Exploring errors in college mathematics courses.

Abstract

In a mathematics course for prospective elementary teachers, one university mathematics professor discussed his students' errors in a way that valued yet extended their mathematical thinking. He used errors to help students reconceptualize mathematics problems, explore contradictions, and pursue alternative strategies.

Introduction

Deciding how to deal with student errors is often difficult. Many teachers worry that discussing errors will make the student who made the error feel embarrassed, while others argue that discussing errors could introduce confusion into an otherwise cohesive discussion. This paper provides examples of how one university instructor navigated a discussion of errors in a way that supported his students' growth in mathematical understanding while preserving the value in his students' reasoning.

One team of researchers found that exploring errors in particular ways (see a, b, and c, below) was one of four features of mathematical discourse that made a difference in students' growth in understanding of fractious (Kazemi & Stipek, 2001, Stipek et al., 1998). In classrooms with the greatest growth in students' understanding, teachers used errors as a way for students to: a) reconceptualize problems; b) explore contradictions; and c) pursue alternative strategies.

For example, a fifth grade teacher created opportunities for her students to explore contradictions by verifying whether solutions were correct (Kazemi & Stipek, 2001). She responded to a student's mistake, which was thinking that 6/8 and 1 1/8 were both correct solutions to a problem, by encouraging all students to explain whether and why 6/8 and 1 1/8 were or were not equivalent. By encouraging the students in the class to reason about the two solutions, the teacher provided an opportunity for the students to engage in mathematical inquiry. Mistakes were used as opportunities for students to clarify and extend their thinking.

In contrast, Kazemi & Stipek (2001) found that in the classrooms with lesser growth in student understanding, errors were either ignored until a satisfactory solution was offered, or the teacher rather than the students provided the reasons that strategies were mathematically incorrect. In so doing, these teachers reduced the opportunities for students to reason mathematically. In high- growth classrooms, mistakes were not only viewed as a normal part of learning, but teachers pressed students to critically analyze both appropriate and flawed student reasoning.

This paper serves to contribute to the knowledge base about discourse, and in particular, the discussion of errors, by focusing on a university mathematics professor's class; most research on discourse has focused on K-12 classrooms (e.g., Nathan & Knuth, 2003; McClain & Cobb, 2001, Sherin et al., 2000, Kazemi, 1998, Yackel & Cobb, 1996). Although there are principles about discourse that university mathematics professors can garner by reading about elementary teachers' classrooms, the fact that university professors teach prospective elementary teachers adds a dimension to the art and science of teaching that merits its own examination.

The goal of this paper is to provide a lens for university mathematics professors to examine their own classroom discourse, and in particular, how they respond to prospective elementary teachers' errors. Excerpts of discussions in one university mathematics professor's course are provided below. The excerpts are followed by comments about how the professor's pedagogical actions when exploring errors are comparable to those of the elementary teachers whose students showed the greatest growth in mathematical understanding. The university mathematics professor, Dr. DiMarco (a pseudonym, abbreviated DM in the excerpts), was teaching a first-of-four semester-long mathematics course for prospective elementary teachers. The course focused on problem solving, number sense, and multiplicative reasoning.

Squareness Problem Discussion Exploring Contradictions

Squareness Problem

A new housing subdivision offers rectangular lots of three different sizes:

75 feet by 114 feet, 455 feet by 508 feet, and 185 feet by 245 feet.

If you were to view these lots from above, which would appear most square? Which would appear least square? Explain your answers.

Prior to the discussion, students had solved the Squareness problem for homework. One goal in providing this problem was for students to see ratio as an appropriate measure for characterizing the relative squareness of rectangular housing lots. However, many students incorrectly adopted an additive strategy for approaching this problem (that is, they subtracted the width from the length and searched for the smallest difference between the dimensions). Consider how you might respond to a student who has used this additive strategy. The excerpt below provides an example of how Dr. DiMarco explored a student's strategy of reasoning additively about the problem.

DM: (Reads the Squareness Problem to the class.) ... The question is: which one is the most square? Which is the least square? How would you know? Does anyone want to just share? (Looks to Ann, who has raised her hand) Yes?

Ann: I kind of just did it where I subtracted the largest from the smallest one like we did A minus B. Like I took A as 114 minus 75.

DM: Okay, so let's look ... (DM writes the dimensions of the lots and the values of the differences on the board--39, 53, 60) So this is the difference. So what does that tell you? What do you think that's telling you?

Ann: Well, I kind of did it where whichever one would be like close, like the same, because a square has all the same lengths on all the sides, so I kind of just said well whichever one has the least difference.

DM. Okay, so you were looking for the smallest difference, all right? Does everyone understand what she did? She looked at the squares. She got the difference. And the least difference you would think would be the most square. Okay, why would you think that? I mean it's good reasoning. It's the idea that if it really were a square, what would the difference be? If you really had a square piece of property and you subtracted, what would the difference be?

Sts: Zero.

DM. Zero. So, you realize if it's a real square and you get the difference it's zero. So the thought is well, if you get a number that's close to zero, then that means that the sides must be pretty close together.

Dr. DiMarco did not respond to this incorrect response by either ignoring it or explaining why it is wrong, as might have occurred in a low-growth classroom. Instead, he examined the features of the response that were reasonable and involved some sense-making on the student's part. He did not focus on the error but rather on the kernels of reasoning that supported her strategy. For example, he highlighted the fact that Ann made use of what she knew to be true about squares (that all sides are equal) to inform her choice of strategy. In an interview with Dr. DiMarco, he indicated that he consciously attempted to support her reasoning for two reasons. First, he noted that the student had indeed used some reasoning that she could build on and he thus wanted to help explicate the features of her strategy that could help support her (and her likeminded classmates') growth in understanding. He also indicated that prospective elementary teachers often do not like math and do not feel confident in their use of it; so, he sought to provide a learning environment that was supportive of their efforts (M. DiMarco, personal communication: interview, February 17, 2003). This discussion continued, and the next segment included discussion of an additional strategy that Dr. DiMarco placed on the board. Eli suggested examining the ratios of the two (unequal) sides, and checking to see which ratio was closest to one. Dr. DiMarco wrote down the ratios of the sides of the rectangular lots (.66, .90, and .76) in a column next to the differences of the sides. He then began the interactions below.

DM: Let's just stop and talk about the two different methods; there are two nice methods. What is the difference between the two methods that we're talking about right now?

Bob: Additive.

Sue: Multiplicative.

DM: Additive and multiplicative. One of them is doing an additive kind of comparison and the other one is doing a multiplicative comparison, and that's important. We have two numbers. We can look at them additively and we can look at them multiplicatively.... Does it make sense to do both of these? Will they give us the same information? Is one better than the other? What do you think?

Dr. DiMarco invited the students to compare the two strategies. He used the differences in the strategies (and the fact that they provided different answers to the question of which one was more square--the additive strategy indicated that the first lot was most square, while the mnltiplicative strategy indicated that the second lot was most square) to not only support mathematically appropriate ways of reasoning about the problem but to push them to mathematically justify their ways of thinking. Because the two strategies yielded different solutions to the problem, the students were invited to mathematically explore the contradictions in the solutions. The incorrect answer was again not sidestepped and ignored; rather it became an integral component of the mathematical discourse. Without the discussion of the additive approach to the problem, the opportunities for rich mathematical inquiry would have been diminished. Similar to the teachers in high-growth classrooms in Kazemi's and Stipek's study, Dr. DiMarco used errors as a way to explore a contradiction. The two strategies provided a forum for pushing students to mathematically justify the strategy that was most appropriate in this context.

Thread Problem Discussion Reconceptualizing Problems and Pursuing Alternative Strategies

Thread Problem (assigned as homework) Anh uses 3/8 times as many spools of thread in one month as Jack. Anh uses 2 spools of thread. How many spools does Jack use? Who uses more thread? How much more?

In class, a student asked a question about how to solve the problem, and Dr. DiMarco initiated a discussion about it. Ron shared an algebraic solution to the problem while Amy shared a drawing she used to solve the problem. Midway through the discussion of her solution, Amy stumbled on her explanation. Dr. DiMarco asked if another student who used a drawing similar to Amy's could describe Amy's strategy with the class. Eva volunteered.

Eva: (points to a rectangle that Amy had drawn earlier) This is how many spools Jack uses. It's broken up into eight pieces because [Anh] uses three-eighths of what he uses (Eva draws another rectangle and cuts it into eight equal pieces. She then draws a rectangle that is three-eighths as large as the first rectangle and cuts it into three equal pieces, see figure one.) See issue website http://rapidintellect.com/AEQweb/sum2004.htm

DM: Does everyone understand what's going on here?

Eva: This is just representing the ratio. [Anh] uses this much (points to the smaller rectangle) when [Jack] uses this much (points to the larger rectangle).

DM: A misinterpretation could be, so [Jack is] using 8 spools, is that right?

Eva: Yeah, but it's not really. This is just like to represent the different ratios.

DM: This is another thing you might try to do. What could be a misconception someone might have? One misconception is that you drew this diagram so I see 8 spools and 3 spools. Is that right? Is that what is trying to be conveyed there?

Sis: No.

DM: No, it's not 8 spools and 3 spools, so why eight? What's this? What's important here?

In this discussion, Dr. DiMarco anticipated a difficulty prospective elementary teachers might have in interpreting the drawings and he thus raised this potential difficulty in an effort to help the prospective elementary teachers explore and understand alternative strategies. In so doing, he wanted to help them recognize that, as teachers, they would need to attend to their elementary students' approaches to problems. With his actions, Dr. DiMarco indicated that it was not appropriate for prospective elementary teachers to gloss over a strategy that they did not understand. In other words, they needed to understand the strategy so that they could not only better understand the mathematics but also learn to make sense of others' strategies, a skill that would be important when they became teachers. In this discussion, his exploration of potential errors indicates that he pressed his students to understand an alternative strategy, which, in turn, could help them to reconceptualize the problem. Eva continued to describe her strategy, using the information that Anh had 2 spools of thread, to determine the number of spools of thread Jack had. She then used her drawing to show that every one and a half sections (see shaded part of small rectangle in figure 2) was equivalent to one spool of thread, and thus Jack had 5 spools of thread and some extra. She struggled but eventually described how she knew that the extra (see shaded part of large rectangle in figure 2) represented one-third of a spool. See issue website http://rapidintellect.com/AEQweb/sum2004.htm

DM: Okay, now I am going to say that it makes sense to the people that are following. I am sure there are some people who are sitting there saying, "I don't have a clue what's going on because I still don't know what a spool is. I don't know what a box is." (laughter).

Don: I'm confused. (Points to the larger rectangle) Is that whole section you broke into eight parts--is that one spool? Or is that eight [spools]? ... Because when [Eva and Amy] first said it, I thought that was one spool broken into eight pieces.

Eva: No.

Don: That's why I said, what the heck is going on?

DM: Again, this is why it's important [to have these discussions] and it's good to have a problem that's causing all of this [puzzlement].... You want to analyze what causes problems in situations like this. These are problems some of you have and [they are also problems your] students will have....

As Dr. DiMarco anticipated, at least one student struggled to make sense of the drawings and how they related to the problem. Although we do not know whether Don would have shared his misunderstandings with the class without Dr. DiMarco's prompts, I hypothesize that at the very least, the prompts provided a safe way for Don to share. This session occurred near the beginning of the semester, and classroom norms were still becoming established. Dr. DiMarco used this opportunity to explicate the norm of acknowledging confusion and working through it. By initiating conversation about the potential confusion students might have, Dr. DiMarco was able to press his students to stretch beyond their own ways of reasoning. Additionally, he provided a way to discuss the strategy so that students could reconceptualize the problem and pursue an alternative strategy.

Summary

Examining data on student learning was beyond the scope of this study, but previous research supports the productiveness of teachers' discourse practices, similar to those described in this study, on students' mathematics learning (Thompson & Thompson, 1996, Yackel & Cobb, 1996, Stipek et al, 1998). In particular, Stipek et al. (1998) found that exploring errors to help students reconceptualize problems, explore contradictions, and pursue alternative strategies promotes a high degree of conceptual understanding. The examples included in this paper advance the discussion about exploring errors because the episodes take place in a university, rather than K-12, mathematics class. Thus, discussions surrounding the two problems were shared to provide details about the ways in which one professor explored errors to push his students' thinking while valuing their ideas. Because this professor was teaching prospective elementary teachers, his comments during discussions reflected his desire for them to be able to understand other prospective teachers' reasoning because as teachers they should be able to understand their elementary students' reasoning. His actions provided a model that his prospective elementary teachers could strive to emulate. Dr. DiMarco discussed errors in productive ways by anticipating the errors that students might make, explicitly stating the mathematically sound aspects of students' errors, and providing students opportunities to explore and discuss the errors. With these features he was able to value their thinking, orchestrate significant mathematical discussions, and provide opportunities for the students to extend their thinking.

References

Kazemi, E., and Stipek, D. (2001) Promoting conceptual thinking in four upper-elementary mathematics classrooms. Elementary School Journal, 102 (1), 59-80.

Kazemi, E. (1998) Discourse that promotes conceptual understanding. Teaching Children Mathematics 4, 410-414.

McClain, K., & Cobb, P. (2001) An analysis of development of sociomathematical norms in one first-grade classroom. Journal for Research in Mathematics Education, 32 (3), 236-66.

Nathan, M J.; & Knuth, E. J. (2003) A Study of Whole Classroom Mathematical Discourse and Teacher Change, Cognition and Instruction, 21 (2), 175-207.

Sherin, M. G, Louis, D., & Mendez, E. (2000), Students' Building on One Another's Mathematical Ideas, Mathematics Teaching in the Middle School, 6 186-190.

Stipek, D., Salmon, J., Givvin, K., Kazemi, E., Saxe, G., & MacGyvers, V. (1998) The value (and convergence) of practices suggested by motivation researchers and mathematics education reformers, Journal for Research in Mathematics Education, 29, 465-488.

Thompson, A. G. & Thompson, P. W. (1996) Talking about rates conceptually, part II: Mathematical knowledge for teaching, Journal for Research in Mathematics Education 27, (1), 2-24.

Yackel, E., & Cobb, P. (1996) Sociomathematical norms, argumentation, and autonomy in mathematics, Journal for Research in Mathematics Education, 27, 458-477.

Lisa Clement, San Diego State University 

Lisa Clement is an Assistant Professor of Mathematics Education and a co-director of a graduate program specializing in Mathematics Education.