By Elise Lockwood, Contributing Editor, Oregon State University.
When I teach classes for pre-service teachers, I typically have the students read and discuss a math education article about the teaching or learning of content they may eventually teach. This may include research articles (in journals such as Journal for Research in Mathematics Education, which typically report on research studies), or practitioner articles (in journals such as Mathematics Teacher, which offer practical insights without necessarily being rooted in rigorously conducted research).
Recently, however, I have also started to have students in more traditional postsecondary mathematics classes (not just those designed for pre-service teachers) read math education articles. Last term, for instance, after discussing counting problems in an advanced mathematics course, I had my students read an article by Batanero, Navarro-Pelayo, and Godino (1997) about effects of implicit combinatorial models on students’ solving of counting problems. Through such readings, my students can be exposed to research on students’ thinking about the very postsecondary content they are learning. I am always pleasantly surprised by the rich discussion such readings stimulate, and this made me reflect on the value of having students read such articles, even in their “pure” mathematics classes.
Both research and practitioner papers about math education can elicit valuable ideas and points of discussion from which math students can benefit. In this post, I make a case for three potential benefits of having students occasionally read math education articles in their math courses.
Math education papers can help students learn more about a particular concept
There are many math education researchers who focus on student thinking about specific mathematical concepts. This research tends to be qualitative in nature, allowing for the investigation of subtle mathematical details. There are a number of research methodologies that reflect this kind of work, including developing conceptual analyses (e.g., Thompson, 2008) conducting teaching experiments (e.g., Steffe & Thompson, 2000), or creating hypothetical learning trajectories (e.g., Simon & Tzur, 2004). While I will not outline the specifics of each methodology here, the point is that significant work is being undertaken to better understand students’ understanding of a variety of mathematical topics. I believe that having students read such articles at appropriate times could help them learn more about the mathematics they are studying.
As an example of the mathematical insight that can be gained from reading such papers, consider Swinyard and Larsen’s (2011) study in which they had students reinvent the formal definition of limit via a series of carefully chosen tasks. Ultimately, these researchers shared findings about how a pair of students thought about the formal definition, and they identified two central challenges that arose for students: “(a) students relied on an x-first perspective and were reluctant to employ a y-first perspective; and (b) students struggled to operationalize [that is, to clearly articulate] what it means to be infinitely close at a point” (p. 490). They then investigated ways in which students might handle these challenging ideas, providing in-depth discussion about details of the formal definition.
I contend that having students in an Advanced Calculus course take the time to read, unpack, and understand this paper would help them develop a more solid understanding of the formal definition of limit. They needn’t focus on the methodological details of a given study, but rather they can engage with the results and reflect on what those results might mean for their own mathematical understanding. There are countless similar examples in other domains, such as linear algebra (e.g., Wawro, 2014), abstract algebra (e.g., Cook, 2014; Larsen, 2009), calculus (Dorko & Weber, 2014; Oehrtman, 2009), discrete mathematics (e.g., Annin & Lai, 2010; Lockwood, 2013), proof (e.g., Weber & Alcock, 2004), and many, many more. The point is that the work that researchers have done to unpack deep conceptual issues may help students better understand subtleties about a concept they are learning.
Math education papers can help students think more about others’ thinking and learning processes, facilitating reflection on their own thinking and learning
Another benefit to reading carefully chosen math education articles is that students can think more about how they, and others, think. For many students, doing mathematics can be an isolating activity, and they might not naturally reflect on how others think about or approach a problem. There can be two extreme aspects of this phenomenon. For students for whom math comes intuitively, it may be easy for them not to think about others’ thinking at all. They may (even without realizing it) assume that theirs is the only and best way to approach a problem, and that everyone else probably thinks of the problem in the same way. Less confident students, on the other hand, may assume that their thinking about a concept (including any confusion they have) must be unique to them, and that everyone else understands the concept perfectly (this belief is particularly prevalent among female students – see David Bressoud’s November 1, 2014 Launching’s column).
As we know, though, there are many different ways of thinking about a given problem, and chances are good that many students in a class will have the same conceptions or misconceptions about a particular problem or idea. Many math education articles contain data and results that could make students more aware of others’ thinking. For example, strong students may be surprised and intrigued to see that, in fact, people genuinely struggle with concepts that they find trivial. Hopefully (through some well-facilitated discussion) they could become more empathetic with fellow students and recognize that there may be alternative approaches to a problem. For students who feel less confident, it may be encouraging and empowering to realize that other students also struggle or have the same questions that they have. It might be less isolating for them to identify potentially confusing issues, and to be able to face them head on.
As an example, I recently had students read Yopp, Burroughs, and Lindaman’s (2011) paper about one teacher’s understanding of the decimal equality .999…=1, and a number of them said that the teacher in the article reflected many of their own (incorrect) ways of thinking about decimal representations. I feel that this empowered some of them to realize that they were not alone in their ideas, and it also encouraged them to be more reflective about the unproductive notions that they had held.
Math education papers can help create a dynamic class environment
Finally, there are benefits to having students think explicitly about ideas related to the teaching and learning of mathematics and then to discuss them with others. I have had students say that they loved papers they have read, and I have also had students strongly disagree with papers they have read. In either case, I view their responses and their critiques as something positive that contributes to their overall mathematical development. I want students to be able to think critically about ideas, and by reading math education articles, they are invited to think hard enough about an idea to evaluate and critique it. Even an unpopular article can spur thinking and discussion that stands to benefit the students.
Even more, such passionate responses can help create a dynamic learning environment in which students feel free to share their ideas and opinions. By facilitating reflection on an article, we can introduce an extra dimension to the mathematics under consideration. The emphasis can shift away from questions of “What do I understand or not understand?” to questions of “What is hard about these concepts?” or “Why do we all struggle to learn this?” Such discussion can provide a counterweight to some of the more solitary and isolating aspects of doing mathematics.
Concluding thoughts
I am not suggesting that any math education research paper is appropriate for a given situation. Indeed, having students read math education articles requires some skilled facilitation of discussion, and it takes a certain level of mathematical maturity and buy-in for reading articles to be beneficial for students. For instance, I would not recommend that Calculus I students read Swinyard and Larsen’s paper about limit – they simply are not ready to consider the mathematical intricacies presented in the paper. However, that paper could be extremely useful for Advanced Calculus (or Introductory Analysis) students who have seen the formal definition and are in the process of thinking more deeply about it. (For an example of a paper that could be read by Calculus I students, see Trigueros & Jacobs (2008)).
I also acknowledge that engaging with math education research articles should be done with care, and we must not recklessly draw conclusions about mathematical or pedagogical ideas based on cursory readings of a few papers. The ideas discussed here are not meant to suggest an overhaul of existing classes but rather are meant to serve as a supplemental activity. An instructor who wants to explore this idea could experiment with incorporating one paper in a course at first, or maybe two.
To summarize, there is value in exposing math students to papers in mathematics education. Having students read carefully chosen papers in a math class has the potential to effectively enhance students’ mathematical knowledge, improve students’ understanding of others’ thought processes, and contribute to a more dynamic classroom environment.
References
Annin, S. A., & Lai, K. S. (2010). Common errors in counting problems. Mathematics Teacher, 103(6), 402-409.
Batanero, C., Navarro-Pelayo, V., & Godino, J. (1997). Effect of the implicit combinatorial model on combinatorial reasoning in secondary school pupils. Educational Studies in Mathematics, 32, 181-199.
Cook, J.P. (2014). The emergence of algebraic structure: Students come to understand units and zero-divisors. International Journal of Mathematical Education in Science and Technology, 45(3), 349-359.
Dorko, A., & Weber, E. (2014). Generalising calculus ideas from two dimensions to three: How multivariable calculus students think about domain and range. Research in Mathematics Education, 16(3), 269-287.
Larsen, S. (2009). Reinventing the concepts of group and isomorphism: The case of Jessica and Sandra. Journal of Mathematical Behavior, 28, 119-137.
Lockwood, E. (2013). A model of students’ combinatorial thinking. Journal of Mathematical Behavior, 32, 251-265.
Oehrtman, M. (2009). Collapsing dimensions, physical limitation, and other student metaphors for limit concepts. Journal for Research in Mathematics Education, 40(4), 396-426.
Simon, M. A. & Tzur, R. Explicating the role of mathematical tasks in conceptual learning: An elaboration of the hypothetical learning trajectory. Mathematical Thinking and Learning, 6(2), 91-104.
Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education (pp. 267-307). Hillsdale, NJ: Erlbaum.
Swinyard, C. & Larsen, S. (2012). Coming to understand the formal definition of limit: Insights gained from engaging students in reinvention. Journal for Research in Mathematics Education, 43(4), 465-493.
Thompson, P. W. (2008). Conceptual analysis of mathematical ideas: Some spadework at the foundations of mathematics education. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepulveda (Eds.), Proceedings of the Annual Meetings of the International Group for the Psychology of Mathematics Education, (Vol 1, pp. 45-64). Morelia, Mexico: PME.
Trigueros, M. & Jacobs, S. (2008). On developing a rich concept of variable. In M. Carlson & C. Rasmussen (Eds.), Making the Connection, (pp. 3-14). Washington, DC: MAA.
Wawro, M. (2014). Student reasoning about the invertible matrix theorem in linear algebra. ZDM The International Journal on Mathematics Education, 46(3), 389-406.
Weber, K. & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56, 209-234.
Yopp, D., A., Burroughs, E. A., & Lindaman, B. J. (2011). Why is it important for in-service elementary mathematics teachers to understand the equality .999 = 1? Journal of Mathematical Behavior, 30, 304-318.