|
Items in cart [0] |
Questions? Call 888-624-8373 |
PAPERBACK Free Resources |
||||||||||||||||
|
||||||||||||||||
|
The following HTML text is provided to enhance online readability. Many aspects of typography translate only awkwardly to HTML. Please use the page image as the authoritative form to ensure accuracy. its learning. An experienced Chinese teacher said You should see a knowledge “package” when you are teaching a piece of knowledge. And you should know the role of the present knowledge in that package. You have to know that the knowledge you are teaching is supported by which ideas or procedures, so your teaching is going to rely on, reinforce, and elaborate the learning of these ideas. (Ma, 1999, p. 18) To see a topic to be taught as part of a package of knowledge, rather than in isolation, requires a way of thinking that may not be common in the United States. When U.S. elementary teachers were asked how they would respond to a student's mistake in calculating 123 × 645, they focused on the given problem. When Chinese teachers were asked the same question, about 20% made comments such as This mistake should have happened when students learn multiplication by two-digit numbers. The mathematical concept and the computational skill of multidigit multiplication are both introduced in the learning of the operation with two-digit numbers. So the problem may happen and should be solved at that stage. (Ma, 1999, pp. 45-46) This reflects a general principle in the organization of knowledge packages: Not all topics receive equal emphasis. Some are considered key pieces, and teachers take particular care that students understand them. Two-digit multiplication is a key piece for the three-digit multiplication package because the simplest form of the “moving over ” idea involved in multidigit multiplication occurs in the two-digit case. Such attention to an idea in its first and simplest form allows teachers to pay less attention to later and more complicated forms. One Chinese teacher in the study put it this way, “To tell you the truth, I don't teach my students multiplication by three-digit numbers. Rather, I let them learn it [on] their own.” Figure 1 shows a model of the knowledge package for subtraction with regrouping derived from interviews with Chinese teachers. [Here “regrouping” includes more than “borrowing.” Instead, the intended meaning is that some digit in the base-10 representation needs to be decomposed to make the computation. For example, in computing 15 − 7, one can't simply work with the digits 1, 5, and 7. Instead, one needs to decompose the 1 as 10 ones and group some or all of the 10 ones with the 5 (e.g., 2 ones might be grouped with the 5).] The topic under discussion appears in a rectangle surrounded by other topics that occur in the curriculum (in ovals), and basic principles (in rectangles with rounded edges). Key pieces of the package have thick borders. The central sequence in the subtraction package goes from the topic of addition and subtraction within 10, to addition and subtraction within 20, to subtraction with regrouping of numbers between 20 and 100, then to subtraction of large numbers with regrouping. “Addition and subtraction within 10” is addition with sums of 10 or less and subtraction with minuends of 10 or less, which don't require carrying or regrouping. For example, 10 − 4 = ? has a minuend of 10, and requires no regrouping. A related addition problem 4 + 6 = ? has the sum of 10, and requires no carrying. “Addition and subtraction within 20 ” is addition with sums between 10 and 20 and subtraction with minuends between 10 and 20. Three levels of subtraction with regrouping problems are related to this central sequence:
|
|
|||||||||||||||
