Problem-Solving Strategies in the Applied Math Classroom

Problem-Solving Strategies in the Applied Mathematics Classroom

Deborah Roberts November 4, 2003

Committee Members: Dr. Richard Tondra, Major Professor Dr. Irvin Hentzel Sarah Lubienski Table of Contents

Chapter One Introduction 3

Chapter Two Literature Review 8

Chapter Three Modifications 20

Chapter Four Analysis and Results 25

Chapter Five Discussion 29

Appendix 32

Bibliography 43

2 CHAPTER ONE

INTRODUCTION

Ever since I began my teaching career, the question that arose most often in my classroom was: “Where will I ever use this?” It has often been my response that whether or not the student uses a particular skill, the logic and problem-solving skills they are learning will have lifelong applications. Eventually

I began to take a closer look at my glib answer and to analyze just how serious I was about my comments. Was I really providing my students with the skills that would help them to be successful in the future? I began to examine more closely how mathematics education may have changed over the past thirty years, and how this relates to the general focus of my curriculum and teaching style.

Background

My first recollections of my own math education experiences are positive ones. I remember races on the board to see who could do multi-digit multiplication problems the fastest, or verbal quizzes where again speed was the key. Since these things came easily for me, I continued to gain confidence in my math abilities. In recent years, I find myself reflecting on the effect these types of competitions had on the class as a whole. The same two or three students were always vying for the championship, and I wonder how those who understood the algorithmic process perfectly, but were unable to work at that top-level speed, viewed these whole proceedings.

By the time I tackled algebra and geometry in high school, I found that my mathematics courses took more effort than most of my other studies. However, I

3 was willing to work hard enough to get that ‘A’. I began to regard math as my most challenging subject and envied those for whom it seemed to come more

‘naturally’. My last two years of high school, I had an instructor who would get very animated when explaining a new concept, and I enjoyed his excitement in the subject, but still felt somehow that I was missing a piece of the puzzle.

Although I enjoyed finding the pieces to make everything work out logically, I still felt that I was barely grasping the basic essentials and was always on the verge of being lost.

When I began to look at career choices, I found that education seemed like a good fit for me. However, I didn’t even consider the field of mathematics— after all, it was my hardest subject! Instead, I considered English, social studies

—there were so many interesting topics out there. Along the way, I was required to take a mathematics class, College Algebra and Trigonometry (pre-calculus), as part of my class work. I was rather surprised to find myself intrigued all over again. I enjoyed that class so much that I decided to tackle calculus, as well as to consider picking up a minor in mathematics.

I had several lengthy discussions with my college professor about this, and she gave me a piece of advice that I’ve always remembered. She said that for most mathematicians, (those who aren’t incredibly naturally talented), knowledge and problem-solving skills come as a result of being a ‘plodder’. This idea began to simmer in my mind as I considered the implications. Was it possible that I had been looking for memorizable skills and handy little algorithms that I could apply at will rather than digging to the heart of problem-solving

4 concepts? If I had some of those ‘plodder’ tools, would I be able to tackle more difficult concepts in ways that would allow me to break them down into something more familiar or relate them to something I already knew? Did I possess the skills to learn the language and logic of mathematics in a way that would enable me to look at this as a viable career choice?

I began to pursue my mathematics studies in earnest and decided to make this my major degree area. Calculus seemed fairly logical, as did discrete mathematics and most of my other classes. There were a few, like abstract algebra, that really taxed my self-confidence. Fortunately, it seemed like most of the class was in the same quandary, and we found ourselves meeting almost daily to discuss homework assignments or new concepts. I began to see a real benefit in the sharing of ideas and approaches as we struggled with the more difficult material.

About this time I took a course on teaching methods for mathematics. I am embarrassed to admit that I was totally unfamiliar with the National Council of

Teachers of Mathematics or the National Math Standards. Suddenly, the emphasis was not just on the curriculum itself, but also on the thinking processes involved in attaining mathematical knowledge, as well as how students could apply this knowledge to real-world applications! One of the basic principles of the NCTM standards (p. 21) states: “Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.” How much more exciting it would be if students were confident that they had the tools to not only master the skills presented to them in a math

5 classroom, but also to apply these same skills in their everyday life. This was what my college professor was trying to explain to me about being a ‘plodder’. I vowed to be the kind of teacher who challenged and inspired her students to new levels of achievement.

Overview

I am currently in my tenth year of teaching high school mathematics and science. Somewhere along the way, the real world has intruded and I find myself weighed down with six different classes to prepare for, papers to grade, and other life responsibilities all demanding attention. I have begun to take a closer look at my teaching styles as well as the strengths and weaknesses of my students, and am determined to prioritize a little better.

For several years now, I have taught classes in applied mathematics.

These students are for the most part non-college-bound individuals. Many go on to attend vocational or technical schools, and some will probably enter the work force directly after high school. I’m concerned that these students are often under-achieving in their academic pursuits and have low motivation for scholastic endeavors. Several of them are resourced for learning or behavior disabilities.

Short attention spans for more formal types of instruction are common and the students tend to prefer active types of learning activities.

Problem

My curriculum includes many word problems that are applications of vocational fields: business, agriculture, health/medicine, or mechanical. I began to ask myself several questions:

6 What mathematics skills do high school students not interested in a four- year college degree need before entering the workplace?

What skills are our local industries looking for in their employees?

Is the applied/practical mathematics curriculum we offer in Corning High

School successful in teaching these skills?

As I researched this topic, I began to see some consistent themes emerge from interviews and studies on the issue. One of the concerns of many employers was that new staff often lacks the skills to deal competently with problem-solving situations that arise on the job. They felt that high school mathematics preparation for job skills should provide students with the ability to

“use logic to draw conclusions from available information. Students must problem solve, apply rules and principles to new situations, and use questioning, inquiry, and the scientific process” (Day, 1997, p. 37).

I began to seriously assess my teaching curriculum and techniques to see how I could find new ways to motivate my students while also enhancing their problem-solving skills. This is a rather broad scope, and so I will not try to encompass all of these aspects of my classroom in this paper. More specifically,

I have decided to focus on the issue of teaching problem-solving skills, recognizing at the same time that part of the battle comes from sparking student motivation.

7 CHAPTER TWO

Assessing the Need

Before deciding what types of curriculum changes to implement into the classroom, it would be in order to assess more specifically the concerns and needs of the business community. There seems to be a common consensus from many sources that skill requirements in the labor market have escalated much faster than the schools have improved (Murnane and Levy, 1997). The problem is not so much that workers are less educated than they have been in previous generations, but that the demands of the current workplace are more sophisticated than in the past. To compete effectively today, American workers must employ skills at a ninth-to-twelfth-grade level (Scott and Span, 1996). The concern here is that there is increasing disparity between the skills students learn in the classroom and those that are applicable on the job site. There is also evidence to show that those jobs requiring a higher level of math skills also tend to yield a higher financial return as opposed to salaries provided by lower-skilled jobs. So what are the math skills that would better prepare our high school students for success in their future career choices?

The SCANS report issued by the U.S. Department of Labor in June 1991 specifically dealt with the question of what skills they wanted schools to provide their students with for success in the work place. This commission spent twelve months talking to business owners, public employers, and the people who manage employees daily in an effort to answer this question. Their first assessment was that good jobs depend on people who can put knowledge to

8 work. New workers must “be creative and responsible problem solvers and have the skills and attitudes on which employers can build” (SCANS, 1991, p. 3).

Traditional jobs are changing and new jobs are created everyday. High paying but unskilled jobs are disappearing. Employers and employees share the belief that all workplaces must "work smarter" (SCANS).

Although the SCANS report included the need for basic competencies in reading, writing, arithmetic and mathematics, speaking, and listening, they also reported the need for solid thinking skills. These skills include thinking creatively, making decisions, solving problems, and reasoning. One of the problems, as the writers of the SCANS report see it, is that employers have never clearly told educators what students need to know and be able to do in order to succeed.

Therefore, this commission strove to put together a report that would define the skills needed for employment, propose acceptable levels of proficiency as well as ways to assess these proficiencies, and develop a strategy to implement these skills into our nation’s school curriculum. The commission maintained that

SCANS research verifies that what we call workplace know-how defines effective job performance today. This know-how has two elements: competencies and foundation (SCANS).

The basic competencies that would be expected from any high school degree program would include the ability to perform basic computations. The problem is that students may have difficulty transferring the algorithms taught in the classroom to real-world problems in the workplace. Employers expect students to also have the ability to solve practical problems by choosing from a

9 variety of mathematical techniques. The concern arises as to whether schools have focused too much on the algorithms, and not enough on the more fundamental issues of quantities and their relationships (Bracey, 2001).

This leads to an assessment of thinking skills, and a look at whether schools are equipping their graduates with the ability to combine ideas and information in new ways in order to achieve goals. One of the thinking skills that the SCANS report specifically targets is that of problem solving. The commission suggested that students’ problem-solving skills needed to include “the ability to recognize problems and devise and implement a plan of action to resolve the problem” (SCANS, p. 35).

The National Council of Teachers of Mathematics has also targeted the issue of problem solving as worthy of being included in their standards that were published in 2000. They suggest that teachers should “enable all students to apply and adapt a variety of appropriate strategies to solve problems while monitoring and reflecting on the process of mathematical problem solving”. The goal is to equip students with the knowledge and tools that enable them to approach and solve problems beyond those they have studied in their curriculum.

In other words, teachers should strive to help students “develop a broad repertoire of problem-solving (or heuristic) strategies” (NCTM Standards, 2000, p.335).

Many researchers have concurred with the SCANS report’s stand on the importance of problem solving in today’s workplace. Although a few will warn against teaching problem-solving skills at the expense of a solid knowledge base

10 (Sweller, 1989), most commentaries support the agenda that the mathematician’s main reason for existence is to solve problems (Halmos, 1980).

Lawson breaks problem solving strategies into three main categories: task orientation strategies which influence the dispositional state of the student, executive strategies which are concerned with planning and monitoring cognitive activity, and domain-specific strategies with include heuristics and other procedures developed for organizing and transferring knowledge (Lawson,

1989). Lawson maintains that there is evidence to suggest that training in the use of different types of general problem solving strategies will positively affect performance in both mathematics and other curriculum areas. He then goes on to suggest that this training will also lead to a greater ability to transfer similar problem solving strategies to other areas. In summary, Lawson advises that general problem-solving strategies can have a powerful influence on an individual’s success in a variety of situations as long as that individual is also in possession of a well-organized knowledge base.

Problem-Solving Strategies as Related to Learning Taxonomies

Before developing problem-solving strategies for the classroom, it would be advantageous to review some of the more popular theories on learning taxonomies. An understanding of how students learn would provide an instructor with a better foundation upon which to base an instructional problem-solving strategy.

Piaget divided a child’s learning into stages of development, regarding the student’s intellectual development as an outgrowth of their physical age. He

11 maintained that by the age of 11 to 15, a child was ready to begin the period of formal operations. During this time, he determined that children begin “to reason realistically about the future and to deal with abstractions…or ideas about qualities and characteristics viewed apart from the objects that have them”

(World Book Encyclopedia, 1986). Since the age of students commonly found in the high school classroom ranges from about 14 to 18, for the purposes of this paper, we will assume the students have all reached the period of formal operations.

Bloom’s taxonomy divides learning into three domains: cognitive, affective, and psychomotor. The cognitive domain looks at an individual’s knowledge and intellectual skill development, and is divided into six major categories. Bloom maintained that each category must be mastered before an individual could move on to the next level. These six categories include: knowledge, comprehension, application, analysis, synthesis, and evaluation (Bloom, Mesia, and Krathwohl, 1964). Many of the concerns put forth from employers is that while workers may have the knowledge and comprehension of a concept, they often fall short of the ability to apply this concept to a new situation, or even more, to analyze the scenario, making inferences from the present situation to use in future situations. This reinforces Lawson’s concern about transfer of knowledge.

The SOLO taxonomy (Structure of Observed Learning Outcomes) is somewhat similar to Bloom’s taxonomy. It was developed by Biggs and Collis in

1982 and is also based on the concept that students learn with advancing

12 degrees of complexity, dividing this learning framework into five stages. The problem-solving competencies that we are concerned about would fall under the last two stages. The first of these is the rational level, where students are able to see the significance of the parts of their knowledge base as it relates to the entire concept. The last stage is the extended abstract level. This is the most important stage in terms of competencies in the modern workplace as it deals with an ability to make connections and transference of previously learned ideas and principles to new situations that arise (Atherton, 2002).

All of the aforementioned taxonomies have a common theme. It seems obvious that the concerns expressed by the SCANS report could be narrowed down to a concern that we are not preparing our students to enter the world of work at the more advanced end of the learning spectrum. Rather than stopping with rote memorization of common mathematical algorithms, we need to be focusing on guiding students to the upper levels of learning, which allows for the synthesis of ideas, and the transfer and application of a sound knowledge base to new situations and problems.

Span and Overtoom-Corsmit performed an investigation to assess the ways that gifted children solved mathematical problems as opposed to the average student. Their conclusion was that the more talented students immersed themselves in the details of the problem, using their analytical abilities to form hypotheses and a systematic approach to finding the solution (Span &

Overtoom-Corsmit, 1986). Average students took a more random approach to looking for possible solutions, and seemed to lack the tools to tackle the problem

13 in a higher-level manner. The question becomes whether there are specific strategies than can be taught to students that will enable them to use some of these higher-level skills that more advanced students already possess.

Problem-Solving Strategies

I began to study various approaches on implementing problem-solving strategies into the classroom. Burton (1987) suggests the use of an investigational, enquiry-based style of teaching. Even in my college methods classes I recalled the emphasis on using open-ended questions and similar discussion tools to generate higher-level thinking processes. What I was looking for, however, was a comprehensive set of strategies designed to encourage students to synthesize prior knowledge with their own creativity in order to allow them to successfully find solutions to new problem situations. In other words, I wanted to help provide my students with the tools and confidence to deal with new situations that might occur in future real-world situations, whether in the workplace or otherwise.

Lynn Steen addresses this question in an article in Educational

Leadership in which he discusses the necessity for a broader curriculum focus that puts greater emphasis on patterns and relationships, spatial reasoning, observation and conjectures, and genuine problems, rather than spending so much time on more common algorithms like fractions, graphing by hand, or two- column proofs. He pushes for a larger picture of mathematics from kindergarten through high school, with more experiences on the themes of “chance and change, shape and dimension, and quantity and variable” (Steen, 1989, p.19).

14 Halmos suggests that the answer is in a more in-depth approach to curriculum work, emphasizing the quality of the exploration over the quantity of material covered (Halmos).

A change in curriculum focus would necessitate a change in teaching practices. Steen suggests the use of more active learning processes that incorporate real-life problem-solving situations presented with instructional variety, as well as some means of assessing the success of the various approaches (Steen). Susan Forman suggests that while real-life problems are usually fairly concrete, they are not always as straightforward as one might think.

She also concurs that the use of this type of problem can challenge students to use higher-level thinking skills while making connections between the mathematics they’ve previously learned and the world around them (Forman,

1995-96). She reminds us that workers in what used to be called “blue-collar” jobs are now expected to have the skills to use their own problem-solving strategies while working in an environment that incorporates a high level of technology dealing with large amounts of data.

Benander, Cavanaugh, and Rubenzahl developed a group of problem- solving activities for their classrooms at a community college in Massachusetts.

Their first focus was on developing a sense of cooperation and confidence among the students in approaching problems. They used both group-building exercises as well as initiative problems to accomplish these goals (Benander,

Cavanaugh, & Rubenzahl, 1990). These activities included games that involved the students physically while at the same time developing reasoning skills. The

15 activities also included student self-evaluations of both their success and their sense of confidence at the end of the activities. They reported that students expressed a reduction in anxiety levels in approaching math problems, a higher confidence level in their abilities, and a more positive attitude towards math education.

Barba and Rubba approach the incorporation of problem-solving strategies as a procedural task analysis whereby tasks or problems are broken down into pieces. These pieces are then analyzed to determine which bits of information are the most important and what the relationship is between them

(Barba & Rubba, 1992). They suggested the use of audio or videotape so that the teacher can then analyze the student’s declarative, procedural, and structural knowledge base. The teacher can then look for misconceptions or areas in the student’s problem-solving skills that need remediation.

The Workplace Literacy Project funded by the U.S. Department of

Education specifically targeted lower-achieving math students. Rather than concentrating on applied math problems only, however, this program “blended conceptual approaches to mathematics with problem-solving exercises that were anchored in the students’ world” (Woodward, 1999, p.75). One of the instructors involved in the pilot project, Terry Wilson, focused on using common technologies such as calculators and software programs to collect and analyze real data, rather than pencil and paper skills more often seen in the mathematics classroom. The students used simple calculators for all the fraction work, and instead put their time into learning to use spreadsheets and computer graphics to

16 make pie charts and other pictorial representations to display their results. The students then presented their results orally to the class, verbally expressing their understanding of the problem and the results. Students were excited to approach their math in a manner that they felt closely mirrored skills they would use in the workplace later on.

Another discrepancy in approaches to problem-solving strategies is whether we are teaching students particular rules or algorithms that they learn by rote, or whether we encourage them to take commonsense approaches to analyzing new problems. The challenge appears to be how to integrate these two facets, so that students have a strategy in mind for where they are headed in their problem-solving efforts, and then have the algorithms in their repertoire of skills that they can use to finish the job. Booth suggests that if we teach students how to switch back and forth between strategies, so that they are comfortable implementing the most practical system for the problem rather than the most basic or elementary strategy they know, we can help them increase their level of cognitive functioning greatly (Booth, 1981). She maintains that many students cling to their earliest problem-solving strategies basically because they are comfortable with them and haven’t compared them to the efficiency of more recent and advanced methods they now know. If we can use the students’ own valid commonsense approaches and integrate them with more ‘mathematical’ or logical approaches, we can effectively broaden their problem-solving capabilities.

Ruth Parker maintains that a good portion of the difficulties of students

(and later adults) in mathematics is that too often teachers view mathematics as

17 a “collection of abstract skills or concepts to be mastered” rather than regarding mathematics as an understanding of patterns and relationships (Parker, 1991, p.

443). She suggests that we must make a clear connection between classroom mathematics and real world mathematics. In other words, as the NCTM standards state: students must see mathematics as “an integrated whole, explore problems and describe results using graphical, numerical, physical, and algebraic and verbal mathematical models or representations” (NCTM

Standards, p.84). Ms. Parker encourages teachers to develop classrooms where students learn to work both cooperatively and independently on real-world based problems, stressing that students should be assured that there are many approaches to any problem rather than just one right strategy. In effect, the math classroom should resemble the environment one might encounter in the workplace.

Parker had several recommendations to implement these concepts into the mathematics curriculum. First of all, she suggested presenting open-ended problems or situations that would encourage the students’ independent mathematical reasoning without leading them to a particular solution. Secondly, she asked students (and collaborative groups) to share their ideas with others in the class, verbalizing their thoughts and reasoning. Lastly, she encouraged students to try more than one approach to the problem, and stressed that there could be several correct ways of solving a problem (Parker).

Parker also had specific suggestions for staff development. She recognized that it is difficult to change the goals in mathematics curriculum

18 without providing teachers with new methods to do so. She encouraged professional development projects that would last from three to five years and be available to all mathematics faculty. She also recommended that new assessments be written that would allow teachers to evaluate the broader goals of the current NCTM standards. According to the NCTM (1991, p. 1),

“Mathematical power includes the ability to explore, conjecture, and reason logically; to solve non-routine problems; to communicate about and through mathematics; and to connect ideas within mathematics and between mathematics and other intellectual activity. Mathematical power also involves the development of personal self-confidence and a disposition to seek, evaluate, and use quantitative and spatial information in solving problems and in making decisions.”

It would seem imperative, therefore, to implement specific strategies into the curriculum that would encourage and develop these problem-solving skills.

Ina Miller suggests some techniques in an article for Lifelong Learning that summarize many of the ideas presented previously by other authors. These include acting out the problem, constructing a table, making a model, or drawing a picture, looking for a pattern, making an organized list, working backwards, and writing an equation to represent the problem (Miller, 1986). I decided to try some of these specific strategies with the students in my classroom to see if I could increase their level of comprehension and competency in working with problem solving.

19 CHAPTER THREE

Classroom Strategies

The class that I had targeted for my research was my Applied

Mathematics I classroom. I chose to specifically strive to improve the students’ self-reliance and competency as problem-solvers. As mentioned in my introduction, these students often lack self-confidence in their mathematics abilities and are sometimes difficult to motivate. If the research I had studied holds true, it seems of utmost importance to provide these students with both the confidence and the skills to enable them to be successful in their future workplaces. I decided to implement the changes over a period of two units of study. I would then compare the results from the unit tests from this year’s class to those from last year’s to see if there were measurable signs of improvement.

Unfortunately, my class sizes range from fifteen to twenty students, so it may be difficult to call this a viable sample group.

The units that I focused on for my classroom implementations were entitled “Working with Shapes in Two Dimensions” and “Working with Shapes in

Three Dimensions”. These materials are part of a CORD series for applied mathematics. The first unit dealt with identifying common geometric figures and calculating their perimeters and areas. The students also were challenged to find solutions to work-related problems that involved these figures (Appendix A). The second unit worked with geometric shapes in three dimensions: cylinders, rectangular solids, cones, and spheres. They learned to calculate the surface area and volume of these figures as well as solve problems involving them.

20 The strategies that I implemented into my classroom came in a couple of different forms. The first strategy I used was to challenge the students to do some problem-solving activities in an informal setting or as more of a game-type activity. The purpose was to encourage them to use a variety of problem-solving strategies in a non-threatening environment. The second approach I used was to give them a list of strategies to help them break down problems into workable pieces, and then to require them to demonstrate the steps that they followed when they used these strategies as part of their required work.

While looking for activities to supplement into my curriculum, I came across a text written by Dolan and Williamson entitled Teaching Problem-Solving

Strategies. It contained several fun activities that were designed to challenge and strengthen students’ logic abilities with more informal activities. One of the activities that I used was entitled “Selecting a Model” (Appendix B). This activity involved using either drawings or actual physical objects to come up with models for the situations presented. I gave this activity to my students individually, but encouraged them to discuss options or ideas openly within their table groups.

Assessment for this activity was pass/fail. I wanted to see the results of their brainstorming regardless of accuracy. I observed that the students were quite involved and animated in their discussions and efforts on this activity, and several times we stopped to discuss questions or approaches as a class.

The second activity I used from this text was called “Regular Polygons in a

Row”, and was designed to help students explore the concept of perimeter

(Appendix C). I had the students work in pairs on this activity to encourage

21 further team building, and again graded this activity on a pass/fail basis. I was surprised by how intense the students got by the end of the activity, as they tried to find the correct pattern that would give them an algorithm for finding the perimeter of any number of polygons placed end to end.

Another adaptation I made to my curriculum was to write a list of strategies on the board that I left up for the entire time that we spent on these two units (several weeks). These strategies were designed to help the students become comfortable with organizing their own logical thinking processes as new problems or situations arose. These strategies included:

1) What does the problem want to know? 2) What information do you have? 3) Can you draw a picture to represent the problem? 4) What mathematical processes will you need to do to get where you want to go? 5) Can you write a formula to present your problem? 6) Solve!!!

We spent quite a lot of time modeling these strategies by doing problems as a class using this approach. I was amazed to realize that many times students were off track because they got caught up in the language of the problem and forgot to look at what exactly the problem was asking for. I encouraged them to write down information as they encountered it, making a list of all data whether they knew if it would be relevant to the solution or not. I also strongly recommended a picture wherever it was applicable. When I stopped to help a student with a question, I often responded first with: “Where’s your picture?”.

Then we coud begin a discussion of what they were trying to do.

22 We also spent a great deal of time discussing the nuances of language and what that means in terms of mathematics. If they are to find the price per square foot of carpet, what mathematical operation does that represent? If they only sell one-third of their tickets, what does that mean in terms of numbers?

What clues will help them distinguish between perimeter, area, or volume problems?

Once they had assessed what they needed to do, I asked them to write down the formula or expression that they will use. Many in this class struggle with pencil and paper mathematics, but I wanted to emphasize the problem- solving and logic aspect of the process rather than the ability to crunch numbers.

Therefore, I allow full use of calculators, while also spending time discussing how to decide if an answer is reasonable. I encourage the students to look at the calculator as a tool to supplement their own logic and skills. Every time they reach a solution, I ask them to look at it and decide if it seems reasonable. If lumber costs $3.30 per board foot, would it make sense that fifteen board feet would cost $.22? Once the students got in the habit of checking their own work over, it was amazing how many times I would hear them say: “Well, that can’t be right!”, and back to the beginning they would go.

Another change I made was in the way I presented some of our lab activities (Appendix D). I began to ask more specific strategy-oriented questions on the lab papers that I handed out at the beginning of the lab and we would go through the first few questions as a class. This seemed to help the students get

23 a better overall understanding of what the activity was about so that they could develop their own plan of how they wanted to solve the problem.

Finally, I adjusted the way I assessed these activities. Although for my final research analysis I went back to unit test scores for comparison, as we engaged in some of these new activities I found myself using a more holistic approach to assessing the students’ work. I found a text published by the

National Council of Teachers of Mathematics entitled How to Evaluate Progress in Problem Solving that outlined some suggestions. I tended to use a generalized holistic approach that was less time-consuming than some other forms of assessment and took into account my considerable class load. The holistic approach allowed me to take into consideration other aspects of the students’ solutions than just the correct answer, while still being efficient enough to be manageable on a day-to-day basis. Since I wasn’t using a more specific rubric, I made it a point to make comments or suggestions on students’ papers when applicable. I also revisited the more challenging problems in class discussions, both during and after homework assignments.

Overall, I made a concerted effort during this time to conduct my classroom in a manner that continually encouraged the students to be creative, but yet logical, in their problem-solving approaches. I made a conscious effort to use open-ended questions in our discussions, to encourage multiple approaches to the same problem, and to provide an atmosphere where the students felt comfortable sharing their thoughts and ideas.

24 CHAPTER FOUR

Assessing the Modifications

For a formal comparison of the effect of the modifications I implemented into my curriculum for this quarter, I used only the unit test scores as compared to test scores from the same unit from the class the year before. The second unit was rather long, so in the past I have tested over this unit once mid-chapter, and then a final time at completion of the unit.

I took a look at comparisons of both the test scores from the two units in which I had made modifications, as well as the previous six unit tests (Figure 1).

In this manner, I could better ascertain whether class score discrepancies were related to the differences in teaching methods or just different student abilities.

Tw o-Year Comparison

100 90

e 80 g a

r 70 e v 60 A

'01-'02 Class t

s 50 e 02-'03 Class T

s 30 a l

C 20 10 0 1 2 3 4&5 6 7 8A 8B Unit

Figure 1

Two-Year Comparison At first glance, it would appear that there was some improvement after the 100

80 e g a

r 60 e v A

t s e T

40 s s a l C modifications were implemented. Although the20 2002-03 class also scored better

Unit on the unit 6 test, which was before I had implemented the modifications, I was

25 encouraged by the fact that they consistently continued to score better than the class from the year before on the remaining three unit tests, the ones after I’d implemented the modifications. This class had scored slightly better than the previous year’s class on the first four unit tests on the graph, but the discrepancy between the two classes scores increased in the later units, and continued to rise until the final unit test showed an improvement of over ten percent from the previous year.

I decided to run the statistical test for two independent populations to see if it would support my claim that the test scores had improved. I first needed to perform an F-distribution test to see if the variances were equal. The assumptions made for this test are that the populations are independent and approximately normal. I charted my data for the previous year’s class as well as the recent year’s class to see if they approximated a normal curve (Appendix E and Appendix F). I thought the first year’s data was very close to a normal curve, but I was somewhat concerned with whether the second chart could be considered normal or if it was slightly skewed to the right. I did perform the equal variance test on these sets of data. My calculations were as follows:

2 2 Ho: variances are equal (s1 = s2 )

2 2 H1: variances are not equal (s1  s2 )

N1 = 15 N2 = 15

x1 = 74.378 x2 = 81.933

s1 = 12.147 s2 = 11.932

2 2 F = s1 / s2 = 1.0364

26 In a two-tailed F test with 14 and 14 as my degrees of freedom and using

 = .05, the rejection region is F > 2.9829. Since my test statistic does not fall in the critical region, I do not need to reject my hypothesis and I can consider my variances equal. I went on to do a two-tailed t-test analysis. For this test my data was as follows:

Ho: 1  2

H1: 1 < 2

2 sp = 144.961

t = -1.6602

With the two-tailed t-test with 28 degrees of freedom, I will reject my null hypothesis if t < -2.048. Since my test value did not fall within the critical region, I could not reject my null hypothesis and therefore could not statistically support my assertation that my modified curriculum increased the test scores of my class.

Before I quit with the statistical analysis, I decided to also compare my modified curriculum’s scores with those from the same class for the first part of the year before I had implemented the modifications. I again ran an F- distribution test to see if my variances were equal. My data was as follows:

2 2 Ho: variances are equal (s1 = s2 )

2 2 H1: variances are not equal (s1  s2 )

N1 = 15 N2 = 15

x1 = 77.2933 x2 = 81.933

s1 = 11.99862 s2 = 11.932

2 C F = s1 / s2 = 1.0111

27 In a two-tailed F test with 14 and 14 as my degrees of freedom and using

 = .05, the rejection region is F > 2.9829. Since my test statistic does not fall in the critical region, I do not need to reject my hypothesis and I can consider my variances equal. I went on to do a two-tailed t-test analysis. For this test my data was as follows:

Ho: 1  2

H1: 1 < 2

2 sp = 144.961

t = -1.0260

With the two-tailed t-test with 28 degrees of freedom, I will reject my null hypothesis if t < -2.048. Since my test value did not fall within the critical region, I could not reject my null hypothesis and therefore, again, I cannot prove that my class showed improvement.

Due to the fact that I teach in a small school, my test sample was very small. Even if the statistical analysis had been in favor of my hypothesis, it would probably not be valid to draw too many conclusions from this. I turned then to other observations to continue my assessment of the modifications I was implementing.

28 CHAPTER FIVE

Further Discussion

There were several observations that I think bear mentioning. First of all, the fact that the data from the scores after the curriculum modifications seemed skewed slightly to the right indicated to me that there were fewer students in the lower skills area. In other words, although the overall scores only showed a slight improvement, the growth in the lower third of the class was more marked. I would speculate that perhaps those that scored in the top third from the beginning of the year already possessed competent problem-solving skills, whereas those who struggled with these concepts reaped greater benefits.

Another observation I would like to point out is that I felt that the class as a whole exhibited a greater willingness to participate in activities, classroom discussions, or other challenges presented to them. They were more willing to

‘risk’. I felt they were gaining confidence that they had some tools with which to break down situations into workable pieces, and did not let a fear of failure keep them from tackling new situations. I tried to reinforce this behavior by positively responding to all efforts and by giving credit for exhibiting sound problem-solving logic rather than just a correct answer.

The modifications that I was making in my applied math classroom carried over into other areas of my teaching. I also teach chemistry and physics, and I found many times I would use the same problem-solving approaches to their curriculum as I did to the applied mathematics. I soon came to the conclusion that no matter what the level or topic of study, these same skills were important

29 for the students to find success. My upper classmen, juniors and seniors, were often dealing with the same confidence issues as my freshmen and sophomores.

Many of them will soon be going to college or out into the workplace, and it is my job to see that they have the skills to be successful.

One of the areas that I would like to explore further is the use of rubrics to assess these problem-solving skills. I found several interesting approaches in my research, and I plan to try other methods in addition to the generalized holistic approach I used most often during my study. This past summer I helped evaluate ICAM math tests that our area high school juniors had taken as part of their state assessment requirements. The evaluation involved the use of an analytical rubric that divided the problem into several phases and assigned points according to the level of understanding the student exhibited in solving the problem. I found I did not always agree with the rubric we were required to follow in our assessment, and thought it would be interesting to devise my own for some of my classroom testing to see if I could do better!

I find myself recalling one of those defining points in my teaching career when I had a student who was helping his dad and uncle harvest come excitedly to class one day. He shared with us that they were hauling the corn to the bins, and were debating how many more bushels of corn they could get into the bin where they were currently unloading. They knew about how many cubic feet it took per bushel, but were estimating how much space was presently left. My student had just finished studying cylinders and cones, and recognized the shapes in the bin. He quickly pencil and papered out the correct volume for the

30 remaining space. He was quite proud of making the connection and actually being able to use something he had learned, and of course, his teacher was exceedingly proud of him also!

The point is this: We have to make math education applicable to daily life and then insure that the students have the skills and confidence to transfer their mathematics knowledge to these real-life situations. Carmel Schettino (2003) put this very well in a recent article for the Mathematics Teacher journal. She holds that “an educator must first commit to the premise that helping students develop their ability to solve problems independently is the major goal of mathematics education”. She maintains that this “entails teaching students that they have the freedom to solve problems with a set of given tools and knowledgeable guidance and that the goal is to further develop their mathematical toolkit”.

I completely concur with Ms. Schettino’s position. As the National Council of Mathematics reminds us: “A major goal of high school mathematics is to equip students with knowledge and tools that enable them to formulate, approach, and solve problems beyond those that they have studied” (NCTM, p.335). This is the same theme that appears in the Department of Labor’s SCANS report. My goals as a mathematics instructor must include the desire to provide my students with the skills and confidence to meet this challenge. This entire project has broadened the scope of how I perceive myself as a mathematics/science teacher. I would now venture to say that the curriculum itself is not the main focus of my instruction, but rather the desire to provide my students with the skills and assurance to approach future undertakings with confidence.

31 Appendix A

Area Problem Examples (from the CORD curriculum)

1) Mary trains for cross-country by running around the block 5 times each evening. The block is a rectangle 500 feet long and 250 feet wide. a) How far does she run when she runs ONCE around the block? b) Find the total distance that she runs each evening. c) Convert the distance she runs to miles. (5280 ft. = 1 mile)

2) A furniture factory builds rectangular tables that are 5 feet long and 2 feet 8 inches wide. They put oak trim all around the edge of the table. This month they will build 85 tables, and the trim costs 56 cents per foot. How much will they spend on trim this month? Ask if you need help understanding the steps of this problem.

3) A biologist is studying insect populations in a field. In a test square 10 feet on a side, he finds 30 greenbugs. The whole field is 450 feet long and 300 feet wide. He wants to estimate the total number of greenbugs in the field. Follow these steps: a) Find the area of the test square. b) Find the area of the field. c) How many of the test squares would fit in the entire field? d) Approximately how many greenbugs are in the field?

4) A gardener has a rectangular garden that is 120 feet long and 80 feet wide. She is planting the whole garden with a cover crop of rye grass, and she is supposed to use one pound of seed for every 1000 square feet. Find the area of the garden, and then find how many pounds of seed she should use.

5) A map of Ms. Brown’s land shows a large pond (almost circular) inside a rectangular field. She wants to plant the field to pasture, so she needs to know the area of the field, not counting the area of the pond. a) Measure, and use the scale given to find the lengths she needs to know. Scale: 1 inch = 300 feet b) Find the area of the field and pond. c) Find out how many square feet of grass she will plant.

32 Volume Problem Examples (from the CORD curriculum)

1) A loading chute brings grain into a bin as shown in the drawing. The bin is a cone. How many bushels of wheat will the bin hold? (There are 0.8 bushels per cubic foot.)

2) A spherical water tank has a radius of 10 feet. It lets the water flow out of the spigot at 2.5 ft.3/sec. How many minutes will it take to drain the tank?

3) A company has a shipment of 100 mirrors arriving in boxes which are 3 feet long, 15 inches wide, and 6 inches tall. They want to store them in a space that is 8 feet long, 6 feet wide, and 5 feet tall. They want to know if all the boxes will fit. Follow these steps. a) Change all the units of the dimensions of the box to feet, and find the volume of one box. b) Find the volume of all 100 boxes. c) Find the volume of the storage space. d) Answer clearly: will all the boxes fit in the storage space?

4) A construction company is pouring a sidewalk that is 120 yards long, 6 feet wide, and 9 inches thick. Concrete will cost $35 per cubic yard. They want to know how much all the concrete will cost. Follow these steps. a) Change all the units to yards. b) Find the volume of the sidewalk. c) Find the cost of the concrete.

5) A backhoe has a bucket that holds about 5 cubic feet. The backhoe operator knows that he can dig about 30 buckets an hour. He needs to dig a trench that is 400 feet long, 6 feet wide, and 2 ½ feet deep. He wants to estimate how long it will take him to dig the trench. a) How many cubic feet per hour does he dig if he digs 30 buckets per hour at 5 cubic feet per bucket? b) What is the volume of the trench? c) How long will it take him to dig the trench?

33 Appendix B

34 35 36 Appendix C

37 38 39 Appendix D

Measuring Soft Drink Cans and Cartons

Problem: How much wasted space is there in a 12-pack carton of pop?

Procedure: What will you need to know to figure out this problem?

What formulas will you use?

Calculations:

Solution:

Critical Thinking: Can you think of a design that would make better use of the carton’s shape? (Draw a sketch to demonstrate.)

40 Appendix E

41 Appendix F

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