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Move Smoothly from Whole Numbers to Decimals with a Grounding in the Quantities Behind the Numbers

Help children get to the point with these mathematical activities…

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Writing in the book Enhancing Primary Mathematics Teaching, Malcolm Swan makes the following observation: ‘Children should never be regarded as ‘blank slates’. When they first encounter a new idea, they need time to assimilate it and relate it to concepts they already know. They build on their own conceptual framework.’

Malcolm and his colleagues at Nottingham University have done much research into the impact of pupils’ existing conceptual frameworks on learning, including the challenges pupils face in the move from working with whole numbers to decimal numbers, particularly as they sometimes erroneously apply generalisations that apply to whole numbers but do not transfer to decimals.

A typical erroneous reasoning is longer-is-larger, based on the generalisation, correct for whole numbers, that the more digits in a number, the larger it is. Comparing, say, 11,102 and 9,998, we know that 11,102 has to be larger, irrespective of the actual size of each number or the digits in them – any five-digit whole number is always going to be bigger than any four-digit one.

The logic that pupils then apply to decimals is that the decimal point acts as some sort separator of two whole numbers, and ‘longest-is-largest’ must then still apply – 0.235 must be larger than 0.5 because it has more digits.

Contradictory as it might sound, other pupils can hold the exact opposite view, that longer-is-smaller – the more decimal places a number has, the smaller it must be. A logic behind this is that 0.235 is smaller than 0.56 because the former has ‘thousandths’ in it, and thousandths being very small must make that number automatically smaller than one that does not contain thousandths.

Still others come to believe that shorter-is-bigger: 0.5 must be greater than 0.72 because 0.5 is in tenths and 0.72 is in hundredths and tenths are bigger than hundredths.

Insights into understandings

Given that these, and other alternative understandings, are easy to acquire but difficult to change, do national test results indicate anything about pupils’ understanding of decimals? Compare two addition calculations from the 2016 arithmetic paper. Adding large whole numbers, a question posed:

89,994 + 7,643 =

An impressive 100% of all pupils taking the test attempted the question and 90% got the correct answer.

For adding decimals, the same paper had:

15.98 + 26.314 =

99% attempted this, with 88% giving correct answers – near enough to the results for whole numbers to suggest that understanding of decimals may be close to matching understanding whole numbers. That’s good news.

A couple of reasoning paper questions look at the understanding of ordering. One question asked pupils to imagine someone putting these numbers onto a number line:

511 499 502 555 455

Asked which number is closest to 500, 94% answered correctly (with 100% attempting the question). Identifying the furthest from 500, all the test-taking pupil answered the question, with 84% getting it correct. (Why furthest should be more difficult than closest is puzzling, but it may be that having written down 499, pupils continued to think ‘on that side’ of 500, thus choosing 455.)

Ordering decimals meant writing the following from smallest to largest:

0.78 0.607 5.6 0.098 4.003
Again, it is encouraging that all the pupils attempted the question, but now the success rate was down to 70%. Not a terrible result, but it does indicate that around a third of pupils were not secure in ordering decimals and, I suspect, answered the question on the basis of the types of misconceptions raised above. (Incidentally, research indicates that had the decimals in the question all been given to two decimal places then the success rate would have been much higher, so this, for me, is a good diagnostic question.)

As Malcolm points out, much of the difficulty with decimals arises from not recognising ‘the ten-ness’ of a decimal, possibly as a result of decimals being linked to fractions (which bring their own set of misapplied conceptions) rather than treating them as an extension of the place-value system. So here are some examples of tasks that might help pupils deepen their understanding of decimals and the ‘ten-ness’ of them. As the activities are suitable for being adapted for different ages, I’ve indicated how each activity could be done with whole numbers and then how to develop it into decimals.

Four counters

Watch a video of this idea here. Each pupil (or pair of pupils) needs a hundreds, tens and ones board, and four counters.

Putting the four counters on the board creates different numbers, with a blank space being recorded as a zero. For example, 202 and 121 look like this:

Ask the pupils to make and record the largest possible number and the smallest (400 and 4). Challenge them to try and make at least ten numbers between these two limits.

When pupils have created a set of numbers, ask them to draw a blank number line from 0 to 400 and place all their numbers on it.

Initially learners may only put the numbers on the number line in order, without attending to where they are in relation to each other. For example, a pupil may put 4, 13, 310 and 400 on the line thus:

The conversation to have is about how the difference between 4 and 13 compares with the difference between 310 and 400, and what this means for their position on the number line.

The task is easily adapted to be easier or harder: change the number of places – tens and ones only, or include thousands – or change the number of counters. It can also be extended. If pupils work with two counters to find all possible numbers, then three, then four, can they predict and check the number of possible answers for five counters?

Taking this into decimals, initially, the place value board might be extended only to include tenths. Before finding ten numbers with one place of decimals, I would still get pupils to find ten whole numbers as above.

The reason for this is to be able to have a conversation about what is the same and what is different between the two sets of answers. For example, 400, 310, 13 and 4, compared with 40.0, 31.0, 1.3 and 0.4. The key idea here is that the order of each set of numbers is the same, and that the first set are all ten times larger than the second set.

Taking this into two places of decimals, some pupils may find it helpful to record all the numbers to two places, for example, 1.3 as 1.30. A key idea to discuss now is which digit in, say, 1.04, 0.13 and 0.03 has the most effect on the size of the number: irrespective of the number of decimal places, it is the first non-zero digit to the left.

Can you make…?

Now each pupil needs a hundreds, tens and ones board, and five counters.

Present a series of challenges for the class to make by putting all five counters on their board. For example, the largest even number, the smallest even number, the largest multiple of five, the smallest multiple of five, the largest two-digit number and the smallest two-digit number.

Challenge pupils to make up three more like this. The challenges are easily extended into decimals as follows: the largest number with two decimal places, the smallest with one decimal place, the largest decimal number with three decimal places, the largest decimal number with hundredths but no tenths, the number nearest to two, a number that rounds to 1.5.

When might you find…?

This is a whole-class discussion activity asking pupils to think about when they might actually encounter numbers in real life. Put the numbers up on the board one at a time and discuss the situations in which that number might occur in an everyday (or imaginary) situation. Ask pupils to jot down three different circumstances for each number before discussing them with the class.

What real-life situation might these numbers describe: 10, 100, 1000. Repeat this with the following decimals: 1.10, 1.1, 2.5, 0.004

Smart maths

The inclusion of an arithmetic test at the end of primary schooling no doubt reinforces the popularly held view that the main aim of work in number is about getting the answers to calculations. But it is important to make sure that the ability to calculate is underpinned by a good sense of quantities behind the numbers involved and what a calculation might represent.

A strong sense of quantity, scale and place value cannot and should not be presumed on the basis of getting correct answers to calculations. After all, the calculator on my phone can give me right answers but has no sense of what that calculation might be representing – and surely we want our pupils to be smarter than a smart phone?

Swan’s influence

At the time of writing this I’ve just heard of the death of Malcolm Swan, professor of mathematics at Nottingham University. Primary school teachers may not have directly come across Malcolm’s work as much of it was directed at secondary mathematics, although his reach did extend into primary. Even if you haven’t heard of him directly, it’s possible that your teaching has been influenced by his work. For example, if you’ve ever asked pupils to decide whether a statement is always, sometimes or never true, or given them a card sorting activity, then the activities can probably be traced back to Malcolm’s work. And my work has certainly been hugely influenced by his.


Mike Askew is adjunct professor of education at Monash University, Melbourne. A former primary teacher, he now researches, speaks and writes on teaching and learning mathematics. You can find him at mikeaskew.net or on Twitter at @mikeaskew26.

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