In this case, the mistake (or whatever we call it) isn’t about what the student wrote, but what he said.

At the end of class, I asked my 3rd Graders to write a story problem for 13 x 2 and hand it in. As he was leaving, a boy handed me this slip and apologized for it.

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“Why are you apologizing?”

“Because my story is for 2 x 13, not for 13 x 2.”

Commentary:

The big lesson here is that the order matters in multiplication, as it does with addition (for most young kids 9+2 is much easier than 2+9) and as it does for algebra (4 + 2x = 10 is not the same as 10 = 2x + 4). Each of these problems has a different flavor for people who are beginning to get comfortable with these types of problems. Saying that two problems are “the same” is a substantive mathematical claim, and it needs to be taken with the seriousness that all mathematical claims require.

  • Howard Phillips

    It also depends on how you read 2 x 13

    It can be two lots of thirteen: two times 13

    or

    two multiplied by thirteen: two, thirteen times

    The problem in your algebra example is real but different. It is natural to write from left to right as an expression is constructed, but afterwards the use of the = sign states algebraically that the two sides evaluate to the same number regardless of the value of x

  • Hao Ye

    Can we also acknowledge that 13 people with 2 eyebrows each is more reasonable than 2 people with 13 eyebrows each? 🙂

  • There’s no logical reason that 2 x 13 has to be interpreted as 2 groups of 13 or 13 groups of 2. In fact, information I found online indicates that there were 19th century texts that made the *smaller* factor the multiplier by definition.

    One thing that I try to avoid saying is that multiplication (even of positive integers) *is* repeated addition. However, I accept that the answer to such problems can be obtained by repeated addition, at least as far as pure calculation is concerned. That doesn’t make repeated addition the same thing mathematically as multiplication, in my view and that of some other people, including Stanford mathematician Keith Devlin, who has written multiple columns on the question over the years.

    Given the commutative property of multiplication for real numbers, I don’t see much point claiming that 2 x 13 must be interpreted as 2 lots of 13 or vice versa. However, if you’re in certain real world situations, there are clearly meaningful differences: 13 2-lb sacks of potatoes is very different from 2 13-lb sacks from a practical perspective, though not from a total lbs of potatos-perspective. Stretching it a bit, one five-year-old car is dramatically different from five one-year-old cars, yet the same in terms of car-years. 🙂

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  • Sniffnoy

    Yes, absolutely! The first time someone showed me the rectangle, demonstrating that multiplication is commutative, I was amazed. This was certainly not something I had come up with on my own. It’s not something that can be left implicit.

    One thing I note here: Going by what the student says, they seem to be interpreting n*m as m added to itself n times. Whereas in ordinal arithmetic, or if you’re doing number theory from the foundations and defining multiplication recursively, the usual convention is that n*m is n added to itself m times. Since ordinarily people learn about the commutativity of multiplication and internalize it long before they unlearn it for other purposes, these don’t normally come into conflict; but I still think it would be nice to use a convention matching the usual mathematical one.

  • Chauntelle

    This is very interesting why the order matters in multiplication. Even though there is no difference of result, most young kids believe that 13*2 is easier than 2*13. Since multiplication is the only this that is commutative, it does not matter by an orders. However, what I found interesting thing at the same time is with the story problem. Rather than saying, “2 people each person has 13 eyebrows how many eyebrows are there?”, “from 13 people each person has 2 eyebrows” makes sense. Even though the story seems awkward, it still comes up with the same result. When the child apologized “because his sotry is for 2*13, not 13*2), this indicates student is not aware of multiplication as commutative. Looking at the multiplication, this can be written as different methods such as two times thirteen, two multiplied by thirteen or two, thirteen times which does not actually matter with the order of multiplication.

  • Cheng(Tiffany) Tao

    It is a really interesting story that will happen when children learn multiplication at the beginning. I think the children has already understood how multiplication works that he knew the total number of two people’s eye brows is two people times the number of one people’s eye brows. But the student didn’t understand that A*B equals B*A which we can help him to understand from another way. In this case, two people each one has 13 eye brows can be also solved by addition which is 13+13. By going back to addition level that have two of 13 add up, we can then help children find 13*2 equals 2*13. To help student learn that multiplication is commutative, we can transfer different interpretations for 13*2 and 2*13. For example, there are 13 people and each one can get 2 pizzas, to get the total number of pizzas we use 13 times 2. Also, we can say that teachers delivered 2 pizzas for each person and there are 13 people in total, to get how many pizzas we need we use 2 times 13. They mean the same thing here which can also help children get the idea that 13*2 equals 2*13. So does the story that student wrote that is also the story for 13*2.