Here is a tweet I haven’t stopped thinking about for a couple of months.
Any tips around a young lad with ASD who cannot get his head around estimation? He just cannot see that it would be â€˜nearlyâ€™ or â€˜aroundâ€™ something when he can clearly work the answer out. My gut says give him something else to do, but if anyoneâ€™s come across this in the past….
— Ashley Booth (@MrBoothY6) September 11, 2019
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.
- Here’s a beautiful children’s book on exactly this topic. [via Julia McNamara]
- Contemplate then Calculate is an instructional routine that cleverly blocks calculation by only showing a mathematical structure for a limited time only.
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.
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.”
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.
Joel offers an example of this kind of estimation exercise.