## Calculators? For or Against?

2 July 2018

The EEF’s recent guidance report Improving Mathematics at Key Stage 2 and 3 makes no bones about the benefits of calculator use in the classroom: “The evidence suggests that using a calculator does not generally harm students’ mental or pencil-and-paper calculation skills. In fact, studies have shown using a calculator can have positive impacts, not only on mental calculation skills, but also on problem-solving and attitudes towards maths”.

The same studies (Ellington, 2003 and Hembree and Dessart, 1986) also showed that calculator use can have positive effects on students’ attitudes to mathematics.

Some teachers will instinctively disagree; as we learn about the importance of rehearsing the retrieval of key facts to build fluency it’s easy to jump to the conclusion that missing any opportunities for mental arithmetic practice would be a bad thing.

I’ve long been in favour of exploiting the investigational potential of technology in mathematics (if you haven’t experimented with Desmos, for example, I think you’re missing a great trick) but I still worry when I see my A-Level or GCSE students reaching for a calculator to perform a calculation that they could easily do in their heads. Yet these students are neither lazy nor weak in their arithmetic skills: they are adopting a strategy which is sound for several reasons.

Firstly they are conserving the precious resources of their short-term memories by subcontracting the job of calculation to a microchip while they focus on higher-order concerns and, secondly, they are minimising the risk of getting a crucial examination question wrong by making a simple misstep in calculation.

There is, however, a valid concern that students who reach unthinkingly for a calculator could be harming their mathematical development.

There’s a nice thing that happens when you look at division by 7 expressed as a decimal: 1/7=0.142857142857142857142857142857…

It is no surprise to anyone who has ever played with “bus-stop” division (or any other sensible written method) that any fraction will eventually recur in its decimal form. However, sevenths do something that not all prime divisors do. Try calculating 27 as a decimal. There are plenty of ways you could do this – add together two lots of 0.142857142857142857142857142857…, or set up a bus-stop with 7.00000000… as the dividend:

2 |7.0000000000

You could even pick up a calculator, but that has shortcomings due to the inevitable rounding in a finite calculator screen, and also because you aren’t involved in the mechanics of the calculation so the reason for the recurrent pattern is not made explicit.

I’ll make it easy for you:

2/7=0.285714285714285714285714285714…

Did you see what happened? Here’s another:

3/7=0.428571428571428571428571428571…

And another:

4/7=0.571428571428571428571428571428…

And so on. Always the same digits. Always the same order.

You may well think that the occasions in life when it is useful to be able to recite the decimal form of 4/7 at the drop of a hat are few and far between. I seem to find quite a few but it could fairly be said that I need to get out more. What is interesting about this example is that it prompts another question: does this trick only work for sevenths, or would it be true for other numbers? Maybe it’s true for all prime numbers?

Here’s where grabbing a calculator, or typing the calculation into google (which is more than happy to double up as a calculator) comes in handy. Quickly we see that it doesn’t quite work for 13ths, 17ths etc. and a question that could be pursued in any Key Stage 2,3,4 or 5 classroom is: “Why?”. I’ll leave that question with you to puzzle on, but I think you’ll find that a calculator does not help you get to the answer.

So careful use of calculators and encouraging students to self-regulate their use of them can reveal interesting patterns which might aid problem solving, conserve working memory and lead to more positive attitudes to mathematics.

*References:*

Ellington, A. J. (2003). A meta-analysis of the effects of calculators on students’ achievement and attitude levels in precollege mathematics classes. Journal for Research in Mathematics Education, 34, 433-463.

Hembree, R., & Dessart, D. J. (1986). Effects of hand-held calculators in precollege mathematics education: A meta-analysis. Journal for research in mathematics education, 17(2), 83-99.

*Posted on 2 July 2018*

*Posted in: Latest Research Evidence*