Literature on development of algebraic thinking

Question

I'm developing a course that focuses on the transistion from arithmetic to algebraic thinking, particularly in grades 5-8. We will do this through focus on the common core. I'm also putting together a collection of suggested readings from the math education literature. I would be interested to hear your suggestions for suggested readings.

Answer 1

Early algebra research necessarily deals with the development of algebraic reasoning and questions like "what is algebra" and "what counts as algebraic thinking and reasoning?" And my own readings on early algebra have helped me to focus on what about algebraic thinking are students developing, apart from the manipulation of symbols. This is where, I think, some of the early algebra research may have more general interest, since some of the authors deal with what students are capable of that is connected to algebra well before they are doing what we are familiar with in an Algebra 1 course.

A book collecting some different research on Early Algebra is:

Kaput, J. J., Carraher, D. W., & Blanton, M. L. (2007). Algebra in the early grades.

There is a review of it in JRME (Chazan & Edwards, 2010), and someone is sharing the review here as a PDF (in case you don't have access).

From the review:

In addition to these direct assaults on the rhetorical challenge posed by early algebra, one of the strengths of the volume is the richness of the presentations aimed at helping readers appreciate how early algebra is not the sort of activity that a reader might associate with a high school Algebra 1 course. In particular, the seven chapters in Part II offer theoretical arguments for children’s capacity to learn algebra early and empirical studies of children’s algebraic reasoning in a variety of different elementary-level classroom and curricular contexts. These chapters provide illustrations of what early algebra curriculum and instruction look like, as well as existence proofs of children’s capacity to do this work. (p. 204)

Even though you're not talking about algebrafying early mathematics, and depending on what sort of things you want students to consider and even argue about, this book (along with other early algebra research) may provide perspectives that spark thoughtful discussions and reflections.

Answer 2

In case you haven't already seen these, I'll suggest articles by Luis Radford and Jean Schmittau. Both are influenced by Vygotsky, but take quite different approaches to algebra. Most of Radford's articles are available as pdfs on his website.

Radford, L. (2010). Algebraic thinking from a cultural semiotic perspective. Research in Mathematics Education, 12(1), 1-19.

pdf at http://www.luisradford.ca/pub/22_RME2010Algebraicthinkingfromaculturalsemioticperspective.pdf

Abstract: In this article, I introduce a typology of forms of algebraic thinking. In the first part, I argue that the form and generality of algebraic thinking are characterised by the mathematical problem at hand and the embodied and other semiotic resources that are mobilised to tackle the problem in analytic ways. My claim is based not only on semiotic considerations but also on new theories of cognition that stress the fundamental role of the context, the body and the senses in the way in which we come to know. In the second part, I present some concrete examples from a longitudinal classroom research study through which the typology of forms of algebraic thinking is illustrated.

Schmittau, J. & Morris, A. (2004). The development of algebra in Davydov’s elementary curriculum, The Mathematics Educator.

Also available as a pdf at http://math.nie.edu.sg/ame/matheduc/tme/tmeV8_1/Schmittau.pdf

Abstract: A comparison of the development of algebra in Davydov’s elementary mathematics curriculum with the approach to algebra advocated by the National Council of Teachers of Mathematics in the US reveals striking differences. Rather than developing algebra as a generalization of number, Davydov’s curriculum develops algebraic structure from the relationships between quantities such as length, area, volume, and weight. The arithmetic of the real numbers follows as a concrete application of these algebraic generalizations. The instructional approach, while similar to constructivist teaching methodology, emanates from a very different theoretical perspective, namely, the findings of Vygotsky and Luria that cognitive development is enabled by overcoming obstacles for which previous methods of solution prove inadequate. In a study in which the entire three-year elementary curriculum of Davydov was implemented in a US school setting, children using the curriculum developed the ability to solve algebraic problems normally not encountered until the secondary level in the US.

UPDATE: I’d like to add a comment on the takeaway from the Schmittau and Morris article. The Davydov curriculum is very unconventional, and unlikely to be widely adopted in the United States. However, there are important insights that we could gain from it. I’ll highlight two of them. First, Schmittau points out in this and other articles that traditional instruction does not make as much use of what Vygotsky calls psychological tools as Davydov’s curriculum. Psychological tools are graphics, tables, language and other ways of making ideas concrete. Psychological tools turn thoughts into objects that we can reflect upon, manipulate and consciously apply when appropriate.

Second, Davydov’s curriculum develops a thorough understanding of concepts. When I first read the article, I thought, “Wow, they spend an excessive amount of time on one topic.” The students work on different aspects of a concept in a concrete way. Now I ask, ”Is this a good example of what we call rigorous instruction? Is this what the opposite of a ‘mile wide and inch deep’ curriculum looks like?” Concepts are developed deeply, in a logical progression. The zone of proximal development is moved forward so other higher level concepts can be taught. And it works. At first, students in the Davydov curriculum seem to be behind students in our traditional curriculum, as the Davydov curriculum builds a solid foundation. But they overtake students in traditional classes, seemingly performing at levels that we may consider to be developmentally impossible. Vygostsky says, “Learning leads development.” The secret may be a deep understanding of concepts that prepares students to move to the next level.

Answer 3

Although these two NCTM books cover K-12 (there are items specifically directed at middle school level) they have ideas related to how to develop algebraic thinking: The Ideas of Algebra, K12, 1988 Yearbook, A. Coxford and A. Shulte, editors, and Developing Mathematical Reasoning in Grades K-12, 1999 Yearbook, Lee Stiff and Francis Curcio, editors.

Answer 4

I'd suggest literature on students' understanding of the equals sign, e.g., 1. "Concepts Associated with the Equality Symbol" by Kieran. http://link.springer.com/article/10.1007/BF00311062 2.A Longitudinal Examination of Middle School Students' Understanding of the Equal Sign and Equivalent Equations, Alibali et al. http://www.tandfonline.com/doi/abs/10.1080/10986060701360902#.VK3Rx4rF_dc 3. "From an operational to a relational conception of the equal sign," Molina http://www.researchgate.net/publication/46593005_From_an_operational_to_a_relational_conception_of_the_equal_sign._Thirds_graders_developing_algebraic_thinking