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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

Mind and Body Tools

Humans have developed a wide assortment of mind and body tools to aid and extend the capabilities of their minds and bodies. Writing, mathematics, and ICT are powerful mind tools. ICT-based tools may contain or embody a substantial amount of human knowledge and intelligence.

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A body tool.

The following diagram presents the central theme of this Web page.


At the center of the diagram is a team consisting of one or more people that is faced by a problem to be solved, as task to be accomplished, or a question to be answered. The Problem Solving Team has three types of aids:

  1. Mind tools, such as writing, mathematics, books, and ICT.
  2. Body tools such as the plow, microscope, telescope, and bicycle.
  3. Education and training in use of the mind and body tools, and in working with them and other people on the team.

David Perkins, a professor at Harvard University, uses the term "Person Plus" in discussing this situation. The formal and informal education and training that a person receives, along with the tools that are available to the person, create a Person Plus. A teacher in our formal educational system plays a major role in the education of Person Plus.

Mind and Body Tools Overlap

Even the simplest body tools contain, carry, or represent some of the knowledge and skill of their inventors and producers. The spear, perhaps with a fire-hardened point, may have been independently invented by a number of people. But, most people learned about the spear, and how to use it, from others in their tribe. During their lives they were able to make use of the invention,perhaps even without ever constructing a spear.

It is fun to think about the knowledge inherent in the design of a paper clip or a dinner fork. These tools are well designed and have served people for many years.

An automated machine in a factory can be thought of as containing more knowledge than a tool that is not automated. In that sense, the automated tool is both a body tool and a mind tool. This type of analysis leads us to conclude that mind tools and body tools overlap.

The degree of overlap depends on how carefully one defines the terms "mind tool" and "body tool." For the purposes of this workshop, a careful definition is not necessary.

For any given tool, one can think about one's current level of expertise and what it might take to move up the expertise scale in use of the tool.

For example, it takes only a modest amount of time and effort to learn the rudiments of using a hoe. But, it takes a long time to become really skilled in its use.

Activity: Think of the capabilities and limitations of mathematics as a mind tool. In small groups, brainstorm ideas of what would make this tool easier to learn how to use, easier to use, and a more powerful and useful tool.

Simple Electronic Calculator as a Mind Tool

A simple handheld electronic calculator typically has a least five built-in functions: addition, subtraction, multiplication, division, and square root. To calculate the square root of a number, one keys in the number and depresses the square root key. The calculator "knows" how to calculate the positive square root of a positive number.

It is difficult to think of such a handheld calculator as being intelligent in the sense that a human is intelligent. The calculator has no consciousness; it has no concept or understanding of what it (itself) is or what it is doing. The calculator contains a mathematical algorithm (a detailed step by step set of instructions that has been proven to work) for calculating a square root. The electronic circuitry in the calculator can rapidly and accurately carry out this algorithm.

Now, think about a student learning about square root and how to calculate a square root. There are three things for the student to learn:

  1. What is a square root -- the underlying concepts and why one might want to calculate a square root. Square root as an important mathematical idea.
  2. How to calculate a square root.
  3. Uses of square root in representing and solving problems both in mathematics and in many other fields.

In the "good old days," learning to calculate square root by hand, by use of logarithms, or by use of math tables was typically taught in second year high school algebra. The by-hand algorithm that was typically taught requires a reasonable level of computational competence in estimation, multiplication, and subtraction. The algorithm has some of the characteristics of the by-hand long division algorithm that is typically taught in our schools, so there may be some transfer of learning from the long division algorithm to the square root algorithm.

This topic has disappeared from most second year high school algebra courses. Students learn how to use a calculator. And, the learning time that is saved can be used to help students learn items #1 and #3 on the above list.

Activity 1: In small groups, discuss the pros and cons of the change in instruction about square roots that has occurred in our curriculum. Then, think about possible uses of square root that a student might learn before being enrolled in second year high school algebra. In the past, the math curriculum tended to avoid such topics in the earlier grades, since students lacked the ability to calculate square roots. Students could understand the concept and some uses, but they could not master the computational algorithm.

Activity 2: In small groups, discuss the concept that students might learn how to use a calculator rather than learning how to do paper and pencil long division of multi digit numbers. Make sure that your discussion includes a focus on concepts and applications.

ICT as a Mind and Body Tool

We know that ICT is both a mind tool and a body tool. For example, a computerized library stores information, can assist you in finding and retrieving information, and can assist you in using the information. Moreover, all of this can occur through the use of a telecommunications system.

In some sense, Artificial Intelligence (AI) can be thought of as a field in works to develop combination mind and body tools that contain, represent, and make use of a significant amount of knowledge.

This is an important idea. Even though early tools such as the spear and bow/arrow contained or represented knowledge, much of the skill and knowledge to effectively use the tool resided in the human user. Computer and Information Science (which includes the field of AI) allows us to develop tools that are "smarter" or "more intelligent" than tools of the past.

Examine the diagram given below. It is a simplified 4-step model of using mathematics to solve a problem.


My informal studies suggest that our K-12 formal mathematics education system spends about 80% of its time and effort on students learning the vocabulary and the processes to do Step 2. This leaves about 20% of the time for Steps 1, 3, and 4. We now have computer systems (indeed, handheld battery powered calculators) that can carry out Step 2 for the full range of algorithms that students study in the K-12 math curriculum.

This leads to two fundamental questions:

  1. If a computer or other ICT tools can solve of significantly help in solving a problem (task, question) that we typically teach students about in our formal educational system, how should the curriculum, instruction, and assessment in our formal educational system change to accommodate this situation?
  2. What new things should be added to the curriculum to reflect ICT as an aid to problem solving, task accomplishing, and question answering? What should be deleted? What changes in order of topic presentation should occur?

Activity: In small groups, brainstorm answers to one of these two questions.

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Problem Solving. Humans:


Pose, represent, and solve problems

Pose, represent, and accomplish tasks

Pose, represent, and answer questions

Develop mind & body tools to help do the above


These are repeating themes in all academic disciplines. And, of course, they are very important in math and math education.


Focusing on IT-Related Mind & Body Tools

Our specific focus in the remainder on this presentation is on IT-related mind and body tools that relate to mathematics and mathematics education.


Think About a Tractor


Tractor and Agriculture

Agriculture existed for 5,000 years before development of the tractor

Infrastructure needed to support tractors

Formal and informal education needed

Changes tractors produced in agriculture

Appropriate technology



Note to self: Need to have increased emphasis on human-developed languages -- writing, math, science, ICT. But also perhaps music?

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