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大学（教育研究）とか ，親馬鹿とか，和歌山とか，とか，とか．

Order-of-Multiplication Dispute: Links

Dispute cases


Articles

Literature

My opinion is ...

  • A-1. Two factors are commutative. By the commutative law of multiplication, 5 x 3 = 3 x 5 holds.
  • A-2. A "dealing-out" operation can swap the roles of multiplicand and multiplier; we can write "5 apples/round x 3 rounds = 15 apples".
  • A-3. We can write "5 x 3" by arranging the apples in a rectangle shape.
  • A-4. We can recognize the number of dishes as the multiplicand (or the basic quantity) and the number of apples per dish as the multiplier.
  • A-5. With counters attached, "5 x 3 apples" and "3 apples x 5" are equivalent, and so are "5 dishes x 3 apples/dish" and "3 apples/dish x 5 dishes".
  • A-6. In other countries, they write the multiplicand and the multiplier oppositely or do not think that the order matters.
  • B-1. In this case, the number of apples per dish is the multiplicand while the number of dishes is the multiplier.
  • B-2. "5 x 3" and "3 x 5" differ in meaning, although the products are the same.
  • B-3. "5 x 3 = 15" shows the opposite situation regarding the numbers of apples and dishes.
  • B-4. "5 x 3 = 15" leads to 15 dishes but not apples.
  • B-5. With counters attached, "5 apples x 3" and "3 apples x 5" are different, and so are "5 apples/dish x 3 dishes" and "3 apples/dish x 5 dishes".
  • B-6. Writing expressions in consideration of language and cultural differences is educationally valuable.
  • I disagree with all of A-1 to A-6.
  • Just the same, I would not approve of all the reasons B-1 to B-6. I agree with the first two reasons under the condition that the apple problem and others are on the test after the pupils learn B-1 and B-2 properly in class. The condition seems to be met on the grounds of lesson plans developed by teachers as well as textbooks which are well-organized.
Towards Order-of-Multiplication Dispute (English Version)

It may be useful to focus attention on where the commutative law appears in an argument. Those who complain about the incorrectness of the red-pencil seen in imgur's picture often think that although a x b and b x a are different literally, a x b = b x a should hold by the commutative law. On the other hand, people for it seem to understand that even if they accept the commutative law of multiplication, a x b and b x a represent different things.

Multiplication in classes

My blog articles

My blog articles (partially in Japanese)

Who is most active on this topic?

Follow @flute23432, who finds and introduces examples of multiplication from different countries as well as provides a critique of the tweets which suggest bad points of Japanese math education.