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

Introduction and Goals



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

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 Computer-Assisted 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:

  1. 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?
  2. 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.
  3. 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 L-12 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:

  1. 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 higher-order 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.
  2. 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.
  3. 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:
  1. Math as a language. Pay particular attention to the teaching and learning of both the oral and written aspects of this language.
  2. 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.
  3. 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 K-12 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 PreK-8 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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