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I love that phrase: rote understanding.
Exactly.
I was interested to see that Barry was taught "making ten" when he was in grade school:
The “making ten” method is included in the math program used in Singapore—a nation whose fourth and eighth graders have consistently obtained the highest scores in international math tests. Specifically, in Singapore’s Primary Math textbook for first grade, the procedure for adding by “making tens” is explained. Of particular importance, however, is that the procedure is not the only one used, nor are first graders forced to use it. This may be because many first graders likely come to learn that 8 + 6 equals 14 through memorization, without having to repeatedly compose and decompose numbers in order to achieve the “deep understanding” of addition and subtraction that standards-writers—and the interpreters of same—feel is necessary for six-year-olds.I have a question about the teacher's explanation of the number 6:
“Making tens” is not limited to Singapore’s math textbooks, nor is it by any means a new strategy. It has been used for years, as it was in my third-grade arithmetic textbook, written in 1955...
“So if we can partner 9 to a number and anchor 10, we can help our students see what 9 plus 6 is. So we’re going to decompose our 6, and we know 6 is made up of parts. One of its parts is a 1 and the other part is a 5.How do mathematicians think about whole numbers?
Do they see them as "made up of parts"?
Or as decomposable into parts?
(Or both --- ?)
To me, "made up of" and "decomposable into" seem like two different things.
Another question: if 6 is "made up of parts," is 6 one of the parts?
Is 0?
I bet right this minute there are kids all over America who are royally confused by the ramifications of making ten.
7 comments:
Thanks for the link to my article. Of further interest is Katharine Beals take on all this at her OILF blog
There are several issues blended together here: mathematics, the technology of the decimal system, and human cognition.
Yes, mathematicians do think about decomposing whole numbers into parts in various ways--the best example being prime factorization--but they are, in general, no more interested in rearranging for groups of ten than for groups of nine or eleven.
Rearranging for groups of ten is a convenience technique that is only relevant to the "decimal system" technology for representing numbers, not especially relevant to the numbers themselves. Mathematicians, when thinking about math, as opposed to ordinary daily activities, don't think of 27 as two tens and seven ones. "27" is not the number, it's just one of countless ways of representing the number. (Well, probably countable to Georg Cantor.... Apologies for the parenthetical math joke.)
To a mathematician, 27 is a point on the number line (THE number line, not a physical representation of it), or the number you get if you add one to 26, or the product of three 3s, or the "cardinality" (count) of a group of 27 sheep, but none of these are representations, nor are they base ten. They aren't base anything. They are the number twenty-seven itself.
Representations such as the decimal system or algorithms such as long division are mathematically valid but are technologies chosen for their usefulness in daily human life, not for any mathematical significance. They are human cultural artifacts.
And a technology such as regrouping into tens is chosen for its usefulness in helping the human cognitive system work more easily with the human technology of the decimal system. Computers, for example, don't regroup (even in binary), because it wouldn't help them. Their number processing is implemented differently from human brains, so they require different convenience technologies.
The properties of numbers are independent of human cognition or machine design, and those are what mathematicians usually care about. The technologies of working with numbers are of great value, too, and need to be learned, and breaking numbers down in various ways (and yes, 6 can reasonably be defined as one of the "parts" of 6) is mathematically valid, practically useful, and a reasonable thing to teach kids to do.
"'27' is not the number, it's just one of countless ways of representing the number. (Well, probably countable to Georg Cantor.... Apologies for the parenthetical math joke.)"
You started it, but ... 27 can be represented in an uncountable number of ways ... even for Georg Cantor.
-Mark Roulo
"Rote understanding" -- that is BRILLIANT. I think it's my new favorite phrase.
As with Barry, I was taught these things when I was in K-6 in the 50's. I also remember how one of my son's teachers made a big deal about it as if this was a key to understanding. No. It's just one thing and it works better for some than others. Modern K-6 educators believe that they have just discovered understanding after years of rote teaching. They haven't done their homework.
It's rote understanding because teachers just want a process and not a feedback loop. And while they wait for the yearly state test results for feedback, what are they doing day-to-day? Trusting the spiral. That's specifically what Everyday Math tells teachers to do - trust the rote process. Trust engagement. Trust that the "active learning" in class actually gets the job done. When you let all students get into different levels because of full inclusion, teachers can't provide 'N' different feedback loops. They just walk around class helping out here and there - as a rote process - taking no responsibility for making sure understanding is achieved on an individual basis. Their understanding of math understanding is rote because they don't know enough math to have a real understanding.
"I bet right this minute there are kids all over America who are royally confused by the ramifications of making ten."
They are probably confused by the pedantic (rote) approach. I remember a unit in Everyday Math that was called "What is the One?" which had to do with fractions. EM talks about finding and using your own algorithms, like the lattice method, but do teachers ensure that individual students achieve any particular goal? No. EM moves right along and assumes that if they throw in some "Math Boxes" that students can catch up or fix themselves. It's a rote process. There is no explanation of understanding that can change math into a rote process.
The fundamental problem in K-6 is full inclusion. Modern educational pedagogy is built around justifying that goal, and talk of understanding in math is just cover for lower expectations. CC defines no STEM path in K-6. This is not just about slight differences of opinion over what understanding means. K-6 educators can't have both a rote full inclusion process and some sort of best math understanding.
As with Barry, I was taught these things when I was in K-6 in the 50's. I also remember how one of my son's teachers made a big deal about it as if this was a key to understanding. No. It's just one thing and it works better for some than others. Modern K-6 educators believe that they have just discovered understanding after years of rote teaching. They haven't done their homework. lokai germany , lokai usa
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