Math Maturity Short Workshop





Abstract
This document was developed by David Moursund for use as a handout in a 1–3 hour workshop on Math Maturity. The length of the workshop is quite flexible, since it is not necessary to "cover" all of the material.

The key idea is to facilitate small group discussions, sharing, and building personal ownership. Math maturity is what lasts as details of the math one has studied fade quickly or over the years. It is habits of mind and ways of thinking mathematically. Eleven components are listed and briefly discussed. A “Food for Thought” discussion question is provided with each of the 11 components. These are designed to be used in small group and/or large group discussions. A references section provides a number of related documents that are available free on the Web.

MM1. Communicate in the language of mathematics
A very important component of an increasing level of math maturity is growing ability to communicate mathematics and math ideas orally and in writing using standard notation, vocabulary, and acceptable style. An oral communication between two people can be thought of as an exchange of word problems. If the topic is related to and/or involves math in some way, then the two participants are involved in creating and solving math word problems.

Here is a more mathematical way of describing an increasing level of the communication component of math maturity. Consider a situation in which a person is using oral language, gestures, written languages, pictures, and diagrams, to communicate a math problem or math related problem to another person. The idea is to communicate the problem carefully and fully, so that the receiver of the communication can then bring his or her math knowledge and skills to bear in attempting to solve the problem. This type of communication requires a high level of precision on the parts of the two participants.


 * Math Maturity Food for Thought. On a scale from 1 to 10 with 10 being the highest, rate yourself in ability to effectively communicate math orally. Then rate yourself in ability to effectively communicate math using written language. What did your elementary school and middle school teachers do to help you develop your math oral and written communication skills?

MM2. Learn to learn math and help others learn math
Learning math means learning with an appropriate combination of memorization and understanding. Some key ideas include constructivism, metacognition, and reflective thinking. Learning to learn math includes learning to make effective use of the various aids to learning that are available, such as teachers, peers, books, and computer. It also includes leaning to make effective use of one’s overall learning knowledge and skills, and one’s specific math learning knowledge and skills. Increasing math maturity is evidenced by increasing ability to be a self-sufficient intrinsically motivated learner who learns math with understanding. Increasing math maturity is also evidenced by increasing ability to work with people having varying levels of math knowledge and skills, and to help them learn math.


 * Math Maturity Food for Thought. When was the last time you used a hard copy library or the Web to look up something about math content? Can you read math well enough so that you can learn math by reading a freshman university textbook in math? If so, when and how did you develop this skill?

MM3. Generalize from a specific example to a broad concept
Mathematicians often start from a specific example of a problem, and go on to represent, define, and solve a broad category of closely related problems. For example, one might start with a concrete example of a problem involving a specific equilateral triangle, and develop results that solve this type of problem for every equilateral triangle. Increasing math maturity includes getting better at identifying a general class of problems from a specific example, in solving the general class of problems, and in making use of a solution to a general class or problems to solve specific instances of the problem.


 * Math Maturity Food for Thought. Name some general classes of math problems that elementary school students are expected to learn to solve. Which of these require higher-order thinking and creative problem solving, as contrasted with using memorized algorithms?

MM4. Transfer of learning
One aspect of learning math is to learn a variety of strategies (algorithmic and heuristic) that are useful in attacking a broad range of math problems, and to learn to develop such strategies on one’s own. Another aspect of learning math is to learn to think and reason mathematically. Increasing math maturity is evidenced by getting better at transferring "transfer of learning" or applying one’s math knowledge and skills into other areas of math and into math related areas and problems in disciplines outside of mathematics. Progress is shown by increasingly being able to apply one’s math knowledge and skills to challenging math-related problems and problem situations that one has not previously encountered.


 * Math Maturity Food for Thought. When you observe elementary teachers teaching math, what emphasis do you see on teaching for transfer to challenging math-related problems and problem situations that one has not previously encountered?

MM5. Multiple, varied representations
Children begin their leaning of math well before they reach the “concrete operations” phase of cognitive development. This early math learning is rooted in verbal, tactile, and visual representations of specific concrete objects and events. Increasing math maturity is evidenced in increasing ability to deal with generalizations and less concrete examples. For example, there is a difference between working with three blue toy cars and two red toy cars that one has physically sitting before one’s eyes, and doing the same thing with pictures in a book or pictures of cars in one’s mind’s eye. At a much higher math level, increasing math maturity includes getting better at moving back and forth between the visual (e.g., graphs, geometric representations) and the analytical e.g., (equations, functions) math representations.


 * Math Maturity Food for Thought. Do you have several different mental models for addition and for multiplication? Are these models equally useful in picturing and understanding working with natural numbers, integers, fractions, and decimal fractions?

MM6. Math problem solving and proofs
Solving math problems and proving math theorems lie at the very heart of mathematics. Increasing math maturity is evidenced by progress in being able to provide solid evidence (informal and formal arguments and proofs) of the correctness of one’s efforts in solving math problems and making proofs. This is a specific type of communication in the language of math.


 * Math Maturity Food for Thought. What does the term mathematical proof mean to you. Name an elementary school grade at which you would like to teach, and explain what you would like the term mathematical proof to mean to students at that level.

MM7. Math-related word problems
Represent (model) verbal and written problems in any discipline as mathematical problems. Recognize when a word problem might make effective use of math in attempts to solve the problem. Increasing math maturity is evidenced by increasing knowledge and skills in representing word problems using the language of mathematics, solving the resulting math problem, translating the results back into the language and context of the original word problem, and checking for accuracy and mindfulness of the math results in light of the context and meaning of the original word problem.


 * Math Maturity Food for Thought. What are the sorts of challenges do the students you are preparing to teach face as they attempt to solve word problems?

MM8. Math is a human endeavor
Math is more than just solving math problems and making math proofs. Our accumulated math knowledge represents considerable human creativity over thousands of years. Math is part of our culture. Math is fun. Math is part of the games we create and play. Math is part of the beauty of our world. Increasing math maturity is evidenced by increased understanding or and participation in these various aspects of the overall discipline of mathematics.


 * Math Maturity Food for Thought. Can you give examples from your own life when you found math to be fun or beautiful? In your math teaching, what might you do to help your students learn about the fun, joyfulness, and beauty of math?

MM9. Math content
As you think about the math maturity components listed above, notice that few specific math content topics are mentioned. One needs to know some math in order to be able to demonstrate increasing math maturity. But, increasing levels of math maturity are not dependent on gaining some specified and widely agreed on collection of math content.

Of course, great effort has gone into the development of various curricula and various approaches to learning math. There is a reasonable amount of agreement as to math topics that students should learn something about in elementary school, secondary school, and in undergraduate programs of study. However, it is depth of understanding that is the key idea.

Math is an old and vast discipline. It has great breadth and depth. It is a growing field in its own right, and its uses in other disciplines continue to grow.

The leaders and educators in each discipline design and construct curriculum in the discipline. They want students to gain some of the important knowledge and skills in the discipline and to lay a foundation for future learning within the discipline.

Thus, part of one’s increasing math maturity is increasing breadth and depth of mathematical knowledge and skills. However, because of the great breadth and depth of the discipline of math, a mere listing of topics to be studied is a poor approach to improving math maturity. Quoting George Polya from http://mathematicallysane.com/analysis/polya.asp:


 * To understand mathematics means to be able to do mathematics. And what does it mean doing mathematics? In the first place it means to be able to solve mathematical problems. For the higher aims about which I am now talking are some general tactics of problems—to have the right attitude for problems and to be able to attack all kinds of problems, not only very simple problems, which can be solved with the skills of the primary school, but more complicated problems of engineering, physics and so on, which will be further developed in the high school. But the foundations should be started in the primary school. And so I think an essential point in the primary school is to introduce the children to the tactics of problem solving. Not to solve this or that kind of problem, not to make just long divisions or some such thing, but to develop a general attitude for the solution of problems. [Bold added for emphasis.]

In essence, Polya is stressing the need to gain increased skill in representing and solving problems using the math that one has learned. This transfer of learning—moving from one’s math content knowledge to being able to effectively make use of the content in representing and solving problems—is a fundamental aspect of increasing math maturity.


 * Math Maturity Food for Thought. What does the term math problem mean to you? What do you want it to mean to students you plan to teach in elementary school?

MM10. Mathematical intuition
As one’s knowledge of and experience in using math grows, one’s math intuition grows. Herbert Simon, a Nobel Prize winning polymath, defined intuition as “frozen analysis.” He noted that in any disciplines where one studies and practices extensively, a subconscious type of intuition is developed. This intuition may well be able to quickly detect an error that one has made in math thinking and math problem solving, very quickly decide a way to attack a particular type of problem, or provide a “feeling” for the possible correctness of a conjecture.

As an example, look at a student’s statement that 5 + 8 = 40. At a subconscious level your brain might say, “something is wrong.” It might next tell you, “the number 40 is way too large.” Your experience and math teaching intuition might tell you, “perhaps the student multiplied instead of added.” Through grading lots of student papers, you have developed some math intuition that makes you into a faster paper grader.

For a deeper view of math intuition, read Henri Poincaré 1905 paper, Intuition and Logic in Mathematics, available at http://www-history.mcs.st-and.ac.uk/Extras/Poincare_Intuition.html. Quoting the first paragraph:


 * It is impossible to study the works of the great mathematicians, or even those of the lesser, without noticing and distinguishing two opposite tendencies, or rather two entirely different kinds of minds. The one sort are above all preoccupied with logic; to read their works, one is tempted to believe they have advanced only step by step, after the manner of a Vauban who pushes on his trenches against the place besieged, leaving nothing to chance. The other sort are guided by intuition and at the first stroke make quick but sometimes precarious conquests, like bold cavalrymen of the advance guard.

We see such intuition in other areas, such as in chess. An accomplished chess player can glance at a board position and have a “feeling” for the threats and opportunities that the position represents.


 * Math Maturity Food for Thought. On a scale of 1 to 10, with 10 being the highest, how would you rate your current level of math intuition? In the informal and formal math education that you have had up to this point in your life, can you give examples of where there was an emphasis on increasing your level of math intuition?

MM11. Computers and other math tools
All of the above needs to take into consideration the various tools that have been developed to aid in representing and solving math problems and problems in which math can be a useful aid to their solution.

Calculators and computers are powerful examples of such tools. These tools are useful both in representing and solving math problems and also in learning math. Moreover, computational mathematics is now a one of the major subdivisions of the overall field of mathematics. Thus, increasing levels of math maturity are evidenced by increasing knowledge and skills in making effective use of Information and Communication Technology both as an aid to representing and solving math problems and as an aid to learning math. See http://iae-pedia.org/Two_Brains_Are_Better_Than_One.


 * Math Maturity Food for Thought. Calculator and computer technology can be used as an aid to teaching math. These tools can also be part of the content that students learn. Explore your ideas on the following question:


 * If a calculator or a computer can readily solve a type of math problem that students are learning about in school, what should they learn about solving that type of problem “by hand” and what should they learn about solving that type of problem using the calculator and computer technology?

Summary
To summarize the list given above, a mathematically mature adult has the math content and math tools knowledge, understanding, and skills to accomplish the math-related activities in his or her overall set of adult responsibilities and problem-solving challenges.

Here are three questions to ask yourself and/or to discuss with others:


 * 1) How well does your current level of math maturity fit your current needs and your overall aspirations and plans for your future? Think about your understanding of math maturity and arguments to support your answer to the question.
 * 2) How does your current level of math maturity compare with that of your peers and with other people you know?
 * 3) What are some of the things you do that help increase the level of math maturity of students, your children, and other people you interact with?

Outline for a Sample Workshop
The materials given above were used in a 1:15 presentation to about 40 students who were in their last term of a graduate program in elementary education, leading to teacher certification. I used the following outline during my interactions in this Math Methods course.

10 Minutes. Getting started, interacting with the group, setting the tone for a "light, fun, and not very threatening" interaction. Among other things, we talked about higher-order versus lower-order questioning and learning.For example, the class was asked to provide examples of higher-order topics or ideas in math.

10 Minutes. I secured four volunteers who seated themselves at the front of the room and read the "Communications" section of the handout. They then discussed among themselves using their "whole class voices" what the topic means in words and ideas understandable to their fellow students. They then disused examples of how to help the types of students they are learning to teach to become better at their reading, writing, speaking, and listening levels of math knowledge and skills. From time to time I interrupted to help emphasis the importance and excellence of the examples they were giving.

15 Minutes. The class was divided into groups of 4 to five students. (Each of the volunteers was asked to be a fourth or fifth member of a group, so every group had at least three students who had not initially volunteered.It took nearly five minutes to get the groups organized and to hand out the handouts, and to assign each group to a different one of the math maturity topics in the handout. This left about 10 minutes for a group to:


 * Discuss how to represent the topic in a form understandable to others in the class.
 * Find examples of what one might do in an elementary school math class to engage students in activities that will increase their level of math maturity in the topic area.

25 to 30 minutes. The goal was to spend about 5 minutes per group, with no intent in getting through all groups. The first couple of groups were volunteer groups, and after that I called on groups. In their five minutes, a group was to talk about their discussion of questions 1 and 2 given above. I interacted with each group, asking probing questions to stimulate their presentation and occasionally emphasizing the importance of an idea that had been presented.

Remainder of the time available. First, time was provided for general questions. Then I spent briefly discussing some of the documents in the set of references.

One of the things that emerged during the presentation and discussion is that the students knew very little about transfer of learning and teaching for transfer. No one in the class had heard of the "High-road, low-road" theory of transfer of learning.

The terminology "computational thinking" and the idea of humans and machines working together to solve problems and accomplish tasks has not been part of their education up to this time. I spent several minutes discussing how the disciplines of math and science have been greatly changed by computer modeling and simulation.

I tend to measure the success of a workshop in terms of how much I learn. From this (selfish) point of view, it was an excellent workshop. There was a lot of sharing from the participants, and I learned a great deal from their examples and insights.

This workshop provided no opportunity for a followup interaction with the group in which I might have been able to assess possible long term retention and/or use of ideas covered in the workshop.

Author
This page was developed by David Moursund.