Math Forum Discussions

Jonathan Groves

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

"Computational Thinking and Math Maturity" by Dave Moursund
Posted: Jun 20, 2010 10:42 AM

Plain Text

Dear All,

I have recently completed a first reading of the second edition of
Dave Moursund's book "Computational Thinking and Math Maturity:
Improving Math Education in K-8 Schools." Dave Moursund is a
Professor Emeritus of Education at the University of Oregon with
a PhD in Mathematics and has worked as a professor of mathematics
and as a professor and chair of computer science and information
science at the University of Oregon. This book contains many
excellent ideas about the problems of mathematics education in
America. Though the focus is on K-8 math education, many of the
ideas in his book carry over to high school and early college
math education. The book is available at
http://uoregon.edu/~moursund/Books/ElMath/K8-Math.pdf.

One significant idea in his book is that students need teaching
methods to get them to move to the next cognitive level of thinking,
the thinking about the abstract. Most students struggle with math
because they haven't gotten beyond concrete thinking. Moursund mentions
that only about a third of students graduate from high school having
reached Piaget's Formal Operations level of cognitive thinking in mathematics
and that many math teachers have not either. That is, these students
and teachers lack mathematical maturity. Math education neither today nor
in the past does not and has not helped most students reach the mathematical
maturity they need to succeed in algebra and other math courses with symbolic
reasoning. Chapters 5-7 discuss these ideas in depth. This is something that
I knew all along, but Dave Moursund states it in ways more precisely than
I have done.

It is interesting to note that Moursund has divided what Piaget calls the
Formal Operations level of cognitive thinking into several additional
levels: one level for reasoning in algebra and early college math courses,
another level for advanced undergraduate and graduate math courses,
and a final level for mathematicians. I do believe that Piaget's Formal
Operations level does consist of several additional levels of cognitive
thinking, at least in mathematics. But the fact that only about a third
or so of high school graduates have reached the Formal Operations level
says that only about a third of them have the minimum cognitive ability
to have any chance of learning algebra and other mathematics with
symbolic reasoning (symbols besides those in arithmetic). No wonder so
many students struggle immensely with college mathematics.

Another significant idea he explores is that neither current nor traditional
teaching methods teach students to read mathematics and learn from it
and do not teach them how to become independent learners of mathematics.
I would not doubt this claim in the least bit because none of my K-12
math teachers did anything to teach us how to read a math book and to
learn from it and never taught us anything about how to teach ourselves
mathematics. None of the current courses I teach provide students any
materials on how to do this; that is, these course developers assume that
students already have these skills. And few of them actually do. I try
to provide such resources, but I know I need to try to improve those
resources because they don't seem to be helping much (I say "don't seem"
because I have no way to verify if these students are or are not reading
the information I give them; I doubt most of them are because in many
other cases I've seen students act as if they haven't read something,
but I can't say for sure). And it seems that hardly anyone else addresses
this issue in math education except those who support reform teaching
methods. On top of that, the weak reading skills of students today
make such learning extremely difficult for them.

These two significant ideas give students the necessary general skills
they need to learn mathematics. Without these skills, they cannot learn
mathematics successfully. And, if they forget any necessary math they
need later, they cannot relearn the math for themselves. Finally, all
these specific skills they try to learn in math make no sense to them
because they haven't learned what it means to learn mathematics, to
learn to think mathematically.

Finally, another significant idea he explores is the integration of
computers into teaching. He notes that little has changed in the
teaching methods and curricula of mathematics despite the significant
recent changes in technology because mathematicians and mathematics
educators are still unsure about the best ways to integrate such
technology into mathematics education. I believe such integration
that is sane and intelligent is possible and is necessary because much
of today's mathematical work is done via computers. One concern is
that the integration of computers will cause students not to learn
mathematics successfully and become too dependent on technology to do
their thinking for them; this is certainly a valid concern, but I believe
there are ways to overcome such challenges. One possible way is to
assign computer work that cannot be done successfully without thinking
about the mathematics behind it; that is, if the student tries to
complete the work blindly, the computer gives nothing but elaborate
nonsense. And we should stress this idea repeatedly in class.
Moore and McCabe's "Introduction to the Practice of Statistics" does
a good job of stressing that computers are an aid, but not a substitute,
to thinking and understanding and that using a computer blindly produces
nothing but elaborate nonsense. Another possible way is to give quizzes
and exams where at least some of the questions cannot be answered by
a computer and must be answered using the necessary understanding of
mathematics.

In short, here are the Big Ideas (as he calls them) that Moursund explores
in his book (as quoted from the preface):

1. Thinking of learning math as a process of both learning math content and
a process of gaining in math maturity. Our current math education system
does a poor job of building math maturity.

2. Thinking of a student?s math cognitive development in terms of the roles
of both nature and nurture. Research in cognitive acceleration in
mathematics and other disciplines indicates we can do much better in
fostering math cognitive development.

3. Understanding the power of computer systems and computational thinking
as an aid to representing and solving math problems and as an aid to
effectively using math in all other disciplines.

4. Placing increased emphasis on learning to learn math, making effective
use of use computer-based aids to learning, and information retrieval.

I may write more comments later about Moursund's book.

Jonathan Groves