The response? Enthusiasm from students and parents; skepticism from teachers.
Specifically, about "stacking." (Today's word for how we used to add, subtract, and multiply numbers by placing one number on top of the other.)
Kids love it. And not just the ones on our team. As as friend writes:
When I showed one of my sons how I had learned addition, i.e. the "stacking" method, he was very impressed. "Wow, that's so cool! That works great! I wonder if my math teacher knows about this?" was his innocent comment.
Yes, she does, and she doesn't like it. At least if she resembles the teacher who approached me after math practice yesterday and recounted the dismay she felt when she caught one of her students stacking numbers, thus abandoning the more "meaningful" and "faster" way he used to solve problems.
My co-coach and I tried to explain that the Continental Math League numbers are big enough, and random enough, that Reform Math's methods aren't faster and more meaningful, but inefficient and confusing. It's one thing to add 48 and 39 by reasoning that:
48 is 2 less than 50, and 39 is 1 less than 40, so add 40 and 50 and get 90 and then count backwards by 3 and get 87."
But take one of the problems we did at Continental Math League practice yesterday: 825 - 267. Restricting myself to the kinds of calculation that these second and third graders are able/expected to do in their heads, here's the most efficient non-stacking method I can come up with:
The closest friendly number to 825 is 800, and the closest friendly number to 267 is 250. 825 is 25 more than 800. 250 is 10 more than 260, and another 7 gets you 267. 10 plus 7 is 17. So 267 is 17 more than 250. So subtract 250 from 800. Well, 800 minus 200 is 600, minus 50 more is 550. Then subtract 17 from 25 by counting up from 17. Seventeen plus 3 more is 20 plus 5 more is 25. 3 plus 5 equals 8. Add 8 to 550* to get 558."
*By this point in the problem, how many people remember what they should be doing with this 8?
Anyone with a more efficient non-stacking method for subtracting 267 from 825 (no calculators allowed!) is invited to share it here.
(Cross posted at Out in Left Field).
I'm not familiar with the term 'stacking'--are you saying that's the way to describe the ordinary arithmetic method we all learned in elementary school, where you write the numbers in a 'stack' and then add them up or whatever you're going to do with them?
ReplyDeleteI guess I don't see why teachers would object to a method that illustrates place value so nicely.
--a Saxon-using homeschooler
My kid as a new third grader who had never done pencil/paper subtraction would think like this:
ReplyDeletesay 825-267
first, subtract 225...that gets you 600-42 which is 558.
He most likely would have thought 600-42=500+100-42 for that last step but the 100-42 is simple mental math for anyone that has mastered 'make a ten' and understands regrouping.
Some of his classmates could do it; they were coming from a teacher that used Everyday Math and were strong on the doubles+1, doubles-2 type of thinking.
In teaching our dd, my spouse uses essentially the same algorithm used in soroban (abacus) subtraction:
ReplyDeleteIf a digit of the subtrahend is too large to be subtracted from the minuend, the tens complement of the subtrahend's digit is added to the corresponding digit of the minuend, and then the minuend's digit immediately to the left is reduced by one.
(It's easier than it sounds, and of course with a little practice it is very easy to achieve proficiency. People that train on the soroban generally excel at L-to-R calculations, which is extremely helpful for mental math.)
In your example (825 - 267):
Ones digit: 7 is too large to subtract from 5, so 3 is added to the 5 (giving 8) and the 2 is reduced to 1.
Tens digit: 6 is too large to subtract from 1, so 4 is added to the 1 (giving 5) and the 8 is reduced to 7.
Hundreds digit: 7 minus 2 is 5.
Answer: 558
(I gave this illustration as R-to-L since most of us are used to that way, but with a little practice it can be done L-to-R.)
"Anyone with a more efficient non-stacking method for subtracting 267 from 825 (no calculators allowed!) is invited to share it here."
ReplyDeleteWhenever I see a basic math problem, I never do it just one way. That doesn't mean that I consider the traditional algorithm less important. Even though I use a calculator for many things, doing the traditional algorithms by hand is a good exercise in rigor and accuracy.
Generally, my calculations break into a need for an exact number, or a need for an estimate. For exact calculations, I use the traditional methods or a calculator. For estimates or simple problems, I often do the calculations left-to-right, or use some other method, depending on the numbers.
For 825 - 267, I like to use the count-up technique (I don't know if it has an official name). I think they talk about it in "Arithmetricks".
I want to calculate the difference by counting up from 267. First, I ignore the difference from 267 to 300, and count up to 800. I get 500. Then I add the 33 I skipped and the 25. For adding the 33 and 25, I do it left to right; I add 30 to 20, and then I add 5 + 3. Left-to-right is great for large numbers because I can quit whenever I have enough digits of accuracy.
I guess that for some people, it's a lot faster to see the difference (in this case) between 67 and 100, and then add that to 25, then it is to do the traditional method in your head. My wife uses the traditional method because she can remember the numbers that are generated backwards in her head. I can't do that.
The count up technique is also good with estimation because you easily find the major difference between the two numbers. Then you can spend more time if you need more accuracy.
Thanks to everyone for this interesting variety of methods, which I hadn't considered!
ReplyDeleteNow I'm wondering how well the various non-stacking strategies work with four or five digit numbers, and/or with three or more multi-digit numbers added together.
My guess is that, at some point, any method than the stacking up methods (whether the traditional regrouping ones or niels' abacus subtraction method) will run up against short term memory limitations. But perhaps I'm wrong?
Short-term memory constraints don't seem to be much of an issue in the following cases:
ReplyDeleteCase 1
Case 2
Vedic math and/or the Trachtenberg system might also be promising for mental math, but I haven't delved into either one of those.
Cool-- I hadn't considered this exotic world of mental math tricks. Would love to explore it further, time permitting.
ReplyDeleteFor now, though, in light of the Constructivist orthodoxy I've been bumping up against, my follow-up question is this. Are there methods for performing arithmetic calculations on arbitrary numbers that:
1. don't run bump up against short-term memory limitations
2. children can discover on their own without explicit teaching?
I approach arithmetic much like Steve H does. I often use the counting up technique he describes with larger numbers. However, I consider these techniques to be shortcuts, and I would never expect a kid to use these from the beginning. The traditional "stacking" method really is useful, because it demonstrates place value and it will always work. You really have to have a pretty good number sense to figure out the most efficient way to solve an arithmetic problem if you're going to use the shortcuts, and young kids just aren't ready to do that without a lot of practice. I think the mental math "tricks" are useful and important, but the algorithms should be mastered first.
ReplyDeleteI agree. Tricks are fun and quite useful, but they are not a replacement for the standard algorithms. Besides, there are too many other things to do. Perhaps you could come back to these shortcuts in algebra class to see how they are derived.
ReplyDeleteIt wouldn't be at the top of my list of things to teach. If I did teach them, I would talk about one "Arithmetricks" trick per week.
My guess is that, at some point, any method than the stacking up methods (whether the traditional regrouping ones or niels' abacus subtraction method) will run up against short term memory limitations. But perhaps I'm wrong?
ReplyDeleteDon't forget decimal places. The world does not consist entirely of integers.
"Stacking" also runs up into short-term memory limitations if you try to do it without writing anything down. Trying to compare "mental math" strategies to "written math" strategies is bound to lead to the conclusion that the written math is better. The thing is, all of the mental math strategies that have been suggested (counting up, the abacus approach, etc.) can be reinforced by writing down intermediate results -- which takes the short-term memory problem right out of the game.
ReplyDeleteMy guess is that, at some point, any method than the stacking up methods (whether the traditional regrouping ones or niels' abacus subtraction method) will run up against short term memory limitations. But perhaps I'm wrong?
ReplyDeleteSure, but these methods work better when you don't have pen and paper handy, or can't be bothered. For numbers up to four or five digits - counting decimals as digits! - it is often easier and more convenient to add in your head rather than fussing about looking for scrap paper.
And the standard algorithm isn't very well suited for doing in your head. It's much better on paper than the other methods, but that's only an advantage when you have paper.
There is a great benefit in teaching children how to add on pen and paper. However, that doesn't mean we should ignore and disparage mental math.