Date: Apr 28, 2011 7:11 PM
Author: Alain Schremmer
Subject: Re: The instructional Roles of Mathematics Lectures
On Apr 28, 2011, at 4:50 PM, Christopher Elder wrote:
> As a "freshman" to the field of developmental mathematics my
> viewpoint has been broadened by the rich discussion regarding the
> philosophy and pedagogy of teaching mathematics. Thank you.
> I want to expand upon the mathematics eduator's "toolbox" by
> mentioning discourse theory. I am not saying it is a 'one size fits
> all' option, but it might help lead students to a broader, more
> abstract level of thinking by first presenting mathematical topics
> using real life analogies.
You are right: it "might". But nobody has ever explained how. On the
other hand, before anything else, mathematics is a language that
exists in a paper-world (where we write) and like any language, it
describes the real-world. So, (1) make sure the real-world situation
to be described is familiar and clear to the students and then, (2)
let the students describe it in the mathematical language under
study. Thus, to think in terms of "analogy" is not warranted (the two
belong to different worlds).
> As a brief introduction, discourse theory expands upon the
> knowledge students already have. It is not a lecture but a
> discussion, lead by the instructor, and sustained by the students.
(1) Earlier, I used the term "discussion" for lack of a better word.
(2) The "discussion" in mathematics is very different from a
discussion on, say, the influence of May 68 on Godard.
(3) Developmental students do not have the "facility" to discuss as
in (2). All their previous experience lends them to want to be shown.
(4) Thus in fact, one goal is to get the student to acquire said
facility and the taste for discussing things. But it cannot be taken
as a point of departure.
> Our Algebra with Arithmetic course begins with the order of
> The order of operations, as we know, is an algorithm that must be
> followed exactly
> (however, there is some, yet little room for working in a different
There is also a lot of ways for coding things differently, e.g. with
blank spaces as in x^2 ?1 / 3x + 1 or, maybe, writing the slash
in another color. The standard code works well for mathematicians ...
but not necessarily for beginners.
> The discourse can begin by asking what real life activities use a
> specific process. A student may answer: Making gumbo must start
> with a roux (some may disagree-but a discourse on the culinary arts
> can precede at a different time :) before the seasonings can be
> added, before the vegetables can be added, etc.
Two things are going to happen:
(1) Because I discuss the difference between "2 apples + 3 apples"
where + truly means addition and "2 apples + 3 bananas" where + means
AND, I lose students who decide that I am NOT teaching "math".
(2) Starting from "gumbo" will prevent us from reaching a significant
point by the end of the semester so that we will have to "accelerate"
and the students will correctly characterize the initial
"application" as being a gimmick (They use a somewhat stronger word)
> Since mathematics is a very discipined field,
Yes and no. Mathematics is also an art. And disciplined does not mean
> any analogy breaks down at some point, but it can help them to
> start thinking about the rules and characteristics about that
> particular mathematical topic.
I have seen countless colleagues doing that because "what else?"
> From Schremmer,
> >Isn't there something between the two? Like what most of us did
> when we learned mathematics?
> Again using the anology idea, students can be exposed to a topic by
> first 'exploring' a real life example through a discourse led by
> the instructor or (to allow it to be done at home) through a
> written series of socratic questions.
And of course you know the slave's last response: No, master.
> The exposure to graphs can be shown by giving students a graph of
> (you name it in the science field) and asking them what the
> intercepts mean
One of the hardest things for beginning students is to "read" a
graph, that is to see it as a form of input-output device.
> , what a particular point represents, what the up/down ward slope
> represents. Is the up/down ward slope practically a good or bad
> thing? Students know more about math than they know.
> Students often think that mathematics is rigid, like an overcooked
> square biscuit. However, mathematics also takes creativity, which
> means thinking outside the box. Discourse theory can help us as
> well as our students think outside that box.
I did not really know what Discourse Theory was. So, I googled.
>> Until recently most linguistic study has been based upon the
>> premise that the sentence is the basic unit of expression.
>> However, there is a growing interest and acceptance of "the
>> analysis of discourse or 'text' as basic to understanding the use
>> of language" as opposed to the "more traditional sentence-based
>> grammars."1 The study of text grammar or discourse has been
>> defended on several empirical and grammatical grounds. It is noted
>> that most utterances are more than one sentence and that
>> discourses have more psychological reality than sentences. It is
>> also contended that sentence grammar leaves much ambiguous
>> material whereas in discourse grammar much of the potential
>> ambiguity is eliminated by reference to the surrounding textual
>> matter. Furthermore, sentence grammar cannot adequately explain
>> the "definitivization of noun phrases, pronominalization, relative
>> clauses verb phrases and tense, sentence adverbials,
>> conjunctions . . . . Only a discourse grammar can handle . . .
>> morphological markers at the beginning and end of a text."2
To reiterate my point:
(1) Mathematics has been fairly well dissected, digested,
constructed, etc in books.
(2) Why not start from, and consider, this already existing work? We
do not reinvent cars. We buy them constructed. Then, we may want to
see how they were built.
(3) Where we come in is as "mediators" between the book and the reader.
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