At a professional development session Saturday, keynote speaker Marion Small (@marian_small) remarked that she had a "new rant" of sorts:

*Most of the calculations we do are a waste of time. We should be doing estimating.*

Then, in the secondary breakout session, Robin McAteer (@robintg) and Anne Holness asked what we value in our mathematics classes. One of my items was

*PRECISION*, and I think I was the only person to list that.

So, yes. I already know it's me against the world with this post. Still, maybe you can stick around for my four point case for why

**ESTIMATION ISN'T MATH**.

My reality involves precise calculations |

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__POINT 1: MATH IS PRECISE__

Part of the reason I like math so much (and value precision) is the nature of numbers. Draw an isosceles right triangle. Measure the hypotenuse. Surprise - you CAN'T! No matter how hard you try, you will always be able to go another decimal place, you will always be able to get a result that is more precise... UNLESS you invoke mathematical notation. Unless you use root(2).

From the "Matching Fraction" song parody |

Think about that. You have drawn a finite length that is NOT MEASURABLE. This is awesome. Mind blowing, even.

The counterpoint, of course, is something Marion Small herself mentioned. Define a number line, and where is 10? We find it - but wait. Did you use a ruler? Couldn't it be a millimetre to the right? Doesn't your pencil mark have a thickness that "10" doesn't really have? We're estimating where 10 is. We're estimating all the time. So that hypotenuse probably isn't really a length of root(2). How could it be?

Irrelevant, I say!

**ESTIMATING IS NOT MATH**. Estimating is the HUMAN REPRESENTATION of math. Which certainly has it's uses. But why can't we appreciate the theoretical beauty of numbers? Simply because "you'll never use it in real life"?

The math curriculum seems to be swinging the pendulum over to the point where everything has to connect to students' daily existence. I don't know whether that's because teaching is now more student centred (and they're demanding "when will I ever use this"), whether it's due to the stigma math is getting in the public that is forcing math to "always be relevant", whether it's a prioritizing issue on the volume of mathematical material that now exists, or whether it's something else.

Question: When was the last time you estimated something as root(2)?

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__POINT 2: ESTIMATION ISN'T ENOUGH__

So, you're given a problem, and you estimate the answer. Now what? Well, naturally, you want to know if you were right. You need the EXACT answer, and so you invoke MATH to figure that out. When was the last time you went into a store, looked at your cart at checkout, said "I think this will cost about $30. So here's $30. Bye!" That only happens if you're on the barter system.

Without an exact value to compare against, we have no idea what makes one person's estimate any more, less, or equally valid than another person's estimate. Now, am I saying don't estimate? Heck no! Estimation is an important step! Am I then saying you're never using math AS you estimate? No! What I AM saying is that a PRECISE answer

*requires*the mathematics - the estimate does not.

The problem here is that people aren't very good at this estimating skill. This estimating skill... which generally involves numbers. So, hey, numbers - it's math! Right? "Sure it is," cries the public. And so they demand, "Study more estimation in math! Estimation is the goal! Who cares about that root(2) nonsense, 1.41 is good enough for an engineer!" And that makes me sad.

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__POINT 3: ROUNDING BELONGS IN SCIENCE__

I mentioned this before, after taking the Triangleman Decimal Institute Course - decimals belong in science, not math. They are a product of scientific calculators. They imply some number of significant digits - just like estimation. I claim that this is science because math has the potential for an INFINITE NUMBER of significant digits! A potential that lately seems woefully untapped.

At our PD Day earlier this month, teacher Andrew Cumberland asked whether 1 = 1.00000; I would say of course that's true. So is 1.0 m = 1.00000 m? I would say of course not. There's measurement involved in the second example, and thus an element of extra precision. On the right side, we can now carry more decimals around for our science calculations. (Another teacher posed whether 'm' is actually a variable - a discussion for another post.)

Scaling is important. |

Here's the thing. At it's core, math has AS MUCH PRECISION AS WE WANT. Fractions. Ratios and proportions. Root(2). Meanwhile, estimates trend towards rounding off, towards using some number of decimals or significant digits, and - forgive me - that nonsense isn't mathematics. It's science. It's VALUABLE science, and it's a tool for VERIFYING mathematics, but it's NOT the infinite precision OF mathematics.

Granted, there are cases where we must sacrifice precision. If you solve a proportion and get an answer of "buy 13.458 apples", we likely need to round that off to 14. But that's not estimation. That's discrete math. Which brings me to my last point.

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__POINT 4: STATISTICS REQUIRES ESTIMATES__

I know what you're thinking. If estimation isn't mathematics, but statistics uses estimation, is he saying statistics isn't really mathematics? No.

Let's first talk statistics. It's randomness, sampling, and making predictions - based on scientific data. Thus it is inherently flawed, because you can only be as precise as your instruments are calibrated, and only as accurate as your survey allows. This is why you can analyze down to an almost sure thing, only to have the unexpected occur. In fact, in it's own way, statistics has as much mind blowing power as the concept of drawing a length of root(2).

An I-Q-R is not bizarre,If you know medians... you know medians! |

*data*is flawed, the

*mathematics*is sound. So yes, statistics is mathematics.

When you read in an article that "this survey is accurate plus or minus 3 percent, 19 times out of 20"... there's CALCULATIONS that went into that. Margin of error. Degrees of freedom. And while the statistics may only be generating a "best guess" prediction, this is NOT the same thing as being in your grocery store saying "I think this will cost about $30." (Or in my experience, they're not taught the same way.)

So here's the caveat:

**Estimating is not math**, so long as you can look up the answer and see "oh, I was 0.5 off". Estimating is ONLY mathematical if you are comparing against an UNKNOWN quantity, using calculations to determine HOW PRECISE your estimate is. (Again with the precision!)

Now, does hiding the answer from your students (or yourself) make it

*unknown*? Does that now make estimating it a mathematical process? Perhaps it's the theoretical mathematician in me, but I'd say no. Not unless you are mathematically analyzing your estimate without EVER knowing that "true" answer.

#### FINAL CAVEATS

In case I haven't been clear, here is what I am NOT saying:

Perhaps you've misinterpreted... |

- I am NOT saying that we shouldn't estimate in math class. We do word problems in math even though we don't teach English, and in Geography class they look at graphs. Crossing curriculums is good. Just because estimating is not math, that doesn't mean it isn't handy.

- I am NOT saying we should teach infinite precision in the younger grades. A certain amount of precision is sufficient for understanding daily life. That said, we

*should*call rounding out for what it really is - science - and there should come a point in high school when we nix the decimals, in favour of fractions and ratios.

- I am NOT saying that we should nix the real world applications. Applied mathematics is as valid as pure math, and rounding off is tied into that. But at the same time, not EVERY exploration has to be a hands-on relevant experience, does it? Because that's all I'm seeing. (If math is moving to only hands-on... I'm screwed.)

So that's what I'm NOT saying. What I AM saying is... well,

**ESTIMATION ISN'T MATH**. It's humans attempting to represent math, imperfectly. You're welcome to argue with me in the comments.

Is there a difference between "estimating" and "creating an upper/lower bound"?

ReplyDeleteMy immediate thought is yes, because "creating an upper bound" invokes particular calculations, like the "twin prime conjecture" trying to bound the difference between consecutive primes.

DeleteMy follow up thought is that rounding all your purchases up when you buy groceries, to be sure you have enough money, is then probably equally valid. But I worry then that this is on par with someone not thinking, just stating the blindingly obvious like "You're spending less than $1000".

I think it would help if you gave some examples of what you mean by "estimating."

DeleteFor example, I'd bet Marian Small was imagining something like estimating the value of 31 times 17. And that certainly requires particular calculations, if it's done well. (We'd probably want more students to get that it has to be more than 300 and less than 600, and that it's gonna be around 450, or whatever.) How is that different than the TPC?

On the other hand, I could see a stronger argument from you if it comes to (say) estimating the number of pages in a book. That's certainly an estimation that requires as much knowledge about books as mathematical knowledge. I would still argue for the mathiness of that sort of estimation, but I'm curious to know what exactly you're aiming at with this post.

Fair point. I'm thinking estimating is any time you give up on a precise calculation. So any time you give a trig ratio to four decimal places, or quadratic formula to two. Any time you want to give a 15% tip, yet just grab some change that seems about right. Any time you say that a length is 1.4 cm because the ruler tells you so.

DeleteEstimating 31 times 17 is a bit different (to me) in that someone may not yet have the necessary skill for a precise calculation - so fine. Gaining number sense is important, and in fact in the case of the Prime Conjecture, the world is still working on those skills. But if someone DOES have the skill for a precise calculation, saying the answer is about 450 is not doing math. (Caveat: I buy Chris' point that there are times when "close" may be enough. I'm not sure I can pin down when those times would be.)

I'm not claiming I have a strong argument, indeed I started the post by saying I'm probably on my own here. But that's why I put it up. I may very well be missing something. That said, I do not think estimation is anything more than a signpost saying "there's more to this - look over there".

I think that a part of learning mathematics is the ability to recognize when "close enough" is appropriate and when "exact/precise" is appropriate. So with that conjecture, I'd argue estimation is as equally important as being precise, depending on your needs.

ReplyDeleteInteresting. Okay, I'll buy that argument. I know I allow rounding for certain word problems. But it means that this ability to recognize the "appropriateness" should be coupled with estimation, particularly in statistics. Instead, my impression is estimation is simply being taught for the sake of the skill (or perhaps a bit less cynically, for the sake of improving reasoning with numbers), without any consideration for that broader context.

Delete