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What is Mathematics?

Major Unifying Themes in This Document

Syllabus

Foundational Information

Learning Theories

Mind and Body Tools

Science of Teaching & Learning

Project-Based Learning

Computational Mathematics

The Future

Recommendations

References

 

 Website Author
"Dr. Dave" Moursund

Conclusions, Final Thoughts, and Recommendations

We have a steadily increasing number of computer and artificial intelligence-based mind tools that are far more capable than the human mind. Our Pre K-12 Math Education system is falling woefully behind the times.

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This Website is designed for the use of people who want to improve our formal and informal math education systems. Students, parents, educators, and policy makers can all benefit from thinking about the ideas presented on this Website.

This section of the Website consists of two main components:

  1. A very brief set of recommendations for those who want to do something to improve math education "right now" and who want to spend only a few minutes reading and thinking about what they can be doing.
  2. A more detailed set of recommendations that come from the analysis given in this document. You might want to read these before reading and analyzing the main components of this document that start on the Home Page, Or, you might want to use this section as a summary/overview of the entire Website.

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Brief Recommendations for Immediate Action

These recommendations are supported by the analysis given in other parts of this Website.

  1. Math is a language. Math education is improved by activities that contribute to increased fluency in reading, writing, speaking, listening, and routinely using the language. Thus, our math education should system should place increased emphasis on integrating routine use of math in throughout the curriculum.
  2. Math is a tool that is useful in helping to represent and solve problems in every discipline. Math education can be improved by making math use an activity that is integrated into every subject area and into the learner's all day, everyday routine activities.
  3. Calculators and computers are powerful aids to carrying out math procedures. They are far more capable, faster, and more accurate than people in this regard. Math education is improved by helping learners to make routine use of calculators and computers to carry out math procedures in all subject areas.
  4. People are much better than computers at posing problems, understanding the meaning and importance of a problem, and understanding the meaning and making use of a proposed solution solution to a problem. Math education is improved by increasing the focus on problem posing and conceptual understanding--things that people can do better than machines--and decreasing the emphasis on carrying out procedures--things that machines can do better than people.
  5. Each person has a certain level of mathematical "maturity" (math development, understanding, knowledge, and skills), and this varies widely from person to person. Thus, the meaning of lower-order knowledge & skills and higher-order knowledge & skills varies from person to person; for a particular person it changes over time. Math education is improved by substantially increased emphasis on higher order knowledge & skills (slightly above the borderline between lower-order and higher-order) for each individual learner.
  6. There is substantial evidence that well designed Highly Interactive Intelligent Computer-Assisted Instruction can help the majority of students learn significant components of mathematics faster and better than is being accomplished by our current "traditional" math education system.
  7. Math education is improved by helping all learners to understand and to routinely think about the ideas listed about. Such metacognition is an important aspect of learning to learn and to use math.

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More Detailed Recommendations for Longer Term Action

We have long had body tools (machines to aid our physical bodies) that are far more capable than the human body. We do not attempt to train our human bodies to compete with microscopes, telescopes, cars, trains, jet airplanes, space ships, and other such tools. Rather, we have an informal and formal educational system that helps people learn to make effective use of such tools.

We now have a steadily increasing number of computer and artificial intelligence-based mind tools that are far more capable than the human mind. Many of these mind tools have come into routine use. Indeed, many are embedded in the body tools that we now take for granted and routinely use.

Brain Science and ICT combine to provide knowledge and tools that can help students to acquire contemporary levels of knowledge and expertise in many different fields.

This Website has specifically focussed on the field of Mathematics. Both Brain Science and ICT are powerful aids to learning and using mathematics. Thus, they provide the basis for significant improvements in our math education system.

The overall recommendation of this Website is that we redesign our Mathematics Education system with a focus on helping students to learn mathematics and to use mathematics in an environment in which they routinely have ICT tools available. Curriculum content, instructional processes, and assessment should all reflect and support this human-ICT partnership in understanding, posing, and solving math problems both in the field of mathematics and as they occur in all other disciplines. This will require a substantial and continuing inservice teacher education effort, as well as changes in our preservice teacher education system.

  1. It appears that there may be a substantial and perhaps growing mismatch between the developmental level of students and the level of mathematics they are studying. If so, this is a serious design flaw in the math curriculum. More research is needed, and the currently available research needs to be examined to see what light it throws on this topic.
  2. It appears that our math education system achieves relatively poor results in terms of transfer of learning. Perhaps this is because:
    1. A mismatch between student developmental level and the level of math being taught. Such a mismatch leads to rote memorization with relatively little understanding.
    2. Little effort to teach for transfer or to learn for transfer.
    3. Situated Learning, in which the "situation" is the math classroom and math tests."

    It is not clear whether we have adequate research to make this conclusion. But, it is a researchable question. The current research should be analyzed, and additional research conducted if the current research is inconclusive.

  3. We should be implementing a variety of ideas discussed on this Website, such as Constructivism, Situated Learning, integration of IT both as content and as an aid to instruction and learning, assessment in a hands-on environment, and so on.

Thoughts on Potential Research Topics

This section lists some research topics. These vary in difficulty. Some would be suitable topics for a term project in a course in math education. Some would be suitable for Action Research by a classroom teacher or group of teachers. Some would be good "Capstone Projects" for students completing a master's degree in teacher education. Some would be good for a doctoral dissertation or for research at and beyond that level.

Note: This section is under development. Please send me your suggestions.
  1. Select a grade level and a content area. Analyze the content from the point of view of both general Developmental Theory and what is known about Developmental Theory specific to that content area.
  2. Select a grade level and a content area. Analyze the content from the point of view of capabilities and limitations of ICT as an aid to representing and solving the problems of that content area at the grade level you have selected.
  3.  

Some Thoughts Based on Experience in Presenting This Workshop

This workshop was first presented in a three-hour hands on mode to about 25 people at the NCCE conference on 13 March 2002. This was a mixed audience of elementary and secondary school teachers, with a couple of people from higher education.

  1. Suppose that the participants are convinced that Developmental Theory has produced results and findings that suggest a mismatch between our math education goals and curriculum, and the developmental level of students. So what? How about providing some specific recommendations in this area?
  2. The topic of intelligence (the definition used in these materials) needs more thought. Perhaps 1 and 2 fit together. We are interested in students having an increased level of performance in math, perhaps especially in transferring their math knowledge and skills. If we go back to the five-part diagram in which #3 is "Solve the pure math problem." we see how ICT can make a significant difference. We substitute ICT for a significant part of this #3, and we use the time saved to work on the remaining steps in the diagram. The result will be a significant improvement in students' performance in using math as an aid to solving problems. This also helps to take care of the developmental level situation. Many of the concepts of math can be learned independently of learning the processes of carrying out the related procedures. The processes tend to be abstract, symbol manipulation activities.

    This requires careful thought in redesigning curriculum. Let's practice on triangles. A triangle has three sides and three angles. If one knows some of the side lengths and/or angles, then it may be possible to computer the remaining side lengths and/or angles. One can gain a mental/visual model of triangles. This would include an understanding that there are situations in which three line segments cannot be used to form a triangle.

    … I need to think more, and write more. Why might students want to learn more about right triangles rather than other triangles? What might we want students to memorize, perhaps without proof, such as the sue of the angles is 180 degrees? Why learn about triangles? What might we know about triangles that is useful and applicable in areas outside of math?

    …And then, we have the computer programs that can tell us whether a set of side and angle data defines a unique triangle. If the data define a triangle, the computer program can computer all of the massing data and the area of the triangle.

  3. The same ideas hold for other topics. We have the concept that mathematicians have developed formulas that tell us how to solve certain problems, such as how to find the surface area and volume of a sphere. The concept that plane figures have area and that solids have area and volume are a whole lot different than how to compute such areas and volume.
  4. The Syllabus has now gotten out of hand. The Syllabus Page needs to contain a short syllabus and then links. At the current time it appears that there are about six major components to the Syllabus:
    • Introduction/Overview
    • Craft and Science of Teaching and Learning
    • Curriculum
    • Instruction
    • Assessment
    • Recommendations and Closure

    At the current time, what is occurring to me that this means the addition of six pages. Each will have a brief Introduction and/or Summary-Overview. Then each will consist of the types of structure that in the current Syllabus. I think that each of the numbered topics should have a Recommendation. Thus we would have:

    Topic 1.1

    Activity 1.1

    Implications and Recommendations 1.1

    Topic 1.2

    Etc.

    The Math as a Language topic and an Activity on the Home Page seems like it should be moved into the Syllabus. It makes no sense to have Workshop Activities embedded in the general text of the supporting hyper document.

    The structure given above will generate a large number of Implications and Recommendations. These will then be further analyzed and groups in the recommendations section of the overall Website.

  5. Evidently there is a book with a title somewhat like: The Math Gene. The general question about the brain's allocation of resources to math is interesting to some people. A vaguely related question came up in the workshop on 3/13/02. That was in response to the idea of math as a language. We have good insight into the parts of the brain that are language oriented. How does this fit in with math as a language? It may be that we are merely stuck on vocabulary. When we talk about natural language, we have a particular topic and meaning in mind. When we talk about written language, we use the word "language" again. But it may be that we are using it in the sense of an analogy. Similarly, we talk about math as a language, and we have an analogy that is still further removed.
  6. The section on special education is very weak. the material in the Brain Science part of the Foundations belongs there.
  7. Overall, need to do a better job of tying this site together with OTEC. For example, the OTEC site contains a lot on special education.
  8. Math anxiety. How does ICT and Brain stuff fit into this? Is anxiety increased or decreased when working in an environment that includes ICT?
  9. The issue of concepts versus processes or procedures is easy to raise. Do we have much research on this? We have the Jim fey and Kathy Heide inverted curriculum concepts, which are part of it.
  10. Assessment in math environments in which one has access to hands on facilities. We have research ion hands-on assessment in computing. But what about in math? Presumably this is now a common thing with the calculator, so there must be literature on this.
  11. Learning theory, learning styles, transfer of learning: It appears that math teachers and math methods teachers do not know much about these, and so do not teach much about them to their preservice teachers or incorporate these ideas in their teaching. This seems like a big deal to me. High-Road/Low-Road transfer seems like a theory relevant to math education and both teachers and teachers of teachers should know about it.

 

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