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Sunday, August 16, 2009

Place value is such a fundamental concept that we should ensure the students recognize place value and its significance wherever it occurs. An activity I call “The Bake Sale” highlights place value in the operation of division. I will present just one example. Of course, teachers can have as many examples as groups within the classroom. The groups should not be too large, not more than three of four students per group.

The scenario: They are getting ready for a bake sale. They have a platter of cookies and they want to make sure they will have enough cellophane bags to package the cookies. In today's example, the platter has 173 cookies and they will be packing 6 cookies to a bag. I use beans for cookies and little squares of paper for the bags. So the students would start with 173 precounted beans.

The first concept I want them to see is division as repeated subtraction. They are to remove 6 beans at a time, just as if they were really packing cookies, and place them on a square of paper. As they do so they place a tally mark. Very young children would have a specially designed “worksheet” for recoding each “bag.” For example, a page of squares that the students color as they “pack” each “bag.” When they are through, the number of squares with beans and the number of tally marks or colored squares on the worksheet should be the same.

Older students will want to cut to the chase and simply perform the long division. But one purpose of this activity is to help students see the math behind the procedure, and besides in real life, they really would be subtracting 6 cookies at a time, repeatedly, until there were no longer enough cookies to pack a bag.

They should have 28 bags with 5 cookies left over. Some older students already know that the “real” answer is 28 and 5/6, or maybe 28.83 or ... depending on what decisions they make. Some will be sure that the answer is 29 because they learned to round somewhere along the way. Some of them may believe an answer with a remainder (as in 28 R5) is juvenile, and not as good an answer as some of the other possibilities. Students must always be reminded that math is the servant, not the master.

Later in the activity students will see that the “juvenile” answer is the most useful answer.

Once they have determined the answer, it is time to revisit the standard algorithm with a variation. Rewrite the division problem like this:

The green lines show the place value columns. In a class discussion, we establish that a 2 goes above the 7, not because 6 goes into 17 twice, but because the 7 is in the tens’ place, 6 is going into 170 (17 tens) 20 times. The 2 is really a twenty. Students need to be reminded continually what the numerals really signify as they complete calculations. Otherwise, students are merely manipulating abstract, meaningless symbols.

Because we are writing the division problem with Arabic numerals, naturally each digit and its columns represent a place value. Since 6 roundly goes into 170 twenty times, meaning we can show 20 repeated subtractions in one step, we write a 20, not a 2, over the 173. Since we have filled 20 bags at once with 6 cookies per bag, we have removed or subtracted 20 x 6, or 120 cookies from the platter. We show this very concrete action by subtracting 120 from 173, leaving 53 cookies on the platter. We remove enough cookies to fill eight more bags, that is 48 cookies, leaving 5 cookies on the platter, not enough to fill a bag. We needed 28 bags.

Although not “wrong,” 28 and 5/6, 28.833, 28.83 or 29 have no practical utility in this scenario. Students will have an easier time evaluating the reasonableness of an answer if they are encouraged to keep the context and the numbers together. When they round to 29, they are saying 29 what? 29 bags. By the end of the activity, it should be clear that 5/6 of a bag is not helpful and that typical rounding serves no useful purpose. I require students to write their answers in complete English sentences. The answer to this problem is not “28,” or even “28 bags,” but something like “we needed 28 bags to pack the cookies.”

The finished problem would look like this:

The format looks a little different than the standard algorithm, but the significance of place value is preserved. This type of format did not have a name when I first started using it, or perhaps I mistakenly thought at the time that it was an innovation of my own. I was little surprised when the format began appearing in textbooks as “scaffolding.”

Incidentally, at every opportunity we should insist that students read numerals correctly. Simply reading numerals correctly can prevent confusion. “And” marks the spot between “wholes” and “parts.” Although the answers with fractional parts served no real purpose in this activity, of course there are other contexts where the fractional part is important. In any case, some of the other possible answers would be read “twenty eight and five-sixths,” “twenty eight and eighty three hundredths.” I would use “twenty eight point eight three” only for dictation purposes, not for mathematical purposes.

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A lifelong educator, my educational philosophy could probably be described as liberal traditionalism. Liberal traditionalism is an eclectic approach committed to utilizing innovative practices to help students acquire traditional subject matter. So what exactly constitutes innovative practices and what exactly should be included in “traditional subject matter”? What is the relationship between society and education? More...

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