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Topic: MathTeachers?
Replies: 31   Last Post: Apr 15, 2013 1:17 AM

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GS Chandy

Posts: 8,307
From: Hyderabad, Mumbai/Bangalore, India
Registered: 9/29/05
Re: MathTeachers?
Posted: Mar 27, 2013 12:40 AM
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Dave Renfro posted Mar 26, 2013 1:40 AM (GSC's remarks interspersed and follow):
> Danielle T wrote:

> > Hello....I am currently a pre service math teacher
> working on
> > my masters...I'm working on some lesson plans about
> exponential
> > growth... I was curious from your experience...
> What are some
> > of the common misunderstandings and errors you find
> your students
> > have with learning exponential growth?
> Did you grade papers for the math department when you
> were an
> undergraduate or tutor other students for extra
> money? If so,
> think about the kinds of mistakes you saw most often
> and how
> you might go about preventing them. What about the
> errors
> you saw your classmates make when you took high
> school
> algebra 2 and precalculus?
> Off-hand, the most common conceptual errors I can
> think of
> (algebraic procedural errors, of course, are obvious
> -- x^2
> times x^3 is not x^6, etc.) is not fully realizing
> the rapidity
> of growth of exponential functions and incorrectly
> interpolating
> exponential behavior (e.g. the midpoint of 2^4 and
> 2^8
> is not 2^6). For the rapidity of exponential growth,
> an
> example I often used was that about 42 foldings of a
> sheet
> of paper results in a (theoretical) folded thickness
> equal
> to roughly the distance from the Earth to the Moon.
> [Every
> 10 foldings results in very nearly a 1000 factor
> increase,

An excellent - indeed a DAZZLING - example indeed!
> due to the fact that 2^10 = 1024 is almost 1000.
> Hence,
> 40 foldings results in a 1000*1000*1000*1000 = 10^12
> factor
> increase, so 42 foldings results in a thickness of
> about
> 2*2*10^12 = 4 trillion times the thickness of a sheet
> of paper,
> and it's not hard to show that this is roughly the
> Earth-Moon
> distance.]
> See the attached .pdf file handout for more ideas.
> Dave L. Renfro

There's much useful material there, responding on-point to Danielle T's request - "What are some of the common misunderstandings and errors you find your students have with learning exponential growth?" The connection of exponentials to logs should obviously always be explored.

The .pdf is excellent, I'm certain it would be most helpful indeed for teachers as a reference work to help bring home to students the importance of the exponential function (which is often lost sight of, see below on Albert T. Bartlett).

In particular, your suggestions to Danielle T about exploring through recalling the kind of mistakes she saw while grading papers etc, should be very useful indeed.

Clyde Greeno has provided a number of useful observations relating to the mechanics of teaching about the exp. functions, in particular its relation to the graphic behaviour of the log functions:
> 1) reasons for the graphic behavior of the
> 10-log-function [meaning
> log-sub-10 = log-base-10]
> 2) all other pos-log functions are pos-multiples of
> the 10-log function ...
> & conversely (graphically);
> ln(x) is roughly 3-log(x)
> 3) The "a^x" formula is conceptually difficult to tie
> to the a-log(x)
> formulas. As an intermediary, instructional
> alternative to "a^x", also use
> a-exp(x) ["exponential, base-a" or {exp-sub-a}(x)]
> 4) all pos-base exponentials are horizontal
> distortions of 10-exp(x)
> 5) e^x is roughly 3-exp(x) [i.e. exp-base-3 (x)]
> 6) transformation effects of the parametric families
> entailed in
> a[log-sub-b](x-h) +k and in a[exp-sub-b](x-h) +k
> 7) why not neg-base logs & exps? why not base-1? why
> not base-0?

Each and all of these do deserve elaboration - even if the exigencies and constraints of time in the class prevent the teacher from going through all of the above, it is important that the teacher actually has all of this as 'necessary background'.

Doubtless there are other issues that could also lead to misunderstandings and errors as well.

NOW: if an adequately representative listing of these misunderstandings and errors were to be made, then I believe an 'action plan' could be constructed that could help a great many teachers in their 'mission' of helping students understand 'exponential growth'.

This 'action plan' would come out as one or more models (specifically, Interpretive Structural Models, ISMs) showing how, for instance, "UNDERSTANDING OF 'Item A' would enable (or contribute to) the UNDERSTANDING OF 'Item B'" and so on and so forth

(The models are constructed using the transitive nature of the relationships "ENABLE", "CONTRIBUTE TO", "HINDER" etc, etc).

Such an exercise would, when adequately developed, provide a useful graphical picture for learning (and teaching) purposes.

[Bob Hansen - if he is still present with us these days - may usefully note that these models discussed are 'related to' but are NOT the same as the PERT Charts that he keeps bringing up. (The PERT Chart is predicated on the "PRECEDENCE" relationship, which is not terribly useful to help us understand 'system behaviour'].

Now, MOST IMPORTANTLY, while the math teacher appropriately focusses on the whole variety of issues involved in respect to the 'mechanics of enabling students to understand (and how to use) the exponential function in a variety of ways', I believe it is also most important that the teacher keeps in mind Albert T. Bartlett's profound observation:

"... the greatest shortcoming of the human race is our inability to understand the exponential function" -

This 'shortcoming' seems to have been something 'genetically inbuilt in our psyche. How than to bring home to students the overwhelming importance of this function?

I must confess that (probably because of my own inadequacies and possibly also because of the way it was taught/ presented to us) it was actually years and years after I had studied the exponential function in some detail during my engineering math courses that I actually began understanding something of the real importance of the exponential function in human history (in the sense of Bartlett's observation). I observe that we in general are STILL failing to understand its importance and profound significance hin human history (though we do, admittedly, know a fair bit about its mechanics).

It's a sobering moment indeed when we realise that all of human existence (and the existence of much other plant and animal life on this planet) is so closely tied to our understanding/lack of understanding of this function: our success/failure as a species will depend on this understanding!!


Message was edited by: GS Chandy

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