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Some place emphasis on higher order thinking skills at the cost of lower emphasis on drill/practice. Others disagree. Personally, I agree that higher order thinking skills are necessary for effective application of mathematics to problems, but I do not want to ignore the fact that effective higher order mathematical thinking requires firm understanding of fundamental mathematical principles. Therefore, I would like to combine both… but I'm a bit unsure about the “how to”.

Is there any well-vetted way to find a good balance between both?
If there isn’t, can you provide some hints/tips that might help me?

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One of the benefits of practice is a deeper understanding. I don't care how fast you get the answer, but I hope you are not adding by columns when you add 101+99. And that's why thinking of practice as equivalent to drill may be problematic.

Maria Droujkova talks about how you need to love something to learn it well. If we want kids to learn the times tables, we shouldn't be doing drill (unless a particular kid actually likes it), instead we should make it interesting: have them build a times table with heights of legos or cuisenaire rods showing the heights, have them draw ten-stars, and connect every third number for the threes (for example), search for iconic representations (e.g. a dozen eggs is 6x2, a six-pack of soda is 3x2, see some great flash cards here, though kids could make their own), and so on.

I think students in the U.S. are very used to math classes that focus at lower levels. One way to get a better feel for a good balance is to study how Japanese and Chinese math classes operate. For elementary math, read Liping Ma's book, Knowing and Teaching Elementary Mathematics, comparing U.S. and Chinese teachers.

What course do you have in mind?

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  • $\begingroup$ To answer your question: a college level one. But I'm already more than satisfied with your answer – especially the book references and the hint at comparing Asian and US teaching are really helpful. Thanks! $\endgroup$ – e-sushi Mar 14 '14 at 17:05
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    $\begingroup$ I'd also recommend an understanding of concept analysis through the grades. John Van der Walle's series of books on Teaching Student - Centred Mathematics and wps.pearsoned.ca/ca_ab_vandewalle_elementary_2 $\endgroup$ – HarMath Mar 17 '14 at 20:00
  • $\begingroup$ @HarMath Thanks for that! Btw: if you post a comment and want me to get a notification about it too, you need to add a “@” and my nickname before your message (like I am doing here). See, I would never have noticed your comment if I hadn't detected it by accident. Hope that helps. ;) Anyway, +1 one for your comment and thanks for adding your recommendation. Much appreciated. $\endgroup$ – e-sushi Mar 21 '14 at 16:00
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I've found that a big reason for drill and practice is speed and facility. For instance, if you're doing arithmetic, you could get the same results from the addition tables by counting on your fingers and toes but it's slower. And if you're doing multiplication, you could get the same results by repeated addition, but it's slow. Now that we don't have to add long columns of numbers by pencil and paper in order to do accounting and the like, there's less need for extensive drill in arithmetic than there was a century ago, but adding 3+7 mentally is still faster than punching it into the calculator. It takes practice with a technique to be able to solve common problems quickly rather than having to laboriously work through them each time they are encountered. That facility eases comprehension of higher level, abstract concepts. Another reason is that the ability to comprehend abstract concepts is often based on being familiar with a variety of concrete examples.

I would say that the best way to know how much drill/practice is needed seems to be to decide what you need the speed and facility for. If it's a common technique that will be used over and over again, like solving linear algebraic equations in one variable, then spend some time on drill. If it's as a foundation for more advanced, abstract concepts, then it would be useful to practice until the more specific example is no longer a hard problem. If it's a nice but not essential feature, like Egyptian or Mayan numerals, introduce the concept and move on.

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    $\begingroup$ Teach them to use tools (calculator, computer algebra system) effectively... $\endgroup$ – vonbrand Mar 14 '14 at 13:02
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I enforce Sue's question as for which course is this. I'll assume meanwhile that it is a college one (from calculus to advanced analysis).

Examples and computations. Say you are teaching a multidimensional real analysis. Don't be afraid to open a calculus book and take your time through the computations in class, except that this time you're showing how all the machinery is working. They have seen how each piece was constructed and now they deserve to see the whole structure at work.

If you're dealing with measure theory, even if you start with the abstract spaces and move on to integration don't forget to walk them through computations in $L_2$, $l_2$, $C([a,b])$ and other examples. Show that integration means summation in the nonnegative integers.

The higher order thinking is already present at advanced courses, what makes it fail to click is this lack of "technical" skill that are being able to show/compute explicit examples and motivate the theory using those as models. It's not about speed: it's about concept comprehension.

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Here is what Richard Feynman advised CALTECH students regarding mathematics (section 1-3 of Feynman Lectures Physics):

So, this guy comes into my office and asks me to try to make everything straight that I taught him, and this is the best I can do. The problem is to try to explain the stuff that was being taught. So I start, now, with the review. I would tell this guy, “The first thing you must learn is the mathematics. And that involves, first, calculus. And in calculus, differentiation.”

Now, mathematics is a beautiful subject, and has its ins and outs, too, but we’re trying to figure out what the minimum amount we have to learn for physics purposes are. So the attitude that’s taken here is a “disrespectful” one towards the mathematics, for sheer efficiency only; I’m not trying to undo mathematics.

What we have to do is to learn to differentiate like we know how much is 3 and 5, or how much is 5 times 7, because that kind of work is involved so often that it’s good not to be confounded by it. When you write something down, you should be able to immediately differentiate it without even thinking about it, and without making any mistakes. You’ll find you need to do this operation all the time—not only in physics, but in all the sciences. Therefore differentiation is like the arithmetic you had to learn before you could learn algebra.

Incidentally, the same goes for algebra: there’s a lot of algebra. We are assuming that you can do algebra in your sleep, upside down, without making a mistake. We know it isn’t true, so you should also practice algebra: write yourself a lot of expressions, practice them, and don’t make any errors. Errors in algebra, differentiation, and integration are only nonsense; they’re things that just annoy the physics, and annoy your mind while you’re trying to analyze something. You should be able to do calculations as quickly as possible, and with a minimum of errors. That requires nothing but rote practice—that’s the only way to do it. It’s like making yourself a multiplication table, like you did in elementary school: they’d put a bunch of numbers on the board, and you’d go: “This times that, this times that,” and so on—Bing! Bing! Bing!


P.s. Having some rote skills makes it easier to work on other aspects of problems. That is not same as saying having no intuitions is helpful either. Especially for things you don't grind into muscle memory. But for others better to have almost iconic recall, like "bsq-4ac"; do you really want to be completing the square because you don't remember the quadratic as you go through all of upper math, chemical equilibrium problems, physics everywhere, or EE or control systems (characteristic equation of harmonic oscillator ODE).

P.s.s. I actually find that some ideas come to me AS I DRILL, about the substance of what I am working on.

P.s.s.s. When working a homework set (on a new concept), it can be useful to have some easier, drill type problems for the new equation. Then progress to harder problems with proofs or word problems or combining concepts. Because you have a little more feel for things than just reading the text would give you, before you dive into the nutcracker questions. This is actually a major failing of harder classes, graduate school etc. in that the easier running start problems are omitted. Than again these classes seem to have a very cavalier attitude towards actually training people.

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