Information Age Education
   Issue Number 74
September, 2011   

This free Information Age Education Newsletter is written by Dave Moursund and Bob Sylwester, and produced by Ken Loge. The newsletter is one component of the Information Age Education project. See http://iae-pedia.org/ and the end of this newsletter. All back issues of this newsletter are available free online at http://i-a-e.org/iae-newsletter.html.

Beginning in October 2011, Information Age Education will be publishing a series of newsletters exploring educational aspects of the current cognitive neuroscience and technological revolution. Bob Sylwester and Dave Moursund will provide two introductory articles. These will be followed by a long series of “Guest” articles written by a broad collection of experts in the field.

We encourage you to tell your colleagues and students about the free IAE Newsletter. Free back issues and subscription information are available at http://i-a-e.org/iae-newsletter.html.

Tutoring in Informal and Formal Education Part 2: Tutoring in Math Education


"In the book of life, the answers aren't in the back." (Charles Schulz; American cartoonist best known worldwide for his Peanuts comic strip; the quoted statement is from the comic strip character Charlie Brown; 1922–2000.)

IAE Newsletters #73 and #74 draw on the free book  “Becoming a better math tutor” by Moursund and Albrecht (2011). Note that one of the authors of the book is also an author of the IAE Newsletters.

The first of these two newsletters provides some general ideas about tutoring that are not content area specific. Perhaps the most important idea is that we are currently witnessing a trend toward Information and Communicating Technology playing a significant role in tutoring. ICT can do certain types of tutoring quite well. What is happening is the result of slow but steady progress in developing Intelligent Computer-Assisted Learning systems that can provide individualized feedback. In addition, the Web is an aid to accessing content to meet specific needs of a tutee.

The current issue of the newsletter explores tutoring in math in order to provide insight into discipline-specific issues in tutoring. Each discipline has its own content, pedagogical content knowledge, academic goals and academic standard. In addition, computers are more useful in some disciplines than others. Computers are a very powerful aid to solving math problems.

What is Math?

Arithmetic is part of math, but is an inadequate answer to the question, “What is math?” Similarly, the statement that “Math is a language” misses many key ideas. Another standard answer is that math is the study of patterns. That is an inadequate answer because each discipline is the study of patterns that are used to represent, explore, think about, use, and add to the content knowledge and skills of the discipline. Our brains operate by representing and processing patterns.

A better answer was provided by George Polya, one of the world’s leading mathematicians and math educators of the 20th century. His answer emphasizes that math is a way of thinking and problem solving.

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. (George Polya.) [Bold added for emphasis.]

The content of math is both broad and deep, and is organized into a  number of sub disciplines. At the precollege level, students are exposed  to various aspects of arithmetic, number theory, geometry, algebra, probability, statistics, and logic. The unifying goal is to learn to represent, think about, understand, and solve problems that are amenable to using the language and various components of math.

A math tutor is faced by the challenge of helping tutees learn math content, but the larger challenge is helping tutees gain increased knowledge and skills in thinking and problem solving as applied to math-related problems both in the discipline of math and in many other disciplines.

Math Maturity and Math Habits of Mind

A child’s brain reaches near adult size by age 5 or 6, and the rest of a child’s body reaches near adult size by the late teens. However, substantial growth in a person’s mind continues until the mid 20s, and one’s mind continues significant daily change throughout one’s lifetime. Thus, it is appropriate to consider cognitive maturity (mind maturity) both in general and in the specific discipline of mathematics. The careful and prolonged study of math provides the mind with math content knowledge and skills, and help in gaining a higher level of math maturity. Learning to think mathematically comes through a combination of appropriate instruction and long years of practice with appropriate feedback.

As a mathematician, I (Dave Moursund) learned to view the world through “math-colored” glasses. I learned quite a bit of math content—but I also learned how to think like a mathematician. I have math habits of mind that automatically mathematize the problems and problem situations that I encounter in my everyday life.

Moreover, I am moderately good at transfer of learning of my math content knowledge and my math maturity to other disciplines. Let me give an example. Proof is one of the key ideas in math. In essence, a proof is a rigorous, convincing argument that can be followed and understood by a person with appropriate knowledge and understanding within the content area of the proof.

Suppose that you solve a math problem. How do you know that the results you have produced are correct? Can you convince yourself and others that your results are correct? (Reread the Charles Schulz quote at the beginning of this newsletter.) Proof lies at the very heart of math problem solving and the entire discipline of mathematics.

The Moursund and Albrecht (2011) book provides many examples of how to help tutees increase their levels of math maturity and their math habits of mind. One approach used in the book is based on the work of Arthur Costa and Bena Kallick (n.d.). They have develop 16 habits of mind that they feel cut across the various academic disciplines. Each of these habits can be analyzed from a math education point of view. Here are two examples.

Habit of Mind
Applications in Math Tutoring
1. Persisting.
Stick to it through task completion. Remain focused—keep your eye on the ball. Try alternative approaches when you are stuck.
Don’t give up easily.

This is one of the key ideas in math problem solving. ADD and ADHD students have special difficulties in this area. A great many other math students have not learned the need for persistence in dealing with challenging math problems.

However, be aware that not all math problems are solvable, and that others are beyond a student’s current capabilities. One aspect of learning math problem solving is to develop insight into when to temporarily or permanently give up.

Of course, if a math researcher gives up too early, then important discoveries are not made. Examples: The equation 2x – 3 = 0 is unsolvable in the domain of integers, but is solvable in the domain of rational numbers. The equation x² – 2 = 0 is unsolvable in the domain of rational numbers, but is solvable in the domain of real numbers.
2. Managing impulsivity.
Think before you act, and consider the consequences of your actions before taking the actions. Remain calm, thoughtful, and deliberate. Don’t be driven by a need for instant gratification; with practice, one can learn to control this impulse. 
This habit of mind is applicable both in interacting with other people and in carrying out tasks such as problem solving.
 
In math problem solving, one has a goal in mind. Learn to mentally consider various approaches to achieving the goal. Learn to analyze whether the steps one is taking or considering taking will actually contribute toward achieving the goal.

Students who are driven by the need for instant gratification seem to have trouble in their math studies when they reach algebra. See http://i-a-e.org/newsletters/IAE-Newsletter-2009-24.html.)


Computational Thinking as a Habit of Mind

A human brain and a computer brain have overlapping capabilities. Human brains are much better than computer brains in some areas, while computer brains are much better than human brains in other areas. The term Computational Thinking (IAE-pedia, n.d.) has come into common use as a transdisciplinary important approach to problem solving.

Computational thinking builds on the power and limits of computing processes, whether they are executed by a human or by a machine. Computational methods and models give us the courage to solve problems and design systems that no one of us would be capable of tackling alone. Computational thinking confronts the riddle of machine intelligence: What can humans do better than computers, and what can computers do better than humans? (Jeannette Wing, in Moursund and Albrecht, 2011.)

Genetically, human brains are not changing very rapidly. However, our cognitive capabilities have been greatly increased by the combined artificial intelligence (machine intelligence) and “brute force” power of computer brains continues to grow very rapidly.

Here are three key aspects of computer capabilities to keep in mind:
  1. The size and capabilities of electronic digital libraries such as the Web are growing very rapidly. Such libraries help support a “look it up” approach to education, solving problems, and accomplishing tasks. In their math education, students should be learning to make math-related components of the Web.

  2. Computer systems can solve or greatly help in solving many of the math problems that students learn about in school. (Consider a parallel between machines used to automate physical tasks and machines used to automate mental tasks.)

  3. Computer-assisted Learning is steadily increasing in its capabilities to help students learn and do math.

Final Remarks

Math tutoring is becoming a combination human and computer endeavor. The computer components and computer-assisted communication are of steadily improving capabilities. However, it is evident that the human components are still indispensable.


References

Costa, Arthur and Kallick, Bena (n.d.). Sixteen habits of mind. The Institute for Habits of Mind. Retrieved 7/4/2011 from http://www.instituteforhabitsofmind.com/.

IAE-pedia (n.d.). Communicating in the language of mathematics. Information Age Education. Retrieved 7/3/2011 from http://iae-pedia.org/Communicating_in_the_Language_of_Mathematics.

IAE-pedia (n.d.). Computational thinking. Information Age Education. Retrieved 7/3/2011 from http://iae-pedia.org/Computational_Thinking.

Moursund, David and Albrecht, Robert (9/2/2011). Becoming a better math tutor. Retrieved 9/4/2011 from http://i-a-e.org/downloads/doc_download/208-becoming-a-better-math-tutor.html. Eugene. OR: Information Age Education. If you want to just view the TOC, Preface, the first two chapters, and the two Appendices, go to http://iae-pedia.org/Math_Tutoring

Moursund, David and Sylwester, Robert (2011). Four-part series on stress in education. IAE Newsletter issues 64-67. Access at http://iae-pedia.org/IAE_Newsletter.



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About Information Age Education, Inc.

Information Age Education is a non-profit organization dedicated to improving education for learners of all ages throughout the world. Current IAE activities include a Wiki with address http://IAE-pedia.org, a Website containing free books and articles at http://I-A-E.org, a Blog at http://i-a-e.org/iae-blog.html, and the free newsletter you are now reading.

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