M2N3. Students will understand multiplication, multiply numbers, and verify results.

In the CCSS, multiplication is introduced in Grade 3 in the domain of "Operations and Algebraic Thinking." The first standard in the cluster related to multiplication, the CCSS states the following:

1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.

In the public draft released in spring, 2010, there was a statement about multiplication as repeated addition, just like M2N3a. However, that statement has been removed. Instead, the CCSS focuses on the meaning of multiplication as an operation to find the total amount when objects are arranged in equal groups. Then, in Grade 5, the CCSS states that the meaning of multiplication to be expanded and consider multiplication as scaling (or re-sizing). I think the approach the CCSS has taken is much more appropriate than what the current GPS states. For those of you interested in the discussion of whether or not multiplication is repeated addition, I encourage you to read a series of columns written by a Stanford University mathematician, Keith Devlin (June 2008, July-August 2008, and September 2008).

Today, however, I would like to focus on the implicit idea in the CCSS - and the current GPS does not even touch upon this idea. Toward the end of my previous blog, I discussed the order in which you write multiplication sentences. In it, I made it clear that my preference is to write the multiplicand, i.e., the number of objects in a group, first then the multiplier - I might argue that is THE correct way mathematically. However, the CCSS actually suggests we write multiplication sentences in the opposite order. Thus, 5x7 is interpreted as "the total number of objects in 5 groups of 7 objects each." Although I have stated in the past that what is important is we have an agreement on the order, I have run into several situations recently that revealed writing the multiplicand first is the way to go.

But, let's first start with how we state/write/read multiplication sentence. A common way teachers and students read multiplication sentence is "5 times 7 is 35." However, if the sentence is representing the situation with 5 groups of 7 in each, a mathematical way of reading the sentence is "7 multiplied by 5 is 35." When we use the phrasing, "N is multiplied by M," it is clear that M is the number of groups - that is, N is taken M times. Thus, one surface level issue is that the order in which we read multiplication sentences and how they are written may not align. Some might argue that this is a non-issue. After all, the same thing happens with division, too. We say "35 divided by 7," but we also say, "how many times does 7 go into 35?" When we write division problem on paper, the divisor may follow the division symbol or it may be outside of the long division symbol (thus to the left of the dividend).

To me, however, the issue is fundamental, and writing the multiplier first creates some difficulties in mathematical discourses. Let me share some examples. In the 4th grade CCSS standard, student are expected to understand multiplication of fractions by whole numbers. The CCSS document is very careful to remain consistent with the order, so in the examples they include always have the multiplier in front, such as 3 x (2/5). Once we agree that we write the multiplier first, problems such as (2/5) x 3 are treated in Grade 5. [There is actually a similar distinction with respect to multiplication and division of decimal numbers in the current GPS. Students learn about multiplying and dividing decimal numbers by whole numbers in Grade 4, and multiplication and division by decimal numbers are discussed in Grade 5.]

Then, in Grade 5, the CCSS treats multiplication by fractions: "Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b." So, (2/3) x (4/5) is interpreted as taking 2/3 of 4/5. Thus, we first divided 4/5 by 3 (thus students must have learned how to divide a fraction by a whole number before this topic), then we take 2 groups of it. Thus (2/3) x (4/5) = 2 x (4/5 ÷ 3). Thus, the multiplier, 2/3, gets split around the multiplicand, 4/5. If you write the multiplicand first, taking 2/3 of 4/5 will be written as (4/5) x (2/3) = (4/5) ÷ 3 x 2 = (4x2)/(5x3). This seems to be much easier to connect to the formula (a/b)(c/d) = ac/bd. [Another issue here is why there is no parentheses around q ÷ b in the CCSS. It seems like you must first find the "partition of q into b equal parts," but the order of operations says we go from left to right. Without parentheses, the statement "a x q ÷ b" means multiply q by a, then partition the result into b equal parts.]

Another example is when discussing how to create equivalent fractions. Again, the CCSS is very careful about the order in which multiplication is written. Thus, they say, "Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size." Typically, this idea is discussed as "if you multiply both the numerator and the denominator of a fraction by the same number, the value of fraction remains the same." More often than not, this relationship is written in textbooks as a/b=an/bn - the multiplier written after the multiplicand. But, this notation would not match our agreement on how to write multiplication.

One final example is not explicitly mentioned in the CCSS nor the GPS. However, once we learn division by fractions, we often makes the statement, "division is the same as multiplication by the reciprocal of the divisor." When we use mathematical notations, we will typically write (a/b) ÷ (c/d) = (a/b) x (d/c). However, if division is the same as multiplication "by the reciprocal," that is, the reciprocal is the multiplier, it should really be written as (a/b) ÷ (c/d) = (d/c) x (a/b).

In general, many familiar ways we discuss/write multiplication assumes that the multiplier comes after the multiplicand. Thus, "5x7" should not be "5 times" of 7, but rather 5 "7 times." Perhaps because I am not a native English speaker, I also get a different sense when I hear "5 times 7" in one breath and "5" (pause) "times 7." The former gives me the sense of 5 groups of 7 but the latter makes me think of 7 sets of 5. Anyway, I wish we can eventually agree to write the multiplicand first - perhaps in the next revision of the CCSS.

By the way, some people might wonder about how this idea works in algebra. After all, when we think of simplifying "5x + 3x," it is much easier to think of this as having 5 x's and 3 x's altogether, thus 8 x's. On the other hand, we want students to understand the slope of a linear function, like "3" in y=3x+2, as the "rate of change." A rate, however, is typically amount per unit. Thus, "3x" in this context suggests we have x units of 3. In algebra (and other higher level mathematics), I believe there are actually two competing conventions. One is the order in which we write multiplication and the other is the convention of writing numbers (and constants) before variables in a term. In algebra, I believe, the latter convention wins perhaps because it makes manipulation of algebraic expressions simpler. But, I think it is still very important that students pay close attention to how we write multiplication when they first learn it in elementary grades.

## 3 comments:

Hello Tad,

Thanks for this very clear analysis of the complications of writing the multiplier first.

Regarding the algebra example toward the end, the idea of simplifying 3x+5x, I, too used to think of this as "3 Xs combined with 5 Xs." Only recently did I view simplifying such expressions as applying the distributive property in reverse: 3x+5x = (3+5)x = 8x. Is that "better"? It allows us to be consistent....

Tom,

"better" is a qualitative judgement that depends on the criteria. Whether or not 3X means 3 groups of X or 3 X times depends on the quantity 3X is representing. Because of the algebraic convention, we write the number, 3, first regardless of 3 being the multiplier (the number of groups) or the multiplicand (group size). When we are looking at an algebraic expression like 3X+5X, without any context, then it is really up to us to interpret it in the way that is sensible. In the context of learning combining like terms, thinking each term as 3 X's and 5 X's is probably more meaningful than thinking of 3 and 5 as the rate of change - as in linear functions. From the rate of change perspective, 3X + 5X means the sum of these two functions will have a slope of 8, which is also true but I suspect much more difficult for students to grasp, at least initially. At least that is my current thinking.

Thanks for this explanation. The depth of knowledge required to teach mathematics well continues to astound me.

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