What Makes Algebra Difficult is the Equal Sign – Part 1 of x

Algebra is one of the most researched topics in mathematics education. And most of these studies are about students understanding of algebraic concepts, particularly equations and the 24th letter of the English alphabet. With the volume of studies, one wonders why until now algebra many learners still have difficulty with the subject. I read a remark somewhere comparing the search for effective means of teaching/learning algebra similar to that of the quest for the holy grail.

I’m not about to offer in this post a way of making learning algebra easier. I have not found it myself. But let me offer an explanation why algebra is illusive to many first time learners of the subject. I adhere to the belief that once you know where the problem is, you have solved half of it. Sometimes, it could turn out of course that the solution of the other half of the problem is learning to live with it.

Consider the following familiar symbols we write in our blackboard. I will label each string of symbols, A and B.

What do the math symbols in A and B mean? How does A differ from B? How are they similar?

Let’s start with the ‘visual’ similarity. They both have an equal sign. They both show equality. Are they both equations? The statement 12+4x=4(3+x) is an equivalence. It means that the right hand side is a transformation of the left hand side. This transformation is called factoring, using the division operation. The transformation from right to left is called getting the product and you do this by multiplication.

Would you consider statement B an equivalence? It certainly not. You can test this in two ways. One, try to think of an transformation you can do. Two, you can test a few values of x for both sides of the equality sign to check if it will generate equal values. You will find that only x=-5.5 will yield the same result. This means that statement B is not an equivalence but a conditional equation. They are only true for certain values of x. This is what we commonly call equation.

I have shown that we have used the ‘=’ sign in two ways: to denote an equivalence and an equation. How important are the distinctions between the two? Is it so much of a big deal? Are they really that different? Let’s fast forward the lesson and say you are now dealing with function (some curriculum starts with function). Let f:x?12+4x, g:x?4(3+x), and h:x?2x+1. Their graphs are show below. Note that functions f and g coincide at all points while function h intersect them at one point only.

The graphical representation clearly show how different statements A and B are and that the ‘=’ sign denotes two different things here. Now, if you notice the graphs above, the function notation also use the ‘=’ sign. Is it use the same as  in A and B? Try transforming. Try solving. It’s different isn’t it?  In function notation such as f(x) = 12 + 4x, ‘=’ is used to denote a label or name for the function that maps x to (12 + 4×0. This meaning should be very clear to students. Studies have shown that learners misinterprets f(x) as f times x and tried to solve for x in the equation.

In 13 – 5 =____, what does ‘=’ equal sign mean? Ask any primary school learner and they would tell you it means ‘take way’ or ‘do the operation’. You may be interested to read What Pupils Think About the Equal Sign and Teaching the Meaning of Equal Sign.

I have presented four meanings of ‘=’ in mathematics: equivalence, equation, to denote a name for a function, and to do the operation. My point is that one of the factors that make algebra difficult is the multiple meaning of symbols used. We also use of the word equation to everything with ‘=’. Students need to be able to discern the meaning of these in the context to which they are used if we want our learners to make sense of and do algebra.

In Part 2, I talk about the multiple meanings of the letter symbols as source of students difficulties in algebra. You may also want to read Making Sense of Equivalent Equations and Expressions and Equations, Equations, Equations. If you want some references for Algebra teaching you can try Fostering Algebraic Thinking.