By Elise Lockwood, Contributing Editor, Oregon State University
My research focus is on undergraduate students’ solving of counting problems, and I have worked toward better understanding students’ combinatorial thinking. Counting problems provide excellent opportunities for students to engage in meaningful mathematical tasks and to experience tangible beynefits of being precise and meticulous in their work. In this post, I draw on my experience studying undergraduate students’ combinatorial reasoning to offer examples of “careful” work. There is likely little debate that it is important for students to be organized, precise, and careful as they engage in mathematical activities. Although some students turn in homework assignments that are detailed, organized, and well thought out, others pass over details or do not properly represent ideas. What makes some students (and not others) willing to invest time and effort in detailed and methodical work? How can we help students more amenable to being careful and precise? I believe that these are important questions to consider, and in this post I suggest moving toward emphasizing and characterizing this kind of behavior. In this post, I offer three contrasting examples of students’ solutions to counting problems, which highlight characteristics of careful and precise work.
When solving counting problems, students often simply take a guess at an answer, sometimes remembering (or misremembering) a formula, without being careful about generating a solution that makes sense and that can be justified. However, if students get in the habit of being more careful and methodical about identifying outcomes and carefully considering how to count, they can more easily avoid common counting errors.
Consider three students’ responses to a problem that states: “Fred, Jack, Penny, Sue, Bill, Kristi, and Martin all volunteered to serve on a class committee. The committee only needs 3 people. How many committees could be formed from the 7 volunteers?” This problem can be solved in a straightforward way by selecting three of the seven people to serve on the committee, yielding C(7,3) = 35. For students who are not familiar with binomial coefficients, a common response is to create an organized list that reflects a sum (seen in Student 3’s response below). A common incorrect solution would be to count arrangements (rather than sets) of people, which would be 7*6*5=210. This is incorrect because it counts arrangement of people within a committee, thus overcounting the total number of committees. Below, I compare and contrast three students’ written responses to this problem.
First, notice that Student 1’s response reflects the common incorrect answer, and we see that the student computed the product of 7*6*5. There is no attempt to check or verify the answer, nor does the student appear to consider smaller cases or to list potential outcomes. This response is not completely unreasonable, but it does not reflect particularly careful or precise work.
Student 2’s response provides evidence of an attempt at articulating outcomes and connecting those outcomes to a counting process. However, although the list seems organized in some ways, it is not constructed precisely enough to effectively yield a correct answer to the problem. The 7*3 =21 reflects the total number of listed outcomes, but it does not suggest a counting process that corresponds to the list. Student 2 is more organized than Student 1, but the approach still lacks care and precision necessary to answer the problem correctly.
In contrast to both of the other responses, Student 3’s work demonstrates a precise, systematic list of all of the outcomes. In this case, the careful mathematical work is seen in listing outcomes in a methodical and organized way. Specifically, the way in which the outcomes are listed actually reflects an overall structure (the 15+10+6+3+1) that helps to provide a convincing justification that no outcomes are missing or duplicated. Even more, the solution illuminates the recursive nature of the problem, and the level of detail the student included brings out more concepts and potentially more opportunity for generalization. For instance, one might be able to observe a pattern in the sums and generalize that the numbers of ways to choose three members from n is the sum of the first n – 2 triangular numbers. Student 3’s careful work thus affords opportunities for important and powerful mathematical connections.
What kinds of dispositions or experiences might lead a student to create a list like Student 3’s, and how can we develop those desirable traits? It might be the case that some students are predisposed to certain attitudes towards math, and this makes them more or less amenable to detailed work. Some students may have more stamina than others, and some may be more willing to engage in seemingly mundane activities to solve a problem. Regardless of existing predilections, I wonder if students can be convinced of the value of careful work. Perhaps by providing students at a variety of levels with opportunities in which they explicitly benefit from being precise, we could persuade students that such precision is worth their time.
Students should be given opportunities to see the value of engaging in careful and detailed practice – even if such work might initially seem unnecessary, boring, or overly elementary. Often, such additional effort – the patient, dedicated, and systematic approach that reflects a commitment to being careful and precise in one’s work – will pay off. We should put a concerted effort toward convincing students of this fact. Practically, this could look like giving students problems for which careful work reaps benefits – such as new mathematical insights discussed in Student 3’s work. Many problems have solutions that can be seen through a careful build-up of an argument through a series of smaller cases. Often, in order to develop these cases one needs to be organized, precise, and methodical not only in solving each case, but also in connecting smaller cases back to the original problem. In addition, many particular topics can lend themselves to careful and precise work. I have mentioned opportunities for precise work in counting problems, but I could see similar opportunities to highlight careful work in linear algebra, logic, and certainly more broadly in the development of proof.
The Common Core State Standards for mathematics (CCSSM) lists eight mathematical practices that students should adopt over time. Although the CCSSM present standards for students in K-12 mathematics classrooms, there is a need for precision at all levels of K-16 curriculum. One of these standards for mathematical practice is “Attend to Precision.” In the official description of this practice, the emphasis is primarily on communicating precisely, especially in using definitions, symbols, and units. I would argue that the kind of careful, detailed work I have described above is another potential way in which students can and should attend to precision. Indeed, mathematical problem solving involves a dialogue with oneself, and the written work on the page can be viewed as communication of ideas from the solver “back to” him or herself. From this perspective, work that is organized, carefully done, and precise could help formulate and solidify ideas for problem solvers, facilitating critical reasoning and successful problem solving.