November 2, 2015
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It might be useful to tell students that multiplication can be seen as repeated addition under certain circumstances. But docking a student half credit because he/she used commutativity when that was allegedly not permitted–because not specifically taught yet–is fallacious. You can see why if you simply look at a picture of repeated addition for the stated problem (5X3=…), say using stars:
To get the so-called ‘correct solution’ you repeat the addition of the rows, and to get the so-called ‘wrong solution’ you repeat the addition of the columns. Docking half credit amounted to telling the student they could not add the columns. Unfortunately many children in this position simply assume, in total bewilderment, that math is some inscrutable strange mystical language that they will never hope to figure out.
The student was absolutely correct in his/her answer, whether or not anyone taught that student the concept of commutativity. Prohibiting the use of commutativity in this solution amounts to discounting and even prohibiting the picturing of multiplication, whose far more comprehensive and direct meaning is that of area. The teaching of multiplication as repeated addition dependent on the order in which numerals are written down on paper shrinks–and distorts–the concept of multiplication down to a shadow of what it really is. All this confusion could be avoided if students are encouraged to picture math concepts instead of thinking of them as a matter of reading symbols from left to right and following a ‘strategy’ (one that doesn’t work when dealing with concepts like the area of a circle which cannot be solved by repeated addition anyway).
Furthermore, clearly the intent of the problem was for the student to literally write down the meaning of “5X3,” which in the context of repeated addition was assumed to be “the number 3 taken 5 times”. But this is just an arbitrary linguistic convention. The expression “5X3” could also linguistically mean “the number 5 taken 3 times”. Indeed many of us, when ordering an item, list the item on the left and then put a numeral in a box to the right saying how many of that item we want! So in that case the problem is clearly saying ‘we want 3 fives’. This is perfectly natural. So it is not at all clear that the intended grading of this problem is even assessing use of commutativity, as claimed in many defenses of the grading. The student could simply have a different linguistic interpretation of the expression. And it’s completely appropriate. In my opinion, this student and all other students marked wrong on this problem are owed an apology. And a different way of teaching multiplication that shows what it is conceptually rather than reducing it to linguistically arbitrary ‘strategies’ that don’t necessarily work for all kind of numbers (e.g., irrational or transcendental numbers).
And if you’re worried about non-commutativity, I’m sure that students who are not totally discouraged by repeated addition conventions will be able to deal with it when they get to quantum mechanics and/or non-Abelian group theory. There are pictures for that too. Here’s one for non-commuting operators in quantum theory. There’s a big difference between (1) first opening the window and then sticking your head out; and (2) first sticking your head out and then opening the window.