Mathematics for Teaching Teaching algebra – it pays to start early

## Teaching algebra – it pays to start early

I believe in early algebraization. I have posted a few articles in this blog on ways it can be taught in the early grades. Check out for example  Teaching Algebraic Thinking Without the x’s. All the lessons in fact that I post here whether it is a number or geometry or pre-algebra lesson always aim at developing students’ algebraic thinking. What do research say about early algebraization? How do can we integrate it in the grades without necessarily adding new mathematics content?

“Traditionally, most school mathematics curricula separate the study of arithmetic and algebra—arithmetic being the primary focus of elementary school mathematics and algebra the primary focus of middle and high school mathematics. There is a growing consensus, however, that this separation makes it more difficult for students to learn algebra in the later grades (Kieran 2007). Moreover, based on recent research on learning, there are many obvious and widely accepted reasons for developing algebraic ideas in the earlier grades (Cai and Knuth 2005). The field has gradually reached consensus that students can learn and should be exposed to algebraic ideas as they develop the computational proficiency emphasized in arithmetic. In addition, it is agreed that the means for developing algebraic ideas in earlier grades is not to simply push the traditional secondary school algebra curriculum down into the elementary school mathematics curriculum. Rather, developing algebraic ideas in the earlier grades requires fundamentally reforming how arithmetic should be viewed and taught as well as a better understanding of the various factors that make the transition from arithmetic to algebra difficult for students.

The transition from arithmetic to algebra is difficult for many students, even for those students who are quite proficient in arithmetic, as it often requires them to think in very different ways (Kieran 2007; Kilpatrick et al. 2001). Kieran, for example, suggested the following shifts from thinking arithmetically to thinking algebraically:

1. A focus on relations and not merely on the calculation of a numerical answer;
2. A focus on operations as well as their inverses, and on the related idea of doing/undoing;
3. A focus on both representing and solving a problem rather than on merely solving it;
4. A focus on both numbers and letters, rather than on numbers alone; and
5. A refocusing of the meaning of the equal sign from a signifier to calculate to a symbol that denotes an equivalence relationship between quantities.
These five shifts certainly fall within the domain of arithmetic, yet, they also represent a movement toward developing ideas fundamental to the study of algebra. Thus, in this view, the boundary between arithmetic and algebra is not as distinct as often is believed to be the case.
What is algebraic thinking in earlier grades then? Algebraic thinking in earlier grades should go beyond mastery of arithmetic and computational fluency to attend to the deeper underlying structure of mathematics. The development of algebraic thinking in the earlier grades requires the development of particular ways of thinking, including analyzing relationships between quantities, noticing structure, studying change, generalizing, problem solving, modeling, justifying, proving, and predicting. That is, early algebra learning develops not only new tools to understand mathematical relationships, but also new habits of mind.”

The foregoing paragraphs were from the book Early Algebraization edited by Jinfa Cai and Eric Knuth. The book is a must read for all those doing or intending to do research about teaching algebra in the elementary grades. Educators and textbook writers should also find a wealth of ideas on how algebra can be taught and integrated in the early years. Of course it would be a great read for teachers.  The book is rather expensive but if you have the money, why not? Here are some section titles:
• Functional thinking as a route in algebra in the elementary grades
• Developing algebraic thinking in the early grades: Lessons from China and Singapore
• Developing algebraic thinking in the context of arithmetic
• Algebraic thinking with and without algebraic representation: A pathway to learning
• Year 2 to 6 students’ ability to generalize: Models, representations, and theory for teaching and learning
• Middle school students’ understanding of core algebraic concepts: equivalence & variable”

The following books also provide excellent materials for developing algebraic thinking.