We've been working on strategies for division, but we are not working with long division. The big problem with long division is that it reinforces basic misconceptions of numbers that so many kids have. In working with students for the past 8 years, I've found that the one area that causes the most problems with a child's math skills is basic number sense and place value.
Most children, even those who are weak at math, can easily identify what place value column each digit is in. They can identify 1s, 10s, and 100s with ease. But a great deal of them do not understand the VALUE of the digits. This misunderstanding is encouraged by prescribed strategies such as regrouping, borrowing, long multiplication, and long division. In using these strategies, students are taught to disregard the place value of the numbers, and consistently refer to and think of the digits as the digits themselves, regardless of their value in the number as a whole.
For example, when trying to divide 600 / 26, a student using long division will ask himself how many 26s go into 6, and then how many go into 60. Continually misrepresenting the numbers in this way does damage to their sense of the number. When they get an incorrect answer, they generally don't recognize it because they have no concept of what the numbers really mean.
In the "Math Investigations" program, students are taught a method known as "clustering". In the clustering method, students find ways of breaking the numbers down into more workable pieces that still retain the place value of the numbers.
For example, given the 600 / 26 problem, a student might realize that 26 x 10 is 260. Another 26 x 10 makes 260. When you add the two together, you get 520, which is very close to 600. Since each 260 = 10 x 26, then 520 = 20 x 26. You can then add two more 26s, which makes the total 572. Now, you can add one more 26 to make 598. Since 598 is the closest you can get to 600, you cannot add any more 26s. So, in total you have added 23 groups of 26 to make 598, with a remainder of 2, since you are dividing into 600. Written just with numbers, the solution might look like this:
26 x 10 = 260
+26 x 10 = 260
----------------
26 x 20 = 520
+26 x 2 = 52
+26 x 1 = 26
------------------
26 x 23 = 598
600 / 26 = 23 R 2
The beautiful thing about clustering is that there is no set way to solve any particular problem. The particular steps a student takes is completely dependent upon their own particular skill level. A child who is a little more advanced might be able to skip a few of the steps above, finding their solution like this:
26 x 20 = 520
+26 x 3 = 78
-----------------
26 x 23 = 598
Or they might take it in more steps if necessary, maybe counting by 52s or 104s. Or perhaps simply by doubling.
26 x 2 = 52
26 x 4 = 104
26 x 8 = 208
26 x 16 = 416
26 x 32 = 832 (TOO MUCH!)
26 x 16 = 416
+26 x 4 = 104
----------------
26 x 20 = 520
+26 x 2 = 52
+26 x 1 = 26
-----------------
26 x 23 = 598
The possible steps to get to the right answer are numerous - but through working with many types of problems, student develop a good sense of which strategies work best for which types of problems.
Also, this week, I showed the kids an alternate way of organizing the division clusters that makes it easier to keep organized. The format uses the traditional "long division" notation (I call it a "guzinto"), with an added vertical line down the right hand side. On the right side of that line, you write how many of the divisor you have subtracted, and on the left side you write how much you are subtracting from the dividend.
Again I'll use the same example problem from above so that you can see how this format is related to the other clustering strategy.
Again, many students can simplify this strategy and do it in fewer steps.
As we continue to work with more alternate strategies, I will make posts on the blog to give examples of the strategies we are working with so that you may have a better understanding of what your children are learning. If you have any questions about the strategies or would like any clarification, please don't hesitate to call or e-mail me.
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