A good idea might be coins, using money because kids like money…. The idea of taking a quarter even, and changing it to two dimes and a nickel so you can borrow a dime, getting across that idea that you need to borrow something.
There are two difficulties with this idea. First of all, the mathematical problem in Tr. Barry’s representation was 25 − 10, which is not a subtraction with regrouping. Second, Tr. Barry confused borrowing in everyday life—borrowing a dime from a person who has a quarter—with the “borrowing” process in subtraction with regrouping—to regroup the minuend by rearranging within place values. In fact, Tr. Barry’s manipulative would not convey any conceptual understanding of the mathematical topic he was supposed to teach.
Most of the U.S. teachers said they would use manipulatives to help students understand the fact that 1 ten equals 10 ones. In their view, of the two key steps of the procedure, taking and changing, the latter is harder to carry out. Therefore, many teachers wanted to show this part visually or let students have a hands-on experience of the fact that 1 ten is actually 10 ones:
I would give students bundles of popsicle sticks that are wrapped in rubber bands, with 10 in each bundle. And then I’d write a problem on the board, and I would have bundles of sticks, as well, and I would first show them how I would break it apart (italics added), to go through the problem, and then see if they could manage doing the same thing, and then, maybe, after a lot of practice, maybe giving each pair of students a different subtraction problem, and then they could, you know, demonstrate, or give us their answer. Or, have them make up a problem using sticks, breaking them apart and go through it. (Ms. Fiona)
What Ms. Fiona reported was a typical method used by many teachers. Obviously, it is related more to subtraction with regrouping than the methods described by Ms. Florence and Tr. Barry. However, it still appears procedurally focused. Following the teacher’s demonstration, students would practice how to break a bundle of 10 sticks apart and see how it would work in the subtraction problems. Although Ms. Fiona described the computational procedure clearly, she did not describe the underlying mathematical concept at all.
Scholars have noted that in order to promote mathematical understanding, it is necessary that teachers help to make connections between manipulatives and mathematical ideas explicit (Ball, 1992; Driscoll, 1981; Hiebert, 1984; Resnick, 1982; Schram, Nemser, & Ball, 1989). In fact, not every teacher is able to make such a connection. It seems that only the teachers who have a clear understanding of the mathematical ideas included in the topic might be able to play this role. Ms. Faith, the beginning teacher with a conceptual understanding of the topic, said that by “relying heavily upon manipulatives” she would help students to understand “how each one of these bundles is 10, it is 1 ten or 10 ones,” to know that “5 tens and 3 ones is the same as 4 tens and 13 ones,” to learn “the idea of equivalent exchange,” and to talk about “the relationship with the numbers”:
What I would do, from that point, is show how each one of these bundles is 10, it is ...