Tuesday, May 24, 2011

Addition and subtraction facts

For all their renunciation of "drill & kill," Reform Math programs and teachers at least pay lip service to the importance of knowing your "math facts," and some Reform Math exercises even have exercises that solicit recall of these facts explicitly.

Indeed, there's actually more emphasis than in pre-Reform Math days on so-called "addition and subtraction facts"--facts that I don't remember ever having to learn as such. Why, in this day and age, are people more concerned with addition and subtraction facts than ever before?

The answer becomes clear when you contrast Reform Math with more traditional programs. If you compare both the number of problems per worksheet, and the number of individual addition and subtraction operations per problem, you find that Reform Math has students doing only a tiny fraction of the quantity of addition and subtraction operations that traditional math used to require.

Where a traditional math worksheet might have had 20 or more arithmetic problems, a Reform Math worksheet might have only a handful, with most of the space taken up with words, pictures, and room in which to explain your answer in words or pictures.

Where a traditional math worksheet, starting with second or third grade, would have had multi-digit problems in which students must conduct multiple individual addition or subtraction operations per problem (up to n × m operations in an addition problem involving n numbers with m digits and multiple instances of regrouping), Reform Math worksheets don't routinely involve multi-digit problems until 3rd grade and restrict themselves to fewer digits and fewer numbers added together per problem (in general, n and m are much smaller).

Reform Math programs are especially averse to multi-digit multiplication problems in which both factors have 2 or more digits. While multiplication is the primary operation, there's quite amount of addition involved, both at the end and whenever there's regrouping. Even rarer in Reform Math is long division, which involves multiple instances of subtraction.

The upshot of all this is that Reform Math students have many fewer opportunities to practice even the most basic addition and subtraction operations than traditional math students do. And the irony of Reform Math's "solution" to this problem--how it calls on students to memorize their addition and subtraction facts--is that it must resort to mandating explicitly what students used to accomplish (or "construct") on their own.


Deirdre Mundy said...

You know, we had this class is school that focused on 'drill and kill.' It was a grind. It had no relevance to my life. Sure, the kids with natural ability excelled and went on in the field, but for the rest of us? Endless drills. I'm sure we would have learned much more if they'd just turned us loose and let us 'explore' and 'discover' the techniques on our own.

If only we'd had 'reform gym' instead of 'drill and kill,' maybe I'd be in the WNBA today. Or maybe not. Because maybe what made the good players good was their WILLINGNESS to work hard, in and out of school, to master drills llike dribbling in and out of cones, shooting endless free throws, etc.

And maybe, without the "drill and kill," instead of being a hopeless case who can at least dribble and shoot a little, I'd be so pathetic that all I could do is stand still and toss the ball granny-style at the other kids on the court.

So--- why is drill and kill evil and deadening in Math class, BUT with the same kids, in gym class and on sports teams, a necessary step in learning to excel at the game?

I understand that a lot of journalists didn't like math. I loathed gym. But letting me follow my inclinations (sit on the bleachers and read fantasy) would have resulted in even WORSE outcomes. At least now I have a concept of a bounce pass, even if I still often fail on the execution.

Anonymous said...

Everyday math plus rocket math for drill and kill at my kids school.

Decided to enroll DD in a online Singapore Math class instead of working on math facts directly and it seems to be working...

Rather have her learn this stuff while doing problems.

Deirdre Mundy said...

With my 7 year old we spend a bit of time every day drilling (flash cards, mad minutes, etc.) Basically, she has her math facts down, but she needs to get faster. And she enjoys mad minutes because she tries to beat her old scores, and as she gets better we're adding more problems for each minute.

Of course, we don't really do the "and kill" thing--- drilling for a few minutes at a time scattered throughout the day (including in the car, during chores, etc.) is more effective than spending 20 minutes straight on it......

Annic said...

I think memorizing math facts is a totally different skill than understanding math. Perhaps it depends on how fast you want them to be. My son never got fast at math facts, never could remember off the top of his head what 8 x 7 is, and went on to get an engineering degree. Fortunately, I did not kill his love of math by too much drill. Of course, he could figure out 8 x 7 from 8 x 5 + 8 x 2, mentally, it just took him a longer.

Then there is "mental math" which is different from memorizing math facts. It is more applying strategies (and math facts) to solve more complex problems (not just add, subt, mult, div of 1-digit numbers) in your head. It does not have to be a speed thing.

Learning a skill like dribbling is a lot different to someone who likes sports than memorizing the answer learning how to do 100 such problems in 5 minutes to someone who likes math. Sure the skill is important, but how fast you can spit out the answer is a very minor thing in solving a complex word problem or applying "problem solving skills". Memorizing math facts is not math. It is more like spelling, or memorizing dates, or something like that.

At least, that is my opinion.

Deirdre Mundy said...

Well, I found that for algebra 2, trig, etc, being able to add, subtract, multiply and divide quickly made everything else easier--Oh, and physics and chem too! Balancing equations is so much easier when you don't need to use fingers!

Also, being able to compute quickly and accurately helps you see patterns and connections more quickly.... and comes in handy at the bake sale, too.

Computation isn't the ONLY thing in math, but it's useful-- and often, when I taught, the kids who didn't 'get' algebra didn't get fractions... because their multiplication was so laborious that they couldn't see how things fit together--they got caught up in the minutiae....

I guess I see computation a lot like cooking. You don't have to know what 'dice' means off the top of your head. You can figure it out (smaller than chop... bigger than mince) or look it up each time the recipe calls for dicing..... but if you DON'T instantly see dice and start dicing it makes recipes harder and cooking less fun.

Anonymous said...

Annic, yes, memorizing math facts is a completely different skill than calculation and computation. Math teachers need to be learning the most current brain research. Learning multiplication facts uses the same part of the brain that is used for learning words and memorizing language.

The biggest problem is that children who do not know their multiplication tables can be labeled as weak math students. The reality is that they either have language/memory issues or have just not spent the time learning basic math facts.

Using basic strategies (like if I can't remember 6x7, then do 6x6 and add 6) do NOT reinforce the learning of math facts. Repetition and use is the way the brain learns those.

http://web.mit.edu/newsoffice/1999/math-0512.html (there's more recent stuff, but that's what I found with a quick search)

Connect this to the reality that Asian languages have a stronger connection between numbers and language without people needing to learn new words. E.g., ten-two instead of twelve. Children can only absorb a finite amount of vocabulary.

C T said...

I find it odd that people memorize subtraction facts as such. Once people have addition facts down, they know their subtraction facts, too, don't they?