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Jonathan Groves
Posts:
2,068
From:
Kaplan University, Argosy University, Florida Institute of Technology
Registered:
8/18/05


Re: Is Multiplication Repeated Addition?
Posted:
Apr 7, 2010 5:19 AM


Dear All,
Here is a recent reply that I had written to one who has been involved in the recent debates on mathteach about teaching multiplication as repeated addition:
I certainly agree that students having trouble understanding the conceptual meanings of the basic arithmetic operations is a major concern in mathematics education; their lack of conceptual understanding makes it difficult for them to learn to compute and to use these operations in solving problems and makes learning algebra difficult. I too see some of my remedial students using the wrong operations to solve even simple word problems. For example, some of them will try to multiply or add to solve this problem: There are 1288 seats in a lecture hall and 56 rows of such seats. Each row has an equal number of seats. How many seats are in each row? And I have seen some students this term try to solve the equation 12*x=180 by subtracting 12 from each side. Perhaps some of them had misread 12*x as 12+x, but I wonder if some of them really did believe that subtraction undoes multiplication.
The misreading is apparently an issue as well because on one recent assignment, I had asked students to solve a few proportion problems by doing all the computations with fractions, but a fair number of them converted the fractions to decimals and worked with the decimalseither right at the beginning or somewhere in the middle of the problem. I generally allow flexibility in methods for solving problems, but, in some cases, they should know how to work with fractionsespecially if one or more of those fractions are nonterminating decimals. And I do want them to realize that they cannot always avoid fractionsno matter how uncomfortable they feel about them or how much they despise fractions. Some of them might have seen these instructions to leave everything as fractions but used decimals since they want to avoid fractions like the plague. But I wouldn't doubt that some of them didn't see those instructions. Our book, just like many of the standard commercial textbooks for remedial math students, does not do a good job explaining fractions so that they make sense to the students. I should develop some materials for this purpose since their book contains severe gaps in the logic and conceptual meanings of fractions and their operations.
It is clear to me from this post and your previous ones that your understanding of repeated addition is different from the standard meaning of repeated addition. I take it that my understanding of what is the standard meaning of repeated addition really is the standard meaning since this is the meaning I've usually seen and the meaning that appears to be used by most others in these recent discussions about multiplication as repeated addition. But regardless of what your understanding of repeated addition is or what my understanding is or what any particular person's understanding is, we cannot deny that many students, adults, and even teachers think that multiplication is repeated addition and where repeated addition means to add one of the factors to itself a certain number of times. Even if those people "know" that multiplication of fractions and other real numbers is not repeated addition in this sense, they usually do not know any conceptual meanings of multiplication beyond the natural numbers. And not knowing these conceptual meanings of multiplication beyond N has, probably among other reasons as well, contributed to their confusions about fraction multiplication, real number multiplication, ratios, and proportions. This is the "repeated addition" idea that Devlin discusses, and I'm sure the lack of conceptual explanations of multiplication in Q and in R is the reason that Keith Devlin thinks for why students have later difficulties with multiplication in Q and in R.
The books I've seen for remedial math students spend little, if any, time on developing these conceptual meanings of the operations and instead focus their time on doing the computations. Nothing wrong with computations, but math is far more than computation, and computations make little sense to students anyway if they lack conceptual understanding of these computations. And their lacking of these conceptual meanings helps contribute to their lack of number sense so that adding three positive fractions with one of them being almost as large as 2 and getting a fraction less than 1 does not bother them or that getting 1.6 as the decimal equivalent of 5/8 does not bother them. If it does bother them, none of these students have said anything to me to suggest that it does.
From these discussions and from Keith Devlin's articles and from my own thinking as well, there are multiple ways to think about multiplication that are mathematically valid. Some ways are more useful in certain contexts or for certain purposes than others. Some ways work best for developing conceptual understanding but are awkward or inefficient for computations whereas others are best used for computations only since they don't help really get at the underlying meanings of what the operations are or how they behave. Because different students think in different ways and because none of these explanations are suitable for all purposes and in all contexts, it is best to teach students these multiple ways of thinking about mathematical concepts, whether they are learning multiplication or something else. And students often are not taught much to help develop contextual and abstract and creative thinking, which hurts their abilities to learn problem solving and more advanced mathematics such as algebra. The books I've seen do not do a good job of this.
Scaling is one useful way to view multiplication since it helps us understand what multiplication on R does to numbers and helps us see what it means to take 2/5 of a quantity or to make a quantity 3.44 times larger, for example. When I think more about the MARA (aka MIRA) issue, I see several possibilities for why MARA causes problems:
1. MARA does not work well in helping students develop conceptual understanding of multiplication in Q and in R, no matter what we do. That is, MARA may work for some students but only a small minority of them.
2. MARA can be made to work without hurting students' abilities to develop conceptual understanding of multiplication in Q and in R but that this approach is risky since it is easily misapplied, especially if there is no effort by the textbooks and teachers to help develop in students the kind of reasoning they need to extend their thinking so that multiplication in Q and in R makes sense.
I'm not sure which of (1) or (2) is correct. If (2) is the case, then the problem is that MARA is not developed much beyond that and that conceptual meanings of multiplication in Q and in R are not mentioned or used much. Even if (1) is the case, the teacher who uses individualized instruction might use MARA for those students who can learn it effectively for both the short run and the long run and use alternate approaches for those students who need them. In short, whichever is true, I think the real problem is that the teachers and textbooks do not use a wide enough variety of different ways to understand multiplication, especially multiplication in Q and in R.
Jonathan Groves



