GS Chandy
Posts:
4,348
From:
Hyderabad, Mumbai/Bangalore, India
Registered:
9/29/05
|
|
Re: Multiplication Is Not Repeated Addition
Posted:
Apr 7, 2010 9:02 PM
|
|
Alain Schremmer posted Feb 22, 2010 1:25 AM (my comments interspersed in this colour):
>
> On Feb 20, 2010, at 1:40 PM, Jonathan Groves wrote:
>
> > I think that these ideas you had mentioned didn't
> >arise in the math-teach
> > discussions because the question is primilarly
> > about how to teach what
> > multiplication "is" to kids who are learning
> > multiplication for the first time.
> > The formal approach certainly wouldn't work with
> >them.
>
> Not sure what you mean here. I still depends on what
> set this multiplication is working on.
>
> > Strictly speaking,
>
> Not strictly speaking. It just happens that for
> whatever reason, lack
> of imagination or a sense of ecology, humankind keeps
> recycling words
> and not just in mathematics. To infer from that that
> their meaning is
> one and the same is incorrect.
>
> > I must agree: Multiplication of natural numbers is
> > different from multiplication of positive integers
> > and is different from
> > multiplication of positive rational numbers and so
> > on.
>
> yes
>
> > And different mathematicians have given different
> > definitions over the years.
> >
> > Speaking of definitions, at least one mathematician
> > had raised a fair question
> > when he had asked what a definition really is and
> > what separates a
> > definition of X from an equivalence theorem about X
> > and had asked who is to
> > say that a particular definition about X is the
> > "official" definition of X.
>
> Do you happen to remember who?
>
> > We see different definitions because they had
> > arisen from different problems,
> > and one problem suggests one feasible definition,
> > and another problem suggests
> > a different feasible definition (sometimes this
> > alternate definition is
> > equivalent to the first one and sometimes not).
> > For example, some books,
> > especially older ones, do not require that a ring
> > have a unit, and other books do.
>
> That is indeed another aspect, an important one. We
> work on a problem, mathematical or not and we start by
> using the tools of the trade but pretty quickly we
> customize, including the language of the trade.
>
One problem is that in 'pure prose', it's extremely difficult - perhaps impossible - to keep up with arguments involving relationships between factors in a complex system, because in pure prose most such relationships are left largely ambiguous. I recommend 'prose + structural graphics' (p+sg), a mode of communication developed from the insights of the late John N. Warfield into complex systems - including 'thought systems' such as mathematics.
> > But in teaching or presenting mathematics, we have
> > to start somewhere, and the choice of which definition
> > to use depends on what
> > is being discussed, how the ideas are developed,
> > and even to whom you are teaching.
>
> Ditto
>
> > It appears that the teaching of multiplication of
> > natural numbers as repeated addition has trapped many
> >students into thinking that any form of multiplication
> >is repeated addition.
>
> That is a fallacy. What has trapped the students is
> not the introduction of multiplication as repeated
> addition but the fact that
> the teachers then went on using the same term without
> ever mentioning that they were switching to something
> entirely different.
>
> Very small children know that very different looking
> dogs are dogs.
>
Entirely true: I suspect we do not effectively "teach to the capabilities" that children inherently possess - we too much expect them to memorize, when by-rote memorization is NOT what they do naturally (except perhaps for nursery rhymes and suchlike).
>
> But then they can see it while teachers are
> themselves totally oblivious of the fact.
>
> > So fraction multiplication,
> > for example, is hard for them to understand because
> > the repeated addition idea no longer works.
>
> I disagree;
>
> 1) students have no problem with multiplication of
> fractions if a) it is propery introduced as an
> operation.
> b) students are warned that the name multiplication
> has just be(en?) recycled.
>
To my mind, this is still a bit of a tricky issue: both Jonathan Groves and Alain Schremmer's viewpoints contain some part of the underlying truth.
I don't remember now how I myself as a child dealt with the transition from multiplication as repeated addition (which must have been the way it was taught to me) to multiplication of fractions (when that came), and so on - but I guess I must have coped with no great damage done.
(I don't remember that my teachers handled it as Alain Schremmer suggests above).
> >
> > Is it possible that multiplication of integers or
> > rational numbers is taught too soon (before they have
> > developed their mathematical thinking skills enough to
> > learn that context in mathematics affects
> > which definitions and theorems apply)?
>
> No. It is just a matter of using correct
> interpretations (in the
> sense of model theory)
>
I believe it is ALL (or practically all) a matter of using the correct interpretations in the sense of model theory. I have found Warfield's approach to modeling to be the most usable and fruitful approach that I've seen. Below my signature, I am pasting a plain text version of a .doc file of mine "What is modeling?", which explains Warfield's approach.
> > That is, is the main problem not really
> > about teaching multiplication of whole numbers as
> > repeated addition before
> > teaching multiplication of integers and rationals
> > and reals but about the
> > timing and how it is taught?
>
> Certainly not the former, the latter certainly.
>
Largely true, in my view.
>
> > Related to this question about definitions is the
> >second point you had made
> > here, and it is a valid one. One problem with the
> > teaching of mathematics
> > is that often students are rarely told that
> > different definitions of the
> > same concept exist and that sometimes these
> > definitions are not equivalent.
>
> A bit of confusion here. Why say "same concept"? To
> what extent are multiplication in N and multiplication
> in Z avatars of "the same
> concept"? It is precisely because we do not deal with
> this issue that
> it comes back and hits the students later.
>
I also suspect that we do not adequately use the concept of Venn diagrams to explain things to students (even as children) - now those are models that could be understood at almost ALL levels!
> >
> > Furthermore, the students aren't told that one
> >definition isn't necessarily
> > any better or more correct than another one. And
> >they certainly are not told
> > why these different definitions exist and certainly
> >aren't given even the
> > slightest idea of why one definition might be
> chosen over another.
>
> Here too, I a bit lost. Are you saying that there
> are, say two
> possible definitions of addition in N---which there
> are---or are you
> talking about addition in N and addition in Z?
>
> > It is
> > possible that some teachers in K-12 or at the
> > beginning college level
> > teach this, but I know such teachers are rare. I
> >hate to admit that I am
> > one of the many guilty ones, but I'm thankful I'm
> > aware of that now because I wasn't previously.
> >
> > Mathematicians generally agree (at least according
> > to what I have learned in
> > abstract algebra) that the operations on a subset
> > of a ring R are the same operations
> > in R restricted to this subset.
>
> No. You are just taking the injection for granted.
> This is where the
> category theory viewpoint is nice. See Lawvere.
>
I need to "see Lawvere" myself! Shall do so soonest possible.
> >
> > Thus, if we view the integers Z as a subring of
> > Z[x]*, then the addition and multiplication in Z can
> > be defined as addition and multiplication
> > in Z[x] restricted to Z. Likewise, the whole
> > numbers form a subset of the ring of real
> > numbers
>
> ABSOLUTELY NOT TRUE. The whole numbers do NOT form a
> subset of R.
> This is the very essence of the problem. Again, you
> are ignoring the injections.
>
> > (which is also a field, of course), so whole number
> > arithmetic is, by this idea of definition, real number
> > arithmetic restricted to the whole numbers.
>
> Restriction is indeed the word and it is defined.
>
> > I would be interested to know more about what David
> Tall had taught you.
>
> That I was wrong to say that multiplication of
> counting numbers IS the cardinal of the cartesian.
>
> > Did you read one of his books or attend one of his
> lectures?
>
> Neither, I got into an email argument with him. About
> a year later, I
> started to weaken in my absolutism and a few years
> later, realized I had definitely, totally and absolutely > lost, when I read Sen's "Identity and Violence".
>
Wow! That's the first time I would have suspected this kind of relationship between Amartya Sen's work and David Tall's. I need to investigate.
I must say that reading the documents put up by David Tall have taught me a lot. Am still learning there.
GSC
Background note ? What Is Modeling (Warfield's approach)
==================================
D: A background note: What is modeling?
-- By G.S. Chandy
The Structural Modeling Approach ? and how it is significantly different from any conventional approach
First, a quote from John N. Warfield:
Modeling is a process that begins with human perception. A sequence of the following nature describes the activity of modeling:
1) Perception
2) Storage in the brain
3) Identifying a context within which to place the perceptions, and within which they can potentially be integrated
4) Generating factors associated with that context and with the perceptions that are the focus of attention at the time
5) Identifying types of relations that appear to be associated with these factors in the chosen context
6) Structuring the factors to show how they are interrelated through specific relationships that are representative of the selected types
7) Interpreting the structures produced
8) Associating the factors with algorithms that permit the relationships discovered to be quantified (if they are possible to quantify)
9) Assigning or computing numerical values to/for the factors
10) Interpreting the model-related information for purposes of design or decision-making
(Above paraphrased from ?Structural Thinking?, J.N. Warfield: 1995-96 Essays on Complexity)
The above sequence describes Structural Modeling, the process underlying Interactive Management (and the One Page Management System). Built into the above-outlined Structural Modeling process, when IM or OPMS is used, is an ongoing comparison of model-related information at each stage with the reality on the ground. These comparisons become sharper and more focused as the models evolve and develop over time.
The conventional way (which the IM or OPMS process would not allow at all) is to start at Step 8 or at Step 9 of the above-outlined modeling sequence.
In fact, most discussions between people not using IM/OPMS start out at Step 8 or Step 9, usually leaving out Steps 1 to 7, which are pre-requisite for clear understanding all round! (It is true that there are, on occasion, some context-clarifying remarks made, but these generally lack adequate focus to ensure truly clear understanding all round). Thus, many discussions between people are, in the conventional way, based on sets of ?mental models? that are significantly different from each other because of differing backgrounds of the people holding them. These mental models on which different people are basing their discussions are left entirely unclarified. Because of the differences in context, the very same words spoken by different people could often mean significantly different things. In any case, the context is entirely unclear. This leads to non-understanding, misunderstanding, confusion, and, finally, ineffective or incompetent action.
We are interested in ensuring effective action at every level in the organisation ? starting with the individual. Because discussions in the Structural Modeling process are always based on a significant clarification of the context of each idea and thought contributed to the discussion by each person, subsequent action is much more likely to be effective. (Step 3 of the sequence of Structural Modeling outlined above).
It should be observed that ?Structural Modeling? INCLUDES the ?conventional modeling process?. The conventional ?numerate models? (showing numbers, e.g. how much money, how many copies will be sold, and so on ? on which most people rely to the near-total exclusion of any structuring activity) will develop, in a natural way, as the structure of the interrelationships of various issues becomes clear. The difference is that the numbers developing through the Structural Modeling approach are based on a detailed consideration of all structural aspects of the issue, and will therefore have far higher reliability than the numbers made in the usual approach.
E: References
A: Interactive Management
* "Societal Systems: Planning, Policy and Complexity", by John N. Warfield, Wiley, 1976
* "A Handbook of Interactive Management", John N. Warfield and Roxana A. Cardenas, Iowa State University Press, 1994
* "A Science of Generic Design: Managing Complexity Through System Design", John N. Warfield, Iowa State University Press, 1994
* "Essays On Complexity", John N. Warfield, (Review Copy 1997)
* "A Structure-Based Science of Complexity", John N. Warfield (Review Copy, 1997)
B: One Page Management System
OPMS Prototype Software, 2002 ? freely available ? write to:
gs (underscore) chandy(at)yahoo(dot)com
OPMS Workbook, G.S. Chandy, Private Publication, 1993, again 2000
OPMS Handouts, ILW, latest ? July 2001 -
Various other documents 2001-2008, including a sizable number of PowerPoint presentations, Word Documents, etc
C: General Systems Theory (Background study)
There is an enormous literature relating to GST, the mere listing of which would run into literally hundreds, perhaps even thousands, of pages. Those of you who wish to become Facilitators may like to write to us for the names of some reference and background books on GST.
|
|
