Contributed by:
This book is about math tutoring. It is designed to help math tutors and tutees get better at their respective and mutual tasks.
Tutoring is a powerful aid to learning. Much of the power comes from the interaction between tutor and tutee. This interaction allows the tutor to adjust the content and nature of the instruction to specifically meet the needs of the tutee. It allows ongoing active participation of the tutee.
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Becoming a Better Math Tutor
Becoming a Better Math Tutor
David Moursund
Robert Albrecht
Abstract
"Tell me, and I will forget. Show me, and I may remember. Involve me,
and I will understand." (Confucius; Chinese thinker and social
philosopher; 551 BC – 479 BC.)
This book is about math tutoring. It is designed to help math tutors and tutees get better at
their respective and mutual tasks.
Tutoring is a powerful aid to learning. Much of the power comes from the interaction
between tutor and tutee. (See the quote from Confucius given above.) This interaction allows the
tutor to adjust the content and nature of the instruction to specifically meet the needs of the tutee.
It allows ongoing active participation of the tutee.
The intended audiences for this book include volunteer and paid tutors, preservice and
inservice teachers, parents and other child caregivers, students who help other students (peer
tutors), and developers of tutorial software and other materials.
The book includes two appendices. The first is for tutees, and it has a 6th grade readability
level. The other is for parents, and it provides an overview of tutoring and how they can help
their children who are being tutored.
An extensive References section contains links to additional resources.
Download a free copy of this book from: http://i-a-
e.org/downloads/doc_download/208-becoming-a-better-math-tutor.html.
People who download or receive a free copy of this book are encouraged to
make a $10 donation to their favorite education-related charity. For details on
donating to a University of Oregon mathematics education project, see
http://iae-pedia.org/David_Moursund_Legacy_Fund.
Corrections Copy 9/6/2011 of Version 9/4/2011
Copyright © David Moursund and Robert Albrecht, 2011.
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License.
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Becoming a Better Math Tutor
About the Authors
Your authors have authored and/or co-authored nearly 90 academic books as well as
hundreds of articles. They have given hundreds of conference presentations and workshops.
This is the second of their co-authored books. Their first co-authored book is book is:
Moursund, David and Albrecht, Robert (2011). Using math games and
word problems to increase the math maturity of K-8 students. Salem,
OR: The Math Learning Center.
It is available in PDF and Kindle formats. For ordering information go
to http://iae-
pedia.org/Moursund_and_Albrecht:_Math_games_and_word_problems.
Dr. David Moursund
After completing his undergraduate work at the University of Oregon, Dr. Moursund earned
his doctorate in mathematics from the University of Wisconsin-Madison. He taught in the
Mathematics Department and Computing Center at Michigan State University for four years
before joining the faculty at the University of Oregon. There he had appointments in the Math
Department and Computing Center, served six years as the first head of the Computer Science
Department, and spent more than 20 years working in the Teacher Education component of the
College of Education.
A few highlights of his professional career include founding the International Society for
Technology in Education (ISTE), serving as its executive officer for 19 years, establishing
ISTE’s flagship publication, Learning and Leading with Technology, serving as the Editor in
Chief for more than 25 years. He was a major professor or co-major professor for more than 75
doctoral students. Six of these were in mathematics and the rest in education. Dr. Moursund has
authored or coauthored more than 50 academic books and hundreds of articles. He has presented
several hundred keynote speeches, talks, and workshops around the world. More recently, he
founded Information Age Education (IAE), a non-profit organization dedicated to improving
teaching and learning by people of all ages and throughout the world. IAE currently provides
free educational materials through its Wiki, the free IAE Newsletter published twice a month,
and the IAE Blog.
For more information about David Moursund, see http://iae-pedia.org/David_Moursund. He
can be contacted at [email protected].
Robert Albrecht
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Becoming a Better Math Tutor
A pioneer in the field of computers in education and use of games in education, Robert
Albrecht has been a long-time supporter of computers for everyone. He was instrumental in
helping bring about a public-domain version of BASIC (called Tiny BASIC) for early
microcomputers. Joining forces with George Firedrake and Dennis Allison, he co-founded
People’s Computer Company (PCC) in 1972, and also produced and edited People's Computer
Company, a periodical devoted to computer education, computer games, BASIC programming,
and personal use of computers.
Albrecht has authored or coauthored over 30 books and more than 150 articles, including
many books about BASIC and educational games. Along with Dennis Allison, he established Dr.
Dobb’s Journal, a professional journal of software tools for advanced computer programmers.
He was involved in establishing organizations, publications, and events such as Portola Institute,
ComputerTown USA, Calculators/Computers Magazine, and the Learning Fair at Peninsula
School in Menlo Park, California (now called the Peninsula School Spring Fair).
Albrecht's current adventures include writing and posting instructional materials on the
Internet, writing Kindle books, tutoring high school and college students in math and physics,
and running HurkleQuest play-by-email games for Oregon teachers and their students.
For information about Albrecht’s recent Kindle books, go to
http://www.amazon.com/.
Select Kindle Store and search for albrecht firedrake.
For more information about Robert Albrecht, see http://iae-pedia.org/Robert_Albrecht. He
can be contacted at [email protected].
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Table of Contents
Preface ........................................................................................................... 5!
Chapter 1: Some Foundational Information.............................................. 7!
Chapter 2: Introduction to Tutoring ........................................................ 18!
Chapter 3: Tutoring Teams, Goals, and Contracts ................................. 27!
Chapter 4: Some Learning Theories......................................................... 37!
Chapter 5: Uses of Games, Puzzles, and Other Fun Activities............... 51!
Chapter 6: Human + Computer Team to Help Build Expertise ............ 68!
Chapter 7: Tutoring for Increased Math Maturity................................. 76!
Chapter 8: Math Habits of Mind .............................................................. 88!
Chapter 9: Tutoring “to the Test” ............................................................ 99!
Chapter 10: Peer Tutoring....................................................................... 108!
Chapter 11: Additional Resources and Final Remarks ........................ 116!
Appendix 1: Advice to Tutees.................................................................. 125!
Appendix 2: Things Parents Should Know About Tutoring ................ 133!
References.................................................................................................. 139!
Index .......................................................................................................... 143!
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Preface
Somebody came up to me after a talk I had given, and said, "You make
mathematics seem like fun." I was inspired to reply, "If it isn't fun, why do
it?" Ralph P. Boas; mathematician, math teacher, and journal editor;
1912–1992.
This book is about math tutoring. The intended audience includes preservice and inservice
teachers, volunteer and paid tutors. The audience includes parents and other child caregivers,
students who help other students, and developers of tutorial software and other materials.
Tutors—Both Human and Computer
A tutor works with an individual or with a small group of students. The students are called
tutees. In this book we focus on both human and computer tutors. Nowadays, it is increasingly
common that a tutee will work with a team consisting of one or more humans and a computer.
Formal tutoring within a school setting is a common practice. Formal tutoring outside of a
school setting by paid professionals and/or volunteers is a large business in the United States and
in many other countries.
Underlying Theory and Philosophy
Both the tutor (the “teacher”) and the tutee (the “student”) can benefit by their participation
in a good one-to-one or small-group tutoring environment. Substantial research literature
supports this claim (Bloom, 1984). Good tutoring can help a tutee to learn more, better, and
faster. It can contribute significantly to a tutee’s self-image, attitude toward the area being
studied, learning skills, and long-term retention of what is being learned.
Most people think of tutoring as an aid to learning a specific subject area such as math or
reading. However, good tutoring in a discipline has three general goals:
1. Helping the tutees gain knowledge and skills in the subject area. The focus is
on immediate learning needs and on building a foundation for future learning.
2. Helping the tutees to gain in math maturity. This includes learning how to
learn math, learning how to think mathematically (this includes developing
good math “habits of mind”), and learning to become a more responsible math
student (bring necessary paper, pencil, book, etc. to class; pay attention in
class; do and turn required assignments).
3. Helping tutees learn to effectively deal with the various stresses inherent to
being a student in our educational system.
The third item in this list does not receive the attention it deserves. Many students find that
school is stressful because of the combination of academic and social demands. Math is
particularly stressful because it requires a level of precise, clear thinking and problem-solving
activities quite different than in other disciplines. For example, a tiny error in spelling or
pronunciation usually does not lead to misunderstanding in communication. However, a tiny
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error in one step of solving a math problem can lead to completely incorrect results. Being
singled out to receive tutoring can be stressful. To learn more about stress in education and in
math education, see Moursund and Sylwester (2011).
Some Key Features of this Book
While this book focuses specifically on math tutoring, many of the ideas are applicable to
tutoring in other disciplines. A very important component in tutoring is helping the tutee become
a more dedicated and efficient lifelong learner. This book emphasizes “learning to learn” and
learning to take more personal responsibility for one’s education. A good tutor uses each tutoring
activity as an aid to helping a tutee become a lifelong, effective learner.
An important component of tutoring is helping the tutee become a more
dedicated and efficient lifelong learner. This book emphasizes “learning
to learn” and learning to take more personal responsibility for one’s
education.
The task of improving informal and formal education constitutes a very challenging task. “So
much to learn … so little time.” The totality of knowledge and skills that a person might learn
continues to grow very rapidly.
We know much of the math that students cover in school is forgotten over time. This book
includes a focus on helping students gain a type of math maturity that endures over the years.
The book makes use of a number of short “case studies” from the tutoring experience of your
authors and others. Often these are composite examples designed to illustrate important ideas in
tutoring, and all have been modified to protect the identity of the tutees.
Appendix 1. Advice to Tutees. This material can to be read by tutees with a 6th
grade or higher reading level. Alternatively, its contents can be discussed with
tutees.
Appendix 2: Some Things Parents Should Know About Tutoring. This
material is designed to help parents and other caregivers gain an increased
understanding of what a child who is being tutored experiences and possible
expectations of having a child being tutored. Tutors may want to provide a
copy of this appendix to parents and other primary caregivers of the students
they are tutoring.
The book has an extensive Reference section. For the most part, the references are to
materials available on the Web.
The book ends with a detailed index.
David Moursund and Robert Albrecht, September 2011
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Chapter 1
Some Foundational Information
“God created the natural numbers. All the rest [of mathematics] is the
work of mankind.” (Leopold Kronecker; German mathematician; 1823-
1891.)
All the world’s a game,
And all the men and women active players:
They have their exits and their entrances;
And all people in their time play many parts. (David Moursund–Adapted
from Shakespeare)
Tutors and other math teachers face a substantial challenge. Keith Devlin is one of our
world’s leading math education researchers. Here is a quote from his chapter in the book Mind,
brain, & education: Neuroscience implications for the classroom (Sousa et al., 2010.)
Mathematics teachers—at all education levels—face two significant obstacles:
1. We know almost nothing about how people do mathematics.
2. We know almost nothing about how people learn to do mathematics.
Math tutors and math teachers routinely grapple with these daunting challenges. Through the
research and writings of Devlin and many other people, solutions are emerging. We (your
authors) believe that the tide is turning, and that there is growing room for optimism. This
chapter presents some foundational information that will be used throughout the book.
The Effectiveness of Tutoring
Good tutoring can help a tutee to learn more, better, and faster (Bloom, 1984). It can
contribute significantly to a tutee’s self-image, attitude toward the area being studied, learning
skills, and long-term retention of what is being learned.
[Research studies] began in 1980 to compare student learning under one-to-one
tutoring, mastery learning [a variation on conventional whole-class group
instruction], and conventional group instruction. As the results of these separate
studies at different grade levels and in differing school subject areas began to
unfold, we were astonished at the consistency of the findings and the great
differences in student cognitive achievement, attitudes, and self-concept
under tutoring as compared with group methods of instruction (Bloom,
1984). [Bold added for emphasis.]
Here are two key ideas emerging from research on tutoring and other methods of instruction:
1. An average student has the cognitive ability (the intelligence) to do very well
in learning the content currently taught in our schools.
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2. On average, good one-to-one tutoring raises a “C” student to an “A” student
and a “D” student to a “B” student. Many students in the mid range of F
grades see progress to the “C” level.
These are profound findings. They say most students have the innate capabilities to learn
much more and much better than they currently are. This insight leads educational researchers
and practitioners in their drive to develop practical, effective, and relatively low cost ways to
help students achieve their potentials.
Most students have the innate capabilities to learn both much more and
much better than they currently are learning.
Math tutoring is not just for students doing poorly in learning math. For example, some
students are especially gifted and talented in math. They may be capable of learning math faster
and much better than average students. The math talented and gifted (TAG) students can benefit
by working with a tutor who helps them move much faster and with a better sense of direction in
their math studies.
What is Math?
We each have our own ideas as to what math is. One way to explore this question is to note
that math is an area of study—an academic discipline. An academic discipline can be defined by
a combination of general things such as:
1. The types of problems, tasks, and activities it addresses.
2. Its tools, methodologies, habits of mind, and types of evidence and arguments
used in solving problems, accomplishing tasks, and recording and sharing
accumulated results.
3. Its accumulated accomplishments such as results, achievements, products,
performances, scope, power, uses, impact on the societies of the world, and so
on. Note that uses can be within their own disciplines and/or within other
disciplines. For example, reading, writing, and math are considered to be
“core” disciplines because they are important disciplines in their own rights
and also very important components of many other disciplines.
4. Its methods and language of communication, teaching, learning, and
assessment; its lower-order and higher-order knowledge and skills; its critical
thinking and understanding; and what it does to preserve and sustain its work
and pass it on to future generations.
5. The knowledge and skills that separate and distinguish among: a) a novice; b)
a person who has a personally useful level of competence; c) a reasonably
competent person, employable in the discipline; d) a state or national expert;
and e) a world-class expert.
Thus, one way to answer the “what is math” question is to provide considerable detail in each
of the numbered areas. Since math is an old, broad, deep, and widely studied discipline, each of
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the bulleted items has been targeted by a great many books, articles, professional talks, and
academic courses. The reader is encouraged to spend a couple of minutes thinking about his or
her insights into each of the numbered areas.
Humans and a number of other creatures are born with some innate ability to deal with
quantity. Very young human infants can distinguish between one of something, two of that
something, and three of that something. However, it is our oral and written languages that make
it possible to develop and use the math students learn in school. Our successes in math depend
heavily on the informal and formal education system for helping children to learn and use math.
The language of math is a special-purpose language useful in oral and
written communication. It is a powerful aid to representing, thinking
about, and solving math-related problems.
Our current language of math represents thousands of years of development (Moursund and
Ricketts, 2008). The language has changed and grown through the work of math researchers and
math users. As an example, consider the decimal point and decimal notation. These were great
human inventions made long after the first written languages were developed.
The written language of mathematics has made possible the mathematics that we use today.
The discipline and language of math have been developed through the work of a large number of
mathematicians over thousands of years. The written language of math has made it possible to
learn math by reading math.
Math is much more than just a language. It is a way of thinking and problem solving. Here is
a quote from George Polya, one of the world’s leading mathematicians and math educators of the
20th century.
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.]
Math educators frequently answer the “What is math?” question by discussing the processes
of indentifying, classifying, and using patterns. In that sense, math is a science of patterns.
However, problem solvers in all disciplines look for patterns within their disciplines. That helps
to explain why math is such an interdisciplinary discipline—it can be used to help work with
patterns in many different disciplines.
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Other answers to the “What is math?” question are explored in Moursund (2007). The careful
rigorous arguments of math proofs are a key aspect of math. The language of math and the
accumulated math proofs make it possible for math researchers to build on the previous work of
others. Building on the previous work of others is an essential idea in problem solving in math
and other disciplines.
Helping Tutees to Become Mathematically “Mature” Adults
Our math education system places more emphasis on some of the components of the
discipline of math than on others. During 2010–2011, most of the states in the United States
adopted the Common Core State Standards (CCSS). These include a newly developed set of
math content standards that specify what topics are to be taught at each grade level. Progress is
occurring in developing assessment instruments that can be used to test how well students are
learning the content standards. (CCSS, n.d.)
Students have varying levels of innate ability in math and they have varying levels of interest
in math. Precollege students who have a higher level of innate ability and interest in non-math
areas such as art, history, journalism, music, or psychology, may wonder why they are required
to take so many math courses. They may wonder why they cannot graduate from high school
without being able to show a particular level of mastery of geometry and algebra.
People who make decisions about math content standards and assessment try to think in
terms of future needs of the student and future needs of the country.
Math maturity is being able to make effective use of the math that one has learned through
informal and formal experiences and schooling. It is the ability to recognize, represent, clarify,
and solve math-related problems using the math one has studied. Thus, we expect a student to
grow in math maturity as the student grows in math content knowledge.
Mathematically mature adults have the math knowledge, skills, attitudes, perseverance, and
experience to be responsible adult citizens in dealing with the types of math-related situations,
problems, and tasks they encounter. In addition, a mathematically mature adult knows when and
how to ask for and make appropriate use of help from other people, from books, and from tools
such as computer and the Internet. One sign of an increasing level of math maturity is an
increasing ability to learn math by reading math.
For students, we can talk both about their level of math maturity and their level of math
education maturity. As an example, consider a student who is capable of doing math
assignments, but doesn’t. Or, consider a student who does the math assignments but doesn’t turn
them in. These are examples of a low level of math education maturity.
An increasing level of math maturity is evidenced by an increased understanding and ability
to learn math and to relearn math that one has forgotten. Chapter 8 covers many math Habits of
Mind that relate to math maturity. For example, persistence—not giving up easily when faced by
challenging math problems—is an important math Habit of Mind. A growing level of persistence
is an indicator of an increasing level of math maturity.
The “measure” of a math student includes both the student’s math content knowledge and
skills, and the level of math development (math maturity) of the student. Chapter 7 discusses
math maturity in more detail. Math tutoring helps students learn math and to gain an increasing
level of math maturity.
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An increasing level of math maturity is an increasing level of being able
to make effective use of one’s math knowledge and skills dealing with
math-related problems in one’s everyday life.
The Games of Math and in Math Education
The second quote at the beginning of this chapter presents the idea that “All the world’s a
game…” This book on tutoring includes a major emphasis on making math learning fun and
relevant to the tutee. It does this by making use of the idea that math can be considered as a type
of game. Within math, there are many smaller games that can catch and hold the attention of
students (Moursund and Albrecht, 2011).
You are familiar with a variety of games such as card games, board games, sports games,
electronic games, and so on. Consider a child just beginning to learn a sport such as swimming,
baseball, soccer, or basketball. The child can attend sporting events and/or view them on
television. The child can see younger and older children participating in these sports.
Such observation of a game provides the child with some insights into the whole game. The
child will begin to form a coherent mental image of individual actions, teamwork, scoring, and
rules of the game.
Such observation does not make the child into a skilled performer. However, it provides
insights into people of a variety of ages and skill levels playing the games, from those who are
rank beginners to those who are professionals. It also provides a type of framework for further
learning about the game and for becoming a participant in the game.
The “Whole Game” of Swimming
Consider competitive swimming as an example. You certainly know something about the
“whole game” of competitive swimming, even if you have never competed. People working to
become competitive swimmers study and practice a number of different elements of swimming,
such as:
• Arm strokes;
• Leg kicks;
• Breathing and breathing patterns;
• The takeoff at the beginning of a race;
• Racing turns at the end of the pool;
• Pacing oneself (in a race);
• Being a member of a relay team;
• Building strength and endurance through appropriate exercise and diet.
A swimming lesson for a person seriously interested in becoming a good swimmer will
include both sustained practice on a number of different elements and practice in putting them all
together to actually swim.
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A student learning to swim has seen people swim, and so has some
understanding of the whole game of swimming. The student gets better
by studying and practicing individual components, but also by routinely
integrating these components together in doing (playing) the whole
game of swimming.
David Perkins’ book, Making Education Whole (Perkins, 2010) presents the idea that much
of what students learn in school can be described as “learning elements of” and “learning
about.” Perkins uses the words elementitis and aboutitis to describe these illnesses in our
educational system.
In the swimming example, there are a great many individual elements that can be practiced
and learned. These are what Perkins is referring to when he talks about elementitis.
Even if you are not a swimmer, you probably know “about” such things as the backstroke,
the breaststroke, free style, racing turns, and “the thrill of victory and agony of defeat” in
competitive swimming. You can enjoy watching the swimming events in the Summer Olympics,
and you may remember the names of some of the super stars that have amassed many gold
medals. Many of us enjoy having a certain level of aboutitis in sports and a wide variety of areas.
The “Whole Game” of Math
Most of us are not used to talking about math as a game. What is the ”whole game” of math?
How does our education system prepare students to “play” this game? What can be done to
improve our math education system?
What is math? Each tutor and each tutee has his or her own answers.
Still other answers are available from those who create the state and
national math standards and tests.
Your authors enjoy talking to people of all ages to gain insights into their math education and
their use of math. Here is a question for you. What is math? Before going on to the next
paragraph, form some answers in your head.
Now, analyze your answers from four points of view:
1. Knowing some elements of math. You might have listed elements such as
counting, adding, multiplication, or solving algebra equations. You may have
thought about “getting right answers” and “checking your answers.”
2. Knowing something about math. You may have listed various components of
math such as arithmetic, algebra, geometry, probability, and calculus. You
may have thought about names such as Euclid, Pythagoras, and Newton. You
may have noted that many people find math to be a hard subject, and many
people are not very good at doing math. You may have had brief thoughts
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about your difficulties in working with fractions, percentages, and probability,
or balancing your checkbook.
3. Knowing how to “do” and use math. This includes such things as:
a. Knowing how to represent and solve math-related problems both in
math classes and in other disciplines and everyday activities that make
use of math.
b. Knowing how to communicate with understanding in the oral and
written language of math.
c. Knowing how and when to use calculators and computers to help do
math.
4. Knowing how to learn math and to relearn the math you have studied in the
past but have now forgotten.
Math tutors need to have a good understanding of these four categories of answers to the
question “What is math?” They need to appreciate that their own answers may be quite different
than the (current) answers of their tutees. Good tutoring involves interplay between the
knowledge and skills of the tutor and the tutee. The tutor needs to be “tuned” to the current
knowledge and skills of the tutee, continually filling in needed prerequisites and moving the
tutee toward greater math capabilities.
Junior Versions of Games
Perkins’ book contains a number of examples of “junior” versions of games that can be
understood and played as one makes progress toward playing the “whole game” in a particular
discipline or sub discipline. This is a very important idea in learning any complex game such as
the game of math.
Examples of Non-Math Junior Games
Think about the whole game of writing. A writer plays the whole game of effective
communicating in writing. Now, contrast this with having a student learning some writing
elements such spelling, punctuation, grammar, and penmanship. These elements are of varying
importance, but no amount of skill in them makes one into an effective player of the whole game
of writing.
A child can gain insight into the whole game of swimming. How does a child gain insight
into the whole game of writing? Obviously this is an educational challenge.
Our language arts curriculum realizes this, and it has worked to establish an appropriate
balance between learning about, learning elements of, and actually doing writing. The language
arts curriculum also recognizes the close connection between writing and reading. One can think
of the whole game of language arts as consisting of two overlapping games—the whole game of
reading and the whole game of writing.
Even at the first grade level, a child can be playing junior versions of language arts games.
For example, a child or the whole class can work together to tell a story. The teacher uses a
computer and projection system to display the story as it is being orally composed. The whole
class can participate in editing the story. Students can “see” the teacher playing a junior game of
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editing. Using their knowledge of oral language and story telling, they can participate in junior
versions of writing and editing.
Of course, we don't expect first graders to write a great novel. However, they can play "junior
games" of writing such as writing a paragraph describing something they know or that interests
them. They can add illustrations to a short story that the students and teacher have worked
together to create. They can read short stories that are appropriate to their knowledge of the
world and oral vocabulary.
What does an artist do? Can a first grader learn (to play) a junior version of the game of art?
What does a dancer do? Can a first grader do a junior version of various games of performance
arts? Obviously yes, and such junior versions of creative and performing arts are readily
integrated into a first grade curriculum.
Junior Games in Math
This book provides a number of examples of junior math-oriented games. Let’s use the board
game Monopoly as an example. Many readers of this book played Monopoly and/or other
“money” board games when they were children. Monopoly can be thought of as a simulation of
certain aspects of the whole game of business. Math and game-playing strategies are used
extensively in the game.
Figure 1.1. Monopoly board. Copied from http://www.hasbro.com/monopoly/.
You probably know some things “about” Monopoly even if you have never played it. If you
have played Monopoly you know that there are many elements. You know that primary school
students and still younger students can learn to play Monopoly. This is an excellent example of
“play together, learn together.”
Imagine that children were not allowed to play the whole game until they first gain
appropriate knowledge of the game elements such as:
• Dice rolling, including determining the number produced by rolling a pair of dice and
whether a doubles has been rolled;
• Counting and moving a marker (one’s playing piece) along a board.
• Dealing with money.
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• Buying property, building houses and hotels, and selling property. This includes
making decisions about buying and selling.
• Making payments for landing on property owned by others.
• Collecting payments when other players land on your property.
• Checking to see that one’s opponents do not make mistakes—accidently or on
purpose.
• Learning and making use of various strategies relevant to playing the game well.
• Et cetera. One can break the whole game into a very large number of elements.
Learning to play the game of Monopoly can degenerate into elementitis.
Now, here’s the crux of the situation. In your mind, draw a parallel between learning to play
the whole game of Monopoly and learning to play the whole game of math. In either case the
mode of instruction could be based on learning about and learning elements of. Students could
be restricted from playing the whole game or even a junior version of the game until they had
mastered a large number of the elements.
We do not take this approach in the world of games—but we have a considerable tendency to
take this approach in mathematics education. Your authors believe that this is a major flaw in our
math education system.
Many students never gain an overview understanding of the whole
game of math. They learn math as a collection of unrelated elements.
This is a major weakness in our math education system.
Fun Math, Math Games, and Math Puzzles
One unifying theme in math is finding math types of patterns, describing the patterns very
accurately, identifying some characteristics of situations producing the patterns, and proving that
these characteristics are sufficient (or, are not sufficient) to produce the patterns.
This combination of finding, describing, identifying, and proving is a type of math game.
Junior versions of this game can be developed to challenge students at any level of their math
knowledge and skill. Higher levels of such games are math research problems challenging math
Tutoring Tips, Ideas, and Suggestions
Each chapter of this book contains a section giving tutors or potential tutors specific advice
on how to get better at tutoring. The example given below focuses on creating a two-way
communication between tutor and tutee.
Interaction Starters and Thinking Out Loud
One of the most important aspects of math tutoring is establishing and maintaining a two-
way math-related ongoing conversation between tutor and tutee. This is a good way to help a
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tutee learn to communicate effectively in the language of mathematics. It is a good way for the
tutor both to role model math communication and to better understand the tutees math
knowledge, skills, and weaknesses.
A skillful tutor is good at facilitating and encouraging a two-way math-related dialogue with
the tutee. With practice, a tutee gains skill in such a dialogue and becomes more comfortable in
engaging in such a dialogue. This is an important aspect of gaining in math maturity.
One approach is for the tutor to develop a list of interaction starters. As a tutee is working on
a problem, a tutor’s interaction starter can move the task into a math conversation. The
conversation might grow to a “think out loud” conversation or to a joint tutor-tutee exploration
of various points in solving a challenging problem.
Here are some interaction starters developed by the Math Learning Center (MLC, n.d.) and
Mike Wong, a member of the Board of Directors of the MLC. Your authors have added a few
items to the list.
• How do you know what you know? How do you know it’s true? (The tutee makes an
assertion. The tutor asks for evidence to back up the assertion.)
• Can you prove that? (Somewhat similar to an evidence request. A tutee solves a problem by
carrying out a sequence of steps. How does the tutee know that the solution is correct?)
• What if . . .? (Conjecture. Make evidence-based guesses. Pose variations on the problem
being studied.)
• Is there a different way to solve this problem? (Many problems can be solved in a variety of
ways. One way to check one’s understanding of a problem and increase confidence in a
solution that has been produced is to solve it in a different way.)
• What did you notice about . . . ? (Indicate an aspect of what the tutee is doing.)
• What do you predict will happen if you try … ?
• Where have you seen or used this before?
• What do you think or feel about this situation?
• What parts do you agree or disagree with? Why?
• Can you name some uses of this outside the math class and/or outside of school?
• How might a calculator or computer help in solving this problem?
Final Remarks
As you read this book, think about the whole game of being a math tutor and the whole game
of being a math tutee. What can you do to make yourself into a better player of the tutor game?
What can you do to help your tutees become better players of the tutee game?
Use this book to learn more about the math tutor game. Determine elements of the game that
are some of your relative strengths and some that are part of your relative weaknesses.
Consciously think about and work to improve yourself in your areas of relative weaknesses.
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Use the same approach with your tutees. Help each tutee to identify areas of relative strength
and areas of relative weakness. Help each tutee work to gain greater knowledge and skill in areas
of relative weaknesses.
Self-Assessment and Group Discussions
This book is designed for self-study, for use in workshops, and for use in courses. Each
chapter ends with a small number of questions designed to “tickle your mind” and promote
discussion. The discussion can be you talking to yourself, a discussion with other tutors, or a
discussion among small groups of people in a workshop or course.
1. Name one idea discussed in the chapter that seems particularly relevant and
interesting to you. Explain why the idea seems important to you.
2. Imagine having individual conversations with a student you are going to tutor
in math and a parent of that student. Each asks the question: “What is math
and why is it important to learn math?” What answers do you give? How
might your answers help to facilitate future math-related communication
between the child and parent?
3. Think about games and other forms of entertainment you participated in as a
child. Which (if any) contributed to your math education? Answer the same
question for today’s children, and then do a compare and contrast between the
two answers.
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Chapter 2
Introduction to Tutoring
"Knowledge is power." (Sir Francis Bacon; 1561; English philosopher,
statesman, scientist, lawyer, jurist, author and father of the scientific
method; 1561-1626.)
“When toys become tools, then work becomes play.” Bernie DeKoven.
Tutoring is a type of teaching. Good tutoring empowers a student with increased knowledge,
skills, habits, and attitudes that can last a lifetime.
This book makes use of a number of Scenarios. Each is a story drawn from the experiences
of your authors and their colleagues. Some are composites created by weaving together tutoring
stories about two or more tutees. All of the stories have been modified to protect the identities of
the tutees and to better illustrate important tutoring ideas.
Many students have math-learning difficulties. Some have a combination of dyslexia,
dysgraphia, dyscalculia, ADHD, and so on. If you do much math tutoring, you will encounter
students with these and/or other learning disabilities. Learn more about the first three of these
learning disabilities via a short video on dyscalculia and dysgraphia available at
Special education is a complex field. All teachers and all tutors gain some “on the job”
education and experience in working with students with special needs. A tutor might well
specialize in tutoring students who have learning disabilities and challenges. This book does not
attempt to provide the education in special education that is needed to become well qualified to
tutor special education students.
During their program of study that prepares them for a teacher’s license, preservice teachers
receive some introduction to special education. The regular classroom teacher is apt to have
students who spend part of their school day working with tutors.
Tutoring Scenario
In his early childhood, George was raised by a combination of his parents and two
grandparents who lived near his home. George was both physically and mentally
above average. He prospered under the loving care—think of this as lots of
individual tutoring—provided by his parents and grandparents. He enjoyed being
read to and this was a routine part of his preschool days.
George was enrolled in a local neighborhood school and enjoyed school.
However, his parents learned that George had a learning problem when they
received his end of second grade report card. The teacher indicated that George
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had made no progress in reading during that entire year and was having
considerable difficulty with math word problems.
His parents were surprised by the fact that George actually passed second grade,
and that the teacher had not made a major intervention sometime during the
school year.
A grandparent had heard about dyslexia, and so the parents and grandparents did
some reading in this area. Dyslexia is a type of brain wiring that makes it difficult
to learn to read. And sometimes makes it difficult to learn arithmetic. It was
obvious that George was dyslexic.
Under strong pressure from George’s parents, the school tested George, and it
turned out that he had severe dyslexia. With the help of an IEP (Individual
Education Program) that included a substantial amount of tutoring by reading
specialists for more than a year, George learned to read and more than caught up
with his classmates.
This is a success story. Dyslexia is a well-known learning disability that makes it difficult to
learn to read and that also can make it difficult to learn to do arithmetic. Extensive individual
tutoring leads to a rewiring of the tutee’s brain. This rewiring allows the reading-related
structures in the tutee’s brain to function much more like they do in a student that does not have
Many dyslexic students find the reading and writing aspects of math
particularly challenging. Dyscalculia and dysgraphia are other learning
disabilities that affect math learning.
Two-way Communication
Two-way communication between tutor and tutee lies at the very heart of effective tutoring.
Contrast such communication with a teacher talking to a class of 30 students, with the teacher
delivery of information occasionally interrupted by a little bit of student response or question
Two-way communication in tutoring is especially designed to facilitate learning. Tutees who
learn to effectively participate in such a communication have gained a life-long skill. The tutees
learn to express (demonstrate) what they know, what they don’t know, and what they want to
know. To do this, they need to be actively engaged and on task. They need to learn to focus their
attention. Much of the success of tutoring lies in the tutor helping the tutee gain and regularly use
such communication and attention-focusing skills.
Many successful tutors stress the idea that the tutee should be actively engaged in
conversation with the tutor. The tutor provides feedback based on what the tutee says and does.
Tutoring is not a lecture session.
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Perhaps you have heard of a general type of two-way communication that is called active
listening. Its techniques are easily taught and are applicable in any two-way conversation. See,
for example, http://www.studygs.net/listening.htm. Quoting from this Website:
Active listening intentionally focuses on who you are listening to, whether in a
group or one-on-one, in order to understand what he or she is saying. As the
listener, you should then be able to repeat back in your own words what they have
said to their satisfaction. This does not mean you agree with the person, but rather
understand what they are saying.
Here is a math active listening activity that can be used over and over again in tutoring. Ask
the tutee to respond to, “What did you learn in math class since the last time we got together?” If
the tutee’s answer is too short and/or not enlightening, the tutor can ask probing questions.
Tutors and Mentors
A mentor is an advisor, someone who helps another person adjust to a new job or situation.
The mentor has much more experience in the job or task situation than does the mentee. A new
mother and first-born child often have the benefit of mentoring (and some informal tutoring)
from a grandmother, sister, aunt, or a friend who is an experienced mother. One of the
advantages of having an extended family living in a household or near each other is mentoring
and informal tutoring are available over a wide range of life activities.
Tutoring and mentoring are closely related ideas. Although this book is mainly about
tutoring, mentoring will be discussed from time to time. In teaching and other work settings, a
new employee is sometimes assigned a mentor who helps the mentee “learn the ropes.” There
has been considerable research on the value of a beginning teacher having a mentor who is an
experienced and successful teacher. The same ideas can be applied to an experienced tutor
mentoring a beginning tutor.
Here is a list of five key “rules” to follow in mentoring (TheHabe, n.d.).
1. Set ground rules. This can be thought of as having an informal agreement
about the overall mentoring arrangement.
2. Make some quality time available. For example, agree to meet regularly at a
designated time and place.
3. Share interests. Build a relationship based on multiple areas of shared
interests. Include areas outside the specific area of mentorship.
4. Be available. A mentee may need some mentoring between the regularly
scheduled meeting times. Email may be a good way to do this.
5. Be supportive. A mentor is “on the same side—on the same team” as the
mentee.
Any long-term tutor-tutee activity will include both tutoring and mentoring. The tutor
becomes a mentor—a person who supports the tutee/mentee—in learning to become a more self-
sufficient, lifelong learner. Such mentoring is such an important part of long-term tutoring that
we strongly recommend that such mentoring be built into any long term tutoring that a student
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Peer Tutoring and Mentoring
Students routinely learn from each other. Most often this is in informal conversations,
interactions, and texting. However, structure can be added. For example, many schools have a
variety of academic clubs such as math, science, and robotics clubs. An important aspect of these
clubs is the various aspects of peer tutoring, cooperative learning, teams doing project-based
learning, and other activities in which students “play together and learn together.”
Such clubs often bring together students of varying ages and levels of expertise. This is an
excellent environment for mentoring, with more experienced club members mentoring those just
joining the club. It is delightful to create a club situation in which the members actively recruit
students who will become members in the future and then help them to fit into the club activities.
Math clubs, science clubs, and robotic clubs provide a rich environment
for students to play together, learn together.
In small group project-based learning activities tend to have a strong peer-tutoring
component. In forming project teams, a teacher might make sure each team includes a student
with considerable experience and success in doing project-based learning. In some sense, this
student serves as a mentor for others in the group. A teacher might provide specific instruction
designed to help group members learn to work together and learn from each other (PBL, n.d.).
Think about the following quote given at the beginning of their chapter:
“When toys become tools, then work becomes play.” Bernie DeKoven.
Learn more about DeKoven at http://www.deepfun.com/about.php.
To a child, a new toy can be thought of as a learning challenge. The toy, the child, peers, and
adults may all provide feedback in this learning process. A child immersed in learning to play
with a new toy is practicing learning to learn.
A child’s highly illustrated storybook is a type of educational toy. A parent and child playing
together with this type of toy lay the foundations for a child learning to read.
Some toys are more challenging, open ended, and educational than others. A set of building
blocks provides a wide range of creative learning opportunities. A set of dominoes or dice can
serve both as building blocks and the basis for a variety of games that involve counting,
arithmetic, and problem solving.
Dice
As an example, many students have played board games in which the roll of one or more 6-
faced dice determines a person’s move. When rolling a pair of dice, what is the most frequently
occurring sum? Individual students or groups of students can do many rolls of a pair of dice,
gather data on a large number of rolls, and analyze the data. They may discover that the number
of outcomes of a total of seven is roughly the same as the number of doubles. How or why
should that be?
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In a large number of rolls of a pair of dice, the total number of rolls that sum to eight is
roughly the same as the number that sum to six. How or why should that be?
It is fun to explore patterns in rolling dice. It is challenging mathematics to identify and
explain the patterns. See, for example. http://mathforum.org/library/drmath/view/55804.html.
Geoboard
A wide variety of such math manipulatives are often used in elementary school math
education. They can also be quite useful in working with older students. As an example, consider
a 5 x 5 geoboard. A geoboard is a five-by-five grid of short, evenly space posts. Rubber bands
are used to form geometric shapes on a geoboard. Two examples are shown in Figure 2.1.
Figure 2.1. Two 5 x 5 geoboards, each showing a geometric figure.
Notice that there are exactly four posts that are completely inside the first (W-shaped) figure.
Here is a simple game. Create some other geometric shapes on the geoboard that have exactly
four inside posts. A much more challenging game is to determine how many geoboard-based
geometric figures have exactly four inside posts.
The geometric shape on the second geoboard has five fully enclosed posts. You can see that
the game given above can be extended to finding figures with one completely enclosed post, with
two completely enclosed posts, and so on. One can also explore geometric shapes with specified
numbers of edge posts.
What “regular” geometric shapes can one make on a geoboard? What areas can one enclose
on a geoboard? What perimeter lengths can one create on a geoboard?
There are a very large number of geoboard sites on the Web, and there are many interesting
and challenging geoboard activities. The Website http://www.cut-the-knot.org/ctk/Pick.shtml
contains a computer-based geoboard and a discussion of some interesting math related to a
Television
Television can be considered as a toy. Researchers indicate that it is not a good learning toy
for very young children. Its use should be quite limited and carefully supervised. Passive
television programming lacks the interaction and personalized feedback that is especially
important for very young learners. Children have considerable inherent ability to learn by
doing—to learn by being actively engaged. Passively watching television is not active
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Computerized Toys
Many of today’s toys are computerized. Sherry Turkle (n.d.) has spent much of her
professional career doing research on how children interact with computer-based media and toys.
As with TV, the nature and level of child-toy interactivity is often quite limited. Active child-to-
toy engagement and interaction are essential to learning by playing with a toy.
In Summary
There are innumerable fun game-like activities that one can use to help students learn math,
gain in math maturity, and develop math Habits of Mind. In analyzing a game or game-like
activity for use in math education, think about:
1. What makes it attention grabbing, attention holding, and fun to play?
2. Is it cognitively challenging at a level appropriate to a tutee’s math
knowledge, skills, and development?
3. How does it relate to the overall “whole game” of math or a specific
component of math? If you, as the tutor, cannot identify a clear area of math
that is being investigated, how do you expect your tutee to gain mathematical
benefit from playing the game?
Computer-assisted instruction (now usually called computer-assisted learning or CAL) has
been steadily growing in use over the past 50 years. Quite early on in the development of CAL it
became obvious that:
1. A computer can be used as an automated “flash card” aid to learning. A
computer presents a simple problem or question, the computer user enters or
indicates an answer, and the computer provides feedback on the correctness of
the answer.
2. A computer can be used to simulate complex problem-solving situations, and
the user can practice problem solving in this environment. Nowadays, such
CAL is a common aid in car driver training and airplane pilot training, and in
such diverse areas as business education and medical education. Many
computer applications and computer games include built-in instructional
modules.
One of the characteristics of a good CAL system is that it keeps detailed records of a
student’s work—perhaps even at the level of capturing every keystroke. If the CAL is being used
in an online mode, the company that produced the CAL can analyze this data and use it to
improve the product. Very roughly speaking, it costs about $5 million for a company to develop
a high quality yearlong CAL course and $1 million a year to improve it and keep it up to date.
Over the years, this level of investment has led to increasing quality of commercially produced
CAL materials. This high developmental cost means that the leading edge CAL is not apt to be
available free on the Web unless its development was paid for by Federal or other grants.
The US Federal Government has funded a variety of CAL research and development
projects. In recent years, this has led to the development of the Cognitive Tutor CAL by
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Carnegie Mellon University , and a variety of pieces of software called Highly Interactive
Intelligent Computer-Assisted Learning (HIICAL) systems.
Such systems are taking on more of the characteristics of an individual tutor. They are not yet
as effective as a good human tutor, but for many students they are better than large group
(conventional) classroom instruction. In this book, we use the term “computer tutor” to refer to
computer-as-tutor, in the same way that we use the term human tutor to refer to human-as- tutor.
See https://mathtutor.web.cmu.edu/ for some of Carnegie Mellon’s Cognitive Tutor middle
school math materials. It is targeted at students who are reasonably good at math. Recently
Carnegie Mellon sold much of their Cognitive Tutor materials and business for $75 million to
the corporation that owns and runs Phoenix University—one of the largest distance education
intuitions in the world.
Computer tutors can be used in conjunction with human tutors and/or conventional classroom
instruction. The computer tutor, human tutor, and conventional group instruction combine to
provide a better education.
Tutoring Tips, Ideas, and Suggestions: Every Number is a Story
Each chapter of this book contains a Tutoring Tips example. Most experienced tutors
develop a large repertoire of such examples that they can draw upon as needed. Nowadays, it is
convenient to collect and organize such examples in a Digital Filing Cabinet. See details at
When you think about the number 13, what thoughts come to mind? Perhaps for you the
number 13 is an unlucky number or a lucky number. Perhaps you remember that 13 is a prime
Robert Albrecht, one of your authors, has written an entire book telling part of the story of
each of the positive integers 1-99. The 99-cent book is one of a number of books Albrecht is
making available in Kindle format. (Remember, there is free software that makes it possible to
read Kindle-formatted books on Macintosh and PC computers, on the iPad, and on Android
phones. For information about downloading these free applications, see http://iae-
Albrecht, Robert (2011). Mathemagical numbers 1 to 99. Retrieved
6/3/2011 from
http://www.amazon.com/s/ref=nb_sb_noss?url=search-
alias%3Ddigital-text&field-keywords=Bob+Albrecht&x=0&y=0.
Price: $.99. Other Kindle books by Albrecht are available at the
same location.
Here is a short activity that you might want to try out with a math tutee. In this example, we
use the number 13. Pick a number and ask your tutee to say some of the things they know or
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believe about that number. The idea is to engage your tutee in a conversation about a particular
natural number.
The natural number 13 might be a good choice. Here is Robert Albrecht’s story about 13.
13 (thirteen)
13 is a natural number.
13 is the successor of 12.
13 is the predecessor of 14.
13 is a prime number.
13 is an emirp. (31 is a prime number.)
Factors of 13: 1, 13
Proper factor of 13: 1
Sum of factors of 13 = 14
Sum of proper factors of 13 = 1
13 is a deficient number.
13 is a Fibonacci number.
Triskaidekaphobia is the fear of 13.
Triskaidekaphilia is the love of 13.
An aluminum (Al) atom has 13 protons.
Notice that this “story” includes quite a few words from the language of math. Albrecht’s
book contains a glossary defining these words. Here is a suggestion. One of your goals as a math
tutor could be to help your tutee learn to make use of the Web to find math-related information.
For example, what is a natural number? What is a prime number and why is it important in
math? Who is Fibonacci and why is a certain type of number named after him? Do some very tall
buildings not have a 13th floor? How can that be possible? Are there widely used words that have
exactly 13 letters?
What is a proton? Is there an atom that has exactly 12 protons, and is there an atom that has
exactly 14 protons? Why and how is math used in sciences such as biology, chemistry, and
What can one learn about the number 13 through use of the Web? A recent Google search
using the term 13 produced over 20 billion hits! Suppose a person spent just 10 seconds looking
at a hit to see if it relevant to their interests? How long would it take to process 20 billion hits?
A Google search of the word thirteen produced a little over 72 million hits. Why do you
suppose that the math notation 13 produced so many more hits than the written word thirteen?
Final Remarks
In some sense, each person is a lifelong student and a lifelong teacher. In our day-to-day lives
we learn from other people and we help other people to learn. Using broad definitions of tutor
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and tutee, each of us is both a tutor and a tutee in our routine, everyday lives. As both tutor and
tutee, our lives are full of learning and helping others to learn.
Most of us now make routine use of the Web and other electronic aids to accessing
information. These electronic sources of information can be thought of as Computer Tutors
designed to help us learn and to accomplish tasks we want to accomplish. Thus, readers of this
book are routinely involved in being tutored by both people and computers.
Self-Assessment and Group Discussions
This book is designed for self-study, for use in workshops, and for use in courses. Each
chapter ends with a small number of questions designed to “tickle your mind” and promote
discussion. The discussion can be you talking to yourself, a discussion with other tutors, or a
discussion among small groups of people in a workshop or course.
1. Name one idea discussed in the chapter that seems particularly relevant and
interesting to you. Explain why the idea seems important to you.
2. Think back over your personal experiences of tutoring (including helping your
friends, fellow students, siblings), being tutored, being helped by peers,
receiving homework help from adults, and so on. Name a few key tutoring-
related ideas you learned from these experiences.
3. Have you made use of computer-assisted learning and/or computer-based
games as an aid to learning or teaching math? If so, comment on the pros and
cons of your experiences. What are your thoughts on a computer-as-tutor
versus a human tutor?
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Chapter 3
Tutoring Teams, Goals, and Contracts
"There is no I in TEAMWORK." (Author unknown.)
"No matter what accomplishments you make, somebody helped you."
(Althea Gibson; African-American tennis star; 1927–2003.)
A tutor and a tutee work together as a team. The tutor part of a team may include a human
and a computer system. The tutee part of the team may be just one student, but sometimes it
consists of a small group of students who are learning together.
In all cases, the tutor(s) and tutee(s) have goals. It is desirable that these goals be explicit but
quite flexible. The goals need to be agreed upon by the human tutor(s) and tutee(s). It should be
possible to measure progress toward achieving the goals. This chapter discusses these issues.
Tutoring Scenario
Kim was a fourth-grade student who did not like math. Alas, early in the school
year, her math grade was a D. Kim did better in other subjects. Kim's mother Jodi
was sure that Kim could do much better with a little help, so she hired a tutor who
would come to their home once a week, help Kim do her math homework, and
hopefully help Kim to like math better, or at least dislike it less. Jodi knew that
Kim did well in subjects she liked.
Jodi and the tutor talked. "Aha" thought the tutor, who loved math games. "This is
a splendid opportunity to use games to make math fun for Kim." The tutor
suggested to Jodi that each tutoring gig spend some time playing games as well as
doing the homework. Jodi readily agreed.
Tutoring began. Each tutoring session, Kim and the tutor spent 30 to 40 minutes
doing homework and then played math games. Kim loved the math games. After
a few tutoring sessions, she became more at ease doing the homework because
she knew that she would soon play a game. Better yet, she began trying to do
more homework before the tutor arrived in order to have more time to play
games.
Kim became very good at playing games, including games at a higher math
maturity level than usual for a fourth grader. It became clear to the tutor that Kim
was very smart in math.
Kim and the tutor played many games. Her favorite game was Number Race 0 to
12, a game in which you try to move racers from 0 to 12 on five tracks. (See
Chapter 5 for a detailed description of this game.) To move your racer, you roll
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three 6-faced dice (3D6) and use the numbers on the dice to create numerical
expressions to move the racers on their tracks.
As the weeks rolled by, Kim became better and better at creating numerical
expressions. After a few weeks, she became as good as the tutor in rolling 3D6
and using addition, subtraction, multiplication, and parentheses to create numbers
to move her five racers on their five tracks.
Spring rolled around and Science Fair beckoned. Kim and her mother asked the
tutor to suggest science fair topics. He did. Among the topics was one of his
favorites, making homemade batteries from fruit, vegetables, and metal
electrodes. Kim liked this idea and chose it as her science fair project.
Kim, with great support from her mother, made batteries using apples, bananas,
lemons, oranges, potatoes, and other electrolytes. She experimented with pairs of
electrodes selected from iron, aluminum, carbon, zinc, and copper. Jodi bought a
good quality multimeter (about $40) for Kim to use in order to measure the
voltages produced by various combinations of fruit, vegetables, and metals. Kim
found that copper and zinc electrodes produced the highest voltage using several
fruits and vegetables as electrolytes. Figure 3.1 shows her final project.
Figure 3.1. Science fair project done by tutee with her mother’s help.
This story has a very happy ending. Kim’s Science Fair project was outstanding! And, Kim
became a very good math student! In retrospect, we can conjecture that Kim’s previous home
and school environments had not appropriately fostered and engaged Kim’s abilities in math and
science. The combination of two tutors (mother and paid tutor) helped Kim to develop her
interests and talents in both math and science.
The active engagement of Kim’s mother was a very important part of this success story. Jodi
was an excellent role model of a woman quite interested in and engaged in learning and doing
science. This story also illustrates the power of a team engaged in the tutor/tutee process. The
active engagement of all three members of this tutor/tutee team was outstanding.
This story also illustrates another important point. The tutor had a very broad range of
knowledge, skills, and approaches to getting a tutee engaged. The real breakthrough came via
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games and the Science Fair project rather than through the original “contract” on homework
With the help of the paid tutor and her mother, the tutee became a very
good math and science student.
A parent might use a paid tutor without a formal written contract—the “contract” is an oral
agreement or implied by the situation.
A Scenario from Bob Albrecht’s Tutoring
The mother of a 5th-grade student that I tutored at home for an entire school year
said, “I want my son to have fun.” Wow! (I thought). We can do homework for
part of the hour and play games or do experiments for the rest of the hour.
One day we went outside with the goal of measuring the height of tall objects in
the neighborhood such as utility poles, the top of the tutee’s home, trees, et cetera.
From each tall object, we walked and counted a number of steps, and then used an
inclinometer to measure the angle to the top of the object. We drew all this stuff
to scale and used our scale drawings to estimate the heights of the tall objects in
units of the tutee’s step length and my step length—thus getting different values
for the heights. We discussed the desirability of having a standard unit of
measurement, and then did it again using a metric trundle wheel.
This is an excellent example of “play together, learn together.” It shows the value of a
flexible contract and a highly qualified and versatile tutor.
Tutoring is often a component of an Individual Education Program (IEP). The IEP itself is a
contract. However, this does not mean that a tutor helping to implement an IEP is required to
have a written or informal contract or agreement with the tutee. A similar statement holds when a
tutoring company, a paid tutor, or a volunteer tutor works with a tutee outside of the school
Many schools routinely provide tutoring in environments that fall between these two
extremes. The school provides a “Learning Resource Center” that is staffed by paid professionals
(perhaps both certified teachers and classified staff), a variety of adult volunteers, and perhaps
peer tutors who may be receiving academic credit or “service credit” for their work.
A student (a tutee) making use of the services of a school’s Learning Resource Center or
Help Room may have an assigned tutor to engage with on a regularly scheduled basis, or may
seek help from whoever is available. By and large there are some written or perhaps unwritten
rules such as:
1. Tutors and tutees will be respectful of each other and interact in a professional
manner. This professionalism includes both the tutor and the tutee respecting
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the privacy of their communications. This holds true both for the tutoring and
the mentoring aspects of the tutor-tutee communications and other
interactions.
2. In a school setting (such as in a Learning Resource Center or a Help Room)
each of the tutors (whether paid or a volunteer) is under the supervision of the
professional in change of the Center. The tutor is expected to take advantage
of the knowledge and skills of the Center’s director and so seek help when
needed.
3. The tutee has academic learning goals and agrees to use the tutoring
environment to help move toward achieving these goals. Some of these
academic goals may be quite specific and short term and others much broader
and longer term. Some are math content specific and some are learning to be a
responsible student who is making progress toward becoming a responsible
adult. Here are a few examples:
• I need help in getting today’s homework assignment done.
• I want to pass my math course.
• I want to move my C in math up to a B.
• I need to pass the state test that we all have to take next month.
• I need to learn to take responsibility for doing my math homework and turning in it in
on time.
• I want to become a (name a profession). I need to do well in math to get into college
and to get a degree in that area.
• I want to understand the math we are covering in the math class. Right now I get by
through memorization, but I don’t think that is a good approach.
4. The tutor has the academic knowledge, skills, and experience to help the tutee
move toward achieving the tutee’s academic goals. The desirable
qualifications of a tutor are discussed later in this chapter.
Notice the main emphasis in the above list is on academics. But—what about non-academic
goals? A student may be doing poorly academically due to a bad home environment, due to
being bullied, due to poor health, due to identified or not-identified learning disabilities, and for
many other reasons.
Individual paid or volunteer academic tutors should use great care in—
and indeed, are often restricted from—moving outside the realm of the
academic components of tutoring. They are tutors, not counselors.
A school or school district’s counseling and other professional services may well have the
capacity to deal with such problems. However, individual paid volunteer academic tutoring
should use great care in—and indeed, are often restricted from—moving outside the realm of
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academic tutoring. An academic tutor who senses the need for non-academic counseling,
tutoring, or other help should communicate this need to their tutoring supervisor or employer.
A Lesson Plan
A tutor/tutee team has instructional and learning goals. Before a tutoring session begins, the
tutor creates some sort of a plan for the session. If there are to be multiple sessions, the tutor
creates some sort of unit plan or multiple unit plans.
These types of plans can be quite detailed or quite sketchy, such as a few quickly scribbled
notes. Good tutoring often requires extreme flexibility in adjusting to situations that arise and in
being able to “seize the moment.”
Here is a very rough outline for an individual session lesson plan:
1. Begin. Establish social contact with the tutee. Typically this includes friendly,
non-threatening and non-academic conversation relevant to the tutee. Students
can find tutoring sessions to be stressful. If a tutee seems overly tense and
stressed out, work to reduce the tension and stress levels. Some tutors find that
a little light humor helps. Others find it helps to talk about non-academic
topics of mutual interest.
2. Phase into academics. This might begin with a question such as, “How has
school been going for you since our last meeting?” The question can be more
specific. For example, if the previous tutoring session focused on getting
ready for a math test, the question might be, “Last time we helped you prepare
for a math test. How did the test go for you?” If getting better at doing and
turning in homework is one of the major tutoring goals, the tutor might ask for
specifics on how the tutee did on this since the previous session. The goal is to
move the conversation into academics and gives the tutor a chance to pick up
on possible pressing problems.
3. Session goals. Remind the tutee of the very general goal or goals of the
tutoring sessions. Ask if there are specific other topics the tutee would like to
address during the session. In 1-2, both tutor and tutee get an opportunity to
practice active listening and focusing their attention on the tasks at hand. This
component of the tutoring session can end with a brief summary of the
session’s specific goals and tasks. Notice that the tutor may need to make
major adjustments in the predetermined lesson plan.
4. Content-specific tutoring. This might be broken into several relatively self-
contained activities of length consistent both with good teaching/learning
practices and with the attention span of the tutee. A 30-minute block of time
might be broken into two or three pieces of intense effort, with a “breather”
between pieces. (A breather might be quite short, such as 30 seconds or a
minute. It can be a short pause to make a small change in direction. It might
be asking the question, “How are we doing so far in this session.”) Part of the
breather time might be spent on talking about the value and/or uses of the
content being explored, with an emphasis on transfer of learning.
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5. Wrap up (debrief) and closure. This might include asking the tutee “How do
you think this session went?” Get the tutee actively involved in self-
assessment and tutoring session assessment. The tutor provides a summary of
what has been done during the session, makes suggestions of what the tutee
might do before the next tutoring session, and suggests a possible plan for the
next session.
6. Tutor’s personal debrief. Soon after the session ends, make some case notes
about what was covered, what went well, what could have gone better, and
suggestions to oneself for the next tutorial session.
Qualifications of Tutor/Tutee Team Members
Suppose that a tutor/tutee team consists of a human tutor, a computer, and a tutee. There are
expectations or qualifications that one might expect for each of these team members. A later
chapter will discuss computerized tutoring systems. This section discusses the human members
of a tutor/tutee team.
This section mainly applies to tutoring being done by adults. More detail about peer tutoring
is given in the chapter on that topic.
Qualifications of a Tutee
A tutee is a person. A tutee has physical and mental strengths, weaknesses, interests, and
disinterests. A tutee has a steadily growing collection of life experiences and learning
A tutee knows a great deal about him or her self. This self-knowledge and insight covers
areas such as: friends and social life; interests and disinterests; academic and non-academic
capabilities and limitations; current knowledge and skills; current and longer-range goals; home,
school, and community life; and so on.
A tutee is a human being who is facing and attempting to deal with a
host of life’s problems—both in school and outside of school—and
including having learning problems.
Generally speaking, a tutee is in a math tutoring situation in order to facilitate more, better,
and faster learning of math. Think about a typical third grade class. The math knowledge and
skills of students in the class will likely range from 1st grade (or below) to 5th grade (or above).
Students at the lower end of this scale may be learning math at one-half the rate of average math
students. Students at the other end of the scale may be learning math at twice the rate of average
math students.
Students at the lower end of the scale may receive math tutoring that is designed to help them
move toward catching up with the mid-range students, or at least to not fall still further behind.
Students at the upper end of the scale may receive math tutoring designed to help them continue
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to rapidly develop their math knowledge and skills —and to keep them from being “bored” in the
math components of their education.
Schools throughout the country vary widely in the special services they make available to
talented and gifted students. In situations where schools do little, parents may well provide
special instruction to their TAG children and/or hire others to do so. As a personal example,
Dave (one of your authors) is deeply involved in helping teachers learn to make use of
calculators and computers in math education. His older daughter showed interest in learning
about computers when she was quite young. Through Dave’s help, she became a skilled
computer programmer and computer gamer well before she finished elementary school. She has
gone on to a very successful career as a computer programmer and gamer. Bob (your other
author) can tell similar stories about his son who showed an early interest in computers.
However, the typical student a math tutor encounters tends to be struggling in our math
education system. An in-school tutoring arrangement might begin with an intake interview
conducted by a professional in the school’s Learning Resource Center. In this interview a
potential tutee might make statements and/or ask questions such as the following:
• I just can’t do math.
• I hate math.
• Math scares me.
• The stuff we do in math class is not relevant to my life. Why do we have to learn this
stuff?
• The math teacher makes me feel dumb and picks on me.
• Math is boring.
• I’ve got better things to do in life than to waste time doing homework.
• My parents get along fine in life, and they don’t know how to do this stuff.
After tutoring sessions begin, the tutee may express similar sentiments to the tutor.
Experienced math tutors have had considerable practice in dealing with such situations.
Qualifications of a Tutor
Tutors range from beginners, such as students learning to do peer tutoring and parents
learning to help their children with homework, to paid professionals with many years of
experience and a high level of education. Thus, it is important that the expectations placed on a
tutor should be consistent with the tutors knowledge, skills, and experience.
Tutor qualification areas: math content knowledge, math pedagogical
knowledge, math standards knowledge, communication skills, empathy,
and learning in areas relevant to math education.
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This section is targeted mainly to desirable qualifications of professional-level math tutors,
whether they be paid or volunteers. (A parent, volunteer, or peer tutor can be very successful
without having this full set of qualifications.)
Here are nine qualification areas:
1. Math content knowledge. Be competent over a wide range of math content
below, at, and higher than the content being tutored. Have good math problem
solving knowledge and skills over the range of his or her math content
knowledge.
2. Math maturity. Have considerably greater math understanding and math
maturity than the tutee.
3. Math pedagogical knowledge. Know the theory and practice of teaching and
learning math below, at, and somewhat above the level at which one is
tutoring. This includes an understanding of cognitive development and various
learning theories, especially some that are quite relevant to teaching and
learning math.
4. Standards. Know the school, district, and state math standards below, at, and
somewhat above the level at which one is tutoring.
5. Communication. This includes areas such as: a) being able to “reach out and
make appropriate contact with” a tutee; and b) being able to develop a
personal, mutually trusting, human-to-human relationship with a tutee.
6. Empathy. Knowledge of “the human condition” of being a human student
with life in and outside of school, facing the trials and tribulations of living in
his or her culture, the school and community cultures, and in our society.
7. Learning. A math tutor needs to be a learner in a variety of areas relevant to
math education. Information and Communication Technology (ICT) is such
an area. An introductory knowledge of brain science (cognitive neuroscience)
and the effects of stress on learning are both important to being a well-
qualified tutor (Moursund and Sylwester, October 2010; Moursund and
Sylwester, April-June 2011).
8. Diversity. A math tutor needs to be comfortable in working with students of
different backgrounds, cultures, race, creed, and so on. In addition, a math
tutor needs to be able to work with students with dual or multiple learning-
related exceptionalities, such as ADHD students who are cognitively gifted.
9. Uniqueness (Signature Traits). A math tutor is a unique human being with
tutoring-related characteristics that distinguish him or her from other math
tutors. As an example, Bob Albrecht (one of the authors of this book) is
known for his wide interest in games, use of math manipulatives, use of
calculators, and broad range of life experiences. He integrates all of these into
his work with a student.
Tutoring Tips, Ideas, and Suggestions: Fun with Numbers
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Math contains a large number of “fun” but challenging activities and challenges for students.
A math tutor can have a repertoire of such activities and draw an appropriate one out of the bag
when time and the situation seem right. Here is an example.
Positive Integers Divisible by 3
We know that some positive integers are exactly divisible by the number 3 and others are
not. The number 7,341 is an example of 4-digit number divisible by 3:
7341/3 = 2447
Now, Let’s form other 4-digit numbers from the four digits 7, 3, 4, and 1. Examples include
3741, 1437, 4137, and so on. It turns out that each of these is exactly divisible by 3.
3741/3 = 1247 1347/3 = 449 4137/3 = 1379
Interesting. Perhaps we have found a pattern. Try some other 4-digit numbers formed from
the digits 7, 3, 4, and 1. It turns out that each of the 4-digit numbers you form will be evenly
divisible by 3. [It also works for 2-digit numbers, 3-digit numbers, et cetera.]
Here are some “junior mathematician” questions:
1. How many different 4-digit numbers can one make from the digits 7, 3, 4, 1? This question is
relevant because we may want to test every one of them to see if it is divisible by 3.
Note to tutors: Use a 3-digit version of this question for tutees you feel will be
overwhelmed by the 4-digit version. Your goal is to introduce the idea of careful
counting and a situation in which your tutee can experience success.
2. Are there other 4-digit numbers that are divisible by 3 and such that any number formed from
these four digits is divisible by 3? This question is relevant as we work to find then the
divisibility conjecture might be true. Some exploration will lead you to a conjecture that this
“divisible by 3” pattern works on the variety of 4-digit numbers that you try. Of course, that
does not prove that it works for all 4-digit numbers that are divisive by 3. How many
different 4-digit numbers are there that are divisible by 3? Is it feasible for a person to list all
of these and then test for each one all of the 4-digit numbers that can be made from the
digits? (A computer could complete this task in a small fraction of a second.)
3. Does the divisible by 3 property we have explored for 4-digit numbers also hold for 2-digit
numbers, 3-digit numbers, 5-digit numbers, and so on? Some trials might well lead you to
conjecture that the answer is “yes.” But now, we have a situation in which an exhaustive test
of all possible numbers is not possible. What is needed next is a “mathematical proof” that
the conjecture is correct, or finding an example for which the conjecture is not correct.
4. Explore the following conjectures:
4a. If the sum of the digits in a positive integer is divisible by 3, then the
integer is divisible by 3.
4b. If a positive integer is divisible by 3, then the sum of its digits is
divisible by 3.
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Final Remarks
Being a tutor or a tutee is being a member of a teaching and learning team. A team is guided
(indeed, driven) by goals that are mutually acceptable to the team members. Success depends on
the various team members being committed and actively involved. It also depends of the team
members being qualified to effectively participate in achieving the goals.
Through education, training, and practice, all team members can get better in fulfilling their
particular roles. Effective tutoring over an extended period of time needs to include a strong
focus on the human and humane aspects of the process—on the humans communicating with
each other and working together to accomplish the agreed upon goals.
Self-Assessment and Group Discussions
This book is designed for self-study, for use in workshops, and for use in courses. Each
chapter ends with a small number of questions designed to “tickle your mind” and promote
discussion. The discussion can be you talking to yourself, a discussion with other tutors, or a
discussion among small groups of people in a workshop or course.
1. Name one idea discussed in the chapter that seems particularly relevant and
interesting to you. Explain why the idea seems important to you.
2. Read through the list of nine tutor-qualification areas. If you like, make
additions to the list. In the original or expanded list what are your greatest
strengths? What are your relative weaknesses? What are you doing to improve
yourself in your areas of relative weakness? One of the ideas that David
Perkins stresses in his book about Whole Games (Perkins, 2010) is
identification of one’s weaknesses and spending much of one’s study and
practice time on these weaknesses.
3. In your initial conversation with a new math tutee, the tutee says: “I am not
good at math and I hate math.” How would you deal with this situation?
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Chapter 4
Some Learning Theories
"Give a man a fish and you feed him for a day. Teach a man to fish and
you feed him for a lifetime." (Chinese Proverb.)
"They know enough who know how to learn." (Henry B. Adams;
American novelist, journalist, and historian; 1838–1918.)
A human brain is naturally curious. It is designed to be good at learning making effective use
of what it learns.
People vary considerably in terms of what they are interested in learning, how rapidly they
learn, how deeply they learn, and how well they can make use of what they learn. There has been
substantial research on similarities and differences among learners. A variety of learning theories
have been developed. These help to guide teaching and learning processes and the development
of more effective schools and other learning environments.
This chapter provides a brief introduction to a few learning theories. As an example,
constructivism is a learning theory based on the idea that a brain develops new knowledge and
skills by building on its current knowledge and skills. This theory is particularly important in a
vertically designed curriculum such as math. Weaknesses in a student’s prerequisite knowledge
and skills can make it quite difficult and sometimes impossible for a student to succeed in
learning a new math topic.
Tutoring Scenario
One of my first tutoring gigs was tutoring two 8th-grade girls in algebra. The
three of us met twice a week for the entire school year in the home of one of the
girls.
For the first few weeks, we spent our hour doing the assigned homework. The
tutees did not do the assignment prior to my visit, but waited until I arrived. Then
we slogged through the assignment together.
One day we finished early, so I asked, "Want to play a game?" They said, "OK."
We played Pig [described in Chapter 5] for the rest of the hour, and I stayed on
for a while afterwards because they were having so much fun.
Before I left, I said, "Hey, if you do your homework before I arrive, we can go
over it, and then play games. I have lots of games."
From that day on, they did their homework before I arrived and we went over it.
Because we were not pressed for time, we could delve more deeply into what the
girls were learning and/or could be learning in doing the homework assignment
problems. We always finished with ample time to play a fun math game.
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