Rationale
My rationale for emphasizing the decimal number system is based on the work of Liping Ma in her book Knowing and Teaching Elementary Mathematics. Ma compared elementary school math teachers in the United States and China and found that the Chinese teachers were more successful in teaching addition and subtraction because they taught the regrouping (borrowing and carrying) as aspects of these operations, as opposed to U.S. teachers who often teach these subjects as procedures and facts. This was the missing link I had been seeking in my years of teaching this subject in elementary school. I knew that simply teaching the procedure of carrying and borrowing by crossing out numbers and moving a "one" to the needed column so that addition or subtraction could take place was not enough information. Clearly, my students did not understand what they were doing; hence many were lost, especially in the mystical (to them) realm of subtraction.
When I began to talk about "tens" and carrying and borrowing them from, for example, the tens column to the hundreds column and vice versa, students began to perk up. I was beginning to help them demystify the process. We used manipulatives bundled in tens and worked simple one-digit addition and subtraction problems by adding to a "tens bundle" or subtracting from it. I was on the right track.
However, it wasn't until reading Liping Ma that I understood the teaching of these "simple" mathematical operations was predicated on children understanding some basic ideas about the base ten system. What follows is taken directly from Ma's account (Ma 1999) of her investigation and her subsequent abovementioned book. I reference the subtraction process because that is the focus of this unit.
One of the key concepts that many Chinese elementary school math teachers teach is that of "decomposing a higher value unit." This phrase describes the "taking" step in the subtraction algorithm. For example, rather than teach students that you borrow one from the ten's place, Chinese teachers use the catchphrase, "you decompose 1 ten." Ma goes on to explain that "in the decimal system numbers are composed according to the rate of ten." Therefore ten units are organized or composed into "1 unit of the next higher place value" in addition. In subtraction a unit is decomposed into 10 units of the lower value. (Ma, 1999, p.8). Hence, the fundamental idea of regrouping in subtraction is that of decomposing higher value units into lower value units.
An advantage to teaching addition and subtraction by the process of composing and decomposing higher value units is that subtraction as the inverse operation to addition is implied. And when it is then articulated, students more easily grasp the unmaking or undoing process of subtraction. Therefore, addition involves a process of making or building tens, and subtraction involves the inverse or opposite process of unmaking or taking apart tens.
As well as teaching the standard algorithm for subtraction by decomposing a higher value unit, many Chinese teachers also teach other ways of regrouping. For example in the problem 62 - 45, 40 can be subtracted from 62 and 5 can be subtracted from 22. Or, the 62 can be regrouped as 50 and 12. Then the 5 can be subtracted from 12, 40 from 50, and we get 17.
Or, we can regroup the 62 as 50, 10, and 2. Then we subtract 5 from 10 and get 5, add it to the 2 and get 7, subtract 40 from 50 and get 10 which we would then add to the 7 to get 17.
Although the standard algorithm is usually the preferred method, offering alternatives to students serves to reinforce the decomposing process, shows students that there is often more than one way to solve a problem, and helps strengthen mental math processes.
A taxonomy that is helpful in classifying addition and subtraction problem types is that found in the book Children's Mathematics: Cognitively Guided Instruction. The authors employ a schema that "focuses on the types of action or relationships described in the problems. The taxonomy is as follows:
Join Problems are problems where elements are added to a given set. The action takes
place over time. For example: 4 children brought candy to school. The next day 2 more children brought candy. How many children brought candy?
Separate Problems are those in which elements are removed. They also take place
over time. They are the inverse process of Join Problems.
In both Join and Separate Problems the problems can be one of three types: Result
Unknown, Change Unknown or Start Unknown—the Result Unknown being the answer,
Change Unknown is the second summand in addition or the subtrahend in subtraction, and Start Unknown is either the first summand or the minuend.
Thus, the abovementioned candy problem would be a Result Unknown Problem.
To make it a Change Unknown Problem we could say: 4 children brought candy to school. The next day 6 children altogether had brought candy. How many children brought candy the next day? A Start Unknown Problem could be: The first day some children brought candy. The next day 2 children brought some more candy. Altogether 6 children brought candy. How many children brought candy the first day?
Part-Part-Whole Problems are problems that are static. There is no change over
time and no action. The relationship is among a set and two subsets.
These problems are of two types: Whole Unknown or Part Unknown. For example:
4 boys and 6 girls sold cookies. How many children sold cookies? This is Whole Unknown Problem. A Part Unknown Problem could be: 10 children sold cookies.
6 of the children were girls. How many boys sold cookies?
Compare Problems are problems that treat relationships between quantities. They
contain a Referent Set, a Compared Set and the Difference. Any one of the three can be the unknown. For example: Jake has 3 pretzels. Zach has 8 pretzels. How many more pretzels does Zach have than Jake? This is an example of a Difference Unknown problem. To make it a Compared Set Unknown problem: Jake has 3 pretzels. Zach has 5 more pretzels than Jake. How many pretzels does Zach have? And a Referent Unknown problem using the same data could be: Zach has 8 pretzels. He has 5 more pretzels than Jake. How many pretzels does Jake have? (Carpenter et al., 1999, pp. 7-10).
This taxonomy is helpful in that it demonstrates how to make the action or the relationships in word problems as clear as possible. It shows ways to vary problems and it gives children descriptors for talking about problems and for targeting the part for which they need to solve. In fact, eleven different situations are described above that represent various interpretations of addition and subtraction (Carpenter et al., 1999, pp. 10-12). Although this taxonomy is presented for primary grades, I think it is useful throughout the elementary school years as a way of helping children target what parts of a problem are given and what part needs to be found. It is especially useful when solving missing addend problems. This taxonomy and the Chinese teachers' use of decimal operations will be discussed and used throughout this unit.
Another taxonomy that deserves mention is that of Larry Sowder in an article entitled " Addressing the Story-Problem Problem," from the book Providing A Foundation for Teaching Mathematics in the Middle Grades. Sowder divides the arithmetic world into two types: "Real World" and "Math World". He links "situations in which groups or amounts are put together, either literally or conceptually," as Real World situations, and addition as the Math World situation that handles these real world settings. "Take-away situations" and "Comparison situations" belong to the Real World, and the subtraction process belongs to the Math World. (Sowder, 1995, pp.129-130).
However, missing addends can be linked to both addition and subtraction, because the problems can encompass joining groups or amounts or take-away and comparison situations (Souder, 1995, p. 131). This is where the confusion often sets in for students. Thus, articulating this taxonomy to them and pointing out that missing addends are the anomalies in the cousin family of addition and subtraction can be helpful in aiding students to choose the best of the two processes to solve a given problem.
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