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.

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.

1.
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.

1

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.

1

2.
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

2

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

2

3.
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]

3

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]

3

4.
Becoming a Better Math Tutor

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!

4

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!

4

5.
Becoming a Better Math Tutor

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

5

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

5

6.
Becoming a Better Math Tutor

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

6

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

6

7.
Becoming a Better Math Tutor

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.

7

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.

7

8.
Becoming a Better Math Tutor

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

8

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

8

9.
Becoming a Better Math Tutor

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.

9

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.

9

10.
Becoming a Better Math Tutor

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.

10

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.

10

11.
Becoming a Better Math Tutor

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.

11

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.

11

12.
Becoming a Better Math Tutor

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

12

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

12

13.
Becoming a Better Math Tutor

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

13

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

13

14.
Becoming a Better Math Tutor

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.

14

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.

14

15.
Becoming a Better Math Tutor

• 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

15

• 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

15

16.
Becoming a Better Math Tutor

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.

16

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.

16

17.
Becoming a Better Math Tutor

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.

17

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.

17

18.
Becoming a Better Math Tutor

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

18

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

18

19.
Becoming a Better Math Tutor

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.

19

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.

19

20.
Becoming a Better Math Tutor

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

20

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

20

21.
Becoming a Better Math Tutor

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?

21

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?

21

22.
Becoming a Better Math Tutor

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

22

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

22

23.
Becoming a Better Math Tutor

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

23

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

23

24.
Becoming a Better Math Tutor

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

24

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

24

25.
Becoming a Better Math Tutor

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

25

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

25

26.
Becoming a Better Math Tutor

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?

26

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?

26

27.
Becoming a Better Math Tutor

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

27

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

27

28.
Becoming a Better Math Tutor

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

28

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

28

29.
Becoming a Better Math Tutor

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

29

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

29

30.
Becoming a Better Math Tutor

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

30

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

30

31.
Becoming a Better Math Tutor

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.

31

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.

31

32.
Becoming a Better Math Tutor

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

32

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

32

33.
Becoming a Better Math Tutor

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.

33

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.

33

34.
Becoming a Better Math Tutor

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

34

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

34

35.
Becoming a Better Math Tutor

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.

35

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.

35

36.
Becoming a Better Math Tutor

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?

36

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?

36

37.
Becoming a Better Math Tutor

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.

37

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.

37