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Problems in Fifth Grade Math
Last year, I bought almost all my daughter's text books, so that she would always have a copy at home. The school provides free copies to the students, but they are very big and very heavy, and the children seldom take them home.
This year, I decided to pursue a different policy. We didn't use most of the books I bought last year, so I decided to focus on what I can teach my daughter independently and let the school teach her what they think she should know.
However, we've been having some problems with math, so I may end up getting the math textbook so that I can keep up with the material they are studying in class.
Meet the Authors of the Textbooks
The other day my daughter was struggling with some math problems that she had been assigned as homework. Since she was complaining about it, I offered to take a look. One of the problems looked like this:
18 + 26 = _______
"Well, that doesn't look very hard," I said. "What seems to be the problem?"
I rewrote the problem for her on a piece of scratch paper like this:
"Of course, that's easy!" my daughter said. "But I'm not allowed to do it that way. I have to do it sideways."
This had me baffled. "You have to do it sideways? But what difference does it make?"
"I'll get counted off if I don't do it sideways."
"But the answer will be exactly the same!"
"Yeah. But this is supposed to be mental math. I'm supposed to do it mentally."
"Oh. Well, in that case, you do it in your head. Then you write the answer down."
"No! We have to show our work."
I was at a loss. To me "mental math" means that you don't show your work! If you write it down, then in what way is it "mental"? By that reasoning, all math is mental.
But then I looked at the instructions, and it seems my daughter was right. Here is what it said: "Add or subtract mentally. Use compensation. Show your work!"
Later in the week, I spoke with my daughter's teacher, and the teacher confirmed that my daughter is capable of solving the problems, but not by the method that the class is currently studying. "Well, if you can send home a description of the algorithm, I'll make sure she learns to do it that way," I said.
"Okay. I'll send some instructions on how to use compensation." She was true to her word. The instructions were essentially copied from page 88 of the textbook, so I'll just cite the textbook here:
To use compensation, add a number to one addend to make the addition easier. Then adjust by subtracting the same number from the other addend.
When I finished reading these instructions, I was not very much enlightened. Clearly if you add and subtract the same number from your total, the total will remain unchanged. But what did they mean by "to make the addition easier"?
Fortunately, the textbook also provided an example.Next to the picture of a cute little terrier, there was the following tabulation.
Category Number of Breeds
Terrier ................................................ 28
Toy ..................................................... 23
Which is easier to add: 28+23 or 30 + 21 ?
My initial response was that they were both equally easy to add. But after a little more thought, I came to realize that this was a rhetorical question. The textbook writers had already decided that the easier sum was "30+21". They wanted the student to add two to twenty-eight and subtract two from twenty-three, so as to get the sum thirty plus twenty-one.
Having determined this much, I was now in a position to ask the all important question: why? Because without answering that quesiton, I would never be able to come up with an algorithm that my daughter could use in order to predict which sum the educators think is "easier."
I'm not a mathematician. I don't know very much about numbers. However, it seemed to me that this is not really a math problem. It's a mind reading problem. "Easier" isn't a rigorous mathematical concept. Easier depends on your point of view. So what was the point of view of the people who wrote the text book?
There must have been some assumptions that they were taking for granted and that they refused to spell out for the students. My job, as a parent, was to ferret out those assumptions.
"This has something to do with the fact that they are using the decimal system to do all their calculations," I mused. "It has nothing to do with the numbers 28 and 23 per se."
For instance, in hex, twenty-eight would be 1C and twenty-three would be 17. Adding two to one addend and subtracting it from the other would not get you a number that ends in zero on either side of the addition sign. If we were doing this problem in base 28, then twenty-eight would be 10 and twenty-three would be N and the sum of 10 and N would be 1N.
The educators who wrote the textbook assumed that it's easier to add two numbers if one of them is divisible by ten. Of course, if they used a different base, say base N, where N stands for any whole number, they would think numbers divisible by N were easier to use in additions.
The rule my daughter needs to learn is that she must determine which of the two addends is closest to a number divisible by ten, then she has to determine which arithmetic process, addition or subtraction will get it to equal the number. In the case of twenty-eight and twenty-three, twenty-three is three away from twenty (by subtraction) and twenty-eight is two away from thirty, (by addition). Since two is smaller than three, then the educators want her to choose two and to add it to both addends, and all so that one of them will end in zero.
That seems like a lot of work. Now all I need to do is find an algorithm for determining which of two numbers is closer to a multiple of ten! Then I can give airtight, rigorous instructions on how to solve an addition problem using compensation.
Other parents grapple with the problem
Some children just instinctively know what the teacher means, because they are able to put themselves in the teacher's shoes and realize what her assumptions are. For those children, rigorously defined instructions are not necessary.
However, many children do not know what the teacher or the text book writer mean unless given explicit directions. They have no idea why one sum is supposed to be easier than another. They may have a picture of twenty-eight dogs in their head and another of twenty-three dogs, and they do not know why counting them should be easier if they are rearranged into two different groups of thirty and twenty-one. In particular, if one is a group of terriers and the other is a group of toy poodles, the child might even be reluctant to mix the two groups.
Such children can follow instructions, but only if everything is completely spelled out. They will do what you say, not what you mean. The same is true of computers. Perhaps a good exercise in the pedagogy of elementary arithmetic might be to require all text book writers to first write a computer program that performs the calculation the students are expected to make in the manner that they are expected to make it.
(c) 2009 Aya Katz