Teaching Mathematical Conventions
by Melissa Braaten
You’ve probably heard (or will hear very soon) about the “shift” towards increased rigor in the new College and Career Readiness Standards for Adult Education (CCRSAE). In the context of the math portion of the CCRSAE, the word rigor has a specific definition: increased rigor involves equal emphasis on developing conceptual understanding, procedural fluency, and application. Here I want to focus on an important part of developing procedural fluency: comfort and flexibility with mathematical notation.
Learning to read and write in the symbolic language of math is very much like learning a new language with an entirely new alphabet: not only are the concepts unfamiliar, but the symbols and syntax are full of unknown meaning and nuance. For example, the ubiquitous parentheses that appear as students begin algebra not only take on new meaning in a math context, but can mean several different things, based on their syntax, or placement, in a mathematical expression. Students might encounter
where parentheses are merely indicating multiplication, or they may read
2 + (5 + 7)
where no multiplication is involved, and instead the parentheses are affecting the order of operations. A very slight alteration to the expression above gives us
2(5 + 7)
in which our parentheses are now both grouping and indicating multiplication at the same time.
And (this is the one that really drives my students crazy), they may encounter
4 + (-5)
in which the parentheses do exactly nothing, besides visually separate a negative number.
This is just an example to demonstrate how one of the most commonly used math symbols (especially in algebra and higher math) is riddled with nuance and can present a barrier to understanding, even if students conceptually understand the material.
For this reason, it is critical to help adult students become comfortable and fluent with mathematical notation, and to familiarize them with conventional ways of writing “in the language of math.”
Notation Must Be Explicitly Taught
One of the criticisms of “traditional” models of math education is that they focused too much on explicit instruction in procedural fluency, while neglecting the type of exploration that allowed students to build conceptual understanding. All of this is quite valid. Ideally, algorithms should make sense after a solid conceptual understanding is built. Notation, however, is arbitrary. There is no way to make sense of why a vertical curve is used to indicate multiplication: it is an agreed upon convention, just like the shape of phonetic letters. There is no way to reason from the shape of a letter to its sound—it has to be taught. Similarly, fluency in mathematical notation works best when the symbols, the syntax, and their meaning are explicitly taught and practiced over time. This means teaching and re-teaching, practicing different forms of notation, encouraging students to represent their mathematical expressions in various conventional ways. If a student offers an expression like the following
it might mean asking whether there are other correct ways to represent the same information.
Affirm All Notation that Makes Sense (But Explain Conventions Too)
Just like all other languages, the language of mathematics is about communication. Students who are learning but are not yet fluent in the conventions of this new language will often produce expressions that are understandable and technically correct, but perhaps not conventional. For example, a student may learn that when you multiply a number and a variable, no symbol is required to indicate multiplication – the number and letter are simply put side by side. I find that students commonly understand this notational rule and may write, for example,
s4 + 5
When I see this, I always affirm that they have a correct understanding of the way constants and variables are multiplied, and that their expression makes sense. In addition, I point out that the convention (and I explain this word) is to put the number first
4s + 5
Other unconventional but technically correct syntax I see a lot include expressions like
(4 + 2)5
1x + 2
In each case, I make sure to affirm the correctness of the student’s understanding, but also point out that there is a more conventional way of expressing the information. If possible, I also explain why the convention exists (most conventions in mathematical notation help to avoid ambiguity – for example, s4 could easily be confused with s4).
Teaching and reinforcing mathematical notation and conventions is an important way to ensure that students can apply their mathematical knowledge to a variety of applications. In addition, I have found that it plays a powerful role in reducing math anxiety: when the notation is familiar, students are not “scared off” before they even begin to understand the problem. When they understand what is being communicated in the language of math, they can then engage their problem solving and conceptual knowledge to solve what is before them.
Melissa Braaten is an adult education instructor, as well as the technology coordinator and assessment coordinator, at Project Hope in Roxbury, MA. Melissa has taught ASE and pre-ASE math and reading, as well as ABE writing, computer skills, and health classes.