What is Mathematics?
Major Unifying Themes in
Mind and Body Tools
Science of Teaching &
This Website is designed for the use of people who want to
improve our formal and informal math education systems.
Students, parents, educators, and policy makers can all
benefit from thinking about the ideas presented on this
Conclusions, Final Thoughts, and
We have a steadily increasing number of computer
and artificial intelligence-based mind tools that
are far more capable than the human mind. Our Pre
K-12 Math Education system is falling woefully
behind the times.
here for this Website's search
This section of the Website consists of two main
- A very brief set of
recommendations for those who want to do something to
improve math education "right now" and who want to spend
only a few minutes reading and thinking about what they
can be doing.
- A more
detailed set of recommendations that come from the
analysis given in this document. You might want to read
these before reading and analyzing the main components of
this document that start on the Home Page, Or, you might
want to use this section as a summary/overview of the
to send questions, comments, and suggestions to Dave
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Recommendations for Immediate Action
These recommendations are supported by the analysis given
in other parts of this Website.
- Math is a language. Math education is improved by
activities that contribute to increased fluency in
reading, writing, speaking, listening, and routinely
using the language. Thus, our math education should
system should place increased emphasis on integrating
routine use of math in throughout the curriculum.
- Math is a tool that is useful in helping to represent
and solve problems in every discipline. Math education
can be improved by making math use an activity that is
integrated into every subject area and into the learner's
all day, everyday routine activities.
- Calculators and computers are powerful aids to
carrying out math procedures. They are far more capable,
faster, and more accurate than people in this regard.
Math education is improved by helping learners to make
routine use of calculators and computers to carry out
math procedures in all subject areas.
- People are much better than computers at posing
problems, understanding the meaning and importance of a
problem, and understanding the meaning and making use of
a proposed solution solution to a problem. Math education
is improved by increasing the focus on problem posing and
conceptual understanding--things that people can do
better than machines--and decreasing the emphasis on
carrying out procedures--things that machines can do
better than people.
- Each person has a certain level of mathematical
"maturity" (math development, understanding, knowledge,
and skills), and this varies widely from person to
person. Thus, the meaning of lower-order knowledge &
skills and higher-order knowledge & skills varies
from person to person; for a particular person it changes
over time. Math education is improved by substantially
increased emphasis on higher order knowledge & skills
(slightly above the borderline between lower-order and
higher-order) for each individual learner.
- There is substantial evidence that well designed
Highly Interactive Intelligent Computer-Assisted
Instruction can help the majority of students learn
significant components of mathematics faster and better
than is being accomplished by our current "traditional"
math education system.
- Math education is improved by helping all learners to
understand and to routinely think about the ideas listed
about. Such metacognition is an important aspect of
learning to learn and to use math.
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Detailed Recommendations for Longer Term Action
We have long had body tools (machines to aid our physical
bodies) that are far more capable than the human body. We do
not attempt to train our human bodies to compete with
microscopes, telescopes, cars, trains, jet airplanes, space
ships, and other such tools. Rather, we have an informal and
formal educational system that helps people learn to make
effective use of such tools.
We now have a steadily increasing number of computer and
artificial intelligence-based mind tools that are far more
capable than the human mind. Many of these mind tools have
come into routine use. Indeed, many are embedded in the body
tools that we now take for granted and routinely use.
Brain Science and ICT combine to provide knowledge and
tools that can help students to acquire contemporary levels
of knowledge and expertise in many different fields.
This Website has specifically focussed on the field of
Mathematics. Both Brain Science and ICT are powerful aids to
learning and using mathematics. Thus, they provide the basis
for significant improvements in our math education
The overall recommendation of this Website is that we
redesign our Mathematics Education system with a focus on
helping students to learn mathematics and to use mathematics
in an environment in which they routinely have ICT tools
available. Curriculum content, instructional processes, and
assessment should all reflect and support this human-ICT
partnership in understanding, posing, and solving math
problems both in the field of mathematics and as they occur
in all other disciplines. This will require a substantial
and continuing inservice teacher education effort, as well
as changes in our preservice teacher education system.
- It appears that there may be a substantial and
perhaps growing mismatch between the developmental level
of students and the level of mathematics they are
studying. If so, this is a serious design flaw in the
math curriculum. More research is needed, and the
currently available research needs to be examined to see
what light it throws on this topic.
- It appears that our math education system achieves
relatively poor results in terms of transfer of learning.
Perhaps this is because:
- A mismatch between student developmental
level and the level of math being taught. Such a
mismatch leads to rote memorization with relatively
- Little effort to teach for transfer or to learn
- Situated Learning, in which the "situation" is the
math classroom and math tests."
It is not clear whether we have adequate research to
make this conclusion. But, it is a researchable question.
The current research should be analyzed, and additional
research conducted if the current research is
- We should be implementing a variety of ideas
discussed on this Website, such as Constructivism,
Situated Learning, integration of IT both as content and
as an aid to instruction and learning, assessment in a
hands-on environment, and so on.
Thoughts on Potential Research Topics
This section lists some research topics. These vary in
difficulty. Some would be suitable topics for a term project
in a course in math education. Some would be suitable for
Action Research by a classroom teacher or group of teachers.
Some would be good "Capstone Projects" for students
completing a master's degree in teacher education. Some
would be good for a doctoral dissertation or for research at
and beyond that level.
Note: This section is under development.
Please send me your suggestions.
- Select a grade level and a content area. Analyze the
content from the point of view of both general
Developmental Theory and what is known about
Developmental Theory specific to that content area.
- Select a grade level and a content area. Analyze the
content from the point of view of capabilities and
limitations of ICT as an aid to representing and solving
the problems of that content area at the grade level you
Some Thoughts Based on Experience in Presenting This
This workshop was first presented in a three-hour hands
on mode to about 25 people at the NCCE conference on 13
March 2002. This was a mixed audience of elementary and
secondary school teachers, with a couple of people from
- Suppose that the participants are convinced that
Developmental Theory has produced results and findings
that suggest a mismatch between our math education goals
and curriculum, and the developmental level of students.
So what? How about providing some specific
recommendations in this area?
- The topic of intelligence (the definition used in
these materials) needs more thought. Perhaps 1 and 2 fit
together. We are interested in students having an
increased level of performance in math, perhaps
especially in transferring their math knowledge and
skills. If we go back to the five-part diagram in which
#3 is "Solve the pure math problem." we see how ICT can
make a significant difference. We substitute ICT for a
significant part of this #3, and we use the time saved to
work on the remaining steps in the diagram. The result
will be a significant improvement in students'
performance in using math as an aid to solving problems.
This also helps to take care of the developmental level
situation. Many of the concepts of math can be learned
independently of learning the processes of carrying out
the related procedures. The processes tend to be
abstract, symbol manipulation activities.
This requires careful thought in redesigning
curriculum. Let's practice on triangles. A triangle has
three sides and three angles. If one knows some of the
side lengths and/or angles, then it may be possible to
computer the remaining side lengths and/or angles. One
can gain a mental/visual model of triangles. This would
include an understanding that there are situations in
which three line segments cannot be used to form a
I need to think more, and write more. Why might
students want to learn more about right triangles rather
than other triangles? What might we want students to
memorize, perhaps without proof, such as the sue of the
angles is 180 degrees? Why learn about triangles? What
might we know about triangles that is useful and
applicable in areas outside of math?
And then, we have the computer programs that can
tell us whether a set of side and angle data defines a
unique triangle. If the data define a triangle, the
computer program can computer all of the massing data and
the area of the triangle.
- The same ideas hold for other topics. We have the
concept that mathematicians have developed formulas that
tell us how to solve certain problems, such as how to
find the surface area and volume of a sphere. The concept
that plane figures have area and that solids have area
and volume are a whole lot different than how to compute
such areas and volume.
- The Syllabus has now gotten out of hand. The Syllabus
Page needs to contain a short syllabus and then links. At
the current time it appears that there are about six
major components to the Syllabus:
- Craft and Science of Teaching and Learning
- Recommendations and Closure
At the current time, what is occurring to me that this
means the addition of six pages. Each will have a brief
Introduction and/or Summary-Overview. Then each will
consist of the types of structure that in the current
Syllabus. I think that each of the numbered topics should
have a Recommendation. Thus we would have:
Implications and Recommendations 1.1
The Math as a Language topic and an Activity on the
Home Page seems like it should be moved into the
Syllabus. It makes no sense to have Workshop Activities
embedded in the general text of the supporting hyper
The structure given above will generate a large number
of Implications and Recommendations. These will then be
further analyzed and groups in the recommendations
section of the overall Website.
- Evidently there is a book with a title somewhat like:
The Math Gene. The general question about the brain's
allocation of resources to math is interesting to some
people. A vaguely related question came up in the
workshop on 3/13/02. That was in response to the idea of
math as a language. We have good insight into the parts
of the brain that are language oriented. How does this
fit in with math as a language? It may be that we are
merely stuck on vocabulary. When we talk about natural
language, we have a particular topic and meaning in mind.
When we talk about written language, we use the word
"language" again. But it may be that we are using it in
the sense of an analogy. Similarly, we talk about math as
a language, and we have an analogy that is still further
- The section on special education is very weak. the
material in the Brain Science part of the Foundations
- Overall, need to do a better job of tying this site
together with OTEC. For example, the OTEC site contains a
lot on special education.
- Math anxiety. How does ICT and Brain stuff fit into
this? Is anxiety increased or decreased when working in
an environment that includes ICT?
- The issue of concepts versus processes or procedures
is easy to raise. Do we have much research on this? We
have the Jim fey and Kathy Heide inverted curriculum
concepts, which are part of it.
- Assessment in math environments in which one has
access to hands on facilities. We have research ion
hands-on assessment in computing. But what about in math?
Presumably this is now a common thing with the
calculator, so there must be literature on this.
- Learning theory, learning styles, transfer of
learning: It appears that math teachers and math methods
teachers do not know much about these, and so do not
teach much about them to their preservice teachers or
incorporate these ideas in their teaching. This seems
like a big deal to me. High-Road/Low-Road transfer seems
like a theory relevant to math education and both
teachers and teachers of teachers should know about
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