************************************ From the AMTE listserve, February 3, 2000 ************************************ A big part of the difficulty with decimal multiplication is because we tend to think of multiplication as repeated addition/equal sets. I am beginning to wonder if a part of the problem, though, is that, even though we approach multiplication as repeated addition/equal sets, we no longer emphasize the difference between multiplier and multiplicand.
Here is what I observed in Japanese elementary math textbooks. In Japan, multiplication is introduced as repeated addition/equal sets. However, they emphasize the multiplier/multiplicand distinction, to the point if students write a multiplication sentence with factors in the wrong order, they would be marked wrong.
But, with this distinction, they seem to be able to gradually expand children's multiplication knowledge.
They start with the case where multiplicand is a decimal but multiplier is a whole number. It is not too difficult to extend the repeated addition/equal sets idea to this situation. For example, 3 x 1.4 = 1.4+1.4+1.4 [by the way, in Japan, they write multiplicand first, so they would write this as 1.4 x 3]. At this stage, they also take advantage of the metric measurement. So, a typical introductory problem will go something like:
A juice bottle contais 1.2 liter of juice. How much juice do you have altogether if you had 3 bottles.
They ask children to think about how they can find the solutions, and among the methods children come up with is that 1.2 liter is the same as 12 dl. So, 3 x 12(dl) = 36 (dl), and that's equal to 3.6 liters. So, in the measurement context, they experience "multiplying."
Then, when they start looking at decimal multipliers, they start with the case decimal x whole number first. For example, one textbook series opens this topic with the problem:
Paper tape costs 180 yen for 1 m. How much will the cost if you buy 3.4 m?
As usual, they ask children to think about different ways to figure this problem out. They give a few possible methods in the teachers' manuals: figure out how much 0.1m is, then multiply that by 34; and, figure out how much 34m would be then divide the cost by 10.
They then focus on this second method and discuss (review?) the property of multiplication, a x (nb) = n x (ab) [also (na) x b = n x (ab)], which probably makes good sense in this problem context.
Then, they quickly extend to think about decimal x decimal using this idea.
However, in the example in the book, they have something like this: 2.17 ---(100 times) --->217 x 2.8 --- (10 times) --->x 28 ---- ---- (partial products) (partial products) --------- -------- 6.076 <--- (divide by 1000) --- 6076
Only after this lesson,they introduce the "rule" about locating the decimal point in the product so that the number of digits to the right of the decimal point in the product is equal to the sum of places to the right of decimal point in the factors.
What I find interesting is that this particular Japanese textbook does more "teaching" than just giving a rule than the series Kris' daughter uses. I'm not sure if I agree with this particular sequencing, and I would have probably preferred using more of multiplication as rectangular area idea. But, one thing that strikes me about Japanese textbook is how well organized it is and how they fit all together to form a nice cohesive unit. Moreover, even when they are teaching a procedure, they do seem to try to establish a sound conceptual base.
Just offering my observations.
Kristin Kaul wrote:
> I have a question for this list. > Having just read Liping Ma's book, I was amazed to see an example that > was tailor-made for her book arise in my own home. My 5th grader came > home with problems in decimal multiplication. At the top of the sheet > was the explanation: "the number of decimals in the product is the sum > of the decimals in the factors" - and that was all the teaching they > received. > > My daughter, meanwhile, could see the procedure she was supposed to > follow, but couldn't really understand why 0.19 x 0.08 wasn't = 1.52 > (i.e. she was multiplying 0.19 x 8 = 1.52). In trying various ways of > explaining this to her, so many other underlying principles came to > light that it became difficult to know where to start - clearly they > had to be made concrete first (place value, multiplication and division > by 10's, relation between multiplying by a fraction and dividing by a > whole number..). How do people on this list think this topic should > have been approached, and what fundamental principles drawn on to > support it? *************************************** Jerry P. Becker Dept. of Curriculum & Instruction Southern Illinois University Carbondale, IL 62901-4610 USA Fax: (618) 453-4244 Phone: (618) 453-4241 (office) (618) 457-8903 (home) E-mail: firstname.lastname@example.org