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Topic: What Is Mathematics For?
Replies: 5   Last Post: Aug 18, 2010 9:05 AM

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Jonathan Groves

Posts: 2,068
From: Kaplan University, Argosy University, Florida Institute of Technology
Registered: 8/18/05
Re: What Is Mathematics For?
Posted: Aug 18, 2010 9:05 AM
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Amy and others,

Speaking of a one-size-fits-all approach to learning, I have
recently been told about in another discussion group a blog
on Challenge by Choice:
Students choose their own levels of challenge for assignments
and exams. The levels are green, blue, and black--with black
being the most challenging. There is a lot of information on
this blog post, and I have not had the time to read it all
and think about it. This idea makes sense because different
students come into a course with different levels of ability
and preparation. If all exams and assignments are the same,
then not all students are being appropriately challenged.
Some students will find the work too easy, and others will
find it too difficult.

Placing the decisions in the hands of students helps
encourage students to take charge of their learning. And
students might not trust the teacher's judgment as well as
their own judgment anyway. And students are free to move from
one level to another throughout the course. Thus, if the
students choose a level lower than they are capable of reaching,
they can discover that for themselves and move to a higher level
later. I believe we can encourage such independent thinking
and learning even further by encouraging students to seek
and invent their own problems that they are interested in
solving or to learn whatever mathematics they are interested
in learning. Of course, the students will need some guidance
in making these decisions. We also will need to identify
what mathematics is essential for them to learn so that
the students don't miss learning that mathematics.
That is, the course does not need to move all the way to the
other extreme into a total free-for-all, totally unstructured
course either. I mention that only because my experiences
with discussions in education have shown me that some who
oppose reformist views of math education seem to believe that
all reformists want to push math education all the way to
an extreme as far away from traditional math education as

I strongly agree that teachers are often forced to teach
in ways that still assume a one-size-fits-all approach to
math education (and education in general). Many online courses
at many online schools assume this because we are required to
use their tests and their assignments and their syllabi.
So all students do the same assignments and same exams, regardless
of their major, their interests, their needs, their abilities
and previous preparation in mathematics, etc. When I think
about this, such problems give me yet another reason to oppose
common course structures. I tried to do away with common course
structures at another school but no luck. I wasn't surprised,
especially since a well-respected mathematics professor there
had tried the same thing for years with complete failure.
If they weren't going to listen to him, then it is no surprise
that they wouldn't listen to a lowly graduate student (I was
a graduate student at the time when I was at this school--the
University of Kentucky). But I had decided to try anyway,
and I wish I had thought of this reason at the time as one to
add to my letter.

I believe that there are several reasons we see such problems
persist: administrators and parents do not like drastic and
unfamiliar changes to traditional teaching of mathematics ("What
the heck is this stuff you are trying to do? I didn't learn math this
way!"), many want quick and easy ways to evaluate teachers and
courses--such evaluations are much easier when all courses are run
in similar ways with similar assignments and exams for everyone,
many are not aware of what the real problems are, many want to
cater to students' wishes rather than determining what is best
for students, many others do not want such changes made until they
know how the problems should be fixed--that is, they are afraid of
trying something new when it has not been tested in advance, and others
I have not mentioned.

I like Dan Meyer's idea that we tell students too much.
If we tell them too much, then we basically do the thinking
for them rather than teaching them how to do the thinking for

And I like his presentation on "impatient problem solving."
That is, textbooks often give problems that are quick and
easy to solve. Even if a textbook manages to give a more
involved problem, the textbook breaks it down into multiple parts.
Part a--find this. Part b--find that. Part c--do this. And so on.
Rather than having the students try to figure out what
needs to be done to solve the problem, the textbook lays it
all out in front of them. But many real-world problems
are not like this at all! We need to figure out for
ourselves what we need to do to get the solution we seek.
I take it from your comments that you might have seen
the same video or a version of the same video I am referring to.
For the benefit of others, I will include the link to this

Another serious error is not teaching students how to think
and reason mathematically. Skills in mathematical reasoning
and in problem formulation and problem solving will go much
further than any specific math content they learn. Furthermore,
any specific math content they need to learn will be easier for
them to learn than otherwise. Without any genuine problem
solving skills and mathematical reasoning skills, any math
content they learn they cannot use anyway and will become
totally useless to them. Finally, students end up missing
the point of what mathematics is ultimately about.

Jonathan Groves

On 7/30/2010 at 12:14 am, Amy wrote:

> I agree that teachers should encourage critical
> thinking skills and how to solve problems without
> just memorizing facts and formulas but rather learn
> to interpret and solve problems with their own
> reasoning and like you said "learning to think".
> I love your comment 'We should shed the notion
> of the "one-size-fits-all" approach to teaching
> because students
> are not clones of each other: What works well for one
> student
> may not work well for another student. Maybe some of
> those turned off
> by traditional curricula might like math better
> because they have more
> options that now appeal to them or simply because the
> traditional curricula was taught to them in these bad
> ways."
> I recently watched a video clip of a math teacher
> name Dan Meyer. He made some very eye opening
> comments about the way we teach math these days. It
> is not a "one-size-fits-all" curricula. Why would it
> be? We clearly know that students learn differently
> yet we are still forced to teach in ways that doesn't
> recognize this. In his video, Dan suggests that as
> teachers we should be less helpful to our students.
> Stop giving them all of the information and encourage
> them to plug it into a formula to come up with an
> answer but instead include them in the formulation of
> the problem. We should give them the "bare
> essentials" and let them go from there. Allow the
> students to make mistakes and learn from them, to
> come up with their own formulas and reasoning.
> Students may start to see math in a different way
> when we as teachers stop doing the "textbook" way of
> teaching math and start bringing it to life! As Dan
> put it in his presentation, math makes sense of our
> world, it is the vocabulary for our intuition"

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