I was prompted by a recent tweet to think more about exponentiation. The problem being posed was how to convince students that the laws of indices apply even when the exponent is not a whole positive number.
There is a deep issue lurking here, which is that the more abstract the mathematics, the less likely it is that a concrete, number-based ‘explanation’ is going to be available. Several possible reactions from students need to be considered:
- Superficial obedience to rules,
- ‘Holding onto handles’: be they concrete, visual or numerical,
- Looking for realistic applications,
- Questioning of everything,
- Generalising from limited examples.
Most crucially in this example, though, is the fact that a student will also need to consider whether what works for natural numbers might also work over an extended field.
Whilst it’s not always obvious how maths describes ‘the real world’, In the case of exponentiation it’s a pretty good tool for doing this – rather more useful than, say, quadratics. That’s because it deals with repeated scalings. Or, in other words, scalings of scalings.
Real-world data is replete with examples of exponential growth and decay: duckweed on a pond, dosage effect of medicines, the spread of diseases, population growth, and so on. These situations all offer cases in which exponentiation can be presented in realistic ways.
On a more basic level, when children share out food between more than two people, they frequently display basic exponentiation by applying sequences that involve halving, halving again, and so on until they have close-to-equal shares. This natural propensity has been called ‘splitting’ by Jere Confrey, a mathematics education researcher in the US, and she sees it as a basis for exponentiation.
Confrey found that when students compared arithmetical and geometric sequences, they ‘naturally’ began to talk about the multiplicative growth characteristics as repeated events of splitting. Repeated splitting can provide an image for the backwards reasoning of ‘how many and what sort of splits did I use to get to this number from the starting value?’
These questions can provide a way in to exponential functions and their inverses: logarithmic functions. Confrey’s discrete model of a tree diagram can be used to model situations for which repeated multiplication is appropriate (for example, 33 × 34 = 37 ), where you want to keep on multiplying. However, it’s not so useful for situations that involve splitting, such as non-integer indices.
So how might real contexts display the rules for manipulating indices? We need a situation that evolves continuously, so that the value of t can be a non-integer. I like duckweed on the surface of a pond or, more generally, population growth. Say duckweed on a pond increases in area by a scale factor of 3 every week, this can be modelled with the function
f(t) = a(3t)
where t is the number of weeks and a was the starting area. Even without any knowledge of maths, anyone will intuit that the area doesn’t instantaneously jump up by a scale factor of 3 between each given week – it will grow continuously throughout any given time period.
Suppose we are half way through a week, say t = 3.5 weeks, and the duckweed area at the time is 5 m2. Even though t is not a whole number, we know that one week later on, when t = 4.5 weeks, the area will be 3 times larger, or 3 × 5 m2. From a practical point of view, the rule for growth (in this case, trebling), still holds. Using index notation, this is:
So it looks as if the subtraction rule for indices, which works for natural numbers, will work here too.
Furthermore, if we want to know the length of time it would take for the duckweed to grow between given areas, say from 1 m2 to 6 m2, then we know that the time is something between one week (after which it would be 3 m2) and two weeks (after which it would be 9 m2), but the actual growth at any moment (gradient of the graph) depends on the area at that time. So the time interval we are looking for is not exactly half way between 1 and 2 weeks, even though 6 is half way between 3 and 9.
Understanding that might be tricky – to get away from linear, proportional, assumptions is important when we try to use contexts to enhance students’ understanding. You would need to take 1 m2 as a starting value, so that it represents 30, and then ask ‘what value of t makes 3t = 6?’ To get this value, you can either read it off a graph of f(t) =3t, or use the logarithm function for base 3: log36 = t. None of this looks nice at first, but using digital tools makes it possible and enhances experience of modelling.
You could argue with me and say “but you have assumed 1 m2 was the starting value when it wasn’t”. In that case, instead think of it this way: assume the duckweed started at m2 two weeks ago. Then 1 m2 has to be thought of as (32), and the question becomes ‘what value of t makes
(3t) = 6?’ This is a good moment to compare the two versions: the one that starts now and the one that started two weeks ago. This value of t can be found by trial and adjustment, or through a graphical approach.
The realistic/graphical approach to the concept of exponentiation does not give a smooth route into manipulations, but it does avoid simplistic assumptions. It also indicates that there are non-integer solutions to problems involving exponents, thus setting up the reasons why this matters. The rules for dealing with integer powers clearly need extension across the real numbers.
This argument, however, does not guarantee that the rules stay the same – many standard misconceptions arise from extending rules beyond their correct usage. On the other hand, developing and extending meaning by manipulating a robust notation system is an important feature of advanced mathematics.
On the third hand (!), we cannot just rush into adding and subtracting things and saying ‘these are the rules’: there doesn’t have to be, and cannot always be, a diagrammatic or numerical justification when we deal with abstract ideas. Fortunately, in this case, we can construct situations that bridge from natural numbers to rational numbers and show that the same manipulations that work for whole number indices MUST work also for rational indices to keep the mathematical sense.
For bridging examples, I look at the rule:
(am)n = amn
and use the rules I already know for multiplying natural numbers. Suppose mn = 1 and n = 2, then m must be . I also know that whatever am is, it has to be the square root of a , so an index of must indicate a square root if the notation system works.
Does this work within the notation system, using the rules we already have? Yes, it does. For example, would mean = 32 . So, focusing on the usual rules of number in the indices relates back to their use as indices. This reasoning extends to all unit fractions as values for m. It therefore also extends to all rational numbers, since they can be thought of as mn, where m is a unit fraction.
One question remains: in this notation, are we allowed to have this free choice of numbers for m and n? What if we banned fractions because we don’t know what they mean? I leave it to the reader to think about their response.
Since this is possibly the first place in the curriculum where students need to think about roots other than square and cube roots, it makes sense to do some thinking about what these mean with a variety of roots and base numbers. I have noticed that most textbooks only use 2, 10 or—at a pinch—smaller powers of 3, presumably because those are the powers people might recognise. So it is worth varying the type of question, and using numbers that shed some light on the relationships that are already familiar to students using simpler numbers. For example, Tony Gardiner provides exercises that are worth doing in Extension Mathematics: Gamma (Oxford University Press), on pages 138 and 154.
Does all this provide enough justification for all students to extend the natural number rules over rational numbers? Not in terms of diagrams or concrete models, but it does in terms of numbers and the way the notation works.
I can imagine some readers saying, “I haven’t got time to do all this, we just say here are the rules”, but exponentiation is a ubiquitous idea in contextual occurrences, which I think is worth spending considerable time on. It offers learners an experience in both modelling and in moving towards more abstract ideas.
Anne Watson has two maths degrees and a DPhil in Mathematics Education, and is a Fellow of the Institute for Mathematics and its Applications. Before this, she taught maths in challenging schools for thirteen years. She has published numerous books and articles for teachers, and has led seminars and run workshops on every continent.