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Topic: Algebra & Fractions
Replies: 1   Last Post: Aug 13, 1995 12:55 PM

 Kreg A. Sherbine Posts: 26 Registered: 12/6/04
Re: Algebra & Fractions
Posted: Aug 13, 1995 12:55 PM

On Sun, 13 Aug 1995, Andrei TOOM wrote:
<stuff deleted>
>
> This time I definitely disagree with you, Narge. In this recommendation
> you are mixing quite different levels of mathematical sophistication.
> Students who can understand those logical reasonings, which you are
> proposing, are so sophisticated that they have mastered fractions long ago.
> On the other hand, students who need to study fractions, certainly
> can not yet follow formal arguments. When teaching fractions, you should
> appeal to common sense rather than to formal abstarct properties.
> You should discuss such practical things as dollars, pizzas, apples etc.

<more stuff deleted>
>
> Andrei Toom

Andrei seems to have a conception of math learning, and hence math
education, as a linear process. Experience (mine, anyway) indicates that
this is not the case. For example, some of my calculus students last year
were quite capable of following formal arguments, but they were incapable
of finding common denominators involving algebraic expressions. I'd say
that these folks needed to study fractions.

Also, my understanding of what Andrei writes is that he sees early
mathematical learning as well-grounded in "common sense," or what I would
call everyday reality, whereas higher levels of math, in my understanding
of Andrei's perspective, require students to leave behind the everyday
realities and to begin working in a realm that is strictly mathematical
in the sense of math-in-the-classroom.

An alternative perspective (and the one which I hold) is that students in
the early grades, who use pizza and money to learn about fractions, can
and should also explore deeper mathematical understandings of fractions.
In other words, using pizza and money to introduce the need for fractions
is fine and good, but at some point the fractions themselves, rather than
simply the things to which the fractions refer, should become objects of
discourse. [Incidentally, this shift characterizes reflective discourse
as defined by Cobb, Boufi, McClain, and Whitenack and hearkens back to
Ron Ward's question on the nature of reflection in a constructivist view
of the classroom, be it math, math ed, or otherwise.]

Similarly, students at higher grade levels should always be able to
"downshift," i.e., to relate the formal math that they do to everyday
experience. This can be difficult, particularly with such things as
n-dimensional spaces and abelian groups, but it should also be remembered
that as the level of sophistication of the mathematics increases through
the grades, so (usually) does the level of sophistication of students'
everyday experiences. That is to say that what counts as an everyday
mathematical experience for a fourth-grader, e.g., pizza and money, is
probably different from what counts as an everyday mathematical
experience for a 12th grader, e.g., integers and geometric shapes.

Kreg A. Sherbine | To doubt everything or to believe
Graduate Student | everything are two equally convenient
Vanderbilt University | solutions; both dispense with the
sherbine@math.vanderbilt.edu | necessity of reflection. -H. Poincare