In my last “smart and thoughtful post” (the parlance that edu-pundits use when they refer to each other’s writings), I talked about “understanding vs procedure”. The quote at the end from a teacher in New Zealand seemed to ruffle the feathers of some who took to Twitter to state that they believed otherwise.

In all these discussions of “understanding”, those who believe it is not taught and that students are doing math without knowing math rarely if ever explain what they mean by understanding in terms of how it translates or transfers to problem variations or new areas in math. For example, a student who has learned the invert and multiply rule for fractional division may not be able to explain why the rule works, but may have an understanding of what fractional division represents. The student then uses the latter to solve problems requiring fractional division.

Anna Stokke, a math professor at University of Winnipeg has also addressed the issue of student understanding in math and echoes what the teacher in New Zealand said. She has kindly given me permission to quote her:

When we teach, most of us generally do teach students why things are true but I sure don’t want my students going through the understanding piece every time they solve a problem. What a waste of time! The point, I think, to get across to students is that there is a reason why everything in math works the way it does and you could figure this out if you need to (because you WILL almost certainly forget).

With the consultants I’ve met, who always push this stuff and insist that kids aren’t fluent unless they can explain everything to you, it seems that they themselves just figured out that there are reasons behind procedures in math as adults. Then they’re angry that their teachers (supposedly) didn’t explain all these things to them. They’re certain that they would have liked math more and done better if only their teachers would have focused on understanding. So, their mission is to make sure that all kids are forced to explain their thinking at every step. Pure torture, really.

Funny thing is, that the understanding piece is a lot more difficult for students. They generally don’t like it and it’s something that really comes with much experience and mathematical maturity. It won’t make students like math more if we spend more time on understanding…it will just confuse and frustrate them more. In my experience, I’ve found that students like step-by-step procedures and algorithms more than anything else.

What people in the “understanding uber alles” crowd likely mean when they talk about understanding probably has to do with words. They would probably be happy with words that didn’t ensure that the kids could actually DO the problems: a “rote understanding”.

All the students I tutor have been thoroughly confused by these efforts to teach them understanding. In desperation, their parents forced to pay a tutor.

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Success in math is all about mastery of individual homework p-sets. That’s where all of the so-called constructivism and understanding happens, not with in-class, mixed-ability group work. That’s neither necessary or sufficient, and for many, not engaging or inspiring. The big egos dominate. I discovered and understood almost everything with homework p-sets.

The goal of math teachers is to push and enforce that process. I used to allocate time at the end of class for students to get started on the homework and to ask questions. The hardest part of any task is getting started and I tried to help/push them in that direction. I also thought that weekly quizzes were helpful. It forced them to not wait until the test and I allowed them to throw out two of the lowest grades. I used to tell them that if they did (and understood) all of the homework, they would get an ‘A’ in the class. In my algebra classes, I didn’t want to see words, I wanted them to give me the rule or identity for each step of their work.

Words about conceptual understanding won’t solve all problem variations and won’t transfer to other topics. Conceptual, is well, conceptual. Concepts – “an abstract idea; a general notion.” General notions do not define understanding or ensure any kind of transference in math. There are many levels of understanding and I distinctly remember thinking that I did not fully understand algebra until the end of Algebra II. There is no such thing as pre-understanding before mastery. Concepts are only the first tiny baby step, and they can cause much confusion when confronted with real problem set variations. That process converts concepts to understanding. Conceptual understanding is an oxymoron.

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As a part and parcel of “demonstrating understanding,” my daughter (at least in elementary school, not so much in junior high) had to write out paragraphs describing what she did. Her paragraphs were often given that elusive 4, the type of 4 that teachers “know it when they see it,” and she scored very high on the Common Core test. So, given that, I think that her paragraphs in which she “demonstrated understanding” were what society was looking for.

But those paragraphs never, ever said anything that the math, with properly shown work, didn’t say. Those paragraphs were wordy and cumbersome and each and every time they reminded me how math, pure math, communicates these type of ideas much, much better.

But probably the biggest problem with “demonstrating understanding” is those paragraphs took precious time. So in order to write those paragraphs, my daughter logged a lot less time in working those problems that actually mattered.

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Agreed with you and Anna (of course) and the commenters above.

What these consultants seem to misunderstand is actually something intrinsic to the practice of mathematics altogether: math is supposed to be a tool which lifts the practitioner above menial tasks … and it does so by building a sophisticated superstructure upon elementary building blocks. Learning those building blocks involves some level of menial activity. But its purpose is to bring the user PAST that level of meniality.

It is commonly said (by mathematicians, when teaching math classes) that mathematicians are lazy. This is often a preamble to something that probably looks desperately tedious to the non-mathematician observer, and it is said deliberately, with irony, but also with honesty, because while on one level it’s obviously wrong, on another level it is exactly right.

The mathematician will expend enormous amounts of energy in ESTABLISHING something to be correct, in order to accomplish simple goals. Now it’s quite true that understanding is a critical part of this — the point being that, math being a deductive enterprise, something just ain’t math until it has been shown to have a rigorous basis. This generally takes a lot of work, systematic, mechanical and sometimes abstract cognitive labour. But once it is done, that is all put aside. We call it a “Theorem”. When we use a theorem we DO NOT insist that one goes through the proof from scratch. That’s nonsense, and would negate the whole point of doing the proof in the first place. We do proofs in order to not HAVE TO do proofs when we’re actually using the fruit of our labours.

For example, the Fundamental Theorem of Arithmetic tells us that every positive integer has a unique decomposition of integers into prime powers. Suppose we need to know the GCD and LCM of two numbers like 36 and 105. We can use the FTA by writing 36 = (2^2)(3^2) and 105=(3)(5)(7). Then proceed according to other theorems telling us how the exponents of the primes in these factorizations yield the GCD and LCM. (GCD = 3 and LCM = (2^2)(3)(5)(7) ). The point is — what just happened in that step? Can we trust the answer? You’d have to “understand” what’s going on in order to trust the answer … right? Well, yes in a manner of speaking. But what we DON’T do is go back and prove the FTA. It is ALREADY proved, and we needn’t do so again. We needn’t even think about the proof. All we need is to know that it is done (and it helps, although it isn’t critical to our present task, to have seen or even personally worked through that proof).

You see, proving the FTA is tedious. It’s not a particularly hard thing, but there are a lot of ingredients, and a lot of careful reasoning to answer some potential objections to the conclusion. Yet it MUST be done. And so we do it.

That’s where the irony in the saying arises: it is a lot of tedious labour … and for what? Finding the GCD and LCM of 36 and 105? Wow. But the point is … we have ALREADY done it. We moved heaven and earth, so to speak, to establish that result. Now that it’s done we NEVER have to do it again.

And that is where the honesty in the saying arises. If you NEVER have to do it again, then over the long run, it is a very efficient thing to have done. So you’ll never have to wonder, gee — I took the smaller of the exponents for each prime factor to find the GCD … am I sure that’s the right thing to do? How do I know there aren’t other prime factorizations, which might give different answers? Well. You’re sure because of a THEOREM. It’s called a theorem because it’s already fully established. That’s how math works.

So an engineer learns calculus, and sees the proofs of the results, in passing, when he first learns them. then for the rest of their life, that engineer will USE those results without having to do the proof again. And that’s great — because the proofs of engineering calculus are indeed tedious! But if they really do as these consultants are demanding, and always “show understanding” by giving the full, first-principles demonstration in every task, they will simply not accomplish anything useful.

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I like to let math do the thinking for me. The goal of Math is to make most tasks menial.

It’s one thing for educationalists to claim knowledge of process, but what they are doing is claiming knowledge of, and redefining, math – with NO push-back from high school teachers. That’s wrong.

It’s a real problem that the world changes between K-6 and high school/college. Some make feeble attempts to cloud the issue with “zombie” accusations, but offer absolutely no alternative path with better results. They may point to Phillips-Exeter with their Harkness Table, but that’s an extraordinary stretch and their approach is not necessary. In K-6, students have no choice – AND the curriculum is defined at the low slope level of no remediation in college algebra at the end of high school.

The College Board knows this is a problem. They plan on offering “Pre-AP” classes in ninth grade, but know that it’s almost impossible to fix the low slope expectations of K-8 to get the students from algebra in ninth grade to calculus as a senior. They end up using “weasel words” on their web site and talk of social justice. It’s not social justice to offer only a low slope to K-6 students and then put the onus on students after that to double up in math. While we argue with them about understanding as if they have a clue, their low slope and wide full inclusion goes unchallenged. In our town, full academic inclusion is untouchable, so now the tracking is hidden at home and with tutors. Is it social justice to increase the academic gap and hide tracking in affluent homes where parents say nothing or else risk being called helicopter parents. When we moved our son to a private school for grades 2-5, one teacher had the audacity to tell us that she hoped he will have time to play.

In our town, parents pushed out CMP in seventh and eighth grades because there was a curriculum path gap to geometry as a freshman in high school. Now the big gap is between sixth and seventh grades and many students and parents feel like they’ve been hit on the head with a brick when capable students are put on the low “Groundhog Day” algebra track that leads to no remediation in college algebra – the “distinguished” level of CCSS.

There are fundamental systemic problems here. Best education is not a two-generational process where parents have to fix problems that educators want to ignore.

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Reblogged this on The Echo Chamber.

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