to explain this, but this is not a problem with the interpretation, just something else Aristotle could not explain. Nonetheless, Hussey is correct that for

Aristotle it is not the medium that provides the increased push but possibly something to do with location that provides the accelerated push. An object tries to move towards its natural position and the closer it gets to that place the faster is goes. In a way, this is not unlike more modern field theories (i.e., charged particles nearing each other), but this comparison is not the best. In a field theory, the force is determined by the charges and distances apart, but Aristotle suggests that the weight is also related to the speed, that faster falling objects are weightier. Overall, the philosopher does nothing to help provide us with his ideas of a mechanism for this situation.

2. Ancient Thinkers on Motion after Aristotle

Aristotle’s influence on other philosophers and scientists was significant, though not as universally forceful as some has suggested. Non-scientists used some of his terms, but the physics would not always follow Aristotelian ideas. For example, the historian and

29 essayist Plutarch (early 2nd century CE) described the orbit of the Moon as having two forces: the natural force attracting the Moon to the ground, and a violent force that provided circular motion (Jammer 1957). While it is true that there is the gravitational force between the Moon and the Earth, the force providing circular motion is another reoccurrence of the mistaken assumption of a force in the direction of motion. Moreover,

Plutarch states that the weight is “neutralized” because of the rapid motion of the Moon, so perhaps he means the gravitational force disappears when the Moon is in orbit.

Following Plato’s physics rather than Aristotle’s for falling objects, Plutarch also said that as an object falls it is accelerated downward by the air rushing in from behind and pushing the object down faster (Cohen & Drabkin 1948). Plutarch was familiar with the works of Aristotle, but the essayist was a Platonist and may have not read Aristotle’s work on physics.

Among the scientists that read and commented on the philosopher’s works,

Aristotle’s idea of how a pushed object could continue to move was unsatisfactory even in antiquity. Philoponus (6th century CE), for example, noted that having machines to blow on an object such as a rock had extremely little ability to move it, so it made it all but impossible that the medium an object traveled through was the cause of its continued motion. If air was the source of the continued push on objects, then it should be more efficient to just wave ones hand to launch arrows.

Another explanation, as we know of it from Hipparchus of Rhodes (2nd century

BCE), has striking similarities to how students describe a dissipated force in the case of projectile motion. Hipparchus argued that when an object is thrown straight up, the force of weight pulls down on the object and there is still the upward force acting on the object;

30 that upward force weakens and is then overcome by the gravitational force, and then the object falls back to earth (Lindsay 2009). This is very much reminiscent of a force in the direction of motion that eventually dies off until it reaches maximum altitude. In Section

D of the next chapter, it is found that some students interviewed provided a description very close to that of Hipparchus’s. While such ideas were widely discussed in antiquity, unfortunately, there are no signs of a quantitative treatment of these dissipating, fictional forces in the direction of motion (Jones 2013).

With the fall of classical civilization, scientific pursuits moved eastward, and our last Greek-writing authors on the subject are two commenters on Aristotle, Simplicius of

Cilicia (6th century) and John Philoponus in Alexandria. These two are also our primary source for the other ideas about dynamics in antiquity, and they have their own criticisms of Aristotle’s ideas. Philoponus had shown how the medium cannot be the cause of continued motion of thrown objects and that falling objects fall at about the same rate no matter their weight, while Simplicius showed that an object does not weigh differently at one altitude compared to another in order to explain acceleration of falling objects

(Cohen & Drabkin 1948). It would be Philoponus to first say that an object continued to move forward because of some sort of internalized motive force. “It is necessary to assume that some incorporeal motive force (rhopê) is imparted by the projector to the projectile” (Cohen & Drabkin 1948). This statement comes at the end of antiquity, and because of the spottiness of the historical record in ancient science it is hard to know if this idea came to exist before Philoponus’s time.

3. Medieval Islamicate Physicists

In the 7th century the rise of Islam meant it was Arab-speaking philosophers who had access to regions like Egypt, Syria, and Persia with scientific works, especially those concerning the ideas about dynamics, though this flourishing of scientific investigation

did not take place in the Islamicate until the rise of the Abbasid Caliphate in the 8th

century. The first such person we know of that considered motion in the way who the

Greeks and Romans had was Abū ʿAlī al-Ḥusayn ibn ʿAbd Allāh ibn Sīnā (or simply Ibn

Sina; in the Latin West, he was known as Avicenna), a Persian scholar in the 11th century.

He is said to have been the first to promote the impetus theory in the East. He seemed to follow in the tradition of Aristotle and Philoponus with impetus-like hypotheses, stating

how an object continues to move after a violent tendency (mayl qasri) is imparted.

Similar are the beliefs of the 12th century Spanish astronomer Nur ad-Din al-Bitruji,

Latinized as Alpetragius, and another Spanish mathematician Ibn Bājjah, Latinized as

Avempace. However, unlike Aristotle, Avicenna and others did not believe that the violent tendency dissipated on its own, so an object without external resistance could continue to travel; in a way, this is a forerunner to the notion of inertia (Franco 2003).

In those times, the only alleged challenge to impetus-like ideas that came closer to classical mechanics comes from Abu’l-Barakāt (14th century), a claim that is discussed in

Appendix B. There it is argued that the Arab philosopher instead believed that the total force on a falling object increased as it lost altitude. There is value in comparing the description Hipparchus gave to projectile motion and that provided by Abu’l-Barakat.

When a rock is thrown up, both of these physicists agree that there is an upward force and the natural, downward force. They also agree that the upward force weakens as it climbs.

However, Hipparchus begins to diverge when he says the object reaches its maximum

altitude when the upward and downward inclinations are the same. Then the object falls

over which period the upward force continues to dissipate. When the upward force has

finally disappeared then the only force acting is weight, and so the object has reached is

maximum velocity (Lindsay 2009). This significantly contrasts with Abu’l-Barakat who

believed that the downward impetus continued to increase and sped the object up;

Plutarch, as noted above, had a similar view. As such, the Arab philosopher provides a view of projectile motion that potentially includes constant acceleration, though the

constant nature of said acceleration is not specified.

4. Medieval European Physicists

Eventually the scientific ideas of the Greeks and Romans would return to Catholic

Europe in the High Middle Ages, notably after the conquests of Spain by the Crusaders and their capture of the library of Toledo in 1085. The scholars rushing to Spain would begin to relearn Aristotle, and to help them understand his they had the commentaries of

Aristotle by Ibn Rushd (Latinized as Averroes), often simply called “The Commentator” by medieval scholars. After some theological issues, such as those that lead to the

Condemnation of 1277, scholars such as Thomas Aquinas made it possible to study

Aristotle without it necessarily leading to censure by the Catholic Church. This included the area of physics, which Aquinas also considered. Though the famous theologian did not directly espouse an impetus theory, he did state the maxim that would be common in the Middle Ages: whatever is moved is moved by another (Grant 1974). As such, something needed to continue to push an object that had undergone violent motion.

Natural motion was still a topic of argument. Aquinas thought something external to a falling object propelled it back to earth, which Dun Scotus thought was some sort of internal force.

Once dynamics was receiving appropriate consideration by medieval Europeans, the impetus theory that most PER researchers know of came into being with the writings of Jean Buridan, a professor at the University of Paris in the mid-14th century. In his

Latin treatises (which will also be common in other Latin works of the period), he uses the word inclinatio as a translation of the Greek rhopê and the Arabic mayl; by using this same word for translating the Greek and Arabic shows that it was much the same concept in play from ancient as well as medieval ideas related to impetus. Buridan also provides an understanding of accelerated motion due to weight that is similar to what is found in

Arab commentaries on Aristotle. Buridan considers some of the suggestions for why falling objects accelerate, including how the air is heated while falling and makes it easier to pass through the medium. Rejecting these and other medium-affected hypotheses,

Buridan argued that the impetus of an object increased as it fell, though the weight would remain the same. This had the effect of the object having a greater downward force

(Grant 1974). For the rest of European history, the impetus theory that would be argued would come back to Buridan’s ideas.

While ancient and medieval scientists failed to provide a rigorous mathematical treatment of how an object behaves when no longer in contact with a pushing force, there are several quantitative treatments in ancient and medieval treatises. For Aristotle, the ratio between the applied force and the resistance that was apparently innate to the medium of travel (and for Aristotle, there was no vacuum) would give the velocity; for

John Philoponus, it was the difference in these quantities. On the other hand, the 14th

century scholar and Archbishop of Canterbury, Thomas Bradwardine, applied the

logarithm to Aristotle’s ratio so that if the force and resistance are equal there would be

no velocity (Jammer 1957). More correctly, Bradwardine produced the following ratio by

a process of elimination (believing he had exhausted all other possibilities):

= 푉2 푉1 퐹2 퐹1 � � where F is force, R is resistance of the푅2 medium,푅1 and V is the velocity of the projectile.

Still, there was an awareness of the discrepancies in dynamical theories. For example,

Aristotle’s belief that an object twice as heavy will reach the ground in half the time was

known to be wrong by late antiquity (Pendersen 1993). Nonetheless, all the challengers to

Aristotle’s ideas continued to maintain that a non-zero velocity required a force, leading

to the impetus theory of medieval physics which was only overturned during the

Scientific Revolution, though contrarian discussion takes place with Nicole Oresme (14th

century).

It is also worth considering the methods that the medieval Catholic scholars took

to figuring out their ideas. While they were doing theoretical work, the experimental

methods were not that of the modern scientist. It was largely focused on thought

experiments using examples of things already witnessed or imagined rather than the more

rigorous methods of actually testing. For example, when arguing if an object that is

moving up has to momentarily stop before falling down, natural philosophers thought of

what happens when a fly is traveling straight up while a lance is falling straight down

towards the fly. The thought was that the impact caused an instantaneous change from

upward to downward motion without there even being a moment that the fly had stopped. 35

As one can imagine, having a fly behave in the needed way and be hit by a falling lance

is not something anyone had seen or tested. Other times, though, it was from past

experiences that these thinkers used for their argument. Buridan argued how impetus

could explain why someone travels farther when they run and jump rather than simply

jump in place; the runner already has some impetus, so the additional force of the jump

ensures there is yet more impetus. This use of background knowledge from experience

and interesting thought experiments has been given the interesting title by Grant (2002) of “empiricism without observation.” What this helps to indicate to the modern researcher is that the ideas espoused in this period are based almost completely on the built-up intuitions of the medieval philosopher and not so much from careful experimentation.

5. Physics of the Scientific Revolution

When the new astronomy of the Renaissance began with Copernicus in 1543 in his

treatise De revolutionibus orbium coelestium, the Polish astronomer tried to ridicule the

idea of violent forces on objects in motion, pointing out how there must be an incredible

force on the stars that appeared to revolve about the Earth in a day, far more incredible

than the force supposedly needed to rotate the Earth. Even so, Copernicus did not

perform a complete break from Aristotle’s language and concept of force (Pedersen

1993).

Instead, it was the work of Galileo that would take the common medieval view of

impetus and force in the direction of motion and try to demonstrate its weaknesses.

However, this was only after considerable changes to his own ideas which included the

same impetus theory that he would overturn. Galileo scholars have divided up the

scientist’s career into three periods: his time in Pisa (his early period), his professorship

in Padua, and his later life after his major publications and during his prosecution and

house arrest. In his early period he wrote a treatise De Motu (c. 1590) that he did not

publish. There he reproduced the ideas of Hipparchus about how a thrown object is given

a sort of upward impetus that disappears on the way down (Sharratt 1994). The greatest

voice in the early 17th century for the new physics was, in his early days, arguing for the

ideas he would need to disprove.

That shift is most clearly seen in his Dialogue Concerning the Two Chief World

Systems (1632), where during the second day of the dialogue the impetus view in put into

the mouth of the character Simplicio, who followed the astronomy of the ancient

astronomer Ptolemy and his geocentric model. On the other hand, the character Salviati

takes the view closer to that of Newtonian mechanics (which would not be articulated by

Newton until 1687). The discussion about a man riding a horse at constant pace would

test the idea of motion without an active force. The rider would drop a ball from his hand,

not thrown forward or back, and the two physical ideas would predict differently from

where the ball would land: either behind the rider, as the impetus theory would suggest,

or directly underneath, as the new physics of the time suggested (cf. Figure 1). Simplicio

stated that the ball would fall behind the rider, not unlike what the subjects studied by

McCloskey et al. (1983) had said, and this was the view of Galileo’s rivals that would

come to be ridiculed. The Dialogue thus provides an example of a qualitative rendering of the conception of the results of a force in the direction of motion which exists to this day for many people, even though they are not geocentrists as were Galileo’s antagonists.

However, even in his later works, Galileo would lapse into impetus-concept language. Again in his Dialogue, Galileo talks about how the bob of a pendulum acquires an impeto when it descends, and when it moves up during the second part of its swing it is moved up by a forced motion (moto violenta). In which case, Galileo is transitional from impetus concepts to inertia concepts (Westfall 1972).

Finally, it is Isaac Newton that provided the mathematical framework of classical mechanism and in particular his three laws of force. However, one of the people that had trouble accepting Newtonian physics was Isaac Newton himself. In his papers and correspondences before the publication of his Principia Mathematica in 1687 he spoke of internal forces in objects that go unchanged until acted upon by an external force. For example, in manuscript for his tract De Motu, which Newton sent to Edmond Halley in

1684 that explained the motion of planets in the solar system, Newton stated the following:

I call that by which a body... endeavors to persevere in its motion in a right

[straight] line the force of a body or the force inherent in a body. By its inherent

force alone, every body precedes uniformly in a right line to infinity unless

something extrinsic hinders it. (Westfall 1980; emphasis added)

Such an idea can also be found in Newton’s notes going back to 1664 when he was a 21 years old student at Cambridge. The terminology of impetus also made it into the

Principia, as the phrase vis inertiae, but it took on new meaning. After years of deliberation and reflection, Newton came to the ideas that are associated with him

(Steinberg et al. 1990). Considering the fact that one of the greatest minds in science,

working with other brilliant physicists and interacting with the works of others from the

Scientific Revolution, took decades to finally shake off the impetus theory in favor of the

force laws he is known for, should be an excellent indication of just how deeply-rooted

the naïve concept is and how hard it is to overcome.

6. Conclusions from History of Science

We see a desperate effort to articulate intuitional ideas of motion that reflect the same

sorts of instinctual ideas that modern people have. For well over a thousand years of

articulation and debate, an impetus-like theory of motion was believed even though it was

not understood or modeled well with mathematics. The theory could not be shaken off even by brilliant scientists working very hard and thinking about the natural world up until the early modern period. Those same ideas are ever-present not just in ancient and medieval minds but also in the average student, even if they had attended a college-level physics course. These intuitions are apparently deeply programmed, probably by the creation of heuristics that work well for most things in day-to-day life but break down when examined critically. They also seem unlikely to be the products of culture because of their presence through thousands of years of investigation, from ancient Greece to medieval Persia to Renaissance Europe.

ii. Circular Motion

Related to this incorrectly assumed force in the direction of motion is another imagined pushing action on bodies, in this case dealing with objects moving in a circular path. In

an inertial frame of reference, circular motion is maintained by the application of a force

pointing towards the center (inward) of the circular path created by the object’s motion.

As with Part 2.A.i above, this section will consider the recent literature supporting

the existence of circular motion issues and then historical examples.

a. Modern Examples

When considering objects moving in a circle with uniform speed and at a constant radius

from the center of motion, the net force should only point towards the center of that

circle; that force then has the appropriate name of ‘centripetal’ (center-pointing). There should not be a force pointing outward (centrifugal, or center-fleeing), except in a rotating reference frame, and even then it is not a force due to an interaction. The centrifugal force is otherwise said to not exist or be a fictitious force. However, many students when asked about the forces acting on such an object will say that there is a force pointing outward (Hestenes et al. 1992). Students are also prone to say that there is a force tangent to the circle an object makes, one that points in the direction of the velocity vector. This appears similar to the force in the direction of motion concept.

Another belief that is not uncommon among students is the belief that an object that had undergone circular motion will for some time continue on in a curved path even when the centripetal force is no longer applied (Hestenes et al. 1992). For example, if a ball is tethered by a string and is swung around in a horizontal circle, after the string is cut the ball will continue to move in a curved path. This has been described as a case of

“circular impetus” and has similarities to the case of medieval ideas of linear impetus.

How intertwined the circular and linear momentum cases are in the minds of students is not certain.

One study concerning forces of object undergoing circular motion was done by

McClosky et al. (1980) with university students, many of whom had taken a physics course in high school or college. They were asked to draw the path of an object that had undergone some sort of circular motion and then had been ejected. For example, a marble enters into a curved tube and is then ejected. Another is a ball on a string moving in a circle, and then the string is cut or broken. In most of these problems, a small majority of students did draw the path correctly, but a large minority and sometimes majority provided a curved path of an object when exiting the cause of its circular motion. In other words, there could still be circular motion in the absence of force.

To understand what is happening, consider the case of the ball on the end of a string, twirled around in a horizontal circle. When the string breaks, as seen from overhead, the ball should move in a straight path, which is what 53% of the students in the study drew. However, 30% drew a curved path that closely matched the path of the original circle but slowly straightened out. This appears to be something like a circular impetus concept. But when looking at cases that would suggest the action of a centrifugal force, only 6% of students said the ball would move straight away from the center of rotation when the string was cut; another 6% thought a straight path in-between straight out and straight forward, a situation that seems to combine forward impetus and centrifugal force. It appears then that circular impetus is the more common idea rather than the misconception of centrifugal force.

This study was also followed up a few years later, using several situations and

categorizing student answers as correct, circular impetus, and centrifugal (McClockey &

Kohl 1983). The investigation included 90 university students, one-third of whom having no physics class background, another third having taken physics in high school, and the last third having taken college physics. One scenario included a ball moving through a tube and then exiting, while another included the ball on a string and then released; the students were to choose from several options which best matched the path the ball would take afterward. In the case of the ball on string, 24% provided a circular impetus answer, and only 9% provided a centrifugal answer. Similar results were found in the case of a ball leaving a curved tube. Based on discussions with students, the percentage that would include circular impetus in their strategies to answer the questions was about 35%. The authors of the study also noted that responses improved in accuracy with physics training, but it failed to be statistically significant (p > 0.05) in one experimental case. With 30 students per experiment, this is not surprising. The consistency in these findings shows that circular impetus notions are rather common, while centrifugal tendencies appear to be rare.

The appearance of the centrifugal belief may instead by a result of instructor prompting. diSessa (1988) describes asking a variety of subjects of different ages about how they felt when turning sharply while driving, and this increase the number of individuals that provided more of a centrifugal-like response. diSessa also notes that no

subject spontaneously gave this sort of answer when considering a ball moving through a

curved tube (cf. Figure 3). Perhaps then a centrifugal-like answer is not something

Figure 3. The Curved Tube Problem. The setup (A), the correct response (B), the incorrect response (C).

From Kaiser et al. (1986).

normally considered by students, but instead it is the activation of another model that

comes about when prompted, particularly by a Newtonian thinker.

Circular motion has also been studied in students who are much younger than the

college-age students that PER researchers tend to use. Kaiser et al. (1986) examined students of ages 4 through 12, from preschool/kindergarten to 5th/6th grade, asking them to draw the path of a ball coming out of a curved tube. 20 college students were also included for comparison. The curved paths were made of clear plastic placed on a horizontal wooden base. Students were asked what path a ball going through the tubing would take after it exited the tube (Figure 3). The testing also had two versions of curved

tubing, one that was straight but then bending for make a C-shape, the other making an almost logarithmic spiral. Across all age groups the C-shaped curve produced the greatest number of correct answers, but what is the most interesting result for the purposes of this paper is the correlation with age. The youngest students, ages 4 or 5, answered correctly more than any of the school-aged subjects and on par with college-aged students who had had at least one course of physics either in high school or college. Pre-school students had better than 50% correct responses in both the spiral and C-shaped contexts (and nearly 80% in the case of the C-shaped tube), while fifth and sixth graders were less than

50% in both cases. This means that young subjects have a more accurate model of circular motion than older subjects without instruction.

If the circular impetus theory is strongly correlated with the linear impetus theory, it would suggest that the latter forms at a later point in the life of a person. However, direct testing needs to be done to see if the linear impetus theory has a similar development as the circular impetus theory demonstrated by Kaiser et al. (1986). It would also be important to understand what sorts of experiences students have that lead them away from intuitive ideas of motion that are closer to the Newtonian view and to the impetus theories that dominated ancient and medieval thought. Some of those possible experiences are discussed in Chapter 3.

b. Historical Examples

McClocky et al. (1980) stated that students had a concept of circular motion that conformed more to medieval impetus theories than Aristotle’s idea that without an external force and object stops moving. As noted in the previous section, Aristotle did in

44 fact have something of an embodied inclination in an object that had undergone violent motion, even though ancient and medieval commentators failed to understand him on that point. However, all of those considerations by Aristotle were focused on linear motion, and now it is time to turn to the case of circular locomotion. Unfortunately, circular motion was not given the best deductive treatments by ancient and medieval physicists, and often it is explained using ideas already established from linear motion, especially in the case of impetus. Historically, the ideas of linear impetus and circular impetus were linked for John Philoponus and Jean Buridan, but others such as Giambattista Benedetti

(16th century) thought circular impetus was not real (Pedersen 1993; Drake 1999); if these concepts are in fact intertwined for students, then current research into one case of impetus may provide insight into the other.

1. Circular Forces in Ancient and Medieval Commentary

Concerning the centrifugal force, this does not exist in ancient commentary. Perhaps this is a product of ideas about linear and circular impetus. Consider Aristotle in De Caelo

269b29-270a3 where he talks about the circular motion of the planets. He states that they must not have weight or lightness (forces towards or away from the center of the Earth), but only circular motion. He believed this because the heavens must have natural rather than violent forces acting on them or else they will be unstable and not eternal, and only circular motion can be eternal, but it has the conclusion that there are no centripetal or centrifugal forces.

This may also fit in what is seen in Plutarch when he discusses the material of the

Moon (if it is mostly earth or fire) and why it does not fall down. In his treatise On the

Face in the Moon (in Moralia 923D), Plutarch stated that the weight of the Moon no

longer acts on the heavenly body, that the gravitational force is neutralized because of its

rapid circular motion when in orbit. However, it is not stated that there is one force

resisting another but that the downward inclination of the Moon is gone. Plutarch also

compares this to how a sling with a stone will allow the spinning missile to not fall down

when it is being spun. In such cases, the spinning object is not carried by weight (têv

selênên ouk agei to baros) and the downward inclination is expelled by rotation (hupo tês

periphoras têv rhopên ekkrouomenon). The word used for rotation, periphora, literally

means carried around, and in a circle, so it implies the object in question is forced or

carried (pherô) when it moves with circular motion. So it seems Plutarch is not providing

an early understanding of the centripetal (or centrifugal) force, but rather the other forces

disappear when something is rotated fast enough, that rotation becomes the natural

inclination that carries the object around.

It should be noted that this sort of idea is older, as in another place (Lysander

12.2-4) Plutarch ascribes it to the pre-Socratic philosopher Anaxagoras (5th century

BCE). Now, Anaxagoras and other pre-Socratics had developed the idea of a vortex that

caused the heavy elements to fall towards the center (creating the earth), while other

materials flung out to become the stars and planets. Anaxagoras in particular believed that this vortex was caused by a cosmic principle of mind or nous that literally set the

cosmos in motion (Sedley 2007). Even though terms like force are not used, the model

does imply some intuition about what happens in circular motion. Empedocles (5th

century BCE), for example, provided an illustration of the principle by noting how a ladle

with water in it and spun around fast enough would not have any water fall out (see

Aristotle, De Caelo 295a18-22); this is comparable to the demonstration of spinning a bucket of water around and not having any water leave the bucket. That some ancients produced something like a giant, cosmic centrifuge to explain the world implies that they had some understanding of what tends to happen when objects are spun. However, the stratification of the model is in the opposite order one expects from a giant centrifuge: the most dense or heavy material (earth) should be flung out and lighter materials (fire) towards the center or at least not as far out.

Figure 4. Pseudo-Aristotle’s Idea of Objects Forced into the Center of Rotation. From Winter (2007).

Why these philosophers believed in this order of densities that is in clear

opposition to what Newtonian thinkers understand is probably explained in part by

observations of vortices. This would include water spouts, whirlwinds, tornadoes, and

even the stirring of soup. For example, the author of Mechanica, who claims to be

Aristotle, notes in his 35th question how debris, such as sticks and leaves, in a river eddy

tend toward the center of the eddy (Winter 2007). Similarly, the 4th/5th century BCE physician Hippocrates noted in stirred liquid sediments move towards the middle (Tinger

1974). Pseudo-Aristotle, mentioned above, argued that when a body with spatial extension is moving in a circle, its outer parts are moving faster than the parts closer to the center of rotation; that faster rotation then pushes the object inward (see Figure 4).

The belief that at a greater distance the (tangential) speed is greater is stated in the introduction to Mechanica, and it is also found in Plato, who describes a rational relationship between radius and speed ( = ), in effect reproducing the equation 푉1 푅1 푉2 푅2 = given a constant angular speed (cf. Laws X 893C-D). However, this argument

푣has no휔 ∗regard푟 for object mass and thus fails to explain the stratification of the universe by

mass.

A justification for this can be seen in the work of the most famous of the ancient atomists, Democritus (5th/4th century BCE). He argued that initially the atoms that filled

the universe were in a whirl or vortex, the same vortex concept of the other pre-Socratics,

and in that process the larger atoms were heavier and fell inward to the center of the

vortex. The smaller, lighter atoms were also weighted towards the center, but they were

pushed outward by the heavier atoms. The atoms in the whirl would bump into one

another and refine some into lighter atoms, thus the vortex provided the differentiation of

weights and then the stratification of the cosmos (Vamvacas 2009). According to

Simplicius, the same belief was held by other atomists such as Epicurus (on De Caelo

569.5-6). Now, the ancient evidence is muddled, and some ancient sources such as Aetius

(2nd century BCE) suggest that without the vortex atoms are weightless; perhaps then the weight is a result of centripetal force. However, the best assessment of the evidence suggests that atomists believed that all atoms had weight and tended in some sort of absolute downward direction (O’Brien 1981; Furley 1989).

Given the above, it is hard to show that the pre-Socratics and pseudo-Aristotle had an idea of centripetal force. In the case of river eddies forcing materials inward, it was explained by the outmost part of an object in an eddy going faster than thus pushed more

than in the inner portion of the eddy. This then forced the object towards the center of the

eddy’s rotation. However, the explanation has a force that would not point inward to the

center, but rather it is a force tangent to the circular movement. So, it seems this is not so

much an example of centripetal force but of differential circular impetus. To be more

sure, it would be helpful to examine Empedocles’s use of the a twirling ladle, but

unfortunately the cursory quote from Aristotle is in a context that does not have anything to do with the subject at hand, and this is likely because of Aristotle’s misunderstanding of Empedocles’s work (Tinger 1974). However, we see a repeat of this in Philoponus’s commentary on Aristotle’s Physics (262.8-15) concerning vessels filled with water and spun around fast enough that no water spills out, a point which he ascribes to Democritus.

[F]or the movement and the whirling of the container is faster than the movement

according to nature of the contents and comes round more quickly, before they

can fall out of their proper position. (Lacey 1993).

From this, it appears that it is not a force that keeps water in the container but rather the

vessel (a ladle, bucket, or amphora) has gone around faster than if the liquid had free-

fallen straight down. In other words, the concepts of centripetal or centrifugal force are

not present in pre-Socratic thought. Nor are they found in Aristotle and other ancient

writers. Aristotle and Plutarch even state that an object in circular motion does not have

the sorts of tendencies that would be linear (downward or center-pointing inclinations).

Perhaps because circular motion was natural according to Aristotle, it did not require centripetal forces to keep it going.

The last piece of evidence to consider from antiquity is the work of one of its most famous astronomers, Claudius Ptolemy (2nd century). Now infamous for his geocentric universe, that model was argued based on his understanding of physics, and he argues against the heliocentric and dynamic geocentric model where the Earth rotates once a day to explain diurnal motion. As he explains, if the Earth were spinning, then there would have to be an immense force to keep objects its surface from sliding and falling behind (Toomer 1998; Taub 1993). Ptolemy says how a revolving or rotating

Earth would have to undergo the most violent (that is, not natural) of forces. He also believed that objects in the atmosphere of the Earth would fall behind towards the west if the Earth and its air were spinning, and even the air moving with the Earth was a mere

‘absurdity’ he granted to show another apparent absurdity. For Ptolemy’s arguments to work, they must rely on Aristotelian ideas of force, and they show no signs of centripetal or centrifugal forces. Arguments such as these would not be challenged until the

Scientific Revolution, demonstrating the draw of these intuitions by astronomers over a millennia.

Turning to the Middle Ages, Jean Buridan appears to categorized circular motion as caused by impetus in the same fashion as it does for linear motion (Grant 1971). The impetus appears to act “in the direction towards which the mover was moving the body, either up or down, laterally, or circularly...” The circular impetus idea helped Buridan explain the motion of the heavens so that they could move eternally. In his view, God imparted the initial circular impetus to the planets, an idea also embraced by his student,

Albert of Saxony (Jammer 1957). It appears that this was the consensus view, and thus it was not until Benedetti that this began to change. Even Galileo did not believe circular

motion about the Earth required an outside force (Drake, 1999), so it took a fundamental

rethinking of what circular motion was before forces could come into play.

This is a surprising result finding that the sort of common misconception that

modern physics students have with centripetal and centrifugal forces were apparently

unknown to ancient and medieval thinkers. Rather, it appears that circular motion had its

own sort of impetus or inclination, which of course matches with other common student

misconceptions. This can be seen in the case from pseudo-Aristotle with differential

circular impetuses noted above. It seems then that the force that keeps an object going in

a circle is pulling the said object along in a circular path. This only begins to change at

the cusp of the Scientific Revolution, namely with Benedetti.

2. The Scientific Revolution and Circular Motion

It appears that the move away from circular impetus begins with Giovanni Benedetti of

Venice in the second half of the 16th century, so it is useful to consider his ideas in detail.

He was considering what happens in the case of a rotating millstone and how this worked with Buridan’s ideas of impetus.

Suppose a millstone rested on a virtually mathematical point and was set in

circular motion; could that circular motion continue without end, it being assumed

that the millstone is perfectly round and smooth? [He says no for] there is also the

resistance from the parts of the moving body itself. When those parts are in

motion they have by nature an impetus to move along a straight path. Hence since

all parts are joined, and every one of them is continuous with another, they suffer

constraint in moving circularly and remain joined only under compulsion. For the

more [swiftly] they move, the more there grows in them the natural tendency to

move in a straight line, and therefore the more contrary to their nature is their

circular motion. And so they come to rest naturally; for since it is natural for them

when they are in motion to move in a straight line, it follows that the more they

rotate under compulsion, the more one part resists the next one, and so to speak

holds back the one in front of it (Drake & Drabkin 1969).

For Benedetti’s argument to work, it requires that straight paths are more natural than curved ones, but this is something Buridan and other impetus theorists of that time would

have rejected. Nonetheless, from this passage, there is the appearance of the centripetal

force, that which holds the millstone together. However, there is also an outward force

that is tangential to the motion of the stone.

Benedetti’s insights came about not so much from millstones but an explanation

for why slings were able to fire pebbles with greater velocity than simply by hand (Drake

1999). From firing such shots, it may be obvious to the observer that the pebble travels straight rather than in a curved path, and this may have given Benedetti the impulse to consider linear impetus the more natural. Benedetti, however, illustrated his beliefs with what happens to a spinning top that is cut into pieces; the pieces do not simply fall down but fly off in straight lines in the horizontal plane. He appears to be the first to make this observation and conclude something about the forces needed to keep an object moving in a circle (Drake 1999). But because that impetus is a force, it makes for a sort of outward-

pointing force from the circular path. This is not centrifugal force but a linear impetus, and it required a centripetal force to balance things out in Benedetti’s mind.

Turning to Galileo, who knew Benedetti’s work, he came to a different conclusion. In his early period, as shown in his 1690 De Motu, he believed that there was

no force or inclination acting on a rotating millstone at all, nothing either towards or

away from the center. He referred to the motion as neither forced nor natural, but simply

neutral (Drake 1999). This was written when Galileo still believed in the idea of linear

impetus, which he would later reject. The reliance on linear impetus by Benedetti may

also explain why Galileo later did not take up Benedetti’s ideas about circular motion. It

would seem that Galileo left circles with its own inertia and did not explain it with forces.

Moving forward in time, the problem caused by the centrifugal force idea was

something that the great minds of the Scientific Revolution had to work on. Newton, also

following mathematician and philosopher René Descartes as well as the astronomer

Christiaan Huygens, believed that body perpetually tried to flee from the center of

rotation (Westfall 1980). How this was handled is found in a variety of ways. Descartes,

for example, believed that an object in circular motion had only the centrifugal force. If a

rock was in a sling, the rock would have the tendency to flee from the center in a straight

line. As for moving in a circle, this was the natural motion of the rock in sling—that

circular motion is inertial as was rectilinear (straight) motion. This is in tension with

Descartes’s statement that it is the natural, inertial tendency for an object to move in a straight line if undisturbed, so he appears to be of two minds on this subject. However, to

keep with the idea that the rock would otherwise move along a tangent line, this tangent

vector is decomposed into the centrifugal tendency and the circular tendency which

53 balance out except for the forward component. The centrifugal force also explains why there is tension in the sling (see essays in Gaukroger et al. 2000). In a way, Descartes is a transitional form between ideas of circular impetus, centrifugal tendencies, and modern descriptions of forces on objects in circular motion. It is also an interesting reversal:

Benedetti needed a centripetal force to impress upon the tangential, linear impetus of a bit of matter to explain circular motion, but for Descartes the tangential inertia is a result of the centrifugal and circular (an effective centripetal) forces or tendencies rather than that which is resisted.

In other cases from around the same time, the centrifugal force occurs apparently for a common reason: the need to balance out some other, inward-pointing force.

Christiaan Huygens was the first to introduce the idea of the centrifugal force in 1659, but his first published mention of it came in 1673, after Descartes’s books on the subject. He also introduced the term, calling it vis centrifuga. Like Descartes, the centrifugal force explained the tension in a sling when whirling a stone, but it was designed as a balancing force and not one that produced motion (Meli 2006). Gottfried Leibniz, on the other hand, developed a centrifugal idea that did not have a perfect balancing of the centripetal force. This was used to explain the elliptical paths of the planets. This was attacked by

Newton who tried to have the centripetal force on the planets (from gravitational interactions) be matched by the centrifugal force so there would be a balance of forces based on his third law of motion (Meli 2005). However, this changed from an earlier model with the centrifugal force sometimes being weaker or stronger than the centripetal force as a comet or planet went about the Sun as seen in a letter to John Flamsteed in

1681. By the time Newton wrote his De Motu in 1684, which still had linear impetus, he

had abandoned the centrifugal force. And yet that same force was used to explain rotating

liquids in his Principia (Steinberg et al. 1990).

The debate about the centrifugal force was also charged by attempts to explain the

motions of the planets by a direct, mechanical pressure on objects and not the action-at-a-

distance that Newton would advocate. Much like the pre-Socratic philosophers

mentioned earlier, the forces on the planets were explained by vortices. Leibniz, for

example, created a theory of a vortex that had its speed fall off radially, an inverse

relationship to radius. The vortex then provided a force towards the center that was

proportional to the inverse square of radius, and centrifugal force went as the familiar

relationship. In modern equation format, his expression of the two forces on a body 2 푣in circular⁄푟 motion was

= 푎 푏 푟̈ 3 − 2 where a and b are constants (Meli 2006). In푟 the 푟vortex theory, objects are pushed toward

the center by the whirl of fast-moving fluid (Huygens estimated it rotating at 17 times the

speed of an object at Earth’s equator [Meli 1990]), but bodies also have an inclination to

flee away from the center of rotation. The reason bodies were pushed toward the center of

rotation was because a body will take up some volume in place of the imagined fluid and

will not have the same centrifugal force; thus centripetal (and here gravitational) force is

a result of differences in centrifugal force (Meli 2006). In a sense, the vortex theory was based on Archimedean ideas of buoyancy and would thus be a mechanical hypothesis rather than what Newton would argue with his idea of gravitational force that seemed uncouth at the time.

How Newton came to his final beliefs on the subject were likely stirred by another physicist of his time, Robert Hooke. In a letter sent to Newton in 1679, Hooke wanted

Newton’s opinions about some speculations of objects undergoing revolutions:

And particular if you will let me know your thoughts of that of compounding the

celestiall motions of the planetts of a direct motion by the tangent & an attractive

motion towards the centrall body. (Meli 2005)

Here, “motion” seems to have the equivalent meaning as force, so it appears that Hooke is introducing an impetus-like force in the tangent direction (which Newton would not have objected to at this date) and then a centripetal force as well. This looks not unlike what Benedetti has suggested a century earlier. Newton did not abandon the centrifugal force, and in response to Hooke’s letter he spoke of centrifugal force sometimes overpowering and being overpowered by the centripetal force, a belief repeated two years after this correspondence to explain the orbit of comets. The resolution for Newton came with the application of his third law, making it always the balancing force of the centripetal motion. Now, this would be an incorrect use of the law, but this allowed

Newton to describe planetary motion only by way of the gravitational force (Meli 2005,

2006). It appears Newton was groping with some way of avoiding having to consider the natural tendency to believe in something like the centrifugal force, and it was his

(mis)use of the third law that allowed him to explain celestial motion while also remaining consistent with his other principles of motion. The third law was also how

Newton attacked Leibniz’s theories, but this was not the primary argument to get end the application of the centrifugal force in 17th and 18th century physics.

The resolution of these ideas was through a considerable amount of argumentation. Newton, for example, pointed out how that if the path of an object was straightened the centrifugal force described with Leibniz’s equation would not disappear, and thus there would be a force on an object moving with constant velocity (Meli 1993).

More powerfully, Newton was able to explain the motions of the Earth, Moon, and other planets, as well as comets, with his simpler, centripetal-only theory of gravity, and it did so in a way that did not require a special vortex for each planet (as it did for Huygens and

Leibniz). Moreover, it could explain all of Kepler’s Laws of planetary motion, which

Leibniz could not (Cohen 1987). The debate did not appear to be resolved up until the

1740s and 50s, and the successful prediction of the return of Halley’s Comet in 1758/9 was the “acid test” of Newton’s theories. It appears then that to reach the general acceptance of Newton’s laws of motion, it required the final abandonment of circular impetus, linear impetus, and then arguments showing the failed utility of the centrifugal force, and process that took about 400 years from Buridan’s writings on the subject.

3. Conclusions from History

As such, the beliefs about circular motion were not well expressed by ancient and medieval writers, but what is seen is first the circular impetus idea, especially for Jean

Buridan, and then the problem of centrifugal force, a force designed to counter the inward component of force when an object undergoes rotation. The origins of the centrifugal misconception appear to be the outgrowth of attempts to deal with centripetal forces. At

first Benedetti had a force in the tangent direction on a circular path deflected by the

centripetal force, and then with the rejection of linear impetus new ideas to explain

circular motion emerged that required a balancing of forces but still incorporating the

sense that an object flees from the center as Benedetti had in mind. Later when scientists

such as Huygens, Leibniz, Descartes, and Newton tried to provide a rigorous description

of the situation, they had developed the straight-out centrifugal force rather than the tangential force. Later still, Newton abandoned all of this for the centripetal force acting alone on a body in circular motion.

iii. Sources of Misconceptions

The discovery of the widespread presence of student misconceptions leads to the

question of the origins of such beliefs. The commonality suggests something that is

developed through either biological (i.e., evolutionary psychology, cf. Boyer & Barrett

2005) or experiential processes. The former is unlikely given that modern physicists have

learned to overcome earlier, naïve beliefs, something that seems improbable if these

beliefs were hard-wired at a genetic or developmental level. Moreover, an evolutionary

explanation fails to account for why young children have a model closer to that of

Newtonian mechanics but then transforms into the more common naïve view in later

children and adults. This background information makes a genetically inherited model

untenable, at least for the specific physics misconceptions that are of current interest to

PER, and instead it leaves the lifetime of experience that produces the physical

conceptions that students bring to their high school and college classes, not to mention

the adult population that has not received formal college instruction in the sciences and

the considerable percentage of college students that nonetheless hold onto their

preconceptions.

Asking how such beliefs come about can be a question that helps answer several

others. It could help elucidate the history of science and why it took nearly two thousand

years of work to come up with an accurate explanation of dynamics. Furthermore, such

an answer could indicate when such beliefs have their mental foundations laid. In

connection to this, knowing when such physics beliefs solidify into the forms known to

PER indicates when such concepts should be taught. Knowing when to act may help

avoid having students create these misconceptions in the first place. Lastly, knowing the

sorts of experiences that students use to form their physical conceptions may indicate the

types of lectures or lab experiences that can best help to recontextualize them to better

conform to Newtonian ideas. If such knowledge were in the possession of PER, it could

not only help maximize student understanding of classical physics, but it could

potentially reform physics instruction such that it is unnecessary at the college level,

except for more advanced study; that is, Newtonian physics could become the expected

background of a college student as much as arithmetic and basic algebra.

It is rather shocking to see that there is little in the way of published literature

explaining the origins of student misconceptions. While DiSessa (1993), for example,

tries to explain the structure of such beliefs, he provides no elucidation for why these

beliefs are the ones forming his idea of p-prims rather than others. The closest thing to an explanation for certain student misconceptions comes from McCloskey et al. (1983) where they considered a subject walking forward with constant speed and then dropping a ball. The largest percentage of students (49%) believed the ball would fall behind them

59 and land on the ground behind their feet at some distance; this is in contradiction to what does happen: the ball falls and lands on the ground at the midpoint of one’s stride.

Students who had taken a physics course did better than those who did not by a significant margin: 60% of non-physics class students said the ball would fall straight down relative to the ground, while 40% of physics-trained students made the same mistake. McCloskey et al. suggest that the results are due to perception: an object dropped in such a circumstance appears to fall straight down from the perspective of the walker, so students draw the object falling straight down and ignore any forward momentum. Recontextualizing the problem with a conveyor belt moving the ball forward eliminated this sort of response drastically, though this also depended on how the object appeared to be carried. It was suggested that having objects like this seen in a third- person perspective rather than in first-person helped avoid the visual illusion.

However, the work by McCloskey et al. has not been reproduced with a younger group of subjects, nor have others tried to explain other misconception origins. Chapter 3 in this paper will help to remedy this and begin to tell researchers how such beliefs come about.

B. Assessment Tools

Given the well-documented common misconceptions noted above and elsewhere, instructors and researchers will want to be able to gauge the prevalence of these beliefs in students. In particular, it will be important to researchers to know how much a given tool or pedagogy will affect the changes in a population’s conceptions of physical

phenomena. The most common tool for classical mechanics is the Force Concept

Inventory (FCI), which is discussed in detail. Other established tests are reviewed as well.

i. Force Concept Inventory

a. Creation and Assessment with Factor Analysis

Details of the FCI were first published by David Hestenes, Malcolm Wells, and Gregg

Swackhamer in 1992 (Hestenes et al., 1992), with Hestenes having worked on an

assessment tool about a decade earlier (Halloun & Hestenes 1985a). This inventory of 30

multiple choice questions was designed to look for the most common misconceptions that

were likely to exist in the subjects of kinematics and forces, especially forces that are

fictitious or imply the ideas of impetus. The list provided the misconceptions to measure

and what problem answers were designed to detect them, and is reproduced in Appendix

In order to categorize their test questions and answers, Hestenes et al. (1992)

described six “commonsense categories”: kinematics, impetus (Newton’s 1st Law), active

force (Newton’s 2nd Law), action/reaction pairs (Newton’s 3rd Law), concatenation of

influences (superposition principle), and other influences of motion (kinds of forces). For

kinematics, a question could be one that distinguishes velocity and acceleration. For

impetus, scenarios are given in which an object continues on a trajectory as if the

previous force acting on it continued to push. For active force, questions may ask about

forces on a body with no motion or what happens to motion when not actively pushed.

Concerning action/reaction pairs, the chief question is about the proportionality between

mass and the force acting upon it during a collision. For concatenation, this has a slew of

issues, such as the largest force determining direction rather than the sum of forces or the

last applied force fully determines direction. Lastly, other influences include

misunderstandings about friction, centrifugal force, and gravitational force.

In their paper on FCI (Hestenes et al. 1992), the authors described the

correspondence with their test results and interviews with students, and their categories

were logical combinations for a Newtonian thinker. However, the researchers did not

demonstrate that what they measured was a force concept rather than something else,

such as familiarity with certain scenarios rather than contact with the underlying abstract

physics. To test the FCI, Huffman & Heller (1995) performed a factor analysis, a

statistical tool that seeks out correlations between various test questions to see if they

may be measuring a single phenomenon or ‘factor.’ As they describe it, if someone produced a multiple choice test for velocity and acceleration concepts, that test should have two factors: one measuring velocity and another measuring acceleration. If one

factor exists, then two questions asking about the same concept should have a strong

correlation such that if one question is answered correctly the other will probably be

answered correctly as well.

Factor analysis is done in the following way when the number of factors is

uncertain and errors in data collection (i.e., sampling error) make simple linear algebra solutions untenable (cf. Kim & Mueller 1978; Gorsuch 1983). First, one needs to calculate the Pearson correlation coefficient between any two items. This value, for N members, is

( )( ) = , 푁 ( 1) ∑푖=1 푋푖 − 푋� 푌푖 − 푌� 푟푥푦 푁 − 휎푥휎푦

where and are the average values of all and , and and are the respective

푖 푖 푥 푦 standard푋� deviations.푌� In the case of FCI, the number푋 푌of members휎 would휎 be the number of

students being examined, and items X and Y would be test questions, their averages being

the average number of times the question was answered correctly. For n items, this

correlation data can be placed into an × correlation matrix (in which all diagonal

elements equal 1). 푛 푛

. 푟11 ⋯ 푟1푛 � ⋮ ⋱ ⋮ � 푛1 푛푛 Given this, principal component analysis푟 is ⋯done,푟 in which the variance ( ) is 2 푥푦 maximized in groups of items. The first factor represents the groups with푟 items that

account for the greatest amount of variance; the next factors follow in order of remaining

variance and are uncorrelated with all other factors. The model that represents factor, ,

푗 is 퐹

= 푛 ,

퐹푗 � 푊푗푖푋푖 where X is an item, W is the factor coefficient,푖=1 and n is the number of items. The variance

is maximized usually by computer software packages, such as the Statistical Package for

the Social Sciences (SPSS), and the process tends to produce more factors than expected.

The selection of factors is often a human judgment, and researchers tend to look for

factors that account for at least 5% or 10% of the total variance. Software packages such

as SPSS drop factors with eigenvalues less than 1.0 by default unless changed by the

user. The procedure can also be done by finding the orthonormal eigenvectors of the

correlation matrix such that each eigenvector will be a factor; their corresponding

eigenvalues will then indicate the weight of each eigenvector, so eigenvalues that are

close to or equal to zero indicate that their corresponding factors do not have a notable effect (see chapter 6 in Gorsuch 1983). However, as noted above, this procedure is usually impossible because of imperfections in data collection. The computer-based procedure with the principal factors having eigenvalues greater than or equal to one is a good rule of thumb so long as the number of members N is large, and for less than 40 items (n). If these limits are met, the number of factors is expected to be between n/5 and n/3 (Gorsuch 1983). This would suggest that the six factors expected by Hestenes et al.

(1992) in the 30-item FCI should be a good fit for this sort of factor analysis, so long as a

large enough sample of students is used.

After concluding which are the dominant factors, another step is to consider

which items are most strongly associated with that factor, again using correlations, this

time between a given item and a factor. This correlation is often called “factor loading.”

To represent the relationship between an item X, a factor F, and their correlation r, a

linear combination of n factors is used:

= 푛 .

푋푗 � 푟푗푖퐹푖 The correlation between a factor and item can푖=1 be calculated by taking the eigenvalue of a factor and dividing it by the number of items. The eigenvalues are themselves calculated from the correlation matrix above. Once it is known which factors correlate most strongly

to which items, it is then necessary for the researcher to identify what that factor is, based

on background knowledge and what pattern can be place on the items strongly associated

with a factor.

In their examination of FCI, Huffman & Heller (1995) tested their factor analysis

on groups of high school and university students, analyzed separately, with 145 and 750

students respectively. For the high school group, there were ten, rather than six, factors

discovered, but only two could account for enough of the variance to be significant

(>5%). Those two factors were the third law and kinds of forces. Only three of the 12

questions related to the FCI’s kinds of force actually were identified as a single factor. On

the other hand, the university results indicated only one factor, the kinds of force, and

four of the 12 questions were identified as measuring that factor. Overall, the factor

analysis indicated that most of the questions were not correlated together, and the logical

categories of the FCI authors had little relation to student conceptualization. This means that the FCI tends to not measure a “force concept” but more of a collection of student knowledge (Huffman & Heller 1995).

These results were controversial, which lead to a response from Hestenes &

Halloun (1995) and a retort from Heller and Huffman (1995). The former argued that,

taken as a whole, the test achieves its objectives of measuring student understanding and

weeding out Newtonian and non-Newtonian thinkers. Moreover, Hestenes & Halloun

(1995) argued that factor analysis is not a valid tool to determine the validity of a test, yet

they also contended that the results from the critical paper actually conformed to their expectations and helped show the validity of the FCI; they argued if one looked only at

students who had a decent Newtonian understanding, which they defined as having an

FCI score between 60% and 85%, then the factor analysis would show the desired

clustering around a factor. In response, Heller & Huffman (1995) pointed out that the test

failed to find force concept factors in the intended audience of non-expert physics

students, so saying it works for a particular subset is not exactly an endorsement of the

test. Nonetheless, in the select group who passed the somewhat arbitrary FCI score threshold of 60% there was no improvement in the factor analysis. What this meant is that students who scored less than 60% on the FCI showed no less a coherent Newtonian understanding than students who scored between 60% and 85%. However, this latter paper agrees with the FCI authors: the FCI can still be useful as a diagnostic tool and for evaluating instructional effectiveness by using FCI as a pre- and post-test.

b. Model Analysis and FCI

Since what has been called the “4-H” debate, other developments in the use of FCI have come forward, particularly looking at what may be common student misconceptions rather than testing for singular force concepts (Bao & Redish 2001; Bao & Redish 2006).

Since students do not put together abstract topics as expected by the original 1992 FCI paper, researchers should instead consider how the students develop their ideas after receiving instruction. It may be possible to watch how topically related questions that were uncorrelated at first become correlated. For example, questions pertaining to

Newton’s 1st Law may not all be answered correctly or incorrectly and may reflect the

student’s understanding of the situation through context rather than a general idea. The

transition from novice to expert can be better understood through the process of model

analysis.

Given that knowledge and the activation of such knowledge is greatly

contextualized, it may be the case that different ways of approaching problems exist

because of a difference in the details of the question, even if they fall into a larger group

understood by experts in physics. In that case, students may be applying different models

to different contexts or they may apply a single model to all of the contexts, whether that

model is correct or not. To investigate how student models are applied and how they evolve with instruction, a certain mathematical construction is needed. The following mathematical formalism is summarized from Bao & Redish (2006). Suppose there are i

number of models in solving a set of problems that should be a part of a single concept,

nd such as Newton’s 2 Law. For each model, there is a probability pi that a student will

apply the ith model to a problem. Suppose there are m questions asked that pertain to that

single concept. Then the probability distribution vector for student k can be defined as

푃�⃗푘 푘 1 푘 = 1 = 1 푝푘 푛푘 2 2 푃�⃗푘 ⎛푝 ⎞ ⎛푛 ⎞ ⋮푘 푚 ⋮푘 k 푖 푖 where n i is the number of questions answered⎝푝 ⎠ that⎝ fit푛 model⎠ i by student k. From this, it

is possible to produce a density matrix Dk of student answers if instead of using the

probability distribution vector, vectors of unit length uk are used, which are obtained by

taking the square root of the probability distribution vector .

푃�⃗푘

푘 푘 �푝1 1 �푛1 = ⎛ ⎞ = ⎛ ⎞ 푘 푘 ⎜ 2 ⎟ ⎜ 2 ⎟ 퐮푘 ⎜�푝 ⎟ ⎜�푛 ⎟ ⎜ ⎟ √푚 ⎜ ⎟ ⎜ ⋮ ⎟ ⎜ ⋮ ⎟ 푘 푘 �푝푖 �푛푖 where ⎝ ⎠ ⎝ ⎠

= 푖 = 1. T 푘 퐮푘퐮푘 � 푝푙 푙=1 67

For the density matrix, let us assume there are three models, which can be described as

expert, naïve, and random. If this arrangement is accurate, then it Is to be expected that

untutored students will mostly fill in a naïve response, graduate students and above in

physics should fill the expert response, and the random answers should be rarely chosen.

This will be seen later to be a good assumption. What this means is that the density

matrix Dk will be in square 3x3 form. The matrix is then formed from the unit probability

vector uk:

푘 푘 푘 푘 푘 1 1 1 2 1 3 = × = ⎡ 푛 �푛 푛 �푛 푛 ⎤. ⎢ ⎥ T ⎢ 푘 푘 푘 푘 푘⎥ 퐃푘 퐮푘 퐮푘 �푛2 푛1 푛2 �푛2 푛3 푚 ⎢ ⎥ ⎢ ⎥ 푘 푘 푘 푘 푘 ⎢�푛3 푛1 �푛3 푛2 푛3 ⎥ The advantage of producing this matrix ⎣for a student is that it preserves⎦ the structure of

the probability of that student using a given model. An average density matrix can also be formed for all the students in a class simply through addition of all N student density matrices and dividing by N:

1 = 푁 .

퐃 � 퐃푘 푁 푘=1 With this density matrix, it is possible to perform analysis to see how much of a

class uses a particular model and how often model applications are mixed. Extracting the

eigenvalues and eigenvectors from the density matrix will show what are the predominant

applications of models in a collection of students. This can be shown in the following

way. Given density matrix D with all real, non-negative definite elements (i.e.,

Hermitian, = ), it will have real, non-negative eigenvalues , , … , and 2 2 2 � 1 2 푤 orthogonal eigenvectors퐃 퐃 vl where l = 1, …, w. For the three dimensional,휎 휎 symmetric휎 68

matrix D, w will have a value of 3. Then there will exist a matrix of eigenvectors

= [ ] such that

1 2 3 퐕 퐯 퐯 퐯 0 0 = 02 0 . 휎1 T 0 02 2 퐕퐃퐕 � 휎 2� This follows from the properties of eigenvectors and is휎3 not be proven here.

Now, with the defining equation of D

1 1 1 = 푁 = 푁 = 푁 | |, T 퐃 � 퐃푘 � 퐮푘퐮푘 � 퐮푘⟩⟨퐮푘 푁 푘=1 푁 푘=1 푁 푘=1 the equation can be rewritten as

= 3 = 3 | |. 2 T 2 퐃 � 휎푙 퐯푙퐯푙 � 휎푙 퐯푙⟩⟨퐯푙 푙=1 푙=1 From this equation, it is easy to see that

1 1 = 푁 = 푁 = . T 2 T 2 퐃퐯푙 � 퐮푘퐮푘 퐯푙 � 휎푙 퐯푘퐯푘 퐯푙 휎푙 퐯푙 푁 푘=1 푁 푘=1 This can also be written in bra-ket notation:

1 | = 푁 | | = | . 2 퐃 퐯푙⟩ � 퐮푘⟩⟨퐮푘 퐯푙⟩ 휎푙 퐯푙⟩ 푁 푘=1 Continuing from these definitional properties of matrices and their eigenstates,

define the scalar value as the agreement between the kth student’s model vector uk

푙푘 and vl, the lth eigenvector푎 of D. This means

= = = | . T 퐓 푙푘 푘 푙 풍 푘 푘 푙 Combine this with the previous푎 equation퐮 퐯 to give퐯 퐮 ⟨퐮 퐯 ⟩

1 1 1 = 푁 ( ) = 푁 ( ) = 푁 = , T T 2 퐃퐯푙 � 퐮푘퐮푘 퐯푙 � 퐮푘 퐮푘퐯푙 � 푎푙푘퐮푘 휎푙 퐯푙 푁 푘=1 푁 푘=1 푁 푘=1 which yields

1 = 푁 .

퐯푙 2 � 푎푙푘퐮푘 휎푙 푁 푘=1 This shows that an eigenvector of D is the weighted average of all the individual

student model vectors (uk), and it is weighted by the agreements between the eigenvector and the single student model vectors ( ). As for the interpretation of the eigenvalues,

푙푘 note that the left-multiplication of any of푎 the eigenvectors vl by itself gives

1 1 | = 푁 | = 푁 = 1. 2 ⟨퐯푙 퐯푙⟩ 2 � 푎푙푘⟨퐯푙 퐮푘⟩ 2 � 푎푙푘 휎푙 푁 푘=1 휎푙 푁 푘=1 This means

1 = 푁 , 2 2 휎푙 � 푎푙푘 푁 푘=1 so the lth eigenvalue is the average of the squares of the agreements between the lth

eigenvector and the individual students model vectors. This means that the eigenvectors and eigenvalues can tell us about the average model states of the class and which models

or mix of models are the most dominant. When performing analysis, if there should be an

eigenvector with a large eigenvalue compared to the others, this will contain the

dominant features of the single student model vectors and thus be the most telling about

the mental model state of the class.

To observe the state of the class, the eigenstate data can be graphed to form a

model plot (see Figure 5) as follows: Suppose that two models are dominant, which is the

running assumption for this research (the expert and naïve models are most common, and 70

Figure 5. Model Analysis Graph. From Bao & Redish (2006).

other choices are rare), then we can take a class model state eigenvector (vm) and plot its components into a two-dimensional space. The orthogonal axes represent the probability that one of the two models is used by the class. A point on this graph, called the class model point (labeled B), represents the probability of each model choice. The probability that a given mental model i will be used is = , which will always be non- 2 2 푖 푚 negative; the values i = 1, 2 will provide the vertical푃 and휎 퐯 horizontal푖푚 components of B. The

limit of the probability is 1, so the class model point must be below the line + = 1.

1 2 The better the two models represents the class’ mental state, the closer the 푃point 푃B is to

the upper boundary. The effects of a third model (v3m) that is not considered will create

some distance away from reaching the upper limit, which can also be described in the 71

graph. The effect of the third model will be its probability, , so we can measure 2 2 푚 3푚 that distance from the upper limit to be (1 ). Because휎 퐯 v3m is small, the 2 2 푚 3푚 expression can be approximated as simply the 휎eigenvalue− 퐯 . The distance d between the 2 푚 model point and the boundary then can provide the eigenvalue휎 of the model state. This

can be estimated using the above approximation and geometry:

1 2. With this setup, the probability2 space can be defined into three general regions: 휎 ≅ − 푑√ the Model 1, Model 2, and Mixed Regions. The closer the position of B is to (0,1), the

more it represents the use of Model 2 and is in the Model 2 Region, and similarly when B

is closer to (1,0) and the Model 1 Region. To graphically divide these regions, lines with

slopes 1/3 and 3 are drawn; this arrangement provides half of the probability space to the

Mixed Region. The space close to the origin can also be described as the Secondary

Model Region; if point B is found here, it would mean the two primary models fail to

account for a total of at least 40% of the probability of what models are used. This region

is separated using the line corresponding to + = 0.4. By plotting the initial class

1 2 model point as well as the class point after instruction푃 푃 it is easy to see the evolution of the

average class model state, which we hope involves moving from naïve (Model 2) to

expert (Model 1). Plotting the initial and final class points for different teaching methods,

it becomes easy to compare which curricula moved students towards expert model usage.

Before returning to the utility of the FCI, it is necessary to point out another feature of model analysis compared to factor analysis. If students are applying a single model, either the naïve or expert, in all contexts, then factor analysis should see the appropriate questions clustering as was proposed by the authors of FCI. However, if a population is mixed in their use of models, then there will be little to no clustering 72

besides that from chance, and factor analysis will detect no dominant factors; it may well

generate as many factors as there are questions because of the different contexts the physics is placed into, each activating mental models differently. This would explain why

Heller & Huffman (1995) did not see the significant factors that were expected by the makers of FCI for the intermediate-scoring students; this population is perhaps the most likely to have a mixed use of models and be the most context-dependent. Model analysis, however, will be able to provide structural information in either case, and therefore this tool can better measure students transitioning in their use of mental models through the learning process.

Now, when it comes to the use of FCI in model analysis, the multiple choice answers will need to have not only an expert answer, but each question will also need to have good distractors, answers that will likely fit into the common naïve student models.

Fortunately, the FCI question answers include good distractors as indicated by interview data, though future tests could provide more questions with adjusted contexts to better extract student thinking (Bao & Redish 2006). The same can be claimed for other assessment tools.

ii. Other Inventories

While other mechanics misconception inventory tests have not received the level of scrutiny or use by researchers as FCI, for completeness a survey of most of these inventories is provided. In form they are similar to the FCI as they are multiple choice tests with possible answers that relate to common misconceptions. This could make them

73 useful for model analysis, and it is likely that students would fail to conform to expert categories when a factor analysis is performed.

Prior to creating the FCI, some of the same researchers had created a tool known

Mechanics Diagnostic Test or MDT (Halloun & Hestenes 1985). This was a precursor to the FCI test that was published several years later. Using 28 questions, MDT had questions more abstract than those used for the FCI but covered the same general conceptual areas. The validity of the test was evaluated by comparison with interviews with students, and there were very strong correlations with student response on multiple choice questions and in interview answers. Compared to FCI, the MDT multiple choice questions do not have as many options and they do not seem to have equally good distractors.

In the year the FCI was published, the Mechanics Baseline or MBT was released

(Hestens & Wells 1992). The MBT has three conceptual categories: kinematics (linear and circular), general principles (force laws, energy, momentum), and other forces

(gravitational and friction). The MBT has 26 questions but covers even more physics concepts than the FCI, namely energy and momentum. In addition, the MBT has graphical interpretation in several of those questions, and this may provide another source of confusion for students who may have difficulty interpreting graphs. For that, another diagnostic tool exists, the Test of Understanding of Graphs in Kinetics or TUG-K

(Beichner 1994).

There also exist multiple choice tests focusing on energy concepts, such as the

Energy Concept Inventory or ECI (Swackhamer & Hestenes 2005), or energy and momentum such as the ECS (Singh & Rosengrant 2003). Besides classical mechanics,

there are tests for electricity and magnetism and for thermodynamics. In addition, there

are tests focusing on modern physics concepts, but so far there is less research based on

them because most PER research has focused on Newtonian concepts or E&M.

With the considerable number of other tests currently on the market for use by

instructors, there is likely the desired tool to do research on undergraduate or high school

physics students. An instructor can use these tools to see where their students have

misconceptions and how well they are overcome them. However, these tests may not be

as effective for a younger audience. Research on the tests’ utility with children may be necessary here if one wishes to investigate physics concepts for students that may have

difficulties with the vocabulary, among other potential limitations.

CHAPTER 3. STUDENT PHYSICS MISCONCEPTIONS: TESTS AND INTERVIEWS

A. Previous Research and History of Science

In the preceding chapter, the previous research into student misconceptions about basic

dynamics was discussed, as was the history of physics concepts before the modern era.

The greatest focus was laid upon the concepts of linear and circular impetus. This was a

sort of imbued force into an object that was traveling either in a straight line or circular

path. What seems to be the major point in common in these ideas is that something is

necessary to push on in order for the motion to remain after the initial interaction put

something into motion. For example, Aristotle believed that after throwing a rock the

projectile used the air to push itself forward until that imbued force dissipated and then

the rock came to rest. After Aristotle, figures such as John Philoponus (6th century),

Avicenna (11th century), and Jean Buridan (14th century) supported an impetus concept that did not require that the projectile to be pushed by anything to remain in motion, but rather the essence of the violent force that caused it to move remained and continued to

push that object forward. Even scientists like Galileo and Newton (17th century) struggled

against the notion of requiring a force in order to remain in motion.

Among modern-day physics students, the same ideas are reproduced, sometimes in striking similarity to natural philosophers from the past. In several studies from North

American and Europe over several decades, students are reported to commonly believe that there is a force vector on objects in their direction of motion even if it is

inappropriate. Interviews with students confirm that they are not just adding in the

velocity vector but they actually believe there is as sort of continuous push acting on the

object as it moves. This imbued impetus force is also described to decay and stop after

the object is not being pushed by a force external to the body. Because of this, students

believe that when walking with a constant velocity and carrying a ball, the dropped object

will fall behind them rather than at their feet. Students may describe tossed objects in a

fashion similar to that of Hipparchus (2nd century BCE) who said that the upward impetus

on an object is first greater than the gravitational force, weakens as it climbs, matches

that of weight at its high point, and then decays away as the object reaches its maximum

free-fall speed. Others may understand that there is a continuous acceleration downward,

but again the force downward becomes stronger, as Jean Buridan intimated. Other than in

some particulars, it seems there is a good mapping of current linear motion

misconceptions to past theories of dynamics.

In the case of circular motion, there are two general categories of misconceptions

that are notable from the history of physics: circular impetus with properties like that of

linear impetus, and centrifugal force as an actual cause of acceleration or due to an

interaction rather than a result of the chosen reference frame. The circular impetus belief

seems to have existed in the days of the earliest natural philosophers, the pre-Socratics such as Anaxagoras and Democritus (5th century BCE), carried on by Aristotle (4th

century BCE) and an ancient writer pretending to be Aristotle, and well into the medieval

period, being reproduced by Jean Buridan who effectively introduced impetus to the

Latin-speaking and reading world. Throughout these millennia, there was no sign of

either a centripetal or centrifugal force.

The earliest citation of a centripetal force came from Giovanni Benedetti, and he

argued that circular impetus was not natural as was linear impetus. In his system, a piece

of a rotating object has a tendency to move in a straight line, while an inward-pointing

force continues to deflect it from a straight path. In a sense, this has a center-fleeing force involved—the linear impetus—and it may make sense out of the arguments that came later in the history of science. The earliest use of the true centrifugal force comes from

René Descartes and Christiaan Huygens in the 17th century. The force was used to

explain the tension in a rock sling when the projectile is spun around. Descartes would

have an additional force following the circular path, while Huygens would have a proper

centripetal force. When Isaac Newton worked on his physics, he also included the

centrifugal force, but he has to inappropriately apply it to planetary motion such that it

would not affect his calculations. Later still, the centrifugal force would be considered a

fictitious force and result from using a non-inertial reference frame rather than an

interaction.

Turning to modern physics students, the notions of circular impetus and

centrifugal force reappear. Many believe that an object undergoing circular motion, such

as when a ball moves through a circular tube, it continues to move in the same curved

path as the original motion, straightening out as distance traveled increases. In other

words, this is a case of dissipating circular impetus. Centrifugal force, on the other hand,

appears to be less commonly cited by students, and it appears that it requires some sort of

prompting to come to the conclusion that it is acting on a body undergoing circular

motion. It may be significant to understand this, as one misconception may be natural

while the other is brought about by prompts from the instructor of the contextualization

of the questions asked of students.

While there is a considerable amount of literature showing the existence and

nature of force and motion misconceptions among high school and college students, the

research about the origins of and relationships between these beliefs held by incoming

physics students is not well-explored. McCloskey et al. (1983) is one of the few studies

that looked into the potential cause of the belief in linear impetus concepts, but there has

not been any significant follow-up on this sort of research. Kaiser et al. (1986), who looked at circular motion issues, noted how younger students did not have the circular impetus beliefs nearly as strongly as older students, so this shows there is likely an

evolution in the mental models of aging students that leads them towards the better-

known misconceptions. What experiences may be causing this are unfortunately not

explored in the literature as of now; perhaps it is a natural development, and perhaps it is some aspect of the current educational process.

B. Research Questions

There appears, then, to be a considerable amount of overlap between student misconception and pre-modern scientific ideas, especially among ancient and medieval natural philosophers. Such a phenomenon demonstrates that the beliefs are not likely a product of modern, Western cultural or educational norms, but rather it is something more innate to how people experience the world. Moreover, these beliefs are not easily corrected by a few weeks of direct instruction; the impetus concepts are strongly

ingrained by the time students take physics in late high school or college, if they ever

receive instruction.

If the sources of these misconceptions were more clearly identified, it could be

possible to design a constructivist approach to help students recontextualize the

experiences or abstractions to better conform to Newtonian dynamics. It is also possible

that continued research will inform us when it is best to educate students in these matters

and avoid or minimize the emergence of these misconceptions in the first place.

However, in order to understand the beliefs of students before instruction, it is necessary

to see how many misconceptions are actually in play. If two or more misconceptions are

related or are following from a common premise (perhaps a common p-prim), then this

will help focus on what needs to be corrected and how concentrating on that one point

can help avoid other appearances of related misconceptions.

Along with naïve physics concepts, the students will learn in the course of their

class about Newtonian concepts. While the students may be able to regurgitate Newton’s

laws of motion, they may not actually apply those laws or see how they are pertinent to

their understanding of physics in daily life. Still, in a homework or test problem students

may know how to use the more correct physics concept, but perhaps they do not apply

that sort of thinking beyond the context of the classroom. Part of what needs to be

considered is what are the interactions and activations of the mental physics models that

students have.

How the student activates the use of a model based on prompts from another person, especially a teacher, or a particular circumstance must be considered (cf. diSessa

1988). For example, it was noted how circular impetus concepts were the prevalent belief

80 before the early modern period, and it is also natural for students to believe this as well.

However, with the demise of circular impetus scientists had created the centrifugal force, which could explain the tautness of a rock sling when spun. In other words, the more natural belief was abandoned and a new, yet still erroneous belief, replaced it. Perhaps the way questions are posed to students will activate a belief such as the centrifugal force.

All of this background leads to the following questions:

1. What are the experiential origins of student misconceptions of forces and motion?

2. How related are these misconceptions? Do they have a common origin or are they

of separate origin?

3. How are naïve and expert physics models activated in the minds of students?

4. How does instructor prompting affect how students answer physics questions?

C. Experimental Designs

In order to attempt to answer these questions and provide context for future research related to them, two major experimental interventions were designed for this purpose.

The first was the use of a four-question multiple choice test that dealt with one- dimensional motion and constant force. This test was followed by an interview with the student about those questions. The second intervention was also an interview, but this had

`questions related to the feelings of motion that the student reported experiencing when undergoing a form of constant acceleration they were likely to have experienced.

i. 1D Motion Test and Interview

a. Multiple Choice Test

The first activity that a student was assigned was a test with four questions. Each question

had several multiple choice answers, which would exhaust all of the logically possible answers, though there were only two answers that were of particular interest: one corresponding to the correct, expert physics answer, and one corresponding to a common misconception.

The questions were chosen and designed to capture the naïve beliefs about how force is proportional to velocity rather than acceleration, which include situations that would be very familiar to the student’s experiences, though idealized (i.e., no friction).

The first two questions concerned the force necessary to have a sled either maintain speed or constantly accelerate where the ground was level and without friction. Because these pushing situations are likely part of the student’s real world experience, they are likely to recall the feeling of pushing a sled or something similar, though they may be confounded without considering the effects of friction or the lack thereof.

The misconception these sled questions are likely to elicit is that to maintain speed there is the requirement of a constantly applied force. Moreover, there is a proportional relationship between the applied force and speed. However, the questions are not designed on their own to detect what happens to an object when it is no longer being directly pushed, though later in the experimental process the interviewer could ask for further information (more on procedure below). The next set of questions is better geared for the question of what happens to objects moving without direct contact.

1. A sled moves on ice. Friction is so small that it can be ignored. A force can be

applied on the sled.

Which force would keep the sled moving toward the right and speeding up at a steady

rate (constant acceleration)?

A. The force is toward the right and is increasing in strength (magnitude).

B. The force is toward the right and is of constant strength (magnitude).

C. The force is toward the right and is decreasing in strength (magnitude).

D. No applied force is needed.

E. The force is toward the left and is decreasing in strength (magnitude).

F. The force is toward the left and is of constant strength (magnitude).

G. The force is toward the left and is increasing in strength (magnitude).

The last two questions deal with a ball or marble thrown straight up by a person at

ground level in ideal conditions (i.e., no air resistance). The student considers two

situations: when the ball is traveling upward, and when it is falling back down. In each of

these cases, the student is asked about the direction of the net force on the ball and

whether it is increasing, decreasing, or remaining constant in magnitude. Like the sled questions, the student is very likely to have had experiences very similar if not identical to this, which means it may also activate the naïve physics model of the individual being tested.

For a tossed object, the misconception that is most likely to come into play is the impetus notion—that is, there is a force imbued in the traveling body pointing in the

83 direction of motion and is proportional to the speed of the object. In the case of the ball traveling upward, the impetus is upward and strong enough to overcome the body’s weight until it reaches maximum height. The impetus theory held that the upward- pointing force would decay in strength until it has dissipated enough for weight to become the dominant force and cause of its downward velocity. When traveling downward, the ball may still have a weakening upward impetus, it may have an impetus that is now pointing downward, or there is only the gravitational force; the history of science shows that all of these options are possible.

3. A glass marble is tossed straight up into the air. After it is released it moves upward, reaches its highest point and falls back down again. Ignore any effects of air resistance.

While the marble is moving upward after it is released, which one of the following choices indicates the total force acting on the marble?

A. The total force is down and constant.

B. The total force is down and increasing.

C. The total force is down and decreasing.

D. The total force is zero.

E. The total force is up and constant.

F. The total force is up and increasing.

G. The total force is up and decreasing.

H. None of the above is correct.

In order to have the greatest chance of looking at the naïve view of the student, each individual is given the four-question, multiple choice test without the researcher nearby or able to ask or answer questions. Using these questions about common real- world experiences without additional prompting for reasons will provide the best chance of the student producing their most intuitive physics thoughts.

b. 1D Force Question Interviews

However, the answers themselves may not tell us how the students actually thought about the problem, and they will not show us how students would have answered if they were also required to explain themselves. This leads to the second part of the use of this tool

(Table 1 describes the process). When the student has completed the test as well as a background survey, the experimenter conducts an interview with the student. The interviewer asks the students about their answer choice, especially the physical reasons they thought their choice was the best option. In the process of answering the question, the student may change to a different answer; this final answer is recorded along with the initial answer. Questions that the interviewer will ask include what physics laws or concepts the student used and extension questions related to the problem. For example, the interviewer could ask the student what happens to the velocity of the sled if he or she pushes twice as hard or stops pushing all together; in the case of the tossed marble, the interviewer can ask about the forces acting on the marble at the high point of the toss. On top of this, the interviewer asks about what sorts of experiences the student has had that are isomorphic to the questions asked; this questioning was focused on the sled problem

Table 1. 1D Force Question Interview Process

What Interviewer Does What Student Does Reason for Design

Let student answer question without Chooses one MC Get the initial beliefs of the student explanation on their own answer

Allows researchers to see how test answers Asks student to explain choice of Provides reasoning correspond to student thinking rather than answer behind their choice just hypotheses that could be consistent with answer choices

Provides concepts Asks about what physical laws or they used, See how the student deliberated or if they concepts they used to answer to Newtonian or ran from intuitional beliefs question/thought were applicable otherwise

Asks student for an example of Provide relevant Info to help uncover origins of naïve beliefs situation from their own experience examples

Final answer Explanations may cause student to rethink Finds final answer by student provided problem and provide a different answer

as the ball toss was too common and general to relate to other experiences. At no point are the students told the correct answer.

The procedure of asking about student reasoning and seeing how students change their minds can reveal whether they have more than one competing mental physics model that applies to the given situation. The difference in the use of those models can also be compared to the use of System 1 and System 2 reasoning. During the interview where the student is required to explain their answer, they are far more likely to use System 2 or deliberative reasoning as opposed to System 1 or instinctual thinking. System 1 tends to be used unless there is an unusual or not well-understood situation, so the questions on their own without instructor prompting are likely to elicit System 1 thinking. System 2 is also used when there are conflicting ideas, so it will come into play if there are two competing models being used by the student when explaining their answer to a force question (Kahneman 2011). 86

While the interview questions about the sorts of experiences that students can

compare to the test questions, this design is not able to probe for the origins of

misconceptions, but more for their activation and about how students transition in their

thinking when forced to explain themselves. To uncover where their concepts come from

and how intertwined they are, an additional intervention was devised.

ii. Experience of Forces and Motion during Constant Acceleration a. Constant Acceleration in a car

The consistency of naïve physics concepts similar to impetus are likely due to the common way that people experience and interpret their interactions in the world.

Moreover, these beliefs are most likely produced not through long reflection but rather are intuitions built up over a lifetime and the cultural background. To get at how an individual feels a force is applied or how he or she feels when moving in a certain way has the potential to explain how these physics concepts come about. However, the prompting by a physics instructor may influence an individual to use Newtonian concepts and disguise their intuitions, or that person may use the language of a physicist but the terms do not accurately map to their own model of the way the world works. An experimental design must avoid this and allow the individual to use the language that is natural to them.

This intervention is also based on an interview system, and the questions are chosen that relate to cases of constant acceleration. There are three cases that each interview subject is asked about: forward acceleration, backward acceleration, and circular motion at constant speed. These situations are put in the familiar context of an

automobile that either is flooring the engine, violently braking, or turning. In each case

the question implies a strong acceleration to maximize the memory of how the student

felt when something like this had happened.

All of the questions are provided to the student verbally. No paper or images are

provided to the student initially, but if necessary the students are allowed to draw

something to help themselves explain or visualize the situation. Otherwise, all of the

information is administered and collected verbally.

In each case, the students are asked how they feel when they experience motion

when the car is accelerating. The instructor does not use terms such as ‘force’ unless the

students use it themselves. This way the student is not encouraged to apply vocabulary

that may not mean the same to them as it does to a physicist. Such a design also means

that the interviewer is not prodding them to explain things in terms of force, thereby providing a false suggestion of a feeling of fictitious forces.

1. Car Undergoing Forward Acceleration

The student has the following scenario read to them:

Imagine you are sitting in your car, and you are initially at rest, and then you slam

down on the gas pedal. Describe how you feel.

Because there is the natural tendency to move back into one’s seat when violently

accelerating forward, the expected naïve answer includes a force pointing backward. On

the other hand, the Newtonian answer should be that there is a forward-pointing force. If

88 a student claims that more than one of these forces to be acting on him or her, he or she is asked about the net force to gauge which is considered stronger.

2. Car Undergoing Violent Braking

In the second scenario, the students are asked about their feelings when they are driving forward and then undergo violent braking.

You are driving straight forward, and then you slam on the brakes. Describe how

you feel.

The interviewer may also mention that the student is wearing a seatbelt so there is no focus on what happens if the driver flies out of the seat and the car itself.

The predicted misconception that a student may provide is that there is a force pushing forward because the student is still moving forward, perhaps even moving forward relative to the car. On the other hand, the student may provide the Newtonian answer as well. If forces are indicated in both the forward and backward direction, the student is asked about the net force. This will help determine the comparative magnitude of the fictitious force.

Another aspect of investigating the forward-pointing fictitious force is performed with students who believed such a force was acting on them. The student is asked if this force is stronger when braking at higher speeds; for example, the student is asked if the force is stronger when braking at 60 mph than at 30 mph. This question may help refine

our understanding of the fictitious force and whether it clearly connects to another

variable, in this case speed.

3. Car Undergoing Circular Motion at Constant Speed

The last case has a car moving in a circle at constant speed.

You are driving at constant speed, and you make a hard left turn. Describe how

you feel.

From previous research, the common misconception that the student is pushed radially

outward from the center of rotation is expected—that is, there is a centrifugal force acting

on the person. The correct, Newtonian answer should speak only of a radially inward

force—the centripetal force. However, if there is the possibility of something like circular impetus that the students may have in their minds, it is necessary to consider more than one dimension for forces when analyzing student responses. Nonetheless, as in the

previous cases, if more than one of these forces is indicated, the student is asked to say in

what direction is the net force.

b. Data Collection of Feelings of Force

When the student is asked each question, the individual may not instantly settle on one

answer. Moreover, the interviewer wishes to be thorough and make sure the student has

considered all of the feelings of force experienced in the above scenarios. Also, in the

deliberative process, the student may decide that a certain force is not acting but instead

is illusory.

1. Student Interpretation of Forces and Motion

To make sure that the students believe they are experiencing forces rather than being

convinced by the interviewer of their presence, the interviewer does not use words such as ‘force’ until the student does. If the students say they move or are moved in a certain

direction, the interviewer asks about the cause, if any. This process ensures that only the

student indicates to the researcher that there is a fictitious force rather than the interviewer inducing the interview subject to think that this is how the feelings should be

explained.

The interviewer records what force or forces the students believe is or are acting

on them after each iteration of considering a question. For example, a student may say

that when braking there is only a forward-pointing force, but then may say there is a

backward-pointing force in addition. Each of these iterations is recorded, providing a

record of instinctual thinking about the problem and then any deliberative thinking that

may change their answers. Also, fictitious forces are categorized as considered real or

imaginary by the student; that is, a student may recognize that a feeling of force is not

real but only apparent. The instructor does not say that a force may only be apparent is an

option so as not to prompt the student unnaturally from believing in the potency of a

fictitious force. The procedure of the interview is summarized in Table 2.

As the students provide their answers, the interview asks about what he or she

thinks the cause of the forces, both for real and fictitious. If students say that some force

Table 2. Feelings of Force Interview Process

What Interviewer Does What Student Does Reason for Design

Asking about experience without use of word Asks how student feels when Specifies if pushed ‘force’ so student uses own language rather than undergoing certain constant or if travels in physics language that may not map to their true acceleration certain direction understanding

Says what pushes Looking at student explanations of force may Asks about sources of forces student reveal how they interpret fictitious forces

Asks about direction of net Indicates which Provides a qualitative look at the magnitude of the force (if more than one force way the net force feelings of fictitious forces was specified by student) points

is only apparent and not real, the interviewer may ask why they think that they feel its

effect. The common vocabulary is recorded and later analysis looks to see if certain

words have a similar meaning to all of those that used them to explain fictitious forces.

For example, a student may say that “momentum” provides the forward-pointing force

when braking. In that case, do students who say “momentum” pushes forward think that

it is stronger when braking at higher speeds?

2. Deliberative and Quick-Thinking Students

After the data are collected, we look to see where students answer these questions quickly and without second-guessing themselves and where students are unsure and move back and forth several times before they reach their final answer. If a student answers all three scenario questions without providing a different final answer to any one, that student is categorized as ‘quick-thinking’. For a student who answers one question but changes the

answer more than twice is categorized as ‘deliberative’. The student is put into this

category even if they are deliberative over just one question. This categorization will help

to split students who are likely relying on System 1 and System 2 thinking. Quick- 92

Test: 1D Force Questions Interview: 1D Force Does student Questions change Student on their own answer? Student provides no Students explain explanation to answers answers

Real-world experiences similar to test questions

Interview: Experiences of Force & Motion

Ask students how they feel under three cases of constant acceleration and the sources of forces they think they feel.

Interview: Constant Interview: Constant Interview: Constant acceleration in a car acceleration in a car acceleration in a car

Case 1: Accel Case 1: Circular Case 1: Accel Forward Backward Motion

What is What is direction direction of average of average force? force?

If forward What is force, is it direction stronger when of net stopping at force? higher speeds?

HS Physics

How many classes? Were there labs, demos?

Figure 6. Flow Chart of Experimental Process

thinking students are most likely running on intuition, while deliberative students are

considering multiple possibilities and are using System 2 reasoning. The deliberative

students are mostly likely going back and forth between the naïve physics model and the

classical, Newtonian physics model.

3. Methods of Analysis to Force and Motion Questions

The first experiment yields student answers before and after an interview session, so

these multiple choice answers lend themselves to pre- and post-test analysis with the normalized gain function. It is also useful to map out the transition of student answers from pre- to post-test. This way it is possible to discover how the use of mental models

transitions when there is instructional prompting.

As for the second experiment, we will consider the correlations between the final answer each student gave to the constant acceleration questions. The statistics are calculated in the following way, taken from Sheskin (1997), using 2x2 contingency tables or matrices and chi-squared testing. A 2x2 contingency matrix, which has two variables with two possible states, can be viewed in the following way:

= 1 = 2

= 1 푦 푦 푡표푡푎푙

푥 = 2 푛11 푛12 푛푥1 푥 푛21 푛22 푛푥2 푦1 푦2 Here n is the total number푡표푡푎푙 of subjects푛 tested, 푛and it is 푛also the sum of each column

( + ) or of each row ( + ). With this, it is possible to determine the

푦1 푦2 푥1 푥2 correlation푛 푛 value using the phi coefficient푛 푛

= . 2 2 94 휒 휑 푛 Phi can have a value between -1 and +1, where the closer to one the stronger the relationship between the variables. Here, χ2 is the chi-squared test value. It can be calculated in the following way:

( ) = 푚 . 2 2 푂푖 − 퐸푖 휒 � 푖 푖=1 퐸 This is for m matrix cells, and for each cell there is the observed value Oi and the expected value Ei. The expected values act as the test for the null hypothesis. The expectation values assume that there is no dependence or relationship between the values.

This means that the chances of being in any matrix element are the multiplication of probabilities for having each condition. For example, the probability of being in the x=1, y-1 state ( ) is the probability of the x = 1 state ( ) multiplied with the probability of 푛푥1 푛11 푛 the y = 1 state ( ). Then multiplying this probability by the total number of subjects n 푛푦1 provides the number푛 of subjects in each category. The matrix for expected values appears as follows:

= 1 = 2

= 1 푦 푦 푛푥1푛푦1 푛푥1푛푦2 푥 = 2 푛 푛 푛푥2푛푦1 푛푥2푛푦2 푥 While the equation above will work푛 fine for large푛 enough samples, it is inexact because the equation is for a continuous chi-squared distribution, but the observations are discrete—there is a whole number of observations. For enough sampling points, the discrepancy is insignificant, but it is necessary to check whether in the cases studied this is a problem. Yates (1934) provides one form of correction, but it is only useful if one of the expectation values is less than 5 and the total number of observations is small. Using

the correction will underestimate chi-squared and overestimate p-values and is considered

far too conservative by researchers (Delucchi 1983; Ruxton & Neuhauser 2010). Even for

small numbers of observations there are not more false positive or type II errors when

using the normal chi-squared formula (Thompson 1988); so no corrections are used here.

This also provides for a useful advantage when using the above equation for χ2. It is

possible to directly calculated φ in the case of a 2x2 matrix with this expression (Sheskin

1997):

= ; 푛11푛22 − 푛12푛21 휑 푥1 푥2 푦1 푦2 From this and knowing the total number�푛 n푛 of 푛subjects,푛 chi-squared can be very easily

calculated. To then test for statistical significance, the equation for determining a p-value

is given by Pearson (1901) as

2 푧 = ∞ − , 2 푛−1 휒 ∫ 푒 2푧 푑푧 푧 푝 ∞ − 2 푛−1 where n is the number of degrees of freedom∫0 푒 , which푧 푑푧 is the number of values that are free to vary in the matrix. The p-value is the probability that the results were found even though there is no correlation between the variable; that is, it is the probability of obtaining the results given the null hypothesis. Most researchers use a standard table of chi-squared distribution values, degrees of freedom, and p-values (cf. Fisher & Yates

1948). Fisher (1922) also provides an exact measurement of the p-value for 2x2 matrices, which can be written out as

! ! ! ! = . ! ! ! ! ! 푛푥1 푛푥2 푛푦1 푛푦2 푝 푛11 푛12 푛21 푛22 푛

Note that the calculation does not need to find χ2, so it will not be biased as Yates feared with the chi-squared test. When comparing p-values between Fisher’s exact test and the chi-squared value for the collected data (see results below), the difference was negligible, providing confidence that the χ2 values are not artificially high.

For the 2x2 contingency tables used here, there is only one degree of freedom.

This is because in the tables all of the column and row sums were already known; for example, it was already known how many students correctly answered a force question and how many students were tested. So, by already knowing each column and row sum, only one matrix element determines the remaining elements, meaning there is only one degree of freedom. Given that, a chi-squared value of 3.84 will provide the threshold for statistical significance (p < 0.05). However, to better avoid a false positive, it is desirable to have a p-value of less than 0.01, which would correspond to a chi-squared value of

6.64.

Given statistically significant results, then a comparison of the correlation factors is done. Usually a φ value of 0.4 or more is considered a strong correlation. Using this correlation coefficient information, we are able to tell if there appears to be a common factor relating two misconceptions together. In particular, if the naïve beliefs about circular motion are unrelated to the naïve beliefs about linear motion, then there should not be a strong correlation factor between the questions about linear and circular motion forces. In addition, there may be asymmetries in the contingency matrices that may be revealing. For example, perhaps being correct about forces when accelerating forward does not affect the probability of being correct about forces when turning; on the other hand, being wrong about forward forces may greatly increase the chances of being wrong

in the circular motion case. Such an asymmetry would indicate that one misconception is

more strongly correlated to another misconception, but one Newtonian answer may not

connect well to the related, Newtonian answer. This would suggest further that the

correlation is due to there being a common misconception and not the correct and

consistent application of Newton’s laws.

c. High School Experience

After all the questioning about accelerating in a car, the student is interviewed concerning

his or her experiences in high school physics (if physics was taken) or science classes in

general. This background information helps to establish what level of experience in

classical physics students are likely to have had, and this can be indicative of the

effectiveness of previous instruction. The interviews are designed to determine how much

physics the student has had (number of semesters or quarters, how much material was

covered), what were their instructions and lectures like, how often were demonstrations

done by the instructor, and how much of a lab component was there to their class. The

interviewer also tries to find out the work-load and background of the high school

teacher: does he or she have another, non-physics class to teach? is he or she trained in

physics or in another field?

Ultimately, these interviews about high school physics can help to demonstrate

the persistence of pre-Newtonian thinking even among those who have studied classical

mechanics. Moreover, the physics learned previously in high school and concurrently in

college (all the interview subjects are college students) is not the physics of the mind of the student; that is, Newtonian thinking is alien and remote to the average student. The

98 background information can also be informative to alert current high school teachers about the common instructional methods in use and provide an anecdotal consideration to their efficacy.

D. Results

The tests and interviews described in the methods section were performed with 67 students taking their first semester of calculus-based physics at OSU. They ranged from first-year to senior-level, though most were on the younger end of the range (freshmen, sophomores). All 67 students performed all of the tasks asked of them. The tasks took on average around 15 minutes and no more than 25 minutes.

i. 1D Multiple Choice Answers a. Pre- and Post-test Results

In the first part of this experiment, students were given four multiple choice questions that dealt with simple linear kinematics and the forces in play. The first set of two questions was about pushing a sled on ice such that friction was negligible. The first question asked about the force needed to keep the sled moving at constant speed, while the second question asked what force was need to have the sled speed up at a constant rate. The second set of two questions concerned a ball thrown straight up. The student first had a question about the forces acting on the ball while it was traveling upward (and after it had left the throwing hand), and the second asked the same question about the ball after it had reached maximum height and was falling down. The initial and final answers to those questions are provided in Table 3.

Table 3. Student Answers to 1-Dimensional Force Questions. Correct answers have an asterisk (*).

A B* C D E F G Q1 Initial Answers 58.21% 37.31% 0.00% 0.00% 1.49% 0.00% 2.99% A B* C D E F G Q1 Final Answers 41.79% 58.21% 0.00% 0.00% 0.00% 0.00% 0.00%

A B C D* E F G Q2 Initial Answers 0.00% 55.22% 1.49% 37.31% 0.00% 5.97% 0.00%

A B C D* E F G Q2 Final Answers 0.00% 29.85% 0.00% 68.66% 0.00% 1.49% 0.00%

A* B C D E F G Q3 Initial Answers 49.25% 7.46% 2.99% 2.99% 4.48% 0.00% 32.84%

A* B C D E F G Q3 Final Answers 53.73% 1.49% 7.46% 2.99% 2.99% 1.49% 29.85%

A* B C D E F G Q4 Initial Answers 65.67% 31.34% 0.00% 0.00% 1.49% 0.00% 1.49%

A* B C D E F G Q4 Final Answers 76.12% 23.88% 0.00% 0.00% 0.00% 0.00% 0.00%

From this table, it is apparent that there is a considerable amount of movement towards the correct, Newtonian answer to questions by the end of the interview session.

There is also a drastic reduction of answers that conform to neither the expert or naïve physics model (which are considered ‘random’ to distinguish from the other two expected types of answers). The transitions are mapped out in Figure 7 (for the sled questions) and

Figure 8 (for the free-fall questions). Regarding the normalized gain, there is a significant value of about 50%. This is notable quantitative evidence for the change in student thinking about problems before and then during an interview that forces students to

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explain their reasoning even when the instructor provides no feedback concerning the

accuracy of their answer.

When moving on to qualitative assessment of the student responses, first consider

a student who changed all of his or her sled-pushing answers from naïve to Newtonian

during the interview process. When one such student began answering the accelerating-

sled question, that individual said that he was initially confused as to what would happen.

However,

If I’m pushing it at a constant rate, on the ice with no friction, if I had constant

strength it would just keep the object moving at a constant rate.

When the interviewer asked the student if he had used any laws or principles from physics, the answer was negative; moreover, the student said that no principles seemed to apply to the situation. The student also indicated that he was going on intuition. When moving on to the question for which the sled moved with constant speed, the student was not sure if a constant or decreasingly strong force was needed. The interviewer then asked what would happen if no force was applied. After realizing that the sled was already moving, the student said that no force needed to apply to keep it going. When asked about the answer to the accelerating-sled question, the student changed it to constant strength. This whole process took about 4 minutes and much deliberation.

Now, compare this to a student that correctly answered all of the test questions and did not change his or her answers during the interview. For one subject talking about the accelerating sled,

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... because the force equals mass times acceleration, with a constant force it’s a

constant acceleration. So it’s speeding up at a steady rate.

When asked about personal experiences that relate to such a situation, the student could not think of an example. With prompting, the student did talk about ice-skating and pushing someone on ice but noting it is very difficult to apply a constant force. It seems then the student did not have an experience that could map well to the test question scenario.

These results already indicate that a considerable number of students seem to think about these questions differently when first answering them and then when explaining them (see Figure 6). The first results appear to come from beliefs based on real-world results. Confirmation of this also came from the interviews in which students were asked to provide experiences that corresponded to the questions. When considering the first two questions with the sled, students related these circumstances with experiential events such as pushing a shopping cart or a sled in the real world. In the experience of students who provided the naïve answers known to PER, those students said that to maintain the speed the object needed a constant force; without it the object would slow and then stop. Moreover, if they wanted the object to speed up at a steady rate they needed to push continuously harder rather than maintain a constant force. Those who initially gave the naïve result but changed their minds to the correct result identified the factor of friction that had caused them provide a mistaken answer. Commonly, students would say they did not notice that the question said that there was no friction.

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Questions 1 Questions 2

Pretest Posttest Pretest Posttest

37.3% A A A A 22.4% 1.5% 28.4%

B* 34.3% B* B B 28.4%

C C C 1.5% C

1.5%

35.8% D D D* D*

3.0% E E E 1.5% E 3.0%

1.5% F F F F

G G G G Figure 7. Answer Transitions for 1D Force Questions with Pushed Sled

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Focusing on the questions concerning a ball thrown up and then free-falling, the initial answers students provided were more spread out when the ball is traveling upward but more consistent when traveling downward. First, we look at the sorts of reasoning students provided for the most commonly-given wrong answer (that the force is upward and decreasing). When asked what forces are acting on the ball (or marble) when traveling up, those forces are: “The force you apply when you toss it up, and also the gravity pulling down.” At the high point of travel, the student said that there is no longer an upward force, and when falling there was only the constant gravitational force.

When asking another student, the response to the question about forces when traveling up was the following:

Well, you got gravity pulling it down, then the initial force up, I guess ... I guess

I’m not too sure about that [the upward force] ... I guess it would still feel it. It

needs a force to go up, right?

Again, the student thought the upward force gets weaker, and at the top the upward force has completely dissipated. As the marble falls down, the student initially thought that the downward force was increasing, but the student also said the only force was “gravity” and that was constant. After thinking about it, the student said: “Force is mass times acceleration, so I guess it [the total force] wouldn’t be increasing.” Thus, the student switched from an intuitive answer, but then applied a Newtonian force law after

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Questions 3 Questions 4

Pretest Posttest Pretest Posttest 46.3% 62.7% A* A* A* A* 3.0%

10.5% 3.0% B B B 20.9% B

1.5% C C C C 1.5% 1.5%

D D D D 1.5%

3.0%

E 1.5% E E E

F F F F

29.9% G G G G

Figure 8. Answer Transitions for 1D Force Questions with Thrown Marble

105 considering what was happening to the falling marble. However, this realization did not cause the student to change the answer to the question about the marble traveling upward.

While it was common for students to say that there was an upward impetus-like force acting on the ball when traveling upward, students were less consistent when it came to the magnitude of that force when at the marble was at its maximum altitude. Of the 22 students who initially said there was an upward but weakening impetus force, more than half (13) said that the force was zero at the top of flight, but the other, smaller half said it was equal to gravitational force (8) or uncertain (1). The results are more fully presented in Figure 8. This is worth comparing with past data. As seen from the historical examples noted in Part 2.A.i.b, the record has physicists thinking that the upward force was equal to the weight downward. However, this may be because these thinkers had considered the question for a longer time and believed that the net force and not just the upward impetus that needed to be zero when the object is at its highest point. The diversity of student answers, on the other hand, suggests that the upward impetus/force is simply correlated to the velocity, but further realization makes this untenable—with zero upward force, there is a net force downward, but there should then be a non-zero velocity.

ii. Experiences of Motion and Force under Constant Acceleration

The second phase of experiments with the students interviewed was to try and see how they experienced the sensations of force and motion. Starting with a car at rest and then performing a hard, forward acceleration, one student described his feelings in this way:

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It feels like I’m pushed backward against the chair, but that’s not, not ... now that

I think about it, that’s not due to a force, that might be due to just, I think the term

for it is ‘jerk’? We really haven’t been over it.

After the interviewer defined ‘jerk’ for the student, he decided there wasn’t actually a force pointing backward, but instead just relative motion in the car. The same thought process for this individual took place in the other two cases of acceleration (backward and circular), though in the case of braking, he felt more strongly that there was a forward-pointing force, though he then tried to remain consistent with his previous answer to forward acceleration.

While the interviewer avoided using the term ‘force’ until the interview subject did so as to not influence the student’s vocabulary, a conversation may have the student directly use the term ‘force.’ For example, in the case of braking:

I: You’re driving your car and high speed. You have your seatbelt on because

you’re a safe driver, and you slam down on the brakes. Now describe how it

feels in the moments of acceleration. Or the moments of braking.

S: I feel a force pushing me forward.

I: And backward, anything?

S: Hmm, no.

The forces felt by the students are categorized as either real (Newtonian) or fictitious

(naïve). Also, the fictitious forces were categorized as considered real or fictitious; that is,

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Table 4. Forces Identified or Felt by Students under Constant Acceleration.

Fictitious Force Fictitious Force Fictitious Force Real Force Felt, Real Felt, not Real Total Felt

Forward Acceleration 77.6% 52.2% 35.8% 88.0%

Backward Acceleration 82.1% 62.7% 17.9% 80.6%

Circular Motion 53.7% 71.6% 19.4% 91.0%

a student may say that they feel a push in a certain direction but come to think it is not

actually a force or push. These fractions of students providing these sorts of answers each

of the three cases of acceleration are shown in Table 4.

In each case, over 80% of students said that they felt a fictitious force acting on

them even if they convinced themselves it was not real. In the case of circular motion, it

was over 90% and the majority thought the fictitious force was real. On the other hand, a

majority of students did correctly identify the actual force acting on them, but that was a

small majority in the case of circular motion. The feeling of the fictitious force in all

cases was either comparable to the number that identified the real force in play or more

students felt the fictitious force than noted the real one. Again in the case of circular

motion, more students felt the fictitious, outward-pointing force and thought it to be real

than did students who said there was an inward-pointing force.

The magnitude of these fictitious forces can also be examined using the interview

data. Of the students who were asked about the direction of the net force, approximately a quarter of students in each of the linear acceleration cases (forward and backward) had the net acceleration in the wrong direction (28.4%, 23.9%), either opposite the correct direction, or zero with very few believing the net force to be zero. To highlight the oddity

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of this, note that in the case of linear forward acceleration, the car was said to initially be

a rest and is speeding up in the forward direction, and yet 16 of the 67 students said that

they were accelerating backward. In the case of circular motion, nearly half (46.3%) gave

the net force in the wrong direction, but a considerable fraction (14.9%) simply could not

determine which direction the net force pointed. These results suggest that the fictitious

forces students claim to feel are on the same order of magnitude as the real forces moving them, sometimes exactly equal as seen in the cases of claimed zero net force where the real force balances out the fictitious force.

a. Forward Acceleration

To better interpret these results, let us qualitatively consider some experts from the

interviews. When the car was accelerating forward, one student explained his feelings in

the following way:

You should be feeling a force dragging you, like, to the back ... You feel the car’s

movement [forward] ... motions, like, going to the front, but like, no, I don’t think

so [no force pointing forward] ... You at first feel the force pulling you back.

When asked how he feels after a few seconds of this acceleration in a car, the student said

I would say you feel, like, you feel the force jerking you back, and then it kind of

goes back to normal position, maybe, like during that point you feel like you are

pushed to the front.

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The student also said that forward force only lasted a short period of time. It would seem then that the student feels pushed back into his seat; only after some time does he feel any push forward, but only to return to a sense of normality. The student also talked about having a sense that the car is moving forward, but the student does not seem to believe he can feel that force. Interestingly, the forward force is completely uncorrelated to forward acceleration; one returns to a more “normal” position in the seat of a car when the acceleration is weaker and a passenger is not so compressed into the seat, so less acceleration means that the student is moving forward a little bit with respect to the car seat. The force is thus correlated with motion with respect to the car and not the non- accelerating reference frame (i.e., the road).

b. Backward Acceleration

Turning to interview data from the case of a car braking, one student said that there was a forward-pointing force acting on him when stopping, but the backward-pointing force was under a more interesting constraint. The student said of the backward-pointing force, that you do not feel it

until you slam on the brake again. I mean, like, not until your car has, like,

stopped. So like you feel ... you should be feeling like your whole body is moving

to the front, but until your car comes to a complete stop you don’t actually feel,

you don’t actually feel like that force dragging you to the back.

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Now, this student was not asked about the feeling from the seatbelt, but another student

who claimed there was only the forward-pointing force was asked about what the seatbelt was doing. That individual claimed that she felt the belt on her, but it was applying a pressure rather than a force.

c. Circular Motion

As expected with the case of circular motion, there is a strong misconception that there is some sort of fictitious force. As the statistics from above show (see Table 4), circular

motion had the greatest number of students reporting feeling a fictitious force and the

fewest number identifying the actual force in play. The student quoted at the end of the

last section gives this story: “You should be feeling your body is moving away from your

turning.” When asked about the existence of any forces pointing inward, the response

was negative. Another student who was asked about circular motion did come to believe

there was an inward-pointing force to account for the motion of the car, but that

individual did not actually feel the force.

Returning to the student who did not identify an inward-pointing force, he also

said he felt no forces acting forward or backward. This suggests that he does not feel the

straight-forward impetus force as seen in the cases of forward and backward acceleration

noted above. The reason for this is likely because there is no motion with respect to the

car forward or backward, so long as the car is not speeding up. However, other students

do say that they feel pushed forward as well as outward, or that there is a net force that is

out- and forward-pointing.

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Your upper body is kind of like, since you going towards the left, it’s pulled

towards the right because of centripetal, centripetal force ... You do feel pushed

forward, and that is by the car seat.

Why some students do believe there is also the forward-pointing force while turning is hard to explain. Perhaps it is because the experiences they have do not include constant speed when making such turns. When someone makes a turn, the car may be slowing in order to make the turn safely. Because students already assign braking with a forward- pointing force (as did the above quoted student that believed there was a forward- pointing force when turning), then the act of slowing down while turning may provide an impression of being pushed to the side while also moving forward because of the effect from slowing down. On the other hand, perhaps it is related to the other misconceptions of force with linear acceleration. This is considered in below.

d. Correlations between Accelerated Motion Answers

If it is the case that the outward feeling of force when traveling in a circle is a result of the belief in a forward-pointing impetus, then there ought to be a correlation between those who have such a belief in linear impetus and thus who feel an outward, centrifugal- like force. On the other hand, if the outward force students feel is in fact pointed straight away from the center of rotation, then there is no a priori reason for linear and circular force misconceptions to be strongly correlated. We also would not expect forward and backward acceleration to be less strongly correlated than backward acceleration and

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circular motion feelings of force if these concepts are not related to the same impetus-like

belief.

In order to examine this, contingency matrixes were composed to see how

strongly the linear force answers students gave when accelerating in a car were correlated

to the feeling of an outward force when turning. The first matrix (Table 6) and correlation

comes from how often students are correct or incorrect about both cases of linear

acceleration (forward and backward) and how often that aligns with correct or incorrect

answers to the forces acting on them during circular motion. The next two matrices and

correlations are for the individual cases of linear acceleration. Table 7 looks at forward

acceleration and circular motion, while Table 8 looks at backward acceleration and circular motion. Table 5, on the other hand, looks at the correlation of feelings in the two linear cases. This can similarly demonstrate that there is an underlying idea or p-prim in linear motion, and it can act as a comparison when looking at the relationship between linear and circular motion feelings of force.

As can be seen, in every case there was a significant correlation (φ > 0.5), and the statistical significance is not in doubt (p < 0.001) so long as there was no bias in the sampling of students. Already, this indicates that there is a relationship between linear and circular feelings of force, in that they rest on a common factor. There is the potential for this result to have been confounded by always asking the questions in the same order, so a reproduction of the experiment with the order of acceleration questions randomized will ensure the robustness of these results.

There is also an interesting characteristic of these tables when considering the anti-diagonal elements. The feature worth highlighting is that the elements in the first row

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Table 5. Contingency Matrix for Forward and Backward Linear Acceleration.

Linear Backward φ χ2 p

Correct Incorrect 0.512 17.3 < 0.001

Correct 19 11 Linear Forward Incorrect 5 31

Table 6. Contingency Matrix for Forces during Linear Acceleration and Circular Motion

Circular φ χ2 p

Correct Incorrect 0.706 25.9 < 0.001

Linear Correct 14 5 (Both forward and backward) Incorrect 2 31

Table 7. Contingency Matrix for Forces during Forward Acceleration and Circular Motion

Circular φ χ2 p

Correct Incorrect 0.510 17.4 < 0.001

Linear Correct 15 15

(Forward) Incorrect 2 35

Table 8. Contingency Matrix for Forces during Backward Acceleration and Circular Motion

Circular φ χ2 p

Correct Incorrect 0.660 28.7 < 0.001

Linear Correct 15 8

(Backward) Incorrect 2 41

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are much closer in value than those in the second; in particular is the case of forward

acceleration correlated with circular motion: if a student correctly figured out what was

happening in the case of forward acceleration, they were no better than chance at

answering the case of circular motion correctly. It appears that the case of circular motion

is more difficult for students to interpret. The situation also exists when considering the

correlation between forward and backward acceleration: correctly answering what

happens in forward acceleration hardly guarantees a correct response in the case of

backward acceleration. This feature of the matrices is less significant when both linear

acceleration cases are correlated to circular motion, but it still remains noticeable.

Why this is the case may be due to the fact that linear motion had been discussed

at length in class by the time these interviews were done, but circular motion either had

not been covered or it had much less time devoted to it than in the cases of linear

acceleration. In other words, the interviewees may have been more likely to provide an

expert physics answer where they had had the greatest amount and the most effective

instruction. On the other hand, if a student was wrong about any linear motion case, he or

she was almost certainly going to be wrong in the case of circular motion. This means

that the naïve answers of linear and circular motion are the things that are so strongly

correlated. That naïve answer correlation is also the greatest between backward and

circular motion. If there is a common naïve physics factor involved in these cases, it is

most strongly represented in backward acceleration and circular motion.

To better determine what that factor may be, again note the correlation values.

The strongest correlation occurs when students provide consistently correct or incorrect answers to linear motion forces, but there is a notable difference when comparing the two

115 cases of linear motion and their separate correlations to circular motion forces. The φ value for backward/circular acceleration is notably larger than for forward/circular acceleration. This correlation between backward and circular motion forces is even strong than between backward and forward forces. The main difference is that in the case of backward acceleration the forward sensation is in line with the misconception of linear impetus, while under forward acceleration this is not the case. The correlation data imply that there is a similar concept in play between circular and backward acceleration, and this certainly is not the idea of a centrifugal force. The discussion section below will consider this further.

e. Deliberative vs. Quick-Thinking Student Responses

To establish how this force is more the product of built-up intuition, it is instructive to look at the students who barely changed their minds when considering these problems and those who struggled to come to a final conclusion. To categorize these students, an interview subject who did not change from his or her initial answer in any of the three cases is called a “quick-thinking” student. An interview subject who changed his or her answer more than twice for any answer are called a “deliberative” student; he or she is put into this category even if classified as deliberative for one question, but the analysis here considers only the questions he or she was deliberative about. The quick-thinking students are most likely to act from intuition built up over their lives rather than their recent instruction in Newtonian mechanics; the quick-thinkers (except for one student) also had no high school physics instruction. Deliberative students go back and forth

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Table 9. Number of Deliberative and Quick-Thinking Student Answers to Acceleration Questions.

Forward Accel Backward Accel Circular Motion

Quick-Thinking Naïve 9 10 11

Deliberative Naive 3 2 4

Deliberative Expert 11 1 2

Quick-Thinking Expert 4 3 2

between what they feel and what they can convince themselves is real; during the interviews they are more likely to refer to lecture materials.

Among the 67 students interviewed, 13 are categorized as quick-thinking, and 20 are deliberative. The number of students in each category and their answers to the acceleration questions is given in Table 9. Of the quick-thinkers, 69.2% of students incorrectly identified the forces acting on them in all three cases; 15.4% were mixed in their accuracy, and the remaining 15.4% were correct in all three cases. Clearly then the intuition of these students is strongly biased to the fictitious force in the direction of motion. As for the deliberative students, 78.6% correctly identified the forward-pointing force as the only force acting on them in the horizontal direction, compared to 42.0% of all the interviewees. In the cases of braking and turning at constant speed, the percent providing the Newtonian responses dropped to 33% in each case, though this matches or barely supersedes the average response of all the interviewees. As expected, deliberation tends to produce Newtonian answers more often than initial intuitions.

The results in Table 9 also indicate the mental model state of the students. The first column with linear acceleration is the most useful here because most of the deliberation took place for this question; if the student deliberated, it was done in the case 117 of forward acceleration, which then influenced their answers for the other two questions and avoided further deliberations. Nonetheless, there is still evidence of mixed use of models. Quick-thinkers became more and more likely to provide a naïve answer to backward or circular motion than for forward acceleration. Very few quick thinkers consistently used a Newtonian model throughout the interview process. Deliberative students, on the other hand, tended towards Newtonian answers, but again with backward and circular motion there is more reliance on naïve physical beliefs. It seems that the deliberative students provided the greatest level of mixed use of naïve and expert models, while quick-thinking students were more consistent, though they also had a small tendency towards naïve physics. This suggests that individuals who are relying on one model are more likely to provide their answers without deliberation, and individuals who are mixed with their model usage require more deliberation before coming to an answer.

This shows that most quick-thinkers, those who are most likely to be utilizing

System 1 reasoning, have created a mental model of how forces work which they do not see as problematic or in contradiction with what they have been taught; Newtonian concepts simply do not come into play when talking about their real-world experiences.

However, deliberative students realize that something is amiss and thus use System 2 reasoning far more. That use of System 2 is likely because the students have two competing models in mind, and it takes considerable reflection to favor one over the other. Overall, one can conclude that the physics taught in class proves to have significant difficulty in causing some people with naïve concepts to realize that what they believe in, in fact, contradicts classical mechanics; Newtonian physics is more alien and is not incorporated into the worldview of the student, at least not without significant effort on

118 the part of teacher and student, and even when the student has learned Newtonian physics, its application is resisted by prior beliefs.

In summary, the interviews found that subjects actually feel pushed by forces that are not due to an interaction. Perhaps it is no wonder then that students find physics instruction so alien, considering they literally feel differently when undergoing acts of acceleration in the real world. This can in part explain where the ideas of impetus have come from; the question about how a student feels when braking is an excellent indicator of the impetus concept with a strong force pointing forward. However, another layer of complexity is found in the case of forward acceleration and the feeling of being pushed backward, something not accounted for by the medieval impetus concept. Then again, the ancient world may not have had as much experience with hard forward acceleration; after all, the primary source of strong forward acceleration today is the automobile, and antiquity had little in the way of providing this circumstance with regularity. Still, the backward-pointing force when accelerating forward seems to be related to the apparent movement back into ones seat, and so a relative velocity with respect to the car may account for this and still remain consistent with the concept of impetus. Further explorations will be necessary.

iii. High School Physics Experience

The group of students examined here were generally not new to a mathematical,

Newtonian treatment of motion and mechanics. Over 80% of the subjects said they had had physics before coming to OSU, usually in their junior or senior year of high school.

Most of the students were from the United States, but a few came from China and had

119 more pre-university instruction in physics. Students also said they had some laboratory activities during their high school course or courses. And yet a substantial proportion had failed to become Newtonian thinkers as indicated by their answers to the three acceleration questions.

To understand this in part, the students were asked about their high school experiences as best as they could remember. For a substantial number this was well over a year in the past, so the survey of answers is incomplete, and it is not be possible to talk about trends with much authority when it comes to the common labs or demonstrations performed by the instructor. Nonetheless, the interview questions can provide some useful information about how they were instructed and what sorts of issues may exist in high school physics instruction. The results may also be supplemented by Sadler & Tai

(2007) and their analysis of high school student experience with science and math and later success in science college courses.

E. Discussion

i. Experiential Origins of Common Student Conceptions of Force and Motion

The second experiment was able to bring out the common force misconceptions when there is an accelerating system. Apparently the fictitious forces identified by the students were in the direction of relative motion with respect to the accelerating vehicle. When accelerating forward, the students feel pushed back into the seat, while they feel pushed forward when braking. Moreover, the fictitious forces were believed to exist while in a number of cases the Newtonian forces were not identified. A substantial number of

120 subjects also believed that they felt a force that they reasoned did not exist based on their understanding of Newtonian physics.

The identification of feelings that students thought were forces, especially after the student reasoned that there was not really a force, shows that there is a basic level connection between the sensation of force and of relative motion. This may be explicable for the reason that when an object moves forward relative to another, a force is usually involved; this is almost a necessity in everyday experience because of friction. The tight correlation between force and motion in most experiences are thus a likely contribution to why there is such a strong association between the direction of motion and force.

ii. Relationship between Some Force Misconceptions

From the contingency tables, there is an obvious and significant correlation between the misconceptions that students have in linear motion acceleration cases as well as between linear and circular acceleration. Moreover, the asymmetry in the tables show that having a Newtonian answer in one cases of acceleration only provides about a chance level of providing a Newtonian answer in another; conversely, if a misconception is given in one case, it is likely to appear in the other. This suggests that there is a singular underlying factor or related collection of factors in play rather than unrelated p-prims.

However, this may seem odd given that in the case of circular motion the expected misconception is that the student feels pushed away from the center (a centrifugal force), and this is not the same as a force in the direction of motion.

Considering the history of pre-modern science though would suggest that the centrifugal force concept may not be the natural misconception. As noted before, the impetus theory

121 had a notion of a force in the direction of circular motion with no force pointing towards or away from the center. This was then attacked in the 16th century, and the first reconstruction of circular motion had a linear impetus and a center-pointing force to account for the change in path. Also, PER experiments seem to recover a circular impetus idea more often than the centrifugal concept. Perhaps then circular motion misconceptions need to be reconsidered.

From the interviews of students when describing the forces acting on them when turning, a number of students simply stated that they felt they were pushed outward (to the right) when making a left turn in a car. However, some students noted that they should be going straight forward. During interviews, students would say that they be moving in a straight line if they were not undergoing circular motion. In one interview in particular this is clearly the case. The student went back and forth whether there is an outward-pointing force.

S: I have momentum going forward, so I feel like I’m [moving?] right when I’m

still just going forward, or attempting to.

I: So are you saying there is a force to the right [outward]?

S: Um, no ... Wait ... I’m not sure. (talks softly about momentum) um, there’s no

force. Well, yeah, but... no, there’s no force acting on me to the right.

I: OK, and you were saying that, you were saying that momentum is not a force.

S: Well, I’m not entirely sure at this point. To be perfectly honest.

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I: Well, let’s go back to when we were braking for example, and you were saying

there was a forward force acting on you [student said that “momentum” was

cause of forward-pointing force during braking].

S: OK, so I guess it is a fo[rce?], ah, yeah, sure. So momentum is pushing me to

what I perceive to be the right of the car, and the car is pushing me to the left.

Here, then, we see how the rightward- or outward-pointing fictitious force is due to forward “momentum” or forward impetus, and that it is perceived to be to the right of that person in the car.

Furthermore, in an interview quoted in the previous section, the students said that they felt they were pushed forward when they were turning. While it is possible to explain this by considering if the car is slowing down and thus the students feel

themselves move forward as they did when braking, it requires knowing how they

normally drive or are driven. Given the background information and other interview sessions that show individuals who also claim there is a forward-pointing force with respect to the car, perhaps the misconceptions from circular motion are better explained by linear motion misconceptions, as is strongly supported by the correlational data.

This makes sense when again considering the previous research and historical data. Up until the Scientific Revolution, there is no talk of either a centripetal or centrifugal force. Rather there was notion of circular impetus. When looking at the force with an infinitesimal range, that force would appear to be pointing tangent to the circle.

When natural philosophers such as Benedetti come along, the centripetal force is introduced to replace circular impetus, but that centripetal force resists not a centrifugal

123 force but the linear impetus of an object undergoing circular motion. Newton and others introduced a centrifugal force in order to balance center-pointing forces, so the inauguration of centrifugal force was because of changes to mathematical physics rather than because of a common, ancient belief. When looking to modern research, the test by

McClosky et al. (1980) with a ball on a straight path that breaks found that very few (six out of 47 students) showed the ball flying centrifugally or with a centrifugal component.

On the other hand, Galil & Bar (1992) find that as many as 70% of novice students believe that an object undergoing rotation has a force acting in the tangential direction.

This suggests that it is not a centrifugal force error that most students have, but something else.

However, this would be at variance to what is shown by Galili & Bar (1992), in which over 40% of all but one subgroup talked of an “outward” force acting on a coin siting on a record spinning with constant angular velocity. A closer examination of the test design, though, suggests that their results can be consistent with the thesis being advanced here. In the category of outward forces, the authors included along with

“centrifugal” and “outward” words such as “inertia”, “friction”, and “moving”.

Moreover, the students who talked of “centrifugal” force had not had that used in their high school physics class, so the language is not that of the student but is appropriated from unknown sources. This means that the students’ use of the term may have little to do with what the instructor or physicists thinks it means. Moreover, the data are gathered from a qualitative-answer paper test, in which the students apparently list and describe the forces acting on objects; it is not said by the authors that the students actually drew force vectors, and interviews apparently were not performed. In this way, the forces

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categorized as “outward” may in fact point closer to the tangential direction. This is

implied by the use of the words “inertia” and “moving” since the coin is moving in the tangential direction rather than in the at a right angle to tangent (in the centrifugal direction).

Still, this may not account for all of the answers students gave that included an outward force. Another possibility is that the centrifugal force was included to balance the centripetal force. A considerable fraction of students in the same study said that there was an equilibrium of forces, so the inclusion of the centrifugal force may be there to balance out another force that students expected, one pointed towards the center (cf. diSessa 1993). From the data presented it is impossible to say how many of the answers for a radially outward force can be accounted for using these complications of interpretation proposed here. Nonetheless, it is enough to allow for the possibility that the centrifugal force is not interfering with students’ beliefs except in abstract situations.

Instead, believing that a body will have a force tangent to the path of rotation would still cause that body to flee from the center. Even if the student believes in circular impetus, which is also seen for most of the students who had misconceptions studied by

McCloskey et al. (1980), the lack of the cause of circular motion still means that the body is increasing its distance from the original center of rotation. In the case of someone in a car in circular motion, the belief that there is a force that points forward and tangent to the path of motion means that students feel they are pushed in a direction that takes them away from the center and thus outward and to the side of the car.

Moreover, while the correlation between backward and circular acceleration is high, it is not very large, and this may be due to the fact that many students have the idea

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of circular impetus. In this case, the students will still feel that they are moving away

from the center of motion when rotating, and when the centripetal force is gone the path

they take is still curved but straightening and increasing its distance from the center.

Again looking at the results from McCloskey et al. (1980), about 30% of students

provided circular impetus answers when an object is no longer forced to move in a circle

(it still has a curved path), and such a fraction of students may well account for the a

significant portion of the remaining students that have circular motion misconceptions

but not because of linear impetus issues.

What this indicates is that the misconception that has been termed “centrifugal”

(i.e., Hestenes et al. 1992) is not accurate; the force is not pointing away from the center, but instead other force concepts lead to a result that appears similar. However, this does not mean there is no true centrifugal misconception. When drawing free-body diagrams,

it is not rare to see a student draw a force vector straight out from the center. Why this is

cannot be explained with the data collected from this experiment.

Now, if correct, the experiment does not distinguish well between those students

who may believe they are pushed straight outward along a tangential path and those who

believe they are pushed along a circular path that would normally straighten out. That is,

it is difficult to examine the difference between students with a linear impetus or a circular impetus concept to interpret what happens when a car is turning. In either case the students would normally increase their distance from the center and seem to experience a centrifugal effect (though not a true one). The best chance for distinguishing

them is comparing those who say there is a true centripetal force or not acting on them.

Benedetti, as noted before, explained circular motion with linear impetus and centripetal

126 force in contrast to the circular impetus theory. However, the design of the current experiment does not collect data in such a way to make this analysis possible, so future experiments will be required to see if students more commonly understand circular motion in terms of linear or decaying circular impetus.

iii. Gut vs. Brain: Mental Models in Conflict

Both of the experimental setups contributed to seeing how deliberations affect student answers to physics questions, and in particular how prompting may cause an individual to use a different mental model then they would normally apply to a given context.

a. Effects of Prompting by Instructor

The pre- and post-testing of students about their answers to 1-dimensional force questions show how differently students will answer and consider questions when first approaching them and then when forced into explaining themselves. On simply the point of raw score improvements, the final answers of students when interviewed about their answers provided about a 50% normalized gain, and this without ever telling the student they were actually correct or not. The answer transition figures above map the movement from random answers often to the correct, Newtonian answer, and there is also a significant move from the expected misconception to Newtonian answers.

From the interviews, it is seen that until asked, students do not consider applying some overarching principle to what happens when something is maintaining speed or constantly accelerating. That is, they seemed to be operating on some base-level intuition and not something more analytic. This is expected given that the student is more likely to

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use System 1 reasoning until they have to think about the reasons or run into some

difficulty in their answer; only then does System 2 reasoning come into play.

b. Mixed Models

The transitions from the first part of the first experimental instrument show that students

in their first year of college physics carry with them more than one model of the world:

they have their intuitive senses and the classical physics that they are taught. Given two

models for the same sorts of physical systems, there is bound to be times that they will

conflict in the minds of the student. This is shown with the case of quick-thinking and

deliberative students from the second experiment (see Table 9).

In the case of quick-thinking students, the model used was very consistent: if students used a Newtonian model in one case, they were likely to use it in the other two; if they used a naïve physics model in one case, they were very likely to use that same model in the other two cases. The consistency in the use of one model is expected for quick thinkers as they are using fast, System 1 reasoning and there is no conflict of thoughts. In the case of deliberative students, there is less consistency of results. These individuals are using System 2 reasoning and apparently are going back and forth between naïve and Newtonian models when interpreting their experiences of force. That is, the students are fighting between the intuitive thinking of how they feel (their “gut”) and the learned physics (their “brain”).

There is also a preponderance of quick-thinkers providing naïve physics answers, while there is a smaller number of quick-thinking classical model students. Deliberation is also more likely to produce a Newtonian answer from the student. This shows the

128 expected sort of transition of a class, moving from naïve to expert conceptions of physics problems with the largest portion in a transitional zone in which their use of models is mixed. This is all in-line with the assumptions of model analysis.

iv. Final Assessment

The experiments conducted with the students taking their first semester of college physics provide insights into their modeling of their physical intuitions. First, we note that students are likely to conceive of a scenario in terms of their naïve conceptions; with prompting, it is usually necessary to have them consider the underlying principles and then the classical physics involved. The lack of use of some overarching principle by the student appears to be consistent with the p-prim model. Moreover, the second experiment showed that those intuitions are built up by students in their belief that they feel a force in the direction of relative motion even when such a force is fictitious; this was most pronounced in the case of backward acceleration and circular motion of a car. The feeling of motion is interpreted as a force, and this is likely to give rise to the millennia-old impetus notion that students seem to come to naturally. The most plausible reason for this association between force and motion is that in the normal run of experiences an individual is likely to require a force in order for there to be motion, thus motion is correlated to the feeling or expectation of force.

The nature of the misconceptions under acceleration indicates to a relationship between linear and circular force beliefs. The strong correlation with the naïve physics idea of a forward-pointing force when braking and the outward-pointing force when undergoing circular motion suggests that they may be caused by the same underlying p-

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prim, especially when considering the history of science and previous work in PER that

uncommonly found the centrifugal force idea. Perhaps then the centrifugal force belief is

a response by the students when their previous model is challenged and starts to be

reassessed.

Lastly, the experiments revealed how the intuitive models possessed by each

student works either independently or in competition with the classical models students

are taught in class. However, through deliberation, students move towards Newtonian

answers more often than naïve ones, so there is a transition from naïve to expert as

predicted by model analysis. This is most clearly seen with the interview of student

reasoning for 1-dimensional force questions and the comparison between quick-thinking

and deliberative students when interpreting feelings of force.

v. Future Research

While the experiments above put light on some of the origins of misconceptions, new avenues of investigation seem apparent. The ideas behind circular motion misconceptions seem to be strongly related to the same ideas of linear impetus, while the centrifugal force appears to be a different sort of misconception. Future experiments should be designed to see which sorts of individuals believe in centrifugal force and how many in circular

impetus forces based on their levels of physics education. It is also necessary to see if the

context of the questions provokes circular impetus or centrifugal force ideas. Perhaps it is

when students are pushed to explain things in terms of Newtonian-defined forces that the

centrifugal force appears.

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More broadly concerning these misconceptions, knowing how they are related to

feelings of motion, it would be helpful to instructors and curriculum designers to know

when these ideas develop. As noted before with circular impetus, younger students are

less prone to have this idea than middle school students and about as well as college

physics students. If circular impetus is related to linear impetus, then perhaps they

develop around the same time, or perhaps linear impetus has to develop first. This line of

inquiry will require looking beyond the high school and college population, but if done it

may make for easier instruction when students finally have classical mechanics

instruction.

Considering that instructors will have many students with naïve physics models, greater detailed examination of the activation of naïve and expert physics models will be useful. This would be especially helpful for educators developing a curriculum to more likely activate Newtonian thinking about physics problems at a conceptual level. A greater understanding of how to trigger such thinking and how the mental models will be in conflict for most students until they become experts will help in creating the most useful instructional methods. Because the first experiment only asked students about their reasoning and this induced a greater use of classical physical principles, then approaches such as this rather than only directly telling students they are in error may make for the best form of guidance.

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CHAPTER 4. TECHNOLOGICAL ENHANCEMENTS

A. Previous Research

The addition of new technologies into the classroom has always had two goals: to free up the instructor for other tasks, and to teach the material more effectively. With instructors

able to avoid monotonous tasks (i.e., grading), they can instead focus on tactics better

geared to helping their students. Technology that can deliver information or otherwise

enhance the instructor’s efforts is also desired. It is important to understand which frequently used technologies are used in a classroom setting as well as their effectiveness.

The 20th century saw plenty of additions to the classroom, from the film projector,

to the television with pre-recorded videos, to the computer (usually only one per

classroom). In these cases, the new technology can deliver the information that the

teacher would otherwise lecture about, and it usually has greater visual or auditory

appeal. However, the addition of these technologies did not change the nature of the

classroom in a fundamental way (Polman 2000). Instead of a teacher speaking, it was a

speaker in a box or on a screen. The format of the classroom remained the same rather

than conforming to inquiry-based methods, which have shown to be more effective.

There is also nothing to help tailor the pace of the material to particular students, and

there is limited interaction with the new medium and student inquiries than with the

teacher providing instruction. With new technologies also come new issues in making the

technology work; educators may not know how to use a computer software package, for

132 example. As such, there is an increased expense in teaching materials without the sorts of changes to the classroom needed to improve educational effectiveness, while additional hazards to teaching are introduced.

Instead, an approach may be taken that does fulfill educational goals and change the classroom dynamic for the better. One major example is the use of computer programs that run interactive simulations that individuals can use to achieve some defined purpose. Consider the physics simulation done by Zhou et al. (2011) with a simulator of forces and circular motion. The goal was to apply a constant force using a joystick in a direction so as to move a ball in a circle. As there is the common misconception that the force to maintain circular motion itself points in a circular or tangential path (as noted in the previous chapters as well as Zhou et al.), applying that belief in the physics simulator will demonstrate that applying a force in such a direction fails. However, with a few attempts (especially if they have had some lessons in centripetal force), students learn that the force should be directed towards the center of the circle. The group of students who played in the virtual environment had a notable change in their pre- and post-test mean scores with an effect size about double that of students who only did solving problems about circular motion.

Experiments such as this one indicate that this change to the student-centered approach, at least when it comes to strongly-held misconceptions, are likely to improve educational gains. On top of this, students working in such virtual environments have a positive view of such activities, especially compared to traditional homework problems.

Simulations or virtual environments can also allow for limitations in the parameter space of what one can do so as providing some level of scaffolding in the direction of

133 understanding the concepts that an instructor wants to teach. Other research indicates that virtual environments allow for more testing than with physical lab experiments and are otherwise preferable (Grayson & McDermott 1996). As such, we know that technology can change the dynamics of the classroom and be a more effective way of teaching.

Along with the changes to the traditional teacher-student setting, the addition of technology has allowed the classroom to extend beyond its normal confines because of the Internet. With an appropriately tailored website, it is possible for students to continue their education at home or generally outside the traditional classroom. This can increase the amount of time on task without burdening the instructor, and this would be a significant boon to learning.

One indication that increased access to educational material in a non-classroom environment comes from studies of those who watch educational television. In the United

States, one of the major channels devoting time to educational programming for children is the Public Broadcasting Service (PBS), part of the Corporation for Public Broadcasting and partially funded by the federal government. Unfortunately with television, there are many other options for viewing, so students may opt for entertainment over school lessons or otherwise dilute their time watching television with material that is not helpful to education. However, for young children there was a positive impact on their knowledge of subject matter so long as they watched some educational programming, even if they watched plenty of other non-educational television. However, the effects became less notable for older children (Wright & Huston 1995). Part of that reason may be because of how much educational material exists for younger rather that older children and teenagers. Nonetheless, this research suggests that Internet websites devoted to the

134 same things could be helpful even though there are plenty of distracting options to view on the World Wide Web.

When it comes to avenues of access of content knowledge, there have been several services that have emerged in recent years. At the college level, several university-taught classes are recreated for free at Coursera; the lectures are in video format, occasionally there is a question pop-up during or at the end of a lecture video, there is homework, quizzes, tests, and projects, but the process is automated. For some courses, there are also chat sessions with the professors in a course, so there can still be interaction between the instructors and their students. Focused more at the high school level, there is the video library from Khan Academy, which contains hundreds of worked examples primarily in the areas of physics and mathematics. It also works on a method of student drills. In addition, Khan Academy has homework problems supplemented with helpful videos and hints when the student is struggling on a question. Like Coursera, the videos are free to anyone interested, but there is not a timetable for when videos are released and can be watched; the students can begin at any point they want and at any chosen rate. Khan Academy is also running pilot studies in classrooms to work in tandem with the traditional classroom setting.

However, like the addition of new technologies, there is still the same fundamental ways of instruction. The videos are still lecture-based, and there is little in the way of inquiry-based explorations. Khan Academy (2013) is aware of this, and in their fact sheet they support and have used project-based learning approaches in summer camps and hope to incorporate inquiry approaches into their more press-worthy innovations. Khan Academy is also aware of the need to avoid rote memorization and

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instead create deep learning and skill transfer, pointing to project-based learning and peer discussion as an avenue to achieving that goal.

There is also the issue of measuring the effectiveness of these online initiatives.

While there have been pilot studies in schools, currently, Khan Academy (2013) only notes how on their own metric that their methods have been improving from year to year.

To date, there is no published evidence in education research journals that the use of

Khan Academy is more effective than traditional methods, let alone by how much. But instead of looking at the particular methods of Khan Academy or Coursera, it should be worthwhile focusing on the effectiveness of the elements used by these sorts of programs.

There are two domains to consider: the use of video lectures and worked examples, and the electronic version of homework sets.

Providing a lecture as a video allows the student to listen and watch what the professor had taught at any desired time and repeatedly. This flexibility allows students to go over material from lecture at a more controlled pace, but this cannot replace lecture itself. With the video, interaction is not possible, so the student cannot ask questions as they potentially could during a lecture. Moreover, if there is an activity in the class done with the students, then that interaction is also lost.

Of greater interest to this thesis is the use of worked examples and solutions to homework problems. Already there is research that indicates the utility of providing students the work it takes to complete a problem. For example, Mason & Singh (2010) ran an advanced undergraduate class in quantum mechanics for physics majors, and after their homework had been turned in the worked solutions to those problems were provided, in the hopes of their use for studying would help them as they could learn from

136 their mistakes. Questions that were used in a midterm exam were used in the final exam, making it possible to analyze student improvement into solving the same quantum mechanical problems. Besides the traditional lecture format, the students had the additional help of worked homework solutions and their graded exams as study aides.

The average scores on these repeated questions actually became slightly worse. However, this was not true across all levels of ability. High-scoring students on the midterm questions performed better on the final, while low-scoring students on the midterm did about as well or worse. The regression was explained by the low-performing students using a different technique to solve the problem than they had before, but the new technique was not relevant and was instead related to materials that came after the midterm. In other words, they never learned from their mistakes and went on to make new ones.

These results, and others like them (see Mason & Singh 2010 for references), suggest that providing the answers is not a sufficient condition to increasing the size of a student’s ZPD, and thus worked solutions do not act as useful scaffolding. What this also suggests is that initiatives such as Khan Academy may be of limited utility given how their problems are done. This can be made explicable if one takes into account how the worked solutions tend to not have the motivations provided for how the answer was reached. A solution will have one line of advancement towards the resolution of the problem after the next, but it may not be clear why that step was taken in the first place, let alone the series of steps to get to the expected answer.

If this is the case, then changes to how homework solutions are given could make them more useful to students. A solution set that not only provides the steps to solving a

137 problem, but a solution that explains the underlying concepts involved and how they apply may have a positive effect on student understanding. However, even if true, student enthusiasm can be a limiting factor. If students are attempting to pass a class via rote memorization, then worked solutions even with motivations provided will not make much of a difference.

Turning to how homework is administered, there has been a significant turn to using computerized assignment software packages. These programs automate when students receive homework problems, what those problems will be, when they are due, and, most importantly, scoring them. This process has the obvious advantage of minimizing the time that professors or teacher assistants would have to grade homework by the traditional methods of turned-in physical paper solutions to homework. With physics classes at major universities having hundreds of students for a single lecturer in one term, the reduction of this workload is appreciatively desired; the benefits are the time saved for other tasks, and students can get feedback faster than with the traditional route. Moreover, with online homework systems, it is possible to instantly tell a student the submitted answer is in error and potentially provide hints, giving enough scaffolding to help the student understand the problem, which cannot be done as well with just marking an answer or step in a problem as incorrect.

However, the implementation of computerized homework assignments needs to be examined for its effectiveness in enhancing student learning. The history of the use of computers in the classroom has not indicated that their addition is sufficient to improving the classroom (Mayer 2009). Fortunately, there have been efforts to study and create computerized homework sessions that act as tutors and as tracking methods for student

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progress by teachers. Programs such as MasteringPhysics and others have intelligent

tutoring algorithms, so the advice provided is related to the answers the student provides

and the likely mistakes they made given their answer (Pritchard & Morote 2004). This is

becoming true for qualitative questions as well, such as interpreting graphs in the ALEKS

software package (Hardy 2004). As for the overall effectiveness, the major homework software packages have shown to help students more than was done with traditional written homework assignments based on normalized gains in physics concept assessment tools such as FCI and MBT (Morote & Pritchard 2002). This success is also found in more advanced physics topics than motion and energy, including electromagnetic and quantum mechanics (Sadaghiani 2012; Kohnle et al. 2012; Singh 2008). There is also evidence of better skill transfer when using these tools with their built-in tutoring

algorithms (Warnakulasooriya & Pritchard 2005).

Certain laboratory exercises may also be more effectively done in a computerized

setting, avoiding distractors with issues when using real equipment; this may be why

some electronic and optics labs induced greater learning in a digital format compared to

the same labs using physical equipment (Finkelstein et al. 2005; Martínez et al. 2011).

However, this situation may not be universally true, so further research is necessary to

show what sorts of labs are best done using a simulation (Chini et al. 2012). What can be

generalized is that computer programs that are designed with tutoring software that is

theoretically and experimentally-based will be a boon to students over traditional

homework and human tutoring alone. Nonetheless, current systems cannot take over for

instructors or human tutors with their greater flexibility in explaining and responding

(VanLehn 2011).

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For the purposes of this paper, the distinctions of the various computer-based

homework software packages is not necessary as the same homework system and

problems were given to students in both the traditional setting as well as in the

experimental group. The software package that was used was WebAssign, which was developed at North Carolina State University. Most questions require a numerical answer to a physics calculation, and that answer has a certain tolerance of 1% from the accepted

value. Usually students are given ten attempts per question to input the correct answer.

When an incorrect answer is given, a small hint may be provided. Other questions may be

multiple choice or free response. The latter are not automatically graded and simply tests

to see if anything was written; however, it is possible to manually grade these answers,

but that was not done during the semester when the experiment to be described later in

this paper was done.

Returning to the lecture space, there has also been the increased use of new

technologies in the classroom either for displaying information or for increasing student

interaction. Many lecture halls have moved from just chalk boards to projector screens,

either using an overhead projector or LCD projectors connected to a computer. The use

of computers and projectors had the added ability of presenting multimedia, from pictures

to animations to videos. These changes may allow information to be efficiently presented,

but they do not constitute an adjustment in pedagogy in themselves.

One additional technology that has the ability to change the classroom dynamic is

the use of an audience response system, better known as a clicker, a device that allows

students to vote or choose among options given by the lecturer (Caldwell 2007). For

example, the instructor can post a question to the class with multiple choice answers; the

140 students use their clicker device to pick an answer, and the results are tallied up. The instructor can then determine quickly what the status of the class’ understanding of a problem or concept is. This can prompt the instructor to present more to the class or encourage students to talk about the problem. Such interactions could help make the classroom less instructor-centered and more student-centered, especially if the questions encourage student-to-student communication about the subject. Overall, clickers allow the classroom to become less static by using these interactive devices, while providing real-time information about the state of understanding of the student body.

As for the effectiveness of clickers, the studies surveyed by Cadwell (2007) show that students tend to do better, earning higher grades in the same course with the same teacher. Moreover, there were the benefits of higher attendance levels for lectures, a lower course attrition rate, and students usually have positive attitudes about the tool.

Lasry (2008) questions if the technology is the real boon or instead the use of peer instruction. In his study, Lasry found that similar results in student gains in understanding when flash cards were used to vote as when using clickers when teaching the same course. It is worth noting that this was done with classes of about 40 students, and even in this case the use of flash cards was less efficient in tabulating results than with electronic clickers. For lecture halls filled with hundreds of students, this method of flash cards may be too cumbersome, and with class time at a premium the extra cost of clicker devices may well pay for itself in efficiency.

This survey of past and current uses of technological enhancements to instruction demonstrates that it is the use of technology, rather than the technology itself, that will help in improving student learning. With any future advancement in the use of computer

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software or digital learning programs, the point of pedagogy must be considered before

other factors such as processing power or screen resolution. There is also the point of

optimizing the tools we currently have, to which research is on-going and is a part of this

study.

But given we know how to apply the technology, what sorts of changes can be

made in the current instructional environment that can help teachers and students? This is

discussed in the following sections, including the details of the experimental group with

technological additions. The next chapter will discuss how effecting these additions were

to the classes.

B. In-Class Additions

In this section, we describe two notable additions to the standard lecture format:

demonstrations of concepts and clickers. The former was used in all five sections that

were taught in one semester of physics, and the same demonstrations were provided in all

cases. Clickers, on the other hand, were used only for the experimental group and one other class section; however, that other class did not use the same clicker questions, and

did not have access to the online innovations used by the experimental group discussed

below, and did not have the same pedagogical format. This last point will be of greatest interest for the experiment and its results.

i. Demonstrations

In the majority of the physical demonstrations of a concept the instructor would perform some task for the rest of the class to observe. A professor may bring up a conceptual

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point for consideration, or it may be a proof of some mathematical solution. In general, it

is designed to link the physics concepts and equations to an observable experience for

everyone in the classroom.

One example is how to aim a projectile to hit a target in free fall; the launcher,

pointed straight at the target’s location at time zero, is set to fire at the same moment as

the target falls. Should the timing mechanism work correctly and the launcher pointed at

the right angle, the projectile and falling target hit each other and make an audible clang

for all to hear. These sorts of demonstrations are set up in advance by the staff and have

the same demonstrations in all first-year physics classes, usually on the same day. The only case of students doing the demonstration was with rubber balls attached to strings, swung around and then let go to hit their target (in this case, the lecturer); students were asked to consider at what moment they needed to release the string in order to hit the desired target.

Throughout the semester the frequency of demonstrations had considerable flux.

Sometimes there would be more than one demonstration in a week (there were three one- hour lectures per week), while some weeks had no demonstrations at all. Some subjects, such as relativity, could not practically have a demonstration in-class, but this problem does not characterize the Newtonian mechanics that is of interest for this paper. While the machines used for the purposes of demonstrations may not always work or may need to be adjusted during lecture to work, the operations were mostly successful. Also, because of the large classroom size, holding around 250 students, with some considerably far away from the lecturer and the equipment, the room also had video cameras with optical

143 zoom; the video from those cameras was projected onto screens at the front of the room to make some of the experiments easier to view.

Along with physical demonstrations, some lectures included videos of a phenomenon or a concept being applied in popular media format, such as from a movie or TV show. However, this was primarily done in two sections of physics taught by the same lecturer who was not using the other classroom enhancements described below.

These video demonstrations have the same issue as physical demonstrations in that they may be passively viewed, but they were less prone to mechanical failures. On the other hand, technological issues, such as video compatibility, may come into play, but these kinks were worked out well in advance, minimizing this potential drain on instruction time.

The key advantage of such demonstrations is that the mathematical formalism of the class can be placed into an empirical context for the students to observe and understand. The demonstration may also allow for related conceptual questions to be formulated and answered. However, the activity is often done passively, so the activity may fail to induce the desired metacognition. There are limitations to exploration or discovery beyond what the lecturer provides, and so demonstrations are still generally teacher-centered. However, an instructor could engage the class using prompting questions or use the next technological implementation described below.

ii. Clickers

While demonstrations were used by all instructors, clickers were used only by two lecturers, one of whom was in charge of one of the control groups. In the experimental

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group, innovations were made that were not used in the other lecture section. In

particular, the standard clicker used by OSU students was one provided by the physics

department in the class. These clickers have 10 lettered buttons for answering multiple

choice questions and transmitted to the lecturer’s computer via radio transmission

directly from the clicker to an antenna plugged into the computer. The most significant

limitations of these standard clickers was that responses could only come from multiple

choice and there were not as many clickers as there were enrolled students in a lecture

section.

In the experimental group, the clickers used an Internet connection to a server

operated by the research group. The clicker website could be accessed by almost any

mobile device, such as smart phones, laptops, and smart pads. A wireless router was

provided to increase access to the server, though the lecture room had two other sources

of Wi-Fi. Because mobile web-enabled devices have become common and usually carried around by students, this should increase the number of possible participants who

could answer clicker questions. A poll conducted at the beginning of the semester found

that over 80% of the students had web-enabled mobile devices that they would bring to

class, which in theory means that far more students could participate than with the limited

number of clickers.

Web-based clickers also had a user-interface advantage to the standard clicker.

With the normal clicker, a small bulb would light up when a button is pressed; however,

it would be uncertain to a student if their signal was received by the clicker antenna. This

means that many students may have answered a clicker question but not had their answer

recorded. With the web-based clicker, the screen would clearly say that their answer had

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been submitted. However, the overall web-based access to the clicker system had its own user-interface issues, which are discussed below in Part 5.E.i.

Along with the increase in potential users, the questions could take other forms than multiple choice. It was possible to have free response questions in which students could type their answers. Software was created so that the most common answers would

be listed in descending order of how many provided that response, and this could then be

displayed to the class in multiple ways. Because of this flexibility, a greater variety of

possible questions could be implemented.

With the web-based clickers also came an adjustment to pedagogy. In order to

make lectures more student-centered, the results from the answers to a clicker question

were displayed immediately after the results were tabulated. If a significant majority of

the class failed to provide the correct answer, the lecturer would instruct the students to

talk among themselves for about a minute or two about what they thought was the correct

answer and why. The instructor may also provide prompting thoughts or questions to help

guide student conversations towards the best answer. After this, the same question would

be administered and the new results tabulated and displayed. One or two iterations were

sufficient to get a majority of students to choose the correct answer. However, in the

deliberative process the lecturer could also provide hints, so discussions were more likely

to move in the direction of the answer desired by the instructor. This procedure was done

both for multiple choice and free response questions. This method helps create a more

student-centered classroom, while the instructor still provides some scaffolding to the

students but only when the class is misunderstanding something.

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C. Outside Additions

In order to extend the available learning materials to students, the experimental group was provided with further instructional additions. With a server already established for using clickers, students in the experimental section were able to log into a website to access online materials, including the daily lectures, slides, and homework solutions.

i. Lecture Videos and Slides

Each lecture was recorded by a single, static video camera positioned to capture the projected slides and the blackboard for work done there by the lecturer. The lecturer would also be in this same framing. Audio was picked up by the camcorder’s microphone, so it may also capture nearby student movements and sounds, so generally the audio did not have great fidelity. Fortunately, this seemed to be a minimal issue as the speech by the lecturer was understandable most of the time. As for visual quality, it was sufficient to be able to read most things written on the blackboard, though it would not pick up the projection screen’s images very well.

To maximize the accessibility of the video lectures, the video files were made into two formats, one for normal desktop or laptop computers (.avi format), and another for mobile devices (.ogg format). In addition to these file formats, the video was processed to run at 10 frames per second (FPS), making the files a fraction of their normal size. These technological considerations made the videos accessible on most any device a student was likely to use and with files small enough that they would not have to be burdened by loading times or memory issues. After some initial problems with the server and student

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access, the system was able to provide the video of the lectures to everyone who wanted

to watch them.

Along with the videos were the slides from the lectures. The slides originally were

made in PowerPoint, but they were saved into the more universally accessible PDF

format. Before uploading, minor changes were made to some slides, such as fixing a typo. With the slides, students could either watch a lecture with clear images of the slides or review the slides on their own.

Because the server was operated by the research group and required students to enter their username and password, a record of who watched or downloaded what files, how many times, and (in the case of videos) how long they watched were all recorded.

This is important for the analysis done in Chapter 5.

With these additions, a student would have access to the same sorts of lecture

materials as everyone else, but this would be completely teacher-centered, though it could

be used at the desired pace of a student. The lectures would also not allow for someone

watching (or re-watching) to interact with clickers or fellow students. This means that the

videos and slides were not a substitute for actual lecture attendance and participation.

While this means that there was no fundamental changes to pedagogy, instead the video

and slides would ensure the student knew what the material to be considered and how the

lecturer chose to have it incorporated, thus extending the lecturer’s ability to present the

materials. Nonetheless, it was another addition that had the potential for creating

something that would better engage a student’s metacognitive activities.

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ii. Homework Solutions

While video of the lectures can only repeat what was said and done and (we hope) already seen in person by the student, one item that a student would not have been able to see without using the server was video solutions to the homework problems. A set of homework solutions would only be released after the due date of the corresponding homework set from WebAssign. As with the video lectures, more than one file format was provided to maximize accessibility.

Video solutions were usually only a few minutes long, and they would go through a homework problem in a fashion similar to that from Khan Academy. A digital pen or

“smartpen” was used to record what was written digitally, and the pen would also record what was said while writing and explaining the solution. The software package for the smartpen had additional packages for PDF readers so it was possible to watch what the pen recorded along with the audio. However, to make sure that this material was accessible to students who likely did not have the additional software for their PDF reader, the PDF video was played and screen capture software created a standard format video. These videos were then put into the same format as the lecture videos and placed online for students in the experimental sections.

Most solution videos for one problem lasted a matter of minutes, with few being longer than 10 minutes in length. Homework questions included conceptual problems as well as algebraic or “plug ’n’ chug” problems. All of the problems in a homework set would be given a video solution.

In appearance, the video solutions had a similar look to those from Khan

Academy, though the latter had a greater variety of colors for the words and pictures

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Unlike Khan Academy, however, there were some differences in pedagogy. When setting up a problem, it was emphasized what concepts are needed to understand and set up a problem. Additionally, the common misconceptions that a problem may incorporate were specified and explained how it would lead to a non-physical conclusion. Illustrations of what were happening in problems or something analogous was also provided where necessary to further elucidate the solutions.

In this way, the concepts were highlighted, the misconceptions were brought to light, and something in the experience of the student was used as a comparison for what was happening. The homework solutions provided more than just the work needed to find the answer; the solutions explained why they were set up the way they were, as an expert would do it, and to provide contrast with others ways of thinking about the physics such that the student can reflect on the problem and its underlying concepts. While the solutions were still in effect video lectures (and thus teacher-centered), the design was set to create an impetus for metacognition. This should then make the homework problems along more effective to learning, and the solution design should be an improvement on the straight-forward presentation of how the answer was reached, such as was described for the quantum mechanics course above (Mason & Singh 2010).

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CHAPTER 5. TESTING METHODS AND RESULTS

A. Standard Physics Class

For the fall semester of 2012, first-year calculus-based physics had five different lecture

sections with four different lecturers (one lecturer had two periods to teach). Only one

section was in the experimental group, the remainder acting as the control group. All

sections had the same homework problem sets and the same laboratory experiments.

Quizzes and tests were created by the individual lecturers for their section, though they all covered the same materials at about the same time; on a quiz day, all sections should receive a quiz on the same topic, such as projectile motion or collisions. A given quiz from one section would have one grader to make sure the same judgments were made for all problems on the quiz and thus enacting a fairer grading scheme; similarly, a particular exam question would have one grader, while another exam question would have a different grader.

Other than the lecturer, all of the sections were the same with the exception of the changes to the experimental group. Those changes are described below.

B. Additional Lecture Materials

During lecture, an Internet-based clicker system was instituted for the experimental group. Questions were in two formats: multiple choice and free response. A given lecture period could have between one and five different questions asked of the class. The

151 procedure is done in the following way: the question or problem is projected onto the main viewing screen, showing the possible answers if it is a multiple choice question.

After the question is displayed, the lecturer opens up the clickers to voting, and the students who are logged into the clicker question server choose the poll that is currently open. The students cast their vote, and a counter is displayed showing how many have voted along with a timer saying how long the poll has been open. A poll is open usually for less than two minutes, but enough voters to be representative of the class need to enter in an answer. The lecture room holds over 200 students, but on most days there is closer to 100 students attending. With this class attendance, at least 20 voters are desired, and there was a push to get at 30 or so. Then the poll is closed, and the results are displayed on the screen. If the question is multiple choice, then a bar graph displays how many gave a particular answer. If the question is free response, then a list of the answers provided by the students is projected with the most common answer on top. The number of students providing such an answer is also displayed. Some answers students give will be equivalent but not counted as the same because of spelling, capitalization, etc.

In many cases the class may have a majority give a particular wrong answer or they are evenly split among two or more options. If this is the case, the lecturer asks the students to talk among themselves for about a minute about the question and what they think is the right answer. This usually influences student results, as can be seen when the students are then polled again after the discussion has ended. If the majority of students still fail to form a consensus on the right answer, then the lecturer explains how to reach the correct solution. In this way the students have to reflect on the lecture material,

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All of the results of the polls are recorded on the server from where clicker questions are run, including if the same question is re-opened to polling. In this way the lecturer can have a record of what problems the students show they struggle with, and a similar or the same question may be asked in a future lecture.

C. Online Materials

The experimental group was given several additions to the lectures online. The slides for each day’s lectures were posted for download. Along with lecture slides were the videos of the lectures themselves. Two video formats were provided to make them watchable on either mobile devices or full-size computers. The camera for the lectures was placed so as to capture the lecturer, the main blackboard on which the lecturer may write things, the demonstration objects (if any), and the screen on which the PowerPoint slides were projected. The camera would not provide good resolution of the slides, but they are visible enough so that a student with the slides on their one computer can follow along nonetheless.

The other video addition was the homework solutions. One video covers a single question from a homework set, going through not only how to solve the problem but also what concepts are being used. The solutions include the writing of the solution as well as audio explanations. These homework solutions were posted after a homework assignment was due and was only accessible to students in the experimental group.

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D. Assessment Methods

The primary tool used for assessment purposes was the FCI. The test was given to students on three different occasions. In all cases students were to put their answers into an optical answer sheet or “bubble sheet” to be run through a scantron device. The first time was in the students’ first recitation, before any considerable lecture materials had been presented beyond the syllabus and the very basics of kinematics. No time limit was given to the students except of the length of the recitation period, approximately 45 minutes. The FCI questions had an additional 14 questions added for the purposes of research by another group and are not considered here.

After this were two posttests. The first was performed shortly after the first midterm examination. This midterm test focused primarily on basic Newtonian mechanics from kinematics to rotational dynamics. This included projectile motion, forces, energy, and momentum. The first posttest was administered during laboratory time when group work was normally assigned, and it was administered about seven weeks after the pretest. Unlike the pretest, there were no additional questions to the FCI.

However, three versions of the test were provided. One version was the standard, 30- question FCI, while two others were a slightly shorter version of the total number (14 out

of 30) of FCI questions and in differing orders. These shorter tests also had an additional

13 questions that were not found in FCI. This was part of an examination of how many

FCI questions are needed to assess a student’s conceptual understanding properly and if

other test questions could substitute equally well or better in testing for misconceptions.

The third use of FCI acted as a delayed posttest. This was administered in lab

after the second midterm and approximately one month after the first posttest. This

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second midterm focused on rotational mechanics, fluids, thermodynamics, and oscillations—all topics not covered by the FCI. The timing of the delayed posttest was also four weeks before the final exam, which was cumulative and thus would have likely increased student studying in the concepts tested by FCI. This delayed posttest could then examine the retention of previous material from the semester even when the students were studying and focusing on fairly unrelated physics material. In this case, again a shortened version of the FCI was used in two versions. All tests had 14 questions, though in differing orders. In addition, there were four different versions of a basic math quiz testing students’ abilities in proportional, arithmetic, and calculus-based reasoning. This was to see if there was any correlation with math abilities and the abilities to understand physics and complete such problems. The four versions of the math quiz had mixed ordering of questions.

To analyze the data from the three uses of FCI, the percent correct was calculated for each student. Between the pretest and either posttest the normalized gain was calculated. Those normalized gains could then be averaged for each section, and in this way a comparison between the control and experimental classes could be assessed.

Originally 1055 students in all sections took the FCI pretest. 987 took the first

posttest, and 216 were in the experimental group. This drop is in part because of lower

attendance in the lab period than in the first recitation, and in part because of people

dropping the class after the first midterm. For the second posttest, 845 participated.

Again, lab attendance was lower, especially as the semester was ending and not all labs

needed to be attended for earning all of the possible lab points. Also by this point more

students had dropped the class. Additional tests could not be analyzed because the student

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failed to leave any identifying information (student number, name, etc.) or their test version, so it is not possible to calculate the normalized gain for them. For the first posttest, 625 students provided the necessary information; for the second posttest, 439 students provided the same information. Overall, only 164 students took all three FCI tests and provided identifying information. Of those 164, 44 were in the experimental group.

For the purpose of comparing the experimental and control groups, the pretest-to-

first-posttest evaluations receive the most attention. This provides the necessary evidence

for how effective the experimental methods were in increasing learning gains, and it is

for these first two tests that we have the most data. In order to compare sections and how

much each population of students increased their physics understanding, the normalized

gain is used. Such a calculation compares how much a student already knew and how

much that they did not know was learned by the time the posttest was administered. The

normalized gain is defined in the following way (Hake 1998):

= . 100% 푝표푠푡푡푒푠푡 − 푝푟푒푡푒푠푡 푔 This equation has a problem if the pretest score− 푝푟푒푡푒푠푡 is 100%, so in those cases the student

score is not included into the average calculation. After all, with accurate knowledge that

is examined by FCI when entering into the class, it is a priori impossible to teach them something and have it register on the same test. However, such a circumstance was exceedingly rare.

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E. Difficulties in Implementation

New technological additions are always prone to unforeseen issues when first used, from

glitches in the machinery or software to problems actually being able to use the

technology at all. A few of the issues that came up during the implementations are

highlighted here and what problems this may have for the analysis of their effectiveness.

i. Clickers

After three weeks the Internet-based clicker system was the only way for students to answer questions put up by the lecturer. Besides a few issues with some mobile devices at first, apparently at the user end more often than not, any Internet-enabled device had the ability of accessing the server to answer clicker questions. However, reaching that server was not made user-friendly. First, the student would have to logon to the wireless Internet router, which required a password. Then the students would need to log into the OSU web server, again using a username and password. Finally, the students would need to logon to the clicker question server with yet another username and password. These three hurdles, each taking time, limited the willingness of students to actually answer clicker questions. This would be most pronounced for the first clicker question asked. A student who is not already logged in would need to go through the entire process described above, but in that time the question may already not be available to answer as the lecturer has closed the poll.

The unfriendly nature of this interface is most apparent when comparing the first few weeks of class with the standard clickers provided by the physics department and the

Internet-based system that was used later. In those first weeks, most of the available

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clickers would be picked up by the students before lecture began, and it was not

uncommon to have over 100 students answer a question. When the Internet-based clicker system was implemented, it was a struggle to get 50 students to answer the question and often using more time that before with the older system. This lowered participation needs to be understood to avoid it in the future.

To examine this lower participation in clicker voting, an anonymous survey was

given to the students to perform, to which 43 students in the experimental lecture group

answered. These students were diverse in how much they used the clicker system, from never (6) to always (1). The students were asked why they may not use the clicker system, why they thought others did not use the clicker system, and they could leave comments to explain their answers more fully. Some had simple connection or software issues that did not allow them to access the server or the questions. When students tried to logon they may overwhelm the server, limiting the number of students that would potentially answer a question; this was also noticed when I walked around the lecture

room to see if students had issues. A few pointed out that by the time lecture began their

mobile devices had been on for much of the day and their batteries were drained.

Numerous students worried that the answers were not anonymous; they had to login with

their username, so all of their answers were recorded. Many students also thought that

their fellow students were too distracted by the web itself. By having a laptop or smart

phone out already to peruse the Internet, it becomes easy to access social media or other

websites having nothing to do with the lecture content and making the period wasted time

for teaching.

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Perhaps the most noteworthy social issue that the clicker system brought in,

which was also mentioned by several of the students who answered survey questions, was how students answered the free response questions. Early on it was not uncommon for

one or two students for a given question to type in profanity or suggestive comments.

During lecture the professor would ignore them, and filters were installed. Nonetheless, the creativity of students who would rather find ways of displaying inappropriate message to the class was beyond what any filters could have considered in advance.

Fortunately the vast majority of students would legitimately try to provide an answer to the free response questions, but the distractions were noticeable to the student body even if the lecturer chose to ignore them.

The problems of distractions by the Internet and by the inappropriate responses of students are perhaps the biggest issues for continued use of Internet-based clickers, along with others that can be overcome with better network throughput to and from the server and a better user interface. Several solutions for a future attempt to get past the more serious problems present themselves. To keep students on task, it could be possible to have the router set up such that the only website accessible to the student is the clicker server. This may also alleviate issues of wireless data throughput since students will not be able to access unrelated music, videos, and social media outlets that can be data-heavy when downloaded. Concerning inappropriate comments, a more sophisticated filter may be helpful, but again students will discover other ways around the problem, such as purposeful misspellings of vulgarity or using other non-alphabetic characters in place of letters which can still be understood, such as with the Internet “alphabet” of leet (or l33t).

It may be necessary to intercept these answers manually before they are displayed or

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simply not have free response questions anymore, leaving only multiple choice questions

that cannot be used to project distracting commentary.

As for generally overcoming the problem of students not wanting to use the

clickers and answer questions, it may be worthwhile making their participation a part of

their class grade. For example, if students had to answer 75% or more of the clicker

questions to receive full participation credit, that should encourage student use of

clickers. This was something that the lecturer wanted, but such decisions about what can

count for points in a class are decided at an administrative level outside the control of a particular lecturer. Hopefully in the future it is possible to amend this grading system and allow for participation to count, perhaps for 10% of the final grade in order to maximize lecture attendance and participation. Should this happen, then it will be necessary to minimize the issues students have accessing the clicker system itself lest they lose points because of technological hurdles to which they are not at fault and cannot overcome.

ii. Homework Solutions

When classes began, not all of the technology was set up, in particular the server and database for the videos of lectures and homework solutions. This created a delay in publishing the first solution sets, which made the watching of the first videos less likely until the first midterm encouraged the students to use the video solutions as a study guide.

When the server was ready to publish video solutions, the format of the video files was not always what was needed on a particular device. For example, the database may not have the video optimized for mobile devices accessible, leaving only the standard video

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format which mobile devices could not display. However, these issues were resolved

within a week and well before the first exam.

Another limitation was placed on how much the video solutions were viewed

because of administration concerns. The discovery that all of the homework solutions

were being placed online was disconcerting to other lecturers. This was because this

would mean the homework solutions could be copied and put online for future students to

use. In other words, the worry was that having the solutions given as such would mean

future physics classes could not use the same homework problems else the students could

simply search the Internet for the answers rather than try to answer them and learn

something in the process. Because of this worry, new video solutions for several weeks

were not produced, though the solutions sets that were put online remained there for

student consumption.

This hiatus eventually ended, but this meant that not all of the solutions were

created and put online before the end of the semester. Fortunately, all of the video

solutions to problems pertaining to questions asked on the FCI were put up in a timely

manner (after the initial delays). As for the concerns about the homework solutions

existing online because of the implementation of video solutions, it is worth pointing out

how the established system limited this and why it is not a defeater to future use. First,

only students enrolled in the experimental lecture period could access these videos. Once the semester was over, their access to the website with the videos was suspended.

Second, there was no way provided by the website to download the videos to the hard

drive of the students’ computers. Perhaps a student could find a way of doing this, such

as an application run from the web browser, but this would at least mean there was a

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barrier to download the video and uploading them elsewhere. Lastly, there already exist

websites that have solutions to homework problems in many physics textbooks.

However, these require either uploaded a text or PDF file or typing the solution into a

format that can be accessed on those websites. With the solutions in video format, it

would mean a student would have to do a considerable amount of work at no direct

benefit to themselves or any of their fellow students; the video solutions were only posted

after the homework due date had passed so it could not be used to give homework points

without doing any work. The overall point is that there already exist websites that provide an outlet for students to cheat, and the video format of the solutions is not conducive to disseminating these cheats.

Nonetheless, even with the use of video solutions justified and all solutions pertaining to FCI questions were available to students in a timely fashion, the delays and hiatus in putting up new content may well have limited how much students came to use this tool. Numerous times students would email the lecturer or me about putting up new homework solutions, but with the delay it meant that many would not find the resource all that helpful or dependable. This may well then produce a ceiling on usage. In the next section it is shown that the consumption of the video materials was not great, and the

problems mentioned above may have been a contributing factor.

F. Results

For the purposes of comparison, students who had taken the FCI pretest and those that

took the identical FCI test for the first posttest are compared, leaving out those students that took a similar FCI test but not with all the same questions or in the same order. To

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compare the control groups from the experimental lecture group, the normalized gain

calculation is used.

In the control group, the normalized gain was 22.58% (SD 20.77%), while the control group had a gain of 34.71% (SD 20.57%). These results were found to be statistically significant using a chi-squared test ( 0.01), and the effect size is also notable (d = 0.58). The new gain moved up more 푝than≈ half a standard deviation from the control group. Looking at individual lecture periods in the control group, those periods each had a lower normalized gain than the one experimental lecture period. The best individual lecture period in the control group had a normalized gain of 27.15% (SD

22.23%), which is still more than a third of a standard deviation below the experimental group. This best control group period was also performed by an experienced physics

lecturer, who also taught another lecture period with a comparable normalized gain; the

remaining two control lecture periods were done by others with less experience in

teaching first-year physics and had other duties besides teaching. The results of all

sections and the control group average are provided in Figure 9.

Turning to the delayed (or second) posttest, the analysis here is a bit more

ambiguous because the same version of FCI was not used during the second posttest but

instead shorter versions. These faster versions of the test were also done during the first

posttest, and it was found that the 14-question version of the FCI was strongly correlated to the full-length, 30-question version. When comparing the questions that a given form of the short FCI test had in common the full FCI, the percentage of times the students would answer the question correctly was very much the same; the correlations between short-test questions and their corresponding full-test questions was R = 0.82 for one short

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Figure 9. Normalized gains for each section and the control group average for physics

taught in Fall 2012. Error bars represent the standard error.

test and R = 0.971 for the other, extremely high correlations, but not unexpected.

Moreover, the short FCI tests had average percent correct results very close to the full

version, within a couple of percentage points. As such, using the shorter and full-length

FCI tests for analysis ought not to give biased results to a significant degree.

Using all of the data from all test versions given during the first and second posttests, but only for students who completed all the pretest and the two posttests (and provided the needed identifying information), the experimental group had a small but negative change in normalized gains; the normalized gain from the pretest to the second

(delayed) posttest was about 6 percentage points lower than the normalized gain from the pretest to the first posttest. There was also decay in normalized gains for the control group, though there was a much smaller deficit than that seen in the experimental group.

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Still, the experimental group had a larger normalized gain after the second posttest than

the control group had after the first when their gains were largest.

While already an interesting and positive result, it is possible to learn even more

from the FCI test by observing the evolution of what models the students used. This is

done through model analysis (see Part 2.A.ii.a). In this case, the two models that should account for most of the class’ mental states are the Newtonian model that is taught in class and the common, naïve model. The FCI test provides several answers to their multiple choice questions with good distractors that conform to the naïve physical models that have been noted in earlier research. Using this, it is possible to plot how often students use the correct and naïve models before and after instruction. Moreover, because the FCI’s authors believed that there were six conceptual categories in their test

(kinematics, impetus force, active force, action/reaction pairs, concatenation of influences, other forces), it is possible to watch particular categories evolve.

The model analysis of the control group (see Figure 10) shows that the students at the beginning of the course were already in the mixed model region for most physics concepts, and two categories fit into the low corner of the model 1 (Newtonian) region.

This shows that the class already had a fairly good understanding of mechanics, likely in part due to prior instruction in high school. However, the total use of either the naïve or

Newtonian model was rather low, sometimes not accounting for half of the responses of the students (the remained going to answers considered random, perhaps due to some misinterpretation of the questions). Because of this large group of random model answers there is a lot of space for the class to move as those with random answers can reinterpret the questions and provide Newtonian or naïve answers.

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Figure 10. Model Analysis of Control Group.

In the case of the control group, it is apparent than a large portion of the random answers filled into the two models. However, a considerable portion of that evolution was into the naïve model. In the starkest case, the kinematic questions went from 49% random answers to zero, but the probability that the class would provide Newtonian answers was virtual unmoved—in fact, it was slightly worse. This indicates that almost all of those that gave random answers then began to provide naïve answers. On the other hand, conceptual categories such as active force and impetus saw an evolution that was almost totally in favor of the Newtonian model. Nonetheless, almost all of the conceptual categories of the FCI ended in the mixed model region.

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Figure 11. Model Analysis of Experimental Group.

Compare this now to the experimental class (Figure 11). For the pretest, the class

is almost identical to the control group, so there does not appear to be an issue of

sampling or selection bias. As for the evolution of the class, in every case the FCI

conceptual categories in the posttest are in the Newtonian model region. There is also

movement away from giving random answers, though this is a slighter effect than seen in

the control group. What this suggests is that students who came in with a naïve physics

model of mechanics were more likely to understand Newtonian physics concepts, while

students who gave random answers had little evolution towards either naïve or

Newtonian conceptions.

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Figure 12. Correlation between Normalized Gains and Pretest Scores for the

Experimental Group.

This leads to the question of how equitable were the effects of the experimental group’s new teaching methods and tools. That is, were students that were initially weak in physics concepts having the greatest difficulty in learning? To examine this, the pretest scores were compared to the normalized gains and the correlation was determined (see

Figure 12). A statistically significant correlation was found (p < 0.05), but the effect itself was rather small; the correlation value R was only 0.25, which is fairly low and only able to account for about 6% of the variance of the normalized gains. So while weaker students had a harder time improving their understanding of Newtonian physics concepts, the effect is not large.

Given the overall effectiveness of the experimental group in learning compared to the control groups, now it is necessary to see if the classroom and online additions were of any significant help affecting student normalized gains. Correlations between normalized gains and the amount of time video homework solutions were watched and

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between normalized gains and number of times clicker poll questions were answered

were calculated. In all cases, there were no statistically significant correlations, and the R

values were very small and negative. When comparing students who watched at least one

minute of homework video solutions to those that watched less than a minute (usually

zero), the students who had watched videos had an average 2.7% deficit in normalized

gain than those who did not use videos. Similarly, students who used clickers had an

average 4.9% deficit in normalized gains compared to those who voted ten times or less

throughout the semester (there were 130 Internet-based clicker questions asked during the

semester in the experimental lecture section). Comparing students who used both videos and clickers and those who used none of the additional tools, the tool users had an average 6.9% lower normalized gain than those who used none of the additional learning tools.

The lack of a positive effect from the use of the tools and normalized gains requires explanation, especially because there were no other significant differences between the control and experimental sections. In the case of watching homework solutions, an absence of effect is likely a product of a lack of use of the tool. A majority of students in the experimental group never used the homework solutions at all, while those that did watch barely made use of them. Among only those students that watched any of the video solutions, the average total time watching them was just over 12 minutes. This may be the length of a single video solution and at most four, while 58 video solutions relevant to FCI questions were made, providing over 5 hours of content.

The median total time watched was a little more than 6 minutes, meaning half of the

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students that watched video homework solutions only saw one or two homework

problems solved at most.

It is also worthwhile to consider the population that did watch those videos. Much

like the deficit in learning gains among students who used video solutions, the pretest

scores of the students who did use the videos had an average 4.3% deficit compared to

the students who did not use the tool. This suggests then that the population that used the

videos were those who were struggling the most, but these were also the students who

actually failed to use the tool for any significant amount of time. In other words, those

that needed the most help looked to the provided tools to assist them, but they failed to

utilize the tool to any significant level. As such, the negative result is indicative of the self-selecting population that knew it needed additional help but did not schedule any significant use of said assistance. With such minimal use and a possible selection bias, whether video solutions would help or hurt cannot be determined from the given data.

Turning now to the clicker results, the count of how often students voted may not be indicative of how much the tool helped them. As noted earlier, various issues caused students to be reluctant to vote on an answer for a given clicker question. Even with a hundred students in the lecture room, it was a struggle to get a third of them to cast a vote. However, it was still possible for students in the lecture room to answer the questions posed to them and just not submit them. Moreover, if those that did vote provided answers that had incorrect answers in the majority, then everyone were instructed to talk with their neighbors about what they thought was the correct answer. In addition, the correct answer was provided in the end to the whole class, so there was

170 continuous feedback between students and teacher even if they did not leave an electronic artifact for data collection purposes.

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CHAPTER 6. CONCLUSIONS

A. Summary of Findings

As there were two major aspects of research for this paper, there are two major sets of

results to consider. First are the results from students about their preconceptions about

physical concepts and their variance with Newtonian mechanics. Second are the results

from changes in lecture to help overcome these preconceptions.

i. Misconception Origins

The misconceptions, such as a net force always acts in the direction of motion, has shown up in the literature for millennia, starting with Aristotle and up into the modern era when examining students entering college. In an effort to try and understand where these ideas come from, interviews were done with several dozen physics students in their first term of university-level physics. In the case of the force in the direction of motion, when students were given abstracted scenarios of moving objects they had a tendency to assign this nonexistent force, but when questioned about it they often reverted to their college lessons and gave a Newtonian answer. During interviews about how they feel pushed and pulled under various cases of acceleration, students often claim to feel the fictitious force, sometimes believing it to be real, sometimes thinking it is an illusion. The more they thought about it, the more likely they are to believe that fictitious force to not actually exist.

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The key finding here is that students have built up an instinctual feeling of the fictitious force that seems to conflate velocity and acceleration. When a car is braking, one’s body goes forward unless stopped by friction in the seat or the seat belt, so that feeling of being flung forward becomes interpreted as a force acting in the forward direction. Because the feeling has become so instinctual by the end of a student’s teen years it takes directed thought to overcome such a belief. To get that directed thought, though, will require particular teaching methods that provide feedback to let them know when they are slipping into thinking as a medieval natural philosopher did.

ii. Lecture Changes

For the Fall 2012 semester of first-term physics, an experimental group of students entered into a lecture series that had several additions intended to help students overcome these misconceptions. The first was an Internet-based clicker voting system, which during lecture was used to have students answer conceptual questions. The second major addition was access to the lectures online, accessible either on a computer or mobile device. In addition to the lectures, video solutions to the homework were provided. These solution videos not only showed the correct procedure but also included explanations of the concepts and likely ways that a student may fall into a trap when trying to solve the problem.

To discover how the class improved, the FCI test was administered to all sections, both in control groups without these additional tools and the experimental group that did have access. The results were that the average normalized gains of the experimental group were of a considerable effect size compared to the control group and statistically

173 significant, moving the class learning average over half a standard deviation. Using model analysis, it was shown that in all of the conceptual categories of the FCI that students moved from a mixed use of naïve physics concepts to used Newtonian ones. The control group had much less evolution from naïve to Newtonian, while those who gave random results tended to give naïve results in the posttest. The experimental group had less effect in changing random results, though there was some improvement.

It is possible to be skeptical of this result if one considers the Hawthorne effect, by which studying the students will affect their thinking and answers. In other words, because the students know they are being studied, they could be providing the sorts of answers the researchers want. While the reality of the effect has been controversial (cf.

Adair 1984), it will not explain these results. First, the testing was done with all groups, so they will all experience equal levels of the Hawthorne effect, if any. Second, the students received no feedback concerning FCI results and were thus blinded. This is considered by Parsons (1974) to act as a sufficient defense against the Hawthorne effect.

On the other hand, the related issue of novelty in the classroom produces a response is mitigated; while the experimental group used a clicker system, so did the first control group which did not perform even at the control average (perhaps because lectures were early in the day). So it is unlikely that simply having clickers will move up the average learning gains of the class, which leaves the change of pedagogy when using clickers.

The advantageous results from the experimental group were found to benefit students with a stronger background in Newtonian physics conceptions than weaker students, and students that provided random results had the weakest change in a positive direction. Students in the experimental group, on the other hand, seemed to have greater

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benefits to those that gave random results, though it did not necessary mean that they

became Newtonian thinkers.

A possible explanation for why instruction in the control group was more helpful

to students that gave random answers is that the control group lectures were more along the lines of direction instruction. This teacher-centered method is effective in getting out factual information at a good pace, and weaker students are probably those with the least amount of background in physics or science in general. Direct instruction would provide them with background information that would then allow students to better understand what the questions on the FCI were asking. With this, students could then provide a correct, Newtonian answer, or they could provide a naïve answer, now understanding how the question relates to their own concepts of forces and motion. Being given the information in an upfront fashion, there is less of a cognitive load on working memory and allows the new information to be absorbed more readily. However, direct instruction has difficulty in overcoming misconceptions; it may fill an empty vessel, but it may not get past a filter. Long-term retention is also less likely, as the second posttest revelaed.

The experimental group, on the other hand, had a more student-centered approach, which led to more reflection or metacognition during lecture. Clicker questions and inter-student conversations about concepts during lecture would help facilitate that.

As such, those who gave naïve answers to physics questions were given the opportunity to reflect on how their beliefs differed from the mechanics taught at the college level, so they could improve towards being Newtonian thinkers. However, those students who

gave random answers did not have as much background information, and the student- centered approach may not have provided them with the needed knowledge to reflect

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upon. Without the needed background knowledge the lecture period would present

information to them for the first time and then ask them to think deeply upon this new

knowledge. Such a situation is likely to overtax the working memory of such students

and be of minimal help.

Concerning the effectiveness of the particular tools used to help the experimental

class, the results are not as obviously indicative and require considerable interpretation.

The video homework solutions were not shown to have any positive effect, and the

students who used the videos at all were, on average, weaker physics students, both in their starting point and in their normalized learning gains. However, this poor result cannot act as evidence against the utility of video solutions because they were used so little by so few. Fewer than half of any students watched the videos, and half of those students watched less than seven minutes throughout the semester. With such minimal

use there is no plausible way the videos would have any effect, positive or negative. As such, it cannot be said with this data if video homework solutions as designed are helpful, let alone how much.

As for the clickers, there was similarly no significant correlation with the number of times students voted and what their normalized gains were. Student voting was, much

like the video solutions, not often done, and in a given lecture it would be difficult to

have 30 students cast a vote on a given question. The reasons students did not vote were

varied, but a very common reason given by the students were technical issues: battery life

on their web-enabled devices, problems accessing the Internet, issues logging into the

clicker question server. With small numbers of people who are able to vote on a clicker

question and the oddities of the sampling that this problematic system used in lecture

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would provide, statistics about how well students are doing in either group using or not using the clickers is not meaningful; there is no reason to assume the sampling will have a normal distribution, and with small numbers of participates random fluctuations cannot be eliminated.

On the other hand, the voting system still has an important role in the lecture, and that is the ability to provide quick and broad feedback between the lecturer and the students. While more students proving answers to poll questions will better reflect the classroom’s mental model state, a small number can still be reflective if, say, 10% or more provide answers. This way the instructor can know with enough confidence that the students understand a concept or if there is a need for further discussion. Moreover, the feedback between students and lecturer is helpful even to those who did not vote; they still had the question and the time to think about it, and the professor is still there to provide the right answer. Furthermore, the classroom discussions that take place after a question is not correctly answered by a majority of the students mean that everyone has the opportunity to reflect upon the concept that is involved in the question, and this will not have any of the technical limitations that the clicker system may have.

In sum, the clickers and the particular way they were used are likely to be the primary reason that the experimental group had the greater normalized gains compared to the control group. The effect of the video solutions on student understanding cannot be determined, but because they were virtually unused they likely did not act to change the conceptual understanding of physics for the students.

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B. Future Goals

While some insights into the origins of physics misconceptions were brought about through the interview of college students, there are numerous uncertainties. In particular, there is a lack of research into when misconceptions such as the force in the direction of motion become cemented into the minds of people. The only data that could be related to this concerns not linear motion and forces but the trajectory of objects coming out of curved tubes; this research showed that pre-teen children went from giving correct answers to incorrect answers as they became older, at which point it took a college course to return students to a state when they could answer the question correctly more often.

While the trajectory of an object leaving a curved tube could be indicative of the force in

the direction of motion and the old impetus theory, it cannot be certain if these concepts

are intertwined in the minds of young and teenage children. They may be based on a

common p-prim or on different ones.

The study of linear misconceptions done here with college students needs to be adapted and performed with a younger collection of subjects. By observing a younger population it may also be made clearer as to what experiences they have that lead to the misconceptions seen in adults. Considering that these misconceptions are seen in great antiquity it is unlikely due only to modern experiences, such as the feeling of being driven in an automobile. Perhaps a careful examination of ancient sources as well as modern observations can provide the necessary information for what are the most prominent experiential sources for physics misconceptions. Such knowledge could help inform educators when it may be necessary to provide instruction to prevent the misconceptions from becoming fixed and then hard to overcome at the college level. The

178 experiences involved may also dictate what sorts of lessons, computer software, or toys could be manufactured to create fixed Newtonian thinkers. However, lesson plans and the like must wait until research provides the needed details of what causes the naïve views and at what point in the lifetime of children.

As there will be considerable time between the success of such research and implementation at the K-12 level, the first-year college physics lecture experience will remain a staple item for university classes. Concerning the changes to the classroom that proved to be effective, there are aspects that need scrutiny. As mentioned in the results in

Chapter 4, the experimental lecture system had less of an effect on students who gave random answers than did the control group. It was suggested that the direct instructional methods in the control group are a probable cause for helping students who gave random answers. In that case, future research ought to focus on helping those students who failed to evolve. Perhaps this could be alleviated if a way to encourage the weakest students to come in to lecture with as much content knowledge as possible could be arranged to provide them the best chance to reflect upon that knowledge.

One possible way to help weak students is to provide an incentive to read the material that will be shown in lecture before coming to class. New programs such as

LearnSmart from McGraw Hill, for example, provide ways for the students to encounter the materials they will see in lecture while also providing feedback to them (cf. Griff &

Matter 2013). This software has been piloted here at OSU during the Summer 2013 term for physics, and future implementations of something like this before students come to lecture may help those who would otherwise struggle to absorb the material in lecture and then provide thoughtful feedback during clicker questions.

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Students need to be encouraged to want to use the tools provided to them,

especially the video solutions. Administration guidelines cannot allow for points that

would go towards the students’ grades in one section of physics and not others, so the

most direct incentive to make students want to use the video materials is unlikely to

happen. Besides, if students only have to play the video to earn points it is relatively easy

to cheat in such a system (just run a video in the background and pay attention to

something else), so this incentive method has a weak chance of providing the desired

effect on student learning gains. Should future classes use the video solutions to an appreciable amount, then it will be possible to measure their effectiveness in increasing normalized learning gains. Perhaps it is small, and programs like Khan Academy will have crippling limits to how much they can help; on the other hand, perhaps they are helpful to students with naïve physics concepts. Only future research can say either way.

Turning to the classroom, changes to the efficiency of using the clicker system should increase classroom participation. More importantly, changes to the web-based clicker system are needed to provide a way of making sure that students stay on task.

With the infinity of distractions that the Internet can provide, future classes that use web- enabled clicker devices need to find a way of reducing distractions. Even the best instruction methods will not affect a student who is occupied watching entertaining videos rather than focusing on the content of the lecture. At the very least, the ease of using the clicker system should be improved so students are less likely to pull out their web-enabled device, become frustrated in trying to access the polling website, and move on to the distractions that are now a screen-touch away.

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Lastly, the testing of students with the FCI could be altered in such a way to make the process more efficient. Shortened versions of the test were performed for both the first and second posttests, and the shortened tests from the first posttest correlated very strongly with the full version of the FCI. Perhaps then using the shorter version will make the testing process easier on all parties without sacrificing analytical research on test data.

A streamlined version of FCI has the additional possibility of being used multiple times throughout the semester and the evolution of the students’ conceptions can be more finely measured over time.

These future lines of research and changes to how the tools should be used will be a considerable exploration program in its own. Still, other lines of research about physics class innovations will also intersect with this program. For example, the ways laboratories are conducted could benefit from innovative techniques or software programs that avoid the distractors that may exist when using physical equipment. In which case it is necessary to think broadly on how changes may affect the way students are taught.

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APPENDIX A: FCI QUESTION CATEGORIES

The following list of misconceptions that are captured by the FCI test from Hestenes et al. (1992), which was their Table II, is provided here. However, numerous mistakes were made. For example, numbers were given as possible mistaken answers for the multiple choice even though all choices were letters (A, B, ect.). Moreover, in numerous cases an answer that would have captured a misconception was the correct answer to the FCI question. Some of these mistakes may have been simple typing mistakes, and they may have placed correct answers in the list to compare with misconceptions but then forgot to edit out the correct responses as misconceptions. As such, this is a reconstruction that will differ from what is found in Hestenes et al. (1992) but ought to be closer to the desired result.

0. Kinematics Kl. position-velocity undiscriminated 20B,C K2. velocity-acceleration undiscriminated 20A; 21B,C K3. nonvectorial velocity composition 7C 1. Impetus I1. impetus supplied by "hit" 9B,C; 22C,E; 29D; 30B,D,E I2. loss/recovery of original impetus 4D; 6C,E; 26A,D I3. impetus dissipation 5A,C; 8C; 16C,D; 23E; 27E I4. gradual/delayed impetus build-up 6D; 8D; 24D; 29E I5. circular impetus 4A,D

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2. Active Force AFl. only active agents exert forces 11B; 15B; l8D; 22A AF2. motion implies active force 29A AF3. no motion implies no force 12E AF4. velocity proportional to applied force 25A; 28A AF6. force causes acceleration to terminal velocity 17A; 25D AF7. active force wears out 25E 3. Action/Reaction Pairs AR1. greater mass implies greater force 2D; 13B; 14B AR2. most active agent produces greatest force 13C; 14C 4. Concatenation of Influences CI1.largest force determines motion l8A,E; 19A CI2. force compromise determines motion 4C, 10D; 19C,D; 23C; 24C CI3. last force to act determines motion 6A; 24B; 26C 5. Other Influences on Motion CF. Centrifugal force 4C,D; l0C,D,E Ob. Obstacles exert no force 2C; 9A,B; 12A; 13E; 14E Resistance R1. mass makes things stop 29A; 23A R2. motion when force overcomes resistance 28B,D Gravity G1. air pressure-assisted gravity 9A; 12C; 17E; 18E G2. gravity intrinsic to mass 5E; 17D G3. heavier objects fall faster 1A; 3B,D G5. gravity acts after impetus wears down 16D; 23E

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APPENDIX B: THE CASE OF CONSTANT ACCELERATION, GRAVITY, AND

ABU’L-BARAKAT

In Part 2.A, various historical examples of ancient and medieval descriptions and explanations of dynamics were provided, especially as they related to the fictitious force pointing in the direction of motion and related to the concept of impetus. Throughout the pre-modern scientific literature, there is a continuity of use of the idea of the need for there to be a force for there to be motion, and increased force is required to have increased speed. However, there exists one alleged challenge to impetus-like ideas that comes closer to classical mechanics. Purportedly, the 12th century Jewish Arab philosopher Abu’l-Barakāt Hibat Allah ibn Malkā al-Baghdādī claimed that constant

force leads to constant acceleration (Franco 2003; Pines, 1970). This is marked as the

first time that force was made proportional to acceleration rather than velocity.

However, the text this is based on, the Kitāb al-Mu’tabar (“The Book of What

Has Been Established by Personal Reflection”), is somewhat ambiguous. On a careful reading and when compared to pre-modern statements and modern student responses, the text does not seem to support the contention that the Arab philosopher was ahead of his time. According to Abu’l-Barakat, an arrow that is fired upward is given a “violent tendency” or violent mayl, much like Aristotle’s violent force and using the same vocabulary as Avicenna for his impetus theory. Avicenna in turn uses the term mayl the same way Philoponus described the motive inclination of an object that underwent

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violent motion. Now, Abu’l-Barakat believed that as the arrow rises up its violent mayl

weakens, and eventually it is overcome by the natural force of gravity, a natural mayl; at

this point, the notion is similar to that of Hipparchus discussed before in Part 2.A. When

the object falls, Abu’l-Barakat claims, it undergoes an addition of force as “one mayl

after another” is applied. In other words, he seems to suggest that the force acting on the

object is becoming stronger (more force/impetus/mayl is being applied) as it falls, thus

accounting for its acceleration.

In antiquity there was already the awareness that a falling body accelerates, and

explanations for this were varied (for sources, see Cohen & Drabkin 1948). Hipparchus

of Rhodes (2nd century BCE), for example, claimed that because there was still the remainder of the force that threw an object up or held it up in place before falling, this meant that the object would more slowly fall and then speed up as this upward force decayed into nothing; then the object was at maximum speed. This was not position embraced by others; it directly contradicted by Alexander of Aphrodisias (3rd century),

and Simplicius of Cilicia (6th century) said that it was universally agreed that falling

objects underwent acceleration. Strato of Lampsacus (4th/3rd century BCE) believed that

the acceleration was more continuous than Hipparchus did. He pointed out how dropping

an object close to the ground made an imperceptible thump, while holding an object up

and dropping it at a greater height made a loud thud. This was proof that the object hit the

ground at a greater speed and thus underwent acceleration. To rationalize such

acceleration, there were several explanations. For Plutarch (early 2nd century), a falling

object would produce a void behind its path, and the inrushing air would push the object

faster and faster. Aristotle, on the other hand, believed that as a heavy object came closer

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to the Earth it would become heavier. This was believed to be the case because as an

object fell it would have a smaller column of air beneath it pushing it up; in a sense, in

was argued that as the object fell there was less of a buoyant, resistant force. To Aristotle, this means that there was less resistance to downward motion, so with the same natural inclination towards the ground it would be as if the net force on an object were increasing. Simplicius, however, seems to have found this an unconvincing explanation as he suggested that the apparent weight of an object did not differ whether it was on the ground up in a tower or over a precipice. Nonetheless, he agreed with previous natural philosophers that the downward force must have increased as the object fell.

This idea of an increasing downward force as an object falls is not that dissimilar

to what some interviewed students said: after a ball starts to fall, the gravitational force is

constant but a force in the direction of motion points downward and increases (cf. Table 3,

Question 4, Answer B in Section 3.B). This is also what the Latin scholars Jean Buridan

and Albert of Saxony (14th century) believed, and it is how they interpreted Abu’l-

Barakat: the impetus of an object increased as it fell, and impetus was in direct relation to

force or inclinatio (Gutman 2003). With this background, it does not appear to be prudent

to believe that Abu’l-Barakat was saying the downward force was constant and thus

providing the necessary acceleration. Rather, he seems to suggest that further sources of

downward inclination or mayl were being added to the object, such as how Plutarch said

the air was pushing more and more as it fell down with increasing speed.

Moreover, the constant force reading of the Arab philosopher would lead to a

stark inconsistency: if constant force equated to constant acceleration, then this leads to

the obvious expectation of no force leading to constant speed; this is inconsistent though

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with Abu’l-Barakat’s statement that a violent mayl is imparted to a launched arrow that weakens and thus slows down—a direction relationship exists between the mayl and the speed. Additionally, for Abu’l-Barakat an object has only one natural mayl (gravitational) and it apparently remains constant as the force due to gravity does, so the successive adding of mayl indicates that the total mayl, and thus force, is increasing and not remaining constant. So it appears that Pines’s (1970) statement that Abu’l-Barakat

“seems to anticipate in a vague way the fundamental law of classical mechanics, according to which a continually applied force produces acceleration” is not accurate, at least no more than Plutarch or Strato. With this potential exception looking unlikely, this leaves the Arab philosophers and scholars agreeing with their ancient counterparts that an object undergoing motion has some force acting upon it to keep it moving, and falling objects have an increased net force acting on them as they fall.

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