What do you think of this story of a mature third-year student on one of our BSc chemistry degree courses who said: "I can't re-arrange equations. I've got this far by remembering examples from lectures and textbooks but I know it's not going to get me through my final year."? She had scoured books to find examples of every possible arrangement of any equation that had been used and then tried to remember them all.
Her parents had moved a lot during her education and some fundamental pieces were missing from her maths. Over the next five minutes, I explained that subtraction was the opposite of addition and division was the opposite of multiplication. This was the missing piece. We then looked at how an equation must give the same number on each side (that's what the equals sign means).
I then gave her an equation and asked her to rearrange it. Once she got the answer, she looked at me in disbelief and said: "But it can't be that simple. I've been running from this for 20 years!"
The next day, I overheard her with a group of my younger students. "It's easy, look," she was saying. "If you want to get rid of the plus five on this side, since minus is the opposite of plus, just subtract five; but you have to do it to both sides because what you do to one side you have to do to the other, otherwise it won't balance."
The best teacher is one who has only just discovered the truth for themselves.
University lecturer (name and address supplied) A
Your story confirms that the most effective teachers are those who continue learning themselves.
There are two fundamental concepts to understand about equations. The first is that the equals sign means that the expressions on each side of it are equal in value. The second is that whatever we do to the expressions on one side must be done to the other, otherwise the two sides will not be of equal value.
For convenience, let us refer to the left hand side as LHS and the right hand side as RHS. So, whatever we do to the LHS, we must also do to the RHS. It can be helpful to think of the LHS and RHS as two pans on a balance which must be kept level whatever we do to them.
There are many valid operations we can apply on both sides of the equals sign. The ones we choose are determined simply by whether they will help us reach the answer.
You mention "opposites" and this is exactly what is happening mathematically when we apply operations and their inverses. Equations involving one variable, always called x, can be thought of as a collection of operations (or functions) applied to that variable so "solving the equation" can be viewed as the process of undoing the effect of those operations. "Undoing the effect" means finding the inverse of the operation seen in the equation.
For example, in solving 5x - 7 = 8, we need to undo the effect of applying - 7 to 5x and this is achieved by applying the inverse of - 7 (which is +7) to both sides.
First, decide which side is for letters and which for numbers.
5x-7 = 8 (largest x on LHS)
5x-7+7 = 8+7 (-7+7=0)
5x = 15
5xV5 = 15V5 (5V5=1)
x = 3
This idea of inverse operations is fundamental to solving equations.
With algebra it is so easy to get the wrong idea at an early stage. I think that both at the start and later on, students need to spend a lot of time verbalising what operations are involved and why.
As you say, the best way to gain a good understanding of any topic is to have to teach it to someone else. Until this happens in every maths classroom, many people will continue to think of algebra and maths as an incomprehensible mumbo-jumbo. I wonder whether attitudes have really changed since the 17th century when the mathematician John Dee was feared as a magician, was imprisoned and had his house ransacked.
Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses. Email your questions to Mathagony Aunt at firstname.lastname@example.orgOr write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX