While researching the history of division recently, I came across this old photograph of a Hospital School in Vienna, 1921. If you look carefully you see the students attempting to calculate 6975 ÷ 235.2. Obviously about 100 years ago we did not have calculators available to students to help them work out their division. Notice also that there is no real-life link, just the algorism waiting to be solved. The students have pen and paper ready and they also seem to be listening to the student at the front who may be trying to explain to the rest of the class how he will work it out.
My worry is that this sort of experience still happens today. We no longer need to calculate such an example using pen and paper as we have a calculator on our phone or computer that can do this for us. We also need a real-life context so that this calculation makes sense to our brain and is not just a meaningless number.You could ask your Stage 3 students to brainstorm for 1 minute with a partner as many different examples as they can. For example, there is a length of timber that is 6975 cm long and we need it cut into 235.2 cm lengths. How many lengths will you get? Or you have 6975 mL and you need to pack it into containers that each hold exactly 235.2 mL. How many containers will you need? But notice how this actual calculation is pretty obscure. It is not something you will come across every day.
And we still need to estimate our answer so that we can check we pressed the correct buttons. For example, 6975 is reasonably close to 7000. And 235.5 is closer to 250 than 200, if we round to the nearest 50. There are 4 groups of 250 in 1000 so there will be 7 x 4 = 28 groups of 250 in 7000. So I would expect to get an answer round about 28.
When I press 6975 ÷ 235.2 on my calculator it shows 29.6556122. I know this rounds up to 29.66 as a 2-place decimal or to 30 if I want no remainders. So my estimate was a useful check that my calculation was correct. Division estimates are a little tricky as you don’t need to round to the nearest 100, 1000 or 10 000 but to the nearest easy multiple. In this case 7000 was OK but if your calculation was 4530 ÷ 72 then an effective estimate would be 4200 ÷ 70 which you can then calculate mentally. 72 was rounded to the nearest 10, but then 4530 was rounded to an easy-to-think-about multiple of 7 and 42 is closer to 45 than 49 is to 45 so 4200 was a suitable number to round down to. I can then mentally calculate 4200 ÷ 70 to get 60. The calculator shows 62.9 so my estimate was in the right ball park.
So, what do we want our students to learn?
This is a difficult question to answer today as the world changes so rapidly. We are not sure what the world of our students will look like in 20 years time. But we are pretty sure that, in terms of number calculations, estimation is a vital skill and an understanding of what happens to numbers when you round up or down is a vital part of this skill. Once we go beyond our basic number facts and extended number facts, we no longer need to do the actual calculations ourselves. Those pages and pages of complex number calculations no longer have a central role in our primary classrooms.
We want students who are confident problem solvers, who can work co-operatively and explain processes. We want students who can tackle relevant, real-life problems, who can persist and not give up easily. Is this the focus in your maths sessions?