Maths teachers, what’s your favourite topic to teach?
A couple of years ago I ran a series of polls on Twitter in which hundreds of specialists answered this exact question. It will probably come as no surprise to you that indices finished in the top four, along with surds, trigonometry and quadratics.
I’m not sure I’ve ever met a maths teacher who doesn’t love teaching indices. It’s a delightful topic. On the surface it seems so straightforward: indices are simply a concise, efficient space saver.
Who would want to write 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 when you could write 210? Even the most reluctant mathematician can’t argue with that.
It wasn’t until the 17th century that mathematicians started using raised numerals to represent powers. In 1637, index notation as we now know it was popularised by Descartes. He wrote: “ … and aa, or a2, in order to multiply a by itself; and a3, in order to multiply it once more by a, and thus to infinity ; …”
We now teach our students to use this notation fluently, as maths teachers have done for centuries.
Index, exponent, power
Students first see index notation in Year 5 when they are introduced to square and cube numbers.
In their Key Stage 2 SATs they may be asked questions such as “Work out 52 + 10” and “Write down a square number between 15 and 20”.
When our students first work with algebra at Key Stage 3, we pick up this topic directly from where the Key Stage 2 curriculum left off. Moving from 32 to a2 and then onto higher powers, we generalise – teaching our students to interpret the meaning of an where n is any positive integer.
At this point, where clear definitions and consistent use of vocabulary are absolutely vital, I have to admit that I’ve been making a mistake for the entire time I’ve been a maths teacher.
See, I’ve been teaching my students that the raised number is called a power, index or exponent. Recently I found out that’s not actually correct. Thankfully, I also discovered that the majority of maths teachers say the same thing I do.
In fact, maths textbooks have been calling it a power for decades. It’s so commonplace that it’s become totally acceptable to refer to the raised number as a power. It’s not a power though. The whole thing – a base with an index – is a power.
I’m going to make a concerted effort to take more care over this vocabulary in future!
One of the things that we like about teaching indices is that it’s relatively easy to explain the underlying concepts, perhaps until we reach fractional indices where it gets a bit more abstract. But the rest of the topic – index notation, index laws and even negative indices – is fairly straightforward to explain.
It also leads somewhere very important: the study of calculus at A level requires a sound understanding of indices.
However, the apparent simplicity of the underlying concepts means that we risk rushing through the content. And this is where we go wrong. We underestimate the complexity of the notation to novice learners.
Think about it: no tens are used in the calculation of 210. No wonder it’s confusing! Our students are used to being given clear symbolic instructions (x for multiply, + for add and so on), but here there are two numbers and no operation symbol. It’s new territory.
The big misconception
A failure to develop fluency in using index notation from the start leads to perhaps one of the most fundamental and frustrating misconceptions we see in secondary mathematics: the dreaded 32 = 6 and its algebraic partner in crime, a x a = 2a.
If you see your students making mistakes like this, you’ll be pleased to know that it’s not just you! This misconception has existed forever – maths textbooks from the late 1800s warn gravely that “the beginner must be careful to distinguish between coefficient and index”.
So the question is, how do we avoid it? Part of the problem is that index notation is often covered as a bit of an aside, popping up briefly in the middle of our introduction to algebra in Year 7. It would be far better to treat it as a topic in its own right.
Starting with some lessons on square numbers (try the delightful Rainbow Squares from MathPickle), we should then spend some time looking in depth at index notation – where it came from, why we use it, how we say it out loud, what it means, how it varies, where the common misconceptions arise, and how we can use it to solve problems.
The aim, amongst other things, is to develop absolute fluency in using index notation from as early as possible.
There are loads of great resources we can use in these lessons. You can download a new set of activities at teachwire.net/tsindexnotation. Let’s tackle that big misconception head on!
10 ingredients for teaching index notation in depth
Talk about the history, the background, the etymology, the cross-topic and cross-subject links. Tell a story or perhaps discuss a real life application.
Be consistent and crystal clear about definitions, notation, and how we say things out loud. Don’t be afraid to use repetition and chanting.
Extensive examples and non examples leave no room for confusion. Address misconceptions head on.
Isolation of skills
Don’t cover more than one new thing at a time.
Lots of it!
Think carefully about the questions you use to assess your students’ understanding.
This is a great opportunity to revisit negative number arithmetic. In fact, most topics are a great opportunity to revisit negative number arithmetic
Depth not speed
Don’t rush stuff you think is simple or obvious. It’s no good for anyone to move on too quickly.
Don’t shy away from using calculators in Year 7. They work really well here.
Once they’re confident working with powers, your students will enjoy exploring problems, patterns