Math Tutoring



Becoming a Better Math Tutor

This IAE-pedia entry contains the first Table of Contents, Preface, first two chapters, Appendix 1, and Appendix 2 of the free 145 page book:


 * Moursund, David and Albrecht, Robert  (9/2/2011). Becoming a better math tutor. Eugene, OR: Information Age Education.

A PDF of this book is available for free download at http://i-a-e.org/downloads/doc_download/208-becoming-a-better-math-tutor.html.

A Microsoft Word file of this book is available at  http://i-a-e.org/downloads/free-ebooks-by-bob-albrecht/220-becoming-a-better-math-tutor-2.html.



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 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.

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.

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            Normal.dotm  0  0  1  9  54  University of Oregon  1  1  66  12.0         0  false      18 pt  18 pt  0  0    false  false  false                           moursund@uoregon.edu.

Robert Albrecht
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 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 starshipgaia1@msn.com.

Table of Contents
Preface

Chapter 1: Some Foundational Information

Chapter 2: Introduction to Tutoring

Chapter 3: Tutoring Teams, Goals, and Contracts

Chapter 4: Some Learning Theories

Chapter 5: Uses of Games, Puzzles, and Other Fun Activities

Chapter 6: Human + Computer Team to Help Build Expertise

Chapter 7: Tutoring for Increased Math Maturity

Chapter 8: Math Habits of Mind

Chapter 9: Tutoring “to the Test”

Chapter 10: Peer Tutoring

Chapter 11: Additional Resources and Final Remarks

Appendix 1: Advice to Tutees

Appendix 2: Things Parents Should Know About Tutoring

References

Index

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. XE &quot;math maturity&quot;  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 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

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.
 * 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 XE &quot;habits of mind&quot;, 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 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.


 * 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.

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.


 * 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.


 * 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 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 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 XE &quot;games:Monopoly&quot; XE &quot;Monopoly&quot;  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/. [Note to readers: The figure containing a picture of Monopoly board  is not included in this IAE-pedia version of the chapter.]

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 XE &quot;play together, learn together&quot; .”

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.


 * 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 researchers.

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 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, XE &quot;Wong, Mike&quot; 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.

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.

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 http://www.youtube.com/watch?v=fhpOoj7VqcE.

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 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 dyslexia.


 * 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 asking.

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.

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 receives.

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 and 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.).

Toys
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?

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. XE &quot;games:Geoboard-based games&quot; 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. One of the figures is shaped like the letter W and the other is shaped like the letter X. The figure is not shown in this IAE-pedia version of the chapter.

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 geoboard.

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 engagement.

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-as-Tutor
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) 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 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 http://iae-pedia.org/Math_Education_Digital_Filing_Cabinet.

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 number.

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-pedia.org/IAE_Kindle_Books.


 * 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 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 physics?

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 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?

Note for Tutors
An article for tutees starts on the next page. It has a Flesch-Kincaid reading level of about grade 5.5. It is printed using a slightly larger type size than the rest of the book.

Some of your tutees will be able to read and understand the content. Others will lack this level of knowledge and skill to read by learning. So, here is a three-part suggestion:


 * 1) If you are working with a tutee you believe can benefit by reading this document, provide a copy to the tutee and suggest (encourage) the tutee to read it. You might, for example, provide a tutee with a copy during the first tutoring session. Explain that you want the tutee to read a certain part of the document before the next tutoring session and to bring it to the session. Suggest that the tutee mark places in the document that require further explanation. This will give you a chance to find out if the tutee will take the personal responsibility of reading the document. Whether the tutee follows through or not, the next tutoring session can spend some time discussing ideas in the document.
 * 2) If you are working with a tutee with reading skills below the level this document requires, then spend some of the tutoring time helping your tutee learn the most important ideas in the document. Come back to these ideas from time to time—indeed, some could be revisited in every tutoring session.
 * 3) Provide the tutee’s parents or other adult caregivers with a copy of the article.

'''A handout for tutees begins on the next page. Remember, make use of this handout only if you are confident that your tutee has an adequate level of reading ability'''. Your goal in the tutoring sessions is to help your tutee gain in math content and math maturity knowledge and skills. However, be aware that a tutee may be having math difficulties partly because of reading difficulties.

Avice to a Math Tutee

 * “Help me, Obi Wan Kenobi.” Princess Leia from a Star Wars episode.

A tutor is a special type of teacher. A tutor often works with just one student at a time. This makes it possible for lots of interaction between the teacher and student. Such interaction helps a student to learn faster and better.

A tutor’s student is called a tutee. This article will help you (a tutee) learn to work with a tutor. Being a tutee is a lot different then being a student in a large class. A tutee can ask questions at any time during a tutoring session. A tutor can provide individual help when a tutee gets stuck. A tutor and a tutee are a team, working together to help a tutee learn.

Some Important Ideas
You know that school helps students learn reading, writing, and math. These are all hard subjects. It takes a lot of brainpower to learn to do reading, writing, and math. If you can read this article, it proves that you have the needed brainpower!

Some people learn faster than others. Some people are good at learning in large classes. Some learn best when they receive one-on-one help.

This article is about being a math tutee. Here are three really important goals to hold in mind:


 * 1) Your goals as a math tutee are to learn some math and to get better at learning math.
 * 2) To learn math means to learn how to do math. It means learning to use math to help solve problems in everyday life and in all areas that you study.
 * 3) Math is a language. You can learn to read, write, speak, listen, and think in the language of math. You can learn to understand math. All of these activities are part of math maturity. Math tutoring will help you to increase your level of math maturity.

Why Learn Math?
Math is required in schools. Have you ever wondered why? After all, there are many other things that students could spend their time learning.

What is math? Think about this question. Can you give several different answers? Are your answers the same as your friends would give? Your math tutor will help you explore different answers to the “What is math?” question.

Teachers and parents give a number of different answers as to why math is required in school. Mainly, they say you need it because it is useful. They say things like:

1. You need it in your life outside of school. You use math in dealing with money. You use math to measure and think about time, distance, speed, length, area, weight, and so on. You use math in sports and in many computer games.

2. Math is important in many different school subjects. For example, math is used in all of the sciences. A type of math called statistics is useful in all subjects that gather data and work to figure out what the data means. Graphs are often used to help explain such data.

3. You will need to use math as you do adult things like run a household, hold a job, and be a responsible citizen. You need math to understand using credit cards, borrowing money, making interest payments, and saving for retirement.

There are other possible answers. For example:

4. Math is an important part of human history and achievement. Thousands of people helped develop the math you study in school.

5. Many people have found that math is fun. They enjoy playing games and solving puzzles that involve math.

6. Mathematicians and many other people think of math as having beauty, somewhat like there is beauty in art, dance, and music. Think about the beauty in the math types of patterns of nature found in flowers, sunsets, rainbows, and clouds.

A math tutor can help you explore possible answers to the “why do I need to learn math” question. Some students are satisfied with answers such as “because the teacher and my parents tell me I have to learn it.”

However, life is much more than just doing what adults tell you to do. Part of growing up is learning to provide answers for yourself. You, personally, need to find answers that are meaningful to you. Think about how math is part of your life now, and how it will always be part of your life.

Your Goals as a Tutee
In your first tutoring session, your tutor may ask questions such as:


 * “Why are you here?”


 * “What is math?”


 * “What do you want to get out of these tutoring sessions?”


 * “What can I (your tutor) do to make these sessions useful to you?”

Your tutor is looking for answers that will help him or her to best serve you. Of course, how you respond is up to you. Here is a type of answer to the first question that is not helpful.


 * “I am here because this is third period. I am scheduled for tutoring during third period.”

That is a smart aleck type of response. It disrespects your tutor.

Here are some more useful types of answer.


 * "I need help on my math assignments. I get stuck on some problems and the teacher does not have time to answer my questions."


 * “I need to do better on the state math test.”


 * “I am way behind other kids in the math class. I need help catching up.”


 * “I need to pass the math course in order to graduate.”


 * “I want to raise my math grade to a B (or, to an A).

Notice that each of these better responses states a goal. It is possible to measure progress in these types of goals. Both you and your tutor can tell if you are getting your math assignments done and turned in on time. You and your tutor can tell if you are getting better at answering the types of questions used on state math tests. You and your tutor can tell if you are catching up with the rest of the class.

Item 6 in the previous section said that there is fun and beauty in math. Thus, you might have goals such as:


 * “I want to learn some fun math things.”


 * “I want to learn to see the beauty in math.”


 * “I want to learn how math has helped change history.”

In summary, your tutoring sessions should be goal directed. You can help set the goals. You can help change the tutor’s goals so they better fit your needs and interests.

Your Tutor’s Goals
Your tutor will have goals. These can be divided into three categories:


 * 1) Goals related to meeting your needs.
 * 2) Goals related to being a professional, successful tutor. If a tutor is being paid, the tutor must meet the needs of the supervisor or employee. A volunteer tutor also needs to perform in a professional manner. Both paid and volunteer tutors need to gain personal satisfaction in their work.
 * 3) Goals related to creating a professional, mutually respectful learning environment. Both the tutor and the tutee need to focus their attention. They need to be on task.

Being a Responsible, Attentive Student
Notice the third of these goals. You need to pay attention and to be on task. Your tutor needs to help you pay attention and to help you stay on task. Many students find that it is hard to stay on task when they are learning math. You can get better at this through practice!

Tutoring can help you learn to become a more responsible student. A responsible student puts energy into learning, both in class and outside of class. A responsible student does required reading and written assignments. A responsible student turns in homework on time. A responsible student comes to class and tutoring sessions with the needed tools (such as pencil, paper, and books).

A responsible student pays attention in class and while being tutored. You can learn to detect when you are not paying attention–when your mind is wandering from the learning task. With some instruction and practice you can learn to focus your attention on a learning task.

Notice that the word math is not used in the two paragraphs given above. Learning to be a responsible and attentive student is useful in studying any subject area. It is also applicable to the jobs and other tasks you will undertake in the future.

Tutoring Sessions are Different Than a Graded Class
Your tutor does not give you grades! Your tutor does not decide whether you pass or fail a subject. Being a tutor is quite different than being a regular teacher.

The math you are learning now builds on the math that you covered in the past. As a math student, you face two major challenges:


 * 1) You might not have learned some of the topics the teacher assumes you know. There are lots of reasons for this. For example, you may have changed schools, and your previous teachers did not cover the topic. You may have missed school because you were sick. The teacher may have covered a topic but you did not understand it.
 * 2) You have forgotten some math you learned in the past. You have probably heard the expressions, “Use it or lose it.” It is easy to forget math that is not part of your everyday life.

You and your tutor working together can figure out what you have forgotten or never learned. Your tutor can help you learn topics that you did not have a chance to learn. Your tutor can help you relearn topics that you have forgotten. It is important to gain skills in relearning topics you have forgotten.

Computer as Tutor
A computer’s brain is a lot different than a person’s brain. A computer can have a certain type of smartness. It is called artificial intelligence or machine intelligence. A computer can do some things much better than a person.

Computers can do certain types of tutoring. For example, a computer tutor works well in drill and practice situations. The computer tutor asks a question, accepts an answer, and provides feedback on whether the answer is correct. A drill and practice program of this sort can help a tutee gain speed and accuracy in math facts and mental arithmetic.

The computer tutor can keep records on what you know well and what you are struggling with. It can present review questions to help you maintain your skills. It can keep records that show your progress from week to week.

In a math class the teacher helps you cover material in a textbook. The teacher gives explanations and provides examples. Now, think about a computer version of the book and what the teacher does. A computer can use video to explain a topic and give examples. A computer can check for understanding by asking you questions and immediately processing your answers.

In addition, such a computer system allows you to move at your own pace. It is easy to stop and look at a video again. You can listen to an explanation as many times as you want. You can take a sample quiz as many times as you want. You can be in control!

Many companies are working to make better computer tutors. This is an important part of the future of math education. But, a computer is not a human being. There are many things that a human tutor can do that a computer tutor cannot do. A human tutor and a computer tutor working together can be a powerful aid to learning. As you talk to your tutor, find out whether you will get a chance to work both with your human tutor and with a computer tutor.

There is quite a bit of free tutorial software available on the Web. Here are a few examples.


 * A variety of lessons at http://www.math.com/students/practice.html.


 * AAA Math arithmetic lessons at http://aaamath.com/. See the long list of topics in the menu on the left of the Website.


 * Arithmetic quiz at http://www.thatquiz.org/tq-1/.


 * Kahn Academy (n.d.) Watch. Practice. Learn almost anything—for free. Retrieved 6/14/2011 from http://www.khanacademy.org/.

You can search the Web for more free tutoring sites. You need to tell the search engine your particular interests. A search on arithmetic practice will lead you to a huge number of arithmetic tutor sites. A search of algebra practice will provide you with lots of algebra help. You might want to have your tutor provide you some help in learning to do such Web searches.

Final Remarks
Tutoring is a valuable learning experience. With the help of a tutor you can learn faster and better than you do in a class with many other students. That is because you can get immediate help on any difficulties you are having. You can talk over what is going well and what is going poorly.

Sometimes a student gets to have just a few tutoring sessions, and there are quite specific goals. For example, 10 tutoring sessions may help prepare you for a state test. Tutoring twice a week for a full year can add greatly to your math knowledge and maturity.

We encourage you to share this article with your parents and others. Spend time talking with them about your tutoring experiences.

Note for Tutors
An article for parents, grandparents, and other adult caregiver starts on the next page. It has a Flesch-Kincaid reading level is about grade 8 or 9. It is printed using a slightly larger type size than the rest of the book.

It is useful to think of a tutoring team as having a number of potential members (components):


 * 1) The tutee. Tutoring centers about a tutee, and the overall goal is to help improve the education of the tutee.
 * 2) The “lead” tutor. This may be a paid professional, a volunteer, a same-age peer tutor, or a cross-age peer tutor.
 * 3) Parent, grandparents, guardians, and/or other responsible adult. They may help provide both informal and formal tutoring.
 * 4) The overall environment and in which the tutee lives, and people within the environment that have routine contact with the tutee. This include siblings, close friends, school counselors, personnel in religious institutions, and so on.
 * 5) Computers, audio and video materials, edutainment, and other aids to learning.

You are familiar with the often-quoted statement, “You can lead a horse to water, but you can’t make it drink.” Similarly, we can provide a tutee with many and varied aids to learning, but we cannot make a tutee take appropriate advantage of these opportunities. One of the challenges of being a tutor is that there are so many different “players” in the game. The tutor often has little control over what the various players do.

This particular Appendix focuses on the adult caregivers in a tutee’s life. It may well be that the only contact a tutor has with these people is through the tutee. (Of course, if you are doing the tutoring in a tutee’s home, that is a different matter.) If it seems appropriate to you, consider providing your tutees with a copy of the document that starts on the next page, and request that it be given to the tutee’s parents and/or other appropriate people. (Feel free to rewrite the document to better fit the situation.) The next time you meet with your tutee, you can inquire as to whether the document was delivered and whether there was any reaction to it. From time to time you can talk to your tutee about his or her home environment as it relates to the tutoring going on and what your tutee is learning.

Advice to Parents of a Child Who Is Being Tutored in Math

 * The document is from Appendix 2 of the book:


 * Moursund, David and Albrecht, Robert (2011). Becoming  a better math tutor. Eugene, OR: Information Age Education.

A tutor is a special type of teacher who works with one student or a very small number of students at a time. A student who is being tutored is called a tutee.

This article is written for parents and others who are responsible for the day-to-day care of a child who is being tutored in math. We will use the expression “your child” even though you might be a grandparent, foster parent or other caregiver. The article will give you some insights into:


 * 1) What goes on in a tutoring session?
 * 2) How tutoring helps a tutee.
 * 3) What you can do to help your child benefit from the tutoring.

What is Math?
Many people think of math just in terms of arithmetic. However, today’s students also learn about algebra, geometry, probability, and statistics. The emphasis is on thinking, understanding, and problem solving.

While rote memory is important, the thinking and understanding are more important. We want students to be able to use their math knowledge and skills in everyday life. We want them to recognize when math can help solve a math-related problems such as shopping, borrowing money, and planning for one’s future. We want them to understand the graphs and charts used in newspapers and other publications.

At the current time, school math focuses on preparing for tests. However, this is a very narrow-minded approach to math education. Outside of the math classroom, math does not consist of True/False and Multiple Choice Tests!

Tutoring can help a tutee on the routine drill and practice types of homework. But, a good tutor does much more. A good tutor helps a tutee to understand math and to get better at solving challenging math problems. A good tutor helps a tutee develop habits of mind that are needed to do well in math.

The Bottom Line
Tutoring works! With the help of a tutor, a student can learn faster and better.

Your first thought might be that tutoring is a simple thing. A tutor and tutee get together regularly. The tutor helps the tutee on homework ad on getting ready for tests.

However, there is much more than this. For example, how can you tell if your child has a well-qualified tutor? How can you tell if tutoring is helping your child?

You are an important member of your child’s tutoring team. In the “big picture,” it is helpful to think of five parts of a tutoring team:


 * 1) The tutee. Tutoring centers about a tutee, and the overall goal is to help improve the education of the tutee.
 * 2) The “lead” tutor. This will likely be a paid professional, or a volunteer. If the tutoring takes place in a school, the tutor will have a supervisor.
 * 3) Parents and/or other responsible adults. They may help provide both informal and formal tutoring. In addition, they help to create a home environment that is conducive to learning.
 * 4) The tutee’s environment. This includes the overall environment and in which the tutee lives and the people in it. It includes your child’s home environment, siblings, close friends, school counselors, personnel in religious institutions, and so on.
 * 5) Technology. This includes computers, audio and video materials, books, and other aids to learning. It also includes entertainment television, music, and computer games.

Think about your roles. You are both a team member and you help to create the environment in which your child learns. A later section of this document will provide you with some advice.

Who Needs and Who Gets Tutoring
If we use a very broad definition of tutoring, then infants receive a lot of tutoring. In a normal home environment they receive lots of personal attention from caregivers. They learn their native language(s) through one-on-one help. They learn about their culture by being immersed in the culture and through individualized feedback from many different people in the culture.

Moreover, reading to your children, playing games with your children, working with your children on learning sports, and so on are all examples of tutoring. In some sense, whatever a parent is “into,” such as music, reading, athletics, and so on gets communicated to their children.

However, let’s talk about tutoring that is a supplement to schoolwork or a major component of schooling. Each child is unique. Our school system groups students into classes of perhaps 20 to 30 or more students. A teacher of such a large group of students cannot know the individual needs of each student. The teacher does not have the time to identify the specific strengths and weaknesses of each individual student. The teacher is required to cover the curriculum specified by the school, school district, and state.

New learning is built on what a child already knows. Often a student has not gained the prerequisite knowledge and skills needed to successfully learn a new topic. Instruction on the new topic is so far over a student’s head that the student is confused, easily discouraged, and unable to keep up with the class.

There are many reasons why a student is not well prepared to learn a particular new topic. The student may have missed school when required previous knowledge and skills were being covered. The child may have changed schools, and the previous school may not have covered the topics.

The child may have specific learning challenges that can slow down progress in school. Here are a few examples:


 * Vision, hearing, general health, nutrition, and other problems that have not been identified and/or adequately addressed.


 * Dyscalculia is a particular type of math-learning challenge. Students with dyscalculia have difficulty learning to do arithmetic.


 * Dyslexia, a particular type of reading problem. Many dyslexic students have trouble in learning math. This is partly because math is a language. Learning and doing math requires reading and writing in the language of math.


 * ADD (Attention Deficit Disorder) or ADHD (Attention Deficit Hyperactive Disorder). This makes it difficult for the student to pay attention to (concentrate on) a topic being taught.

The list can be expanded. What is important for you to know is that a great many students can benefit from individualized help in school. A tutor can help a student to identify learning gaps and to address them. A tutor can provide individualized feedback to a student in a timely and personalized manner.

Tutoring is not only for students having learning difficulties. Students who are mentally or physically gifted often receive tutoring or special small-group instruction. This is common in the performing arts—for example, for a student taking music lessons.

Tutoring and/or very small classes are sometimes made available to very gifted students. Your school district may have a special program for talented and gifted students. Students who are especially gifted in math can easily learn math at twice the pace of an ordinary math class.

A Highly Qualified Tutor
A tutor is a teacher. As a parent, you will want to know about the qualifications and experience of your child’s math tutor. Here is a list of some possible qualifications that a good math tutor might have:


 * 1. Math content knowledge. Have good math problem solving knowledge and skills over the range of his or her math content knowledge.


 * 2. Math education experience. Have considerably experience in helping students learn math. If your child has particular math-learning challenges, you want a tutor who is experience in dealing with such challenges. In any case, you want a tutor who understands both the theory and practice of teaching and learning math.


 * 3. Math Standards. Know the school, district, and state math standards below, at, and somewhat above the level at which one is tutoring.


 * 4. Communication skills. 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.


 * 5. Empathy. Knowledge of “the human condition” of being a student with a challenging life in and outside of school. A good tutor can help a tutee build self-confidence as a learner.


 * 6. Learning. A math tutor needs to be a learner in a variety of areas relevant to math education. Important areas include computer technology and brain science. Computers are an important tool in tutoring.


 * 7. 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.

Roles of a Parent of a Child Being Tutored at School
Remember that we are using the term “parent” to include grandparents and other regular caregivers.

If your child has an Individual Education Plan (an IEP) at school, then you have a legal right to know details of tutoring your child receives through the IEP.

In any case, parents should know both the purpose of a child being tutored and the outcomes that can be expected. Parents should interact with their children in a manner that helps support the tutoring.

Here is an example. You ask your child, “What did you do in school today.” You are probably used to answers that don’t say much. This does not help you or your child.

Now, change this. You might say, “Hi Pat. Today you worked with your math tutor in school. Tell me about some of the things you and your tutor did.”

Don’t give up if you get a “nothing” type of answer. Ask some specific questions. “Tell me something that you learned. Explain it so that I can understand it.” “Give me an example. Here is a pencil and piece of paper. Show me.” What are some uses of this math? Can you give me an example from what you do outside of school?” Your goal is to engage your child in a conversation about math and learning math.

This type of routine math-related interaction is very good for your child. It can also be very good for you, as it will help you learn what your child is learning. It will give you insight your child’s math learning challenges and successes.

DO NOT spend time telling your child how difficult math was for you! You want to create a positive math-learning environment. Your child can learn math, and you what to help and encourage your child. You want to celebrate the successes your child is experiencing.