Using mental math vs. paper and pencil

The example of decimal division I found recently made me think some more about how we teach our students the use of mental math.

For example, how would you want your (current or future) high schooler to find the answer to 5 × 24 or maybe 6 × 71 and such like?

a) I wish he'll use pencil and paper, put 5 under 24 and use the algorithm.
b) I wish he will go 5 × 20 and 5 × 4 in his head and add those.
c) I wish he grabs the calculator.
d) I wish he asks me.

Also think WHICH of the four options is the most efficient 'tool' for finding the answer? After all, just like a carpenter, a good problem solver knows which tools are available and which one is the best to use for a particular task.

Now let's back up in time when the student is in third grade. How are we encouraging third graders to do the same problem? As you may realize, kids usually spend lots of time mastering the multiplication algorithm (the pencil & paper method) and go thru tons of practice problems similar to my examples.

But, what if we FIRST let the third-graders do such problems mentally, and only afterwards taught the algorithm?

Wouldn't that make them more apt in using their head for such problems instead of the long algorithm?

(And they'd also understand the logic behind the algorithm better, I think.)

I tried to check and I didn't find any online activities for this exact thing. I even tried searching for 'distributive property quiz' (since the mental calculation is based on distributive property) but that only brought up algebra type things.

Obviously you can just practice by calling out numbers to your students, but if you want to have a worksheet, you can make such worksheets here: choose multiplication, horizontal, value 1 to be from 2 to 9, and value 2 to be from 11 to 99 - and hit SUBMIT.

You can make easier ones, too, if you choose value 1 to be from 2 to 5, say. Or, choose value 2 to be from 10 to 100 with step 5.

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Comments

Anonymous said…
Hi! I thank you for having this blog. I enjoy reading it. As a mom who is not fond of math yet finds herself teaching it, I wish you would do a post explaining exactly what logarithms are. My daughter is not satisfied with my explanations. She is 11 years old.
Maria Miller said…
I assume you're meaning ALgorithms, not logarithms...

Algorithm is a fairly mechanical step-by-step procedure to solve a problem.

For example, the multiplication algorithm I was talking about in the blogpost is this:

76
x 8
--------


Go 8 x 6. Write the ones digit of that number under 8. Record the tens digit somewhere. Then go 8 x 7. Add to that result the tens digit from before. Now write the result of that just directlyin front of the number you already wrote under 8.

You can even 'teach' or program the computer to do this algorithm or step-by-step procedure.
Anonymous said…
Try using key words like "problem solving using decomposition" or problem solving, decomposing numbers." Constructivist programs like Investigations Math tend to use these terms.
-Patricia
Anonymous said…
I did actually mean logarithms. My daughter wanted me to explain them, and all I could come up with was that it is another way to talk about exponents. We searched the web, but I didn't think about looking on your website. Perhaps I will do that.
Michael Weiss said…
I guess I'm a latecomer to your blog, but:

In answer to your original question ("How would you like your child to solve 5 x 24?"):

I would be thrilled it my kids multiplied 24 x 10 and then cut the answer in half. This kind of ad hoc approach doesn't get taught in schools, but they illustrate a comprehensive understanding of the underlying meaning of the symbols far better, in my opinion, than do most of the other approaches.
Anonymous said…
I hope this helps:

A good way to conceptualise logarithms is that asking what the logarithm of a certain number is, is like asking how many digits the number has. For example, (assuming base 10, of course) log(100) = 2, log(1000) = 4, and so on.
The caveat is that (a) it's not LITERALLY the same. It's off by one. log(1) = 0, not 1. log(10) = 1, not 2. And (b) asking what the logarithm of a number is is a little more general and useful than asking how many digits because: you can only have whole number amounts of digits, but if you have a number in between, you can get a fractional logarithm. Eg, log(100) = 2, and log(1000) = 3. Well, log(31) is approximately 2.5. And the (c) third caveat is that you can have logarithms in other bases, not just 10. Such as base 2, which rather than telling you how many times you'd have to multiply by 10 to get your number, would tell you how many times you had to double. So LogBaseTwo(8) = 3 and LogBaseTwo(65536) = 16. Changing from one base to another just involves multiplying by a constant, which is useful. E.g, if you have to double something 10 times, you must have quadrupled it.... 5 times (The constant to go from base 2 to base 4 is 1/2... which is LogBaseFour(2)).

Finally, the most useful property of logarithms is that the more you increase the number you're taking the logarithm of, even a lot, the more and more slowly the answer will grow, as you can see. This is just the opposite of exponential growth, where you start doubling and before you have a moment to breathe you can't see a thing anymore.

...And that is how I first remembered the meaning of logarithms back when I learned them in school: every time I saw the word "logarithm" I mentally replaced it with "digit level".
Maria Miller said…
Thank you for the detailed comment on logarithms... it would almost better belong to my post on logarithms.

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