Estimation Isn’t Just Calculating Badly On Purpose

Here is a tweet I haven’t stopped thinking about for a couple of months.

I think it’s possible we should cut the student some slack here.

If the student has all the tools, information, and resources necessary to calculate an answer, we should be excited to see the student calculate it. Asking students to do anything less than calculate in that situation is to ask them to switch off parts of their brain, to use less than their full capacity as a thinker.

If we treated skills in other disciplines the way we often treat estimation in math …

… we’d ask students to spell words incorrectly before spelling them correctly.

… we’d ask students to recall historical facts incorrectly before recalling them correctly.

Estimation shouldn’t ask students to switch off parts of their brains or use less than their full capacity as thinkers. It should ask them to switch on new parts of their brains and expand their capacities as thinkers. Estimation tasks should broaden a student’s sense of what counts as math and who counts as a mathematician.

Estimation and calculation should also be mutually supportive in the same way that …

… knowing roughly the balance of yeast and sugar in bread supports you when you pour those ingredients exactly.

… knowing the general direction of your destination supports you when you drive with turn-by-turn directions.

… knowing the general order of your weekend schedule supports you when you carry out your precise itinerary.

Engaging in one aspect of mathematics makes the other easier and more interesting. That’s what Kasmer & Kim (2012) found was true about estimation. When students had a chance to first predict the relationship between two quantities it made their later precise operation on that relationship easier.

If we want students to develop their ability to estimate, we need to design experiences that don’t just ask them to calculate badly on purpose.

Create tasks where estimation is the most efficient possible method.

Take that worksheet above. Give students the same sums but ask them to order the sums from least to greatest.

Students may still calculate precisely but there is now a reward for students who estimate using place value as a guide.

Create tasks where estimation is the only possible method.

This is the foundation of my 3-Act Task design, where students experience the world in concrete form, without the information that word problems typically provide, without sufficient resources to calculate.

“Estimate the number of coins.” Estimation feels natural here because there isn’t enough information for calculation. Indeed, estimation is the only tool a student can use in this presentation of the context.

Meanwhile, in this presentation of the same task, there is enough information to calculate, which makes estimation feel like calculating badly on purpose.

Estimation isn’t a second-class intellectual citizen. It doesn’t need charity from calculation. It needs teachers who appreciate its value, who can create tasks that help students experience its benefits.

BTW

Featured Comment

William Carey:

One thing I love about calculus is is proceeds from estimation to exact calculation, and there’s no way to justify the exact calculations without working through the estimation first. We often think of mathematics as a discipline that proceeds deductively from perfect truth to perfect truth, but there are whole swaths of mathematics where the best way forward is to work from an answer whose incorrectness we understand towards an answer whose correctness we don’t yet understand.

Mark Betnel:

I agree with you, but I think it’s interesting to turn your non-math examples into better activities that reflect what we’re trying to do with “good” math estimation tasks.

Mr. K references Fermi problems, which fall really nicely in the category of “tasks where estimation is the only possible method.”

Theresa Clifford:

At the beginning of the year, I fill four jars around the room. One with M&M’s, one with eraser caps, one with cotton balls, and one with paper clips. They are all allowed a guess for how many in each jar. They enter their answer and their name on a slip of paper and place it in a collection jar. Whenever we come to a question where I want them to estimate first, I remind them of what they did when they first looked at the jar. I don’t tell them how many in each until the winter break – the suspense is awesome. Then in January I start with four new jars.

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I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

19 Comments

  1. Reply

    I was just going to suggest something like the pennies activity. Something where the oder of magnitude is so immense that one HAS to estimate it. …OR something thats incomensurable.

    But I think my favorite theme of this post (and many of Dan’s posts) is, don’t ask students to switch of parts of their brain. In my mind, its the opposite of “think like a mathematician”.

    Thanks Dan, I love it!

    • Just to add,
      Something I thought very interesting while peruzing the thread: There seem to be 4-5 ‘types’ of estimating.

      1) ballparking an incalculable large number, like estimating how many people in a crowd or how many pennies in the pyramid.

      2) eyeballing an indistinguishable measurement, like estimating how much is left in a 2-liter bottle.

      3) predicting some unknowable future quantity, like estimating how many people will read this post

      4) rough but speedy calculation, like quickly estimating your grocery bill so you can have your cash out and ready (obsolete example? lol)

      And this 5th one that I think is more rounding than it is estimating, but maybe that is a subset of estimating, so…

      5) Approximating an incomensurable value, like estimating how old I am at the exact moment that I post this!

      I also found the wikipedia article on Fermi problems facinating. Not to mention the Kasmer & Kim article Dan linked to. I never realized how rich the feild of estimation is–which I am somewhat ashamed to admit considering that as a stats teacher I spent a lot of time froleking in that feild.

      I’ve got a lot of reflecting and pondering to do. :)

  2. Reply

    Bravo Dan. Mathematicians love exact answers. I have a feeling this student will correctly solve the equation 7x = 40 instead of giving the wrong answer x = 5.71!

    As you mentioned there is a place for estimating, but just a place.

  3. Reply

    I love the story of Fermi and the sheaf of papers tossed in the air.

    Estimating tools:

    7^2 = 50
    5 miles = 8 km
    Earth circumference = 40,000 km
    2^10 = 10^3
    pi = 3 + 5%
    metric is 10% more than imperial (for tons and meters, at least. liters too, if you’re not being picky)

    • Nice reference here. Fermi problems seem to fall in the category of “tasks where estimation is the only possible method.”

  4. Reply

    Featured Comment

    One thing I love about calculus is is proceeds from estimation to exact calculation, and there’s no way to justify the exact calculations without working through the estimation first. We often think of mathematics as a discipline that proceeds deductively from perfect truth to perfect truth, but there are whole swaths of mathematics where the best way forward is to work from an answer whose incorrectness we understand towards an answer whose correctness we don’t yet understand.
    • Powerful. Elaborate a little so I know we’re somewhere in the same neighborhood? i’m thinking about limits in particular. Approximating towards very large and very small numbers. What else?

    • Sure – a couple that I do with students are recreating the method of exhaustion to approximate pi (where you know your estimate will be wrong, and in what way, but there’s literally nothing else to do). We also do the volume of a pyramid inscribed in a cube (with duplo at first, then via estimation, then analytically). First we estimate, then we approximate, then we refine.

      I also run through a bunch of estimation and approximation techniques with the kids for data modeling in pre-calc. So, RANSAC, least sum of squares, and the like. Once we have tools to evaluate our estimates, they become much more powerful.

  5. Reply

    Back in slide rule days estimation was a required skill. You had to estimate the magnitude of an answer or at least have an idea what the answer should look like. I remember for many students this was very difficult and took a lot of practice. The introduction of the calculator sort of blew that needed skill out of the water. It is still useful to be able to look at a problem and have an idea where the answer should be in case of a math/calculator error but is the classroom time required to develop this skill something to worry about? I still have my students think of estimations but it is sort of a by product of working the problem with calculator/computer.

  6. Reply

    I agree with you, but I think it’s interesting to turn your non-math examples into better activities that reflect what we’re trying to do with “good” math estimation tasks.

    For the spelling one: “I’m going to say a word out loud and use it in a sentence, and then I want you to make your best guess for the spelling of it. After we’ve gotten all your best guesses down, you’re going to look up these words in a dictionary. We’re working on your intuition about the spelling of words you’ve just heard, so you are able to use resources like a dictionary to get more information about them later.”

    For the history one (perhaps with different events): “Without looking up exact dates, try to put these historical events around World War II into chronological order. Think about which events were causes and which were effects of the others in making your timeline. Then look up the exact dates to make a new timeline so you can compare with your neighbor. Discuss which ones you each got in the wrong order and the relationships between these events.”

    I wonder if these might be good Family Math Night activities to help families understand what we’re trying to do with the “do it wrong first” math activities.

    • Love the mental exercise you’re proposing, Mark.

      In both cases, I’d recommend a) imposing a time limit that discourages a student from perseverating too much on precision, b) having students compare answers with each other, understanding that under that time limit, it’s likely that each student had an idea that might be useful to the other.

  7. Reply

    The question is difficult. It is concerns a real person.

    I’d like to say something about estimation from the perspective of a quantitative career.

    Since machine computation came along, it is far more useful than than long number arithmetic. How many times a year, or in a lifetime for that matter, must you make a 6-significant-figure calculation by hand?

    A computer calculates better and far faster than we do because it is an automaton. It seems hard to us because we aren’t, not because it is intrinsically difficult. Significant estimation is a conscious-mind activity.It requires we know what we are thinking about. It requires us.

    Perhaps we would like math more if we had greater opportunity to come at it from the human side .

  8. Reply

    Estimation is also useful after an exact calculation has been performed. I often ask my students to perform a “sanity check.” For example, if you calculate a speed and get a number larger than the speed of light, your answer is insane and must be wrong.

  9. Theresa M Clifford

    November 26, 2019 - 9:25 am -
    Reply

    Featured Comment

    At the beginning of the jar I fill 4 jars around the room. One with m&m’s, one with eraser caps, one with cotton balls, one with paper clips. They are all allowed a guess for how many in each jar.
    They enter their answer and their name on a slip of paper and place it in a collection jar. Whenever we come to a question where I want them to estimate first, I remind them of what they did when they first looked at the jar. I don’t tell them how many in each until the winter break – the suspense is awesome. Then in January I start with 4 new jars. Not sure what I’m doing in those ones yet so if anyone wants to suggest anything….. Thanks Dan for reminding us about estimating. It’s so important and our kiddos are so afraid of it.
  10. Reply

    Hey Dan –
    Can the Penny Pyramid problem be solved by computing the sum of consecutive squares from 1 to 40?
    (2n + 1)n(n + 1)/6
    (I found this formula on The Math Page: https://www.themathpage.com/Arith/asquares.htm#sum.)

    Granted, solving this by mental math is out of reach of all but the few savants in the crowd, so estimation is by for the preferred approach.

    I’m a longtime fan. Thanks for your work!
    –Don

    • Yep! It can be solved with that formula, with a spreadsheet, with sigma notation – through lots of means all of which depend on the student knowing information about the pyramid that isn’t evident from the photo. It has a square base. 40 layers. Each layer has 13 stacks. Etc.

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