On 6/25/09, Elementary Teacher wrote:
> I teach elementary school, and I have always thought
> multiplication was just repeated addition. Can you explain
> to me why it's not?

I think the first thing you need to differentiate is between
the computational procedure and the concept. There is nothing
wrong with repeated addition as a computational procedure to
calculate the product of whole number multiplication. It is a
useful computational strategy, and it works with whole
numbers, but it is not the concept of multiplication. Just as
counting up is a good computational strategy for addition,
but it is not the concept of addition.

I brought up this topic as an example of how we in this
country focus on the computational procedure and ignore the
concepts. That was the finding of the video study of three
countries that was the subject of the book The Teaching Gap.
In the other two countries, Japan and Germany, teachers spend
a lot more time on concepts than we do in the United States.
The authors, Stigler and Hiebert, argue that it is a cultural
script—demonstrating procedures and having students practice
the procedure—it is what we think of as teaching, so it is
difficult to see that we are ignoring the concepts. If the
students can get the right answer, then they understand,
right? They know the procedure but we never taught them the
concepts. Procedural knowledge without conceptual
understanding is shallow. The procedures are easily
misapplied or forgotten when there isn’t conceptual
understanding to guide their use. The evidence is our
mediocre performance on international tests, and the amount
of remedial work that needs to be done. I did some volunteer
math tutoring at a community college. It is really amazing
the number of students who do not pass the placement exam for
Math 100. It is amazing how many students fail Math 100 and
other courses.

In my opinion, the consequence of teaching multiplication as
an additive operation is that students do not have the
foundation for dealing with multiplicative relationships—
ratios, proportions, fractions, decimals, percents. I would
like to quote from the National Mathematics Advisory Panel’s
final report. The panel of experts was formed by President
Bush in 2006 and their final report was issued in March of
2008.
-- --
So what is the concept of multiplication? As I said in my
prior post,
> multiplication is scaling. It is using an intermediate unit
> size, a change is scale. I’ll give an example: 3 six-packs
of
> juice is 3 x 6; 6 is the intermediate unit size and 3 is
the count
> of how many sixes there are. The product is the conversion
of the
> units back to the standard unit 1. Multiplication always
involves
> transforming the units, in this case from 3 six-packs to 18
cans.
> Addition is joining like units. The units must be the same
size.
> Scaling is closely related to proportionality. Rational
numbers
> require paying close attention to the unit size, what is
> considered the whole or 100%. When we treat multiplication
as an
> additive operation and don’t develop the concept of
multiplication
> as a multiplicative operation, we set up our students for
> difficulty as they progress in math topics.

Here is how a Russian curriculum by Davydov introduces
multiplication. Students are asked to find how many tiny cups
of water can be served from a large container. They start
pouring the water out and counting how many tiny cups there
are, transferring the water into another large container. But
it is tedious and difficult to pour into such a tiny cup. The
teacher asks them to think of a better solution than pouring
into the tiny cup. The next day the teacher suggests using a
large cup. They pour 8 tiny cups into the large cup, so they
know that each large cupful is 8 tiny cups at once. They
record 7 large cups, and he teaches them to write it as 7 x
8. They don’t compute the product right away. They do more
exercises measuring different things with a larger unit,
measuring how many smaller units are in the larger unit, and
writing the result as 4 x 5, for example. The teacher wants
them to understand the concept of using a larger unit size
and counting in that unit size. Then later they compute the
product using repeated addition. You see, repeated addition
is a good computational strategy, but the concept is changing
the unit size and counting in a larger unit size.

Let me just say that with multiplicative relationships, the
scale changes all the time. If I compare 2 with 8, 2 is 25%
of 8, and 8 is the whole. But if I compare 8 with 2, 8 is
400% of 2, and now 2 is the whole. Multiplicative comparisons
are relative. The reference quantity, what is the whole or
100%, changes. In additive comparisons, the difference of 2
and 8 is 6. I can say 2 is 6 less than 8 and 8 is 6 more than
2, but the scale is always the standard unit 1, and the
difference is always 6 units. Even with signed numbers, the
scale is still the standard unit 1. (If I multiply with the
standard unit 1, then I don’t change the scale. Hence, 1 is
the identity element for multiplication.)

We use different units all the time. It is not a strange
notion. Take the number 325. We have three digits and
therefore three units: hundreds, tens, and ones. The 3 is the
count of the number of hundreds (3 x 100); the 2 is the count
of the number of tens (2 x 10), and the 5 is the count of the
number of ones (5 x 1). Beginning with multiplication, we
should help students consciously think in terms of different
unit sizes, because that is a skill they need with all
multiplicative relationships. Also, the same number of units
on different scales are proportional. Thinking in terms of
scaling helps children develop an intuitive understanding of
proportions.

Share This Post | Report This Post

Next Post >>

Posts on this thread, including this one