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  1. Abstract
  2. Introduction and Goals
  3. Craft and Science of Teaching and Learning Mathematics
  4. Curriculum Content
  5. Instructional Processes (Pedagogy)
  6. Assessment
  7. Closure

Abstract of Syllabus for a Workshop or Short Course

This section contains both a Short Abstract and a Longer Abstract.

Short Abstract This provides a brief summary of the workshop or short course.

Longer Abstract This provides an indepth overview of the workshop or short course.



Short Abstract

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.


Longer Abstract

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:

  1. 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.
  2. Expertise in the Craft (art and practice) of being a teacher and in teaching the specific content topics and general content areas being taught.
  3. 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 broaden:

  • The range of content areas within their repetoire.
  • 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.

Content Knowledge

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 Learning

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 areas.

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 other.

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 tools (languages).

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