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Foundational Information



Via learning, you move from having a beginner's novice level of knowledge and skills, up to your current level of expertise.

1. Strands

Much of the material for a workshop or short course based on this Website is divided into strands. Within a strand there are a sequence of instructional materials and activities designed to help a person gain increased expertise along that strand.

One might think of a person moving through a sequence of levels as they move along an expertise scale in a strand. The diagram given below suggests that we are exploring many different topics (areas in which an expertise scale is applicable) and that we want to help people move up one or more levels in a number of different expertise areas.

In the diagram there is a basic inner core, perhaps based on what we assume all readers know.In a workshop or short course, there might be an initial 1-hour of the getting all participants to have some common core of knowledge and insights.

After that, one could perhaps go to any of the topics, studying it at the first level. After studying a topic at the first level, one might be able to go to the 2nd and 3rd levels on it. Of course some of the 2nd and 3rd levels might need to draw on other 1st level knowledge. The point is, the materials given in this syllabus and Website are highly flexible. For the most part, the materials need not be presented and/or studied in any specific order. Readers/users will develop orders that best fits their needs.



 2. Expertise

Expertise is a unifying theme in this Website. Over the years, you have gained knowledge and skills in many different areas. In each of these areas you have moved from having a beginner's novice level of knowledge and skills, up to your current level of expertise.

Your job as a teacher is to help your students move up various expertise scales. As as teacher, you have a relatively high level of expertise in how to do this. You have gained your current level of expertise through many years of formal and informal education and experience. This workshop or short courseis designed to build upon your current expertise in a constructivist manner, and to add a small increment to your overall expertise as a teacher.

The content listed in the syllabus ranges over many different topics. Some you already know quite well, and others will not interest you. Thus, you will likely skip over some topics, spend only a modest amount of time on other topics, and perhaps explore some topics in depth. In any case, hold the idea of expertise and of increasing our expertise in mind.

From time to time, do some metacognition (think about your cognition). One of the findings from Brain Science is that such conscious, directed metacognition can be a significant aid to learning.

Activity: In small groups, discuss what you do to help your students understand the process of gaining increased expertise within an area. Such a discussion might be wide ranging and include "the forest and the trees" as well as metacognition and self assessment.

3. Constructivism

Constructivism is a unifying learning theory in this syllabus. Even an absolute novice/beginner approaches a new learning domain with previous knowledge and skills. The youngest child has build-in innate knowledge and skills that function at a subconscious level and contribute to survival. As children grow older and as they develop their innate abilities, they are able to bring more and more to a new learning task.

As a teacher, you are well aware of the wide range of differences that your students bring to your classroom. Part of your job is to help design curriculum, instruction, assessment, and the overall learning environment so that it meets the individual needs of each of these different learners. You have an intuitive, functional understanding of constructivism. However, you are also aware that it is impossible to devote your full attention to each individual student, and it is impossible for you to develop and implement an individual educational program for each of you students.

This is an important realization. Research proves that such individualization, implemented by a skilled tutor, is far more effective than our traditional whole class approach to education. However, our society is not able to afford this level of individualization of instruction for most students. Over the 5,000 year history of formal education in subjects such as the 3 Rs, a lot has been learned on how to deal with classes of 20 to 30 or more students. While such learning environments are less than ideal, for most students they are far better than not having access to a formal education system.

While formal and informal instruction may occur in individual, small group, and larger group settings, learning is an individual thing. The human brain is naturally inquisitive and has an innate ability to learn. Learning goes on all the time. As a teacher, part of your job is to focus your students' attention on the topics that you want them to learn. You make use of your knowledge of various learning theories and how the brain learns to try to facilitate the learning that you want to occur. Over your teaching career you have gained increased expertise in doing this -- and you realize that you still have a lot of room for personal growth in this area.

Activity: In small groups, discuss your understanding of constructivism as a learning theory, and how it applies to both you and your students. What (if anything) do you do to help your students understand constructivism as it applies to them, individually? What are good arguments in favor of and against students coming to understand constructivism and how they build upon their current knowledge and skills?

4. Potential Expertise and World Class Expertise

You are familiar with the idea of a person being "world class" in a particular endeavor such as writing, research, or various sports. There has been quite a bit of research on the combination of innate abilities and efforts that are needed for a person to become world class in a particular endeavor. Very roughly speaking, it takes a minimum of 10 years of very hard work, accompanied by an appropriate combination of formal and informal education and training, for a person to begin to approach their potential within an area. The starting point may be quite young, as we see in gymnastics, ice skating, and golf.

In academic pursuits, the starting point may be at a somewhat older age. However, in many areas of science research, the most brilliant work of a researcher is often done before the age of 30.

The game of chess has been extensively studied both from a human learner point of view and from an Artificial Intelligence (machine intelligence) point of view. It takes 10-15 years of very hard work (50 to 80 hours a week, or more) for a person to achieve their potential as a chess player. Maintaining the resulting level of expertise requires continuing hard work.

In 1997, a computer beat the reigning human world chess champion in a six game match. Five years later (2002) we had a super computer that is 1,000 times as fast as the computer that beat the human world chess champion. A chess program running on today's microcomputers can easily defeat most people who know how to play chess. Ten years from now a microcomputer will likely be able to beat the world's (human) chess champion.

Here are two very important points to hold in mind as you continue your mathematics teaching career:

  1. It takes a great deal; of time and effort for a human to achieve their potential in even one narrow area of expertise.
  2. In an increasing number of narrow areas of intellectual expertise, computers can out perform humans. (We have mentioned chess as an example.)

Activity: In small groups, discuss the educational implications to the two points mentioned above. If you like, focus the discussion on mathematics.

5. Expertise in a Multi-Strand Field

Most of us are not destined to become world class in our chosen avocations. Our interests and our innate abilities are wide and varied. We do not have the desire or opportunity for specialization at an early age, with years of striving to be the best that we can be within a narrow specialization area.

As you know, there are many possible goals for mathematics education. The following diagram gathers together three rather general purpose aspects of mathematics as expertise scale strands. In this diagram. "contemporary standards" might be viewed as the standards that are currently expected for a teacher at your grade level and in the math-related subject areas that you teach.

Now, think about the three strands in this diagram. As a person who teaches math, your current levels of expertise on these three scales might well vary. Here is a possible sample pattern.

You have achieved these varying levels of math expertise while you have been achieving varying levels of expertise in many other fields. For example, it may well be that your people skills, organizational skills, public speaking skills, teaching skills, and piano playing skills are all higher up on their respective expertise scales than are your math expertise levels.

Activity: This activity is designed for small group discussion and then whole group debriefing. Disciss the following:

You are reading this documente because you want to become more effective in helping your students to learn mathematics. Thus, you can imagine examining various aspects of your math expertise versus the goals of math education, your insights into helping students learn mathematics, you skills as a teachers, and so on. You might well decide that you could improve the effectiveness of you math teaching by taking a course on Assertive Discipline, a course on Multicultural Education, or an advanced course in mathematics. Instead, you have chosen to spend some time learning about ICT and Brain Science as they apply to mathematics education.

Who should make the decision about how you should spend your professional development time?

While you contemplate possible answers and arguments within this area, ask the same question from the point of view of your students. Why is it all right for curriculum specialists, textbook authors and publishers, and teachers to tell students what mathematics they must learn? Why should we have a uniform, state-mandated form of assessment over math topics selected for use throughout the state? Think about this from a constructivist point of view, and from the point of view of the many and varied interests of your students.