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Here is my reply to Sybilla and Karen’s post on October 18, 2012:
Dear Sybilla and Karen,
Maybe you are misreading my misgivings: I’m actually asking for your help.First, it is nice to see that you have already thought through the different methods/algorithms displayed in the figure and discussed at:
(For other readers of this forum, Method 4 is the one I discussed in my previous post above.)
In the curriculum I’m helping to create, we will use Methods 1 and 2 (and a combination of both of them) to help explain multiplication by a 2-digit number in the curriculum. (We will probably avoid Method 5 for the same reasons you state in the discussion.) The plan is to help students understand multiplication in several ways (area, partial product, distributive property, etc.) and, as a capstone to that experience, to learn a standard algorithm that is “accurate and reasonably fast” as required by the K-8 Publishers’ Criteria for CCSS-M.
The statement “accurate and reasonably fast” seems to imply Methods 3, 4 or 6 (with Method 2 as a slightly slower backup). Students also need to learn an “accurate and reasonably fast” division algorithm. Thinking of a particular set of addition, subtraction, multiplication, and division algorithms as grouped together into a system of algorithms, Methods 4 and 6 stand out for their ability to embed into their corresponding division algorithms without confusion (at least without confusion in regards to the embedding–Method 6 has other problems as you know).
In my previous post I was basically begging that a figure similar to the one at the link above be included in the NBT Progressions together with a corresponding figure for division algorithms. The discussion below the figure is insightful and I hope can be incorporated as well (together with similar descriptions for the division algorithms).
Having Method 4 clearly articulated in the NBT document together with how it embeds into its corresponding division algorithm will help tremendously any attempt to move to a system of algorithms that makes sense with regards to place value.
With warmest regards,
ScottHi Bill,
I discussed the multiplication algorithm on page 14 of the NBT Progressions document with a number of mathematicians. Many pointed out how completely baffled they were with the “44” and “720” appearing on separate lines. (How do you get 44 from any partial product of 94 and 36?) I explained that there was a mini-version of the lattice method built into the algorithm (one needs to read the numbers down and to the right to give 54 tens and 24 ones), but agreed that changing the font size of the 5 and 2 was bad form—at a minimum they should be the same size as the 44 so that it is easier to recognize the “54” and “24.” Changing the font size does not alleviate the problem I discussed in my first post, however.
Finally, while talking with a Chinese mathematician about this problem, we realized that this algorithm was actually an incorrect modification of the standard Chinese multiplication algorithm.
In the standard Chinese algorithm, the first line under the 36 would contain a small “2” and the second line would read “564.” The 564 would then accurately represent the sum of the partial products. The process would go like this: 4 x 6 = 24 = 2 tens 4, put the 4 in ones column and put a small 2 in the tens column (where it is now). Next, 9 tens x 6 = 54 tens. Add the 2 tens to get 56 tens and write 56 in the hundreds and tens column to get 564. In this algorithm, the “2” is only a reminder and isn’t used after 564 is established (which is why it written smaller).
The standard Chinese multiplication algorithm is as fast, efficient, and as error free as the standard U.S. multiplication algorithm (with extra benefit of placing “carries” in the correct place value locations). Also, and this is the main point of this post, it embeds very nicely into the division algorithm without the big concern I had in the first post above. (Students still have to be careful with recording small numbers versus big numbers, but now at least, “small” and “large” mean something.)
I understand why you might be loathed to make a major change to the NBT Progression document, but in this particular case I really think it is warranted. The algorithm as presented doesn’t make sense (again, the 44 means?) and cannot be nicely embedded into the long division algorithm.
As of now, the multiplication and division algorithms in the NBT Progressions document are not strong enough to risk switching away from the standard U.S. algorithms in my curriculum (or even to suggest it as an alternative to the standard U.S. algorithm). Frustrating!
Best,
Scott
