I Think I’ll Just Calculate the Product, Thanks


I’ve said it before, regarding perfectionism, and I’ll say it again, this time regarding estimation: that word…I do not think it means what you think it means. After our last estimation post became a polemic, with opinions piling up in our inboxes on both sides of the argument, I wanted to take a moment and add my perspective on it. Specifically, I want to make sure we’re talking about the same educational construct.

First, estimation isn’t rounding – at least, it’s not rounding the output of a problem. Rounding is the mathematical truncation of a decimal number from more place values to fewer. Rounding outputs – results of mathematical problems – is a fairly valuable skill, especially when significant digits come into play. If I’m working with raw data that comes in from one source to the nearest tenth, and from another source to the nearest thousandth…well, I’m going to have to report up to the client in tenths; that’s the greatest level of precision I received in the combined sets of values. Or I might want to know about how many unit sales of a product were made last month; I can take that output to the nearest hundred thousand, please. No one’s debating the relative merits of learning to round output numbers.

Second, estimation also isn’t prediction. Prediction is a guess of an unknown future number. It’s where I call my wife from the grocery store and ask about how many people might be coming to the dinner party; if it’s more than twenty, maybe I’ll need two bags of ice. If less than twenty, probably one. I don’t know the total, and I likely won’t have the final data in hand until the last time the doorbell rings. I’m asking for an educated guess as to what might happen. We all use predictive skills, every day, and no one’s talking about taking a baseball bat to the honored tradition of a rough guess.

Third, estimation isn’t sampling and extrapolation, either. If I’m killing time with a friend in Lower Downtown Denver and we decide to try and figure out how many people walk past our cafe storefront in a day, we could sample the number that walk by in a ten-minute period and multiply out through the business’ operating hours. That’s not estimation by the definition of modern elementary-school math. Sampling and extrapolation have obvious uses in day-to-day life, most notably for those of us engaged in using past data to model possible future events.

Finally, estimation isn’t a reasonable guess as to a number – the ubiquitous jar-full-of-gumballs problems tossed out frequently. Guessing what a number might be, without any attendant input information (“this jar is twenty centimeters tall and has a circumference of 14 centimeters”) is just that – a guess. We know there aren’t two hundred billion gumballs in the jar, and we know there aren’t twelve. It’s somewhere in between, and we can probably use our own experience in life to get within a hundred or so of the answer. Guessing how many gumballs are in a jar doesn’t have a great deal of use in life, either, but it does form the basis of a whole category of retail and Facebook contests.

Estimation1, for those not currently teaching an elementary school-aged child, is a different animal. It’s where we provide a very typical, straightforward math problem to someone trained either to solve the problem by hand, or use a calculator to come up with the right answer. We then ask them to do neither, but rather just kinda take a rough shot at it, neither a true guess (which would have no inputs, and require no work, but also have no value) nor an answer precise to any specific level (which would require actual work, but yield a precision result). Instead, we’re asking a child to combine the worst aspects of both: do some work to yield a result with little value.

Estimation problems go like this: estimate the product of 347 and 13.

What my kid is supposed to do is decide that 347 is pretty close to 350, and 13 is pretty close to 10, and come up with an estimate of 3,500. Doesn’t that sound neat and orderly? It’s got to be around 3,500, because both of my estimate numbers were pretty close to the original numbers – right? But that’s not actually true. The answer isn’t anywhere near 3,500 – in fact, it’s off by more than a thousand. The right answer is 4,511.

Now, in what universe is the 3,500 estimate valuable? Take science, law, engineering, medicine, and business right off the board. If the bridge cables have to weigh less than 4,000 pounds in total to keep the bridge aloft, and I turn in my ‘estimate’ of 3,500, I’ve probably sent an entire rush hour’s worth of commuters into Lake Washington. If the patient must receive no more than 4,000ml per month of medication, I’ve probably put him in a pine box. There are professions that need the right answer, and only the right answer, and the ‘estimate’ was nothing more than a waste of a few seconds that could have been used finding a calculator. Worse yet, we encouraged you to do this by telling you that you’re good at estimation in elementary school. You’re not. None of us are. And the last thing we want is to encourage professionals to keep after this lunatic waste of time as they enter their chosen fields of work.

So maybe it’s useful in real life. Except it’s not. ‘I use estimation all the time at the grocery store,’ I read in the comments on the original estimation post. Really? Are you keeping a running log in your head of how much your grocery purchases cost? That seems like a great deal more work than totting them up on a calculator as you go. If, that is, you care about the pre-checkout cost – since the store is going to provide you an itemized receipt after the fact. Maybe what you meant was approximation – “there’s sixteen people on the soccer team, but not all of them show up every week; still, CapriSuns come ten to a box, better get two boxes.”

So why are we doing this? There’s no question estimation is quick, but it’s also wrong, and it provides us with a false sense of confidence in our own estimation abilities. Sometimes that’s mostly harmless, as Douglas Adams would say; and sometimes, as Kathy pointed out in her post, it’s potentially deadly. I suppose my question is this: what sort of mathematical questions do we encounter in day-to-day life that we’re both unwilling to calculate through and comfortable with our answer being wildly wrong? If we’re looking for an alternate form of math to teach our kids, one that emphasizes mental calculation, why not replace estimation with something useful – like Vedic math? At least they’d end up with the right answer in exchange for their time and effort.


1 Ironically, we’re not even using these terms correctly; what modern math calls estimation is really approximationEstimation is, technically, making an educated guess – ‘how many gumballs do you think are in this jar?’  – while approximation is making a measurement or count specific enough for a given purpose. Guessing that the flagpole on top of a building is ten feet tall is an estimation. Measuring that flagpole as being ‘about’ ten feet tall is an approximation.  This isn’t helped by the fact that Merriam-Webster defines ‘estimate’ as ‘to judge approximately.’ But I won’t even get into that here.


5 responses to this post.

  1. So, estimation in the grocery store would be something like, “which one is the better deal”, And you’re right, we should just whip out our phone and calculate. Estimation is really not relevant nor efficient. I remember in my young 20’s as a new adult ‘estimating’ my bills with my new apartment–well, let’s just say I’m glad I’ve changed my ways. Love both your posts, as always.


  2. What skill is it when you ask a student, “Does this answer to your calculation make sense?” Does it differ depending on the type of calculation?


    • It’s worth knowing that multiplying hundreds by tens – as in the example above – should yield thousands, not hundreds or hundreds of thousands. But that’s place-value awareness, not estimation. If I tried to use estimation to determine whether my calculated answer of 4,511 made ‘sense,’ the answer would be no, other than to indicate that I should be in the ones-of-thousands for my answer. But I already knew that if I know place value.


  3. Posted by cocoder on December 3, 2012 at 7:27 am

    I agree that estimation doesn’t have a lot of value as a stand alone skill but that doesn’t mean it’s useless. For example, it’s a way to come up with a starting point in a division problem. For a fifth grader who struggles with math (like the kids I tutor), learning to estimate can turn a problem that is overwhelming into something that is doable. They see 13421/471 and they shut down, it’s too hard. By showing them that 1342 rounds to 1000 and 471 rounds to 500 so we can ESTIMATE that the first number in the quotient will be in the VICINITY of 1000/500 which is 2 (easy peasy) then they at least have a starting point. But, you have to take it to the next step and teach them how to determine if the estimate is correct and what to do if it’s not. In my experience, that’s when the light bulb over the head goes off for the kids and they start to understand WHY they’re doing what they’re doing.

    Here’s where the problem comes in…as a tutor, I can work with a small group of kids on a division problem and see whether or not the concept of estimation is working for them. For me, it’s a contextual evaluation. I think the problem with the type of estimation problem you give as an example is that they are teaching it as a standalone concept so they can test kids on it. Unfortunately, that gives it a level of perceived importance it doesn’t merit, especially if the teacher is teaching to the test.


    • It has become a standalone concept, I agree. It’s no longer a screwdriver in the box (“this can be handy as a very rough check”); it’s become a mathematical concept in and of itself, and that’s very dangerous, because it teaches kids that they’re good at estimating. They’re not. Nobody is. Casinos count on that fact. (http://forgeover.com/articles/2009/10/11/humans-suck-at-estimating-modern-odds).

      And today, there’s so many other options for kids who have trouble starting out. I’ve got a dyscalculic daughter (H) and I’ve found it necessary to try double-handfuls of alternative tools. Lattice multiplication is great for kids who have initial ‘problem panic’ (including H) because it gives them something calming to do (sorting the problem into a lattice) that they can gain confidence with. Vedic math is also great for this, and so is Japanese multiplication (kind of a lattice variant). They’re fun and visual and engaging and they ALL have in common that they offer a precise answer. I’m hugely in favor of teaching means for checking our work; checking work is a habit we need to get into. But spending this vast amount of time teaching estimation as that method is wasteful, dangerous and silly.


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