Symbol sense is an essential part of mathematical literacy. It is the understanding that undergirds effective symbol manipulation, and perhaps more basically the ability to interpret and create algebraic expressions. Symbol sense, like number sense and operation sense, is not learned so much through listening to a teacher. Rather, it grows as one gets practice generalizing numerical relationships, translating functional relationships into formulas, and reading algebra. The latter is the topic of this post.
Successfully reading algebra does require some clarity on the conventions that are accepted in the international institution of mathematics. Because these are conventions, they can only be transmitted by explicit instruction. (This is unlike, for example, the properties of proportional relationships, which can be explored and discussed in different contexts. Students can “discover” some of the properties, as well as be introduced to them by the teacher.)
Here are some examples of conventions in the writing of algebraic expressions, and therefore in reading them.
1. The Three Meanings of Minus
In front of a number, minus means negative. For example, -2.
In front of any expression, minus means the opposite of. For example:
-(y + 1)
In between two expressions, minus means subtract. For example:
2 – 3
2 – (y + 1)
Of course, for this to be meaningful, students must understand the concepts expressed by the symbol. (For example, that the opposite of a quantity is that which you add to it to get zero.) In other words, it is pointless to try to have students memorize these ideas if they don’t have a need for them.
A good question to discuss is: “Is -x negative?”
On some calculators, there are different keys for minus as “opposite” and minus as “subtract”. That is inconvenient, perhaps, but it does provide an opportunity to have this discussion.
On the meanings of minus, see my Algebra Lab Gear: Basic Algebra, Lesson 3.
2. Sum or Product?
What is the result of this calculation: 2 + 3 · 5? If you just read from left to right, you get 25. If you start with the multiplication, you get 17. Discussing examples of this type shows that a convention is needed. And, of course, the convention is to multiply before you add, so the “correct” answer is 17. Nix the Tricks, an excellent booklet by Tina Cardone and others, has a good discussion of this and recommends GEMA (grouping, exponentiation, multiplication, addition) instead of PEMDAS. Including division as part of multiplication, and subtraction as part of addition makes a lot of sense, and deserves a conversation with your students.
Once this has been established, a good exercise is to show expressions to your students, and ask: is this a sum or a product? For example:
x + 5 → sum
5x → product
4(x + 8) → product
4x + 8 → sum
… and so on, with increasingly complicated expressions. This is an essential prerequisite to talking about distributing and factoring, which really have no meaning without this foundation.
On “sum or product?”, distributing, and factoring, see my Algebra Lab Gear: Algebra 1, Lesson 3.
Reading algebra is not just an issue for beginners. For example, older students sometimes get confused when reading algebraic expressions that include radicals or absolute values. Again, it is essential to discuss the meaning along with the “grammar”. (See for example this activity: GeoGebra | TI-eighty-something.)
Part of what I’m saying in this post is that it’s a mistake to put all your eggs in the “discovery” basket. Discovery, more or less guided, is excellent in many cases, but when it comes to arbitrary conventions, you will need to just tell students what they are. The pedagogical questions are: when is this telling appropriate? what sort of practice is interesting to help learn this particular convention? I have often found that it is helpful to make connections with manipulative or electronic environments (as you see in some of the above links), but you will have to figure out what will work in your classroom and curriculum.
One last point: do not overgeneralize! For example, a mathematician who is clueless about math education once wrote about my Algebra: Themes, Tools, Concepts: “Some facts students are led to find are important, such as commutative, associative and distributive laws. But I felt they would simply waste time by the lengthy explanations and explorations. These laws are like ‘a red signal light means you have to stop.’ Nothing more.” That is ridiculous. There is much interesting work to be done to help students understand these laws, and treating them as arbitrary conventions would be doing a huge disservice to them. (Read my reply to this man here.)