In the previous post a comparison was made between the math curriculum currently taught in grades 3-6 in my province (NL, CA) and the one used in the 1970s. It was noted that, while there’s an assumption that the scope of the content has been significantly expanded, the evidence does not support it. In fact, the present curriculum is not significantly different in terms of content from its predecessors. It was also pointed out though, that, perhaps the real issue is not with *what* is taught but, rather, with *how* it is done.

Besides the (erroneous, as it turns out) assumption that the current curriculum includes too much material—it’s often been called “a mile wide and an inch deep”—there’s also the assertion that current teaching practice is based on what some call “discovery learning.”

This is wrong on at least two levels:

- There is a misunderstanding about what is meant by the term.
- The teaching methodologies recommended today are not significantly different from the ones that were in vogue when most of today’s adults went to school.

This post, and the next, will justify the above statements. First, the misunderstanding. When many people hear, in the media, “the current curriculum advocates a *discovery learning* approach” they conclude that the teaching and learning of mathematics is left to the students’ own ingenuity; that students are expected to learn everything essentially unassisted. This, in turn, leads to a vision of frustrated students vainly trying to glean, in a noisy classroom, that which mathematicians took many centuries to build. Nobody can argue that this is anything but silly. After all, for just about everyone, learning mathematics is difficult and frustrating at the best of times. A situation in which students are left to discover all of it themselves it is nothing short of impossible.

That said, the above is not what is expected anyway. So, what is?

Let’s look at some of the language used in the curriculum guides, starting with a sentence right from page 1 of grade 4: “*Students need to explore problem-solving situations in order to develop personal strategies and become mathematically literate. They must come to understand that it is acceptable to solve problems in a variety of ways and that a variety of solutions may be acceptable*.”

At first blush–focusing on the term “explore”–this can be taken to mean, “leave the students to it. Anything goes.” That’s not the case, however. Contrary to popular opinion this statement does not mean that students must explore novel situations *unassisted*. It means, rather, that in the real world problems are complex and rarely boil down to a situation in which a pat algorithmic single solution can be implemented. The world is not a series of predefined story problems, but, rather a complex set of interacting priorities, situations and limitations and students need to be given something more than just a few multiplication facts to memorize. But there’s more–students also need time. Simply presenting a “problem” situation is not enough. When confronted with a novel situation sufficient time and support needs to provided so that the student clearly understands what it is they are facing. Too often, “problem” situations are briefly presented and then the student is led immediately to THE solution. While this might seem to make a degree of sense–after all the best solution to anything is generally far from obvious–the fact remains that jumping too quickly from problem to solution typically results in learning that is superficial at best.

When it is said that students need to explore the situation it is not meant that they should be left to stumble erratically to a solution, but, instead be allowed sufficient time so that they clearly understand the situation and then be led, through various means, to an acceptable solution that results in the greatest possible growth mathematically. It is expected that a sufficient degree of guidance needs to be given by the teacher and that this will vary somewhat from student to student owing to: differences in maturity level, knowledge background, previous mastery level and so on. It’s not simple.

There should be no doubt that the current math curriculum favors understanding over rote memorization. Consider, for example, this quote from the grade 3 guide, where multiplication is first introduced: “*Strategies for multiplication facts are a focus in Grade 4 and should not be the emphasis in Grade 3. The focus here is on understanding the meaning of multiplication and division and one to the other.*” Fine, understanding is key.

I know what you’re thinking. “Hey, but surely you’re not saying that a basic knowledge of math facts is unimportant. That’s stupid on about as many levels as I can think of. Besides not being to do everyday calculations, how about being able to do further math? After all, how do you expect someone to factor *x*^{2} + 14*x* + 48 if they don’t know 6 x 8 = 48?” As a side – note, one quick way to arrive at the solution to the above is to find two numbers whose sum is 14, the middle coefficient and whose product is 48, the last term. they are 6 and 8 so the solution is (*x* + 6)(*x* + 8)

Admittedly, you might be led to think that multiplication facts are unimportant by merely skimming through the guide. Consider this quote, for example, which comes from page 17 of the grade 4 guide: “*By decreasing emphasis on rote calculation, drill and practice, and the size of numbers used in paper and pencil calculations, more time is available for concept development*.” So what’t it saying? After all, to say the least, this one is open to interpretation. On the one hand it could easily be taken to mean that rote memorization of things such as multiplication facts, is not something that should be happening at all (a contention that I most certainly DO NOT agree with, in case you are wondering). On the other hand, you could choose, as I have, to take this to mean that teachers should ensure that rote memorization is *not seen as the one and only thing that is emphasized*.

It’s likely that this is closer to what is intended in the guide as evidenced by a quote on the same page, “*There should be a balance among mental mathematics and estimation, paper and pencil exercises, and the use of technology, including calculators and computers. Concepts should be introduced using manipulatives and be developed concretely, pictorially and symbolically.*” If this is true the real issue, then, is in finding that balance, not of eliminating the less fun activities, such as memorization. In other words it’s important that students learn math facts but doing this in the absence of other things is pointless.

It’s really too bad that the term “discovery learning” has made its way into the popular vocabulary, especially in light of the fact that the word “discovery” does measure greatly in how math, and for that matter just about everything, is taught. Perhaps, with that in mind, it’s wise then that we distinguish between two different uses of the term:

- (the bad use) Discovery is what happens when students arrive independently and essentially unassisted, at some major conclusion. This is something that is very difficult to cause to happen in a classroom. Sure, some students may be able to make it on their own but they are very mich the minority.
- (a better use) Discovery is what happens when cognition happens, through whatever means—presentation, direct teaching, performing various teacher guided learning activities and, yes, guided discovery. This is something that is difficult to manage but doable by a teacher who is suitably prepared. What it involves is a teacher who knows when to take the reigns and when to hand them over.

There’s one more thing. In no part of any guide is a “discovery learning” approach recommended. That part is just a misinterpretation of the real intent–to balance “the basics” with understanding and application.

There’s still the second issue: that is, the notion that the situation was radically different a generation ago.

They weren’t. Next post.

You know your stuff, Maurice!!