# Articles I wish I never finished reading, Dept.

Yet another in a long line of articles that caricatures traditional math teaching as rote learning, lacking conceptual understanding:

Traditional middle or high school mathematics teaching in the U.S. typically follows this pattern: The teacher demonstrates a set of procedures that can be used to solve a particular kind of problem. A similar problem is then introduced for the class to solve together. Then, the students get a number of exercises to practice on their own.

For example, when students learn about the area of shapes, they’re given a set of formulas. They put numbers into the correct formula and compute a solution. More complex questions might give the students the area and have them work backwards to find a missing dimension. Students will often learn a different set of formulas each day: perhaps squares and rectangles one day, triangles the next.

The article goes on to say that this “rote process” is seldom questioned, seems to make math easier, and students find such kind of teaching to be very satisfying.  I’ve written before about textbooks from previous eras that did indeed provide explanations for why various algorithmic procedures and formulas work.  Areas and volumes are explained in terms of multiplication for figures such as rectangles and triangles. Areas of quadrilaterals such as parallelograms and trapezoids are also explained; I recently showed the derivation of same to my 7th grade math class. To say that teachers fail to teach understanding is misleading.  Some teachers may not, but many do.  A math teacher I know clarified his position on understanding versus procedures

A few years back I started explicitly telling my students “I don’t care if you understand it, provided you can do it” when they complained that they “didn’t understand”. I tell them that when their exam papers are marked there are no marks for “understanding”. I follow that up with saying that understanding will inevitably follow in time, provided that they could do the skills, but that it would not follow if they couldn’t do the skills.
Now that isn’t to say that I don’t teach the reasons for things — I teach invert and multiply explicitly, but I also explain why it works. What I don’t do is fret about whether they understood my explanation, and I don’t let them not do something because they “don’t understand”. I most certainly do not try to teach understanding of a procedure to a student who can do it accurately.
Some students find that truly liberating — they can get on with learning the Maths without any pressure to have to understand the whole picture first. Most just do what they always have done, which is do what the teacher asks them to do and not worry about understanding because they never have (most kids really don’t want to understand very much). Every now and then I have a student who refuses to learn a new skill until they “understand” it — and that causes problems, largely because they learn so unnecessarily slowly as a result, which I find difficult as a teacher.

I happen to teach the same way, and I suspect others do too but cannot be too vocal about it given the current atmosphere surrounding math education these days.  The author of the paper, however, seems to think that understanding is rarely taught.He provides what he believes is a telling example: Students are taught how to distribute multiplication across addition (e.g., 3(x+5)= 3x+15, with the result that it becomes so ingrained (because of the “rote” nature of the procedure) that when they see an equation such as 3(x+5)=30, they will distribute rather than divide both sides by 3 to get the simpler equation of x + 5 = 10.  The author claims “but a child who learned the distribution method might have great difficulty recognizing the alternate method – or even that both procedures are equally correct.”

Actually we do show them when we come to such equations so they can recognize when it’s advantageous to divide, and also (not mentioned by this author) when it is not.  But he would rather disparage the teaching of the distributive rule because it may interfere with some later “conceptual understanding” of equations which is easily handled by–dare I say it–direct instruction.

He concludes his essay with

If we really want to improve America’s mathematics education, we need to rethink both our education system and our teaching methods, and perhaps to consider how other countries approach mathematics instruction. Research has provided evidence that teaching conceptually has benefits not offered by traditional teaching. And students who learn conceptually typically do as well or better on achievement tests.

This song has been sung in the US for the past 30 years; he’s not the first person to sing it and he likely will not be the last.  The US has floundered with various types of teaching conceptual understanding in the lower grades. How’s that been working out for everyone so far? Oh, he has that one covered: It’s because we’re doing it wrong:

As an education researcher, I’ve observed teachers trying to implement reforms – often with limited success. They sometimes make changes that are more cosmetic than substantive (e.g., more student discussion and group activity), while failing to get at the heart of the matter: What does it truly mean to teach and learn mathematics?

What he and other writers of this type of math ed pabulum never seem to get around to doing is seeing how students who do succeed in math have done it.  No mention is made of the practice they get at home, or of tutoring they might receive and how such tutoring is conducted.  Rather, we get articles that extol programs that challenge students with difficult “rich” problems such as what is done in the Russian School.  But rarely do we get a glimpse of the practice such students do using what he would call “rote methods”.

And of course parents want the traditional methods it’s because they just want what we had when we were growing up or that our kids are somehow different than others.

Yeah, OK, whatever you say.

## 5 thoughts on “Articles I wish I never finished reading, Dept.”

1. SteveH

Sigh. I’m sorry to sound like a broken record. It’s not possible to do well in math with rote understanding, and nobody has shown better results with reform math or whatever they call it these days – “anything but traditional math?” If they did, I would be their biggest supporter. So what is it about traditional math and individual textbook homework problem sets that is not rote learning – that creates STEM-prepared students? We STEM parents know. Ask us. Our kids are your best students.

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2. Chester Draws

that when they see an equation such as 3(x+5)=30, they will distribute rather than divide both sides by 3 to get the simpler equation of x + 5 = 10. The author claims “but a child who learned the distribution method might have great difficulty recognizing the alternate method – or even that both procedures are equally correct.”

But both procedures are not equally useful.

If you have a standard 14 year old and you give them 7(x + 5) = 30, then they are going to stuff it up if they don’t distribute first. x + 5 = 30/7 is not something you want them to be attempting. Whereas 7x + 35 = 30 produces no such problems, despite yielding the same fractional negative.

The divide first method is usually more difficult and much more prone to error. Why would you even want them to know about it? I have nothing but derision for teachers that show students methods that aren’t universally applicable so they have “choice”. They don’t need or want choice, they need and want to get the answers correct with a reliable method. The time lost teaching a trivially useful technique would be much better spent getting the ones that they do need properly organised in their heads.

I teach all my students that normally the first thing you do in any algebra solving problem is get rid of fractions and brackets. Then you can see what you have. They then have their minds freed of what to do first — remove brackets and fractions — and that leaves more brain power for the hard bits.

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3. SteveH

In a traditional approach to algebra, you learn that there is no one way to solve anything, even though pedagogues really push ideas of order of operations. Learning this is not an understanding issue. It’s a practice issue, where mastery of problem sets give you plenty of chances to solve problems in different ways. Practice for SAT also teaches you to look for tricks and short cuts, but that is neither necessary or sufficient. Practice, practice, practice on problem sets is the solution. That level of understanding is only driven by individual practice on problem variations, not transference of a few in-class group projects covering general ideas. Words are not understanding.

There are ways to talk and provide proper and more abstract algebraic understandings, but most of these rote pedagogues don’t have a clue. In the end, the only way to create proper understandings is via lots of individual practice on problem sets. Practice is not just about speed.

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4. I learned from Don Cohen that solving equations and other problems more than one way is a great way to “prove” your answer and learn a few new tricks. I’m open to embracing all possible tools from pencil and paper to string and pushpins and scissors to computer programming. However, it’s the last tool that’ll pay off with the most power in the long run. Functions and programs can be extended and applied to seemingly unrelated problems. Just my 2 cents.

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