Math Points: Reform – a
Duet??
by Jack Rotman
Note: The name reflects a
normal paradox in education – we must work with objects that have no real
dimensions, like a point in space. The
name also reflects my belief that every reader needs to start off assuming that
a writer has no real dimension; that is, assume that the writer does not know
anything, and judge by the validity of the presentation whether there is
something “real” to be learned.
Are
we playing a duet, or are we playing two solos which sound similar even though
they are written in a different key?
Stay with me while I explore this musical analogy for the relationship
between K-12 mathematics and college mathematics.
In
our ‘standards’ document, Crossroads in
Mathematics, AMATYC makes this statement:
The standards included in
this document reflect many of the same principles found in school reform [for
example, see NCTM Curriculum and
Assessment Standards for School Mathematics, 1989 …] (page x)
On
the same page, relative to the foundation courses, the Crossroads document says:
Courses at this level should
not simply be repeats of those offered in high school.
If
reform courses in college should not be a simple repeat of high school courses,
what is the desired relationship? In
particular, how should a reformed high school mathematics curriculum relate to
the college mathematics curriculum?
Now,
we could approach these questions from a variety of viewpoints; for example, we
could set out a detailed analysis of the content and methods used in some
sample classes at each level (high school, college). However, I want to maintain the musical analogy: Are the two curricula, as experienced by a
listener (student), seen as a duet or as two solos?
Actually,
I must confess that I have not done
this; to do this, I would need to develop an assessment instrument that could
be used on a sample of students who have experienced the two curricula. However, I want to mention a student’s
history as a way to hint at the nature of the students’ general experiences.
In
my college’s service district, there are several high schools that have a
reputation as having reformed their mathematics curriculum. One of them happens to be the high school in
the district that I live in. A 1997
graduate of this high school came to my college this year, and I have had a
chance to follow her progress.
Her
high school mathematics included two years of algebra and one year of
geometry. She completed them, but I got
the impression that math was not an area of high confidence for her. When she entered my college, she took our
placement test – which placed her into our first algebra course. I asked her how she felt about this, and she
indicated that she thought she really ought to be in the intermediate algebra
class instead of the introductory one, but that the situation was okay with
her.
As
she proceeded in the course, she found a number of areas challenging. She ended up with a passing grade (2.5), and
enrolled for intermediate algebra in the second semester. This course was very challenging for her,
and she was not able to pass; currently, she is repeating the intermediate
algebra this semester.
Now,
I know that the high school curriculum and the college curriculum for this
student had some things in common – both used the graphing calculator, both
used a lot of applications, and both focus on communicating
mathematically. I also know that there
are some differences, such as the fact that her high school algebra did very
little with factoring or rational expressions, while the college courses
covered an average amount for a college course; however, this is not very
helpful, since she had trouble before either topic.
I
don’t consider her grades to be an indicator that there is a “problem”. My interest now is in considering how
students such as this experience the two curricula. I have not asked her questions about her experience, and one
student is obviously not a guarantee of an entire group of students. However, I am left with the impression that
this student did not see much harmony between her high school experience and
her college experience.
Even
if this is true, and even if this is true for many students, maybe there is still
not a “problem”. Maybe the goals of
high school mathematics and of college mathematics are so different that we
should expect students to “repeat” material, and maybe not even pass the
course; maybe we should expect two solos.
I know, in fact, that the goals are quite different, as evidenced by the
types of assessment that are employed.
But,
let’s stay with the musical analogy:
Do students expect to hear a duet?
Would they be comfortable with a high school mathematics experience that
was a solo, and comfortable with a college mathematics experience that was a
solo? I would contend that, in the
absence of extremely strong messages to the contrary, students expect a duet –
they expect a very direct connection between their high school and college
experiences. I think they find it
discouraging when they must “repeat” material – “I already had this in high
school”. I believe that students
wonder, if they think about it, whether math teachers really know what they are
doing.
If
this musical analogy is accurate – that students experience two solos in high
school and college mathematics, but expect a duet – then the question
becomes: What should be done about
it? An obvious answer is that there
really should be a duet; however, I think this is wrong – I think there are
sufficient reasons for the goals of the two curricula to be very
different. If there needs to be two
solos, then the next answer would be that we need to make sure that students
don’t expect a duet; this is the path I think we need to take.
This
would be a problem for both teaching groups to address – high school and
college. High school mathematics
students need to be told what the goals of that experience are; I would even
assess the students on whether they understand this. In college, we need to also communicate our goals – and maybe
assess the students on whether they have understood those goals.
A
duet is a wonderful experience, just as solos are wonderful experiences. Each has their own place and beauty, and I
enjoy both of them – as long as the performance matches the requirements.
I
also need to make an apology. In my
very first try at this, I made a mistake in a URL. If you tried to get the paper I mentioned last time, the address
I gave you did not work. Now, if you
are very persistent, you could find the correct address by good searching
techniques. In case you just want to
‘look in the back of the book for the correct answer’, here is the address that
worked the day I wrote this:
http://act.psy.cmu.edu/ACT/papers/misapplied-abs-ja.html
(This
is the paper by researchers at Carnegie Mellon University-- John Anderson, Lynne Reder, and Herbert
Simon with the title Applications and Misapplications of
Cognitive Psychology to Mathematics Education. )