Improving Math Education

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"The reason most kids don't like school is not that the work is too hard, but that it is utterly boring." (Seymour Papert; South African/American mathematician, computer scientist, and educator; 1928-.)
"If you don't know where you are going, you're likely to end up somewhere else." ” (Lawrence J. Peter; American educator of “Peter's Principles” fame; 1919-1990.)


This document is intended for all people who have an interest in improving our math education system. However, it is specifically targeted toward preservice and inservice teachers, and their teachers. So, when I say "you" in this document, I am talking to teachers, teachers of teachers, parents, and others who are committed to understanding some of the important ideas in this document and taking an active role in improving our math education system.

The message in this document can be summarized by the statement: They, we, and you can improve our math education system.

The task of improving our math education system is so large that it will take they, we, and you working together to make significant progress. There is no "magic bullet" and no single approach that will accomplish what we want to accomplish.

Rest assured that you are not alone in your desire to help improve math education. For example, a 6/19/2014 Google search for improving math education produced more that 43 million hits. The top-ranked hits on Google change from time to time, and many have enduring value. Right at this moment, the top hit is a 2014 article by James Stigler and James Hiebert based on ten years of work they had done on the Trends in International Science and Mathematics Study. The study was sponsored by ASCD, formerly the Association for Supervision and Curriculum Development.

Second on the current list is Annual Perspectives in Mathematics Education 2014: Using Research to Improve Instruction, a book published by the National Council of Teachers of Mathematics.

Third on the list is a U.S. Department of Education document, Improving Math Performance.

My point is that there are many thousands of organizations and individual people working on the task of improving math education in the United States and in other countries. You need to spend time familiarizing yourself with the literature and both the people and the groups working to improve math education. This book will touch on a large number of important ideas

The work has been going on for a long time. Perhaps you have heard of the terms:

A. Reading, writing, and arithmetic. The world's educational systems have a very long history in which schooling focused on rudiments of reading, writing, and arithmetic. Back at the time of the American Revolution, before we had a United States of America, most people living in this region were illiterate and ill numerate.

In 1778, Thomas Jefferson brought before the Virginia Legislature A Bill for the More General Diffusion of Knowledge. His bill (quoted below) was not approved!

At every one of these schools shall be taught reading, writing, and common arithmetick, and the books which shall be used therein for instructing the children to read shall be such as will at the same time make them acquainted with Graecian, Roman, English, and American history. At these schools all the free children, male and female, resident within the respective hundred, shall be intitled to receive tuition gratis, for the term of three years, and as much longer, at their private expence, as their parents, guardians or friends, shall think proper. [Bold added for emphasis.] See

We have come a very long way since then. You may enjoy reading some of this history.

Back when I (David Moursund) was graduating from high school in 1954, math education for all students was about the same through grades 1-8. Ninth graders then took either General Math or Algebra 1. Those who took Algebra 1 then had available Geometry, Algebra II, and some combination of Trigonometry and Solid Geometry. Calculus had not yet made its way into the precollege math curriculum in my home town.

In those "good old days" about 20% of high school graduates went on to college. Sputnik had not yet stirred up U.S. science and math education.

B. New Math. Quoting from the [ Wikipedia article New Math:

New Mathematics or New Math was a brief, dramatic change in the way mathematics was taught in American grade schools, and to a lesser extent in European countries, during the 1960s. The name is commonly given to a set of teaching practices introduced in the U.S. shortly after the Sputnik crisis in order to boost science education and mathematical skill in the population so that the perceived intellectual threat of Soviet engineers, reputedly highly skilled mathematicians, could be met.

New Math was rather short-lived and not particularly successful with the great majority of students. Quoting from the reference mentioned above:

In 1973, Morris Kline published his critical book Why Johnny Can't Add: the Failure of the New Math. It explains the desire to be relevant with mathematics representing something more modern than traditional topics. He says certain advocates of the new topics "ignored completely the fact that mathematics is a cumulative development and that it is practically impossible to learn the newer creations if one does not know the older ones" (p. 17). Furthermore, noting the trend to abstraction in New Math, Kline says "abstraction is not the first stage but the last stage in a mathematical development" (p. 98).

C. Modern (Progressive) Math Education. The next major change effort was to take the "New" out of New Math and produce a more modern and more widely acceptable math curriculum. This too, however, proved to be contentious. Quoting from A Brief History of American K-12 Mathematics Education in the 20th Century:

Mathematics education policies and programs for U.S. public schools have never been more contentious than they were during the decade of the 1990s. The immediate cause of the math wars of the 90s was the introduction and widespread distribution of new [newer] math textbooks with radically diminished content, and a dearth of basic skills. This led to organized parental rebellions and criticisms of the new math curricula by mathematicians and other professionals.
In some respects the education wars of the 1990s have little to distinguish them from earlier periods. There is nothing new about disagreements over the best ways to educate the nation's school children. The periodic waves of education reform from the nation's colleges of education are more similar than they are different. The American education establishment has consistently advocated a progressivist education agenda for the bulk of the 20th century, and the mainstream views of the education community have enjoyed a commanding influence on public schools.2

D. Common Core State Standards Math. The Common Core State Standards initiative was developed by the Council of Chief State School Officers and the National Governors Association Center for Best Practices. Quoting from the initiative:

For years, the academic progress of our nation’s students has been stagnant, and we have lost ground to our international peers. Particularly in subjects such as math, college remediation rates have been high. One root cause has been an uneven patchwork of academic standards that vary from state to state and do not agree on what students should know and be able to do at each grade level.
Recognizing the value and need for consistent learning goals across states, in 2009 the state school chiefs and governors that comprise CCSSO and the NGA Center coordinated a state-led effort to develop the Common Core State Standards. Designed through collaboration among teachers, school chiefs, administrators, and other experts, the standards provide a clear and consistent framework for educators.

Quoting from Common Core State Standards Initiative:

For more than a decade, research studies of mathematics education in high-performing countries have concluded that mathematics education in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country. To deliver on this promise, the mathematics standards are designed to address the problem of a curriculum that is “a mile wide and an inch deep.”
These new standards build on the best of high-quality math standards from states across the country. They also draw on the most important international models for mathematical practice, as well as research and input from numerous sources, including state departments of education, scholars, assessment developers, professional organizations, educators, parents and students, and members of the public.

For more about CCSS, see the free book Common Core State Standards for Education in America (Moursund & Sylwester, March, 2013).

Note that CCSS is not a U.S. federal initiative. However, funding for the project has come from a variety of sources, including the U.S. federal government. The student assessment component of CCSS Math is being carried out by two groups—both funded by the National Science Foundation. See: [ Smarter Balanced Assessment Consortium and Mathematics Assessment Project.

For Reflection and Discussion. This document contains a number of "For Reflection and Discussion" inserts. These are designed for use in a workshop or a course that covers the topic of improving math education. Many readers will find it beneficial to reflect on possible answers to the questions. Here is a question to get you started. "What are several things that you expect to learn by reading this document?" In educationese, such a question is an advance organizer. It is designed to get your mind working on where you are now and where you want to go.
For Reflection and Discussion. Consider the following quote: "Be the change you want to see in the world." (Mahatma Gandhi; Indian scholar and political activist; 1869-1948). Consider what this might mean in terms of what you, personally, can do to improve math education. This quote focuses on the "you" part of the "they, we, and you" that need to work together to improve our math education system.

What Is Math?

"God created the natural numbers; all the rest is the work of man." (Leopold Kronecker; German mathematician; 1823-1891.)
“Mathematics consists of content and know-how. What is know-how in mathematics? The ability to solve problems.” (George Polya; Hungarian and American mathematician; 1887-1985.)

A Little Bit of History of Math

Quoting from the Wikipedia:

Long before the earliest written records, there are drawings that do indicate a knowledge of mathematics and of measurement of time based on the stars. For example, paleontologists have discovered ochre rocks in a cave in South Africa adorned with scratched geometric patterns dating back to c. 70,000 BC.[2] Also prehistoric artifacts discovered in Africa and France, dated between 35,000 BC and 20,000 BC,[3] indicate early attempts to quantify time.[4]

Throughout their history, humans have faced the need to measure and communicate about time, quantity, and distance. The Ishango Bone artifact shown below is perhaps 25,000 years old.


Quoting from

At one end of the Ishango Bone is a piece of quartz for writing, and the bone has a series of notches carved in groups. It was first thought these notches were some kind of tally marks as found to record counts all over the world. However, the Ishango bone appears to be much more than a simple tally. The markings on rows (a) and (b) each add to 60. Row (b) contains the prime numbers between 10 and 20. Row (a) is quite consistent with a numeration system based on 10, since the notches are grouped as 20 + 1, 20 - 1, 10 + 1, and 10 - 1. Finally, row (c) seems to illustrate for the method of duplication (multiplication by 2) used more recently in Egyptian multiplication. Recent studies with microscopes illustrate more markings and it is now understood the bone is also a lunar phase counter. Who but a woman keeping track of her cycles would need a lunar calendar? Were women our first mathematicians?

The picture given below shows Sumerian clay tokens whose use began about 11,000 years ago. Such clay tokens were a predecessor to reading, writing, and mathematics.


Quoting from the document History of Mathematics:

Every culture on earth has developed some mathematics. In some cases, this mathematics has spread from one culture to another. Now there is one predominant international mathematics, and this mathematics has quite a history. It has roots in ancient Egypt and Babylonia, then grew rapidly in ancient Greece. Mathematics written in ancient Greek was translated into Arabic. About the same time some mathematics of India was translated into Arabic. Later some of this mathematics was translated into Latin and became the mathematics of Western Europe. Over a period of several hundred years, it became the mathematics of the world

Mathematics as a formal area of teaching and learning was developed about 5,200 years ago by the Sumerians. They did this at the same time as they developed reading and writing. The development of reading, writing, and formal mathematics allowed the codification of math knowledge, formal instruction in mathematics. It was the start of a steady accumulation of mathematical knowledge.

Math in Our Information Age World

To understand some of the fundamental issues in discussions about improving math education, one needs to have an understanding of what math is and why it is so important to people in our Information Age world. This section provides a brief answer to the question, "What is math?" To get your mind headed in that direction, reread the two quotes at the beginning of this section.

Presumably all readers of this document have studied math for many years, beginning well before they started school, and probably continuing on into at least some college math. So, each of you have your own ideas on what math is.

Perhaps for you, the number line is an important part of math. Perhaps in your mind's eye you can "see" the integers "stretching" from far to the left on the number line, passing through -2, -1, 0, 1, 2 and continuing far to the right on this line. You can "see" fractions, and you know something about other numbers on the line that are called irrational. You know about doing arithmetic on the numbers of the number line. You know about some special categories of numbers, such as odd and even integers, prime numbers, and so on. For you, the number line is a rich source of interesting, fun aspects of math.

When I have my "math educator hat" on, I enjoy asking people to answer the "What is math?" question. I have asked this question of a wide range of children and adults—including teachers. I get a very wide variety of answers, most including some statement about numbers and solving math problems.

Think about how you would answer the question, "What is math?"

  • What would you say to a 2nd grader, versus what would you say to an 8th grader?
  • What would you say to parents and other adults who have children in school?
  • What would you say to a preservice math teacher who is preparing to teach in elementary school, middle school, or high school?

I find it quite a challenge to create useful, honest answers. Let's take the situation of what we want a preservice math teacher to know. Consider whether it could be helpful for this person to memorize Kronecker's statement, "God created the natural numbers; all the rest is the work of man."

I find that even Kronecker's simple statement is an intellectual challenge. Does "natural number" mean "integer"? (No.) Is zero one of the natural numbers? (No.) Is the number zero the same as "nothing"? (No.) Do the words "number" and "numeral" mean the same thing? (No.) Math is a complex discipline that has a language (the language of mathematics) that facilitates very precise communication in and about math.

For Reflection and Discussion. What was your response to my set of four questions that each has "no" as an answer? What was the basis for your agreement or disagreement with each of my answers? Have you developed a "habit of mind" that leads you to question assertions that other make and make use of resources such as the Web to check on the accuracy of such assertions? Do you think students of math (and other disciplines) should blindly accept the assertions of their math teachers and math books, or perhaps they should question these assertions and learn to do library-based research on the history and correctness of the assertions? One of the really important goals in math education is to help students develop such habits of mind.

Many years ago, I memorized the statement, "Algebra is the language of mathematics." I wonder what that really is intended to mean? Recently I thought about math as a language, and I developed a Web page titled Communicating in the Language of Mathematics. There I explore such ideas as learning to read, write, speak, listen, and think in the "language" of mathematics. I enjoy exploring such questions, and then sharing my insights with others.

Many people have attempted to provide simple answers to the question, "What is mathematics?" They face a fundamental difficulty in communicating their answers because it tends to take considerable knowledge of math in order to understand their answers.

An answer is generally developed in order to communicate to a particular group of people and for a specific purpose. This is not a "one size fits all" question and does not have a "one size fits all" answer.

For example, consider the following materials quoted from Principles and Standards for School Mathematics published by the National Council of Teachers of Mathematics (2000):

Attaining the vision laid out in Principles and Standards will not be easy, but the task is critically important. We must provide our students with the best mathematics education possible, one that enables them to fulfill personal ambitions and career goals in an ever changing world.
Principles and Standards for School Mathematics has four major components. First, the Principles for school mathematics reflect basic perspectives on which educators should base decisions that affect school mathematics. These Principles establish a foundation for school mathematics programs by considering the broad issues of equity, curriculum, teaching, learning, assessment, and technology.
Following the Principles, the Standards for school mathematics describe an ambitious and comprehensive set of goals for mathematics instruction. The first five Standards present goals in the mathematical content areas of number and operations, algebra, geometry, measurement, and data analysis and probability. The second five describe goals for the processes of problem solving, reasoning and proof, connections, communication, and representation. Together, the Standards describe the basic skills and understandings that students will need to function effectively in the twenty-first century. [Bold added for emphasis.]

The chances are that the math education you received at the precollege level was based on the above statement from the NCTM. Now, the NCTM is supporting the Common Core State Standards Mathematics Initiative. Fifteen years from now we will be able to see how well this new approach to math education is standing the test of time.

The NCTM focus is specifically on math education. Notice the last line of the quoted material. The NCTM wants students to have a math education that will serve their needs in the 21st century. When you read such a statement, do you think about who will decide what will serve the future needs of students? Who, among us, is such a good futurist that he or she can determine and then effectively teach for these future needs?

Here is a still different approach to discussing what math is. George Polya was a 20th century world-class mathematician and math educator. The Goals of Mathematical Education is a 1969 talk that he gave to a group of elementary school teachers. Quoting from his talk:

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.

In this statement, Polya is talking both about problem solving throughout the field of math, and also about the use of math in solving problems in other disciplines. He is also talking about “the right attitude and to be able to attack all kinds of problems.” This is a statement about math maturity, rather than about knowledge of any specific math content.

Thus, although he does not specifically address the question of "What is math?" Polya tells us what he believes math education should be. Math education should focus on helping students get better at solving math problems.

So, math is a discipline of study in which one learns to solve (math) problems. If the instruction and learning are well done, learning to solve math problems carries over to dealing with a wide range of problems from other disciplines. Indeed, math is important partly because it is so useful in helping to represent and solve problems in many different disciplines.

For Reflection and Discussion. What does the term "math problem" mean to you? Can you give a definition that meets your needs as a teacher, and the needs of the students you teach? Do you and your professional colleagues share a common definition? What are similarities and differences between problems in math versus problems in other disciplines of study?

Math Proofs Lie at the Heart of Mathematics

Many mathematicians agree that the very heart of mathematics lies in mathematical proof. Proofs in Mathematics provides a discussion of this idea. It also provides a number of examples of important mathematical proofs that require relatively little background in math. A number of other interesting, amusing, and challenging examples are provided.

Quoting from the Wikipedia entry Mathematical Proof:

In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. A proof is a logical argument, not an empirical one. That is, one must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture.

The Wikipedia article includes a discussion of a number of different methods of mathematical proof.

Math is Representing and Solving Math Problems

The diagram below captures the essence of many different math problem-solving situations. I like to use this diagram when I talk about math problem solving.


  1. Problem posing and problem recognition to produce a Clearly Defined Problem;
  2. Mathematical modeling;
  3. Using a computational or algorithmic procedure to solve a computational or algorithmic math problem;
  4. Mathematical "unmodeling";
  5. Thinking about the results to see if the Clearly-defined Problem has been solved; and
  6. Thinking about whether the original Problem Situation has been resolved. Steps 5 and 6 also involve thinking about related problems and problem situations that one might want to address or that are created by the process of attempting to solve the original Clearly-Defined Problem or resolve the original Problem Situation.

In steps 1 and 2 a person works to understand a problem situation and makes a decision as to whether it might be useful to attempt to solve the problem using math. A person deciding to take a math-oriented approach to resolving the problem situation attempts to represent or model the problem situation using the language of mathematics. This math modeling leads to having a math problem that may or may not be solvable, and that may or may not be solvable by the person attempting to solve the problem.

In step 6, the person who has a solution to the math problem extracted when dealing with step 1 checks the degree to which the results achieved are relevant to the original problem situation and decides whether the overall process has been useful in trying to resolve the original problem situation.

The great majority of K-14 math education is focused on students learning to do step 3 using paper and pencil algorithms. Step 3 is what calculators and computers are best at. Thus, the great majority of math education at the K-14 levels is spent helping students learn to compete with calculators and computers in areas that are not well suited to the capabilities of a human mind but that are well suited to computers.

For Reflection and Discussion. Carefully examine your own personal position on helping students learn to use calculators and computers to do step 3 in the math problem-solving diagram above. In this assessment, think about what gaining speed and accuracy in by-hand (pencil and paper) in doing step 3 contributes to a student's understanding of math and of math problem solving. Give arguments for and against allowing students to use calculators and computers in doing math homework, on math in-class seat work, and on math tests.

Students vary tremendously in how long it takes them to gain speed and accuracy in using pencil and paper to do the types of calculations and procedures that calculators and computers can do much more rapidly. A fundamental issue in math education is how much emphasis should be placed on having students develop such speed and accuracy. This is part of what the Math Education Wars are about.

Here is still another approach that can be used as one explores the "What is math?" question. Math is a very old, deep, and broad academic discipline and human endeavor. Each academic discipline can be defined by a combination of general things such as:

  • The types of problems, tasks, and activities it addresses.
  • Its accumulated accomplishments, including results, achievements, products, performances, scope, power, uses, impact on the societies of the world, and so on.
  • Its history, culture, language, and other modes of communication—including notation, special vocabulary, and gestures.
  • Its methods of teaching, learning, assessment, and thinking. This includes what it does to preserve and sustain its work and pass it on to future generations.
  • Its tools, methodologies, and types of evidence and arguments used in solving problems, accomplishing tasks, and recording and sharing accumulated results.
  • The knowledge and skills that separate and distinguish among people with varying levels of expertise in the discipline, such as (a) a novice, (b) someone competent to accomplish the task, (c) a local or regional expert, and (d) a national or world-class expert.

Thus, one can examine each of the bulleted items from a math and math education point of view. You can self-assess on your own personal strengths and weaknesses in the general areas that are listed. You can compare and contrast your understanding of the discipline of math versus your understanding of other disciplines. You can think about what aspects of the discipline of math various of your students might want to learn better than they learn other aspects.

Math is Creating and Using Math Models

The previous section mentioned math modeling. The Math Forum@Drexel is a good source of information for teachers and others interested in mathematics education.

The Math Forum (1999). Ask Dr. Math. The Math Forum@Drexel. Retrieved 12/2607:

Quoting from a philosophical question sent to the Forum that included the following observation:

You cannot say 2 + 3 = 5, because 2 is not three is not five. How can two things (2 and 3), neither of which is identical to 5, be identical to five if they are united? 5 in itself is also an independent "being." If it weren't, it indeed could exist as a collection of at least two other 'entities'.

Here is part of the response:

Mathematics deals not with reality, but with an abstraction of reality: a "model" of just one aspect of the reality we use it to describe. For example, a number such as 2 doesn't represent any particular pair of things, but the idea of "two-ness." Man has found through long experience that things can be counted, and that the resulting numbers accurately describe one aspect of reality: if I counted two apples yesterday, and nothing has been done to them, then when I count them again there will still be two. The number doesn't tell us their color, or how they taste, or who owns them; but it describes something about them that is true of any pair of apples. It is as much something I perceive about them as is their color or taste; but when I talk about numbers, I am abstracting one property from the rest, thinking only of the "two-ness" and ignoring the "apple-ness."

The response provides a good way to think about the abstractness of math and the idea of math modeling. A number such as 2 is a mathematical model. The number 2 is a model of a certain property of a collection that happens to contain two apples, or a collection that happens to contain two people, or a collection that happens to contain two candy bars. Mathematical modeling is a very important and fundamental aspect of math, but is it a relatively abstract idea. Many people teaching and/or learning math do not have a good grasp of the concept of math modeling.

Two Cultures of Mathematics

Another way of thinking about "What is math?" is discussed in the following article:

Gowers, W.T. (The two cultures of mathematics. Retrieved 6/30/2014 from Quoting from the article:

In his Rede lecture of 1959, entitled "The Two Cultures", C. P. Snow argued that the lack of communication between the humanities and the sciences was very harmful, and he particularly criticized those working in the humanities for their lack of understanding of science. One of the most memorable passages draws attention to a lack of symmetry which still exists, in a milder form, forty years later:
A good many times I have been present at gatherings of people who, by the standards

of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold: it was also negative. Yet I was asking something which is about the scientific equivalent of: Have you read a work of Shakespeare's?

I would like to argue that a similar sociological phenomenon can be observed within pure mathematics, and that this is not an entirely healthy state of affairs.

The "two cultures" I wish to discuss will be familiar to all professional mathematicians.

Loosely speaking, I mean the distinction between mathematicians who regard their central aim as being to solve problems, and those who are more concerned with building and understanding theories. This difference of attitude has been remarked on by many people, and I do not claim any credit for noticing it. As with most categorizations, it involves a certain over simplification, but not so much as to make it useless. If you are unsure to which class you belong, then consider the following two statements.
(i) The point of solving problems is to understand mathematics better.
(ii) The point of understanding mathematics is to become better able to solve problems.

For a great many years it was common to divide math into two categories: "pure math" and "applied math." Some research mathematicians concentrated most of their work in only one of these two areas, but some were quite prolific in both areas.

G.H. Hardy is a very famous mathematician who liked to think of himself as a pure mathematician. However, eventually some of his "pure" theories turned out to be quite useful in applied math. Quoting from the Wikipedia:

He is usually known by those outside the field of mathematics for his 1940 essay [A Mathematician's Apology], which is often considered one of the best insights into the mind of a working mathematician written for the layman.

Here are some links to additional answers to the question, "What is math?"

For Reflection and Discussion. Select some age group of students, such as fifth grade. Ask what they think math is. Then, do the same thing for "typical" adults. What insights does this activity give you into the current success of our math education system?
For Reflection and Discussion. Think back over your own studies of math. Where did you receive explicit instruction on the topic "math modeling?" If you asked various students and adults what they think math modeling is, what kinds of answers do you expect you would receive?

Math is a Discipline of Study

A discipline (a organized, formal field of study) such as mathematics tends to be defined by the types of problems it addresses, the methods it uses to address these problems, and the results it has achieved. One way to organize this set of information is to divide it into the following three categories (of course, they overlap each other):

  1. Mathematics as a human endeavor. For example, consider the math of measurement of time such as years, seasons, months, weeks, days, and so on. Or, consider the measurement of distance, and the different systems of distance measurement that developed throughout the world. Or, think about math in art, dance, and music. There is a rich history of human development of mathematics and mathematical uses in our modern society.
  2. Mathematics as a discipline. You are familiar with many academic disciplines such as archeology, biology, chemistry, economics, history, psychology, sociology, and so on. Mathematics is a broad and deep discipline that is continuing to grow in breadth and depth. Nowadays, a Ph.D. research dissertation in mathematics is typically narrowly focused on definitions, theorems, and proofs related to a single problem in a narrow sub-field in mathematics.
  3. Mathematics as an interdisciplinary language and tool. Like reading and writing, math is an important component of learning and "doing" (using one's knowledge) in each academic discipline. Mathematics is such a useful language and tool that it is considered one of the "basics" in our formal educational system.

To a large extent, students and many of their teachers tend to define mathematics in terms of what they learn in math courses, and these courses tend to focus on #3 above. The instructional and assessment focus tends to be on basic skills and on solving relatively simple problems using these basic skills. As the three-component discussion given above indicates, this is only part of mathematics.

Even within the third component, it is not clear what should be emphasized in curriculum, instruction, and assessment. The issue of basic skills versus higher-order skills is particularly important in math education. How much of the math education time should be spent in helping students gain a high level of accuracy and automaticity in basic computational and procedural skills? How much time should be spent on higher-order skills such as problem posing, problem representation, solving complex problems, and transferring math knowledge and skills to problems in non-math disciplines?

Every discipline of study can be analyzed in terms of lower-order knowledge and skills versus higher-order knowledge and skills. Educators in each discipline are faced by the challenge of deciding how much emphasis to place on lower-order and on higher-order at each phase of the educational process.

In math education, for example, students typically learn counting numbers (words) one, two, three, and so on well before they enter kindergarten. However, their understanding of the number line is minimal.

Thus, our formal educational system is faced by the task of helping students come to know and understand the number line. Ask yourself, what constitutes lower-order knowledge and skills and what constitutes higher-order knowledge and skills in terms of learning about the number line? This is not an easy question to answer. However, you probably agree that students need to learn some things about the number line when they are in kindergarten or the first grade, and that they need to continue to learn more about the number line as they proceed through subsequent years of schooling.

However, there is more to math education than just learning about the number line. Thus, students need to be gaining lower-order and increasingly higher-order math knowledge and skills in a variety of math areas as they proceed through the math education curriculum. This situation tends to create a delicate balancing act, especially because some students can move much more rapidly than do others along the higher-order knowledge and skills pathway.

The math education system in the United States does not do as well as the systems in a number of other countries.

Here are some paragraphs from a 2005 American Institutes for Research report:

Despite a widely held belief that U.S. students do well in mathematics in grade school but decline precipitously in high school, a new study comparing the math skills of students in industrialized nations finds that U.S. students in 4th and 8th grade perform consistently below most of their peers around the world and continue that trend into high school.
The study, “Reassessing U.S. International Mathematics Performance: New Findings from the 2003 TIMSS and PISA,” focused on students in the United States and 11 other industrial countries that participated in all three assessments: Australia, Belgium, Hong Kong, Hungary, Italy, Japan, Latvia, Netherlands, New Zealand, Norway, and the Russian Federation. U.S. students consistently performed below average, ranking 8th or 9th out of twelve at all three grade levels. These findings suggest that U.S. reform proposals to strengthen mathematics instruction in the upper grades should be expanded to include improving U.S. mathematics instruction beginning in the primary grades.
Countries that score well on items that emphasize mathematical reasoning (a higher-level skill) also score well on items that require knowledge of facts and procedures (a lower-level skill), suggesting that reasoning and computation skills are mutually reinforcing in learning mathematics well. Compared to other countries, students in the United States students do not do well on questions at either skill level. [Bold added for emphasis.]

Pay particular attention to the last of the paragraphs. It suggests that the current balance in US schools between lower-order and higher-order math knowledge and skills is not as effective as it might be.

Beauty in Mathematics

Relatively few K-12 teachers study enough mathematics to be able to understand and appreciate the breadth, depth, complexity, and beauty of the discipline. Mathematicians often talk about the beauty of a particular proof or mathematical result. Do you remember any of your K-12 math teachers ever talking about the beauty of mathematics?

G. H. Hardy was one of the world's leading mathematicians in the first half of the 20th century. In his book "A Mathematician's Apology" he elaborates at length on differences between pure and applied mathematics. He discusses two examples of (beautiful) pure math problems. These are problems that some middle school and high school students might well solve, but are quite different than the types of mathematics addressed in our current K-12 curriculum. Both of these problems were solved more than 2,000 years ago and are representative of what mathematicians do.

1. A rational number is one that can be expressed as a fraction of two integers. Prove that the square root of 2 is not a rational number. Note that the square root of 2 arises in a natural manner in land-surveying, carpentering, and navigation problems.


2. A prime number is a positive integer greater than 1 whose only positive integer divisors are itself and 1. Prove that there are an infinite number of prime numbers.

We all know that prime numbers are important in the simplification of fractions. In recent years, very large prime numbers have emerged as being quite useful in encryption of electronic messages.

The Problem of Change

"Humans are allergic to change. They love to say, 'We've always done it this way.' I try to fight that. That's why I have a clock on my wall that runs counter-clockwise." (Grace Hopper; American computer scientist and computer educator; 1906-1992.)
"It is not the strongest of the species that survive, nor the most intelligent, but the one most responsive to change." (Charles Darwin; British scientist and developer of the theory of evolution; 1809-1882.)

Math education has carved out a large niche in our overall education system. It is one of the "big three"—reading, writing, and arithmetic. It has long been accepted that basic literacy and numeracy are core to a good education.

Over time, it has become clear that arithmetic is too narrow a term. This represents a major change in math education. For example, even in the earliest years of schooling, we want students to learn about measurement, pre-algebra, geometry, data gathering and analysis, searching for and representing mathematical patterns, and so on. So let's agree to use the term math rather than arithmetic as we explore ways to improve our math education system, even at the lower grade levels. Arithmetic computation is only part of what math education is about, even in the PreK-5 grade levels.

Many people assert that our math education system is not as good as it could or should be. Of course, this type of assertion is not unique to math education. Each component of our informal and formal educational systems is open to analysis, criticism, and suggestions for improvement. This is and should be an ongoing process, because our world is changing, and the educational needs of people are changing as well.

There are many change agents in our world. For example, increasing population and increasing consumption contribute to a problem of sustainability. This is a huge and growing problem that faces all of us.

Here are some current rapidly progressing areas of technological research and development that are all very powerful change agents:

  • Genetics. Study of the human and other genomes.
  • Nanotechnology (Study of the very small; manipulation of individual atoms and molecules.)
  • Information and Communication Technology (ICT).
  • Cognitive neuroscience (brain science).

In terms of math education, ICT—which includes computers—is part of the force for change. For many years, the amount of "compute power" that one can buy for a dollar has been doubling in less than two years, and this pace of change seems likely to continue for a few more years. As compared to the first commercially-produced computers in the early 1950s, today's microcomputers are more than a billion times more cost effective.

Moreover, computer systems are getting "smarter"—their level of artificial intelligence is increasing. Thus, computer systems can solve a wide range of the math-related problems that previously people worked to solve using by-hand methods and tools that are much less powerful than today's and tomorrow's computers and other ICT facilities.

In summary, our math education system needs to appropriately educate students for life in a world that faces major and continuing problems and change. Computers and other ICT are a major change agent in math and math education, as well as in the world at large.

For Reflection and Discussion. Give some examples of where it seems to you possible (reasonable, likely, worth exploring) that changes in technology have changed what students need to be learning in math.
For Reflection and Discussion. What is your current knowledge and understanding of roles of math in each of the four technological change areas listed above? Does this level of knowledge and understanding meet your personal needs and the needs of your students?

The Change Problem, More Carefully Stated

"All progress is precarious, and the solution of one problem brings us face to face with another problem." (Martin Luther King, Jr.; American minister and civil rights activist; 1926-1968.)

In a clearly defined problem, there is a clearly defined initial situation and a clearly defined goal. It is not enough to say that our current math education system is not as good as we would like it to be, and that the goal is to make it better.

Instead, we need to have good and clear information on the current status of our math education system, and we need clear goals. As we struggle to get a good understanding of the current status of our math education system, we begin to appreciate that we are faced by an immense, complex problem situation. As we struggle to set goals, our understanding of the complexity of the challenge grows.

Perhaps you are familiar with norm-referenced testing and criterion-referenced testing. Quoting from the second of these two links:

For example, if the criterion is "Students should be able to correctly add two single-digit numbers," then reasonable test questions might look like "2 + 3 = ?" or "9 + 5 = ?" A criterion-referenced test would report the student's performance strictly according to whether or not the individual student correctly answered these questions. A norm-referenced test would report primarily whether this student correctly answered more questions compared to other students in the group.

It is easy to make up a test and see how a group of students perform in comparison to each other. Indeed, the test can be used with different groups, so some sort of comparison can be made among the groups.

But what questions should one put on the test? It is difficult to make up a math test where there is wide agreement on the specific topics and questions to be on the test. It is also difficult to make up a test that is fair to all the test takers, reliable, and valid. Here is a short quote from the linked article:

Test takers often have limited options in when, how, or why they are taking the test, and may feel victimized in the process. The purpose of this paper [on fairness] is to focus on the test taker and to consider how all parties in the test process (test sponsor, test developer, test administrator, and test taker) have a role to play in ensuring fair testing practices and valid test results.

Notice the emphasis on the test taker. People trying to improve math education often forget the goal of helping students to gain increased and appropriate knowledge and skills in learning and using math, both now and in the future.

Thus, for example, we can make up a paper and pencil arithmetic computation test that contains problems such as 387 x 2593 and 693/87. We can administer the test to students throughout the world. Then we can make statements about the mean and standard deviation of test scores for different groups of students in different countries.

However, who cares whether students can do such paper and pencil calculations relatively rapidly and accurately? "Ah, there's the rub." What criterion does one use in a criterion-referenced test? If a specific teacher or school system knows the criterion being tested, and teaches quite specifically to that criterion, the teacher's or school district's students are apt to score high as compared to students whose education is not specific to the criterion.

This would likely be good, if there was good understanding and widespread agreement of what constituted "the" right things to know and be able to do in math. However, this is an impossible task. Consider, for example, cultural differences, differences in standards of living, differences in tools that can aid one in doing math, differences in natural math ability, differences in interest level, and so on. Math education cannot be a "one size fits all."

Let's take a more complex example. The ideas variable, function, and equation are very important concepts in math. As math educators, we want students to gain some understanding of these topics. What should we teach about them, and how should we measure the nature and extent of student understanding that comes from this teaching? Are these universally important math concepts so that it is reasonable and fair to measure the quality of a school, school district, state, or national math education system by a criterion-referenced test on these topics?

Well… let us simplify the situation. Consider a quadratic (polynomial) function of one variable and its associated quadratic equation. We now have a much simpler environment in which to explore variable, function, and equation. In this environment we can explore real and complex variables, factoring, graphing by hand, completing the square to solve a quadratic equation, use of the quadratic formula to solve a quadratic equation, graphing using a calculator or computer, maximum and minimum values of the function, and solving an equation using a calculator or a computer. We can study uses of quadratic functions and of solving quadratic equations, both in the discipline of math and in other disciplines. Clearly, this is a potentially very rich teaching and learning environment.

What do we want students to learn, and how can we assess this learning in a criterion-referenced manner? Will the criteria we pick be appropriate to the specific student or group of students being tested?

How long will various parts of this learning be retained? This is called long-term residual impact. The big three in terms of assessment are formative assessment, summative assessment, and long-term residual impact assessment. Thus, for example, what is the typical adult's understanding of "mathematical variable" two years or more after she or he is no longer in school? Quoting from the Encyclopaedia of Technical and Vocational Education:

Goal 13: Formative, summative, and residual impact evaluation. Implementation plans for information technology shall be evaluated on an ongoing basis, using formative, summative, and residual impact evaluation techniques. Formative evaluation provides information for mid-programme corrections. It is conducted as programs are being implemented. Summative evaluation provides information about the results of a programme after it has been completed, such as a particular staff development programme, a particular program of loaning computers to students for use at home, and so on. Residual impact evaluation looks at programs in retrospect, perhaps a year or more after a program has ended.

As we think about improving our math education system, we need to think in terms of the long-term residual impact of the many hours of schooling that focus on math education. What can the curriculum content, teaching process, and assessment process do to help prepare students for their math needs in the future? What can we do to facilitate and promote transfer of learning, both to areas outside the math class and to a student's future math needs?

For Reflection and Discussion. Select a math idea that you consider to be very important, such as variable or function. Think about what the idea means to you from a math point of view. Then think about what the idea means to a typical adult from a math point of view. Finally, think about what you would like a typical adult to understand about variable, function, and other math really "big ideas." Are these ideas carefully and clearly taught in school in a manner so that there will a high level of long-term residual impact? What could be done in school to increase the long-term residual impact of student learning of these ideas?

The Major Stakeholders

"Adults are obsolete children." (Dr. Seuss, Theodor Seuss Geisel; American writer; 1904–1991.)
"Leadership: The art of getting someone else to do something you want done because he wants to do it." (Dwight D. Eisenhower; 34th U.S. president; 1890–1969.)

It has become relatively common to talk about student-centered education. Clearly, students are the most important stakeholder group in our math education system. We need a math education system that prepares students to meet their own math education needs now and in the future.

However, the math education system includes many other stakeholder groups. Some play major leadership roles in trying to improve our math education system. Examples include parents, teachers, teachers of teachers, school administrators, school boards, state and federal education agencies, state and federal governments, businesses, and employers. There are so many stakeholder groups, and they have such diverse points of view, that it is very difficult to get all, or even most of them, to work together in a collaborative manner.

This current document cannot be the "be all, end all" for the complex problem of improving our math education system. Remember, the specific focus is on preservice and inservice teachers, and the teachers of these teachers. The suggestions given later in this document are targeted specifically toward these stakeholder groups.

However, we need more general background before looking at possible specific things that preservice and inservice teachers, and the teachers of these teachers, might be doing.

For Reflection and Discussion. In your opinion, who are the one or two most powerful stakeholder groups in math education? How does this situation affect the quality of our math education system?

Two Thought-Provoking Question Areas

"When people cannot see the need for what’s being taught, they ignore it, reject it, or fail to assimilate it in any meaningful way. Conversely, when they have a need, then, if the resources for learning are available, people learn effectively and quickly." (John Seely Brown & Paul Duguid, The Social Life of Information, 2000.)

Here are two challenging and thought-provoking math education question areas that are relevant to people of all ages. Begin by asking yourself the questions and forming mental answers. Then, in the future, explore how other people of all ages think about and answer these questions.

1. Can you do and use math at a level that meets your personal current needs and the current expectations you have for yourself? What about needs and expectations you believe you may have in the future? This is a question related to intrinsic motivation.
2. Can you do and use math at a level that meets the current needs and expectations of various stakeholder groups such as parents, our schooling system, potential employers, politicians, our government, and so on? What about needs and expectations that they may have in the future? This is a question related to extrinsic motivation.

These are student-centered questions that focus on being able to "do and use" math to meet one's personal needs and the needs of others. They concern both the present and the future. They concern intrinsic and extrinsic motivation. They concern helping students learn to self-assess their own education. Wow! That is a lot to think about.

One way to increase your insights into these questions is to think about the same questions for the other two basics of education—reading and writing. Are your students well versed in self-evaluation of their reading and writing skills?

You and your students may need to gain an increased level of expertise in self-assessment. Reading, writing, and math provide an opportunity to do a compare and contrast in self-assessment. Teachers can ask themselves, "Am I teaching in a manner so that my students are learning to self-assess?" "Can my students clearly see the progress they are making and how this progress is of value to them, both now and in the future?"

The two question areas tend to emphasize intrinsic motivation on the part of the learner versus extrinsic motivation being provided (or forced) by those with an interest in or stake in the student learning. The issue of intrinsic versus extrinsic motivation is a challenge. For many students and teachers, our math education system (as well as other parts of schooling) tends to become a battleground. As you think about ways to improve our math education system, think about ways that will help reduce the confrontation (battle) between students and their teachers. Other battles include parents versus their children or their children's teachers, and politicians versus school systems.

For Reflection and Discussion. Self-assessment is very important to learners, whether they are learning in a school setting or on their own in an informal learning environment. Think about your own experiences in helping yourself and others to learn in informal and formal math settings. How much emphasis was placed on learning how to do self-assessment and to take increased responsibility for one's own learning? What could have been done to help you to become better at self-assessment in math?
For Reflection and Discussion. As you teach and/or in other ways help others to learn, how much emphasis do you place on intrinsic motivation of the learner, and how much emphasis do you place on you and others providing strong extrinsic motivation? Give some examples to help clarify your point of view on this topic.

This student-centered approach to assessing the effectiveness of our math education program suggests one possible way to improve math education. The suggestion is to help students to understand their current and possible future math education needs, and help them learn to self-assess their knowledge, skills, and use of their math education.

A student-centered approach to education places considerable importance on helping students to become intrinsically motivated and to act upon their intrinsic motivations. Of course, this raises an interesting question. What is the relationship between externally encouraging, inspiring, and in other ways cajoling students, and helping students to develop their own intrinsic motivation?

Six Types of Math Education Goals

"An individual understands a concept, skill, theory, or domain of knowledge to the extent that he or she can apply it appropriately in a new situation." (Howard Gardner; American psychologist and educator; 1943-.)

This statement from Howard Gardner is applicable to a person's expertise in any discipline. Roughly speaking, it divides education into two categories: 1) learning with good understanding, in a manner that allows one to tackle new problem situations; and 2) learning with relatively little or no understanding, which only allows one to solve the specific problems that one has previously encountered and studied. Sometimes the first type of learning is called "education" while the second type is called "training." However, the two types are thoroughly intertwined. Both are an important part of schooling. The challenge is in reaching an appropriate balance to fit the needs of widely varying students who are studying a wide range of disciplines.

In improving math education, we need to think about current math education goals along with possible deletions and additions to such goals. Here are five general types of math education goals. Different groups focusing on improving math education tend to have different areas of emphasis on goals that they feel should receive more emphasis. The "more emphasis" is usually some combination of Content, Pedagogy, and Assessment. So, we can look at: A) the Common core State Standards Math Initiative; B) NCTM and other versions of math education used over the years from roughly 1970 until the current time; C) "New Math of the 1960s; D) the "Old Math" of the preceding half century; and the "Still Older Math Curriculum" dating back to the development of reading and writing about 5,000 years ago. From a historical point of view, reading and writing were accompanied by written notation for numerals, and eventually led to the development of the "language" and discipline of math.

In all cases, math education leaders and others have made major efforts to improve our math education system. We should not belittle their efforts. Rather, we should continue our research and development on what works well and what does not work so well. We need to ensure that the changes we make will help prepare today's students for their next half century or so of life.

1. Standards-based

A stakeholder group, state government, state organization, national government, national organization, or international organization draws up a list of math topic-based and performance-based standards. For example, see the work of the National Council of Teachers of Mathematics and the Common Core State Standards Math Initiative. Remember that a particular combination of Curriculum Content, Pedagogical Processes, and Assessment must necessarily reflect compromises among various stakeholder groups. It is fruitless to hope that we will discover one best approach to math education that meets the needs of the diverse students in our school systems. Think about what we are learning about how the human brain learns math and other disciplines, how it uses what it learns, what it retains over time, how it relearns when the need arises, how all of this is affected by changes in technology and societal values, and so on.

For Reflection and Discussion. Reflect on the issue of who should determine math Curriculum, Pedagogy, and Assessment. Some countries use a top down approach, with the national governments and "experts" operating under the authority and power of the government accomplishing the task. Others favor local control—even to the level of each individual family being allowed to set the standards and perhaps take on the educational responsibilities for their own children. What are your thoughts about these two extremes?

2. Attitudinal

Math educators want students to have and to continually demonstrate a positive attitude in their knowledge and skills in learning and using math. Math educators are unhappy when they hear an adult claim, "I can't do math and I hated math when I was in school."

It is easy to state that we want students to develop a good and healthy attitude toward math. However, we have yet to develop a good research-based theory of how to develop positive attitudes toward math in a manner that will be successful with most students.

For Reflection and Discussion. Just for the fun of it, think about some area (such as food, clothing, music, art, and so on) in which you have strong personal opinions. What caused you to develop a good attitude or a bad attitude toward specific food, clothing, music, art, and so on? Is there anything particularly wrong about having a bad attitude toward certain forms of food, clothing, music, art, and so on? Now, do some reflective thinking to compare and contrast between these situations and the bad attitudes that many adults have toward math.

3. Math Habits of Mind

Costa, Arthur & Kallick, Bena (n.d.). Sixteen habits of mind. The Institute for Habits of Mind. Retrieved 7/4/2011 from

The table given below contains the Costa and Kallick's collection of 16 habits of mind. It also contains a 17th habit of mind contributed by David Moursund. These are general-purpose, interdisciplinary habits of mind they define as follows:

A "Habit of Mind” means having a disposition toward behaving intelligently when confronted with problems, the answers to which are not immediately known. When humans experience dichotomies, are confused by dilemmas, or come face to face with uncertainties, our most effective actions require drawing forth certain patterns of intellectual behavior. When we draw upon these intellectual resources, the results that are produced through [their use] are more powerful, of higher quality and greater significance than if we fail to employ those patterns of intellectual behaviors.

The left side of the table given below includes a number of quotes from Costa and Kallick. That reference is abbreviated (C/K). The right side of the table provides brief analyses and comments from a tutoring in math education point of view.

Habit Discussion
1. Persisting. Stick to it through task completion. Remain focused—keep your eye on the ball. Try alternative approaches when you are stuck. Don’t give up easily. This is one of the key ideas in math problem solving. ADD and ADHD students have special difficulties in this area. A great many other math students have not learned the need for persistence in dealing with challenging math problems.

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

Of course, if a math researcher gives up too early, then important discoveries are not made. Examples: The equation 2x – 3 = 0 is unsolvable in the domain of integers, but is solvable in the domain of rational numbers. The equation x2 − 2 = 0 is unsolvable in the domain of rational numbers, but is solvable in the domain of real numbers.

2. Managing impulsivity. Think before you act, and consider the consequences of your actions before taking the actions. Remain calm, thoughtful, and deliberate. Don’t be driven by a need for instant gratification; with practice, one can learn to control this impulse. This habit of mind is applicable both in interacting with other people and in carrying out tasks such as problem solving.

In math problem solving, one has a goal in mind. Learn to mentally consider various approaches to achieving the goal. Learn to analyze whether the steps one is taking or considering taking will actually contribute toward achieving the goal.

Students who are driven by the need for instant gratification seem to have trouble in their math studies when they reach algebra. See

3. Listening with understanding and empathy. It is difficult to listen with empathy and with a goal of achieving understanding. “Some psychologists believe that the ability to listen to another person, to empathize with, and to understand their point of view is one of the highest forms of intelligent behavior.” (C/K) A math tutor needs to learn to promote two-way conversations with tutees, and needs to learn to listen with understanding and empathy. A tutee’s math learning problems often extend far beyond the math classroom and the math content being taught in the classroom.

It is common to start a tutoring session with the question, “How’s it going with you?” You are seeking “deeper” insight into the trials and tribulations in your tutee’s life that might be affecting math-learning performance.

4. Thinking flexibly. Think before you act, and consider the consequences of your actions. Don’t let your thinking and approaches to dealing with the world get stuck in a rut. Entertain new ideas; learn to change perspectives, generate alternatives, consider a broad range of options, and understand other’s points of view.

It is interesting to see students who are studying algebra struggle with problems that are readily solved without the use of algebra. They get stuck in an algebra rut. Descarte’s development of analytic geometry through combining ideas from Euclidean geometry and algebra represents a major change in perspective. Folk math (Maier, n.d.) is math developed by people to solve the math problems and accomplish the math-related tasks they encounter in day-to-day life. Their approaches are often far different than use of school math.

5. Thinking about your thinking (metacognition). You have the ability to think about what you know and what you don't know. You have the ability to think about your thought processes. You can develop increased skill in being aware of your own thoughts, strategies, feelings and actions, and their effects on others. In any discipline, metacognition is a powerful aid to cognitive development. Math education stresses the idea of “showing your work.” What they are really trying to get at is having students understand and explain their thinking as they solve math-related problems and accomplish-related tasks. This is a useful activity. However, be aware that students (especially TAG students) can make math intuitive leaps that are beyond their power of explanation. Much research consists of working to explain and understand a “hunch” or “intuitive leap.”
6. Striving for accuracy and precision. "Routinely check your work and thought processes. “To be craftsman like means knowing that one can continually perfect one's craft by working to attain the highest possible standards, and pursue ongoing learning in order to bring a laser like focus of energies to task accomplishment.” (C/K.)

One of the great strengths of the discipline of math is the careful statement of problems and tasks, and providing details of solving the problems and accomplishing the tasks. This is done in a manner that allows the steady accumulation of math knowledge and skills so that others can build on them with confidence.

In math education, we stress: “check your answers.” What we are really trying to teach is the concept that math can be done with a level of correctness that both the doer and others can act on and build on the results with confidence.

7. Questioning and problem posing. “Effective problem solvers know how to ask questions to fill in the gaps between what they know and what they don't know.” (C/K) Effective questioners are inclined to ask a range of questions such as: “How do you know that?” What evidence do you have?” Problem posing (asking relevant, interesting, challenging questions) is an important habit of mind. Problem posing and problem solving lie at the heart of math and many other disciplines. Math is a powerful aid to representing and helping to solve problems in many disciplines. Thus, one aspect of good math instruction is helping students do the transfer of learning required to make effective use of their math learning in new settings. A tutoring environment can provide the time and individualization needed to help make math useful and relevant to a tutee’s life outside of the math classroom.

Math tutoring also provides an environment in which a tutee can learn to ask meaningful math related questions far beyond the traditional one of: “Is my answer right?”

8. Applying past knowledge to new situations. Stress learning to make effective use of what one has learned. Know what you know, and learn to apply it in new, different, challenging problem situations. (C/K)

From your author’s point of view, Costa and Kallick fail to emphasize applying the accumulated knowledge and skills of others, such as one can readily do making use of the Web. They do not emphasize Computational Thinking (IAE-pedia, n.d.) as a desirable habit of mind. See the next main section of this current chapter.

One of the most important ideas in math problem solving is building on the previous work of oneself and others. This “building on” is a type of transfer of learning that has been done by one’s self and by others. One gets better at it by long and concerted effort to solve challenging math problems. Our current math education system is weak in helping students learn to read and understand math so they can make effective use of the accumulated math that can be assessed on the Web and through other library resources. A tutor with network access can work on this aspect of math education as “targets of opportunity” arise in tutoring sessions. Our math education system also is weak in helping students to understand that the use of tools such as calculators and computers is a way of storing, using, and accessing accumulated math knowledge and skills.

Remember, a computer is both a way to access math knowledge and a way to automate a wide range of math procedures.

9. Thinking and communicating with clarity and precision. “Language and thinking are closely entwined. Like either side of a coin, they are inseparable. When you hear fuzzy language, it is a reflection of fuzzy thinking.” (C/K) Be clear! Strive for accurate oral and written communication. The written and oral language of mathematics allows of precise communication in the discipline of mathematics. However, it takes a great deal of study and practice to learn to precisely communicate in this language. A tutoring environment allows an ongoing oral and written math communication between a “native math speaker”—in this case, a well-qualified math tutor—and a novice in the field. See “Communicating in the language of mathematics” available at (IAE-pedia, n.d.). Writing using the language of math should begin early elementary and be given a high priority every year. (See
10. Gather data through all senses. Pay attention to—be alert to, be actively involved in—the world around you and the steady stream of input provided through your senses. Math word problems are a standard part of the math curriculum. Many good math word problems are embedded in the world outside of mathematics. They challenge a student to draw on his or her world knowledge and understanding both in representing the problem mathematically and in checking to see if the results of the math problem-solving work make sense. See “Word problems in math” (IAE-pedia, n.d.).
11. Creating, imagining, and innovating. “All human beings have the capacity to generate novel, original, clever or ingenious products, solutions, and techniques—if that capacity is developed.” (C/K) The challenging problems and tasks in each discipline “call for” creativity. Creative people are often uncomfortable with the status quo. Robert Sternberg’s three-component of definition of multiple intelligences includes creativity as one of the three components. (See Humans have considerable innate creativity, and this creativity can be fostered through instruction, encouragement, and use.

In math, there are many different ways to solve a problem and/or make a proof. Posing math problems, conjecturing math theorems, and solving math problems, and proving math theorems provide outlets for mathematical creativity. And, of course, tutoring and teaching provide great opportunities for creativity.

12. Responding with wonderment and awe. Quoting Albert Einstein, “The most beautiful experience in the world is the experience of the mysterious." A dedicated lifelong learner follows a pathway strewn with new experiences, excitement, beauty, wonder, awe, and enjoyment. Some people who study math come to appreciate the excitement, beauty, wonder, and awe of the discipline. Alas, most don’t!

One might point to issues of nature versus nurture, and argue that those who are not endowed by nature with great natural math ability cannot come to appreciate math as a human endeavor that is full of fun, creativity, excitement, beauty, and so on. Others argue that our informal and formal math education system (the “nurture component of the puzzle”) is responsible for so many people growing up claiming they hate math and could never learn to do math. A good math tutor directly addresses these issues and helps tutees personally experience some of the wonder and awe of math.

13. Taking responsible risks. Be flexible and venturesome. “Flexible people seem to have an almost uncontrollable urge to go beyond established limits. They are uneasy about comfort; they ‘live on the edge of their competence’. They seem compelled to place themselves in situations where they do not know what the outcome will be.” (C/K) Quoting Bobby Jindal from the (C/K) document: “The only way to succeed is to be brave enough to risk failure.”

Quoting (C/K): “Some students hold back [participating in] games, new learning, and new friendships because their fear of failure is far greater than their experience of venture or adventure. They are reinforced by the mental voice that says, "if you don’t try it, you won’t be wrong" or “if you try it and you are wrong, you will look stupid”.

Our math education system teaches students to believe that math problems have “right or wrong” (correct or incorrect) answers and solution processes. A student of math frequently fails in attempts to solve a specific problem or accomplish a specific task.

Such mistakes and/or failures are an essential component of a good learning environment. They provide a basis for self-analysis, more learning, and persistence.

However, they can also “beat down” a student. Rather than failure and mistakes being a useful learning experience, they become something to be avoided. One way to avoid failure and mistakes is to withdraw—don’t participate. A slightly different result is a decision to participate at a minimal level in a manner that reduces one’s risks.

In a tutoring environment, a tutee is expected to show and explain his or her work. A good tutor uses this situation to help a tutee learn to learn from errors in logic/analysis and errors in computation/symbol manipulation. The tutor can help the tutee fill in gaps in math knowledge and skill that come from forgetting or from not learning needed prerequisites. The following quotes capture important ideas:

“I have not failed. I've just found 10,000 ways that won't work." (Thomas Edison)

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

“What is your radius of exploration? How far from your comfort center are you willing to wander? As you meander and encounter wonders, how many possibilities do you squander because of baggage you carry? Travel light – carry a few essentials. Make room in your backpack for new experiences, new tools, new toys.” (Laran Stardrake)

14. Finding humor. “[Laughter] has been found to liberate creativity and provoke such higher level thinking skills as anticipation, finding novel relationships, visual imagery, and making analogies. People who engage in the mystery of humor have the ability to perceive situations from an original and often interesting vantage point.” (C/K)

Lighten up. Look for and enjoy humor. Be able to laugh at one’s self.

There are, of course, many mathematical jokes. A recent Google search of math jokes produced about 1.7 million hits. Many require some insight into math in order to be humorous. For example, what makes the following math joke funny?

Teacher: "Who can tell me what 8 times 7 is?" Student: "It's 56!"

Teacher: "Very good! And who can tell me what 7 times 8 is?" Same student: "It's 65!"

Humor is an important component of education. (See “Using humor to maximize learning” at

A math tutor might want to make a collection of math jokes suitable to the math level and interests of his or her tutees. These could be “doled out” one or two per session to help lighten up the tutoring sessions.

15. Thinking interdependently. Think in terms of collaboration in work and learning. “Human beings are social beings. We congregate in groups, find it therapeutic to be listened to, draw energy from one another, and seek reciprocity.” (C/K) In work, play, and learning, engage with others, depend on others, and be dependent on others. Many people view learning and doing math as solitary activities. They view math-oriented people as somehow a different (and somewhat peculiar) breed.

That is a misconception. The (C/K) quote to the left certainly applies to mathematicians. Like specialists in other academic disciplines, they like to interact with their peers, and they often work together to learn and to do their research.

Math project-based learning and team-based problem-based learning are useful approaches to engage students as social beings who are learning and doing math together (PBL, n.d.). In math tutoring, a 3-component team (human tutor, computer tutor, and tutee) brings together and draws on each member’s unique capabilities in addressing a math teaching/learning problem.

16. Learning continuously. It is the nature of a human brain that is it continually engaged in learning and in processing what it has learned. We are all lifelong learners. Some of us develop a habit of mind that focuses our learning capabilities toward gaining an increasing level of knowledge, skill, and expertise in areas that serve our individual interests and needs, and help us to better serve needs of others. In our lifelong quest for knowledge and wisdom, we each in our own way “boldly go where no one has gone before.” Math is a huge and vibrant discipline, with great breadth and depth. The world we live in is alive with math-related problems, tasks, and challenges. Thus, in our everyday lives we routinely encounter opportunities to see math in use, to use math, and to learn math.

Successful math education helps students learn to view the world through “math-colored” glasses and to effectively deal with the math-related problems and tasks that they encounter in their everyday lives. It helps them to develop math-related habits of mind. It starts them on a livelong pathway sets them on a path of using, maintaining, and strengthening these habits of mind.

A good math tutor role models such behavior and helps his or her tutees to develop such behaviors. A math tutor might want to make such transfer of learning from the math classroom to the rest of the world be an integral component of tutoring sessions.

17. Computational Thinking as a Habit of Mind. Quoting Jeannette Wing from the Computational Thinking document:

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

Genetically, human brains are not changing very rapidly. However, our cognitive capabilities have been greatly increased by the development of aids such as reading and writing, libraries, and the STEM disciplines of science, technology, engineering, and mathematics. The combined artificial intelligence (machine intelligence) and “brute force” power of computer brains continues to grow very rapidly.

Here are four key aspects of computer capabilities to keep in mind:

A. The size and capabilities of electronic digital libraries such as the Web are growing very rapidly. Such libraries help support a “look it up” approach to education, solving problems, and accomplishing tasks.

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

C. Telecommunications systems facilitate widely dispersed teams of people and computers to work together.

D. Math is a human endeavor. Computational thinking is a human activity in which a human thinks about how to make effective use of Information and Communication Technology in representing and solving prolbems.

4. Math Cognitive Development

Math educators want students to make steady progress in moving up a math cognitive developmental scale, such as a Piagetian math cognitive developmental scale. The ultimate goal is to get a large percentage of students to function at a formal operations cognitive developmental level in math. Examples of math cognitive developmental scales are available in Moursund, D.G. (June, 2006), Computational Thinking and Math Maturity: Improving Math Education in K-8 Schools. Access at

The following scale was created (sort of from whole fabric) by David Moursund. It represents his current insights into a six-level, Piagetian-type, math cognitive developmental scale. Note that the first four levels correspond to the traditional Piagetian stage theory. The last two levels can be considered as postformal stages. Quoting from

Postformal stages have been proposed. Kurt Fischer suggested two. Michael Commons presents evidence for four postformal stages: the systematic, metasystematic, paradigmatic and cross paradigmatic (Commons & Richards, 2003; Oliver, 2004).
Stage & Name Math Cognitive Developmental Scale
Level 1. Piagetian and Math sensorimotor. Birth to age 2. Infants use sensory and motor capabilities to explore and gain increasing understanding of their environments. Research on very young infants suggests some innate ability to deal with small quantities such as 1, 2, and 3. As infants gain crawling or walking mobility, they can display innate spatial sense. For example, they can move to a target along a path requiring moving around obstacles, and can find their way back to a parent after having taken a turn into a room where they can no longer see the parent.
Level 2. Piagetian and Math preoperational. Age 2 to 7. During the preoperational stage, children begin to use symbols, such as speech. They respond to objects and events according to how they appear to be. The children are making rapid progress in receptive and generative oral language. They accommodate to the language environments (including math as a language) they spend a lot of time in, so can easily become bilingual or trilingual in such environments.

During the preoperational stage, children learn some folk math and begin to develop an understanding of number line. They learn number words and to name the number of objects in a collection and how to count them, with the answer being the last number used in this counting process.

A majority of children discover or learn “counting on” and counting on from the larger quantity as a way to speed up counting of two or more sets of objects. Children gain increasing proficiency (speed, correctness, and understanding) in such counting activities.

In terms of nature and nurture in mathematical development, both are of considerable importance during the preoperational stage.

Level 3. Piagetian and Math concrete operations. Age 7 to 11. During the concrete operations stage, children begin to think logically. This stage is characterized by 7 types of conservation: number, length, liquid, mass, weight, area, and volume. Intelligence is demonstrated through logical and systematic manipulation of symbols related to concrete objects. Operational thinking develops (mental actions that are reversible).

While concrete objects are an important aspect of learning during this stage, children also begin to learn about objects that are not concretely available to them. This learning comes from words, language, pictures/video, and so forth.

For the average child, the time span of concrete operations is approximately the time span of elementary school (grades 1-5 or 1-6). During this time, learning math is somewhat linked to having previously developed some knowledge of math words (such as counting numbers) and math concepts.

However, the level of abstraction in the written and oral math language quickly surpasses a student’s previous math experience. That is, math learning tends to proceed in an environment in which the new content materials and ideas are not strongly rooted in verbal, concrete, mental images and/or a clear understanding of somewhat similar ideas that have already been acquired.

There is a substantial difference between developing general ideas and understanding of terms such as conservation of number, length, liquid, mass, weight, area, and volume, and learning the mathematics that corresponds to these terms. These tend to be relatively deep and abstract topics, although they can be taught in very concrete manners.

Level 4. Piagetian and Math formal operations. After age 11. Starting at approximately age 11 or 12, thought begins to be systematic and abstract. In this stage, intelligence is demonstrated through the logical use of symbols related to abstract concepts, problem solving, and gaining and using higher-order knowledge and skills.

Math maturity supports the understanding of and proficiency in math at the level of a high school math curriculum. Students begin to show an understanding of math-type arguments and proof.

Piagetian and Math formal operations includes being able to (1) recognize math aspects of problem situations in both math and non-math disciplines, (2) convert these aspects into math problems (math modeling), and (3) solve the resulting math problems if they are within the range of the math that one has studied. Such transfer of learning is a core aspect of Level 4.

Level 4 cognitive development can continue well into college, and most students never fully achieve Level 4 math cognitive development. (This is because of some combination of innate math ability and not pursuing cognitively demanding higher level math courses or equivalent levels on their own.)

Level 5. Abstract mathematical operations. Moving far beyond math formal operations. Mathematical content proficiency and maturity at the level of contemporary math texts used at the college upper division undergraduate level in strong programs, or first year graduate level in less strong programs. Good ability to learn math through some combination of reading required texts and other math literature, listening to lectures, participating in class discussions, studying on your own, studying in groups, and so on. Solve relatively high level math problems posed by others (such as in the textbooks and course assignments). Pose and solve problems at the level of one’s math reading skills and knowledge. Follow the logic and arguments in mathematical proofs. Fill in details of proofs when steps are left out in textbooks and other representations of such proofs.
Level 6. Mathematician. A very high level of mathematical proficiency and maturity. This includes speed, accuracy, and understanding in reading the research literature, in writing original research literature, and in oral communication (speak, listen) of research-level mathematics. Pose and solve original math problems at the level of contemporary research frontiers.

The age ranges in a general-purpose Piagetian cognitive developmental scale are approximate, with significant variations due to nature and nurture. The variation is likely much larger still for math cognitive development. General cognitive development is "thinking" development, and it does not refer to any specific knowledge and skills in particular areas, while math cognitive development refers to math-like and math logic-like thinking, and is dependent on making progress in learning and thinking in the language of mathematics.

One of the challenges in developing and implementing a good math education system is the extreme variation in the math cognitive development students at a particular age level.

For Reflection and Discussion. It is relatively easy to compare the specific basic math knowledge and skills of two students. There, one focuses on specific topics that the two have likely encountered in school. How does one go about comparing the math cognitive development of two students? Organize your own thoughts on this question, and compare/contrast your thoughts with those of some of your colleagues. How does one go about teaching a vertically structured discipline in a manner that fits the varying levels of math cognitive development (and, personal interest) of students at a particular age or grade level?

5. Math Maturity

It is helpful to think of math maturity as a mathematician's way of talking about math cognitive development. Math educators want their students to steadily gain in their level of math maturity, a term most often used at an upper high school or college level. Often it is used to help describe the prerequisites for a course in computer science or mathematics. The prerequisite is more than just having passed certain math courses. It is being able to learn math, think mathematically, and solve math-related problems at a level that depends on understanding and making relatively fluent use of math.

Quoting from the Wikipedia: "Mathematical maturity is a loose term used by mathematicians that refers to a mixture of mathematical experience and insight that cannot be directly taught, but instead comes from repeated exposure to complex mathematical concepts." Still quoting from the Wikipedia, other aspects of mathematical maturity include:

  • the capacity to generalize from a specific example to broad concept
  • the capacity to handle increasingly abstract ideas
  • the ability to communicate mathematically by learning standard notation and acceptable style
  • a significant shift from learning by memorization to learning through understanding
  • the capacity to separate the key ideas from the less significant
  • the ability to link a geometrical representation with an analytic representation
  • the ability to translate verbal problems into mathematical problems
  • the ability to recognize a valid proof and detect 'sloppy' thinking
  • the ability to recognize mathematical patterns
  • the ability to move back and forth between the geometrical (graph) and the analytical (equation)
  • improving mathematical intuition by abandoning naive assumptions and developing a more critical attitude

A considerably expanded discussion of Math Maturity is available in the following free book:

Moursund, D., & Albrecht, R. (2011). Using Math Games and Word Problems to Increase Math Maturity. Eugene, OR: Information Age Education. Download the PDF file from Download the Microsoft Word file from

Increasing Math Maturity

From time to time I rethink what I have written about math maturity. Recently I thought about four levels of knowing some math. I can use the following 4-point scale for myself, and I can help my students to make use of it for themselves.

  1. I know some memorized math facts, algorithms, and procedures. I can recognize some math problem situations in which I can apply these math facts, algorithms, and procedures. When I accurately identify such a situation and accurately apply my facts, algorithms, and procedures, I get a correct answer.
  2. I can recognize some math problem situations in which my memorized facts, algorithms, and procedures do not exactly fit. In some of these cases, I can break the problem situation into two or more sub-problem situations (represent the original problem situation as two or more sub-problem situations) that I can deal with using my memorized facts, algorithms, and procedures. This allows me to deal with some "unfamiliar" math problem situations.
  3. In a math problem situation that is not adequately handled by 1 and 2, I can sometimes figure out what I need to know and do in order to deal with the math problem situation. This may involve figuring out or looking up (finding from some source other than myself) some additional facts, algorithms, and procedures, and ways to use them.
  4. In math-related problem situations in disciplines other than math, I am steadily increasing my ability to do 1, 2, and 3.

I believe that math educators can learn something useful by thinking about a parallel between students learning reading and writing, and students learning math. In teaching reading and writing, there is a balance among the ideas 1-4 above that is quite a bit different from the balance achieved in teaching math. There are many reasons for this. One is that small errors in doing a sub-part of a math problem typically lead to incorrect results in trying to solve the overall problem. Small errors in spelling, grammar, understanding the meaning of a word, and so on typically do not have such profoundly incorrect results in reading and writing.

I also find it helpful to compare and contrast math education with art education, music education, physical education, and other curriculum areas in which learners can self-assess their levels of performance and can assess the levels of performance of their peers. What can math educators do to make this type of self and peer assessment receive more emphasis in math education? Would doing this improve math education?

That is, much of math is taught at a level of abstraction to which a student is not able to attach "real world" meaning and understanding. Much of reading and writing is not taught at such a level of abstraction. We do not expect children to read and write at the level of a professional lawyer!

One of the really important aspects of math is its precision and very carefully applied logic. The importance of such precision and logical argument varies from discipline to discipline. It is, for example, quite important in the sciences, and we know that math is quite important in the sciences. In education, we have quantitative studies and qualitative studies. The quantitative studies are more math-like and make more use of the types of precision and logical argument developed in math. The qualitative studies often better fit the "real world" nature of people, teachers, students, and our overall education environment.

These types of thoughts typically lead me back to cognitive development theory. My personal belief is that much of the math curriculum is taught at a level of abstraction, and assumes a level of math cognitive development, that is quite a bit above that of an average student. And much of how we teach math is not well designed to improve the math cognitive development level of students.

6. Computers in Math and Math Education

As noted earlier, computers are an important component of the math field and an aid to using math in many other disciplines. Computers are also an aid to teaching and learning math. Some of the key ideas are summarized in other IAE-pedia articles under the heading of Computational Thinking and under the general idea of Two Brains are Better than One. Computers can solve a wide range of the types of problems that we currently teach students to solve using paper-and-pencil methods. In addition, ICT is a big help in distance learning and in accessing information.

As we work on ideas of how to improve math education, we need to consider all five categories of goals listed above. In addition, we need to explore ways of assessing whether the changes we are thinking about making will lead to improvements in the math education of our students.

For Reflection and Discussion. Many people claim that the typical adult understands and uses math at about the 6th to 7th grade level. This is the level of just beginning to move into Level 4, Formal Operations. Think about yourself and the adults you know. Argue for or against this "6th to 7th grade level" conjecture.
For Reflection and Discussion. Here is a conjecture. Starting at about the 6th to 7th grade level, math education tends to be designed to "push the envelope" in terms of a student's level of math cognitive development or math maturity. Students with reasonably good natural ability in math, and who also are appropriately intrinsically and extrinsically motivated, tend to thrive in this environment. Many other students do not thrive in this environment. Based on your own insights into math education, argue for or against this conjecture.

The "I Can't Do Math" Phenomenon

"We have met the enemy and he is us." (Walt Kelly, American cartoonist, 1913-1973; statement by comic strip character Pogo in 1970.)
“When you are up to your neck in alligators, it's hard to remember the original objective was to drain the swamp." (Adage, unattributed.)

Let's continue with the question: "Can you do and use math at a level that meets your personal current needs and the current expectations you have for yourself?" Ideally, the great majority of students and adults who have been educated by our math education system would say "yes." We want and expect that our math education system will provide adequate and appropriate opportunities for most people to meet their personal math education needs and expectations.

However, many adults in the United States claim that they are not very good at math. Indeed, it is common to hear statements such as: "I was never much good at math" or "I can't do math" or "I hate math." Such statements, however, do not necessarily mean that the respondents' math knowledge and skills do not meet their personal needs and expectations. It may well be that our math education system tends to be successful at meeting the math needs of a great majority of learners, but at the same time leaves many learners with a feeling or attitude that they can't do math or that they hate math.

Like many others, I have spent time thinking about the "I can't do math, I hate math" phenomenon. I have formed a number of conjectures that might help us to better understand this situation.

Here is an example of some of my thinking. Some people are better at math than others, and some people have a greater interest in math than others. Math is but one of many disciplines that a student might study. Perhaps the math standards that are being set are beyond what many students can achieve or beyond what they are intrinsically motivated to achieve. Perhaps the standards are well above what many students believe to be their own personal math knowledge and skill needs and expectations.

This type of thinking leads me to conjectures such as:

  1. Perhaps the average human brain is not up to the learning tasks specified in the various national and state math standards.
  2. Perhaps the amount of time devoted to the math teaching and learning task is inadequate to the teaching and learning task defined by state and national math standards.
  3. Perhaps the methods of instruction, the materials being used, the informal and formal instructional environments, the nature and extent of teacher preparation, and so on, are not up to the teaching and learning tasks defined by state and national math standards.

I am sure that many readers of this document can add to this list of conjectures. Each of these conjectures is a researchable question. If even one of these conjectures is correct, knowing this will be quite helpful as we work to improve our math education system.

The next three sub-sections explore these three conjectures.

Math Expectations Beyond Average Human Capabilities

"Now here, you see, it takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!" (Lewis Carroll, pen name for Charles Dodgson; English author, mathematician, logician, and photographer; 1832–1898.)

The undamaged human brain is genetically wired for learning some math and math-related knowledge and skills. For example, very young children have a little number sense, such as being able to distinguish between two of an object and three of an object. Toddlers readily learn to orient themselves in their spatial environment, finding their way around different parts of a house. Such spatial skills are essential to a hunter-gatherer life style in which people had to forage for food and then find their way back to their clan.

Now, think about a child learning words for numbers. As an example, I have a young grandson who is quite bright. He can say in order the words one, two, three, … up to about sixteen. However, his understanding of these words is quite limited. He has some working understanding of one and two, and perhaps three. There is a large difference between being able to say words and having an understanding of what the words mean. This, of course, is true for both math words and non-math words.

By the time an average child enters the first grade, the child has developed a reasonable level of skill in using the number counting words to be able to say the number of objects in a small set. The child can do simple addition, such as 2 + 5 through a process of counting. Quite a few children have learned counting on either through their own discovery or through being explicitly taught by the time they begin the first grade.

Unschooled Children

There has been a reasonable amount of research on the math knowledge and skills of unschooled "street urchin" kids and on craftspeople who need to use some math in their work. They tend to develop the money-related, measurement-related, and other necessary math-related skills to survive and function in their environment.

The types of observations given above support the idea that students are capable of learning quite a bit of math. In terms of street urchins and craftspeople, the evidence is that they learn the math they need to use in order to solve the math-related problems they face.

Gene Maier's article on Folk Math gives interesting examples and argues that perhaps much of the math we are teaching in our schools today is irrelevant to the lives of most students.

Unreasonable Expectations

American Diploma Project Algebra II End-Of-Course Exam: 2008 Annual Report discusses a new attempt to determine how well students are learning Algebra II. Quoting from the report:

In the spring of 2008, nearly ninety thousand students across twelve of the fourteen states in the partnership took the ADP Algebra II end-of-course exam for the first time. This report, released as the scores from the first administration are reported to students, their teachers and their parents, provides an overview of the test as well as exam results from each of the participating states. Most states offered the test on a pilot basis this year, giving many educators a first look at the expectations of a rigorous Algebra II course assessment. States are in the process of developing policies for the use of the test. and how it will fit into their still-evolving high school assessment and accountability systems. And while most of the states in the partnership require, or plan to require, students to take Algebra II in order to graduate, these new requirements are being phased in over time and do not apply to most of the students who took the exam this year. Consequently, this year’s participation rates in Algebra II courses and in the ADP end-of-course exam vary from state to state. Therefore, comparisons of state results are neither meaningful nor appropriate at this time.

The results of this test suggest a huge mismatch between what the testers think students should be able to do and what the students were actually able to do. Average scores for various groups of students on the Constructed Response part of the test was 10.2%.

Ten point two percent! My reaction was, "You have got to be kidding!" I wonder what the exam makers thought about that score!

Here is a personal story:

Math educators know that it is easy to design math problem-solving tests that all but the most capable students will fail. Let me share a personal story on this. At the college level, there is a national Putnam Competition. It is a six-hour math test consisting of 12 problems. I was a math major in college and a very good math student. I took the Putnam test during my junior and senior years. I don't remember the fine details, but my general memory is of being able to solve one problem each year, and to have made some progress on a few of the other problems. In terms of the way the tests were graded, my scores were probably under 25%. Still, I suppose I ranked in the upper half of students taking the test. Evidently the problems are so hard that at least half of the students taking the test are unable to solve any problem correctly.

My conclusion is that if one sets math education standards based on being able to solve novel, challenging problems that are solvable with the math that one has studied, almost all students would fail. Since such a level of failure is not a good way to design and run a school system, math standards are set much lower than this level.

However, it is possible to teach specifically for gaining expertise in the types of problems used in the Putnam competition and in other national or international competitions. When I started teaching at Michigan State University shortly after having completed my doctorate in mathematics, I found that one of the faculty members searched out the best freshmen students with an interest in math and put them into a rigorous problem-solving training program. The Michigan State University Putnam team consistently placed in the top ten in the country as a consequence of this approach.

In summary, if we take students who are talented and interested in math, and give them lots of special tutoring in small groups, we can produce some very high math performers. This approach to math education makes a school (or country) look good in national and international competitions, but it is not a good measure of the success of our overall math education system.

For Reflection and Discussion. Think about the math that an average person uses in their everyday life. What parts of this can be "picked up" in a learn by doing mode, learning from one's peers, colleagues, and fellow workers, versus what parts require formal schooling?
For Reflection and Discussion. What are your personal thoughts about the results on the Algebra II report discussed above?

Multiple Intelligences

"An intelligence is the ability to solve problems, or to create products, that are valued within one or more cultural settings." (Howard Gardner; American psychologist and educator; 1943-.)

Likely you are somewhat familiar with various theories of Multiple Intelligence. Howard Gardner is well known for his work in this area. Logical/mathematical is one of the nine areas of intelligence Gardner has identified. See

People vary considerably in their "Math IQ," including how well and how fast they learn math, and how well they can use the math they have studied. Thus, the bottom five percent of students in our schools, measured in terms of Math IQ, learn math less than half as fast as do average students, tend to learn it poorly, and tend to peak out at about a 4th to 5th grade level or at a still lower grade level. At the opposite end of the math learning scale is a group of approximately the same size made up of students who learn math at least twice as fast and a lot better than the average students. Some students in this group go on to achieve doctorates in math and/or go into careers that make good use of math knowledge and skills.

This type of data raises the question of what math we should require students to learn "with understanding" in school. It is easy enough to prescribe math curriculum to be taught at various grade levels. We are currently making major changes in both the content and process via the Common Core State Standards Mathematics Initiative. It is another thing entirely to teach this curriculum so that students gain long-term knowledge, skills, and understanding.

George Polya was both a great math researcher and a great math educator. Here is a repeat of part of the George Polya quotation given near the beginning of this document:

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.
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 (Polya, circa 1969).

For Reflection and Discussion. Should a relatively low "Math IQ" keep a student from being about to graduate from high school? As you ponder this question, consider the fact that the average adult in the United States functions at approximately the ninth grade level in math. See

The same type of statement can be applied to any discipline of study. Each discipline is defined by the types of problems it has solved, the types of problems it tries to solve, its history, and so on. What distinguishes math from other disciplines is the breadth and depth of the discipline, and the "exactitude" of preservation and communication of results.

In math education we want students to learn to solve certain categories of problems. Thus, for example, we want students to learn to use the four basic arithmetic operations on integers, fractions, and numbers representation in decimal notation. But, that is not enough. We want students to transfer this knowledge and skill into problems in non-math disciplines. We want them to be able to use their arithmetic knowledge and skills to effectively deal with problem situations that they have not encountered before.

Addition, subtraction, multiplication, and division are, in some sense, mechanical skills. Machines have been built that can perform such operations much more rapidly than people. However, today's calculators and computers are not good at detecting errors in math-related reasoning in a problem or deciding whether math is going to be useful in attacking a certain"novel" problem.

Math is a difficult subject because it has great breadth and depth, because it is useful in so many disciplines of study, and because it requires students to deal with challenging problems that they have not previously encountered. Remember, however, that one can make a silimar comment about any well-established academic discipline. Perhaps the key issue is that we do not select other really important disciplines such as interpersonal skills, the fine and performing arts, physical education, and many other disciplines, require students to take a multi-year sequence of steadily increasingly difficult courses, and threaten them with not graduating from high school unless they accept and complete such challenging studies.

How does one educate students to solve novel problems they have not seen before? We know it is possible. However, we also know that students vary considerably in their interest in and ability to learn to solve novel, challenging math problems. For these and other reasons, our math education system has adopted standards that are far below what Polya would like.

Cognitive Development

Another approach to whether math education expectations are being set too high is to look at research in cognitive development. Piaget is probably the best known name in the field of cognitive development, but many other researchers have worked on understanding the changes going on in a human brain as it moves toward maturity. The changes are a product of both nature and nurture. A combination of native ability together with good learning opportunities and environments can produce research mathematicians and other people who excel in using math to help represent and solve problems in their careers.

My exploration of math cognitive development and our math curriculum suggests that there may be a large disconnect here. In brief summary, it appears that quite a lot of the math being taught starting about about the 6th or 7th grade is well above the math cognitive developmental level of a great many of the students. Areas falling into this category include percentage, ratio and proportion, geometry proofs, algebra, and probability.

I have written about this topic elsewhere. For example, see:

This mismatch between math education expectations and math cognitive development leads to many students taking a "memorize with little or no understanding, regurgitate, and forget" approach to learning math. It helps to explain some adult statements about their math education.

Math educators have long realized that a moderately rigorous high school geometry course tends to separate students into categories. This separates those who "can really do" math from those who cannot.

Here is a Paigetian-type geometry cognitive development scale developed by Diana and Pierre van Hiele about 50 years ago. The layout and numbering in this scale is similar to that used in the Math Cognitive Developmental Scale earlier in this document. There is no sixth level on the van Hiele scale.

Stage & Name Geometry Cognitive Developments
Level 1. (Visualization) Corresponds to Sensorimotor on a Piagetian scale. Students recognize figures as total entities (triangles, squares), but do not recognize properties of these figures (right angles in a square).
Level 2. (Analysis) Corresponds to Concrete Operations on a Piagetian scale. Students analyze component parts of the figures (opposite angles of parallelograms are congruent), but interrelationships between figures and properties cannot be explained.
Level 3. (Informal Deduction) Corresponds to Concrete Operations on a Piagetian scale. Students can establish interrelationships of properties within figures (in a quadrilateral, opposite sides being parallel necessitates opposite angles being congruent) and among figures (a square is a rectangle because it has all the properties of a rectangle). Informal proofs can be followed but students do not see how the logical order could be altered nor do they see how to construct a proof starting from different or unfamiliar premises.
Level 4. (Deduction) Corresponds to Formal Operations on a Piagetian scale. At this level the significance of deduction as a way of establishing geometric theory within an axiom system is understood. The interrelationship and role of undefined terms, axioms, definitions, theorems, and formal proof is seen. The possibility of developing a proof in more than one way is seen.
Level 5. (Rigor) Students at this level can compare different axiom systems (non-Euclidean geometry can be studied). Geometry is seen in the abstract with a high degree of rigor, even without concrete examples.

The work of the van Hieles provided solid evidence that the traditional and somewhat rigorous high school geometry course was over the heads of a great many high school students. The attempts at rigor and proof place such a course at the beginnings of Level 4. It may well be that this is above the level that about 2/3 of students have reached by the time they finish high school.

Moreover, the same type of analysis can be applied to the situation of the "algebra for all" movement and trying to push younger and younger students into a rigorous algebra course. It appears that ratio and proportion, computations with and manipulation of fractions, and the abstraction of algebra are quite far above the math cognitive development level of many of the students to whom we are trying to teach algebra.

This math cognitive development mismatch between students and beginning algebra and geometry courses is (in my opinion) a disaster for many students and their teachers. Many teachers realize that they cannot fail a large percentage of students in such courses. Thus, many math teachers are forced to "water down" these algebra and geometry courses so most students can pass. It is common that this passing grade can be achieved by some combination of (1) constructing single topic or very limited topics quizzes that most students will be able to pass, (2) offering extra credit activities, (3) grading for participation in class discussions, and providing similar activities that allow students to add to their total point score. The points gained may thus allow a student to fail every comprehensive test and still pass the course. Both the students and the teacher fall into this pattern of behavior based on the goal being that students pass the course rather than that students gain knowledge and skills that meet current and future personal needs and the needs of potential employers.

Forgetting What One Has Learned

There is another really important aspect of human mental capabilities. People forget much of the material they learn in school. They forget content that they learn in a memorize and regurgitate mode. They forget material that they do not understand. They forget material that they do not use in their future. Some good research has been done in the area of forgetting and how to teach/learn to decrease the amount of forgetting. Here is an example:

Rohrer, Doug, & Pashler, Harold (2007). Increasing Retention without Increasing Study Time. Retrieved 2/18/08:

Quoting the abstract of this paper:

Because people forget much of what they learn, students could benefit from learning strategies that yield long-lasting knowledge. Yet surprisingly little is known about how long-term retention is most efficiently achieved. Here we examine how retention is affected by two variables: the duration of a study session and the temporal distribution of study time across multiple sessions. Our results suggest that a single session devoted to the study of some material should continue long enough to ensure that mastery is achieved but that immediate further study of the same material is an inefficient use of time. Our data also show that the benefit of distributing a fixed amount of study time across two study sessions—the spacing effect—depends jointly on the interval between study sessions and the interval between study and test. We discuss the practical implications of both findings, especially in regard to mathematics learning.

I found this paper particularly interesting because it addresses the issue of forgetting. Students "forget" most of what is covered in their math courses. Since we know this, it behooves us to address the issue. At the current time, our math education system is doing a poor job of addressing this challenging problem.

Inadequate Instructional and Learning Time: Breadth Versus Depth

"The saddest aspect of life right now is that science gathers knowledge faster than society gathers wisdom." (Isaac Asimov; Russian-born American author and biochemist; 1920-1992.)

Even in a single discipline such as math, there is far more content knowledge than a person can learn in a lifetime. The collected math content is growing quite rapidly. Thus, we need to be wise in our selection of what math we put into the math curriculum at various grade levels. We must try to avoid information overload, and we must seek an appropriate balance between breadth and depth Appropriate answers on these issues vary with the student. One size does not fit all.

We know that if more time is devoted to math education and if students are taught by highly qualified math teachers in "reasonably sized" classes, then average math performance can be substantially improved. There is quite a bit of research literature on the value of individual tutoring and small classes.

The essence of this situation is individualization both in content and in feedback. Constructivism is a theory focusing on how learners build new knowledge and skills based on their current knowledge and skills. A good learning environment stretches a student by placing a major focus just above where a student currently is.

One way to think about constructivism is from a student's expertise level point of view. What he or she already knows can be considered as lower-order knowledge and skills. In some sense, it is "easy stuff." What the student is trying to learn is considered to be higher order knowledge and skills. It is "new, hard stuff."

This is, of course, an over-simplified representation of the very important idea of lower-order versus higher-order knowledge and skills. However, see the diagram below. From a student learning point of view, the "sweet spot" is the large dot. To push the level of a student's expertise above where it currently is, we need to place considerable teaching/learning efforts above the student's current level of expertise.

LO-HO dot.jpeg

As with "reading recovery" programs, there are "math recovery" programs in which students who are falling far behind their classmates are given extra math instruction in very small classes. Math recovery programs tend to be aimed at primary school students who are rapidly falling behind their fellow students in numeracy. We know that individual tutoring and intensive extra help in small classes can make a significant difference.

Unfortunately, such an approach is expensive and is unlikely to be used as a general approach to improving the math education being received by average students.

Probably you have heard the "inch deep, mile wide" characterization of the math education system in the U.S. This statement about depth versus breadth comes from an analysis of U.S. performance on international assessments. In countries where students tend to perform better than in the U.S. the curriculum has less breadth and greater depth.

This is an aside.

My mental model of what is being talked about is a cylinder that is a mile in diameter and an inch in height. Or, it could be a rectangular solid, with a base that is a mile by a mile square, and a height of one inch. Perhaps these are the types of mental images that the writers of the "inch deep, mile wide" description had in mind. In any case, I understand the concept of depth versus breadth in math. What I don't understand is measuring depth and breadth using inches and miles. (Use of the metric system would not be an improvement.) In my opinion, someone made a very poor choice of words in trying to quantify or measure breadth and depth in math.

Here is one conjecture of how we got into the great breadth, little depth math education curriculum. The possible goals and topics for the curriculum were developed by large teams of people who were working to make most of their team members happy. The compromises led to having many, many topics in the curriculum—far too many to treat in any reasonable depth.

This conjecture is only part of the situation. Textbook adoptions are generally done by teams. If a book is to be sold in a number of states, it has to include every topic that each state lists in its math benchmarks or standards. Thus, many topics are included in a book in order to help ensure adoption in a variety of states, even though many of the topics do not fit the needs of all states.

One way to write such a book is to make many of the topics somewhat independent of each other. That is, instead of having depth in the curriculum, have breadth of relatively independent topics.

Breadth of somewhat or almost completely independent topics has another value. The instruction can be built into relatively independent pieces. Students memorize with little understanding, pass a test over a small amount of material, and then move on to the next topic that depends very little on the previous topic. Teachers cover the book, students pass the tests and the course, and many students learn very little of lasting value.

The coverage approach is moderately successful. Our math education system has had many years of experience in writing texts and teaching courses that follow this coverage model. Our math assessment system and this math content and instructional approach have been aligned. However, a coverage model is not well designed to help students learn to solve challenging, novel problems.

The Common Core State Standards (CCSS) initiative discussed earlier in this document has an increased focus on depth and a lessened focus on breadth.

For Reflection and Discussion. Analyze your precollege math education experiences from the point of view of learning to solve novel and challenging problems versus learning to memorize and regurgitate math facts and to solve problems that were nearly identical to those that you had practiced on repeatedly. As you think about this topic, think about how much or how little you actually remember from such coursework.

Inadequately Researched Curriculum Content and Instruction

I have recently read research that suggests that overall learning is significantly improved by spending less time doing a long set of problems that are essentially all the same, and instead doing a mixed variety of the types of problems one has studied in the past.

When I read such material, I wonder whether it is based on carefully done research. I also wonder if the ideas will ever be widely implemented in our math education system.

Consider the basic question: Can our current math education system be significantly improved without extending the amount of time spent on math education in schools? Perhaps there is some easy "fix" such as better books, more required homework, or harder and more frequent tests. Another widely supported fix is merely the use of once or twice a year high-stakes tests, with students who don't pass the tests not being allowed to graduate from high school.

It is easy to think of ideas that might improve math education. However, it is less easy to think of ideas that have characteristics such as:

  1. Being researchable. One can design and implement research that is likely to produce convincing evidence of the value or lack of value in a proposed way to improve math education.
  2. Being scalable. This means the idea can be widely implemented if the research supports doing so.
  3. Being fiscally sound. This means that wide-scale implementation can be done at a "reasonable" cost—at a cost that is within the current funding plus any extra money that funding agencies might be willing to provide.
  4. Being strongly supported by the teachers who will implement it and the parents of the children who will be educated using the new curriculum and teaching methodologies.

For example, consider the idea of providing each student with a highly qualified math tutor. This is certainly researchable, and such research has already been done. However, it is not scalable (there are not enough highly qualified math tutors), and it is not fiscally sound (it would cost too much).

But wait... Perhaps it is possible to develop a Highly Interactive Intelligent Computer-assisted Learning (HIICAL) system that is as effective as a well-qualified human tutor. Progress in developing computer-based tutors is certainly occurring. For example, see If we could develop such software, then we would be able to satisfy all three criteria.

Does this sound good to you? What possible flaws in this approach occur to you? Here is a researchable question. What are the long term effects of learning math in a HIICAL environment? Here are a few difficulties that occur to me:

  • Education is a social, human endeavor. Might this much HIICAL warp a student's social development?
  • Is HIICAL math equally appropriate and effective for students from different cultures, income levels, races, religions, and so on?
  • Is HIICAL acceptable to parents, teachers, politicians, and other stakeholders groups?
  • Is this much computer use healthy for children?
  • If children learn most of their precollege math in a HIICAL environment, how well will they do as they go on to college and take math courses that are not taught using HIICAL?
  • If elementary school teachers are not directly involved in teaching math, how well will they do in using math in the other disciplines they teach? Might the teaching of the uses of math in other disciplines decrease in effectiveness?

The point is, it is not easy to think about the range of possible effects of a change, and then to carry out research on whether these effects will actually be produced and whether they will be positive or negative.

For Reflection and Discussion. While I was a college math student and later a college math professor, I noted that many faculty members make use of colored chalk. And, of course, teachers at all levels sometimes use colored chalk or colored white board markers in teaching math and other subjects. Analyze the following idea for improving math education: In order to improve math education, the U.S. Federal Government should provide an adequate free supply of good quality colored chalk and colored marking pens for all math teachers to use when teaching math.

Ideas for Improving Math Education.

"It is common sense to take a method and try it. If it fails, admit it frankly and try another. But above all, try something." (Franklin D. Roosevelt; 32nd President of the United States; 1882–1945.)

When Roosevelt became President in 1932, he and the country faced many problems. The status quo was not acceptable. A number of different ways of addressing these problems were explored, and some were implemented.

Nowadays, our education system is placing increased emphasis on research-based changes that have research-based evidence indicating they are likely to improve our education system. Sometimes such changes are successful, and oftentimes they are not. Our educational system is highly resistant to change, and it is difficult to achieve long-term high-fidelity implementation of research-based changes.

Every person is a researcher. We all routinely encounter personal problems (tasks, decision situations) and take action. We observe the results of our action, we learn from these results, and we often share our results with others. That is a form of research. In fancier terms, it is the foundation of "action research."

Some Personal Thoughts and Feelings

"Learning without thinking is labor lost; thinking without learning is dangerous." (Chinese Proverb.)

A number of the ideas in this document are shaped by my experiences as a university faculty member, first in Math, then in Computer Science, and then in Teacher Education. During my long academic career, I have had the opportunity to interact with and learn from many different students, teachers, and other people. I have come to believe in a need for some major changes in our educational system. The following list contains my current recommendations.

  • Empowerment of students.. I believe that our educational system needs to substantially increase its emphasis on students learning to take more responsibility for their own education. We want students to become independent, self-sufficient, lifelong learners. It is possible to increase emphasis on this, starting in the earliest grades, and gradually increasing the emphasis as students progress through school. As students learn to take more responsibility for their own learning, they are given more power to shape what it is they will learn and how they will demonstrate this learning.
  • ICT tools integrated into problem solving throughout the curriculum. I believe that the content of our curriculum needs to better include the use of Information and Communication Technology tools to represent and solve problems in each discipline that students study. We are spending a lot of time in school teaching students how to do by hand things that computers can do far faster and more accurately. Students and their teachers need to learn about Computational Thinking, people and computers working together to solve challenging problems and accomplish challenging tasks.
  • Authentic content, instruction, and assessment. I believe that our educational system can be improved by better aligning content, instruction, and assessment, making them more "authentic" with respect to today's world and what the world is apt to be like when students become adults. Much of our assessment system is not authentic (Wiggins, 1990). Consider how education would be changed if students were tested in an open book, open computer, open computer connectivity environment, and were educated to work and be assessed in such an environment.
  • Focus on student expertise. I believe that students can develop increased levels of expertise in areas that fit their own needs and interests, and in areas that are specified by others such as parents, teachers, potential employers, and so on. Many areas of potential expertise fall into both categories. Education can be improved by improving the quality and timeliness of feedback that is available. Good feedback provides information about current levels of student expertise and also what to do to improve this expertise.
  • Distance learning and computer-assisted learning.. I believe that all students should be learning how to learn in a distance learning environment and in a computer-assisted learning environment. These two environments are gradually merging. I expect that we will see continuing success in developing Highly Interactive Intelligent Computer Assisted Learning (HIICAL) systems that are delivered over computer networks. Already, within parts of the curriculum, HIICAL systems lead to significantly better learning results than are obtained by an average teacher working with a class or 20 to 30 or more students.
  • Communicating across the disciplines.. I believe that each well-developed discipline has its own special vocabulary, symbology, and aids to effective communication. Although they draw upon natural language and the general vocabulary that people use all the time, there are many discipline-specific aspects of communication. Thus, for example, when people are trying to say what math is, they often say math is a language, or perhaps they say that Algebra is the language of mathematics. I believe that our math education system is not very successful in helping students learn to read, write, speak, listen, and think in the language of mathematics.
  • Transfer of (math) learning. College_Student’s_Guide_to_Computers_in_Education/Chapter_6:_Learning_and_Learning_Theory I believe that transfer of learning is one of the most important ideas in education. In terms of math education, each major math idea (topic) can be taught in a manner that facilitates transfer of learning (and transfer of use) to other disciplines. The high-road/low-road theory of transfer is especially suited to this task. Moursund's free book Introduction to Using Games in Education contains an extensive treatment of high-road/low-road transfer of learning and a long list of problem-solving strategies that can be transferred across disciplines using this theory.

Each of these seven ideas can be a starting point for exploring what we are doing in math education and how well we are accomplishing our goals in various aspects of math education.

For reflection and discussion. Which one of the seven ideas given above do you feel is of most potential importance in improving math education, and why? Do you support the idea strongly enough to do something about it? If so, what can and will you, personally do?

Things Individual Teachers and Teachers of Teachers Can Do

“Never doubt that a small group of thoughtful committed citizens can change the world: indeed; it's the only thing that ever has.” (Margaret Mead; American cultural anthropologist; 1901-1978.)
"What do you want to contribute?" (Peter Drucker; American author and business management consultant; 1901–2005.)

When you view yourself as a math teacher or teacher of math teachers, what are your "signature" traits? What distinguishes you from other teachers of math? What do you want your students to learn that is an important aspect or part of you—such as your math and math education beliefs, your uses of math, or your understanding of math?

Remember, math is a human endeavor. What can you do to bring this "human endeavor" to your students? One way is to share your personal knowledge and insights into math as an important part of your life.

For example, perhaps one of your major interests and areas of expertise is ancient history. Then you might want your math students to learn some of this ancient history of math and math education. Your math signature trait might be that you occasionally "slip in" interesting tidbits from ancient history and how they are related to math and math education.

Or you might have a special interest in mental math computation. You strongly believe that being able to do mental math calculations rapidly and accurately is both fun and important. So you share some of your "tricks" with your students and you routinely demonstrate your own ability in this area.

My personal belief is that math education is improved by having math teachers who have some "special" interest in math—some signature traits that they bring into their everyday teaching to help personalize and add interest to their students' math learning environment.

This section is the start on a long list of quite specific things that one person—for example, a teacher or a teacher of teachers—can do to improve our math education system. As you browse this list, look for things that you are already doing. Think about how well they are working. If some work really well for you, share them with your fellow teachers.

Look for ideas that you would like to try. Select one and try in a reflective manner—in a personal action research manner. How can you tell if your implementation or use of the idea is making a positive difference? If it seems to you that it is making a positive difference, how can you communicate this to your professional colleagues?

Possible "Signature" Areas of Expertise

If you have a math education "signature" or trait, think about its role in your math teaching. Work on ways to make it an important aspect of what your students learn from you. If you don't have any such traits, develop one and do personal action research on its use in your math teaching. Some possible areas are given in the next few entries of this list.

A key idea in improving education is to have students, teachers, parents, and others involved as a "community of practice." As noted elsewhere in this document, math is a human endeavor. Teachers can improve math education by helping to get their students and others involved in the human activities of math, and in math as a community of practice.

  1. Calculators are a "fun" area for a signature level of expertise. There is, of course, the inexpensive 6-function solar-powered calculator with memory. Your expertise might include understanding solar batteries, so you can help your students learn about solar batteries. Your expertise might include understanding the memory features in a calculator (the M+. M-, MR, and MC) and how they are used in problem solving. Your expertise might include understanding how a calculator can calculate a square root. Your expertise might include understanding the repeat feature, used in repeated multiplication or addition. Your expertise might include understanding the calculator number line, and how it differs from the real number line. And, of course, there are a huge number of different types of calculators. See
  2. Origami.jpeg
    Paper folding, as in Origami or in making paper airplanes, has many geometry aspects. There are many Origami and paper airplane sites on the Web. In terms of learning math, think about symmetry, spatial sense, following a detailed step-by-step set of directions, and producing a product that one can be proud of and share with others. Could this be one of your math teaching signature areas of expertise?
  3. Model making. Perhaps you like to make physical or virtual (computerized) models of boats, airplanes, bridges, robots, and so on. Modeling is a fundamental aspect of what math is all about, and why math is so important. There are many important aspects of math that can be illustrated by reference to physical and virtual modeling and simulation.
  4. Math puzzle problem of the week. Each week, post a math puzzle problem on your classroom bulletin board or on the website you use with your classes. See Chapter 4 of Moursund's free book Introduction to Using Games in Education. Also, encourage your students to find and post such challenging puzzles.
  5. Geoboard.jpeg
    Geoboard. A geoboard is an inexpensive and versatile math manipulative. At a beginner's level it can be used in creating "named" geometric figures such as square, triangle, rectangle, and so on. However, there is a large amount of math that can be taught in a geoboard environment. A teacher can become a school-wide or district-wide expert in use of the geoboard in helping students at a wide range of grade levels to increase their understanding of math. See, and Explore a circular virtual geoboard at
  6. Math manipulatives—physical and virtual. As indicated in the geoboard listing, many math manipulatives come in both physical and virtual (computerized) versions. A math teacher might well have the use of math manipulatives as one of his or her signature areas of expertise. Math manipulatives help a learner to make many problems more concrete. They help in the visualization of problems as a learner makes progress in working using the "mind's eye." The National Library of Virtual Manipulatives is an excellent resource. Read an excellent article by Doug Clements on physical versus virtual manipulatives. Quoting from Clements' 1999 article: "Students who use manipulatives in their mathematics classes usually outperform those who do not, although the benefits may be slight. This benefit holds across grade level, ability level, and topic, given that use of a manipulative 'makes sense' for that topic. Manipulative use also increases scores on retention and problem-solving tests. Attitudes toward mathematics are improved when students have instruction with concrete materials provided by teachers knowledgeable about their use."
  7. Uses of math in (name a discipline). You know, of course, that math is quite important in many different discipline areas. Many math teachers have held non-teaching jobs (often with a high level of success) before deciding to become teachers. Others have hobbies in which they have developed a high level of expertise. Photography is an excellent example. If you are such a teacher, explore your current level of insight into roles of math in a non-math discipline where you have a lot of knowledge and experience. Add to that knowledge as seems appropriate. Then make use of these "real world" applications of math as you help your students to learn math. This might become one of your signature areas of expertise in teaching math.
  8. Math quotations. A quote can be thought of as a very short story. Often a short quote captures an important event in history. Quotations are often taken out of context, and this tends to make the author's intended meaning hard to discern. On your math bulletin board or website, post a math quote every one or two weeks. Select ones that you would like your students to understand. A few days after posting, spend a little time in class discussing possible meanings and current relevance, or have your students write on such questions in their math journals. The IAE-pedia has two good sources of math quotations: Math Education Quotations at and Quotations Collected by David Moursund at You also can do a Web search on math quotations or on math education quotations. You will find other useful sites such as and
  9. History of math. Each discipline has its own history. What do you think your students should be learning about the history of the development and use of math? You may find this to be a fun and rewarding topic to develop as one of your signature areas of expertise in math. See, for example, You might also be interested in a history of math education. How about a sliderule? Once a highly important and indispensable math tool, the sliderule has become "ancient history" during my lifetime.
  10. Computers and math. Many teachers are really "into" computers. It may well be that one of your islands of expertise is the use of computers to help represent and solve math related problems, or uses of computers to help teach math. Perhaps, with some extra effort in this area, you can know more about roles of computers in math and math education than anyone else in your school. You can routinely make use of this computer knowledge and skill in your math instruction. Your students will remember you as the computer math person.
  11. Klondike.jpeg
    Games. There are a huge number of games (video games, card games, board games, role-play games, etc.) that require careful thinking, planning ahead, use of knowledge of probability, and problem solving—all links in math problem solving. Many involve the direct use of math, such as in scoring, buying and selling goods, spatial orientation, generating and using random numbers, and so on. Moursund's book Introduction to Using Games in Education: A Guide for Teachers and Parents is specifically designed to teach problem solving through use of games.
  12. Different cultures and countries. Perhaps you have intimate knowledge of countries, cultures, and languages from countries outside of the United States. One of your math teaching signatures might be bringing in math education ideas from this particular aspect of your background.
  13. Computer animation. Starting in the earliest grades, students can learn to use Kid Pix or an equivalent graphics program, and various other software that permits creating and animating objects. There is a lot of math in this, and many kids find it to be fun. Related areas include animation-type programming languages such as Alice, Logo, [ Scratch, Squeak, and so on. See also LEGO Digital Designer.
  14. Computer Algebra Systems and other powerful math software packages. Examples include Maple and Mathematica.

Ideas from Dan Meyer

Meyer, Dan (March, 2010). TED Talk: Math Class Needs a Makeover. (11:39 video.) Retrieved 8/31/2010 from Quoting from the website:

Today's math curriculum is teaching students to expect–and excel at–paint-by-numbers classwork, robbing kids of a skill more important than solving problems: formulating them. At TEDxNYED, Dan Meyer shows classroom-tested math exercises that prompt students to stop and think.
Dan Meyer asks, "How can we design the ideal learning experience for students?" As a part-time Googler, a provocative blogger and a full-time high-school math teacher, his perspective on curriculum design, teacher education and teacher retention is informed by tech trends and online discourse as much as front-line experience with students.

Meyer has spun off his enlightening message–that teachers "be less helpful" and push their students to formulate the steps to solve math problems–into a nationwide tour-of-duty on the speaking circuit. He summarizes, "I teach high school math. I sell a product to a market that doesn't want it but is forced by law to buy it."

Teaching Math for Social Justice

Here is a useful reference on the topic:

Gutstein, Eric (2007). And That’s Just How It Starts: Teaching Mathematics and Developing Student Agency. Teachers College Record. Retrieved 2/23/09: Quoting from the article:

This article reports on a two-year qualitative, practitioner-research study of teaching and learning for social justice. The site was my middle-school mathematics classroom in a Chicago public school in a Latino/a community. A major pedagogical goal was to create conditions for students to develop agency, a sense of themselves as subjects in the world. My research suggests that students learned mathematics and began to develop sociopolitical awareness and see themselves as possible actors in society through using mathematics to understand social injustices. This research contributes to our understanding of how to create opportunities for students to develop agency in K-12 mathematics classrooms, and may also contribute to our knowledge of developing agency in any subject area.

Miscellaneous Other Areas

1. Ask your students to talk about and/or write about their current knowledge and skills in math relative to their personal needs. As your students talk about or write about this question, have them explain how they know what they are saying is correct. Have them provide specific examples. Use this activity as an aid to approaching the topic of self-assessment. Through this activity, both you and your students can gain increased insight into the quotation:
"When people cannot see the need for what’s being taught, they ignore it, reject it, or fail to assimilate it in any meaningful way. Conversely, when they have a need, then, if the resources for learning are available, people learn effectively and quickly." (Brown, J.S., & Duguid, P. (200). The Social Life of Information.)
2. Develop math area of expertise. Let's use number theory as an example. Number theory is rich in problems and results that are of potential interest to students at all grade levels. Perhaps number theory is one of your islands of math expertise, or an area in which you want to gain greater expertise. Probability is another great area, one that is full of problems that students find both challenging and fun.
3. Analyze your math teaching style, paying particular attention to the ideas of intrinsic motivation versus extrinsic motivation. Specifically raise this topic with your students. Help them to learn about intrinsic and extrinsic motivation. Solicit their insights into their current levels of intrinsic and extrinsic motivation in learning and using math.
4. Math is a language. That means that students can learn to read, write, speak, listen, and think in the language of math. Moreover, math is not a "foreign" language that one may have only limited opportunities to use. Rather, math is a routine part of the everyday activities of all of us. If you teach both math and some other subject(s), think abut how to have communication in math and the use of math be part of the non-math subject(s) you teach. This idea is to increase your students' use of quantitative thinking throughout their daily activities.
As a specific example, suppose you teach elementary school. Try out the following idea as you start the day with your students. With this sort of start, it then becomes appropriate to ask about and/or talk about using math in the various topics being taught during the day.
"Good morning students. When I got up this morning, I noticed it was 6:45. I needed to catch my bus at 7:30. It is an eight-minute walk to the bus stop, and I like to get there a couple of minutes early. That meant I had only 35 minutes to shower, dress, get some breakfast, and leave for the bus stop. I had to solve the problem of using that time wisely in order to get all of the things done that I needed to do."
"Today I am especially interested in math and problem solving. What math did I use this morning? Who has used some math so far today? Please share your examples with the whole class."
"Today we are going to look for uses of math in all of the things we are doing. For example, later today I will ask you to read three pages in your history book. But, before you read these pages, I will want you to figure out how long it will take you to read the three pages. That is a math problem!"
5. Learning to learn math by reading math can be an important aspect of one's math education. Reflect on what you do to help and to encourage your students to learn to read the math book or other written materials your students have access to in your math classes. Develop and implement teaching methods that lead to your students making significant gains in reading, understanding, and learning from written math content.
6. Do you have a personal Math Education Digital Filing Cabinet? This link shows you my personal example. If you are a teacher of preservice or inservice teachers, you can benefit by having such an electronic filing cabinet and helping your students to develop one for their personal use. Part of the goal is to facilitate and encourage sharing and the building of a professional community of math educators.
7. What is your current answer to the student math questions: "Why do we have to learn this? What good is it?" Develop some improved answers and teach them to your students throughout the math course, not just when a student raises the question.
8. Think about your personal approaches to determining a student's level of math cognitive development and/or math maturity. Name some specific math teaching techniques you use to increase the level of math cognitive development and/or math maturity of your students. Help your students to learn the concepts of math cognitive development and math maturity, and to recognize when they are making progress toward increasing their own competency levels.
9. When the original version of this document was being written in August, 2008, hundreds of thousands of people in the United States were in the process of losing their homes because they could not afford to make payments on the mortgage financing plans they had entered into. A variety of different "initial low interest and low payment" arrangements had been used to convince a very large number of people to borrow to buy a home. The mathematics of these deals was often confusing enough to be beyond the understanding of the borrowers. Similar types of statements hold for people borrowing on their credit cards, and then making the minimal monthly payments. My conclusion is that a very large number of people do not learn enough mathematics to successfully deal with the money aspects of their own lives. What can you, as a math teacher, do to help improve this situation?
For Reflection and Discussion. Many people see a large schism between the math they learned in school and the math they use in their everyday lives. Do you experience such a schism? If so, how does this affect your teaching of math, and your students' learning of math?

Video from Judy Willis

See for six short video segments discussing her ASCD book, Learning to Love Math. Quoting from the site:

Dr. Judy Willis, a board-certified neurologist and middle school teacher in Santa Barbara, California, has combined her training in neuroscience and neuroimaging with her teacher education training and years of classroom experience. She has become an authority in the field of learning-centered brain research and classroom strategies derived from this research.
Willis attended UCLA School of Medicine, where she remained as a resident and ultimately became chief resident in neurology. She practiced neurology for 15 years, and then received a credential and master's degree in education from the University of California, Santa Barbara. She has taught in elementary, middle, and graduate schools; was a fellow in the National Writing Project; and is currently an adjunct lecturer at University of California, Santa Barbara.

See her website at

Suggestions from Steven Leinwand

The following is quoted from an Op Ed from Education Week written by Steven Leinwand. It has been widely republished.

Leinwand, S. (1/5/2009). Moving mathematics out of mediocrity. Education Week. Retrieved 8/30/2013 from

The logic for the importance of improving school mathematics programs is reasonably unassailable. The country's long-term economic security and social well-being are clearly linked to sustained innovation and workplace productivity. This innovation and productivity rely, just as clearly, on the quality of human capital and equity of opportunity that, in turn, emerge from high-quality education, particularly in the areas of literacy, mathematics, and science. Applying the if-then deductive logic of classical geometry puts a strong K-12 mathematics program at the heart of America's long-term economic viability.
But the problems with mathematics in the United States are just as clear. A depressingly comprehensive, yet honest, appraisal must conclude that our typical math curriculum is generally incoherent, skill-oriented, and accurately characterized as "a mile wide and an inch deep." It is dispensed via ruthless tracking practices and focused mainly on the "one right way to get the one right answer" approach to solving problems that few normal human beings have any real need to consider. Moreover, it is assessed by 51 high-stakes tests of marginal quality, and overwhelmingly implemented by under supported and professionally isolated teachers who too often rely on "show-tell-practice" modes of instruction that ignore powerful research findings about better ways to convey mathematical knowledge. …
Fortunately, the solutions are as clear as the problems. The answers do not revolve around costly new initiatives. Moving beyond mediocrity does not have to mean new textbooks and supplemental programs, or a slew of new calculators and computers, or jumping on the latest bandwagon of benchmark assessments. Instead, our attention needs to focus on how effectively existing programs are implemented, how available technology is integrated and used to enhance the learning of skills and concepts, and why assessments that steal valuable instructional time must provide relevant information that is actually put to use to inform revisions and reteaching.

Suggestions from Arthur Benjamin

Arthur Benjamin is a mathematician, a mathemagician, and a math educator. His current views are that our secondary math education curriculum is very strongly slanted toward getting students ready for and into calculus. He feels that this is wrong. Instead, we should be focusing on probability and statistics. See his 3-minute 2009 TED Talk at

Suggestions from Michael Fullan & Ben Levin

Fullan, M., & Levin, B. (June, 2009). The Fundamentals of Whole-System Reform. Retrieved 8/14/09:

This article argues that significant improvement of our educational system depends on approaching improvements from a whole-system point of view. Quoting from the article:

Charter schools, Teach For America, and the Knowledge Is Power Program may have their merits, but they are not whole-system reform. The latter is about improving every classroom, every school, and every district in the state, province, or country, not just some schools. The moral and political purpose of whole-system reform is ensuring that everyone will be affected for the better, starting on day one of implementing the strategy. The entire system should show positive, measurable results within two or three years.
We have done this in Ontario, Canada, where we have had the opportunity since 2003 to implement new policies and practices across the system—all 4,000 elementary schools, 900 secondary schools, and the 72 districts that serve 2 million students. Following five years of stagnation and low morale, from 1998 to 2003, the impact of the new strategies has been dramatic: Higher-order literacy and numeracy have increased by 10 percentage points across the system; the high school graduation rate has risen 9 percentage points, from 68 percent to 77 percent; the morale of teachers and principals has improved; and the public’s confidence in the system is up. [Bold added for emphasis.]
This work is by no means completed, but the number of elementary students at low literacy levels has fallen by 50 percent, and the number of schools with a large percentage of students not meeting the high provincial standard has been cut by 75 percent.
Whole-system reform means focusing on a small number of core policies and strategies, doing them well as a set, and staying the course by not being distracted. It must be politically driven by leaders at the very top, such as is the case in Ontario, with Premier Dalton McGuinty. But these leaders also must understand, embrace, and participate deeply in implementation by putting in place a set of fundamental whole-system-reform strategies.

Suggestions from Ken Jensen

The following is quoted from an email message sent to the NCSM distribution list on 11/27/09 by Ken Jensen.

This article referring to a previous distribution to the NCSM distribution list seems to reinforce the need to prepare math teachers with a strong understanding of mathematical concepts as well as the instructional practices necessary to develop the concepts in the minds of students.
I have found that there are two struggles that math majors have when working with students. First, it is difficult for a math major who was able to easily take on these concepts to have empathy for students who struggle with them. Second, many math majors were successful in a procedural based classroom where getting the correct answer was virtually all that mattered. Many therefore struggle with teaching in a problem based classroom where the students are required to explain why the answer is correct and how it applies to the context. The example given from Mr. Fennel's response, "create a basic word problem or number-line representation of a simple fraction or math concept" is classic. Those who grow up in procedure based classrooms will usually struggle to represent meaning in a model such as this.
A teacher's understanding of both math concepts and math instruction is necessary if we are to ever develop the kinds of mathematical understandings that will create a mathematically literate society. This of course is necessary at all levels—elementary, secondary, and post secondary.

Michigan State University Teacher Education Study in Mathematics

A study led by William Schmidt of Michigan State University reports that the United States needs better-trained elementary and middle-school math teachers to compete globally. The Teacher Education Study in Mathematics (TEDS-M) surveyed more than 3,300 future teachers in the U.S. and 23,244 future teachers across 16 countries. Read the full 2010 57-page report, “Breaking the Cycle: An International Comparison of U.S. Mathematics Teacher Preparation.” Quoting from the Executive Summary:

The Teacher Education and Development Study in Mathematics (TEDS-M) examined teacher preparation in 16 countries looking at how primary level and middle school level teachers of mathematics were trained. The study examined the course taking and practical experiences provided by teacher preparation programs at colleges, universities and normal schools. (The study did not include what are often referred to as alternative programs.) Future teachers near the end of their programs were assessed both in terms of their knowledge of mathematics as well as their knowledge of how to teach mathematics (pedagogical knowledge). For the U.S. nearly 3300 future teachers from over 80 public and private colleges and universities in 39 states were involved. Data were collected over two years. The public colleges and universities were sampled and the data were collected in 2007 while the private data were collected in the spring of 2008.
The study reveals that middle school mathematics teacher preparation is not up to the task. U.S. future teachers find themselves, straddling the divide between the successful and the unsuccessful, leaving the U.S. with a national choice of which way to go.
The findings of TEDS-M additionally revealed that the preparation of elementary teachers to teach mathematics was comparatively somewhat better as the U.S. found itself in the middle of the international distribution, along with other countries such as the Russian Federation, Germany and Norway but behind Switzerland, Chinese Taipei (Taiwan – throughout this report) and Singapore.
U.S. future teachers are getting weak training mathematically, and are just not prepared to teach the demanding mathematics curriculum we need especially for middle schools if we hope to compete internationally. This is especially true given that 48 of the states are currently considering the adoption of the more rigorous “Common Core” standards

Technology-Supported Math Instruction for Students with Disabilities

Hasselbring, T., Lott, A., & Zydney, J. (2006). Technology-supported Math Instruction for Students with Disabilities: Two Decades of Research and Development. LD Online. Retrieved 6/30/2010 from Quoting from the article:

Most people would agree that a major goal of schooling should be the development of students’ understanding of basic mathematical concepts and procedures. All students, including those with disabilities and those at risk of school failure, need to acquire the knowledge and skills that will enable them to figure out math-related problems that they encounter daily at home and in future work situations. Unfortunately, there is considerable evidence to indicate that this objective is not being met, especially for children exhibiting learning difficulties. Since the first discouraging results of mathematics achievement reported by the National Assessment of Educational Progress (NAEP) in 1973, there has been little evidence to suggest that mathematics achievement has improved significantly, especially for students with disabilities.…
A variety of technologies are available to enhance students' mathematical competency by building their declarative, procedural, and conceptual knowledge. The remainder of this paper will review these technologies. This review will be guided by the NCTI Mathematics Matrix found at The matrix identifies six purposes of technology use for supporting student mathematical learning, including:
  1. building computational fluency;
  2. converting symbols, notations, and text;
  3. building conceptual understanding;
  4. making calculations and creating mathematical representations;
  5. organizing ideas; and
  6. building problem solving and reasoning.

Research on Studying Math

Roher, D. & Pashler, H.(2007). Increasing retention without increasing study time. Retrieved 6/27/2014 from

This five-page research article includes a discussion of math learning. It begins by noting that most of what one learns in a course is not retained very long. It argues that a change in study habits can make a very large difference in long-term retention.

Roughly speaking, the authors argue that the design of the seat work and homework in the typical math book is poor if one's goal is long-term retention. In the two paragraphs that follow, "Spacers" divide their study time into two sessions with a space in between. "Massers" mass their study time into one concentrated session.

Quoting from the article:

Because the experiments described thus far required subjects to learn concrete facts, it is natural to wonder whether the results of these studies will generalize to tasks requiring more abstract kinds of learning. To begin to explore this question, we have been assessing the effects of overlearning and spacing in mathematics learning. For example, in one experiment (Rohrer & Taylor, 2006), students were taught a permutation task and then assigned either three or nine practice problems. The additional six problems, which ensured heavy overlearning, had no detectable effect on test scores after one or four weeks. In another experiment with the same task (Rohrer & Taylor, in press), a group of Spacers divided four practice problems across two sessions separated by one week, whereas a group of Massers worked the same four problems in one session. When tested one week later, the Spacers outscored the Massers (74% vs. 49%). Furthermore, the Massers did not reliably outscore a group of so-called Light Massers who worked only half as many problems as the Massers (49% vs. 46%).
This apparent ineffectiveness of overlearning and massing is troubling because these two strategies are fostered by most mathematics textbooks. In these texts, each set of practice problems consists almost entirely of problems relating solely to the immediately preceding material. The concentration of all similar problems into the same practice set constitutes massing, and the sheer number of similar problems within each practice set guarantees overlearning. Alternatively, mathematics textbooks could easily adopt a format that engenders spacing. With this shuffled format, practice problems relating to a given lesson would be distributed throughout the remainder of the textbook. For example, a lesson on parabolas would be followed by a practice set with the usual number of problems, but only a few of these problems would relate to parabolas. Other parabola problems would be distributed throughout the remaining practice sets.

Math in Context

There are a number of different names (such as Realistic Math) for this concept of Math in Context. One of the basic ideas is to situate math in a context so that students have an increased chance to detect when they are making errors that lead to grossly wrong answers.

An example of this is provided in the way that scientists do many of their computations. They carry along units of measure in their computations. In essence, the types of problems that involve working with measurements provide a context. Carrying along the units of measure provides an aid to detecting errors in computation and thinking.

A variation on this is for students to learn to add context to a problem. Richard Feynman was a Nobel prize winning physicist. One of his strengths is that he could take a math problem or proposed (conjectured) theorem and visualize it in a meaningful way for the problem or proposed theorem in the context of physics. This then provided him with insight to help check answers produced from solving the problem in order to determine whether the theorem might be a correct theorem.

For a simple example, consider the equation x - 17 = 5. Before solving the equation, a student might think (might be provided with instruction in learning to think):

I think this is an equation involving numbers. My goal is to find a number that is a solution to the equation. I see the numbers 17 and 5. I wonder if an answer might be larger or smaller than 17? Aha! x must be larger than 17, because when I decrease it by 17 I get the positive number 5. … Hmm. I can sort of "see" x as a starting point on the number line, moving left on that line a distance of 17, and ending up at the number 5. …

This "math in the mind's eye" type of visualization and thinking is an importatn aid to learning and understanding.

Here are a few references on Math in Context.

MMM (n.d.). Modeling Middle School Mathematics: Mathematics in Context Video Series. Retrieved 5/18/09:

Mathematics in Context curriculum is designed so that lessons begin with a meaningful context and the math is extracted from that context. The context here involves a ladder positioned against a building, what it does, and for whom and when would a ladder be needed. The class moves outside where they help the school janitor position a ladder. By directing the positioning of the ladder, students see and describe how the angles and distances made by the ladder change when the ladder moves from an unstable position to a safe and stable position- underscoring the mathematical focus of this lesson - the tangent ratio.

Romberg, T. (September, 2006). The Promises of Realistic Math Education. Wisconsin Center for Educational Research. Retrieved 5/18/09:

Students in mathematics classrooms should not be considered passive recipients of ready-made math. Instead, students should be guided toward using opportunities to reinvent mathematics by doing it themselves.
That’s one of the principles underlying Hans Freudenthal’s concept of mathematics as a human activity. Students start with a context-linked activity.

Vos, P. (2007). The Dutch Mathematics Curriculum: 25 Years of Modelling as Part and Parcel of Mathematics Education. Retrieved 5/18/09:

ABSTRACT: In the Netherlands, mathematics education is intertwined with applications as a result of the inspirational work by Hans Freudenthal and his colleagues, who developed a treatise known as Realistic Mathematics Education (RME). In this paper I will present a retrospective on twenty-five years of curriculum revisions in the Netherlands, exemplified by two nation-wide projects that established new routines into modern Dutch mathematics education. The first project established a new mathematics curriculum for the lesser gifted students in grades 7-10. In this curriculum modelling serves as a vehicle for students’ construction of mathematical knowledge. The second project established an annual modelling competition for teams of students at senior secondary level in the social sciences streams.

Math, A Human Endeavor

Likely you have heard the statement that mathematics is a human endeavor. Indeed, there is an important [ mathematics book by that title. Math has a very long history and has made major contributions to the development of the current cultures and societies of our world. This history of math goes back to well before the development of reading and writing (which first occurred more than 5,000 years ago).

What do we want students to know about the history, beauty, the fun and joy, and the human endeavor aspects of math? My 6/20/12014 Google search of history of mathematics produced about 78 million hits. Quoting from the Main Page of the History of Mathematics:

Pythagorean Screenshop.png

For Reflection and Discussion. What was the world like 3,000 year ago? What geometry-related problems led the mathematicians of that time to study triangles containing a "right" angle. Who was Pythagoreas, who lived 700 years later? What was the world like at that time? Did the mathematicians of his time rediscover this theorem, or had the result been preserved for 700 years, passed down from generation to generation? Why is this theorem so important that it is still taught to students throughout the world?

The idea of communication in the language of math is included in the CCSS Math Standards. (See: and However, the general idea of learning to read math well enough to learn math by reading is not often made explicit. Also, it is not made explicit that one of the most important ideas in problem solving is to build on the previous work of oneself and others. In particular, there often is no emphasis on using the strategy "look it up" as an aid to learning and doing math.

Nowadays, the Web is very frequently used as one employs a "look it up" strategy. What do we want students to learn about the use of the Web and other information storage and retrieval devices as part of their math learning endeavors? What do we want students to learn about open book, open computer, and open connectivity as a key aspect of how one does math outside of school settings?

Games in Math Education

There are many good resources on the use of games in education. Here are a couple of free books from IAE:

Moursund, D. (3/6/2012). Play Together, Learn Together: STEM. Eugene, OR: Information Age Education. Retrieved 6/30/2014 from (Microsoft Word file) and (PDF file).

Moursund, D. & Albrecht, R. (2011). Using Math Games and Word Problems to Increase the Math Maturity of K-8 Students. Eugene, OR: Information Age Education. (210 pages). Retrieved 6/30/2014 from (Microsoft Word file) and (PDF file).

Moursund, D. (2007). Introduction to Using Games in Education: A Guide for Teachers and Parents. Eugene, OR: Information Age Education. Retrieved 6/30/2014 from (Microsoft Word file) and (PDF file.

Here is a link to recent 160 page report on games in math education assessment:

Institute of Play. (2014). The real work of a 21st century education. Retrieved 6/30/2014 from Quoting from the of Play website:

Now available for free download or print-on-demand, the first publication from GlassLab, “Psychometric Considerations in Game-based Assessment” explores the opportunities and challenges for psychometricians and measurement experts in using gameplay data to assess student learning, and proposes a design approach that links the process of game design with the process of assessment design.

Distance Education for Students and Teachers

A 8/12/09 Google search of "Cognitive Tutor" OR "I Can Learn" produced over 2.7 million hits. A 6/20/2014 Google repeat of the same search produced over 240 million hits. Highly Interactive Intelligent Computer-assisted Learning Systems (HIICAL) have come a long way in the past five years. In addition, we now have Massive Open Online Courses (MOOCs) capable of handling enrollments of hundreds of thousands of students in a single course.

Think about how the mass production and distribution of books changed educational systems throughout the world. It took hundreds of years for educational systems to become highly dependent on books. But now, a goal in education is to help students learn to read well enough so that they can learn by reading. In the U.S., this is a goal that current educators expect to be reasonably well accomplished by the end of the third grade. Our book-and-teacher-driven educational system uses books as a dominant information delivery system starting by about the seventh grade.

But now, we have intereactive books that include video, audio, simulations, and interactive graphics. We have in place educational systems based on this type of hardware and software that gather and analyze the work of hundreds of thousands of students. This type of research is used to incrementally improve the content, instruction, formative assessment and summative assessment in these courses.

We are all aware that education is a social endeavor. In terms of HIICAL and MOOCs, courses may have a face-to-face component (hybrid courses) and may require or provide for extensive Internet-facilitated interaction among students and with their faculty.

In addition, over the years, we have developed computer systems that can solve a very wide range of the same math problems students are taught to solve manually in K-14 math education. We have the technology and productivity capabilities to provide every student with a powerful Internet-connected computer with good access to this type of math software and to a huge and growing collection of HIICAL and MOOC resources. Quoting from my recent IAE Blog article on Grand Challenges in Math Education:

For more than two decades I have believed that we should provide all students with the computer capabilities of a modern laptop or tablet computer and with good connectivity to the Internet. My goal is to substantially improve students’ quality of math content education and math maturity education for life in our current and future Information Age. Within the discipline of math, course materials and other resources should include:
1. Excellent quality Highly Interactive Intelligent Computer-assisted Learning (HIICAL) math materials for both students and teachers that cover the K-14 math curriculum and other math content that students might want to explore on their own.
2. A substantial library of math-oriented books and other learning resources, including math problem-solving software. The collection of materials would include books, articles, videos, guided access to appropriate websites, and so on, designed to help students learn about roles of math in each of the disciplines they are required to study and/or might be interested in studying. Students need good opportunities to learn about what math they will need if they want to pursue various disciplines and careers after they finish high school.
3. Very good self-assessment materials that include diagnostic tests with links to appropriate remediation and to more advanced and challenging work.
4. An overall math curriculum of study that thoroughly integrates the resources and content from items 1-3 above into the content, formative assessment, and summative assessment of the required math courses.
5. Math teachers who are skilled in working in this 1-4 type of a highly individualized math education environment. See
For Reflection and Discussion. Reflect on your current knowledge, skills, and experience in learning from HIICAL and MOOCs, and in making use of such resources in your teaching. If you feel somewhat "out of date" in these areas, reflect on what you, personally, are doing to rectivity this situaton for yourself.


American Diploma Project Algebra II End-Of-Course Exam: 2008 Annual Report discusses a new attempt to determine how well students are learning Algebra II.

Boldt, M. (10/24/08). State can begin using new graduation tests. Retrieved 10/25/08: Quoting from the article:

An administrative judge's ruling that Minnesota can move ahead with its new high school graduation tests has school administrators worried.
They fear high school students won't have enough time to get the help they need to pass the exams.
Most troubling is the math test. This year's juniors will be the first to take the math graduation test; it will be embedded in the Minnesota Comprehensive Assessments.
Last year, nearly two-thirds of juniors failed a similar math exam.
"This is a daunting issue coming at us. These are almost 70 percent of our students, who might already be accepted to college, who are getting As and Bs ... who might not get a diploma," said Grace Keliher, director of governmental relations for the Minnesota School Boards Association.

Comment by David Moursund: Of course, the test grades will go up as teachers consciously or subconsciously teach to the new test. What is interesting to me is that the expectations in the new test are so different from the expectations of the teachers and the overall math curriculum. This article, in conjunction with the reference American Diploma Project Algebra II given above, provides evidence of a mismatch between the expectations being established and the reality of our current school system.

Borasi, R., & Fonzi, J. (2002). Foundations: A monograph for professionals in Science, Mathematics, and Technology Education. Professional development that supports school mathematics reform. National Science Foundation. Retrieved 10/28/08: Quoting from the Chapter Summary:

No single model of professional development emerges from the many successful examples reported in the literature on mathematics teacher education. Instead, we find many examples of worthwhile experiences that address the multiple needs of teachers engaged in school mathematics reform. In Chapter 1, we identified and discussed these needs, categorizing them as follows:
  • Developing a vision and commitment to school mathematics reform.
  • Strengthening one’s knowledge of mathematics.
  • Understanding pedagogical theories that underlie school mathematics reform.
  • Understanding students’ mathematical thinking.
  • Learning to use effective teaching and assessment strategies.
  • Becoming familiar with exemplary instructional materials and resources.
  • Understanding equity issues and their classroom implications.
  • oping with the emotional aspects of engaging in reform.
  • Developing an attitude of inquiry toward one’s practice.

Cavanagh, S. (3/28/08). Essential qualities of math teaching remain unknown. Education Week. Retrieved 12/9/08: Quoting from the article:

It is one of the most widely accepted axioms in math education: Good teachers matter.
But what are the qualities of an effective mathematics teacher? The answer, as a recent federal report suggests, remains frustratingly elusive.
Research does not show conclusively which professional credentials demonstrate whether math teachers are effective in the classroom, the report found. It does not show what college math content and coursework are most essential for teachers. Nor does it show what kinds of preservice, professional-development, or alternative education programs best prepare them to teach.

Coughlan, S. (5/20/2014). US 'in denial' over poor maths standards. BBC News. Retrieved 6/30/2014 from Quoting from the article:

The maths skills of teenagers in parts of the deep south of the United States are worse than in countries such as Turkey and barely above countries such as Chile and Mexico.
An international study of maths ability in the US shows how individual states would have performed if they were ranked against other countries, using the OECD's Pisa results as a benchmark.
The study also shows that privileged youngsters in the US, with highly-educated parents, are lagging behind similar youngsters in other developed countries.
This analysis, from academics at Harvard and Stanford in the US and Munich University in Germany, punctures the idea that middle-class US pupils are high achievers.

Dickerson, Kelly (11/18/2013). 'I'm Not A Math Person' Is No Longer A Valid Excuse. Business Insider. Retrieved 9/15/2014 from Quoting from the article:

Contrary to popular opinion, a natural ability in math will only get you so far in studies of the subject.
Research published in Child Development found that hard work and good study habits were the most important factor in improving math ability over time.

Fenwick, Chris (n.d.). UCL's University Preparatory Course in Science and Engineering. Retrieved 12/6/07 from Quoting from this website:

Surprising as it may sound the learning of mathematics is not just about learning to 'get the right answer'. It is also (amongst other things) about being able to think mathematically and read mathematically, and then being able to show how you develop your ability, reading and thinking. Consequently, as part of the coursework you will need not only to be able to do the mathematics set but also be able to describe exactly the process by which you went about doing such mathematics.
Hence, throughout the course you will need to demonstrate your developing mathematical thinking, technical and reading ability by :
solving specific mathematical problems
adopting the approach of reading mathematics. This will be done by interpreting technical text, mathematical expressions, solutions to mathematical problems, diagrams, etc...
studying and learning how you go about working on, solving and hence, learning mathematics
Specifically, the course aims to help you develop the following abilities :
  1. the ability to solve appropriate mathematical problems
  2. the ability to construct appropriate mathematical proofs
  3. the ability to read mathematically by interpreting/describing mathematical text, expressions, solutions and/or proofs as appropriate, and demonstrate this through written and/or oral work
  4. think mathematically by identifying mathematical patterns and use these to extend given mathematics
  5. the ability to critically analyse and discuss issues in mathematics, as well as your learning of mathematics
  6. the ability to work individually and in groups on the topic of mathematics
  7. the ability to improve &/or extend any aspect of 1) - 6) above.

Good, Thomas L., Burros, Heidi Legg, & McCaslin, Mary M. (2005). Comprehensive school reform: A longitudinal study of school improvement in one state teachers college. Retrieved 3/14/08 from This provides a nice summary of some of the research on comprehensive school reform.

Kadiec, A., & Friedman, W. (2007). Important, but not for me: Kansas and Missouri students and parents talk about Math, Science and Technology Education.Public Agenda. Retrieved 9/20/07:

Klein, A. (5/30/08). Ed. Dept. says enrollment nears milestone: NCES report projects growth in the number of students through 2017. Education Week. Retrieved 6/3/08 from

Public school K-12 enrollment in the US is projected to hit 50 million for the year 2009-2010. One of the things I found interesting is:
The report also found that the percentage of school-age children with one or more parents who have attained at least a bachelor’s degree has also soared over the past quarter-century. In 1979, 19 percent of students had a parent who had attained that level of education. By 2006, it was 35 percent.
This is interesting to me because this increasing level of education of parents should be leading to an increasing level of student performance in schools.

Klein, D. (2003). A brief history of American K-12 mathematics education in the 20th century. Retrieved 8/13/08 from The following is quoted from this article:

By 1996, the NSF clarified its assumptions about what constitutes effective, standards-based education and asserted that:
  • All children can learn by using and manipulating scientific and mathematical ideas that are meaningful and relate to real-world situations and to real problems.
  • Mathematics and science are learned by doing rather than by passive methods of learning such as watching a teacher work at the chalkboard. Inquiry-based learning and hands-on learning more effectively engage students than lectures.
  • The use and manipulation of scientific and mathematical ideas benefits from a variety of contributing perspectives and is, therefore, enhanced by cooperative problem solving.
  • Technology can make learning easier, more comprehensive, and more lasting.
  • This view of learning is reflected in the professional standards of the National Council of Teachers of Mathematics, the American Association for the Advancement of Science, and the National Research Council of the National Academy of Sciences.

Landers, J. (2/8/2009). Texas school reformers try to learn lessons from Finland. The Dallas Morning News. Retrieved 2/12/09: Quoting from the article:

By the time Finland's children complete the ninth grade, they speak three languages. They have studied algebra, geometry and statistics since the first grade. And they beat the pants off students from just about everywhere else in the world.
In math, science, problem solving and reading comprehension, Finland's 15-year-olds came out at or near the top in international tests given in 2000, 2003 and 2006. Even the least among Finnish students – the lowest 10 percent – beat their peers everywhere else.

Lockhart, P. (2002). A mathematician’s lament. Retrieved 6/20/09:

Mehta, S. (6/11/08). Students likely to fail high school exit exam can be identified as early as 4th grade, study says. The authors use the findings to question the wisdom of spending millions to tutor older students struggling with the test. Retrieved 6/13/08:,0,2483597.story.

This article is relevant to math education, since high stakes testing at the high school level in math is increasingly common in the United States.

Mooney, N., & Mausbach, A. (2008). Align the design: A blueprint for school improvement. ASCD. Retrieved 3/15/08: Takes an appropriate multidimensional approach. Quoting from the Introduction to the book:

The five processes that we use for school improvement are not new to teachers and administrators who diligently seek to raise the bar for their schools. We call these five essentials the blueprint processes.
  1. Establishing a mission, vision, and values that guide the general direction of the school and its future actions,
  2. Using data analysis, which includes both collecting and interpreting data, for better decision making,
  3. Using school improvement planning to guide goals, strategies, decisions, and action steps and to create a working plan for the school,
  4. Reshaping professional development to become the engine of school improvement, and
  5. Differentiating supervision of teaching and learning to monitor how processes are working inside classrooms.

Moursund, D., & Sylwester, R., eds. (March, 2013). Common Core State Standards for Education in America. Eugene, OR: Information Age Education. See for the Microsoft Word document and for the PDF.

Recently, Information Age Education has published a series of ten IAE Newsletters on CCSS. These ten newsletters have been integrated into a book, Common Core State Standards for K-12 Education in America. The set of newsletters has been augmented by an Appendix, Goals of Education in the United States.

This book is an experiment in a new way to put together an edited collection of writings. Chapter 10 of the book contains a brief introduction to a number of topics that are suitable for making into additional chapters. Potential authors are encouraged to contact David Moursund ( or Robert Sylwester ( about submitting a chapter based on Chapter 10 ideas or other ideas of their own choice. The book will “grow” as new chapters are integrated in.

Moursund, D.G. (June, 2006). Computational thinking and math maturity: Improving math education in K-8 schools. Eugene, OR: Information Age Education. Retrieved 9/16/07:

NCTM (2010). Linking research & practice: The NCTM Research Agenda Conference report. Retrieved 5/12/2010:,_Issues,_and_News/Research/Linking_Research_20100511.pdf.

This 56-page report represents the work of a large number of people who participated in a 2008 4-day meeting funded by the National Science Foundation. They worked their way through 350 mathematics education practitioner-generated questions and reduced these to a manageable set, and then down to 10 questions. Quoting from the report:

It is also important to note that the questions presented in this document constitute what some might consider "a" research agenda for mathematics education. It is not our intention to imply that this set of questions constitutes "the" research agenda for mathematics education.

Nodding, N. (2004). Learning from our students. Kappa Delta Pi Record. Retrieved 10/9/08:

This is an excellent article discussing the fact that many students lack the math ability to preform at the math levels our school systems expect of them. Quoting from the article:

We find it especially difficult to consider the possibility that some people have little aptitude for, say, mathematics. The great mathematician Poincare (1956, 2041) asked, "How does it happen there are people who do not understand mathematics?" Yet, he proceeded honestly in a fascinating essay to describe his own difficulty with chess. Despite his incredible mathematical aptitude, he was a poor chess player. He also had little aptitude for drawing and got a zero on the admission test required by the Ecole Polytechnique. Wisely, the admissions committee opted to disregard this score (which, by the rules, automatically disqualified him) and admit him on the basis of his outstanding achievements in mathematics. For some people-including those who try hard-certain academic studies are a mystery and remain extremely difficult.
Instead of reflecting on our requirements and considering ways in which we might build on students' real interests, many of us strive mightily to motivate students and to assure them that they will achieve whatever standard we set-if they just try hard enough. I no longer believe this. The slogan "All children can learn" is popular today, but empty until we say what "all children can learn." Can all children learn to interpret great literature? Can all learn to play the violin well? Can all learn algebra and geometry in a meaningful way?

Polya, George (circa 1969). The goals of mathematical education. Mathematically Sane. Retrieved 9/16/07:

Rohrer, D., & Pashler, H. (2007). Increasing retention without increasing study time. Current Directions in Psychological Science. Volume 16, Number 4. Retrieved 9/19/07:

Science News (2/25/09). Public Schools Outperform Private Schools in Math Instruction. ScienceDaily. Retrieved 2/28/09: Quoting from the article:

In another “Freakonomics”-style study that turns conventional wisdom about public- versus private-school education on its head, a team of University of Illinois education professors has found that public-school students outperform their private-school classmates on standardized math tests, thanks to two key factors: certified math teachers, and a modern, reform-oriented math curriculum.
Sarah Lubienski, a professor of curriculum and instruction in the U. of I. College of Education, says teacher certification and reform-oriented teaching practices correlated positively with higher achievement on the National Assessment of Educational Progress (NAEP) exam for public-school students.
“According to our results, schools that hired more certified teachers and had a curriculum that de-emphasized learning by rote tended to do better on standardized math tests,” Lubienski said. “And public schools had more of both.”

SIAM (n.d.). Moody's Mega Math Challenge Contest. Retrieved 4/25/09: This is an open-ended, realistic, applied math modeling problem focused on a real-world issue. Quoting from the website:

The purpose of the Moody's Mega Math Challenge, in addition to being a contest for the best, brightest and most creative minds, is to elevate high school students' enthusiasm and excitement about using mathematics to solve real-world problems and to increase students' interests in pursuing math-related studies and careers in college and beyond. Moody's and SIAM are interested in improving the pipeline of young people into studies and careers in applied mathematics, and encourage students to participate in this contest as an educational process.
Authentic math contest:

Stansbury, M. (3/3/08). U.S. educators seek lessons from Scandinavia. eSchoolNews. Retrieved 3/5/08:;_hbguid=31475690-290f-4e70-8ce4-2742f7b52b83&d=top-news. Quoting from the article:

A delegation led by the Consortium for School Networking (CoSN) recently toured Scandinavia in search of answers for how students in that region of the world were able to score so high on a recent international test of math and science skills. They found that educators in Finland, Sweden, and Denmark all cited autonomy, project-based learning, and nationwide broadband internet access as keys to their success.
What the CoSN delegation didn’t find in those nations were competitive grading, standardized testing, and top-down accountability—all staples of the American education system.
In all three countries, students start formal schooling at age seven after participating in extensive early-childhood and preschool programs focused on self-reflection and social behavior, rather than academic content. By focusing on self-reflection, students learn to become responsible for their own education, delegates said. [Bold added for emphasis.]

Wiggins, G. (1990). The case for authentic assessment. Practical Assessment, Research & Evaluation, 2(2). Retrieved 9/16/07:

Links to Other IAE Resources

This is a collection of IAE publications related to the IAE document you are currently reading. It is not updated very often, so important recent IAE documents may be missing from the list.

IAE Blog

All IAE Blog entries.

Moursund’s 1987 futuristic visions of math education.

Grand Challenges in math education.

An introduction to college math placement testing.

Education for the future.

Transfer of learning.

21st Century skills.

A technology developmental line, and applications to math education.

A serious problem situation with math word problems.

More about a poor way to teach the solving of math word problems.

In problem solving; think before you act. Above all, first understand the problem!

Some things brain science research tells us about learning and doing arithmetic.

Teaching kids real math with computers (17 minute TED video).

Home and school environment—and games—in the math education of kids.

Large study shows females are equal to males in math skills.

IAE Newsletter

All IAE Newsletters.

IAE-pedia (IAE's Wiki)

Home Page of the IAE-pedia

Popular IAE Wiki Pages.

Math word problems divorced from reality.

An introduction to College Math Placement Testing. Retrieved 6/23/2014 from

Computational thinking. Retrieved 6/23/2014 from

Communication in the language of mathematics. Retrieved 6/23/2014 from

Folk math. Retrieved 6/23/2014 from

Free online math education staff development course. (This is a very detailed syllabus suitable for self-study.) Retrieved 6/23/2014 from

Good math lesson plans. Retrieved 6/23/2014 from

Math education digital filing cabinet. Retrieved 6/23/2014 from

Math education wars. Retrieved 6/23/2014 from

Math maturity. Retrieved 6/23/2014 from

Math problem-based learning. Retrieved 6/23/2014 from

Personal professional development for educators. Retrieved 6/23/2014 from

Problem solving. Retrieved 6/23/2014 from

Transfer of learning. Retrieved 6/23/2014 from

What is mathematics? Retrieved 6/23/2014 from

Word problems in mathematics. Retrieved 6/23/2014 from

I-A-E Books and Miscellaneous Other

David Moursund's Free Books.

Moursund, D. (2012). Good Math Lesson Plans. Eugene, OR: Information Age Education. Retrieved 6/23/2014 from (PDF file) and (Microsoft Word file).

Moursund, D. (Math Maturity Distance Education Course. Eugene, OR: Information Age Education. Retrieved 6/23/2014 from (PDF file) and (Microsoft Word file).

Moursund, D. & Sylwester, R. (2013). Common Core State Standards for K-12 Education in America. Eugene, OR: Information Age Education. Retrieved 6/23/2014 from (PDF file) and (Microsoft Word file).

Sylwester, R. and Moursund, D. (2012). Creating an Appropriate 21st Century Education. Eugene, OR: Information Age Education. Retrieved 6/23/2014 from (PDF file) and (Microsoft Word file).

Moursund, D. (2012). Using Brain/Mind Science and Computers to Improve Elementary School Math Education. Eugene, OR: Information Age EWducation. Retrieved 6/23/2014 from (PDF file) and Word file).

David Moursund's Learning and Leading with Technology Editorials

Author or Authors

The initial version of this document was written by David Moursund.