Math Forum Discussions

Jonathan Groves

Posts: 2,068
From: Kaplan University, Argosy University, Florida Institute of Technology
Registered: 8/18/05

ST Math at Madison Elem. in Santa Ana, CA
Posted: Oct 1, 2010 2:48 PM

Plain Text

Dear All,

I thank Michael Paul Goldenberg for sharing this information about ST Math
that Madison Elementary School in Santa Ana, CA, is using to help their
students learn mathematics. See the article

http://www.districtadministration.com/viewarticle.aspx?articleid=2351

for more information.

The link

http://mindresearch.net/cont/programs/demo/tours/SolvingLinearEquations/progTour.php

gives an example of one of these games that students are playing to help
them learning mathematics.

Games are good ways to help motivate students to learn math and to help
make learning more enjoyable for students. Traditional ways of practice
and drill bore most students and do not give them much of a good reason to
learn math other than possibly as a mere excuse to do work, to occupy
their time with something boring to do, or to please their teachers and
the administration.

I looked over some of this stuff, and this is an interesting approach to helping
students learn mathematics. A visual and intuitive approach to mathematics
without using mathematical language and symbolism up front is a good way to
help students learn concepts and helps motivate the need for mathematical
language and symbolism. Such an approach can also help students learn to see
what mathematical language and symbolism say. I know that students'
confusion about mathematical language and symbolism is a major hang-up for
them. Not only that, many students end up believing that mathematical
language is meant to make math and problem solving more confusing rather
than less confusing.

I can't say for sure whether the rise in test scores is a genuine sign
or something fake, but that rise does indicate that perhaps this ST Math
might have some promise behind it. At least it is worth looking into
to see what is going on and why. It is possible to find a way to fake
success like this, but it is not easy to do; otherwise, we would see
big increases in test scores across the country. I could think of
ways to fake success, but how to do that in ways so that the chances
of getting caught are low I'm not sure.

And it is just one school, and the test scores are based on tests I
wouldn't give. I would like to see tests that dig deeper into the
students' understanding to see if they really understand the math and
to see if their learning is long-term learning or simply the kind of
learning that helps them do well on tests but then quickly evaporates
afterward.

If ST Math has been tried elsewhere, I hadn't heard yet. If not or
if has been tried in only a few other schools, then ST Math needs to
be tested more.

So the signs aren't definite, but these signs look better than what
we usually see or hear about.

Since one purpose of mathematics is translating real world problems into
languages on paper so that we can analyze and solve these problems without
having to carry out physical processes, we should help students to see this
purpose of mathematics. That is, mathematics is supposed to make problem
solving easier rather than more difficult and is supposed to help make
many problems accessible that would be, for practical purposes, inaccessible
without mathematics.

The games I had mentioned here from ST Math are not modeling problems, of
course, and I do not mean to suggest that they are. But these games do
help students develop a way to picture these ideas in their minds, and
these visual representations of the mathematical ideas will help them later
in using the mathematics as a way to understand our world. It is extremely
difficult, if not impossible, for students to use the math outside the
classroom meaningfully if they see the mathematics as nothing but symbolic
manipulation. For instance, how can we model a real-world situation using
mathematics if we see the math as nothing but symbolic manipulation. I have
seen students who struggle with basic word problems in arithmetic, for example,
but who are decent at working purely computational questions (that is,
questions that explicitly mention what computation to do and no context of
any kind is given). They cannot decide what operation to use because they
see the operations in arithmetic in essentially the same way that we see
operations in an abstract algebraic structure but nothing more. That is,
the operations to them are just formal rules for "generating" a number
from other numbers. Games such these, even if they don't include modeling
problems, can still help students develop these meanings for operations
that go beyond just mere symbolic rules.

If students can really see for themselves why we need mathematical language
and symbolism, then they should be more eager to learn it. That is,
mathematical language and symbolism is not, despite what many students
may believe, just there for the hell of it or just there to make their
teachers and mathematicians and others sound brainy. But to help make
this learning effective, we should follow Clyde Greeno's and Alain
Schremmer's advice about identifying the mess in curricular language
and cleaning it up (see my recent post about this). Garbled and
imprecise language hinder many students' learning of mathematics.

As for cleaning up curricular mathematical language, ST Math doesn't
do that, of course. I had mentioned that as something we educators
need to do if we are to help students become fluent with mathematical
language and to help clarify to our own students what we are talking
about. Clyde Greeno and Alain Schremmer and I have noticed some mess
in curricular language. These discussions can be found on Math-Teach
and Mathedcc. For those interested, I can give more details about
where to find some of this information.

I had begun noticing the mess consciously a while ago when I had noticed
that curricular language does not generally make it clear what the
difference is between a fraction and a ratio or what the differences
are when working with real numbers as real numbers (that is, as elements
of an algebraic structure) versus working with real numbers that represent
counts or measurements. We can add any two real numbers, but can we
add 12 liters and 14 meters? The real number division 134/23 makes
sense, but what about 134 miles divided by 23 hours? Does that
make sense? If it does, what does it mean? It certainly cannot have
the same meaning because in, 134/23, both 134 and 23 are expressed in
terms of the same unit; that is, 134 is 134 times 1, and 23 is 23 times
1. Both 1's are the same. But in 134 miles/23 hours, if it even makes
sense, 134 miles is 134 times 1 mile, and 23 hours is 23 times 1 hour.
But 1 mile and 1 hour are not the same thing! Curricular language
does not help here because it does not generally distinguish between
the arithmetic of real numbers (as real numbers) and the arithmetic
of counts and measurements.

ST Math does not teach language and symbolism, so that is up to us
to do. But making language and symbolism clear requires precise
language, but unfortunately curricular language--at least in the
United States--is highly imprecise.

Jonathan Groves