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Replies: 1   Last Post: Apr 7, 2010 5:19 AM

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 Jonathan Groves Posts: 2,068 From: Kaplan University, Argosy University, Florida Institute of Technology Registered: 8/18/05
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 math-teach about teaching multiplication as repeated

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 decimals--either 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 fractions--especially if one or more of those fractions are
nonterminating decimals. And I do want them to realize that they cannot
always avoid fractions--no 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
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
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

Date Subject Author
3/24/10 Jonathan Groves
4/7/10 Jonathan Groves