Analyzing solution methods for combinatorics problems: differences across problems and across teachers
Caddle, Mary C.
2010
- A qualifying paper submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Math, Science, Technology, and Engineering Education. Abstract: This paper describes a small study in which five teachers were interviewed in order to gain insight into their understanding of combinatorics. Combinatorics is a challenging mathematical area that provides learners with a ... read morebasis for a great deal of applied mathematics, including statistics. Through interviews with teachers, this study sought to investigate how the teachers solved combinatorics problems, the methods they used, the issues they raised, and the knowledge they displayed.In particular, the analysis presented here addresses three questions:i. how do the methods used when solving combinatorics problems relate to the mathematical attributes of the question;ii. how do the methods used when solving combinatorics problems in an interview differ across teachers;iii. how does a mathematical problem with unexpected difficulties or subtleties illuminate the differences among teachers.In order to address these questions, this paper will first look at the reasons for focusing on this area of mathematics and will define the specific area that I am addressing. Next, we will look at existing work on teaching and learning combinatorics, including considering some common errors and misconceptions.Following this, the methodology for this study will be described. The methodology section will describe the line-by-line analysis of the transcripts of the teacher interviews, during which each participant was asked to solve and discuss a series of combinatorics problems. The results from this analysis regarding the relationships among teachers, methods, and mathematical tasks are outlined in detail.The subsequent discussion provides both a qualitative look at these first two research questions and a foundation and justification for addressing the third research question, (iii) how does a mathematical problem with unexpected difficulties or subtleties illuminate the differences among teachers. The discussion shows that the differences among teachers extend beyond which methods they choose and whether or not the answers to the questions are simply correct or incorrect, but also include subtleties in how they employ the methods and how they view the utility of each one.read less
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