Presentation: Immigrant Students and Sociomathematical Norms

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Algebraic Thinking in the Early Grades: What Is It?

Author: Carolyn Kieran.

Source: Kieran, C. (2004). Algebraic thinking in the early grades: What is it. The Mathematics Educator, 8(1), 139-151.

The author, Carolyn Kieran, firstly shows the difference between arithmetic thinking and algebraic thinking. She points out students focus on calculating rather than a representation of relations in the shift from arithmetic to algebra. For example, students interpret an equal sign as “a separator between the problem and the solution,” or “a left-to-right directional signal” when computing. (p. 140) She suggests the adjustment is required to develop the way of thinking in algebra with focusing on relations, both representing and solving a problem, doing/undoing, both number and letter, meaning of the equal sign etc.

The author invites us to think about the difference and commonalities of the descriptions of algebraic thinking among several curriculums. While she acknowledges the differences of them, she claims two similarities: an emphasis on “the importance of relationship between quantities”, and a stress on “generalization, justification, problem solving, modeling, and noticing structure.” (p.147)

According to the author, algebraic activities can be categorized into three types: generational, transformational, and global meta-level. Generational activities are making the expressions and equations which contain much of the meaning-building for the objects of algebra. Transformational activities are manipulating activities with keeping equivalence such as operations. Global meta-level activities are “for which algebra is used as a tool but which are not exclusive to algebra” (p.142) such as problem solving, and involve general mathematical processes and activities. These are important especially for the meaning-building in generational activities. The author addresses that the idea that the development of algebraic thinking at the elementary level is the development of these mathematical way of thinking in Korean curriculum, is crucial for the above algebraic activities. She finally claims the global meta-level activities of algebra which involve both for meaning-building and developing-way-of-thinking, can lead mathematics educators to have a vision of algebraic thinking in the elementary level that guides adaptably their students to algebraic activities at the later grades.

I often hear the obstacles that students experience when the transition from arithmetic to algebra, and have experienced my students facing the similar problem when they go to a middle school from an elementary school. One of my students who was not good at mathematics was struggling to cope with equation. He often wrote (added) the operation result of the left side on the right side in equation, and kept trying to solve the equation problem. Therefore, the meaning of an equal sign for him was obviously the same one as the above example in this paper.

The Japanese curriculum does not categorize mathematics into arithmetic, algebra, geometry, and statistics. In fact, I have never seriously thought what algebra is before. Additionally, there are no explicit descriptions about algebra in the mathematics curriculum in Japan while it contains many contents of algebra. I do not think Japanese mathematics teachers exactly understand what algebra is and its features. However, interestingly, Japanese students tend to get higher scores in mathematics in several international academic performance surveys. Thus, my personal questions to myself is whether Japanese mathematics teachers need to understand what algebra and its feature is, to improve their approach (this is not a question for you unless you want to study math education in Japan!).

Questions:

I think there might be other obstacles to transitions in mathematics when students go on to a middle school from an elementary school, to a high school from a middle school, and to a university from a high school. Have you ever faced any other difficulties for your students (or yourself) during the above periods?