Wednesday, 19 November 2014

To use or not to use calculators: a false dichotomy

The issue of whether to use calculators when teaching students maths seems to worry a lot of people.  The argument for is often presented as the use of calculators is part of the modern world and, by having access to answers to calculations rapidly, students are more likely to be able to understand relationships.  The argument against is that they reduce students' calculation skills which can impact negatively on understanding and also that they over-rely on them and are unable to spot errors as they believe the number on calculator must always be the correct answer.

I think that this is a false dichotomy that is caused by a misunderstanding of what mathematics is and the low quality of many of the questions that students are asked to do in mathematics classes.

A question

Here's a fairly typical question: what is 36 × 9?

Do you think students should do this with or without a calculator?

My answer is (to both questions) - I don't care!

This is an absolutely pointless question.  What does it matter what 36 times 9 is?  In the absence of any realistic or mathematical context it is meaningless and there is no reason to calculate this product.

The only mathematics worth doing is mathematics that helps us understand the relationship between numbers/shapes or helps us solve a realistic problem.

Better questions:

How much would a worker earning £9 per hour earn for a 36 hour week?

A rectangle has sides length 36 units and 9 units.  What is the area of the rectangle and will it be a square number?

Both of these questions require the product of 36 and 9; however, in both cases the important skill here is to be able to identify that product is required, once a student has identified that the answer will require a product I would hope that they had a range of strategies available to them and that they can use the most appropriate one.


Strategy 1: using a calculator.

In the 21st century most people (at least in the UK) have a powerful calculator to hand most of the time (in the guise of a mobile phone).  It should be part of students' mathematical education to learn this.

Strategy 2: pencil and paper methods

These are often useful; however, I would argue for developing alternative strategies for processes.  I also think it is important to emphasise methods that aid relational understanding over those that are merely efficient.  For example, of the two presented here I think that box method gives a clearer representation of the 2-dimensional nature of multiplication.

Strategy 3:  in your head

If I wanted to multiply 36 by 9 I would probably do 36 times 10 to get 360 and then subtract 36 to get 324.  There are many other strategies that can be used for this but I think it is important to develop a playful nature with numbers through these kind of mental calculations.

Which is the best strategy?

The three different strategies presented here will be useful in different situations - it would depend on the context.  Once a student has correctly identified that a product is needed I would be happy with them using any of these three but I think students should be encouraged to develop their skills in all of them.

Developing skills in using these strategies

For all of these strategies it is important that students develop their skills in using them and, with any skill, the best way of doing this is through practice.  This practice can either be in the form of drills or play.  The problem with much of the mathematical activity that students are asked to do is that the practice of particular skills, such as multiplying a two digit number by a one digit number, becomes an end in itself as opposed to a useful tool to have in one's mathematical toolkit to solve problems.  Over-emphasis on practising performing mathematical operations is boring whether they are on a calculator, pencil and paper or in your head.

If the process of multiplying two numbers by hand is seen as an end in itself then obviously using technology to do this could be considered "cheating".   I would suggest instead that the goal of the mathematics taught to students should always be explicitly about solving realistic problems or understanding mathematical relationships.  If this is the case then where there is a need to develop and practise strategies for performing processes it will be natural to consider both calculator and non-calculator strategies.

In short - if you are asking whether students should be using a calculator or not it's not that you're asking the wrong question it's that you're asking the wrong questions.

Friday, 14 November 2014

Video: Integrating technology into the teaching and learning of mathematics

This a video featuring Charlie Stripp and me talking about integrating technology into the teaching and learning of mathematics, recorded at this year's MEI Conference.

Friday, 7 November 2014

How to Break Excel

Last weekend was the annual MathsJam Conference.  If you haven’t been (or even if you have) I highly recommend it.  For more details see:

The format is talks of maximum length 5 minutes on anything you find interesting.  This year I gave a talk on “How to Break Excel”.

How to Break Excel

A commonly occurring “error” in Excel happens when you type 1 into cell A1, =A1-0.1 into cell A2 and then drag this down to cell A11.

This problems occurs due to the way Excel stores numbers: it uses floating point arithmetic with 1 bit for the sign, 53 bits for the mantissa and 10 for the exponent.  This means that the number we think of as 0.1 in base 10 is actually stored in binary as:

Similarly 0.9 in base 10 is stored in binary as

This means that every time you use 0.1 in base 10 Excel is actually using:

At each stage in the subtraction Excel rounds to 53 significant binary figures.  There are two places where this introduces an error: when subtracting 0.1 from 0.8 there’s an error in the 53rd binary place and when subtracting 0.1 from 0.4 there’s an error in the 55th binary place.  This results in a total error of 2-53 + 2-55 = 1.38778×10-16

Fractional Powers of Negative Numbers

Another error occurs when you enter =(-8)^(2/3) into Excel .  It gives the result #NUM.

I think this is happening because it is rounding 2/3 in binary to:

As a fraction this is:

The denominator of this fraction is 253, which is even.  Consequently this requires finding an even root of a negative number which isn't real!

But, if you enter =(-8)^(1/3) into Excel it does give the result -2.

I’m not sure why, but I think that a negative number to the power one over an odd number has been hard-coded in as a special case but a negative number to the power of any other number over an odd number hasn't.

Friday, 24 October 2014

“Any teacher that can be replaced by a computer should be”

There’s been a lot of talk on Twitter this week about an app called PhotoMath.  It scans mathematics live and then gives an “answer” to the question.  This is a screenshot of it:

Some comments on Twitter seem to think it’s cheating whilst others think it’s the future (although to be honest most seem to mirror my experience of struggling to get it to read anything sufficiently accurately to evaluate or solve it).

Any question that can be answered by a computer should be

Arthur C Clarke once said “Any teacher that can be replaced by a computer should be”. It’s a great quote and I’d like to offer a variation on this:
Any question that can be answered by a computer should be.

The whole point of mathematics is to investigate the relationships between mathematical objects or solve real problems (I’ll skirt round the definition of “real” here!).  There are many processes that are useful in doing this but where they can be reduced to algorithms (such as solving linear equations, square rooting or dividing) there is no inherent value in performing these algorithms manually as opposed to outsourcing them to technology.

Whilst this app might not be perfect there are many (free) CAS tools that will solve equations and evaluate expressions that are typed in to them.  It doesn't worry me that a computer can solve a linear equation and that a student might “cheat” on their homework.  What does worry me is the excessive focus of many maths questions on getting students to learn algorithmic processes – technology presents an opportunity to refocus the curriculum on the important skills of making connections within the subject and solving problems.