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 2 3 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] 4 ©Copyright by Cheyenne Gordon June 5, 2019 5 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 6 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 7 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. 8 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) 9 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. 10 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