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