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Early Algebra: Research into its Nature, its Learning, its Teaching

Carolyn Kieran, JeongSuk Pang, Deborah Schifter, and Swee Fong Ng
Springer Open
Publication Date: 
Number of Pages: 
ICME-13 Topical Surveys
[Reviewed by
David A. Huckaby
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Several short topical surveys were published to summarize the program of the 13th International Congress on Mathematical Education, which took place in July of 2016. The following is a review of the survey on “Early Algebra,” a topic that encompasses the learning, the teaching, and the nature of algebraic thinking in young students, roughly aged 6 to 12 years.

Aside from a very brief introduction and conclusion, the survey consists of four parts. The first discusses the early research of early algebra, bringing the story into the early 2000s. The second part surveys more recent research and suggests implications for future work. The third part focuses on the nature and praxis of algebra in elementary classrooms, and the fourth part provides a neurocognitive perspective on early algebra.

The discussion of research up to the 2000s consists mostly in the reporting of attempts around the world to understand and foster algebraic thinking in elementary and early middle school classrooms. Studies tended to be isolated and often involved analyzing the effectiveness of a certain country’s typical approach or of a teacher’s or a researcher’s novel approach to introducing algebra and/or fostering algebraic thinking. For those of us not conversant with international trends in education, this section conveniently provides a quick glimpse of practices in various countries. In trying to form a synopsis of these early studies, it is observed that many of the various classroom approaches to early algebra involved students recognizing and articulating patterns and functional relationships, including generalizing and justifying their results.

More recent research has built upon the growing consensus that working with patterns and relations characterizes — or should characterize — early algebraic thinking, and that such thinking is best facilitated through activities that involve generalizing, representing, and justifying. Several recent studies have demonstrated that children can engage in fairly sophisticated algebraic thinking at younger ages than previously thought. Some studies have shown that algebraic thinking does not naturally develop through traditional arithmetic instruction; rather, instruction must intentionally foster algebraic thinking. One is reminded of the van Hiele levels of geometric thought through which students will advance only with active guidance. Attempts have been made to analyze and describe the mental processes involved in learning early algebra; perhaps some of the proposed schemas will be substantiated by future research.

The section on bringing early algebra into elementary classrooms is the most interesting, in part because it features sample problems and activities complete with snippets of classroom transcripts. In concert with the trend across all levels of mathematics education, group work and class-wide discussion are seen as integral to early algebra learning. Indeed, one subsection of this survey begins with these two sentences: “The goal of early algebra is to promote a way of thinking — the habit of looking for regularity, and articulating, testing, and proving rules or conjectures for an infinite class of numbers. This is achieved through classroom interaction around ideas, sometimes in pairs and small groups, but largely through class discussion in which students elaborate their own thinking and engage with their classmates’ ideas.” The second sentence in effect defines early algebra learning as necessarily and predominantly a social activity.

To properly lead these discussions, most teachers need professional development. Teachers must view mathematics not as a collection of recipes for calculating and solving problems but as a way of seeking and examining patterns. They also need to be alert to their students’ mathematical thinking so as to best direct the classroom discussion. Students’ metacognition is important: Reflecting on their own observations is what leads students to generalization, representation, and justification. Teachers should aim to foster such reflection.

The section “A Neurocognitive Perspective on Early Algebra” discusses the results of brain imagining studies and also aspects of cognition in algebra learning. Imaging results performed on adults show that the part of the brain responsible for attention is activated more when people are solving problems using a full-blown algebra approach — complete with letters representing quantities — than when using an approach that is based mostly on arithmetic. Other imaging results remind us that children’s brains work differently than those of adults in many ways, so that results from studies involving only adults should be applied to children with caution. Some reasons why algebra is cognitively demanding are discussed, such as the need to recognize algebraic expressions as legitimate answers and the need to understand the meaning of the equal sign and work with it in different contexts. One gets the impression that researchers are only scratching the surface here. Further research and synthesis are needed.

Encouraging such research is of course one of the goals of this survey. It is a quick read and provides a good feel for the state of affairs in early algebra research. There is plenty of exciting work to be done in this young field, from the testing of existing theories, to further theorizing, to synthesizing present and forthcoming ideas. Current and would-be researchers can take advantage of the extensive bibliography: 129 references along with pointers to four plenary papers that were presented at ICME13 and are suggested for further reading.

David A. Huckaby is a professor of mathematics at Angelo State University.

See the table of contents in the publisher's webpage.