* or at least a small part thereof.

Last week I opened up a thread on what's wrong with many of the homework problems appearing under the banner of Common Core (if you click on the link, make sure to check out the comment section). The post included an example that's been bouncing around the internet for a while.

In this post, I'll drill a little deeper into the problem and discuss what the authors were probably trying to do, how they screwed up and what could be done to fix it.

The offending problem reads as follows:

4. Steven solves 7 x 3 using the distributive property. Show an example of what Steven's work might look like below.

I shared this problem with an engineer, a physicist and a statistician. All three had Ph.D.s and extraordinary analytic chops, but none of them got the problem (the engineer came the closest with a tentative "Surely they don't want you to..."). It wasn't that they didn't understand the mathematics; it was that they understood it too well. What threw them was that using the distributive property was such a silly choice in this context. Once the problem was changed slightly so that the approach made sense, all three got it immediately.

I'll discuss that tweak in a minute, but first, let's talk about what the problem was trying to do.

The idea that the creators are going after both here and in many other problems is that there are different ways of stating the same number. For example, we can think of nine simply as nine or as ten minus one or as three square or as 900% or any other number of ways.

Unfortunately, there is a second equally important part to this idea that the authors generally mishandle or leave out completely: we want to find the form of the number that makes the problem easiest.

It's true that 7x3 = 5x3 + 2x3 = 10x3 - 3x3 but there's no good reason to do the problem those ways. The students have very probably learned their multiplication tables by this point and even if they haven't, it's easier just to add 7+7+7.

So, how can we fix this problem?

For starters, we need to think about what is bookable. Some things are simply better explained in person and left out of assignments the students do on their own.

If we do decide that this is a concept we want to cover extensively in homework, we have to make sure that we are clearly and logically explaining what's going on. This is a particularly weak point in many Common Core based materials. Not only do the worksheets have incredibly inadequate explanations; the books are often even worse.

Finally and perhaps most importantly, we need to set up the problems so that the technique we are trying to teach makes sense.

There is no good reason to use the distributive property to multiply 7×3. There is, however, a not bad reason for using the distributive property to multiply 19×3 and a pretty good reason to use the distributive property to multiply 98×3 and a very good reason to use the distributive property to multiply 998×3.

If you try to do these problems in your head, you will probably find that it's much easier to think of them as

3(20 - 1)

3(100 - 2)

3(1000 - 2)

After tweaking, these problems not only illustrate the technique, they put it in an appropriate context.

This is, of course, a single example but it's consistent with what I've been seeing while helping students from various grades.with their math homework. A large number of the problems I see are so bad as to suggest either extreme carelessness or a profound lack of understanding of what the problems are supposed to accomplish.

p.s. I've also been writing on this topic at West Coast Stat Views for awhile now. You can see a collection of education reform posts here or you can just go to the site and search on 'education'.

The appropriateness of this problem depends on how far along the student is in learning the multiplication tables. At least when I was a kid, many decades back, we learned the multiplication table gradually, starting with small numbers and working our way up. If, for example, this problem was given to students who have learned multiplication up through, say, 5x5, but no farther, then figuring out how to calculate 7 X 3 using the distributive law would be sensible, and a worthwhile exploration.

ReplyDeleteOf course, if it was given to students who already know the whole thing, then it's idiotic.

I can't tell that level of context from what's posted here. (Maybe it's clear in some of the content that's too small for me to read?)

The distributive property is awfully abstract for kids who haven't gotten past 5x5, but even if we ignore that, this is still not the way to go.

DeleteThis kind of semi-open-ended lesson works MUCH better in class. ("let's see how many ways we can multiply seven times three and still get twenty-one").

If we do want to include these problems in homework we have to be clear about what we are asking. This problem has a vague guess-what-I'm-thinking quality. It never even explicitly says that there is more than one possible answer.

Here's how a better version of that question might read:

Since 6 = 1 + 5, the distributive property tells us that 4x6 = 4x1 + 4x5 (try it yourself with a calculator). Using the distributive property, there are lots of ways to multiply 7x3. Find at least two of them.

Extra sneaky bonus question: use the distributive property to multiply 7x3 so that none of the numbers being multiplied is greater than 3 (this is a tricky one but there are lots of possible answers).