Syllabus
 Astract
 Introduction
and Goals
 Craft
and Science of Teaching and Learning
Mathematics
 Curriculum
Content
 Instructional
Processes (Pedagogy)
 Assessment
 Closure

Introduction and Goals


Introduction
Getting Started. Participants are seated
individually or in pairs at computers that are connected to
the Website for the workshop. Introduce myself. Handout the
handouts. Get participants to the Website
http://darkwing.uoregon.edu/~moursund/Math/.
Suggest that participants begin reading the first page of
this Website.
By show of hands, get some information about nature of
the workshop participants, such as what they teach, the
levels at which they teach, and their specific goals in
attending the workshop.
Within every discipline, "Computational" is proving to be
a valuable new approach to combining theory and practice,
and pushing the frontiers of the field. For example, a
Computational Chemist received the Nobel
Prize in 1998 for his previous 15 years of work in
Computational Chemistry. He developed mathematical models
and simulations of chemical processes. As early as 1982,
significant results in Biology were discovered through the
use of Computational techniques.
Personal Note: When I was a math
researcher, I wrote programs that I used to explore my
Numerical Analysis interest areas. In using one of these
programs, I solved two different problems that I expected
to have quite different answers. But, it turned out that
to the level of accuracy of the computer arithmetic, they
had the same answer. This led me to conjecturing, and
then proving, that the two problems had the same answer.
This was in the early 1960s.
Information and Communications Technology (ICT) has
provided a number of valuable tools that support
Computational Mathematics and are a powerful aid to applying
mathematics in all other academic disciplines. The steady
improvements in ICT are creating increasing pressure for
significant changes in mathematics curriculum, instruction,
and assessment.
The past five years has seen more progress in Brain
Science than all of previous time. Much of this has been
possible through the use of computerized imaging systems and
computer modeling of brain processes. We are now beginning
to understand what goes on inside of a person's head as they
learn and they make use of their learning to solve problems,
accomplish tasks, make decisions, and answer questions. Our
current levels of Brain Science knowledge are beginning to
be useful in the design of curriculum, instruction, and
assessment in many disciplines, including mathematics.
The last 20 years have seen a tremendous growth in
understanding of the Science of Teaching and Learning
(SoTL). Now it is relatively common to hear educators
(including math educators) talking about Constructivism, and
perhaps even about Situated Learning Theory. Some of the
progress in SoTL is being incorporated into Intelligent
ComputerAssisted Learning materials in mathematics and in
other disciplines.
Goals of Workshop
The workshop will explore the ideas listed above. This
will be done though a combination of hands on Web
explorations, small groups discussions, whole group
discussions, and lectures.
Specific goals of the workshop include:
 Participants will gain an increased understanding of:
 What is mathematics?
 What are the major goals in mathematics
education?
 What are major weaknesses in our current
Mathematics Education system?
 How can our math education system be
improved?
 Participants will become familiar with Computational
Mathematics, Information and Communications Technology,
Brain Science, and Science of Teaching and Learning as
vehicles for improving math curriculum, instruction, and
assessment.
 Participants and the instructor will share their
insights into current and potential applications of the
above ideas that can be implemented now. There will be
specific emphasis on preservice and inservice teacher
education.
Math Education Foundations
Math and math education have a very long history. Math is
a very large and deep field. There are many potential goals
for math education at the L12 levels.
Activity 2.1 Brief whole group
discussion on "deep" questions such as:
 What is mathematics?
 What are the major goals in mathematics
education?
 What are major weaknesses in our current
mathematics education system?
 What are major barriers to change that leads to
improvement of our math education system?
Ways of Improving Math Education
There are many ways to improve math education. Here are a
few general ideas:
 Curriculum content. Make changes so that the
curriculum content is "better" from the point of view of
those working to improve math education. For example, one
might place more emphasis on higherorder thinking and
problem solving, and less emphasis on memorizing
procedures and developing speed and accuracy in carrying
out these procedures. One might drop or add curriculum
topics. For example, before calculus was developed by
Newton and Liebnitz about 300 years ago, calculus was not
a topic in the math curriculum, even in higher education.
Now it is a topic in the high school curriculum of many
high schools.
 Instructional process (pedagogy). For example,
we can provide students with "better" books; we can
provide teachers with better lesson plans and teaching
materials. We can redesign the school and classwork
physical environments so that they are more conducive to
learning. We can make changes into how time is spent
during math instruction. We can make changes to the
nature and extent of homework. We can provide students
with learning aids such as manipulatives.
 Assessment. In some sense, assessment tends to
drive curriculum content, instructional processes, and
student learning efforts. Thus, changes to assessment may
well lead to changes in student learning. We can analyze
whether the current trends toward more statewide
assessment, and more high stakes testing, are
contributing to improved math education.
Each of these three major aspects of math education can
be examined from the point of view of advances in Brain
Science, Information and Communications Technology, and the
Craft & Science of Teaching & Learning. It is
important to think about improvements in math education from
a scalability (widespread, effective, and continuing
implementation of the change). Here is an example in which
some of these ideas are explored.
An
Example (Math as a Language)
We all know that "natural language" and written language
are both powerful aids to communication and to learning.
Written language (reading and writing) was developed about
5,000 years ago by the Sumerians. At the same time, they
developed the language of mathematics (Logan, 1999).
Reading and writing are so important in education that
they are given considerable emphasis. Indeed, some schools
even go so far as to have a time during the day during which
all students, teachers, school administrators, and school
staff are to be reading. Such schools may place a specific
emphasis on reading in each subject area in the
curriculum.
Activity 2.2 : In small groups (or all by
yourself, if you are working alone) think about and
discuss:
 Math as a language. Pay particular attention to
the teaching and learning of both the oral and written
aspects of this language.
 Similarities and differences between students
learning reading and writing of a natural language
that they have already learned, and students learning
reading, writing, speaking, and listening of
mathematics.
 What are some similarities and differences between
learning math as a language, and learning a second
natural language? What do we mean by fluency in
mathematics? When people learn a second natural
language well, they report being able to think in that
language. Does our math education system bring most
students to a level of fluency that allows them to
think in the language of mathematics?
Activity 2.3: Logan (1999) also argues the
computer and the Internet/Web is also a "language," much
in the same sense that math is a language. What are your
thoughts on this idea? Suppose people came to agree that
this new "ICT language" is quite important for students
to learn. How might this goal be achieved in our K12
education system?
The two activities given above are important thinking
activitys, and they helps to set the tone of both this
Website and of a workshop or course based on this Website.
We will put forth the thesis that our current math education
system can be significantly improved by placing increased
emphasis on mathematics as a language in which students can
gain fluency in reading, writing, speaking, and listening.
An emphasis on reading, writing, speaking, listening, and
using math needs to be built into the entire school
curriculum, rather than be relegated to one specific math
class time during the day.
At the same time, we will put forth the thesis that ICT
needs to be integrated throughout the curriculum if we want
studens to gain fluency in the ICT language.
An
Example: Mathematical Proficiency
Kilpatrick,
Swafford, and Findell (2001) analyze PreK8
mathematics education from the point of view of students
gaining Mathematical Proficiency. They view Mathematical
Proficiency as five interwoven strands:
 Conceptual understanding: comprehension of
mathematical concepts, operations, and relations.
 Procedural fluency: skill in carrying out
procedures flexibly, accurately, efficiently, and
appropriately.
 Strategic competence: ability to formulate,
represent, and solve mathematical problems.
 Adaptive reasoning: capacity for logical
thought, reflection, explanation, and justification.
 Productive disposition: habitual inclination
to see mathematics as sensible, useful, and worthwhile,
coupled with a belief in diligence and one's own efforts.
Activity 2.4: This is a small group activity.
Each person is to order the five items from most
important to least important for student learning of math
at a grade level they are familiar with. Then share the
results and discuss them. Finally, debrief in a whole
class (workshop) setting.
Activity 2.5: This is a continuation of
Activity 2.4, and can be done in small groups or the
whole class (workshop) group. Which of these five items
seems to be getting the most attention in our math
curriculum, and which seem to be getting the least
attention? To what extent are the state and national
assessments targeting each of these five areas of
mathematical proficiency?
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