Transformation of a Math Student’s Learning

By Morgan Mattingly, undergraduate double-major in STEM Education and Mathematics at the University of Kentucky.

Comment from the Editorial Board: We believe that in our discussion of teaching and learning, it is important to include the authentic voices of undergraduate students reflecting on their experiences with mathematics. This article is our first such contribution. We feel it provides a window into many of the subtle challenges students face as they transition to advanced postsecondary mathematics courses, and that it mirrors many of the themes discussed in previous posts. We thank Ms. Mattingly for being the first student to contribute an essay to our blog.

In previous math classes, I was the quiet worker who kept to herself and didn’t know when or how to ask questions. After improving my skills in a problem solving class, that has changed. The group work we did each day allowed me to be around other people who think significantly differently than I do. Being in this environment was difficult at first because I actually had to work through problems with other people, which was somewhat unfamiliar to me. My classmates and I were not just sitting down and reading information about specific math problems. We had to analyze and make sense of the best methods and strategies to use and present our ideas to each other. Confusion would set in when other students introduced different approaches. The only way I could understand their ways of thinking was to ask them to explain. Asking questions in math initially intimidated me, especially because my questions had to be directed to my peers. I did not want them to think that I could not keep up with the material or that I did not belong in the class. But I also did not want to misunderstand major mathematical concepts as a consequence of not asking questions. So I started asking my group members each week what strategies they used in their solutions. Although it may have seemed repetitive to them or obnoxious to have to explain their approaches, it helped me immensely. Through my question asking, I was able to talk and think about math in a unique way. I could compare my peers’ techniques to my own, which further stimulated my interest in the particular subjects that were covered in the class. This skill has been and will continue to be essential in my future relationship with mathematics.

With this question-asking skill came a respect for other students’ speed of thinking. As I was learning to think like my group members in certain situations, I also was learning that these students were thinking at different speeds than I was. In many instances, I would attempt to understand what the problem was asking for and in the meantime, my peers were already halfway to a solution. I knew that I wasn’t misunderstanding anything. I simply was not making connections as quickly as others in the class. It constantly surprises me how swift some students are in accurately assessing problems and understanding what is needed to get to a solution. After working with all different types of students, I have learned to respect and accept that others may be working at faster cognitive speeds than I am. In previous classes, I thought that being quick with my math skills was most important. I have now seen that understanding the material is essential to becoming a fast thinker in mathematics.

Through the homework problems and quizzes in the problem solving class, I realized that math problems require perseverance. The problems that are actually difficult, that actually require a student to think about how to apply his or her mathematical knowledge to the solution of a problem, are the problems that take time. I found myself working hours on different math problems, trying to get closer and closer to the right answer. Oftentimes I went through the trial and error process. Failing in math is not as scary as it once was, because of my experience with this problem-solving course. Sometimes I would successfully solve a problem, while other times I didn’t come close to the solution. Either way, I was able to see that simply working with math was enhancing my problem solving skills. I will always remember Paul Zeitz’s quote in his book, The Art and Craft of Problem Solving [1], that says, “Time spent thinking about a problem is always time [well] spent. Even if you seem to make no progress at all,” (pg. 27). As I encounter future math classes and harder math problems that seem unsolvable, I will keep this quote in mind. Any time spent toying with a problem is enriching my mathematical knowledge.

Understanding Zeitz’s important quote about mathematical thinking prompted me to see the open mind that math problems require. When I took a number theory class in my first year, I was under the impression that all solutions to problems were very obvious and the methods to solve them were evident. Going through a geometry class as a sophomore challenged this belief because I began to see that not everyone knows which method to use immediately to solve or prove a problem. With the problem solving class, this belief was completely put to rest. I have seen firsthand that it occasionally takes experimentation to figure out which method or tool to use in problem solving. Through discussion and group work with my classmates, I noticed that it is not always blatantly obvious that we should draw a picture or use induction or reformulate a hypothesis to find the crux move in a solution. After discovering this, I attempted to open my mind when reading assigned problems. Instead of honing in on one specific method or strategy, I have accepted the fact that one specific method or strategy might not be the only way to achieve my goal.

An open mind in problem solving has allowed me to experience mathematical thinking outside of the classroom. My high school math experiences bred the idea that students don’t necessarily need to think about math outside of the classroom. I brought this idea to college and had no problem passing my classes. But in this math problem solving class, where we were challenged to think in different ways and to explore math on our own, thinking about math strictly inside the classroom was insufficient. One incident that had a deep influence on my problem-solving experience occurred on a walk home from class. In small groups, my classmates and I were trying to determine the difference between permutations and combinations. After working for an entire class period, I did not fully understand the difference between these two basic combinatorics concepts. As I walked home, I reviewed the strategies and methods that I used and compared them with the explanations of my peers. I was still stuck and still frustrated. Upon emailing my professor about my misunderstanding, I realized that although I had encountered an obstacle with the content, I was gaining an invaluable way of thinking. I was extending my mathematical thought processes and interests to outside of the standard classroom environment. I am still constantly left wondering why something in math works. By pondering on my own, whether it is with pencil and paper or not, I am able to see the impact of this class. No class before had prompted me to take my own time to figure out why I did not understand the material. This was a huge win for me as a math student. I was and am still experiencing the effects of struggling with math in a way that will always benefit my problem solving skills.

One of the most interesting proofs that I saw was dealing with the number of subsets of a set with \(n\) elements. Earlier that semester, my probability professor had mentioned that the number of subsets of an \(n\)-element set is \(2^n\). I didn’t think I would be seeing this anymore, so instead of trying to understand why this was so, I just accepted the information. Later on, in my problem solving class, we had a question about the number of subsets in an \(n\)-element set. Obviously I knew it was \(2^n\), but I had no idea why. I eventually understood after I showed interest in the problem through question asking, when another interpretation of all of the subsets of an \(n\)-element set was written up on the board. I saw that another way to write a subset of an \(n\)-element set is by exchanging the actual numbers in the set with 0’s and 1’s. A “0” indicates the absence of a specific element in the subset, while a “1” indicates the presence of a specific element in the subset. For example, the set \(\{1, 2, 3, 4, 5, 6\}\) has the subset \(\{2, 4, 6\}\). This subset can be written as 0, 1, 0, 1, 0, 1, where the 1’s correspond to the numbers found in the subset. Since each space has either the option to be in the subset or not, then each space has two options. Since there are \(n\) spaces, then there are \(2^n\) subsets. By asking questions, expressing curiosity, and actually attempting to understand why the answer was \(2^n\), I was able to see a portion of my growth as a problem solver.

I am left with questions regarding my future mathematical experiences. Instead of simply thinking about mathematics outside of the classroom, I am now wondering how to discover and develop new insights about mathematical concepts on my own. Instead of learning about how to use certain strategies, I am wondering how to present these strategies to a group of peers in an orderly and effective manner. Instead of asking questions that do not necessarily prove to be productive, I am slowly learning how to ask the right questions.

References

[1] Zeitz, P. (2007). The Art and Craft of Problem Solving. Hoboken, NJ: John Wiley & Sons, Inc.