If 80 per cent of all mathematical applications involve estimation, rather than exact answers, why does primary mathematics place so much more emphasis on calculation, asks Mike Askew...
Leaving home in the morning, I decide I haven’t enough cash to see me through the day, so I’ll have to stop at the ATM. It’s not on the way to the station, so I choose to leave home 10 minutes earlier. I’m cooking supper and need to buy groceries on the way home, so draw out £30 on top of what I need for a ticket, coffees and lunch. Four friends are coming round, so I need to adapt that chicken recipe…
Our daily lives are filled with informal calculations to which we do not need to find exact answers: figuring out how much cash we need, deciding when to set off on a trip to arrive on time, figuring out quantities for a recipe when cooking for a different number of people, to name but three. And we often have to check the reasonableness of answers t o calculations that we work out exactly.
The measurements that we do in life are also framed by estimating. Is that bag of onions about a kilogram (I can’t be bothered to check on the scale)? Is that roughly 200 grams of butter? Will walking to the shop get me over my daily target of 10,000 steps? Just as we estimate to check the reasonableness of answers to calculations, we estimate to make sure our measurements are reasonable.
So ubiquitous is our use of estimating that we often estimate without being aware of what we are doing. The American researcher Robert Reys has studied extensively the development of number sense and concludes that over 80 per cent of all mathematical applications involve estimation, rather than exact answers. If that is the case, this raises questions about why primary mathematics places so much more emphasis on calculation than estimation.
One result of the greater emphasis on calculation is that many children come to see estimation as a lesser skill than computation. Some, when asked to estimate an answer, will actually calculate the exact answer and then take a bit off to provide an ‘estimate’. The irony of this is the research evidence shows that helping learners to develop good skills in estimation not only helps them check the reasonableness of their answers, it also fosters better understanding of place value and the nature of the mathematical operations (add, subtract, multiply and divide).
In mathematics lessons, children frequently are presented with situations where good estimation skills would help: about how big will be the product 594 x 322? Approximately where is 5/6 on a number line and how does that compare to 7/8? What volume of water will a jug hold? We could seize upon these opportunities to explore the nature, and skills of, estimation.
Why is more attention not given to estimation?
Part of the reason why estimation may be given scant attention in primary maths is that less research has been carried out into how skills in estimation develop than has research into other aspects of learning about number, particularly counting and calculating. This may be because of the breadth of tasks and situations that come under the blanket heading of estimating. Estimating how much squash to make up for the school disco, the answer to 1567 ÷ 80, or the height of a building have little in common – apart from approximate answers.
Estimation is also complicated by the fact that it often involves two distinct types of knowledge: knowledge of how the number system works and knowledge of the real world. For example, to estimate the height of a building, I may reason that, say, one floor is about twice my height. I’m about 20 cm short of two metres, so let’s say one floor is four metres, and there are 28 floors. Call that 30, so around 120 metres.
Benchmark quantities and numbers
As the example above shows, length can be estimated by using ‘benchmark’ referents. As well as one’s height, knowing that your hand span is around 20 cm, your foot length 30 cm and your arm span (length from the tips of your middle fingers standing with arms outstretched) is around 1 1/2 metres, all provide handy (pun intended) referents. Benchmarks for mass/weight and volume measurements that are helpful for learners to – metaphorically and literally – have to hand include:
> A pack of butter weighs 250 grams
> A bag of sugar weighs
> A teaspoon holds around 15 ml
> A glass holds around 200 ml
> A litre carton of milk
It is also helpful to benchmark fractions, decimals, and percentages. Having a good sense of the equivalence of 1/10, 10%, 0.1; 1/4, 0.25, 25%; and 1/2, 0.5, 50% facilitates working across different representations of fractional amounts.
Strategies for estimating answers to calculations
Choosing a strategy to find an approximate answer is not as simple as being taught to round numbers. Flexible estimating means choosing particular strategies for particular circumstances. For example, take a calculation like 36 x 52 (the exact product is 1872). The strategy most often taught is to round both numbers to the nearest multiple of ten, which in this case gives 40 x 50 and an estimate of 2000. But we could also round both numbers down to the nearest multiple of ten, giving 30 x 50 and an estimate of 1500, or both up to the next multiple of ten, resulting in 40 x 60, that’s 2400.
Which strategy to use is partly determined by why we want an approximate answer. If we want an estimate as close as possible to the precise answer, then the best strategy often is to round only one of the numbers: 36 x 50 yields 1800 (and easily mentally calculated by multiplying 36 by 100 and halving the answer), an estimate closer to 1872 than any of the others above. This works because, most of the time, the less rounding you do the closer the estimate will be to the exact value: 36 x 50 is closer to the exact answer than 40 x 52 as it involves less rounding (and 40 x 52 is going to be closer than 40 x 50). There are, however, exceptions. For example, 39 x 41 is best estimated by rounding both numbers to 40 x 40: the error introduced by rounding one number down helps compensate for the error introduced by rounding up the other number up.
The context of the estimating also matters. If I want to make sure I’ve ordered enough tiles to cover a given area, I want an estimate that will leave a few tiles over, rather than having to order more to fill in a gap.
And, perhaps most importantly, a goal in estimating is ease of calculation – estimating is primarily a mental skill, so ease of handling the numbers is paramount.
1 Early years:more than, less than
Set up a carousel of estimating activities that groups of children rotate around. For example, on one table have a (well sealed) 1 kg bag of sugar or flour and a collection of everyday objects: a book, a shoe, a bag of pebbles. Without using scales, children compare each item with the ‘referent’ 1 kg bag, deciding if they are heavier or lighter.
Another table could have a 500 ml shampoo bottle and a selection of larger and smaller plastic bottles (chosen so their shape does not make it immediately obvious that they could contain more or less). Area could be explored with a carpet or cork tile as the referent and a variety of 2D objects.
After each group has visited each table, assign a group to each table. They choose a measuring tool to check which objects are greater or less than the ‘referent’ objects. Everyone talks together about how they decided which objects were greater or less than the ‘referent’ object.
2 Middle years: what do you think?
Present the children with these two statements of methods of estimating.
Lois estimates the answer to 36 x 49 by rounding both numbers and multiplying. So Lois calculates 40 x 50. Manu estimates the answer to 36 x 49 by rounding one number. Manu calculates 36 x 50. Give the children a series of calculations and ask them to estimate the answers using both Lois’s and Manu’s method. For example:
27 x 9
19 x 44
4 x 99
53 x 48
Discuss with the class: which way to estimate do they think is easier. Why? Which way to estimate they think gives the better estimate?
3 Upper years: Fermi problems
Fermi problems are named after the Italian Nobel Prize winner, Enrico Fermi. He enjoyed playing with problems like How many railroad cars are there in the UK? or How many piano tuners are there in greater Manchester? Clearly such problems cannot be calculated precisely, but Fermi showed that by making reasonable assumptions, sensible estimates and using simple calculations, surprisingly accurate answers can be reached.
Upper primary pupils can get deeply involved in Fermi-type problems such as:
> How many trees are used each year to keep the school in paper?
> How many litres of water does the school use each week?
> Aliens are going to teleport the school away in 30 minutes. How far away could you get in that time? (The aliens have disabled all petrol-fueled vehicles).
Behaviour management: choosing the right words
How to be a singing school
Use the bottle-flipping craze to create good school behaviour, not bad