Abstract of Syllabus for a Workshop or
This section contains both a Short Abstract and a Longer Abstract.
Short Abstract This
provides a brief summary of the workshop or short
Longer Abstract This
provides an indepth overview of the workshop or short
The exponential rate of increase in computer hardware and
telecommunications speed in the field of Information and
Communications Technology (ICT) has continued for many
years, and it seem likely to continue for another 10-15
years. "Computational" (Computational Modeling) has emerged
as an important new way to represent and solve problems in
mathematics, science, and other disciplines.
At the same time, Brain Science and the Craft and Science
of Teaching & Learning (C&SoT&L) have been
making rapid progress. Moreover, the totality of accumulated
mathematical knowledge continues to grow--perhaps doubling
every 10 years. When all of these areas of rapid progress
are combined, they provide a substantial potential for
significant improvements in curriculum, instruction, and
assessment in math education.
This syllabus focuses on mathematics education at the Pre
K-12 levels and its accompanying preservice and inservice
teacher education. The goal is to help workshop and course
participants learn some of the current potential for
significant improvements in math education. This is a
lecture, demonstration, discussion, and hands-on workshop
/or course. It is expected that participants will spend
considerable time exploring the Website resources that have
been brought together to support the workshop or course.
Both Brain Science and ICT are dynamic fields, each
experiencing exponential growth. For example, it is
estimated that the totality of new knowledge developed in
Brain Science during the past five years exceeds that of all
previous work in the field. Many of the capabilities of
various components of ICT systems have been doubling every
two years or less.
Such rapid paces of change far outstrip the pace of
change of our mathematics education system. Thus, there is a
rapidly growing difference between the current curriculum
content, instruction, and assessment in our math education
system, and the potentials that Brain Science and ICT are
bringing to math education.
Content and Pedagogy
A good teacher has an appropriate balance among:
- Expertise in the specific content topics as well as
the general content areas being taught. For example, a
teacher teaching about solving quadratic equations needs
to have knowledge about solving quadratic equations,
about solving the range of equations that occur in
algebra,, applications of solving equations, roles of
computers in solving equations, and so on.
- Expertise in the Craft (art and practice) of being a
teacher and in teaching the specific content topics and
general content areas being taught.
- Expertise in the Science (for example, research on
learning theory and learners) relevant to teaching and
learning the specific content topics and general content
areas being taught.
This is an Expertise Model of of a teacher. A person can
spend a lifetime steadily increasing their expertise in any
one of the three areas listed. They can also steadily
- The range of content areas within their
- Their capacity to work effectively with underserved
students, special needs students, and TAG students,
Teachers face many and varied problems. Increasing
expertise allows a teacher to better recognize, understand,
and address these teaching and learning problems.
ICT impacts each of the three areas of expertise. The
levels of impact vary with the grade levels and the subject
areas being taught.
Math is a very large field. People teaching math at the
Pre K-12 levels have widely varying knowledge and skill in
mathematics. Research strongly supports the need for
teachers of math to know both the specific topics they are
teaching and also how the topics fit into the overall math
curriculum and field of math. In addition, the teacher needs
to know how math is used in other curriculum areas and
throughout our society.
While the content of the Pre K-12 math curriculum tends
to change rather slowly, it does change. Since 1989, the
National Council of Teachers of Mathematics' Standards have
been a powerful change agent. The math curriculum in many
school districts, schools, and individual classrooms has had
considerable trouble accommodating to the changes suggested
by the NCTM.Even the full weight and support of the NCTM has
not been sufficient to bring about a moderately rapid change
in math curriculum, instruction, and assessment throughout
the United States.
Logical/mathematical is one of the Multiple Intelligences
identified by Howard Gardner. We know that people (in this
case, teachers and their students) vary significantly in
their logical/mathematical intelligence and their knowledge
and skills in this field. Teachers teaching math are making
use of their logical/mathematical intelligence, knowledge,
and skills. Students construct their new mathematical
knowledge by using their logical/mathematical intelligence
and building on their previous knowledge of math.
Craft Knowledge of Teaching and
Teachers learn a great deal about the teaching process by
observing their teachers during their Pre K-12 education and
during their teacher education programs of study. However,
teachers also gain a great deal of craft knowledge while on
the job. This type of learning continues to occur throughout
one's teaching career, but the pace of learning is
especially high during the first few years on the job.
Think about this learning of craft knowledge as being
divided into two major phases. The first phase mainly occurs
before a person becomes a teacher. The second phase occurs
on the job, after one becomes a teacher. During the first
phase, the teachers-to-be observes many different teachers.
But, during the second phase, teachers tends to have
relatively limited opportunities to observe other teachers
at the level they are teaching. It is often suggested that
one of the flaws in our inservice teacher education
environment is that many teachers do not get a chance to
observe master teachers (in this case, more highly skilled
crafts people) in action.
Science of Teaching & Learning
In recent years, the Science of Teaching and Learning
(SoTL) has emerged as a vibrant discipline. There is a huge
and rapidly growing collection of research and
practitioner-based knowledge on teaching and learning. Both
preservice and inservice education includes a focus on
helping teachers to gain SoTL expertise.
For example, at one time B. F. Skinner's Behaviorism was
the dominant learning theory in education. Now, there are a
large number of cognitive learning theories that better
represent certain parts of the learning process.
As another example, consider the "science" of helping
students learn math in a manner so that they can then use it
in other fields. Transfer of learning does not just
"automatically" occur. Indeed, Transfer of Learning is a
challenge in the teaching and learning of every subject
area. The Low-Road, High-Road theory of Transfer of Learning
that was developed during the 1980s is a significant
contribution to SoTL.
ICT as a Language and a Mind Tool
Writing (as a language) and mathematics (as a language)
are mind tools--they are powerful aids to the human mind.
The abilities to use both written language and mathematics
are so useful to people that these are "basics" in our
formal educational system. Students study and practice the
"three Rs" year after year in K-12 education and even on
into higher education as they work to develop contemporary
and more advanced knowledge and skills (expertise) in these
Information and Communications Technology builds upon and
extends the three Rs as well as other areas of human
intellectual endeavor. ICT can be thought of as being a
human-developed language and a brain tool, in the same sense
that mathematics is a human-developed language and a brain
tool. And, of course, ICT includes the field of Artificial
Intelligence, which is developing powerful supplements
and/or aids to the human brain. Artificial Intelligence and
Brain Science have contributed significantly to each
From an evolutionary point of view, "... the brain exists
to make better decisions about helping enhance reproductive
success." Gazzaniga (1998, p 5). Writing, mathematics, and
ICT are relatively recent developments in terms of
evolutionary timelines. The human brain has not had the time
to evolve for increased success in learning and using
writing, mathematics, and ICT. The brain can adapt to these
new human-developed tools, but it did not evolve directly
for that purpose. Through education and training, our minds
adapt to and learn to make use of these human-developed