Estimation and Logic

Someone asked me recently if I had any tips on teaching “estimation with friendly numbers”. I had to admit that I don’t have a magical tip that would instantly help – I’ve struggled to get the same concept through to my own son, after all! However, while framing my reply, it occurred to me why estimation might be difficult for a child who is otherwise good at math.

A brief intro first. Estimation is a way to quickly come up with approximate answers to difficult problems in your head. It is an important part of what’s called numeracy. Numeracy is being familiar and comfortable with numbers, in the same way that literacy is being familiar and comfortable with words. If a salesperson tells someone “pay only 4.85 per week over 2 years“, a numerate individual will know quickly that the total payment is “around about 500“. The innumerate person only has the salesperson’s assurance that the weekly payments are a better deal than “300 up front, no more to pay“.

As you can see, innumeracy costs money. One of my favorite quotes is : “There are two kinds of people in the world – those who understand compound interest, and those who pay it“.

So, where does estimation come in? Well, the numerate person in the example above might have thought something like this : “Let’s see, 4.85 is about 5. The number of weeks in a year is 52, that’s about 50. Two years is about 100 weeks. And 5 times 100 is 500, so I’d have to pay about 500” (the exact correct answer, by the way, is 504.40).

So instead of working out 4.85 times 104, you could use estimation using friendly numbers, computing 5 times 100. Translated into English, that means don’t multiply difficult numbers, multiply easy ones.

Now, why is this hard to learn? I think there’s a few reasons.

Firstly, the way math is often taught, there’s an artificial disconnect between the operations and what they mean. Kids get reams of exercises asking them to “multiply this and that”. While this helps them learn the mechanics of multiplication, it also separates the act of multiplying from its meaning. This is both good and bad. It is good if it means you are then comfortably able to multiply in any situation this is called for (areas, discounts, grocery bills, physics, taxes and so forth). It is bad if multiplication becomes an abstract skill that is irrelevant to any aspect of real life. Unfortunately, I suspect, something of the latter happens in many students.

Secondly, people often fix on a single way to solve a problem. Students are taught, for example,

  • “to find the total amount to pay, multiply the weekly payment by the number of weeks”.

Then, when learning estimation, they are taught

  • “to find the total amount to pay, get weekly payment and round it off, also get the number of weeks and round it off, then multiply them”.

This looks more complicated than the solution they already know. In a way, it is more complicated.

Thirdly, it’s rather vague. Yes, you could define carefully the steps, and say “round each multiplicand to the nearest 10 or 100 or 0.1 or whatever ensures there’s only one non-zero digit”. For a small handful of kids, this may even be the best way to teach estimation. For most people, however, the best way to teach it is to engage their intuition. Train them to intuitively find a number that’s near one of the multiplicands – one they find easy to deal with – and multiply that instead. They know best what’s easy for them. It may e different for different kids, despite the uniformity our education system induces. For an extreme example, read about the uneducated Mushar man asked to multiply 35 by 10.

However, “intuition” and especially “vagueness” are not things that sit comfortably with elementary school math. Kids who are good at math probably major on “precision” and “logic” – not very good tools for doing estimation.

When my son was first asked to do estimation problems, he would simply multiply to get the exact correct answer, then get frustrated when his correct answer was deemed wrong. After a while, he learned to multiply, then round off his exact correct answer to get an approximate answer! Teachers, watch out for this loophole! We want the tykes to round then multiply, not multiply then round!

Eventually (after many explanations) I got through to him the point of the exercise – “we want to be able to get an approximate answer quickly. If you use pen and paper, you’re doing it wrongly.” Then explained “round these numbers before you multiply”. Eventually he would get the idea, and start getting estimation questions right…. although I’m sure I haven’t had to explain it to him for the last time!

In summary – to teach estimation, it’s important to emphasize to the child the purpose of the exercise. Explain the kinds of situations when it might be used in real life. Stress the difference between these situations and the times when using a calculator would be more appropriate. Then, at least, the student may see the point of having two methods for solving essentially the same problem. Once they’ve understood that, it will be much easier to grasp the details of the method – rounding numbers then multiplying, and being content with an approximate result.