kitchen table math, the sequel

Last week, my son came home from school with a study sheet for the last big test for the year. The test covered some basic US geography, including the names of the Great Lakes, some facts about the Mississippi River, some facts about the US Flag, and locating some major geographical features. In addition, the students were supposed to be able to name at least 25 of the states, given a list showing only the first letter of each state.

Over the next one-and-a-half weeks, A. and I studied for his test about 15 minutes each night. This studying was the exactly the sort of studying that ed. schools teach as being the most harmful: pure rote memorization.

Early in this process, A. objected to the continued practice of the entire list on the basis that, "I only have to know 25 states, not all the states." My sympathy was notably limited; we studied the whole list. 8-)

By a couple of days before the test, A. was pretty reliably naming all the states starting with a given letter when prompted with that letter. (BTW, "M" is particularly annoying.) By this point, he was starting to think that getting the entire list right was pretty cool.

The day before the test, the teacher announced to the class that any child who could name all the states would get extra credit and a small prize.

Of the 26 kids in the class, 8 named all the states on the test the next day. None of those kids needed to have his or her self-confidence artificially boosted after the test, and they all now have a much better understanding of the value of hard work.

Educators love the false dilemma. One of their favored false dilemmas in math education is saying what they teach is "higher ordered thinking skills" and what was traditionally taught was "merely teaching by rote." Rote is not merely memorization, it is memorization without meaning or understanding. I contend that very little is taught by rote in any subject, even when memorization (or practice to automaticity) is required.

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.