We know that students struggle with abstract mathematical symbolism at all levels, starting in elementary school, and continuing through various levels in college. Students often work with symbols as though the goal were to move them around on the page in order to satisfy arbitrary rules. These are rules they have adopted from their prior instructional experiences without really understanding how, why, or when they work. Students may even believe that it is not possible to understand the rules that they use—they may view the procedures like allowed chess moves, something that experts know and which must be memorized, but that cannot be derived or explained on one’s own.
In our research, we have been interested in trying to understand what core knowledge is necessary for students to work with abstract mathematical symbolism, thinking in particular about ideas that are relevant across many levels of mathematics instruction, but focusing primarily on algebra since it is a core area where students first have to use more complex abstract symbolism. In order to better understand this, we have observed student work in classrooms, interviewed hundreds of students at all levels of college mathematics, from elementary algebra through linear algebra; we have also administered open-ended questions to over 800 students, and multiple-choice questions to thousands of students. We have tried to pay particular attention to both how students tend to conceptualize core ideas of mathematics, as well as how this might differ from how most mathematicians think about these concepts. In this we focus not on mathematical modeling, which has already been the focus of much research, but rather on the less examined area of how students work with abstract symbolic mathematics, used in computational tasks such as simplifying expressions and solving equations.
Abstract symbolic mathematics has a meaning all its own, different from, but equally important in comparison to other skills like modeling. If we want to understand how to best support students in using abstract mathematical symbols, one place to start is to ask ourselves how we can connect the ways that mathematicians use symbols to the way students use symbols, in order to give students equitable access to formal mathematics.
We have identified three core ideas as critical to using abstract mathematical symbolism productively:
1) Syntactic structure: How do students “read” the meaning of the syntax of expressions and equations (and other objects)?
2) Using properties: How do students justify their transformation steps (e.g., when simplifying)?
3) Equivalence: Do students have a well-defined notion of equivalence appropriate to the specific context, and do they connect it to the justification of their computational work?
For the purposes of this blog post, we will explore how just this third core idea might impact student understanding of simplification. We do this through the lens of one student’s work on the following algebra problem: