Math Forum Discussions

frank

Posts: 122
Registered: 1/25/05

Re: Series
Posted: Sep 24, 2008 3:14 PM

Plain Text

> Frank, have you bothered to read the literature on
> this question
> (specifically the sources Keith Devlin offered at
> length in his most
> recent column on the subject) or do you continue to
> preach ex cathedra?

I read all three of Devlin's articles and several of the references that were available over the Internet. The few I found then were tangential to Devlin's main point. Upon reading your reply, I took another look and found that one reference, Nunez and Bryant "Children Doing Mathematics" (which I now find is partially available through Google Books), that made an excellent point: Children innately understand the difference between adding and the idea of "one-to-many". Too much emphasis on the idea that multiplication is simply repeated addition MAY blur this distinction and induce children to add when they should multiply in problems involving proportional reasoning.
>
> You have evidence to support your assertion that "all
> hope is lost" if
> things aren't done your way? Please do present it.
> Professor Devlin
> has had the courtesy to supply references with
> research that supports
> his views, which is more than I can say for you,
> Wayne, Haim, or Pam
> in decrying them. As for Keith Devlin being "loony,"
> I guess unbiased
> readers will look to the source for that
> pronouncement and to Devlin's
> track record to determine who's the loon.

Below I cite quotes from Devlin's references that support my opinion that "all hope is lost" if we follow ONE of Devlin's recommendations. In his first article ("It Ain't No Repeated Addition", the only suggestion Devlin actually made about how to teach multiplication was:

"Why not say that there are (at least) two basic things you can do to numbers: you can add them and you can multiply them... Adding and multiplying are just things you do to numbers - they come with the package."

I should have said that this was a loony suggestion and not rudely implied that Devlin is always loony. (My apologies.) Do you think this is how multiplication should be introduced in lower school?

The rest of the first article and the second article were mainly devoted to explaining why equating multiplication and repeated addition is mathematically incorrect (without providing any clear examples of why it was incorrect or important for K-12 education.) At the end of the second article, Devlin pleads:

"Let me stress again that I am not suggesting we teach children arithmetic the way professional mathematicians view it. Rather, my point is that, however you teach it (and I defer to professional teachers in figuring out the how), don't do anything that is counter to the way the mathematicians do it .... some of your pupils may well end up in universities where they will HAVE to do it the right way "

In other words, don't create any misconceptions in K-12 that might make my job harder.

In his third article (Multiplication and Those Pesky British Spellings), Devlin finally gets around to explaining why K-12 teachers and students (as opposed to university mathematicians) should care whether or not multiplication is exactly the same process as repeated addition: It may make it difficult to separate the concepts of addition and multiplication when students do proportional reasoning. Devlin states that "Arguably Nunes and Bryant know more about multiplication and how to teach it than any other researchers in the world", so he quotes them on the subject of repeated addition:

"The common-sense view that multiplication is NOTHING BUT [Frank's emphasis] repeated addition, and division is nothing but repeated subtraction, does not seem to be sustainable after a careful reflection about situations that involve multiplicative reasoning. There are certainly links between additive and multiplicative reasoning, and the actual calculation of multiplication and division sums can be done through repeated addition and subtraction. [DEVLIN NOTE: They are focusing on beginning math instruction, concentrating on arithmetic on small, positive whole numbers.] But several new concepts emerge in multiplicative reasoning, which are not needed in the understanding of additive situations.

After reading the portion of their book that is available on the Internet, Nunes and Bryant make a great deal of sense to me. Notice that Nunes and Bryant say that multiplication is MORE than repeated addition, while Devlin says that "It Ain't Not Repeated Addition". Nunes and Bryant refer to repeated addition a "common-sense view" (that is incomplete), while Devlin equates it with the level of ignorance represented by the phrase "ain't not".

Do you think Professor Devlin has done a good job of educating the mathematics community about this issue?