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So let me give you an example of how one of the more difficult elementary math topics might be traditionally taught and you tell me if it's learning by rote.
Today's topic will be subtraction with regrouping (tens and ones). An example of such a problem is 66 - 37 = ?. Ordinarily, this topic gets taught after the student has learned how to do (and is firm on) subtraction without regroup (66 - 34 = ?). Let's further assume that the student knows how to do place value addition. This means the student knows how to decompose the number 66 into 60 + 6. In other words, the students knows that the number 66 comprises 6 tens and 6 ones.
Here's how the lesson might get taught traditionally:
Lesson One
I. Model Phase
When you work subtraction problems that use borrowing, you have to rewrite numerals so you have a new place-value addition. I'll show you how the new place value works.
[Write the number 36 on the board]
We're going to rewrite 36 for borrowing. We'll borrow 1 ten from the tens column and add that ten to the ones column.
How many tens do we start with? [point to the 3][students: 3]
I cross out the 3 and write the number that is 1 less than 3. What number is that? [students: 2]
Now I take the ten I borrowed and write it small in front of the 6.
The new place-value addition is 20 plus 16 equals 36.
We still have 36 because 20 plus 16 equals 36.
[repeat with a different number such as 57]
II. Lead Phase (if necessary)
Write the number 56 on the board.
Your turn. Cross out the 5 and write the number above it that is one less than 5. Then write the 1 ten you borrowed small in front of the 6. Raise your hand when finished.
(observe students and give feedback)
Check your work. Here's what you should have.
Everybody, say the new place-value addition for 56. [students: 40 plus 16 equals 56]
[repeat with another example if necessary]
III. Test Phase
[Write the numerals 84, 51, 45, and 72 on the board.]
Rewrite these numerals. Raise your hand when you're finished.
[Write on the board:]
Check your work. Here's what you should have.
Fix up any problems you got wrong.End Lesson
[After the students are firm on the regrouping procedure, it's time to go on to using the procedure to solve subtraction problems]
Lesson Two
[Write on the board:]
You're going to do borrowing. For some column problems, you have to rewrite the top number so you can subtract. For other problems, you just subtract.
Here's how you figure out whether you need to borrow: You read the problem in the ones column. If the bottom number is bigger than the top number, you can't work the problem in that column, so you have to borrow.
Everybody, read the problem in the ones column. [students: 5 minus 5]
Can you work that problem? [students: yes]
So you don't have to borrow.
[Change the problem to:]
Can you work this problem? [students: No]
So you have to borrow.
[Repeat with a few more examples where borrowing is needed and not needed]
Give students a worksheet with the column subtraction problem: 53 -19 = ?
For this problem, you have to borrow because you can't work the problem in the ones column.
Rewrite the top number.
[Write on board:]
Check your work. Here's what you should have.
You'll make silly mistakes when you subtract unless you're careful about reading the new problem in the ones column.
I'll read the new problem in the ones column. 13 minus 9. That's a problem you can work.
You're going to work the problem now. Read the problem in the ones column. [students: 13 minus 9]. Read the problem in the tens column [students: 4 minus 1]
Write the answer to the problem. [check students work]
End Lesson
These lessons are taken from lessons 7 through 9 of Connecting Math Concepts, Level C which I've condensed a bit.
So tell me does anything in this lesson even remotely resemble rote learning?
The comments are open.
4 comments:
It resembles how I was taught borrowing using the text "Arithmetic we Need", copyright 1955. And for your information, the same explanation is offered in a text called Ray's New Arithmetic, copyright 1877.
So in a reprise to your question, tell me, how is the way math was taught in the 60's (and 1800's for that matter) not teaching the concepts?
It is impossible to overemphasize the role false dichotomies and the fixation on supposed "rote" learning plays in the imagination of educationists. They are a major contributor to dumbed-down education. Educationists cannot conceive of committing knowledge to memory other than by "rote". Since "rote" is bad, knowledge is out.
I have a few musings on the subject here
Regarding subtraction with borrowing, constructivists would be proud that I developed my own algorithm. Instead of trying to do something with the minuend (confusing when dealing with a zero) I simply add 1 to the subtrahend. I find this to be must simpler.
"So tell me does anything in this lesson even remotely resemble rote learning?"
No, because the students understand what borrowing 10 from the 10's column means.
The problem is not learning an algorithm. Fuzzies do that with things like forgiving division. The key ingredient to most of their algorithms is that they don't do mulitple steps at once. The logic of the algorithm (except for perhaps the Lattice Method) is clearly visible, like with partial sums. They don't require a solid mastery of the basic facts; adds and subtracts to twenty and the multiplication table.
Traditional (more efficient) algorithms require more calculating in the head and that might seem to be more like magic or "rote" to some. Also, efficiency is not required if you say that the kids can use calculators. What they fail to realize is that all of the traditional long division and multiplication problems are a great way to fully master the basics while covering more advanced material.
I have a better "rote" problem. Convert a mixed number, like 4 7/8 to a single fractional form. My 5th grade son spiraled back to this in last night's EM homework. This is something he saw, maybe once, last year, and there was no preparation for it in class this year. Morons! [Ken, you give these people way too much credit when you try to logically argue the concept of rote.]
A rote method is to take the denominator and multiply it times the integer, then add in the numerator. Put this number over the original denominator. My son got a little impatient when I required him to follow my mathematical explanation:
4 7/8 = 4 "and" 7/8
4 "and" 7/8 = 4 + 7/8
= 4/1 + 7/8
= 8/8*4/1 + 7/8
= 32/8 + 7/8
= 39/8
However, it isn't clear how the fuzzies intend to teach understanding when the kids don't yet know how to add and multiply fractions.
OK, I dug out my son's EM Student Reference Book and am looking under the section on fractions. They start with "Equivalent Fractions"
"Find the equivalent fractions for 3/4."
How do they do this? They use a "Fraction-Stick Chart". You can't get any more rote than that!
Method 2 - Using Multiplication.
"Rename [sic] 3/7 as a fraction with the denominator 21."
"Multiply the numerator and the denominator of 3/7 by 3."
That's it. No explanation for why this can be done. Of course, we all know that "rename" is such a fine mathematical term full of understanding.
Method 3 - Using Division
"If the numerator and the denominator of a fraction are both divided by the same number (not 0), the result is a fraction that is equivalent to the original fraction. To understand [sic] why division works, think about 'undoing' the multiplication. since division 'undoes' multiplication, divide the numerator and the denominator by the same number to find an equivalent fraction"
There you have it. Division is just the opposite of multiplication, which was taught to you in such fine rote fashion above.
"Renaming [sic] mixed numbers as Improper Fractions"
They use pictures to show how the whole number can broken into a group of wholes that are divided into a number of equal parts defined by the denominator of the fraction. Not too bad, execpt that it's based on their rote understanding of equivalent fractions given above.
Apparently, their idea of understanding is not based on mathematical rules like
a/1 = a and a/a = 1. Their idea of mathematical understanding has nothing to do with real mathematical understanding.
You can't have a logical discussion about "rote" since it's just a huge strawman. Rote learning is just bad teaching.
"Rote is not merely memorization, it is memorization without meaning or understanding."
What Mr. & Mrs. NCTM fail to recognize is this: If the teachers themselves have only a rote understanding of math, their instruction will be rote instruction. It doesn't matter how much they dress it up with activities like "...something that would have a little more meaning to them. Maybe have them do some concrete subtraction with dinosaur eggs, maybe using beans as the dinosaur eggs or something." [from Liping Ma's book] Calling a bean a "dinosaur egg" does not make it so, and calling that sort of teaching "higher order thinking skills" does not make it so.
On the other hand, the lesson you described gets right to the essence of the concept, explains it clearly, and shows the students how to do it. I see nothing "rote" about that!
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