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What is Mathematics?

Major Unifying Themes in This Document


Foundational Information

Learning Theories

Mind and Body Tools

Science of Teaching & Learning

Project-Based Learning

Computational Mathematics

The Future




 Website Author
"Dr. Dave" Moursund

Science of Teaching and Learning (SoTL)

The Science of Teaching & Learning is now making major contributions to education. Brain Science and ICT are important components of SoTL.

Click here for this Website's search engine.

Placeholder for a graphic.

This page contains a few very rough draft ideas. Some of the topics that are under development:

Diagrams Illustrating Some Key Ideas

Teaching and Learning Theory

Brain Science/Cognitive Science

Reform Movements

Self Assessment

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Diagrams Illustrating Some Key Ideas

This diagram captures the idea that appropriate teacher education in Information and Communications Technology will lead to better education for students. It can be viewed as a conjecture. Research in this area is not nearly as strong as we would like.



This second pair of diagrams is useful in discussing the "science" versus the "art" of curriculum, instruction, and assessment. I often use these diagrams in teaching inservice and preservice teachers. There, the focus tends to be on instructional component of teaching. Teachers tend to think that good teaching (good instruction) is more of an art than a science. However, they are able to give examples of the science of teaching and how research progress can contribute to better teaching.

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Teaching and Learning Theory

There is substantial research on teaching and learning theories. In this Website, we are particularly interested in those aspects of teaching and learning theory that are directly applicable to math education.

To illustrate, Behaviorism, Constructivism, and Situated Learning are three important learning theories.

  • Behaviorism is a theory that underlies stimulus-response learning. Drill and practice can train the brain to provide quick response to number fact questions.
  • Constructivism posits that students construct new knowledge and understanding upon their current knowledge and understanding. This theory is applicable in all disciplines, but may be particularly important in vertically structures disciplines such as mathematics and science. For example, one might find that a student is having a great deal of trouble dealing with fractions in Algebra. The difficulty may lie in an inadequate understanding of fractions in arithmetic. The student may have memorized (without understanding) rules that lead to correct answers in working with fractions in arithmetic. This lack of understanding may lead to major problems in dealing with fractions in Algebra.
  • Situated Learning posits that long term retention and transfer of learning are significantly improved when learning occurs in meaningful, applications oriented, problem-solving oriented environments/situations that include a significant emphasis on higher-order thinking skills, metacognition, and understanding.

All three of these theories are important in math education. For many years, leaders in math education have been emphasizing that teachers tend to place far to much emphasis on rote memory leaning of math, and not enough emphasis on Constructivism and on Situated Learning.

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Brain Science/Cognitive Science

We now have brain and body tools, and accumulated data, information, and knowledge, that are allowing us to make significant progress in these areas. Look in the Cognitive Science part of the References in:

Some topics for this section include:

  • The human brain is not well adapted to learning and doing some aspects of math, such as arithmetic. (Stanislas Debaene)
  • Mathematical (and other types of) "talents" vary among people. (Howard Gardner)
  • Rote memory versus learning concepts.
  • Benjamin Bloom's "2-Sigma" Goal.
    • One-on-one tutoring seems to produce a "2-sigma" gain in learning. This helps explain the need for a rich and highly interactive learning environment for young children. (See research on Head Start programs.)
    • This means that compared to a control group with a class average at the 50th percentile, the experimental group has an average at the 98th percentile.
    • Students have the capacity to learn to much higher standards.
  • Computer-Assisted Learning
    • A 1994 Meta-Meta-Study found an average Effect Size of about .35-sigmas, and an average reduction in learning time of 30%.
    • There is some evidence that some "modern" ICAL produces Effect Sizes of 1-sigma or more.
    •  Physics Example: There is some evidence that a combination of an emphasis on modeling and on use of Microcomputer-based Laboratory, along with a lot of intensive staff development, can produce a 2-sigma gain in high school physics courses.
  • Transfer of Learning: Teachers and students should be particularly interested in learning that transfers so it can be used in new problem-solving and task-accomplishing situations.
    • Near Transfer vs. Far Transfer
    • Low-Road Transfer vs. High-Road Transfer

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Reform Movements

Pogrow, Stanley, Reforming the Wannabe Reformers: Why Education Reforms Almost Always End Up Making Things Worse, June 1996, p. 656. Phi Delta Kappan.

This is an excellent analysis of reform movements and why most fail. One of the key ideas discussed is the need for a reform movement to be supported by a "technology" defined as follows:
Large-scale reform requires highly specific, systematic, and structural methodologies with supporting materials of tremendously high quality. (Such methodologies are hereafter referred to as a "technology.")

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Self Assessment

We know that feedback is a necessary component in teaching/learning. Feedback can come from many different sources, such as teachers, peers, answer keys in books, computers (for example, in computer-assisted instruction), and ones self. The research on self assessment and on providing feedback to ones self is reasonably strong, The implementation of this research in our educational system is not widespread. In math education, students are highly dependent on others (not themselves) in learning whether they have does their math in a correct manner and produced correct results.

A Google search under Student Self Assessment produces some useful references.

Self-Assessment Methods [Online]. Accessed 12/27/01: . Quoting from this document that focuses on self-assessment of one's writing:

Self-assessment can take many forms, including:
  • writing conferences
  • discussion (whole-class or small-group)
  • reflection logs
  • weekly self-evaluations
  • self-assessment checklists and inventories
  • teacher-student interviews

Structures for Student Self-Assessment [Online]. Accessed 12/27/01:
univclass/selfassess.html. Quoting from this short Website article:

Critical thinking is thinking that assesses itself. To the extent that our students need us to tell them how well they are doing, they are not thinking critically. Didactic instruction makes students overly dependent on the teacher. In such instruction, students rarely develop any perceptible intellectual independence and typically have no intellectual standards to assess their thinking with. Instruction that fosters a disciplined, thinking mind, on the other hand, is 180 degrees in the opposite direction. Each step in the process of thinking critically is tied to a self-reflexive step of self-assessment. As a critical thinker, I do not simply state the problem; I state it and assess it for its clarity. I do not simply gather information; I gather it and check it for its relevance and significance. I do not simply form an interpretation; I check my interpretation to see what it is based on and whether that basis is adequate.

Because of the importance of self-assessment to critical thinking, it is important to bring it into the structural design of the course and not just leave it to episodic tactics. Virtually everyday, for example, students should be giving (to other students) and receiving (from other students) feedback on the quality of their work. They should be regularly using intellectual standards in an explicit way. This should be designed into instruction as a regular feature of it.

There are two kinds of criteria that students need to assess their learning of content. They need universal criteria that apply to all of their thinking, irrespective of the particular task. For example, they should always be striving for clarity, accuracy, and significance. Of course, they also need to adjust their thinking to the precise demands of the question or task before them.

The above-listed document comes from the Critical Thinking Consortium [Online. Accessed 12/27/01:

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