Tuesday, January 29, 2013

"Number sense" with minimal number work

In my recent post on vocabulary, I wondered whether what holds for vocabulary words also holds for math facts:

not just that knowing them fosters higher test scores and broader intelligence, but also that they are best learned not as lists of "math facts," but in the course of doing actual, systematically structured math.  
In an earlier post on the abysmal mastery of math facts seen in children whose only exposure to math is Reform Math, I estimate that Reform Math children are getting less than one tenth the embedded practice with math facts as their traditional and overseas peers are.

It's interesting to see how often Reform Math problem sets skirt the issue of actual calculation in their attempts to nurture number sense. Here below, for example, we have, from 5th grade TERC / Investigations, (1) a problem set that substitutes multiplication for estimation (where Singapore Math would have students do both):


(2) one of many problem sets in a chapter on decimals that simply ask students to put specific decimals in order of size (while ordering of fractions may require actual calculation, ordering of decimals does not).


(3) a multiplication/division "skill check" that almost looks like it's asking for calculations, but carefully STOPS you before you begin:

I'm reminded of those foreign language software programs that promise fluency in a language without making you practice grammar rules. As with language, people would love to think there's a silver bullet for arithmetic mastery and algebra preparedness that avoids the productive drill and practice that so many elemenary school teachers assume is as tedious for their students as it is for them. But what I've seen even with basic number sense suggests that a fair amount of productive drill ("embedded learning" in the best sense of the term) is, for most students, absolutely necessary.

18 comments:

Maya Thiagarajan said...

This subject is something I've thought about a lot. If you want to get good at anything, you have to do it ALOT. If a kid wants to become really good at math, he/she has to practice math and do math alot. We seem to understand this inuitively when it comes to reading because, as a society, we emphasize reading to kids and raising kids in a print/book rich environment. However, the same should apply for Math. If we want kids to love math and become good at it, we should make them do a lot of math and math related activities, and we should raise them in math-rich environments. With regular math practice, they will certainly internalize math facts and number sense in much the same way as readers internalize vocabulary knowledge and grammar sense. An interesting post.

momof4 said...

The absolute necessity of explicit instruction and focused practice, in a structured and hierarchical format, is unquestioned in the area of sports and the arts. One does not advance until specific skills are mastered, whether it's swimming lessons, ballet lessons or piano lessons. Only academics are supposed to flower instantly, at the magic moment, without explicit instruction or practice. No effort is required. Such magical thinking is unconnected to the reality of any form of human endeavor.

Anonymous said...

But momof4, wouldn't a swim team where every participant was encouraged to find his own way to stroke be much more child-centered and holistic?

momof4 said...

No doubt, but I'm not going there. Chuckles

Despite ed-school insistence that any criticism, however well-deserved, will irrevocably damage kids, I don't believe it. Thinking back to summer rec-league swim meets, I remember the first meet after the league added a breaststroke event for the under 8s; all 6 of the boys and 5 of the girls were DQ'd for stroke/touch faults (each team swam 3 kids per event). After a week's hard work on breaststroke, all 6 of our swimmers swam legal breaststrokes at the next meet; they had learned a valuable lesson; doing it wrong has negative consequences and doing it right earns points for your team and maybe a personal ribbon. All of my kids were elite athletes and all had some disappointments; not making a team, not winning a game/tournament, not making meet cuts, not making it into finals etc. Such is life.

GoogleMaster said...

I'm left wondering whether "odd answers" refers to their mathematical non-evenness or to their unusualness. For example, look at the ones that are "some number that is not a multiple of ten" divided by 70 or 80. Is that odd? It might be unusual, for a fifth grader, especially one raised on EM.

Cynthia812 said...

When I tutored a 3rd grader who used reform math, my two biggest issues were 1)Lack of problems, no way to follow with a second problem to see if he really got it, and 2)the massive amount of writing required to "explain your reasoning" in a meaningful, precise, and accurate way. Reform Math fosters a lack of precision that is going to haunt these kids forever.

Allan Folz said...

I must be a horrible parent. When I see my son guessing... er, estimating, I tell him to stop being lazy and work-out the answer.

There's value in that. I remember well my high school Physics teacher and the exasperation we would express when the class would be guessing at answers based on their intuition instead of doing the harder work of computing an answer. Part of the point of high school physics is, "you don't know what you think is so."

When you're an expert and have a basis of experience from which to draw only then does it makes sense to estimate. Conversely, if you are still learning, by definition you don't have the expertise from which to draw an answer and so you should be carefully working out each problem. Ignorance may be bliss, but it's no way to run a country.

As for all this give your reasons... I also remember in high school trying to pass off thin opinion pieces as research reports. I had one teacher in particular call me on it, to which I said I was using my critical reasoning skills. He not so gingerly explained that he didn't give a fig about what I thought or how I reasoned to think it. Further, until I had a PhD after my name no one else did either. So, get to work, find some facts, cite my facts, and may be then, with my assertions properly attributed he'd be interested in reading what I had to say. I'm sure he retired a long time ago. :-)

Anonymous said...

Estimating and guessing aren't really the same thing, and there certainly is value in teaching kids a method they can use to find out if the answer they get when working out a problem is reasonable. I have started making my niece do a quick estimation before doing any math because otherwise she make silly errors and cannot catch them. Checking her work doesn't seem to catch them either, because she makes the same errors again, but in reverse. So working out in advance that 2372 x 94 is going to be CLOSE to 237200 is good, because when she foolishly gets a much larger or smaller number she can automatically self-correct.

momof4 said...

Anonymous: In order to estimate correctly, as in the problem you mentioned, it is necessary to have a certain amount of number sense - and far too many don't have it. Clueless. Visiting a bakery when during a power outage, I had to teach the 4 counter people how to figure sales tax (with calculator) but none realized that an "answer" of $16 for a $10 purchase with 6% sales tax was wrong. They had no understanding that 6% means 6 cents per dollar.

Anonymous said...

I agree. (Same anon. For whatever reason, openId doesn't like me on this machine.) However, I read the comment as not saying "teaching estimation without teaching calculation is wrong, as is failing to give any real time to practice either skill" but as saying "there is never a good reason to teach estimation, and doing estimation instead of calculation is always lazy and morally suspect". And the latter is simply not true.

For one thing, outside of grade school (where the two should always be taught together to back each other up), there are many times where I don't want or need the one right answer and a fast, close enough answer I can work out in 20 seconds is good enough. For example, adding up groceries as I shop to make sure I don't go over budget. If I wanted accuracy I would add up the exact prices and the 8.25 sales tax, but since I don't care I round everything up to the nearest quarter

Anonymous said...

And take ten percent. This gets me an answer higher than my actual bill, which is just fine by me. (Earlier comment got cut off. See what I mean about this thing hating me?)

portland_allan said...
This comment has been removed by the author.
Allan Folz said...

I fear you may have read too much. :-) I suppose I should have stated my son is 9. He was guessing. :-)

...as would be virtually all the kids doing these RM worksheets. Or are there a couple dozen exercises that come before these pages and our blog host is cherry-picking the examples. ;-)

Whether the children are making sincere attempts to estimate or lazy WAG's, without the expertise to compute the answer adeptly by long-hand they are worthless guesses and it's a lazy man's habit to get into. It also makes one easily fooled. That's the point of the high school physics, to show you your intuition is often be really bad, even when it sounds good. You need to sit down and work out an answer.

(I should clarify there's rigorous estimating which is what you're doing at the grocery store, and trying to intuit an answer which is what this worksheet seems to be exercising.)

Cheers.

Anonymous said...

My niece is nine too! And she was taught how to estimate in the first grade, as I was back in the late 80s. At home we are doing Singapore math, and she is in the 4a workbook, and sure enough (I just checked) she is expected to estimate her answer when doing long division or multiplication of two by two digits prior to solving the problem.

If your son is guessing, the problem is actually not with asking him to estimate (presuming, of course, that he is still being asked to calculate) but with the fact that somehow he has not learned how to estimate correctly. It might have not been taught, but it really should have been. Estimating and guessing are two entirely different things. Estimating is a good sanity check. Guessing is, well, guessing.

Jen said...

Yes, one of the few things (maybe only?) that is better with the math now than when I was taught, long long ago, is the estimating.

Most reform math programs actually teach estimating -- the problem is that they don't follow up with enough practice or with the actual calculation.

By third grade at the latest, a child should be able to look at a problem and have a reasonable guess that will be used to see if they made a horrible mistake in the calculation.

Since this aspect of it isn't really emphasized though, you just end up with bad estimators and bad calculators and students who have no idea where it all went wrong!

Jen said...

I keep thinking of the lack of practice in terms of self-esteem.

Remember when everyone thought self-esteem developed from a child being extravagantly praised for every little thing and being shielded from any criticism.

Then everyone (well, everyone doing research) realized that self-esteem actually comes from growth and accomplishment. That is, that practicing something enough to get better at it and noting that growth leads to esteem.

The option for growth and the sense of efficacy (or math self-esteem) is lost without mastery. There is no sense of having learned something that you couldn't do before is lost in the "spiral" and in the "exposure" to many methods without a real working knowledge with any of them.

Just as those overpraised children ended up doubting their abilities (and with higher levels of anxiety and depressive qualities, if I recall correctly) so do kids without mastery of basic skills end up guessing and feeling doubtful when doing what should be simple problems.

Jen said...

Oh heavens. I wish there were an edit button. I plead sinus pain for my many left out words and other errors!

Anonymous said...

Yes, the problem in this is definitely the lack of follow through and practice. On this, I have absolutely no argument.