Your comments do remind me that the multiplication debates that Keith Devlin's articles have stirred up in many places on the Internet (and I do not know all the places!) have not mentioned division very much, including the debates I stirred up in February and in March last year on Math-Teach. One response mentioning division by Michael Paul Goldenberg can be found on his blog at
Many students' problems with understanding division can be traced to the problems with MIRA (Multiplication Is Repeated Addition) since division is often first introduced as repeated subtraction or, as you put it, partitioning. I believe that, if we can get students to understand multiplication and to get teachers to understand the problems with MIRA, then it will be much easier to develop strategies for helping students understand division and for helping teachers to see the problems with traditional ways of teaching division.
The MIRA idea leads many students to falsely believe that multiplication always makes numbers bigger. Even in everyday language, we see that misunderstanding: When rabbits multiply, we sure as heck are not talking about them working a math problem on paper! Students who believe that multiplication always makes numbers bigger will likely be unsettled by the equation (2/3)*300 = 200. And multiplications such as (3/4)*(2/3) = 1/2 are hard to understand with the MIRA mindset because it does not make sense to add 3/4 to itself 2/3 times or to add 2/3 to itself 3/4 times.
Likewise, the division as repeated subtraction or as splitting into groups leads many students to believe that 2 divided by 5 is impossible since we cannot subtract 5 from 2 or because we cannot split 2 into groups of 5 each. That is probably one major reason why I see college students all the time trying to convert 2/5 to a decimal number by dividing 5 by 2 instead of 2 by 5. There is a way to make sense of splitting into groups even in cases like these, but that is difficult for students to imagine--especially when they are still entrenched with the whole number division into groups idea. There might be a way to introduce this idea effectively, but that requires careful timing and explanation so that it does not become a roadblock in their efforts to make sense of divisions such as 2 divided by 5 or division of fractions or decimal numbers.
As for your final comments on division, I would certainly say that such an approach simplifies division. William P. Thurston in his article "Mathematical Education" says the following:
I remember as a child, in fifth grade, coming to the amazing (to me) realization that the answer to 134 divided by 29 is 134/29 (and so forth). What a tremendous labor-saving device! To me, ?134 divided by 29? meant a certain tedious chore, while 134/29 was an object with no implicit work. I went excitedly to my father to explain my major discovery. He told me that of course this is so, a/b and a divided by b are just synonyms. To him it was just a small variation in notation.
I have not confirmed this link since my Internet in the last few months has not allowed me to access Arxiv articles; I cannot determine what the problem is. I had to grab this quotation from a copy saved to my computer. I cannot find an electronic copy anywhere else at the moment so that I can confirms its link before sending it.
However, there is still a need to convert quotients to other forms--namely, decimal number forms.
On 1/2/2011 at 2:48 am, Anne Watson wrote:
> Further thoughts about multiplication. It is great > that Keith is making this work more widely known, but > I have recently been realising that even the book he > refers to underplays the difficulty children have > with division. As mathematicians we often assume > that it is enough to know about reciprocals, and to > see division not as a separate thing but as the > inverse of some other multiplication - I am talking > here about multiplication on real numbers of course. > > But for school students, even those in upper > secondary school, the meaning of division is still > tied up with algorithms they learn to carry it out - > algorithms mainly concerned with chunking numbers and > then performing some kind of repeated subtraction. > The meaning of scaling gets totally lost in the > e algorithm. Furthermore, the meaning of division is > also very strongly modelled by partition, rather than > scaling, and the everyday language used in school to > trigger division enforces this - "how many of these > go into this?" As educators we should not > underestimate how strongly these meanings linger - > meanings that are associated with repeated addition > but not with scaling. > > I have been wondering what school students' > understanding of division would be like if instead of > learning algorithms for dividing they always > constructed rational numbers to represent > multiplicative inverses, and the 'answer' to a > division is then expressed as an ordered pair rather > than a single decimal number. Then the dominant > mathematical action when dealing with 'division' > would be finding common factors to express the > multiplicative relation in its simplest form. > > Happy New Year > > > > > > > > Anne Watson > Professor of Mathematics Education, Department of > Education, University of Oxford (Linacre College) > ________________________________________ > From: Post-calculus mathematics education > [MATHEDU@JISCMAIL.AC.UK] On Behalf Of Jonathan Groves > [JGroves@KAPLAN.EDU] > Sent: 01 January 2011 22:28 > To: MATHEDU@JISCMAIL.AC.UK > Subject: Another Keith Devlin Article on > Multiplication > > Dear All, > > Keith Devlin has written another good article on > multiplication > called "What Exactly Is Multiplication?" > > In this article, Devlin discusses multiplication from > a conceptual > viewpoint: Multiplication is scaling. Actually, > multiplication > as scaling and rotating is a better image to use > since it extends > to complex number multiplication. The multiplication > z_1*z_2 > scales the vector z_2 by a factor of |z_1| and > rotates it > by arg(z_1). > > He also discusses multiplication as an abstract > operation in > which we do not define what multiplication exactly is > other > than that it is a ring operation with certain > properties. > > He also mentions in his article that mathematicians > rarely think > about what multiplication is since these kinds of > questions > do not arise in their work and hence the reason why > he has not > addressed to his readers much about what > multiplication is in > concrete terms. He does mention that multiplication > as a cognitive > process or as a concrete operation is very complex > and cites a > 414-page book by Harel and Confrey devoted solely to > the > complexities of multiplicative reasoning and the > development > of multiplicative reasoning in students. > > I wholeheartedly agree with his comments that the > abstract > mathematical concept of multiplication avoids many of > these > numerous complexities associated with multiplication > as a > concrete operation; I doubt that I could write 400+ > pages > devoted to just multiplication. Perhaps that > explains why > so many students and even teachers struggle to > understand what > multiplication is in terms of concrete meanings. As > Keith > Devlin nicely puts it, > > "That is the whole point of abstraction. Though many > non-mathematicians > retreat from the mathematicians' level of > abstraction, it actually > makes things very simple. Mathematics is the ultimate > simplifier." > > Of course, the catch is that students need time to > reach > this level of abstract reasoning and to learn to > appreciate it. > > Keith Devlin's latest article can be found at > > http://www.maa.org/devlin/devlin_01_11.html. > > > > Jonathan Groves