The Role of Formulas in Students' Solution Strategies in Regard To

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The Role of Formulas in Students’ Solution Strategies in regard to Combinatorics

by Cheyenne Gordon

A THESIS

submitted to

Oregon State University

Honors College

in partial fulfillment of the requirements for the degree of

Honors Baccalaureate of Science in Mathematics (Honors Scholar)

Honors Baccalaureate of Science in Education (Honors Scholar)

Presented June 5, 2019 Commencement June 2020

AN ABSTRACT OF THE THESIS OF

Cheyenne Gordon for the degree of Honors Baccalaureate of Science in Mathematics and Honors Baccalaureate of Science in Education presented on June 5, 2019. Title: The Role of Formulas in Students’ Solution Strategies in regard to Combinatorics.

Abstract approved: ______Elise Lockwood

There are many reasons toward why students withhold such negative mindsets toward the subject of mathematics. This thesis builds upon previously conducted research concerning affect and confidence in mathematical problem solving, confidence that students exhibit about themselves, and combinatorial thinking and activity.

Key Words: Mathematics education, confidence, formulas, discrete mathematics

Corresponding e-mail address: [email protected]

©Copyright by Cheyenne Gordon June 5, 2019

The Role of Formulas in Students’ Solution Strategies in regard to Combinatorics

by Cheyenne Gordon

A THESIS

submitted to

Oregon State University

Honors College

in partial fulfillment of the requirements for the degree of

Honors Baccalaureate of Science in Mathematics (Honors Scholar)

Honors Baccalaureate of Science in Education (Honors Scholar)

Presented June 5, 2019 Commencement June 2020

Honors Baccalaureate of Science in Mathematics and Honors Baccalaureate of Science in Education project of Cheyenne Gordon presented on June 5, 2019.

APPROVED:

______Elise Lockwood, Mentor, representing Mathematics Education

______Thomas P. Dick, Committee Member, representing Mathematics Education

______Elizabeth Gire, Committee Member, representing Physics Education

______Toni Doolen, Dean, Oregon State University Honors College

I understand that my project will become part of the permanent collection of Oregon State University, Honors College. My signature below authorizes release of my project to any reader upon request.

______Cheyenne Gordon, Author

Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant No. 1419973. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

Introduction

Throughout the years, researchers have investigated factors that may help to promote or inhibit student performance on combinatorics (or “counting”) problems. For example, Annin and

Lai (2010) have examined common errors that students encounter while attempting to solve counting problems. Overcounting, for instance, is a subtle, yet impactful mistake to make within combinatorics. For instance, if a student were to count the number of flags with at least 6 blue stripes, overcounting this number would reflect an overrepresentation of the number of such flags that exist, therefore not being an accurate representation of the problem context that is being depicted. As a result, Annin and Lai examined the ways to help combat such errors in order to help improve student performance within combinatorics such as replacing the minimum condition with several cases containing an exact condition. Annin and Lai also recommended that practice is key to student success, because as students become more familiar with the kinds of problems presented to them, they gain a better conceptual understanding of the mathematics involved and are more aware of common errors that could occur.

In addition, Lockwood (2014) has conducted research concerning combinatorics and has explored the nature of counting problems – in particular, counting problems are straightforward to state, but they can have solutions with underlying complexity. In both high school and college students are introduced to counting formulas and problem types, which can be beneficial for students to know, but they can also be a downfall as students can start to rely on memorization.

In other words, it was concluded in one of Lockwood’s investigations that, “Formulas and techniques should serve as tools for students as they think critically about counting problems; they should not become students’ only mechanisms for counting” (Lockwood, 2014, p. 300)

In this thesis, I explore students’ reasoning about counting problems, as part of research that was conducted with the assistance of Dr. Lockwood and interviews that she had previously done with students. Based on prior work (e.g., Lockwood, 2014; McLeod, 1988) and my own personal experience as a student in K-12, I decided to focus upon three themes that I believe describe what I observed while watching the interviews: a focus on formulas, a resistance to showing work, and self-confidence. By exploring these three themes, I have an overall goal to better depict the potential downfalls that students may experience in not just the realm of combinatorics, but mathematics in general. Mathematics, as a subject, is avoided by many, and rarely talked about in a positive manner by students. Through an exploration of these three themes, a better idea of how to help students combat the negativities they encounter while problem solving will emerge. These insights can be utilized to help make mathematics a subject that is viewed as a useful set of skills that students can utilize throughout their lives, no matter the path they choose to follow.

Literature Review

Upon beginning this thesis writing process and deciding upon which themes would best encompass what I observed from the interviews, there were several studies and written research reports that I looked at to gain a better understanding of the research previously conducted concerning this topic of math education and student problem solving in combinatorics in general.

I organize this literature review in two sections, which relate to the themes I will share in the results. First, I discuss literature related to students’ confidence and active issues related to problem solving. Then, I will present literature related to students’ combinatorial thinking and activity.

Research on Affect

One issue in mathematical problem solving is the affect and confidence can have an impact on students’ work. In his paper “Affective Issues in Mathematical Problem Solving:

Some Theoretical Considerations,” McLeod (1988) examined the emotional responses that

students often exhibit when they encounter mathematics problems, and specifically mathematics

problems with answers that cannot be formulated immediately.

When a student encounters a problem that does not give all of the necessary information

to simply plug numbers into a formula, they must engage in problem solving in order to get the

needed information to either plug into a given formula or to reason and justify a deduced

solution. However, many K-12 classrooms consist of students working on homework

assignments and worksheets that include repetitive tasks in which students are basically given

the answer, and they only need to plug the numbers into a formula. That said, when problem

solving is needed in order to solve a problem, students often “get stuck” or do not know how to

proceed when beginning a problem, and they can relate this to the idea that they are not good at

math or are simply “not a math person.” As McLeod states,

Mathematics students often report feelings of frustration or satisfaction when they work on nonroutine problems. These affective responses are an important factor in problem solving and deserve increased attention in research...In mathematics, where students spend so much of their time doing routine exercises, students’ actions are very highly organized. Thus the blocks that inevitably interrupt problem-solving activities may lead to intense emotions (McLeod, 1988, p. 134).

I agree with this point. The feelings of frustration that students often feel are substantial,

and they often create blockages that prevent students from being able to reason through a

problem. By “blockages,” in this context, I mean the occurrence of a mental block that is often

the result of an intense emotion that prevents students from furthering the problem-solving

process as they are unable to think clearly and reason through the problem at hand. Formulas can 11 give students a false sense of confidence in mathematics because it fosters the idea that most, if not all, problems can be solved with an equation. This promotes the idea that mathematics is a black and white subject in the sense of there always be a correct or incorrect answer. As Annin and Lai (2010) observed in their paper “Errors in Counting Problems,” “Many times, no rigid procedures or formulas can be used to solve the problems directly, and students simply do not know where or how to approach them” (Annin and Lai, 2010, p.403). To combat this focus on formulas, Annin and Lai (2010) suggested to present students with a variety of problems in order to develop a better conceptual understanding of the problem type and with common pitfalls that can occur in order to better avoid them. One such common pitfall is overcounting, and can easily cause students to report an incorrect answer. Practice, in this case, is the best way to prepare students for success and to help lead them toward becoming not only more proficient at counting, but at mathematics as well.

Some students, upon encountering this frustration and therefore the blockages that follow suit, will attempt to persist to solve a given problem. This persistence can easily intensify the emotion. As McLeod states, “When students are engaged in trying to solve a nonroutine mathematical problem, they often express a lot of emotion. If they work on the problem over an extended period, the emotional responses may become quite intense, especially for inexperienced problem solvers. Students who fail to reach a solution frequently report feelings of frustration or even panic” (McLeod, 1988, p.134). Reaching an emotional state of panic is a point that necessitates a break, but how often are students told that “hitting a wall” or experiencing a blockage is normal? How often are students told that taking a break is okay, and often an efficient practice to partake in? I would argue that students are not encouraged to take a break or told that it is acceptable to experience some kind of barrier or impasse on a problem. Students are 12 often told not to procrastinate their work, but struggling to solve a problem is often looked down upon, which helps emphasize the negativity associated with the subject. That said, I believe it is important to help students to celebrate when they succeed in problem solving, or encounter the blissful “Aha!” moment. McLeod (1988) mentions this “Aha!” moment by correlating it to a rather intense, positive emotion.

Positivity is something that mathematics lacks from a student-perspective, and that is something that needs to change. Instead of letting students quit when they get stuck, or letting them sit in silence as they perpetuate the feeling of fear of asking for help, it is important for math educators to help establish that getting stuck or asking for help is normal. Not knowing something is normal. Feeling frustrated is normal. These seemingly negative aspects of problem- solving help to create proficiency in mathematics. As McLeod states, “Once students understand that problem solving involves interruptions and blockages, they may be able to view their frustrations as a normal part of problem solving, not as a signal to quit.” (McLeod, 1988, p.138).

Another point of consideration related to students’ affect in problem solving is the confidence that students exhibit about themselves as mathematicians. Kloosterman (1989) conducted on the relationship between self-confidence and motivation in his article entitled

“Self-Confidence and Motivation in Mathematics.” In this work, Kloosterman claimed that students who are confident in their abilities within mathematics are more likely to take math courses when they become optional and are more comfortable while encountering mathematical situations. On the other hand, students who do not exhibit self-confidence will blame failures on a lack of ability, which results in poor motivation because they begin to believe that their abilities (or lack thereof) will never change significantly to the point in which they can overcome 13

the cause of their failures. A lack of self-confidence can become established in students at early points in academic careers such as in elementary school. As Kloosterman states

In many mathematics classes, techniques for solving mathematical problems are presented in a step-by-step sequence. Students have learned to simply follow the steps to produce a right answer. The belief that all mathematics problems can be done quickly and without error may be perpetuated in the elementary school by the consistent use of timed tests in mathematics classes” (Kloosterman, 1989, p. 346).

To me, this quote suggests that students are often taught in such a routine way that they believe

that this is how all math-related problems can be solved. As a result, a preference for formulas,

or simply a methodology to complete the problem as quickly as possible, begins to develop and

if a problem takes a student longer than what is expected, they begin to feel frustration or even

panic. Thus, blockages begin to form and prevent students from efficiently problem solving.

To build on Kloosterman’s (1989) comments on timed tests, I make a couple of

additional observations. Perhaps one of the biggest issues with timed-tests is that even if students

are trying their hardest to solve problems, if they make even the slightest error that affects their

end result, they are marked incorrect. When students are given back tests, they frequently focus

on what they did wrong instead of what they did right. Students will compare with each other to

see how their friends performed, and students who did not perform as well begin to believe that

they are incapable. Even if students were to acknowledge that the solution path they tried to go

down to reach a solution was incorrect, but that they do not have the time to try a different path,

they will be marked incorrect. Tests can be brutal on the confidence that students have

concerning their abilities, and standardized testing definitely does not help this. However, testing

is, and has been, a part of math curriculum for many years, and it may never change anytime

soon. What can change, though, is how teachers interact with their students concerning the

mistakes that they make. Words of encouragement, for instance, may not seem like much, but for 14

some students it can make a significant difference for how they view themselves as

mathematicians. Tests should not be the sole determinants of how capable and proficient a

student is in mathematics, yet many times they are. Students begin to focus on their grades, and

their grades become a reflection of how capable they are.

That all said, the previous research that has been conducted concerning the ideas of the

emotions that students often feel while problem solving, a focus on formulas, and self-

confidence not only raise issues concerning these ideas, but emphasize how important it is to

study them more closely. My recent experience as a K-12 student has made the points made in previous research articles extremely relevant and worthwhile. Thus, I decided to build on these ideas and research articles and put them toward my thesis project in attempt to convey ideas to help make mathematics education not only more worthwhile, but something that students may actually enjoy and appreciate. As a mathematics student, I feel as though I have a strong understanding of how students tend to react to today’s curriculums and ways of teaching.

Moreover, as a future math educator, I feel as though these ideas and themes are worthwhile to examine to not only help myself to better prepare to try and help students to become proficient mathematicians, but to shed light on these ideas and themes as well.

Research on Combinatorics

I now discuss research done on combinatorial thinking and activity. The study I conducted in this thesis is based on students who are solving combinatorics problems. Thus, I want to review some work on students’ combinatorial problem solving in particular. In the realm of combinatorics, sometimes students make shortcuts and quick solutions, and this preference can often easily lead to mistakes and incorrect answers. One potential way to help students be more successful in counting is for them to realize that problem solving does not consist of simply 15

plugging numbers into formulas. As Lockwood discusses in, “Using sets of outcomes to

reconcile differing answers in counting problems” (2014), counting problems can be

straightforward to state but can be surprisingly complex to solve. As a result, counting problems

provide accessible but challenging contexts in which students can engage in meaningful problem

solving. Upon giving students a counting problem, as was observed in the interviews conducted

and examined for this thesis, students may attempt to jump straight to formulas. In this instance, students are not understanding the context of the problem and what is trying to be counted, but rather attempting to configure the numbers in a way that can be plugged into related formulas.

A simple, and valid, methodology for solving counting problems is to create sets of

outcomes. Lockwood (2014) suggests that while creating such might seem to be tedious for some

students, and students may be reluctant to engage in listing, the task is crucial for helping

students to understand combinatorics. Creating sets of outcomes allows students to visually see

what is being counted and how the examined objects correlate to each other. For instance, in the

interviews, students were asked to determine the number of ways three red hats can be given to

five people. Listing out the outcomes can help students to understand the context of the problem

by demonstrating through the listing process that only three people can be given a hat at one

time, and what is being asked is how many different ways can these hats be distributed. This

problem is straightforward, but immensely difficult to complete mentally. Yet, many students

have the preference to try and solve the problem mentally instead of writing out the sets of

outcomes. As Lockwood states,

Formulas, problem types, keywords, and tricky techniques can certainly be valuable tools for successful counters. However, they can easily become substitutes for critical thinking about counting problems and for deep consideration of the sets of outcomes. Formulas and techniques should serve as tools for students as they think critically about counting problems; they should not become students’ only mechanisms for counting.” (Lockwood, 2014, p. 300). 16

Lockwood (2014) worked with undergraduate students, and they found solving

combinatorial tasks to be challenging. This is an important point to consider because it

emphasizes the point that students at all levels struggle with solving counting problems correctly.

Moreover, this struggle is normal, and even okay. Students should never be expected to know

everything, but instead be encouraged to learn and find out how to correctly solve problems.

Following this, students should be encouraged to show their work on paper; as Lockwood states

in “A Set-Oriented Perspective on Solving Counting Problems”, “...for the mental act of problem

solving, a solution to a particular problem represents a way of understanding, but a general

problem solving strategy, applicable across a variety of contexts and problems, can represent a

way of thinking” (Lockwood, 2014, p.32). No two students will ever understand or think about a

problem in the same exact way, and this is another point that should be encouraged. Math

educators should help guide students toward a path of understanding, but never try to dictate

student thinking in the way of telling them that there is one, and only one, way to understand a

given mathematics topic (often times the way that the teacher understands the topic).

Given students’ difficulties in solving counting problems, in this study I want to build on previous work by trying to identify how aspects of students’ affect and confidence emerged in their problem-solving process in a combinatorial context. By identifying these aspects, I hope to develop ideas that can help students to not only see mathematics in a new and more positive way, but to help them believe that they can be successful mathematics students no matter their previous mistakes and negative experiences. These ideas, however, are connected to the important role that teachers hold, and this is a focus that I hope to establish upon building on previous work, identifying aspects of students’ affect and confidence, and developing ideas to help students in mathematics. 17

Methodology

In order to conduct research for this thesis, I watched seventeen videos of interviews that were each approximately one hour in length. These interviews were previously recorded by Dr.

Elise Lockwood as part of an NSF-funded project. She was also the interviewer for each of the videos. To recruit students for the study, Dr. Lockwood sent out an email to students that were taking Calculus and asked for participation in the interviews. The first 20-30 students that responded were selected for the interviews, and they were asked to partake in the study that Dr.

Lockwood was conducting under a bigger grant to investigate students’ combinatorial reasoning.

Each student participated in an individual, one-hour long task-based interview. The only eligibility requirement was that they were enrolled in a Calculus 1 course at the time. To begin each interview, Dr. Lockwood asked students about their major and the mathematics courses they had taken prior to the interview. This last detail was also included in this research report because it correlates to some of the comments that students made, as some students may have taken a math course in the past that introduced combinatorics in some way. Dr. Lockwood wanted to keep track of whether students had taken discrete mathematics previously. She wanted to know whether students were recalling topics they had seen before. The interviews were videotaped and audiotaped, and the students also used Livescribe pens to write down their work during the interviews. These pens allow for the real-time writing and audio to be captured in a pdf, so researchers can see what students wrote if it was difficult to capture in the video.

The tasks that students were asked to complete were all the same, but the number of tasks that students completed varied depending upon how far they got in the process. All interviews began with the task of figuring out how many ways are there to give a group of five people three red hats. This situation then progressed toward finding the general formula for giving hats to n 18 people, which was supported by giving the student the ultimate task of filling out a table that indicated the number of ways to give hats to 0-5 people. Then, students were tasked to find the number of ways to give multiple objects to people such as a blue and green hat to n people (at first, n was a concrete number such as how many ways to give a blue and green hat to 6 people).

Progression through these tasks varied between students as some students became stuck at the first generalization while other students were able to advance further along to other tasks. The progression that each student made was not considered important in this research; it was the work and thought processes depicted by each student that were closely examined. In other words, the approaches that each student took to solving the tasks presented to them was the main focus of this research. Dr. Lockwood’s motivation for writing these tasks were to have students engage in generalizing activity (this was part of a broader project investigating students’ generalization in a variety of mathematical domains, including combinatorics).

To analyze data for my thesis, I first watched all of the videos. While watching the videos, I took notes and pulled out quotes that I believed were relevant to my focus for this thesis. After watching all of the videos, I met with Dr. Lockwood to discuss my overall findings, and then I chose three broad themes to focus on: students tended to a focus on formulas, students demonstrated a resistance to showing work, and students talked about confidence. My collected notes, at this point, were compiled together and categorized based upon which theme I thought they best represented. From what I found, I decided that three of the videos would best represent the themes that I chose to examine. That said, I chose to examine Students 2, 7, and 10 more in depth. Student 7, at the time of the interview, was taking Integral Calculus (MTH 252) and

Physics I with Calculus; he had previously taken a Statistics course at OSU, took math courses at a community college 3 years prior, and stated his experience with high school math. Student 10, 19 whose major is in Energy Systems, was also taking Integral Calculus at the time of the interview, and had previously taken AP Calculus, AP Statistic in high school and MTH 251 while at OSU.

Student 2 is an Engineering major, and at the time of the interview she had previously taken

MTH 111, MTH 112, MTH 251, MTH 252, and MTH 254, but was currently in the process of retaking MTH 254.

Before categorizing each video in regard to these three themes, I wrote descriptions of each video that summarized what occurred in each interview such as the work that the student showed, their conjectures and thought processes, and any interpretations that I may have made concerning certain conjectures, words spoken, or body language. In these descriptions, I also included relevant quotes that stood out to me from each interview that I thought would strongly emphasize the themes I ultimately selected and their relevance to this research. (See below for a screenshot of this work).

Figure 1: A screenshot of a part of a description I wrote for the interview with Student 10.

Then, to select these themes, I considered what I believed to strongly correspond to the factors that help to contribute to students often viewing mathematics negatively. I looked for particular utterances that came up repeatedly for the students. Then, I sorted the videos under the three themes by utilizing the descriptions I had previously written, and decided, after sorting each video, which videos I would select based upon the strength of how well I thought the given videos correlated to the respective theme. For instance, in regard to confidence, Student 2 was an interview that I thought strongly correlated to confidence because she explicitly used the word

“dumb” in several of her statements throughout the interview, which suggested a lack of 21 confidence. Moreover, Student 7 mentioned that they “...used to be good at these problems I think like four years ago…” while Student 10 made a similar remark in regard to previously taking a course that mentioned combinations, “You know, it was killing me, because it was like I know where I was. I knew everything that was happening except for what was on the board”.

Each of these students had striking comments that I believe strongly emphasize and support the theme of confidence that I chose to focus upon throughout this thesis.

After sorting the videos and selecting those that I wished to highlight, I chose certain quotes that I found striking such as the ones mentioned previously. I primarily focused on the students’ dialogue rather than trying to describe exactly what was going on for other factors such as body language. Moreover, I was able to know what each student spoke without needing to play back each video because each interviewee wrote out their work using a pen that had a microphone imbedded in it (the Livescribe pens). Thus, the dialogue from each interview was compiled into documents that I utilized in order to maintain accuracy in regard to what each student spoke about. Moreover, the pens served as a sense of reliability as they pick up on every word spoken as long as it was clearly spoken with enough volume (most students did speak this way unless they were mumbling under their breath concerning their thought process). This also prompted students to speak more clearly more often as opposed to mumbling for a majority of the time as most students do when they are figuring out math problems in the environment of a classroom.

Upon including the selected videos in this report, I described each interview with as much detail as possible, and included the quotes that I selected, as well as included my interpretations on significant details. This writing yielded the results section. Alongside the transcripts that were compiled through the recordings of the pen and videotaping, I also had 22 access to images of the work that each student wrote down. I utilized these images in order to give readers a better understanding of what each student was thinking about while communicating to the interviewer. In other words, the photos (depicted as figures in this paper), gave a visual representation of what was going on in the students’ thought processes.

Results – Descriptions of Three Student Interviews

In this section, I present interview descriptions for three students that comprise my results. I focus on these three students because they provided compelling instances of student work. It is these three videos that I analyzed to uncover my themes. In this section I describe each interview in detail. Then, in the following section I synthesize the results and articulate my three themes.

Student 2 Interview Description

To begin the interview, the student was asked the question, “How many ways are there to give 3 identical red hats to 5 people.” At the start of the interview with Student 2, the student first made a conjecture that there would be 15 total outcomes for giving 3 red hats to 5 people, because 3*5 = 15. To test this conjecture, the student developed a strategy for herself: visualizing the given problem by drawing shapes to represent people and hats. Specifically, she drew circles to represent people's’ heads and triangles to represent the hats. To depict the action of giving 5 people 3 identical red hats, she connected each hat to a person, with a line signifying the action of giving a person a hat. 23

Figure 2 – Student 2’s visualization of the problem.

She also demonstrated that she sought to understand what exactly was going on in the

problem by asking questions such as “Everyone has a different hat each time?”. I interpret that

by asking such questions she wanted to make sure she understood what the question was asking.

Eventually, Student 2 decided that drawing each hat and person combination would be too

tedious, considering that the lines she was drawing would end up overlapping and not be very

neat, so she resorted to making a chart of all possible combinations for each number of hats

being examined.

Figure 3 – Student 2’s chart of all possible combinations.

In the statement below, the student indicated that she wanted to write it visually but thought she should look for an equation.

Student 2: I just want to draw a chart, like I need to see it visually that’s how I see it best like I’m sure there’s an equation that works for it and I could plug it in but like it only makes as much sense as like looking at it. 24

However, it was at this time that she realized that her initial intuition of 3*5 = 15 total outcomes for 3 hats was incorrect because in making this chart, she had listed less than 15 outcomes. After she had physically written all the outcomes, she was struggling to understand why her initial answer was incorrect.

Throughout this process of writing out combinations, the student became visibly flustered with how tedious and lengthy the method was, and she expressed her desire to have an equation by stating that she hasn’t found an equation that solves this problem. From this statement, I interpret that the student was conscious of an equation existing that could solve the problem.

Eventually, the interviewer instead asked the student how many ways 2 identical red hats can be given to 5 people (the questions throughout this interview would progressively change the number of hats given to 5 people). At this point that the student resumed her previous numerical method of listing the outcomes, but she began to organize her combinations by starting with one number and then exhausting all possible combinations that included that number. Upon doing this, the student did not repeat or miss combinations as she previously had. However, this student reverted back to her previous conjecture of 3*5 and made a similar statement for 2 hats: the total number of combinations would be 5*2=10 ways.

To explain why she repeated a conjecture that she already proved to be incorrect, the student blamed her sickness (she was indeed sick at the time of this interview) for being slow, and that she just really wanted the multiplication to work.

Student 2: “That’s not the ten that I estimated because I don’t know why. Lemme figure it out, um, I’m claiming sickness, which is why I’m slow.” 25

At this time, the interviewer asked Student 2 about giving one hat to five people, and the student

immediately said 5 ways. However, just as quickly she said that 5 ways may not be the correct

answer and that she was unsure about the total number of ways.

After this, the interviewer asked about giving hats to 6 people, and the first comment that

the student made was, “I’m sure there’s an equation that works for it and I can plug it in…”.

Instead of solely focusing on this thought, the student resumed the numerical listing method that

she had utilized previously, but she did not use the same organization technique in which she

started with one number and exhausted all possible combinations. Instead, she used a less

systematic method, which presented her with the higher probability of either repeating or missing

combinations, which she began to realize as she was now dealing with more numbers.

Upon having 6 people and having up to 6 hats to distribute, she once again reverted to her initial conjecture of multiplying the number of people with the number of hats being distributed.

For example, for 6 hats, she said:

Student 2: “...But still only up to 6 hats, 6*6=36, that doesn’t do anything for me. Umm...I don’t know! I want to know the equation now!

At this point in the interview, the student was asked if she could determine the general relationship between people who got hats and those who do not get hats (otherwise known as a complementary relationship). The student glanced at the work she had previously done and said that she could not provide any answer to the question. Her response below indicated that she thought there would be “a lot of ways” but did not establish a relationship between the number of hats and the number of no hats.

Student 2: “My chart? So you can’t have more than one hat, but you could not have a hat? That’s a lot of ways I think? Like a lot a lot of ways…”

Interestingly, while talking through her thought process, the student began to use the word

“dumb” more frequently as she talked about herself as a math student. For example, at one point she said, “No, that was dumb, okay so what have I come up with?” and later said, “...Which is probably dumb because I could probably find one [an equation] if I tried hard enough.” I interpret these comments to be a demonstration of a lack of confidence in herself. I interpret that she believed that her thoughts about the problem were not what the interviewer expects.

Moreover, it seemed that the student had reached a point in which she was stuck and more problem solving was necessary, but she did not know how to proceed.

She also brought up, after some more thought about the question concerning the complement relationship, multivariable calculus, and from that point onwards, she visibly became overwhelmed and did not seem able to do any more problem solving. During this time, she made some additional comments giving insight into her process, such as “I wanna know the equation now”, “I feel like there’s so many options that I can’t even get with the stupid chart!”, and “I want to think about it but I just...I don’t know how my brain is supposed to tackle it right now.” It was at this point that the interview ended. I will provide more analysis and synthesis in the Synthesis of Results section.

Student 7 Interview Description

To begin this interview, the student was asked the question, “How many ways are there to give 3 red hats to 5 people?” At first, Student 7 made the conjecture of there being 15 ways, since 5*3 = 15. He said that this was the first thing that came to his mind when pondering this question. Upon stating this conjecture, the interviewer asked the student why they thought 5*3 =

15 represents the total number of combinations for the situation, and the student began to delve into their own thought process in an attempt to work through their justification for their 27 conjecture. I interpret this ponderance to be the student trying to understand what the given problem was asking of them.

Interviewer: So why - why did you say 3 times 5? Student 7: It’s just the first thing that comes to my head. Interviewer: Uh-huh. Student 7: I imagine - I feel like there’s more the different - the total combinations of each hat with each person...or something. That’s what the 3 times 5 would be. Uh, because it would be 3 for the first person, 3 for the second - that’s too much.

At this point, the interviewer clarified the question by stating that each person could get at most one hat. The interviewer clarified that there were only 3 hats, and the question was asking to distribute them to 5 people. In response to this, the student explicitly discussed their thinking about the existence of a formula. The following excerpt suggests that the student knew there may be a formula, but since he couldn’t remember it, he would “start drawing stuff out.”

Student 7: So since I’m kind of blanking on a formula, I’m going to start drawing stuff out.

The student then began to draw out each combination by making dots in a chart, with the dots representing that a hat was given to the indicated person. The student proceeded with this method for a bit, and then he counted up the total number of combinations he had at that point, which ended up being 12. Since the number was near 15, the number he got from his initial conjecture of 3*5, the student believed that his conjecture must be correct and concluded that there were indeed 15 ways to distribute 3 red hats to 5 people. (See Figure 4)

Figure 4 - The student’s initial problem-solving process *Note: the student first concluded 12 combinations, but later on decided it was incorrect and crossed out 12 and instead wrote 10

The interviewer then asked the student to consider the situation in which 2 red hats were distributed to 5 people, specifically asking how many combinations exist for that situation. One of the first ideas that the student suggested was that there would be fewer combinations than if there were 3 red hats.

Student 7: So it would definitely be fewer combinations, wouldn’t it? I’m not sure. Because it’s just 1 - there’s 1 extra moving part sort of, and when you have 3 different hats moving between the 5 people...visually it seems to make much sense that with 2 there would be fewer combinations, fewer possible combinations for each of those to take.

To visually demonstrate what he was initially thinking, the student constructed the same chart and utilized the same system as before of drawing dots to represent which people were given hats. This time, however, the student was more organized with his method. Specifically, the student started with two dots, and then for successive combinations the student only moved one dot to the right until all possible combinations with that first dot were exhausted. For example, if the student started with giving Person 1 and Person 2 hats, they would then move the dot representing that Person 2 got a hat to Person 3, representing another combination, all the way to

Person 5 so that all possible combinations involving Person 1 was exhausted (see Figure 5). 29

Figure 5 - The student’s process for 2 hats given to 5 people. The table on the right was constructed as the student continued with different numbers of hats (the student kept returning to this table)

However, an interesting point is that this student only utilized this method for Person 1; the

student did not do the same thing for other people. Instead, the student just filled in gaps where

he didn’t see positions in the pattern. I find this interesting because it seems as if even though the

student had a valid methodology for solving the problem, they are not wanting to show their

work for all possible combinations for each person. In other words, I infer that the student was

desiring a shortcut and was believing that since they have established a pattern through Person 1,

that they can just quickly fill in the gaps for the remaining people and get the total number of

combinations for giving 3 hats to 5 people. More specifically, quickness seemed to be their focus rather than correctness. At this point, the interviewer asked the student to explain their thought process.

Student 7: I’m not really sure how I got the last one...I need to do more proofreading. Interviewer: But other than that, I mean, I think that was good. Is there a way just by like examining the table or thinking about what the table represents where you can say for sure that you have all of them [the combinations], not too many, not too little? Student 7: I really can’t think of something right now...just totally drawing a blank. I used to be good at these problems I think like four years ago. I went and like memorized a lot of the formulas for basic things like this, and I just completely lost them.

As an answer to the question involving 2 hats, Student 7 decided that there were 11 total combinations in which the 2 hats can be distributed. The student concluded 11 combinations by utilizing the method of exhaustion while also quickly filling in any gaps he saw, but he had to be 30 prompted to conclude that there were actually 10 total combinations because he ended up repeating a combination, leaving him with an extra to be counted. That said, the initial conclusion of 11 emphasizes the problematic nature of his preference for shortcuts and not utilizing the same organization method consistently. By quickly filling in the gaps, he accidentally repeated a combination that he already had written down, but he did not notice his error until it was examined by the interviewer.

The interviewer then asked the student to consider 4 hats being distributed to 5 people. At first, the student decided that he was originally wrong with how he thought of less hats meaning fewer combinations.

Student 7: ...I’m thinking that I was wrong originally, that if I had more hats there are fewer combinations and fewer hats, more combinations...because there’s less - there are fewer empty spots to fill - to move dots around.

The dots that the student was referencing is his method of utilizing dots to represent that a hat was distributed to a given individual. Based on the comments that the student made in regard to him being wrong initially, it seems that the student was relying on the visual depiction of the scenario. I infer this because he referred to the dots and how they move around on the charts he had been constructing. Following up with this statement, the student began to question his previous answers for 2 and 3 hats being distributed, and he pondered the combinations he listed in the charts he had constructed for those contexts. The student began to construct new charts for the scenarios, but, ultimately, he soon gave up and just figured that he had the correct answers

(see Figures 6 and 7).

Figure 6 - The student re-doing his work for 2 and 3 red hats given to 5 people

Figure 7 - The student further re-doing and checking his previous work

Student 7: I don’t have the 3 - no, I do have the 3. I don’t have, what - maybe I’m just imagining it. Maybe I do have all of them. There should be 2 and then 1 somewhere. 2 and the 1 that’s there. I’m missing - no. Never mind. So maybe it’s safe to say that I got them as far as I know.

Overall, through each of the scenarios of different amounts of hats being distributed to 5 people that were presented to the student, the student was regularly visualizing the combinations through constructing a chart and filling in spaces (or positions, as the student called them), with 32 dots. When he considered a given scenario in a general sense, he seemed to picture the spaces he needed to fill.

Eventually, the student was asked the question of how many ways there are, in total, to give red hats to 5 people. This question eliminated the restriction on hats by not focusing on one sole number of hats to be distributed (as was done previously with questions such as how many ways to distribute 3 hats to 5 people). At first, the student did not understand what was being asked of him, but he asked for clarification until he understood that the question was asking for a generalized answer to how many ways hats can be distributed to 5 people. However, the student admitted that he was not sure how to proceed, and the following dialogue is something that I interpret as a lack of confidence in the student’s ability to answer this question.

Student 7: Well, in this case for 5 hats for 5 people you end up with what, 22, including the 0… No, not really including the 0. Interviewer: 22 or 32? Student 7: 30 - oh, my God. I should not be doing this in the afternoon, I’m brain dead...I’m going to think a little more here, a little less stream of consciousness.

The student then examined the table they had previously constructed for 5 people.

Student 7: It would be - adding all the number of ways that I did find with all the different combinations of hats sort of explored the table...I just noticed that the number of ways - the total number of ways that you can distribute 4 hats is half of the total number of ways you can distribute 5 hats. Interviewer: So - so maybe just to kind of hone in, so think about why maybe adding 1 more person would double the number of ways to distribute hats. Student 7: Why would just adding 1 person double the hats? I guess because you have to take all the other combinations you had before and factor in - for using the column structure you’d be adding 1 more sort of dimension where you had to change all of the previous combinations so that they use this slot at one point. Does that make sense? So when adding another person a column would be added to this diagram, and you’d have to take in the example of when it says 3 hats - you have to take that first combination and add a dot at the end. And the second combination add a dot at the end and no, not just add a dot, move one dot, put it over here. So what is it? You’d have that, you’d have - I’m trying to figure out why that makes sense.

Throughout this student’s thought process, I noticed that Student 7 was asking himself questions toward why his explanation makes sense and how it corresponds to the question that was asked of him. I interpret this as the student problem solving by examining the work he had already done through the table, and using that prior work to explain their answer to the question of why adding one person doubled the hats.

Student 7: So what it is is it takes all of these old combinations and allows them all to be shifted over one place and become a new unique combination. So is that right? Because if you do move, for example, what was sort of the end before these 3 dots, move it over to the right and move it over again. You can take that beginning - that’s what I was doing right there, take that one and move that one over so that it’s over 1 column. Interviewer: Okay. And so what is that doing to exactly double everything?

At this point in the interview, the interviewer made a point to clarify whether the student had gotten 22 or 32 combinations for the total number of ways to distribute hats to 5 people. This was an important piece to clarify because the conclusion of 32 aids in the generalization when asked about how many ways are there to give hats to n people. In other words, when the student totaled up the number of ways to give hats to 4, 5, 6, etc. people, a pattern emerged, but that pattern would only emerge if the numbers are correct. The number 22 would not follow the pattern, which makes it important that the interviewer clarified at this time. That said, the notion of doubling was a prompt to get the student thinking about the pattern of doubling as 32 is a power of 2 (as are the other totals when calculated for the other concrete number of people such as 5 or 4 people).

Student 7: I’m not sure why it’s exactly doubling everything. I mean, it - just from the logic I did before, if you’re - if there’s 1 more sort of version of each previous combination that would be doubled. I’m not sure exactly how to describe - how like show and describe how each thing is technically moving over, but that’s what makes sense to me. 34

Interviewer: So are you saying that when you add a person you have all the old cases still and then you’re also creating new cases with a dot in that slot, in that new slot? Is that what you’re saying? Student 7: Yeah. And that opening up that new slow adds 1 more sort of version of each previous combination. Like you can shift it over - like what I was saying like shift it over 1 person or rearrange this one hat per combination and create 1 more combination. Because it gives you 1 more slot to play with.

After this, the student began to play with the idea he was formulating about the combinations by drawing out concrete cases and testing the idea with a set number of people. The case that he specifically chose was for 3 people. He drew a new table, and he started with 0 hats and increased up to 3 hats to demonstrate what would happen to the number of combinations.

However, the student eventually expresses his frustration due to “previously knowing how to do this sort of math.”

Student 7: I’m just - I’m frustrated, because I like - I knew this. I hate - that’s - that’s the worst thing. I’m okay not knowing something, I’m not okay with having known it and just losing it. Interviewer: Yeah. You’re saying you knew the formulas? Student 7: Yeah, like I had worked through all this. Like I’ve taken like a discrete math class at one point, and you sort of play with stuff like this. Yeah, just frustrating. I don’t remember it.

After finishing his work with drawing more tables and tallying up the total number of

ways to distribute hats to 3, 4, and 5 people, the student got 8, 16, and 32 combinations

respectively (see Figures 8, 9 and 10). The student was more successful at this point because he

was more aware of the possibility of duplicating combinations or not accounting for every

combination, so he adopted a more systematic method. Although his charts are not the most

organized, he acknowledged the method of exhaustion by starting with one person and

exhausting all possible combinations when he examined what he could quickly put down as

possible combinations. At this point the interviewer asked, “We want to give either a hat or no 35 hat to any people, can you conjecture how many ways we could do that?” The student examined their work from before and started to work out the pattern at play.

Figure 8 - The student concluded that there were 32 total ways to give 3 red hats to 5 people based off his previous work.

Figure 9 - The student’s work with giving hats to 4 people and his conclusion of 16 total combinations.

Figure 10 - The student’s work for 3 people and his conclusion of 8 total combinations

Student 7: 2, 2, 1 is to 2 squared is 4. 2 cubed is 8 and so on and so forth. What is my relationship there? Okay. That’s - now I see it finally. It’s 2 to the n, isn’t it? There are only 2 possible options for each person, to give them a hat or not to give them a hat. So 2 to the n (see Figure 11).

Figure 11 - The student’s associated work to their 2 to the n conclusion for the general relationship of giving hats to n people

This occurred shortly before the interview ended, and the interviewer asked further clarifying questions of the student to make sure that the interviewer understood what the student was trying to say about the general relationship between the number of people and red hats being distributed to the given number of people. I interpret the previous dialogue from the student as the relief that is often felt after problem solving and finally reaching the finish line in which the problem is solved after spending a good amount of time struggling to piece all of the information and scratch work together. The student had moments of struggle, such as when he indicated by his frustration of not remembering even though he has taken a discrete math course previously, and he ended up more successful in the end because he was able to progress from concrete cases to an overall generalization. Even though the student expressed acknowledgement of a formula and knowing it at one point, the student persevered and eventually came up with an answer to the 37 question even though they previously admitted they were stuck and did not know how to proceed. At this point the interview concluded.

Student 10 Interview Description

The interview began with the interviewer asking the student the following question: “We have 5 people, and I want to give 3 of them red hats. And so in how many ways can I do that?” At first, this student brought forth the idea of permutations almost instantly as they begin to think about the problem set before them and wrote down what they initially thought when they heard the problem (see Figure 12).

Student 10: Oh, permutations. Great. Okay. So 1, 2, 3, 4, 5, 6. 6 different patterns.

Figure 12 - Student 10’s initial idea concerning the problem

From this comment, I infer that this student had seen some kind of probability material before in a previous math class that they had taken; this would most likely be the AP Stats class. The interviewer asked the student to explain their thought process behind their conclusion of there being 6 different patterns for the given situation. The student drew boxes to represent the 5 people, and then marked tallies underneath each box to represent that the “person” in that column was given a hat. Each row represented a different combination of 3 hats given (see Figure 13 below).

Student 10: So I drew out 5 people just to make it easier to individualize it. So then you go - the first 3 people have hats. And then from there you switch out, and you have the first 2, the next, the first 2 in the front, and then this guy has now had a combination will all those.

Figure 13 - Student 10’s initial work after some more explanation

Most of the work that the student had done at this point was starting to draw out the problem in order to visualize it, and then make a mental plan for how he would continue his problem-

solving process to come up with an answer to the problem. At this point, the student decided to

check his work to make sure that he was solving the problem correctly, claiming that it had been

a long time since they had worked with permutations and problems such as this one. The student

proceeded to check their work and verbally told the interviewer that they were making sure that

they were not skipping something. The student also proceeded to fill out a more completed chart

than what he had initially. In general, he utilized a method of exhaustion in which he began with

one combination of people (i.e. persons 1, 2, and 3 getting a hat), and then exhausted all possible

combinations with Person 1 and continuing this method for all 5 people. (Figure 14).

Student 10: Let me just make sure I’m not skipping something. If I go here, here, yeah. Sorry. It’s been a long day (student laughs). Here and here. So that’s another one. Here, here, here, that’s unique as well. So it’s at 6, 7, 8 at this point. Hopefully, that’s all of them. So, 1, 2 - we have him and him. I’ve already done this one, this one is not unique as well. Here, here. It would appear that it’s unique, too. Okay, so 9. I believe 9 is the total.

Figure 14 - Student 10 filling out a chart of combinations for giving 3 hats to 5 people

I found it interesting that the student decided to let it be known that it had been a long day for them, and that he proceeded to laugh afterwards. I interpret this comment and reaction to be a sign of nervousness being exhibited by the student, as they were having to explain their thought process, and they were in the midst of checking their own work to confirm that their initial conclusion was correct. This interpretation was further supported when the student, once again, commented that it had been a long day shortly after speaking the lines indicated previously, saying, “It’s a long day.” I interpret the comment possibly to come from the tediousness that

results from the method of exhaustion; the student ended up missing some possible

combinations, potentially because they did not fully exhaust the total number of combinations for

one given person. For instance, for Person 1, he represented combinations 123, 124, and 125, but

it was not until later that he included 134 and 135, but he was still missing 145. It seems that

organization is a crucial aspect to consider when utilizing this method to solve the problem in

order to ensure that all combinations are accounted for.

The student continued to mull over their thoughts about this problem. As seen in the

excerpt below, the interviewer asked the student to explain the process and soon the student

mentioned other formulas derived from the topic of probability.

Interviewer: And can you explain your process? Student 10: I do the little hats and just go across. I know that you can do the - the probabilities 40

like the, what is it, NPC or NPN? What’s - I’ve forgotten what the shortcut is - shortcut is, but I was really bad at probability, so - as you can tell (student laughing). Interviewer: So there might be some formula that would count that, but okay. That’s fine. Student 10: Permutations, I think, is what it’s called.

I interpret this previous passage of dialogue to be the student exhibiting self-consciousness or a lack of confidence in their ability to solve the problem before them. They were commenting about their capabilities as a student of probability. Moreover, the student was trying to remember the formulas that may have been previously taught to them in the math class in which they were introduced to ideas such as permutations and the idea of probability in general. The student, in this case, was recounting the formula as a shortcut and was not really expressing that they were relying on the formula, which is interesting. I find this interesting because although the student acknowledged the existence of a formula that would act as a kind of shortcut to solving the problem, he still proceeded with his previous method of drawing out a chart.

The student was then asked the question: “How many ways are there to give 2 hats to 5 people?” Similar to what the student previously did with the chart, the student drew out another table in which they drew five boxes to represent five people and put tallies underneath boxes to represent hats given. This time, however, the student has a more organized way to find all of the possible combinations in this situation, as seen in Figure 15.

Student 10: So make sure 2 to the 4, 2 to the 4 - oh, all right. That’s what the pattern was. Okay. So, uh, I totally forgot about this. So these 2 go together. Cool. I guess that way to look at it first off is you know that you’ll get all the combinations with this one, because you start here, and you work your way across. So here, here, here, here. So then you know that there’s 4 in this, and there have to be, because there’s 5 people. You can’t have 1 person have 2 hats. So then we know 4 - 4 here, 4 here, 4 here, 4 here. And that’s how you can total it up. So 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 [possible ways to distribute 2 hats to 5 people].

Figure 15 - Student 10 still utilizing the exhaustion method, but is now more organized

At this point, the student was asked to fill out a table that would give all possible combinations for 0-5 hats given to 5 people. Before beginning work on the table, however, the student made some comments that I want to discuss.

Student 10: I’m just picturing my math teacher who taught me all this stuff: “Seriously, like I told you this so many times”. Interviewer: Was that in high school? Student 10: Yes. I was - I wouldn’t say burnt out on math, but I was like 3 or 4 years ahead of math at one point, and then a slight break so it’s been a long time.

I find these comments interesting because they do not necessarily have relevance to the task that

the student was currently solving, but the student decided to bring it up as they begin working on

filling out the table they were asked to draw out. I interpret the comments as the student still

trying to remember what they were previously taught in regard to probability (hence the

comment of the math teacher telling them so many times), but they were unable to remember

what they were taught such as the formulas. Then, the student remarked that it has been a

considerable amount of time since they had taken math, although they were still taking math they

were 3 or 4 years ahead. I interpret such a comment to be implying that the student believed that

they were good at math, but due to the break they have had, their math skills had declined. These

comments may explain why they were not remembering the formulas that can be applied or

instantly knowing the answers and having some trouble with the problem solving process. 42

After some deliberation, the student concluded that for 0 hats, there would be 1 way, as

everyone could get no hats, and there would only be one way in which this could occur. The

student then concluded that for 1 hat, there would be 5 ways since there are 5 people and each

person gets 1 hat. For 2 and 3 hats, the student utilized their previous work in which they

concluded that both amounts of hats results in 10 possible ways. The student also remarked on

the complementary (symmetric) relationship at play with 2 and 3 hats, 4 and 1 hats, and 0 and 5

hats.

Student 10: Well, if you have 0 - I’ll start from the top. So if you have 0 hats, then there’s 1 way for everybody to get it, because everyone doesn’t have a hat. And if you have 5 hats there’s only 1 way to do it because everyone’s wearing hats. So like these 2 are paired. The same thing for 1 hat and 4 hats - there’s 1 person who has a hat or there’s 1 person who does not have a hat (the student writes the arrows in Figure 16).

Figure 16 - The student representing the complementary relationships via arrows

After completing this table, the student was then asked to fill out a similar table, but this time for

4 total people (that is, the case with 0 through 4 hats).

Student 10: All right. So 0 is still 1 - 1, 2, 3, 4 - that would be 1. So going 4 hats, 1, 2, 3, 4. And then 2, 1, 2, 3 - no, no, no. 1, 2, 3, 4, 5, 6. And - okay. It seems logical.

While going through this thought process, the student was not drawing any new charts or performing any new work; they were relying on their previously made charts that they did for 2 43

and 3 hats given to 5 people. I interpret this action as perhaps being a reluctance to show work because problem solving (and specifically writing down outcomes in this case) can be tedious, and it is possible that the student is viewing the creation of charts to be tedious work. It is also possible that the student favored shortcuts as they had mentioned formulas as shortcuts previously.

By relying on the previous chart, the student simply covered up the 5th column of that chart (as there are no longer 5 people in the problem), and he utilized his drawn out combinations to conclude that for 2 hats there would be 6 ways and for 3 hats there would be 4 ways, as shown in Figure 17.

Figure 17 - Student 10 quickly fills out a new chart utilizing their previous work

Table that depicts figure 17 for clarity # of Hats # of Ways (Combinations) 0 1 1 4 2 6 3 4 5 1

Now, the student was asked to complete a table for 6 people. After spending some time working

on the problem, the interviewer asks the student what he was doing.

Student 10: I was checking to see if there was a pattern, and if there is, there are more data points to get it. So let’s see what this looks like - 1 - let’s see, so we’ve got 2, 3 - this is enough options. I’ll just draw it out. I just wish there was a computer where I could go enter, enter, enter.

I interpret this to be the student expressing his preference for a shorter way of solving the

problem, as computers are known to make work considerably faster, especially tedious math

work/calculations. The student then proceeded to continue working on their chat of giving 3 hats

to 6 people. After some time, the student was having trouble coming up with a conclusion 44

concerning the number of ways this could be done; the student realized that they were missing

some combinations, while they were repeating others (see Figure 18).

Figure 18 - Student 10 working through their chart of giving 3 hats to 6 people

Student 10: So trying to cheat is getting me nowhere. So I’ll try to be super neat this time and get all of them so I can get in a nice group picture. All right. So you go there. I have at least 3, possibly 4 more. So I’ll just put those down, because I know they exist. Skip the first one. Go here and here. Skip the first one. Go here and here. And then go here and here. Now, and if I put one here, it means I go there, there and there. Just here, so there and there. I’m not done. So we go - and then that obviously means I have to have one there. And that means I have to have on here. Now the real question is 1, 2, 3, 4, 5, 6, 7, 8, 8, 10, total at that point. Now 1, 2, 3, 4, 5, 6. 6 to 10 - oh - that’s that. Let’s see. Now, I have 1 here, 4 here, so at least 6.

Eventually, the student concluded that there are 22 ways to give 3 hats to 6 people (see Figure 19).

Figure 19 - Student 10’s completed chart in which he concludes 22 combinations

The interviewer then asked the student about their work, and they had the following exchange.

Interviewer: Why does it - I understand that like verifying that they all have the same number is like assurance somehow, right? Is there any sense of like why that might make sense? Like why should they all have the same? Why is that a reasonable check? Student 10: Because if you’ve - every single - all the people going across, every single one of them will have the same thing happen to the overall if you’re evenly distributing, because that’s what like even means. So every single person should wear the hats in equal amount of times or not wear the hats in equal amount of times. So overall, all of , you know, person 1 or column 1, I guess, even as - all of person 1 should wear the same amount of hats overall. And so then - so, you know, they get old, and then at the beginning it gets - not wearing anything at the very end. And then they just break up, but every single one will wear it at some point. Interviewer: Okay. Okay. Great. Fantastic. So you now have, I think - oh, so I think we can finish this chart. Student 10: Somewhere in here I wrote it down that it’s going to the same increments. 5, 4 - 5, 4- 5, 4 (student leafing through pages) I know it’s somewhere in here. Some of the increments are the same, but I don’t think I can salvage a pattern. The worst part is I know there’s a formula somewhere.

I find it interesting to note that the student mentioned the presence of a formula, yet again. The student, upon explaining their thought process, had not really used mathematical terms and has instead heavily relied on what their drawn visuals represent in regard to their thought process. I suggest that without the visual, it would be difficult to understand what the student was thinking 46

about in terms of the problem. The student proceeded to complete the table for giving 0-6 hats to

6 people (see Figure 20).

Figure 20 - Student 10’s completed table for giving 0-6 hats to 6 people

Table that depicts Figure 20 for clarity # of Hats # of Ways 0 1 1 6 2 15 3 20 4 15 5 6 6 1

The interviewer then asked the student to consider the situation in which there are n people,

asking “How many ways are there to give n people n hats?”

Student 10: Well, if n is even, then you do whatever n divided by 2 is. You add up all those and multiply by 2. So if you can get the pattern to try those things, I know it’s there. And figure out these guys, and you just double it for the second half down below. Because they’re weird. And then if they’re odd, then you obviously have to get the middle guy and then scoot one out. There’s a program on your calculator, which I always used to pass the tests, which I’m doing badly here. Interviewer: Okay. Could you find another expression for the total number of ways to give n hats to - or sorry, hats to n people? Student 10: Okay. So n equals people. So - and the number of hats go from 0 to the equal amount of people, correct? Interviewer: Yeah. Student 10: So you have n plus 1 options. n plus 1. Is factorial an exclamation mark that goes down all the way to 0, is that multiplied, or is that addition? Interviewer: That’s multiplied. Student 10: Oh. That’s not what I want. And there’s no symbol for addition - well, Sigma, I 47

guess (writes the expressions in Figure 21). I forget - okay. I didn’t give myself enough space. So Sigma from 0 to n - it’s bothering me that I don’t know what the mathematical term is for this whole organization thing. I know it’s like NPC or NCP or something like that. I’m sorry. You guys are going to be sitting here for a while. My brain is so dead.

Figure 21 - Student 10 writing expressions involving Sigma

I found the previous exchange to be interesting, as well as worthwhile to include, because it demonstrates the student going through what they know from previous methods to see if something can be applied to help solve the problem he was currently considering. Also, it is interesting that the student states that their “brain is so dead”. I interpret this comment to mean that the student is feeling as though he is taking too much time to think about this problem, and made the statement as a way to “excuse” the time he is taking, as if he should know the answer faster. Previously, the student brought up permutations and how they corresponded to this type of problem involving combinations, and now the student is discussing the relevance of Sigma and factorials. I interpret this thought process to be the student trying to remember the formula for permutations, but can only remember the formula in pieces, such as with factorials. However, the student does not fully understand the concept of each individual piece (i.e. the student clarifies what a factorial consists of), which may explain why he does not remember the formula very well.

As the student continued to struggle with the new proposed question, they began to look at their previous work in which they found the total number of ways to distribute a concrete number of hats to a concrete number of people. The student realized that upon totaling up the 48 number of ways to distribute hats to 4, 5, and 6 people, the totals doubled. However, the student did not really know what to do with this newly discovered information. He wrote some expressions (Figure 22) but did not do anything with them.

Figure 22 - The expressions student 10 wrote

The interviewer then asked the student a new question, which is seen in the following exchange.

Interviewer: So here’s a question. So you wrote out these 4 ways to give hats to 2 people. Could you like use this to construct the ways to give hats to 3 people? Student 10: Yeah, because one person is just on or off. Two people, it’s, you know, all off, all on or then one or the other. 1 person has it; 1 person doesn’t. Um, so I guess if you double the amount of people that can have the hat on, then that’s a no, because it’s just one original time. So 2 people it’s neither - it’s 1, 1 or it’s noth. All right. 3 goes none, 1, 2, that’s 1. He needs 1 by himself. They all need hats since the last one 2, 3, 4, 5, 6. I think 2 more. These 2 don’t have hats. And, let’s see, 3 ,4 ,3 - so these - 2 at this time. So what’s the pattern? Oh, duh. Okay. Because there’s 2 empty, so you - 4 - so every single time they might have options double, because the amount of empty hats and then if covered heads…(writes the work in Figure 23).

Figure 23 - Student 10’s work as he tries to determine a pattern

The interviewer then asked the student to consider a new situation.

Interviewer: Suppose now I want to know the total number of ways to give hats to N people, 49

but now I have not just red hats but red and blue hats that I can give out. Student 10: It’s killing me, though, because I - I studied this. I know - I can picture my math teacher. He’d do this great thing where people would do this in class, and he’d flip pen - like the pen caps of white dry erasers, and I always sat next to Cole, and Cole was always hit in the head, and I would remember one time he got hit right in the ear, and it got stuck in his ear while we were teaching this. You know, it was killing me, because it was like I know where I was. I knew everything that was happening except for what was on the board.

The student, although attempting to solve the problem given to them, was visibly struggling with how to connect the work they had already done to the context of the current problem. I interpret their comments about remembering everything but what was on the board in the classroom in which he learned about probability to be one of the reasons behind this. Throughout the interview, the student had consistently mentioned the existence of a shortcut or a formula, or both, and the student expressed frustration at not being able to remember what he had learned or been taught previously. Although the student eventually began to draw out the problem in order to visualize what was going on, the student utilized shortcuts whenever they could. For example, for giving hats to 4 people instead of 5 people, the student just covered up the 5th column from the work he did for 5 people to get the total number of ways for several situations within the 4- person question. Shortly after this question was asked, the student proceeded to try and solve the problem for red and blue hats, but he never reached a concrete conclusion before the interview concluded.

Synthesis of Results

From the three student videos I chose to examine more closely, I identified three themes

that emerged: these students tended to focus on formulas, the students showed a resistance to

showing work, and the students tended to talk about their own confidence. Each student utilized

different strategies to solve the same set of problems. A group of students, no matter the size,

will not approach and solve a given problem in the same exact way; although there may be a 50

“right answer” in the questions asked, there is not a “right way” to approach the solution.

Moreover, when a student is tasked to solve a mathematics problem with no given resources such as a formula sheet, their problem-solving strategies and approaches are exemplified.

Theme 1: these students tended to focus on formulas

To begin, this overall theme of a focus on formulas was observed from the three interviews previously examined. Some of the students from all of the interviews conducted revealed that they had previous experience in probability; one such student was Student 10.

Immediately upon being asked the question concerning giving three red hats to five people,

Student 10 brought to attention the idea of permutations. As the student progressed through his solution to the question of how many ways are there to give 3 red hats to 5 people, he mentioned the formulas associated with probability.

Student 10: I do the little hats and just go across. I know that you can do the - the probabilities like the, what is it, NPC or NPN? What’s - I’ve forgotten what the shortcut is - shortcut is, but I was really bad at probability, so - as you can tell (student laughing).

This focus on a formula that does exist, but is not remembered, shows how this student was overly focused on knowing the formula, or at least a shortcut as he seems to favor, that it is likely that it inhibited his problem-solving process to some degree. Moreover, the student brought forth permutations, which is not applicable to the questions asked since order does not matter (it draws upon the idea of combinations, instead). As can be observed above, the student came to the conclusion that although they knew that a formula existed for what he was trying to do, he could not remember it, which also led to his comment about being “...really bad at probability”.

Although confidence is another theme to be discussed, it ties in to the theme of a focus on formulas because it shows that this student associated knowing formulas to capability within a certain area of mathematics (in this case probability). However, for this student, knowing 51 formulas is not associated with understanding them, but rather memorizing the symbols associated with those formulas. When a student focused their attention on recalling the symbols they wrote on their note page in a lecture, they were not demonstrating an understanding of the mathematics at play, but of their skill of memorization. Furthermore, when such memorization cannot be attained (or is forgotten, such as for this student), then problem solving may become inhibited as students believe that they are not capable or they may focus on recall more than the problem at hand. This idea is important in regard to educational instruction because although a student may be successful in a given math course, it does not necessarily mean that they will be successful in future math courses that draw upon the skills they learned in previous courses, especially if they just memorized formulas. Often times, a break exists between terms or semesters, and during this break, academics are often times far from the minds of students. That said, the memorization they once focused upon begins to fade and they can only recall seeing a formula that is associated with a given problem, which was the case for Student 10.

By having such a strong focus on formulas, Student 10 exhibited that they relied on the formula and favored it over attempting a different approach, or simply problem solving without a formula. This focus was so strong that the student determined that they were “...really bad at probability”, which I infer was a conclusion made simply because they could not remember the formula. Memorizing formulas is often associated an activity is required within mathematics by many students, and if one is unable to remember such formulas at some later point after they were introduced to them, such as in the case of this student, then they begin to believe the idea that they are not proficient mathematicians. It is something that they begin to internalize as something about themselves academically in regard to mathematics rather than just the fact that they are unable to remember a formula. 52

Student 7 is an explicit example of the idea of memorization beginning to fade as he says,

“I really can’t think of something right now...just totally drawing a blank. I used to be good at

these problems I think like four years ago. I went and like memorized a lot of the formulas for

basic things like this, and I just completely lost them”. This was their answer when the

interviewer asked them about the table that they had constructed to represent combinations of

giving two hats to five people. Moreover, this answer represents an instance of a student

focusing on formulas because they associated their capability of being “good at these problems”

with the fact that they once knew a list of formulas that they had memorized. Furthermore, this

memorization caused them frustration upon solving the given problem as he stated, “I’m just -

I’m frustrated, because I like - I knew this. I hate - that’s - that’s the worst thing. I’m okay not

knowing something, I’m not okay with having known it and just losing it”. If students were

taught to bring their focus on understanding the ideas behind formulas rather than just

memorizing a list of formulas that would appear on a formula sheet, then I believe that students

would find more success in mathematics.

The idea of a desire for a formula inhibiting the problem-solving process was also demonstrated in Student 2’s interview because she became visibly flustered with how tedious and lengthy the process of filling out a chart that represented the different combinations was. She expressed her desire to have an equation by stating that she had not found an equation that solves the given problem. Although she was given the question of determining how many ways there was to give three red hats to five people, which would result in a concrete answer, the student was focused on finding a formula. A problem with this mindset in this situation is that a formula cannot be derived from one given case; the formula that she was referring to would need to work 53 for all cases as formulas represent a generality for the concept it is referring to. For instance, for combinations, the formula is:

nCr = n!r!(n-r)!.

This formula represents the number of possible combinations of r objects from a set of n objects.

The formula that the student desired would take this form since it would give her the answer that she was pursuing, but she would not be able to derive this formula since it cannot be derived from one concrete case, which is the same for all other formulas that exist. However, the student was not aware of this problem, and she increasingly became flustered to the point in which she could not try to solve the problem any longer.

For Student 2, the inability to remember a given formula, or formulas, serves as a strong hindrance in the problem-solving process because they become flustered. This relates to the research conducted by McLeod, who states, “Students have difficulty persisting in problem solving if their reaction is intense and negative, so they tend to quit and reduce the magnitude of the emotion...they assume automatically that getting stuck is a sign to stop or ask for help, rather than a normal part of problem solving” (p. 136-137) Although formulas can prove immensely useful in the sense of reducing the amount of work that is needed to solve a problem, especially if the problems are routine and are plentiful in number, they can also be strong deterrents in problem solving.

Theme 2: The students showed a resistance to showing work

Another theme that I found from the interviews is that the students sometimes displayed a resistance to showing work. I am not saying that the students were not necessarily working hard or willing to work, but they did not want to engage in some of the by-hand, systematic work that can be involved in solving combinatorial problems. Typically this had to do with being resistant 54 to writing thorough lists of outcomes. Student 7 demonstrates this theme because although he utilized a valid methodology for solving the problem (in which he made columns for each of the five people and then marked those that gets hats in a way that exhausts all possible combinations including a given person), he did not see it through to the end. Instead, the student utilized the method for only one person in the questioned five, and exhausted all possible combinations that include one person. From there, he continued to fill in gaps wherever he saw them, deviating from any sort of organization that he previously had. As a result, the student was unable to draw conclusions from his constructed table and could not reason why he concluded that giving two red hats to five people would result in a fewer number of total ways in which they could be distributed compared to three red hats given to five people. Upon shorthanding the work that he was previously doing, the student was unable to support his conclusions and could not effectively utilize his work for future questions.

Similarly, Student 2 demonstrated a resistance to showing work because although she had demonstrated a similar methodology of exhaustion, she did not retain the organization that she held consistently for questions concerning five people. When she was asked questions concerning giving hats to six people, she heavily came upon the problem of repeating previous combinations, which further her flustered reaction as she progressed through the problem solving process. A difference that exists between Student 2 and Student 7 is that Student 2 had more of a focus on formulas that she was very fixated on; she insisted on there being a formula that would help the process go quicker (and she was not wrong), and this insistence ultimately prevented her from progressing very far during the one-hour interview time allotment.

Student 10 also displayed a similar resistance to showing work as the other two students as he stated, “I was checking to see if there was a pattern, and if there is, there are more data 55 points to get it. So let’s see what this looks like - 1 - let’s see, so we’ve got 2, 3 - this is enough options. I’ll just draw it out. I just wish there was a computer where I could go enter, enter, enter” upon completing a table for giving red hats to six people.

Although I acknowledge that as the number of people considered increases, the length of work shown can easily increase, it is important to also note that the discoveries made within mathematics were not all done so with the sole reliance of a shortcut such as a formula or technology such as a computer. Showing one's work is not just a way for students to display to their professors their thought processes as they approach assigned problems, but a way for individuals to keep track of their thought processes and visually see what is going on mathematically in the given problems. Mental math is a difficult skill that some may master, but it proves immensely difficult in the case of real-world problems or word problems such as the ones asked of these students. Answers are not always immediately found, and if students learn to develop the belief that they are, then I believe that students will continue to show a high resistance to showing their work. Although it can be aggravating to partake in a tedious and lengthy solution path, once completed it can be viewed upon as a way to learn from mistakes that can more easily be pointed out (and improved upon), and it can also be utilized as a manner of repetition as it can be looked back upon at a future time. Simply stating an answer does not help those who do not understand the processes involved. That said, from an educational standpoint, encouraging students to show their work would be beneficial because it would help them to become more proficient mathematicians, and understand the processes that they are utilizing rather than just “plugging and chugging” numbers into formulas or a calculator.

Theme 3: The students tended to talk about their own confidence

A third theme that can be observed from the interviews has to do with the students’ comments that reflect their own confidence. Student 2 provided the most extreme case of this theme as she explicitly began to use the word “dumb” as she talked about herself as a math student and the ideas that she had concerning the problems. For instance, she said, “No, that was dumb, okay so what have I come up with?” and “...Which is probably dumb because I could probably find one [an equation] if I tried hard enough”. Student 7’s previously mentioned comment of, “I really can’t think of something right now...just totally drawing a blank. I used to be good at these problems I think like four years ago...” is another instance of confidence coming into play. Moreover, Student 10 told the interview of an interesting story in regard to a previous math class that he took, which relates to the theme of confidence as well, “It’s killing me, though, because I - I studied this. I know - I can picture my math teacher. He’d do this great thing where people would do this in class, and he’d flip pen - like the pen caps of white dry erasers, and I always sat next to Cole, and Cole was always hit in the head, and I would remember one time he got hit right in the ear, and it got stuck in his ear while we were teaching this. You know, it was killing me, because it was like I know where I was. I knew everything that was happening except for what was on the board”.

I now offer some extended discussion of Student 2. Student 2’s comments reflected her lack of self-confidence in herself as a mathematics student because the comments that she was making in regard to the problems demonstrated that she was understanding the context, but she was bogged down by believing that she was “too dumb” to accurately solve the problems. She states that if she tried hard enough, she could probably find an equation that would solve the problem. Although this is another demonstration of a dependence on formulas, there is the 57 underlying resistance due to the fact that the student did not believe in herself. She believed she was “dumb,” and therefore incapable of reaching a solution without the aid of a formula.

Moreover, she even doubts the possibility of her finding an equation as she uses the word

“probably.”

This raises another issue in regard to formulas: if students already have the idea established that they are not proficient mathematicians, then the inability to recall previously memorized formulas will do much more harm. Throughout the interview, Student 2 was making good progress toward reaching a solution, but she kept herself from progressing; she was her own biggest enemy. I do not believe it was the case that she could not solve the problem, but the fact that she believed she could not solve the problem seemed to prevent her from solving the problem. When students begin to believe that they are “not good at math”, then they begin to set themselves up for failure, and Student 2 is a prime example of this.

All three of these students are capable math students; I do not know them or know of how they performed in each of the math classes that they have taken, but just based on the interviews

I know that they are capable. The theme of confidence is important to consider in regard to education because if all students are treated the same, then not all students will be successful.

Moreover, some students are given the opportunity to move ahead of their peers in math classes that are typically offered in upper grades/levels, which can hinder the confidence that their peers who remain in the mainstream courses have, and more importantly can hinder the confidence of those who do not pass a math course and must remain behind. Students easily lose confidence in the subject of mathematics and many wish to never see it again once they graduate high school, yet mathematics exists everywhere in the real world. I believe that confidence should be a focus that educators have for their students in order to help convince students that everyone is capable 58 of succeeding in mathematics. The key point to acknowledge is that some students will need either more support or a different kind of support. Not every student comes from the same exact background nor has the exact same skillset, so it should not be assumed that all students can perform the same given the exact same support system.

Conclusion

To conclude this thesis, I have found that the videos that I chose to examine more closely greatly emphasized the three themes I felt accurately represented a method of approach for how to better help students in mathematics while also combating the negatives often encountered while problem solving. With combinatorics, there are many approaches students can utilize to solve a given counting problem This feature of combinatorics makes it a particularly useful field of mathematics in which to examine problem solving, as it underscores the idea that mathematics is much more than a right or wrong answer. Different students may not approach and solve a given problem in the same exact way. In other words, there is a “right answer” to every question, but not a “right way” to get there.

The first theme that I examined throughout this project was a focus on formulas. In research conducted by Lockwood, it was observed that, “As students encounter complex counting problems in high school and college, they are introduced to a myriad of counting formulas and problem types…but research indicates that counting can become a matter of simply memorizing formulas, identifying keywords, and matching problems to predetermined formula types,” (Lockwood, 2014, p. 298) This observation corresponded strongly to my findings through examination of the videos as Student 10 immediately called to focus on the formula to find the number of possible permutations for a given situation (in this case giving 3 red hats to 5 people). Moreover, this student also mentioned that they had previously taken a probability 59 course in which they were introduced to counting formulas, and I infer that this might explains their immediate attendance to a formula for permutations.

From what was observed in the videos and the acknowledgement of previous research done in regard to the role of formulas in mathematics, I remain aware of the usefulness that formulas provide, but I have found that it provides a more heightened sense of false understanding. Often times, professors, especially in the college setting, will relate to students that memorization is not needed if the process for how the formula is derived is understood.

However, the formula is still given and students may desire to write down each one that they are given on their note sheet come exam time. As Student 7 stated in regard to his own personal experience, “I used to be good at these problems I think like four years ago. I went and like memorized a lot of the formulas for basic things like this, and I just completely lost them.” Once the exam is over, the note sheet is not only thrown away, but the knowledge of the formula and the mathematics surrounding it are thrown away as well.

Note sheets are often a false sense of comfort because if students are able to plug in given numbers into the formula from the problem that they encounter, then they will be able to

(hopefully) solve the problem accurately, but if problem solving is needed, then many times students will encounter difficulties and potential failures. That said, memorizing formulas should not be a favorably looked upon approach to be successful in mathematics. As Lockwood stated,

“It is surprising how many students do not, at the start of a problem, consider what they are trying to count. Rather, students often tend to jump straight to formulas” (p. 300). By having a list of formulas memorized, students are doing just as Lockwood stated: they are glazing over what the problem is asking and what is given, and instead jumping to try and configure the numbers to work in the formulas. Although formulas remain useful in the realm of mathematics, 60

I am doubting the degree of usefulness that they provide students in the long run of becoming proficient mathematicians. Instead of providing students with routine problems in which they directly utilize said formulas, I would recommend teachers to instead have students work on problems that require problem solving (or, in other words, do not give numbers that are directly plugged into a formula) or problems that help students to understand the mathematical processes involved rather than just how the numbers work in the formula.

Alongside the theme of a focus on formulas is the theme of a resistance to showing work.

In combinatorics, specifically, many of the problems that are introduced in the high school and college setting can involve writing thorough lists of outcomes (even partial listing can be valuable). This action is especially useful for students who are being introduced to topics such as probability for the first time because it allows them to visually understand what counting consists of, and what certain problem types are asking for (e.g. when a question asks for the number of flags with at least 4 stripes). Specific mathematical terminology are utilized in these problems and are crucial for students in their paths of becoming proficient mathematicians.

In the videos that I examined for this research, all of the students demonstrated similar methodologies for how they approached the given problems: they utilized charts in which they created columns to mark the different possible combinations. However, although such a method is valid and will lead to accurate results, if done correctly, it can prove tedious. For instance, in the case of Student 7, he started with the creating columns, but after exhausting the number of combinations for one person, he tried to utilize a shortcut by filling in the gaps whenever he saw them, which deviated from any sort of organization that he previously had. As a result, the student shorthanded work that was a valid approach at first, and could not formulate a concrete conclusion that answered the question posed to him. Moreover, due to the shorthanded work, the 61 student was unable to support any conclusion that he did state (that is, he could not justify any of his answers), which strongly inhibited his future work since the problems built on each other.

Similarly, Student 2 approached problems posed with the same column methodology, and she also failed to maintain organization for all five people concerned in the context of the problem. Without consistency nor organization, she quickly, and continuously, made the common error of overcounting, which began to fluster her. As a result, the frustration she felt prevented her from making progress throughout the duration of the interview because the first question posed caused this intense emotion. As McLeod states, “The most common reaction students express is the frustration of getting stuck - a reaction that is frequently intense and negative” (p.136). In the case of this student, McLeod’s statement is affirmed as the student’s frustration was very intense and negative, which was demonstrated through her inability to progress further in the problem solving process once she hit the blockage of getting stuck. The third student considered, Student 10, displayed a similar resistance to showing work by explaining that his thought process was to search for a pattern to make the problem solving process transpire more quickly, “I was checking to see if there was a pattern, and if there is, there are more data points to get it...I just wish there was a computer where I could go enter, enter, enter.”

Overall, although showing work can easily prove tedious and dissuade students in the sense of their preference for formulas or patterns, showing work is crucial for success within mathematics. Showing one’s work is not just a way for students to display to their professors their thought processes as they approach assigned problems, but a way for individuals to keep track of their thought processes and visually see what is going on mathematically in the given problems. A strong reliance on formulas or patterns fosters the belief that answers to any 62 problem can be immediately, or at least quickly, found, which is not the case. However, if students continue to be presented with problems that can be solved by just plugging numbers into formulas or that have answers that can quickly be found, then when they are presented with more complex problems, they will begin to display a reluctance to showing work.

Showing work is also a useful tool for teachers to to determine the points in which students may need more assistance or a guiding nudge in order to overcome the blockage of getting stuck. This point is commonly emphasized in classrooms, yet students are still often given problems that have quick answers and will groan in disappointment when given word problems. Often, the reason that students dislike word problems is that the problems require problem solving and an understanding of the mathematical processes involved; they are not problems that provide all the numbers needed to be plugged into a relevant formula. On the other hand, a reluctance to show work may not be due to a preference or accustomed sense of problems having quick answers, but rather a preference to do mental math. Mental math is a difficult skill that some may master, but it proves immensely difficult in the case of real-world problems or word problems such as the ones asked of the students observed from the interviews.

Regardless of the preferences, it would be beneficial to encourage students to show work in order to show their thought processes and provide the opportunity to reinforce points that may be weak or not understood as well, while also being a immensely crucial skill to strengthen through practice. If students are taught to problem solve more efficiently (and thus not depend on formulas, patterns, or other shortcuts), then students would report less intense, negative emotions as opposed to students that cruise through math classes through a dependency on formulas written on a note sheet. 63

The third theme examined was self-confidence. Often, students will reveal their perspective of themselves as a mathematician, which was displayed from the students observed for this research. Student 2 provided the most extreme case of this theme as she explicitly began to use the word “dumb”, “No, that was dumb, okay so what have I come up with....which is probably dumb because I could probably find one [an equation] if I tried hard enough.” Student 7 mentioned his perspective of himself as a mathematician, “I used to be good at these problems I think like four years ago…”, which showed that he believed he is not good at counting problems anymore. Similarly, Student 10 stated that he “previously knew” how to efficiently do counting problems because he can clearly remember everything that was going on in the environment at the time he was introduced to the concept, but could not remember what was written on the board.

At times, such as with Student 10, it may be the case that students are simply not paying attention and the fault cannot be placed upon the teacher necessarily. In Student 10’s case, he talked about flipping pens and students getting hit with the pen caps from said pens, which is a strong indication that he was goofing around instead of paying attention. However, his comments are still important to consider because a common perspective from Students 7 and 10 was that they “previously knew” or were “previously good” at counting problems. I put such phrases in quotations because I am led to believe that these students may have had a false sense of confidence in their abilities when they were “good” at them. This is meant in the sense that their confidence was most likely derived from the ability to use counting formulas and could easily conclude how many possible combinations existed because a formula told them the answers.

That said, if a student has truly mastered a given skill, or has demonstrated proficiency, then it is unlikely that they will simply forget how to approach the problem. It may take more 64 time than when it was fresh in the mind, but two of these students encountered too many blockages, and when each one was encountered, had a reason for why they could not proceed.

For Student 10, it was because he could not remember. Student 7 claimed that he used to be good at the problems, but is now not good at the problems. I interpret their responses to be the result of them memorizing formulas and having such a strong dependence on having these formulas at hand (e.g. on a note sheet), and their absence exposes the fact that they were never truly proficient at solving the problems. This is not to say that these students are not capable or that they were not capable at the past time they indicated, but does suggest that perhaps they did not fully understand the context and complexities of counting problems. Knowing formulas is different from knowing how to solve problems. Similarly, memorization is different from problem solving; memorization does not make a proficient problem solver as problem solving does not make a proficient memorizer.

This discussion leads to a natural question: How can teachers help to boost the confidence of their students? One solution would be to help students to improve their problem- solving skills. By continuously presenting students with problems that have quick answers (e.g. problems that explicitly give numbers that need to only be plugged into a formula to be solved), there is a false sense of confidence being established. Students believe, during this time, that math may be easy or that they are proficient at the subject. However, when given more complex problems that require problem solving, they falter and begin to believe that they are not proficient at the subject. A problem with students is that many never question why they are suddenly not doing as well on problems, but instead simply accept that they are not a “math person” and never were, despite having that previous confidence. Students 7 and 10 are examples of this phenomenon. 65

Boosting student confidence is important because if students are left to believe that they are not capable mathematicians, then the beliefs will be carried with them for the rest of their lives. The interviews I examined were of undergraduate college students, and because I only examined three interviews I cannot make more general claims about more students. However, my findings do raise the question that if these students demonstrated that they did not view themselves very highly as mathematicians, then one cannot help but wonder how many other college students have similar beliefs.

In conclusion, this research examined three themes that were prevalent in the three student videos examined more closely: a focus on formulas, a resistance to showing work, and self-confidence. From the work that was displayed from each student, there were certain blockages, or the occurrence of a mental block that is often the result of an intense emotion that prevents students from furthering the problem solving process as they are unable to think clearly and reason through the problem at hand, that each one encountered. These blockages are not unique as many, if not all, students will encounter such blockages as they travel down the road to becoming more proficient mathematicians. Upon acknowledging and examining these blockages, the importance of the role of teachers arises because teachers withhold a crucial role toward helping not to prevent such blockages, but to change the reaction that students have toward them.

By encouraging students to show their work, having students complete problems that do not have quick answers, and by aiming to boost the confidence that students have in themselves, I believe that teachers can help to dispel the negativities and stereotypes that exist concerning mathematics. If students are led to believe that they are not capable mathematicians at an early point in their academic career, then such a belief will be carried with them for the rest of their lives and will be hard to dispel because the longer the belief is held, the stronger it will take hold 66 in the minds of students. That said, all students are capable mathematicians; it is not the problem of knowing this, but rather the problem of helping students to believe this.

References

Annin, S. A., & Lai, K. S. (2010). Errors in Counting Problems (Vol. 103). The National Council of Teachers of Mathematics.

Kloosterman, P. (1988). Self-Confidence and Motivation in Mathematics (Vol. 80). American Psychological Association.

Lockwood, E. (2014). A Set-Oriented Perspective on Solving Counting Problems. FLM Publishing Association.

Lockwood, E. (2014). Using Sets of Outcomes to Reconcile Differing Answers in Counting Problems(Vol. 108). The National Council of Teachers of Mathematics.

McLeod, D. B. (1988). Affective Issues in Mathematical Problem Solving: Some Theoretical Considerations (Vol. 19). National Council of Teachers of Mathematics.