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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．

「3×5 vs. 5×3の問題」を読んだ

OoM

気軽に読める解説記事でした．
読んでいきながら，「これはあれだな」と，思い浮かぶ情報もいくつかありました．並べてみます．

  • 交換法則と式の意味

The students in Mrs. Schuster's third-grade class are discussing a question she has set out for them to consider: "Does the order of the numbers in a multiplication sentence affect the answer? Explain why or why not." In order to explore this question, they are generating examples of multiplication sentences and testing what happens when they change the order of the factors. Students know many of the basic multiplication facts but have not yet learned an algorithm for multidigit multiplication.
One student has made a conjecture that the order of the factors does not make a difference --- "the answer is the same no matter which number goes first." Students are agreeing with this conjecture by bringing up other examples that work, such as 3 x 4 = 12 and 4 x 3 = 12. Mrs. Schuster then asks if this conjecture works with larger numbers and suggests they use calculators to check. Students are able to generate many examples to verify the conjecture, but explaining why the products are the same is not as straightforward as carrying out the multiplication.

1. Eddie: Well, I don't think it matters what order the numbers are in. You still get the same answer. But the multiplication sentences are different because they mean different things.
2. Mrs. S: OK, Rebecca, do you agree or disagree with what Eddie is saying?
3. Rebecca: Well, I agree that it doesn't matter which number is first, because two times five equals ten and that's the same answer as five times two. But I don't get what Eddie means about the multiplication meaning different things.
4. Mrs. S: Eddie, would you explain what you mean?
5. Eddie: Well, I just think that the two times five that Rebecca used can mean two groups of five things like two bags of five apples. And five times two means five bags of two apples. Those aren't the same at all.
6. Tiffany: [Hand up, waving] But you still have the same number of apples! So they do mean the same!
7. Mrs. S: OK, so we have two different ideas here to talk about. Eddie says that order does matter, because five times two and two times five can each be used to describe a different situation, like two bags of five apples or five bags of two apples. So the two number sentences mean different things. And Tiffany, are you saying that those two number sentences can't be used to describe two different situations?
8. Tiffany: No, I mean that even though the two situations are different, the answer is the same.
9. Mrs. S: OK, so you're saying that order doesn't matter because the answer is the same?
10. Tiffany: Right.
11. Mrs. S: OK. We need to think about this. In Eddie's statement, order makes a difference in the situation you're describing. In Tiffany's statement, order doesn't make a difference in the answer we get. So when does order make a difference in multiplying two numbers together?
（http://books.google.co.jp/books?id=2NX4I6mekq8C&pg=PA3, 転載元）

1. Connie wants to buy 4 plastic cars. They cost 5 dollars each. How much does she have to pay?
a) 5＋5＋5＋5＝20
b) 4・5＝20
c) 5・4＝20
d) 4＋4＋4＋4＋4＝20
...
The comparative facility of isomorphic over functional properties is even easier to show by considering all four procedures a, b, c, and d. Procedure b is a meaningful concatenation of procedure a. The cost of 4 cars = the cost of 1 car, plus the cost of 1 car, plus the cost of 1 car, plus the cost of 1 car. Expressed formally in terms of the isomorphic property for addition, this is f(1+1+1+1) = f(1)+f(1)+f(1)+f(1), and in terms of the isomorphic property for multiplication, f(4・1)=4・f(1). Procedure d is meaningless in terms of cars and costs. Twenty dollars cannot be 5 cars + 5 cars + 5 cars + 5 cars. Young students apparently are aware of this and never user procedure d. So there is a strong asymmetry between procedures b and c. They are not conceptually the same, although because of the commutativity of multiplication they may be mathematically equivalent.
（Vergnaud: Multiplicative Structures. isbn:0873532651 転載元）

A situation in which there is a number of groups of objects having the same number in each group normally constitutes a child's earliest encounter with an application for multiplication. For example,

3 children have 4 cookies each. How many cookies do they have altogether?

Within this conceptualization, the two numbers play clearly different roles. The number of children is the multiplier that operates on the number of cookies, the multiplicand, to produce the answer. A consequence of this asymmetry is that two types of division may be distinguished.
...
Cartesian products provide a quite different context for multiplication of natural numbers. An example of such a problem is

If 4 boys and 3 girls are dancing, how many different partnerships are possible?

This class of situations corresponds to the formal definition of m × n in terms of the number of distinct ordered pairs that can be formed when the first member of each pair belongs to a set with m elements and the second to a set with n elements. This sophisticated way of defining multiplication of integers was formalized relatively recently in historical terms.
There is a symmetry between the roles of the two numbers here, and hence only one type of division problem. Given that there are 12 possible partnerships, there is no essential difference between (a) being told that there are 4 boys and asked how many girls there are and (b) being told that there are 3 girls and asked how many boys. (In fact, it would be unusual to pose division problems of this type.)
（Greer: Multiplication and Division as Models of Situations. isbn:1593115989 転載元）


少しだけ，記事（3×5 vs. 5×3の問題）の本文より，思うところを書いておきます．「3. 小学校の教科書」というセクションで，6つの教科書会社の小学校2年の教科書を開いてみたとのことですが，2年のみに着目するのでは，啓林館1年の「子どもが 3人 います。みかんを 1人に 2こずつ あげます。みんなで なんこ いりますか。」*2が欠落しますし，学年が上がって，「小数×整数」は4年，「整数または小数×小数」は5年で学習すること（そしてそれぞれの学年でどんなかけ算を学習するか）を，見過ごすことになります．
記事内の異なるページで見かける「マス目」「下駄箱」は，いずれも，アレイもしくはデカルト積(Cartesian product)でモデル化される場面です．上で転載したGreerの記事を（上記引用だけでなく，書籍を）読めば，それは乗除算モデルの一つであり，そのモデルではかけ算1つに対してわり算1つであり，みかんと皿のかけ算の問題は，かけ算1つにわり算2つの他のモデル(Equal groups)が該当するのを確認できます*3．
といった次第で，高名な数学者*4が「掛け算の順序」をテーマに記事を書くと，歴史*5や海外事情に立ち入ることなく，気軽に読めるエッセイが出来上がるのだなと感じたのでした．

（最終更新：2015-01-30 早朝．タイトルを変更しました）

*1:奥付は「2014年12月26日発行」，Amazonでは「2015/1/12」

*2:http://d.hatena.ne.jp/takehikom/20140703/1404313204, http://d.hatena.ne.jp/takehikom/20111117/1321460871．教科書も学習指導案も，2＋2＋2で求めるという流れです．これは，「一つ分の大きさ」と「幾つ分」とを区別して2年でかけ算を学習するための素地となっています．この形式の出題を1年に入れている教科書は，知る限り啓林館のみですが，乗法・除法の素地として，同じ数ずつのたし算やひき算を行うことは，「具体物をまとめて数えたり等分したりし，それを整理して表す活動」という表現で，学習指導要領の第1学年のところに記載があります．

*3:推測ですが，算数教育の書籍で「かけ算の意味は，…である」と書かれているのを，批判的な人によって「小学校の算数で学ぶ/指導する，かけ算の意味は，…のみである」と誤読されてきたのが原因かなと思っています．

*4:同誌p.1にも，名前を見ることができます．

*5:なぜ数学者が「順序がない」などを書けるのかについて，またも推測ですが，古くは高木貞治，そして1970年代に南雲道夫，小島順，田村二郎が著してきた，量と数の関係が，数学者らにとって，過去のものになったからなのかなと感じています．量と数の関係を数学的に記す際には，「量×数でも数×量でもよい」よりも，「量×数で定義し，拡張を図る」を採用することで，シンプルであるとともに読みやすいストーリーを構築できるわけです．この脚注で挙げた方々の著作についてはhttp://d.hatena.ne.jp/takehikom/20120213/1329083228をご覧ください．