The meaning of "product" – 3.OA.1

[Duane]

I’m a little unclear as to the intent of 3.OA.1, “Interpret products of whole numbers”. One reading of it is that, given a number, the students should be able to identify what two numbers can be multiplied to give that number, i.e. finding factors. Another reading, based on the example given in the Standards, is that the students think up a story to match an expression such as 5 x 7. Or is it more general in that students just need to know what a product is, i.e. the result of multiplication? Are any, or none, of these interpretations correct?

Adding to my confusion is the apparent use of the term “product” to describe a mathematical expression. This is the first sentence of 3.OA.1: “Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each.” My understanding (for elementary school math) has always been that the “product” refers to the result of multiplication, not the expression that describes the numbers to be multiplied. Under this definition the first sentence should read, “Interpret products of whole numbers, e.g., interpret 35 as the total number of objects in 5 groups of 7 objects each.” In other words, an expression is not the product, but this is how the original sentence reads to me. Other places in the Standards seem to reinforce this interpretation too, such as p.89 in the glossary where one of the columns is headed “Unknown Product” with the example “3 x 6 = ?” But other places in the standards contradict this meaning too.

Any clarification is appreciated!

 

[Bill McCallum]

Your second interpretation is correct. This standard is the first standard about multiplication and is about understanding the meaning of multiplication. So yes, it’s about the meaning of expression $5 \times 7$. I agree the word product is used with two meanings; as expression here, and as the result of a calculation elsewhere. Do you think this causes confusion? It seems fairly common.

[Duane]

Phew, thanks for that Bill – thinking up stories is much easier to explain!

Using product to mean both “5 x 7” and the result “35” is confusing for me and I can’t imagine using it with kids in this way.  If I write “5 x 7” on the board and ask students what the product is I don’t want them to simply say “five times seven” back to me – teaching terminology is hard enough without throwing this one in!

Knowing that there is an alternative use of “product” in the Standards helps me interpret it from now on, but all the everyday dictionaries (Dictionary.com collates some online versions) and the few specialist math dictionaries (including MathWorld.com) I can access  define product as the result of multiplication. The only exception is Wikipedia. Without a glossary entry in the Standards that explains an additional meaning I can imagine others may find it similarly confusing.

[Bill McCallum]

I don’t think the two meanings are inconsistent; $5 \times 7$ is indeed the result of multiplying $5 \times 7$, and the student who says “five times seven” isn’t wrong (maybe a smart aleck, but not wrong). In fact, now I think about it a little more, I’m not sure it’s correct to say that there are two different meanings; rather there are two different ways of writing the product.

At any rate, the language of the standards is not necessarily the language you would use with students. The standard describes a certain understanding you want students to have, an interpretation of $a \times b$ as the the number of things in $a$ equal groups of $b$ things each. You are probably not going to bring about this understanding by asking a class of third graders “what is the intepretation of $5 \times 7$?” You might simply work with various situations in which products arise in this way.

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