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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

Instructional Processes (Pedagogy)


5. Pedagogy (Instructional Processes)

5.1. Math as a language. In addition to "natural" languages we have writing. mathematics, science, computer programming, and Information and Communications Technology (ICT) as widely used, general-purpose types of languages. (See Math as a Language.)

Our brains are wired for the learning of "natural" languages. To a much lesser extent, our brains are wired to learn a little math. Thus, it requires a large amount of effort over a long period of years to develop reasonable fluency in mathematics. In essence, we are repurposing parts of the brain from what they evolved to do.

The analogy with written language is interesting. A person can communicate effectively in written language (that is, solve a written communication problem) even when they make a substantial number of errors in spelling and grammar. Math problem solving tends to require a much higher level of perfection. The human brain, which can be thought of as an analog computer, is not designed for such levels of perfection. Thus, it is not surprising that people have worked to develop tools that can aid the brain in achieving the levels of perfection that are needed in using math to represent and solve problems.

There has been a substantial amount of research on the teaching of English as a Second or Other Language (ESOL). Some of this research is applicable to the teaching of mathematics. Also, there is an increasing amount of general literature on roles of ICT in ESOL.

Activity 5.1A: In small groups, share your insights into how math instruction might change if it was generally agreed that math is a language and that we should teach math in a manner that leads to fluency in communicating both orally and in writing in this language.

Activity 5.1B: In small groups, discuss some major weaknesses and some major strengths in the analogy between learning math and learning a "natural" language?

5.2. Learning Theories. There are many learning theories. Math education tends to pay particular attention to Behaviorism and Constructivism.

The current tread in math education is to place an increasing emphasis on Constructivism and on higher-order thinking skills. We are beginning to see some acceptance of Situated Learning as being quite important in math education and in the learning of other disciplines. Situated Learning theory suggests that at the current time students are mainly learning to do math only in the environment that is provided in a classroom. Thus, for example, their focus on learning math is to get the assignments done and pass the tests. But assignments and tests are not part of the environment that students will live in after they leave school.

Activity 5.2: In small groups, discuss Situated Learning. For example, discuss "word Problems" versus the types of potential math application environments that one encounters outside of school. (Food for thought: How often does one encounter a word problem when walking down the street?)

5.3 Virtual manipulatives. We are all familiar with the use of hands-on manipulatives in learning and teaching math, especially at the elementary school level. A virtual manipulative is one that exists in a computer and is displayed on a computer screen. There are advantages and disadvantages to virtual manipulatives as an aid to learning and as an aid to solving problems.

Note that using a virtual manipulative as an aid to representing and solving math problems is a form of Computational Mathematics. One general strategy for attacking a math problem is to represent the situation by a picture, diagram, graph, etc. That is, develop an aid to a visual mental model of the problem. Nowadays, this is often done using a computer. This form of Computational Mathematics is beginning to be well entrenched as a tool both for learning and for problem solving.

Activity 5.3A: In small groups, share your experiences in making use of concrete manipulatives and virtual manipulatives in math education. What evidence do you have about the plusses and minuses of using virtual manipulatives?

Activity 5.3B: In small groups, argue for and/or against the idea that learning to make use of virtual manipulatives (either as a student or as a teacher) is closely related to learning about Computational Mathematics.

5.4 Transfer of learning. The High Road, Low Road theory of Perkins and Salomon of transfer is appropriate to mathematics. The research supports explicit teaching for transfer and suggests ways to do this.

Activity 5.4A: In small groups, discuss the value of the Near Transfer, Far Transfer theory of transfer of learning in math education. Just for the fun of it, think about how this fits in with students using math oriented edutainment software in math classes.

Activity 5.4B: In small groups, share ideas and teaching techniques that you use in math instruction to increase High Road and/or Low Road transfer of learning.

5.5 Project-Based Learning and ICT-Assisted PBL. There is substantial research to support the effectiveness of Project-Based Learning in a variety of disciplines. PBL and IT-Assisted PBL are useful in creating teaching/learning environments that are based on the theories of Constructivism and on Situated Learning.

Project-Based Learning is not a learning theory. Rather, it describes a type of curriculum, instruction, and assessment environment. PBL is often a component of, or an approach to, creating a situated and constructivist learning environment. Moursund maintains a Website on IT-Assisted PBL.

Activity 5.5: In the whole workshop group, share experiences you have had with PBL and with IT-Assisted PBL in your math teaching and your math learning.

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