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What is Mathematics?
Major Unifying Themes in
This Document
Syllabus
Foundational
Information
Learning
Theories
Mind and Body Tools
Science of Teaching &
Learning
Project-Based Learning
Computational
Mathematics
The Future
Recommendations
References
Website
Author
"Dr.
Dave" Moursund
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Science of Teaching and Learning
(SoTL)
The Science of Teaching & Learning is now
making major contributions to education. Brain
Science and ICT are important components of
SoTL.
Click
here for this Website's search
engine.
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This page contains a few very rough draft ideas. Some of
the topics that are under development:
Diagrams Illustrating Some Key
Ideas
Teaching and
Learning Theory
Brain Science/Cognitive
Science
Reform Movements
Self Assessment
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Diagrams Illustrating Some Key
Ideas
This diagram captures the idea that appropriate teacher
education in Information and Communications Technology will
lead to better education for students. It can be viewed as a
conjecture. Research in this area is not nearly as strong as
we would like.
This second pair of diagrams is useful in discussing the
"science" versus the "art" of curriculum, instruction, and
assessment. I often use these diagrams in teaching inservice
and preservice teachers. There, the focus tends to be on
instructional component of teaching. Teachers tend to think
that good teaching (good instruction) is more of an art than
a science. However, they are able to give examples of the
science of teaching and how research progress can contribute
to better teaching.
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Teaching and
Learning Theory
There is substantial research on teaching and learning
theories. In this Website, we are particularly interested in
those aspects of teaching and learning theory that are
directly applicable to math education.
To illustrate, Behaviorism, Constructivism, and Situated
Learning are three important learning theories.
- Behaviorism is a theory that underlies
stimulus-response learning. Drill and practice can train
the brain to provide quick response to number fact
questions.
- Constructivism posits that students construct new
knowledge and understanding upon their current knowledge
and understanding. This theory is applicable in all
disciplines, but may be particularly important in
vertically structures disciplines such as mathematics and
science. For example, one might find that a student is
having a great deal of trouble dealing with fractions in
Algebra. The difficulty may lie in an inadequate
understanding of fractions in arithmetic. The student may
have memorized (without understanding) rules that lead to
correct answers in working with fractions in arithmetic.
This lack of understanding may lead to major problems in
dealing with fractions in Algebra.
- Situated Learning posits that long term retention and
transfer of learning are significantly improved when
learning occurs in meaningful, applications oriented,
problem-solving oriented environments/situations that
include a significant emphasis on higher-order thinking
skills, metacognition, and understanding.
All three of these theories are important in math
education. For many years, leaders in math education have
been emphasizing that teachers tend to place far to much
emphasis on rote memory leaning of math, and not enough
emphasis on Constructivism and on Situated Learning.
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Brain Science/Cognitive
Science
We now have brain and body tools, and accumulated data,
information, and knowledge, that are allowing us to make
significant progress in these areas. Look in the Cognitive
Science part of the References in:
http://otec.uoregon.edu/
Some topics for this section include:
- The human brain is not well adapted to learning and
doing some aspects of math, such as arithmetic.
(Stanislas Debaene)
- Mathematical (and other types of) "talents" vary
among people. (Howard Gardner)
- Rote memory versus learning concepts.
- Benjamin Bloom's "2-Sigma" Goal.
- One-on-one tutoring seems to produce a "2-sigma"
gain in learning. This helps explain the need for a
rich and highly interactive learning environment for
young children. (See research on Head Start
programs.)
- This means that compared to a control group with a
class average at the 50th percentile, the experimental
group has an average at the 98th percentile.
- Students have the capacity to learn to much higher
standards.
- Computer-Assisted Learning
- A 1994 Meta-Meta-Study found an average Effect
Size of about .35-sigmas, and an average reduction in
learning time of 30%.
- There is some evidence that some "modern" ICAL
produces Effect Sizes of 1-sigma or more.
- Physics Example:
There is some evidence that a combination of an
emphasis on modeling and on use of Microcomputer-based
Laboratory, along with a lot of intensive staff
development, can produce a 2-sigma gain in high school
physics courses.
- Transfer of Learning: Teachers and students should be
particularly interested in learning that transfers so it
can be used in new problem-solving and task-accomplishing
situations.
- Near Transfer vs. Far Transfer
- Low-Road Transfer vs. High-Road Transfer
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Reform Movements
Pogrow, Stanley, Reforming the Wannabe Reformers: Why
Education Reforms Almost Always End Up Making Things Worse,
June 1996, p. 656. Phi Delta Kappan.
This is an excellent analysis of reform
movements and why most fail. One of the key ideas
discussed is the need for a reform movement to be
supported by a "technology" defined as follows:
Large-scale reform requires highly specific,
systematic, and structural methodologies with
supporting materials of tremendously high quality.
(Such methodologies are hereafter referred to as a
"technology.")
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Self Assessment
We know that feedback is a necessary component in
teaching/learning. Feedback can come from many different
sources, such as teachers, peers, answer keys in books,
computers (for example, in computer-assisted instruction),
and ones self. The research on self assessment and on
providing feedback to ones self is reasonably strong, The
implementation of this research in our educational system is
not widespread. In math education, students are highly
dependent on others (not themselves) in learning whether
they have does their math in a correct manner and produced
correct results.
A Google search under Student Self Assessment produces
some useful references.
Self-Assessment Methods [Online]. Accessed
12/27/01: http://www.eduplace.com/rdg/res/assess/
. Quoting from this document that focuses on
self-assessment of one's writing:
Self-assessment can take many forms, including:
- writing conferences
- discussion (whole-class or small-group)
- reflection logs
- weekly self-evaluations
- self-assessment checklists and inventories
- teacher-student interviews
Structures for Student Self-Assessment [Online].
Accessed 12/27/01: http://www.criticalthinking.org/University/
univclass/selfassess.html.
Quoting from this short Website article:
Critical thinking is thinking that assesses
itself. To the extent that our students need us to tell
them how well they are doing, they are not thinking
critically. Didactic instruction makes students overly
dependent on the teacher. In such instruction, students
rarely develop any perceptible intellectual independence
and typically have no intellectual standards to assess
their thinking with. Instruction that fosters a
disciplined, thinking mind, on the other hand, is 180
degrees in the opposite direction. Each step in the
process of thinking critically is tied to a
self-reflexive step of self-assessment. As a critical
thinker, I do not simply state the problem; I state it
and assess it for its clarity. I do not simply gather
information; I gather it and check it for its relevance
and significance. I do not simply form an interpretation;
I check my interpretation to see what it is based on and
whether that basis is adequate.
Because of the importance of self-assessment to
critical thinking, it is important to bring it into the
structural design of the course and not just leave it to
episodic tactics. Virtually everyday, for example,
students should be giving (to other students) and
receiving (from other students) feedback on the quality
of their work. They should be regularly using
intellectual standards in an explicit way. This should be
designed into instruction as a regular feature of it.
There are two kinds of criteria that students need to
assess their learning of content. They need universal
criteria that apply to all of their thinking,
irrespective of the particular task. For example, they
should always be striving for clarity, accuracy, and
significance. Of course, they also need to adjust their
thinking to the precise demands of the question or task
before them.
The above-listed document comes from the Critical
Thinking Consortium [Online. Accessed 12/27/01:
http://www.criticalthinking.org/default.html.
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