Communicating in the Language of Mathematics

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Mathematics consists of content and know-how. What is know-how in mathematics? The ability to solve problems. (George Polya, How to Solve It, a New Aspect of Mathematics.)

Extra Short Summary

Communication in math involves making use of the processes of reading, writing, speaking, listening, and thinking as one communicates with one's self, other people, computers, books, and other aids to the storage, retrieval, and use of the collected mathematical knowledge of the world. Current precollege math education systems have substantial room for improvement in helping students learn to communicate effectively in the "language" of mathematics.

Short Summary

Math is a huge discipline with great breadth and great depth. The discipline existed before the development of written languages. As you know, reading and writing—including the development of math notation and math vocabulary—have contributed immensely to the discipline of math.

Many people consider math to be a language. It is not a general purpose language, such as English or Spanish. Rather, it is a discipline-specific language. Each discipline has developed specialized vocabulary and its own ways of communication that are specific to the discipline. Consider, for example, music notation and a person learning to sight read music. Perhaps you have seen and heard examples of the language of basketball or the language of football. On televised broadcasts of games one sometimes gets to see diagrams of plays and hear the language used by the coach to communicate plays and give directions to players.

A summary of arguments supporting the idea that math is a language is available in Logan (2000) and a more recent discussion is available on the Web. Certainly, math is an area in which one can learn oral and written communication and can learn to think using the vocabulary, symbols, and ideas of the language of math.

Now, add to this concept the word communication in the field called Information and Communication Technology (ICT). Note that in the United States this field is more often called Information Technology (IT), while the rest of the world tends to use the term ICT. The communication aspects of ICT open up a whole new dimension in every academic discipline.

Logan (2000) argues that the Internet is a language and that computer programming languages (collectively) are language. Clearly the whole field of Computer and Information Science and its applications in ICT strongly overlap with the discipline of math. Indeed, in a number of colleges, Computer and Information Science is organized within the Department of Mathematics. Thus, it is appropriate to consider the language aspects of Computer Science and ICT as we consider communication in math.

Here are some of the more important ideas in this document:

  • Although one can spend a lifetime studying math and still learn only a modest part of the discipline, young children can gain a useful level of math knowledge and skill via "oral tradition" even before they begin to learn to read and write. Oral and tangible, visual communication in and about math is an important part of the discipline.
  • Reading and writing are a major aid to accumulating information and sharing it with people alive today and those of the future. This has proven to be especially important in math, because the results of successful math research in the past are still valid today.
  • Reading and writing (including drawing pictures and diagrams) are powerful aids to one's brain as it attempts to solve challenging math problems. Reading and writing also help to overcome the limitations of one's short-term memory.
  • The language of mathematics is designed to facilitate very precise communication. This precise communication is helpful in examining one's own work on a problem, drawing upon the previous work of others, and in collaborating with others in attempts to solve challenging problems.
  • Our growing understanding of brain science is contributing significantly to our understanding of how one communicates with one's self in gaining increased expertise in solving challenging problems and accomplishing challenging tasks in math (and in other disciplines).
  • Information and Communication Technology (ICT) has brought new dimensions to communication, and some of these are especially important in math. Printed books and other "hard copy" storage are static storage media. They store information, but they do not process information. ICT has both storage and processing capabilities. It allows the storage and retrieval of information in an interactive medium that has some machine intelligence (artificial intelligence). Even an inexpensive handheld, solar-battery 6-function calculator illustrates this basic idea. There is a big difference between retrieving a book that explains how to solve certain types of equations and making use of a computer program that can solve all of these types of equations.
  • We all understand the idea of a native language speaker of a natural language. Students learning an additional language will often progress better when taught by a native language speaker who can fluently listen, read, talk, write, and think in the language, and who is skilled in teaching the language. We prefer that this teacher be fluent in a "standard" version of the language and not have a local accent and vocabulary that would give pause to many native speakers of the same language. The same idea holds in math education. The math educational system in the United States is significantly hampered because so many of the people teaching math do not have the math knowledge, skills, and math pedagogic knowledge—and, most important, level of fluency—that would classify them as being math education native language speakers.

The last bulleted item points to a major weakness in our current math educational system. A great many of our children are being taught math by teachers who are ill prepared to deal with the complexities of being a successful teacher in this discipline. This topic is addressed in more detail throughout the remainder of this document.

Reading & Writing Across the Curriculum

This document is intended mainly for teachers of math teachers and for preservice and inservice teachers of math. Oral and written communication are recognized as being an important part of the core of a modern education. All preservice teachers learn about the need for students to learn to read and write—both in general, and within each discipline they study. Reading across the curriculum is a common theme. Sometimes this expression is taken to mean both reading and writing, but often writing does not receive as much emphasis as reading.

Many students find it is quite difficult to reach or exceed contemporary standards in reading and writing in a natural language such as English. Even for those who graduate from high school and go on to higher education, reading and writing can still be major challenges. Many students find that they have to undertake remedial work in this area when they enter college, trade school, or an occupation.

“Contemporary standards” are usually set so that a significant percentage of students do not meet the standards. Failure to meet standards is often then considered to be the fault of the teacher not teaching well enough or the student being lazy.

I am particularly interested in students learning to read and write in the discipline of mathematics. For example, I am interested in whether typical students learn to read math well enough so they can use this skill to learn math by reading their textbooks and other books. I am interested in whether students learn to write in the discipline of math well enough so they can communicate mathematically (in writing) with themselves, their teachers, and others. An immediate complication is that many people learn more easily through discussion, demonstration, or guided practice than by reading.

As I have explored this topic, I have done some comparing and contrasting with reading and writing in other areas, such as music and chess. I have also thought about how Information and Communication Technology (ICT) is affecting or should be affecting reading and writing across the disciplines. ICT allows experiences akin to personal discussion, demonstration, and guided practice.

While this document focuses on communication in math, it draws on ideas from other curriculum disciplines and lays groundwork for other people to explore the topic of communication in other disciplines. It also explores current and potential impacts of ICT on communication in math and in other disciplines.

Writing Math to Learn Math

The general idea of writing to learn cuts across the curriculum. There is a substantial amount of research and practitioner knowledge about having students write math in order to learn math.

A 5/19/08 search of "writing to learn" returned abut 90,000 hits. A search on writing to learn math returned about 20,000 hits. Here are some examples of available articles.

Totten, Samuel (2005). Writing to Learn for Preservice Teachers. National Writing Project. Retrieved 5/19/08:

The paper is based on a survey study of 104 teacher education programs spread across the United States. The focus is on all preservice teachers learning more about the process of writing across the curriculum. Here is a paragraph quoted from the paper:

The fact that only four respondents require all preservice candidates to take a separate course in process writing indicates that faculty of many colleges of education do not see the value in a course that focuses on writing and writing-to-learn strategies. That another four require such a course only for English teachers points to a belief that a separate course cannot be justified for teachers across the curriculum. Some faculties may still be uninformed about how writing is an integral tool for assisting students to comprehend more deeply and clearly what they are studying; they may be unaware of the research that underscores the value of incorporating writing-to-learn strategies in every discipline. [Bold added for emphasis.]

Jones, Alexandria—Pseudo name (8/21/08). Writing to Learn Math: Let's Play Math. Retrieved 5/19/08: Contains a nice assortment of links to writing to learn math materials.

Options for Writing in Math (n.d.). (Adapted from Marilyn Burns, Writing in Math Class, Math Solutions Publications, 1995.) Retrieved 5/25/08:

El-Rahman, Madiha (n.d.) The Effects of Writing-to-Learn Strategy on the Mathematics Achievement of Preparatory Stage Pupils in Egypt. Retrieved 5/19/08: Here are a couple of examples of activities quoted from the paper:

Pair Share This is a very simple activity to use when the teacher senses that the student does not understand the lesson. He stops and asks them to explain what is giving them trouble. After the students “Free - write“ for a couple of minutes, they share their writing with their classmates. This can help to remove their confusion (Burchfield and others, 1993).
Journal Writing This is a diary-like series of writing assignments. Each assignment is short and written in prose rather than in the traditional mathematical style. The students can write in their journals: daily goals, rational for learning any concepts, and the strategies used to solve problems (Bagley, 1992: 660). It can give both the teachers and the students great insight into a student’s progress (Potter, 1996: 184).

Russek, Bernadette (n.d.). Writing to Learn Mathematics. Retrievedd 5/25/08 from

Correlation Between Language Arts and Mathematics

Boaler, Jo & Staples, Megan (2008). Creating Mathematical Futures through an Equitable Teaching Approach: The Case of Railside School. Teachers College Record Volume 110 Number 3, 2008, p. 608-645. Retrieved 12/8/09 from

This research study looked at the teaching of high school math in three different schools over a period of several years. The researchers were particularly interested in the effects of not grouping students by math ability or previous math performance. Thus, in one of the high schools, all first year high school students began with first year high school algebra and then all proceeded to geometry. Based on the data from the three high schools that were studied, the argument is that this is a good thing to do.

Here is an interesting tidbit quoted from the Boaler and Staple study:

The correlation between students’ scores on the language arts and mathematics sections of the AYP tests, across the whole state of California, was a staggering 0.932 for 2004. This data point provides strong indication that the mathematics tests were testing language as much as mathematics. This argument could not be made in reverse as the language tests do not contain mathematics. [Bold added for emphasis.]

This very high correlation seems to be supportive of Keith Devlin's assertion (in his book, The Math Gene) that anybody who has the brain power to become literate in a natural language can learn math.

Sparks, Sarah D. (2/17/2011). Studies find language is key to learning math. Education Week. Retrieved 2/19.2011 from Quoting from the article:

New research shows a lack of language skills can hamstring a student’s ability to understand the most fundamental concepts in mathematics. A series of studies led by Susan Goldin-Meadow, a psychology professor at the University of Chicago, found that profoundly deaf adults in Nicaragua who had not learned a formal sign language could not accurately describe or understand numbers greater than three. While hearing adults and those who used formal sign language easily counted and distinguished groups of objects, those who used only self-created “homesigning” gestures could not consistently extend the proper number of fingers to count more than three objects at a time, nor could they match the number of objects in one set to those in another set.

Quoting from the Susan Goldin-Meadow article:

Does learning language change the way we think about number? The exact quantities to which words like “seven,” “eight,” and “nine” refer seem so basic it is hard to imagine that we might need the word “seven” to have the concept seven. But evidence from groups who have not been exposed to conven- tional numerical systems suggests that language, particularly the numeral list in a count routine, may be importantly involved in the ability to represent the exact cardinal values of large sets.
The Mundurukú (1) and Pirahã (2) are Amazonian people in rural Brazil whose languages do not contain words for exact numbers larger than five (the Mundurukú) or any exact number words at all (the Pirahã).* Adults in these cultures have not been reported to invent ways to communicate about the large numbers for which they do not have words. In addition, these groups do not display a robust ability to match sets exactly with respect to number, except when a one-to-one correspondence strategy is readily available (e.g., pairing each object in one visible row with a corresponding object in a second visible row) (3). The absence of a linguistic model for representing exact number (in this case, a count list) could explain the difficulties Pirahã and Mundurukú adults have representing large exact numbers (2, 4–6). However, their difficulties could just as easily be explained by the absence of culturally supported contexts in which exact number must be encoded (7, 8). To disentangle these possibilities, we investigated the numerical abilities of individuals who lack a linguistic model for number but who live in the numerate culture of Nicaragua: “homesigners.”

Specific Problem

The specific problem situation being addressed is that U.S. precollege math education is not as good as most American taxpayers would like it to be. It definitely is not as successful as math education in a number of other countries. Possible reasons are many, and many people are dedicated to improving the American system.

Considerable literature addresses the problems of math education and how to improve math education. This IAE-pedia contains some of these documents.

Relevant substantial research has been done in many disciplines such as math, music, chess, etc. K. Anders Ericsson is a world leader in this research field. The link is to a short article on expert’s long-term working memory that summarizes some of the key ideas in teaching and learning. All teachers (and, indeed, all students) can benefit by having some knowledge of this field.

The document you are currently reading focuses specifically on communication in math. This includes looking at some related aspects of ICT, brain science (chunking and expertise especially), and empowering students and their teachers.

This document is written specifically for use in preservice and inservice math education courses and workshops. The focus is on the idea that the discipline of math includes a language that we call the language of mathematics. We want students to learn to read, write, speak, listen, and think creatively in the language of mathematics. In essence, the goal is for students to become mathematicians at the level of the math they have studied. We want them to learn to use effectively their math content knowledge and skill to solve challenging problems and accomplish challenging tasks that are amenable to effective use of the math they have studied—not to mention their learning the math skills they will need to navigate through all the financial and life-decision hazards and opportunities they will face.

The goal of this document is to encourage and support discussion and deep thought followed by constructive action. A good use of this document in a preservice or inservice teacher education course would be to have students read it in advance of a class meeting and form their personal opinions on some of the ideas. During class, students would then share their insights and ideas in small group and whole class discussion. A follow-up activity might be having students continue the discussion in an online environment, write about this article in their math journals, do research on one specific idea in the article that came up during the in-class discussions, or develop some instructional materials that could be used to help implement their ideas.

To support the intended use, from time to time this document contains a question suitable for personal reflection or for discussion in a workshop or class. Here is an example:

For reflection and discussion: Drawing upon your knowledge of yourself and other people you know, analyze your levels of expertise in the areas of reading, writing, speaking, listening, thinking, and problem solving in mathematics. You might find it helpful to use the terms fluency and/or expertise in doing this analysis. Identify your relative strengths and weaknesses. Think about how our math education system contributed to your relative strengths weaknesses. What does your analysis suggest in terms of possible ways to improve our math education system?
Be tolerant of your first draft; you will gain many ideas when you and your colleagues share your first attempts.

Introduction to Reading and Writing

The development of reading and writing about 5,200 years ago was a major milestone in human history. From then on, there has been an accelerating pace of change in societies of the world brought on by the accumulation and sharing of data, information, knowledge, and wisdom. Major accelerating inventions in this process are the printing press, electronic communication (telegraph etc.), the computer, and the Internet and Web. The totality of this accumulation is huge and currently is perhaps doubling every five to ten years. The Web, all by itself, is a virtual library with many times the content of the largest physical libraries on our planet. It is continuing a rapid pace of growth. More content is added each day than a typical person can read in a lifetime.

The development of general-purpose written languages brought with it a start in the development of a written language for the discipline of mathematics. Over thousands of years, the discipline of mathematics and its language have grown and matured. Math educators support the idea of students learning to read, write, speak, listen, and think creatively in the language of mathematics. All of these aspects of communication contribute to representing and solving math problems.

Developing fluency (read, write, speak, listen, understand, and think) in the language of mathematics certainly has some similarities to doing the same thing in a natural language such as English. However, there are considerable differences, and some are discussed in this article.

An example of a similarity is provided by the challenge a high school student or adult faces trying to learn a foreign language. It is not too hard to memorize a large number of words and phrases. However, many second language learners of high school age and older find it is hard to learn to think and gain verbal fluency in a foreign language. Research strongly supports the value of starting to learn a second language at a much younger age and being taught by native language speakers who are skilled teachers.

For reflection and discussion: You know that children are exposed to math as they gain oral communication skills well before they start school. Our school system starts formal math instruction at the earliest grade levels. Thus, we cannot attribute our lack of success in math education to not starting early enough! Why do you suppose that this large amount of math instruction over many years produces such poor results for many students?
For reflection and discussion: There are many disciplines that school students are capable of learning. What is there about mathematics that justifies the many years of math coursework required of all students? Think of possible arguments for having less required coursework in math, thereby freeing up time for more coursework in other areas that might be of more specific interest and importance to some students. In thinking about and discussing this topic, try to give examples of recent times when it has been beneficial to you to draw upon your knowledge of high school algebra and geometry to deal with problems outside of a school setting.

Learning to Read and Write in a Natural Language

The undamaged human brain is genetically "wired" for learning oral and visual communication. Children learn to understand, talk, and think in spoken language supplemented by gestures long before they reach kindergarten. They learn whatever language or languages commonly used in their environment. Thus, children growing up in a bilingual or trilingual home and community environment will become orally bilingual or trilingual.

When children start on the process of learning to read and write, they already have a substantial level of oral fluency. Young children are displaying a high level of creativity and intelligence as they communicate orally. Learning to read and write draws heavily on the ability to create meaningful utterances and understand spoken language.

Young students also have a substantial and growing knowledge of the world. This often provides help in discerning the meaning of a sequence of words or a sentence. Students have considerable ability to extract meaning from context and from pictures. Pictures in story books help students in extracting meaning from the written presentation.

As a child moves through the first few grades of elementary school, the child continues to gain verbal fluency. A combination of informal learning outside of school and the formal schooling adds thousands of words per year to the child's oral fluency repertoire. This steadily growing oral fluency provides a growing foundation for building fluency in reading and writing.

Being around those whose oral and written language fluency is quite a bit greater than the child’s substantially aids the process. Think of this as role modeling. The child can observe and hear oral communication being routinely used. As a child attempts to imitate and participate in this oral communication; immediate feedback is provided by proficient speakers of the language.

In many homes, young children are read to frequently. Research strongly supports that this and other adult role modeling in reading and writing makes a major contribution to children's future linguistic development.

Even with a strong supportive background, most students take many years of instruction and practice to develop a level of reading and writing expertise that meets contemporary standards. Thus, most colleges and universities require entering freshman to take a year sequence in writing. In the U.S., this course is often called English Composition.

The idea of "contemporary standards" is important. magazines and newspapers are written at or below 10th grade reading level. A great many adults who graduated from high school have considerable difficulty reading above this level. (This document’s Flesch-Kincaid readability is approximately 10th grade level. The Gunning Fog index suggests high school completion is needed for reading the document.)

Similar observations have been made about average adult performance in other areas. For example, many adults who graduated from high school function in math at about the 6th to 7th grade level.

Such observations point to the major difference between standards that governments and others want to set, and what is readily achievable by our current educational system. The educational leaders in each academic discipline have created "standards" that they feel students should achieve. While each discipline's standards may appear to be reasonable or desirable when viewed individually, the collected set of standards far exceed what an ordinary student can achieve, possibly in part because the experts may overestimate the needs of the general public. For example, consider the terms a capella (music), undecidability (math), and zugzwang (chess)—all of which have ‘real-life’ implications.

Moreover, while students demonstrate they have achieved a standard by passing a particular test, the reality is that forgetting occurs (in many cases, quite rapidly) so that even in a test-based standards system, relatively few people continue to meet the standards as they become adults.

For reflection and discussion: Why do you think it is so hard to learn to be a good writer, when it is relatively easy to learn to talk in a manner that meets contemporary standards? (Hmm. Does an average high school graduate meet the oral fluency standards that our schools would like to set?) Next, think about the same question for learning math.
For reflection and discussion:We know that a person's knowledge and skills in an area degradates over time if the knowledge and skills are not being used. A different way of saying this is that students forget much of what they (supposedly) learn in school. Think about some personal examples. In what ways does our educational system acknowledge that people forget, and attempt to accommodate the forgetting?

Some Brain Theory: Seven Plus or Minus Two

Written and oral language are aids to thinking. Thinking is sometimes described as "talking silently to oneself." Such thinking allows a person to contemplate various actions and possible outcomes of the actions—without actually carrying out the actions. You probably know some people who "think out loud." In addition, having s research subject talk aloud while solving a problem is a useful research technique to gain insight into a person's thinking process during problem solving.

This section provides some general information about the human brain and some roles of language in thinking and problem solving.

Humans have three types of memory:

  • Sensory memory stores data from one’s senses, and for only a short time. For example, visual sensory memory stores an image for less than a second, and auditory sensory memory stores aural information for less than four seconds.
  • Working memory (short-term memory) can store and actively process a small number of chunks. It retains these chunks for less than 20 seconds.
  • Long-term memory has a large capacity and stores information for a long period of time. Over time, information stored in long-term memory tends to become more and more difficult to remember—that is, to retrieve—if it is not used very often. However, traces of these stored memories continue to exist, and they can be an aid as one relearns what was learned in the past.

When you work to solve a problem, you bring information and ideas about the problem into your working memory. You consciously manipulate this information and ideas. Research on working memory indicates that for most people the size of this memory is about 7 ± 2 chunks (Miller, 1956).

This means, for example, that a typical person can read or hear a seven-digit telephone number and remember it long enough to key it into a telephone keypad. The word chunk is very important. For example, the sequence of four digits 1 4 9 2 can be thought of as four distinct chunks. However, it can also be thought of as one chunk—the year when Columbus discovered America. It can also be represented as two chunks-14 and 92. The point is that appropriate chunking of ideas and information is a powerful aid to overcoming limitations of short-term memory.

The names of the number words in Chinese are, on average, shorter than the corresponding names of the number words in English. In terms of digit recall, 7 English digits are about the same length as 9 Chinese digits. Native language speakers of Chinese have a greater short term memory digit span than native language speakers of English. The referenced article is about the research work of Stanislas Dehaene.

Your brain is very good at learning meaningful chunks of information. Think about some of your personal chunks such as constructivism, multiplication, democracy, complex numbers, transfer of learning, and Mozart. Undoubtedly these chunks have different meanings for me than for you. Moreover, our chunks are of different size. Research indicates that experts in a discipline have more chunks and much larger chunks (in their discipline) than do novices.

As a personal example, my chunk “multiplication” covers multiplication of positive and negative integers, fractions, decimal fractions, irrational numbers, complex numbers, functions (such as trigonometric and polynomial), matrices, and so on. My breadth and depth of meaning and understanding were developed through years of undergraduate and graduate work in mathematics. Others might connect “multiplication” with the pressure of having to learn the multiplication table before a test or recall “go forth and multiply—a paraphrase of various biblical phrases—and related jokes.

Here is another example. You "know" what the number line is. When you think about the number line, your mind probably conjures up some sort of picture, perhaps a line with equally spaced marks on it, and the marks labeled with digits such as … -4, -3, -2, -1, 0, 1, 2, 3, 4, …. You can think of this as a chunk. My number line mental chunk is not the same as yours. Through years of studying and using math, my mental math number line chunk has grown to include rational numbers and irrational numbers. It has grown to include irrational numbers that are called transcendental numbers. Moreover, my number line chunk is closely tied in with chunks about numbers in different bases, different sizes of infinity, some results from the area of math called number theory, complex numbers, and other components of math. You probably modified your conception of “number line” as soon as you were reminded of these other numbers; I remind myself of these numbers as I’m visualizing a number line.

It is useful to think of a chunk as a label or representation (perhaps a word, phrase, visual image, sound, smell, taste, or touch sensation) and a collection of pointers. A chunk has four important characteristics:

  1. It can be used by short-term memory in a conscious, thinking, problem-solving process.
  2. It can be used to retrieve more detailed information from long-term memory.
  3. It serves as an anchor for constructing new knowledge and skills. (It lies at the root of constructivism learning theory.)
  4. It is a key to higher levels of expertise in a discipline. High-level experts in a discipline have a large repertoire of chunks in that discipline. They think and solve problems making use of these chunks. Furthermore, as proficiency in a disciple increases, chunks become bigger. Such a larger chunk can include both information about a problem situation and possible actions to take in attempting to solve the problem.

In terms of communication, chunks and chunking are a critical aspect of how one's working memory communicates with one's long-term memory. A chunk may have a name. As indicated above, the name "multiplication" allows my short-term memory to access (retrieve) a large "multiplication" chunk in my long-term memory. For another example, the name of one of your friends allows your short-term memory to access a chunk of information about your friend.

However, seeing your friend in person, seeing a picture of your friend, of smelling a particular smell can also trigger this information retrieval. No words are used in this retrieval process. This indicates we all have and use an extensive non-verbal language. And, of course, you are familiar with the language of gestures. This language can be very extensive. Think, for example, of American Sign Language.

There has been substantial research on roles of building and using chunks in gaining a high level of expertise in a discipline. One idea that has emerged is that, in some sense, high-level experts are able to use such chunks as a kind of an extension of their working memory. That is, within the area of high-level expertise, these experts are able to function as if they had a working memory that is much larger than “normal.” For instance, driving a route new to you in a city requires concentration. As you become familiar with the route, you are able to mentally construct the route so that you automatically allow for speeds, lane changes, possible trouble, etc., and you are able to devote much of your consciousness to other matters.

In brief summary, creating, storing, and using chunks of information are essential to building a high level of expertise in an area. Such chunking ties in with oral, written, and nonverbal communication and thinking, and it is applicable in every academic discipline. Expertise in a discipline is dependent on having an extensive repertoire of large chunks specific to that discipline.

However, there are many chunks that have interdisciplinary use and value. Suppose, for example, that a person develops a high level of expertise in understanding and making use of careful, logical, rigorous arguments in a discipline such as math. Many of the chunks involved in this type of problem solving in math carry over to other disciplines such as the sciences and law.

For reflection and discussion: Think about some discipline in which you have a reasonably high level of expertise. Identify some chunks in your brain that you use in this discipline. You know that many other people who have lower expertise in this discipline lack entirely or have much less robust chunks. Also, think about whether you make use of this chunk is other disciplines.

Learning Mathematics

The healthy human brain is genetically wired for learning some math and math-related knowledge and skills. Howard Gardner has identified logical/mathematical and spatial as two types of intelligences. Spatial intelligence can be quite important in attempting to solve some types of math problems. Learn more about Howard Gardner at

Very young infants have a little number sense, such as being able to distinguish between two of an object and three of an object. Recent research suggests that perhaps this is an innate ability of infants to sense that something is wrong when they are expecting to see two objects and are presented with one or three of the objects. Stanislas Dehaene, who was mentioned earlier in this article, is a world leader in this type of math-related brain research.

A research experiment might involve showing an infant two objects set on a small stage. A screen comes down in front of the stage and then goes up. There seems to be an innate expectation that the number of objects will not have changed. Researchers can time the increased eye time fixation of a viewer when a change occurs.

Toddlers who can crawl 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. At an early age he could say in order the words one, two, three, … up to about sixteen. However, his understanding of these words was quite limited. At the time, he had 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 they mean. This, of course, is true for both math words and non-math words. The above example suggests that quantity is a relatively abstract idea that is a challenge to learners.

Attributes such as color, size, shape, numerosity (number, quantity) and so on are all learning challenges. Numerosity and other math-related words and concepts have the added challenge in that our contemporary standards tend to expect a high level of perfection.

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 determine and say the number of objects in a small set. The child can do simple additions, 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.

This is a major step in learning math and allows relatively young children to do math at a higher level than people growing up in a hunter-gatherer society whose natural language is mathematically quite limited.

Stanislas Dehaene has shown that the ability to estimate amounts—an innate 'number sense' that human beings have in common with various other species—forms the basis for our mathematical (abstract reasoning) and arithmetic (calculation) abilities. The latter ability does, however, require a well-developed system of symbols—a language system. Evidence for this duality has been found not only in scientific experiments but also in anthropological research. One example is the language of the Amazonian Mundurukú tribe, which has words for numbers only up to five. The Mundurukú are not able to perform precise calculations with larger numbers, but they can approximate and compare larger amounts.

Thus, the average child starting school has a beginning level of understanding of the number line. However, the number line is a quite complex math concept. We expect students to learn about fractions and decimal fractions. We expect students to learn about both positive and negative numbers. We expect students to learn to perform arithmetic on the various types of numbers on the number line. If we go back 4,000 years, only the most learned mathematicians of their time could effectively handle the range of math we are expecting grade school students to learn.

For reflection and discussion: One difference between natural language and the language of mathematics is the degree of precision required in communication. In many situations, small errors in the use of natural language do not destroy the overall correctness or effectiveness of a communication. Explore this idea and its math education implications.

Oral Tradition

This section is a work in progress.

To a very large extent, math is taught using methods that might be described as "oral tradition." Students learn to recognize some math symbols and math words. However, many do not learn to read math at a level that allows them to learn math by reading math. It is only when students reach the more advanced high school math courses that there is a significant emphasis on learning the math by reading the math book and other resource materials. Thus, a great many students graduate from high school with a very limited ability to learn math by reading a math book or other math resource materials.

Math education makes extensive use of "word" or "story" problems. Here, a problem that can perhaps be represented and solved mathematically is presented in a natural language statement that may contain few or perhaps no math symbols and vocabulary. A student must meet the challenge of understanding the problem, translating it into math notation and vocabulary, solving the math problem, and translating the results back into the context of the original natural language statement of the word problem.

This is an important component of math education, since many of the problems that can be addressed using math are not explicitly stated in math notation. Indeed, we now have very powerful Computer Algebra Systems that can solve a very wide range of problems that are stated in math notation. Thus, a major challenge in math education is to prepare students to deal with the tasks of determining when math might be useful in solving a problem and in representing such a problem using math notation.

Comment by Jen Jensen 4/28/09

The following is quoted from an email message sent to the National Council of Supervisors of Mathematics on 4/28/09:

One of the most difficult components to today's problem-based texts such as CMP or Core Plus is the reading component. I have run numerous collaborative coaching cycles with math teachers over the last two years with the goal of understanding how to implement effective reading strategies in a math classroom. These usually involve reading teachers and their expertise is always beneficial. I also look back to one of the original goals of public education and see that our charge is to create an educated citizenry that can participate intelligently in the democratic process.
This being said, I believe that English and Social Studies teachers are critically important in the education of our children and should be paid at the same level of math and science teachers. To me the issue is not the subject being taught but the quality of the teacher doing the teaching. Highly qualified does not mean quality. We need a process for raising the quality of our teaching force-such as the coaching model, and then we need to remove those teachers who refuse to participate and improve their practice.
Ken Jensen
Instructional Math Coach
Aurora Public Schools

Native Natural Language Speakers and Native Math Language Speakers

When children grow up in a bilingual or trilingual natural language environment, they grow up bilingual or trilingual. This idea is often incorporated into schools. Some students get to attend a bilingual elementary school in which the content areas are taught in the student's second or third language. It is highly desirable that the teachers teaching the content areas be native language speakers of the language(s).

We all understand the idea of a native language speaker of a "standard version" of a natural language. We expect the native language speaker to think in the language, know the culture of the people who speak the language, and have a native accent. Moreover, we prefer that this person not have a strongly regional accent and vocabulary. We want learners to be learning a relatively standard version of the language and with a relatively standard accent.

Now, take this idea and carry it over into math education. What might we mean by a "native math language speaker" of mathematics?

First, consider the following quote from George Polya, a world-class math educator and math researcher. In a talk to elementary school teachers, Polya said:

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. (Polya, 1969)

In summary, the term native math language speaker means someone who has a high level of fluency in reading, writing, speaking, listening, thinking, and creative problem solving in the discipline of mathematics. A native math language speaker knows the culture of mathematicians. In short, a native math language speaker is a mathematician.

In the remainder of this article, I will use the term mathematician interchangeable with a native math language speaker.

However, think about an average 6-year-old, an average 12-year-old, and an average 18-year-old native speaker of a language such as "standard" English. The 6-year-old has substantial oral communication and thinking capability in the language, but has a relatively limited vocabulary and is just getting started in learning reading and writing. The 12-year-old has a much larger vocabulary, is still better at oral communication, and has made significant progress in reading and writing. The 18-year-old is still more skilled at reading, writing, speaking, listening, and thinking in his or her native language.

When applied to math, observation leaves us with the situation in which a person might be a high school graduate but mathematically function like an average 12-year-old. In some cases we have 6-year-olds who mathematically function like average 12-year-olds.

Here is one more piece of the math education puzzle. A person who is certified as a teacher has attended 16 or more years of schooling starting at the first grade or earlier. This person knows a tremendous amount about general pedagogy (how to teach), only a small fraction of which was learned in teacher education courses. This person also knows a tremendous amount about how to be a math teacher, and yet may have taken only one or two courses in math pedagogy. Learning on the job is a very important part of becoming a good teacher.

The typical elementary school teacher is a "math teacher" as well as being a teacher of other disciplines. How can we tell if this person is a qualified "math educator?"

Let's take the specific case of an average elementary school teacher who is responsible for teaching math as well as a number of other subjects. We can talk about the number of years of coursework (precollege and college) experience the teacher has had being explicitly taught in math content, being explicitly taught in math pedagogy, being exposed to general pedagogy, and being exposed to math pedagogy.

This is a work in progress, and the question just asked is a difficult one to address. We might be able to quantify the situation with a statement such as, "This teacher functions at the level of an 8th grade mathematician, an average 10th grader in math pedagogy, and an average graduate of an elementary teachers' college program in general pedagogy."
The levels of the three different measures of qualification will change as the teacher learns on the job through classroom experience and staff development. We know that, on average, teachers gain considerably in their overall levels of effectiveness during their first half-dozen years on the job.
I leave this topic for now, still not having given a specific definition of what we mean by an appropriately qualified (well-qualified) math educator for some specific category of students. For example, a person who is a well-qualified math educator for learning disabled students might not be as well qualified in working with talented and gifted students, and vice versa. I have suggested (recommended) that this person needs to be a "native language speaker" in math content, math pedagogy, and pedagogy. But I have not specified the grade levels or age levels or other measures we want to use in each area. I have not specified what might be meant by "standard math," or "standard math education."
Substantial research supports the contention that a major weakness in our precollege math educational system is the relative weakness in the math pedagogy and math problem-solving capabilities of many teachers of math.

A Personal Story

Being a native math language speaker means that one can "do" mathematics. It means having a high level of expertise in solving math problems, recognizing problem situations in which math is apt to be useful, having quite a bit of math knowledge and skills, being able to use one's math knowledge and skills, and being comfortable in the culture of mathematicians. It takes many years of time and a considerable amount of effort to become a native math language speaker (that is, to be come a mathematician).

Here is a personal example. I grew up in a home where both my mother and father taught math in a university. Each was a mathematician. Thus, merely by growing up in this environment, I was given a large boost in moving toward becoming a native math language speaker. However, much more was required.

I did well in math in elementary and secondary school. I then went on to college and majored in math. By the time I finished a bachelor's degree in math, I had a good start on being a mathematician. Four years of graduate work, resulting in a doctorate in math, certainly qualified me to be considered a native math language speaker. I was fluent, with a high level of expertise, in reading, writing, speaking, listening, creatively thinking, and problem solving in math. My four years of graduate work essentially constituted a math immersion program, with all of the teaching being done by highly-qualified mathematicians with a high level of fluency and competence in math.

However, I was not a mathematics educator. I had very little teaching experience during my graduate work, and I had no specific instruction on how to teach math. I had some math teachers who were more effective than others. Indeed, some of them satisfy my definition of math educator. Others were clearly much more engrossed in their research. Some were both excellent math educators and excellent math researchers.

Nor was I a math historian. Sure, I had learned a little math history as I studied math. However, my level of expertise in this component of the discipline of math was minimal.

One Major Math Education Problem

We have now come to one crux of a difficult education problem. Most children do not grow up in a home environment of native math language speakers (that is, mathematicians). Moreover, most students do not have their elementary school math taught by native math language speakers. Indeed, even in their middle school and high school math courses, many students are not being taught by people who would be considered to be native math language speakers. It tends to be the students who take the more advanced math courses who are taught by mathematicians.

One way to attack this problem is by departmentalizing the teaching of math at the grade school level, and requiring that teachers of math at every grade level have at least a bachelor's degree in mathematics. Some of the countries that do well in international math education comparisons do take such an approach.

Another approach is to have math teachers at all grade levels take rigorous and demanding math education workshops and summer courses, year after year after year. Some teachers do this, and indeed develop into native math language speakers.

Note that the goal is not to make such teachers into research mathematicians. Instead, it is to make them into "expert level" math education mathematicians who specialize in teaching math to specific groups of students.

A third approach would be to place much greater emphasis in the math curriculum on students learning to read, write, speak, listen, and creatively think and solve challenging problems in math. Such ideas are often emphasized in both preservice and inservice education for math teachers.

There is a strong parallel between good math communication and the teaching of reading and writing in a natural language such as English. In teaching writing in a natural language, there is a great need for the teacher to read what a student is writing and provide feedback on the content. How well is the intended "message" being conveyed by what the student has written? Of course, teachers also provide feedback on spelling and grammar. But feedback on the content is essential. So it is with math. But, especially at the elementary school level, the math feedback is usually focused only on the correctness of an answer.

The following reference presents research on the mathematics that a third grade teacher needs to know.

Ball, Deborah Loewenberg, Heather C. Hill, and Hyman Bass (Fall 2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator. Retrieved 7/13/2010 from

Compare and Contrast with Music Education

I find it helpful, when thinking about educational strategies in one discipline, to compare and contrast that approach with strategies in other disciplines. I often have selected music and chess as disciplines to compare and contrast with math. This section looks at the language of music and the next section looks at the language of chess.

Music is one of the nine multiple intelligences identified by Howard Gardner. See While people vary in their musical "IQ," most people can gain a relatively high level of expertise in music if they have appropriate opportunities and interest. Most people gain a useful and enjoyable level of music expertise just through their informal exposures to music and opportunities to sing. A "professional" high level of expertise takes many thousands of hours of good instruction and practice.

This same situation exists for the multiple intelligence that Howard Gardner labels as logical/mathematical. Both music and math are, at some level, built into our genes. But it takes a great deal of study and practice to develop this innate capacity to meet contemporary standards of expertise.


Making music includes singing, chanting, humming, whistling, rhythmic clapping, use of drums and other musical instruments, creating music, composing new music, and so on. Music existed long before the development of reading and writing. Music was part of the environment that children grew up in tens of thousands of years ago. Musical knowledge and skills passed from generation to generation via children growing up in a musical environment of role models and being expected to participate.

Oral tradition and making and using musical instruments served the discipline of music well for many thousands of years, up until quite recent times. Even after reading and writing were invented, very few people learned to read and write. Moreover, even for people who did learn to read and write, a decent system of musical notation for the reading and writing of music had not yet been invented. The invention of a good written language for music took many thousands of years. Quoting from the Wikipedia:

Scholar and music theorist Isidore of Seville, writing in the early 7th century, famously remarked that it was impossible to notate music. By the middle of the 9th century, however, a form of notation began to develop in monasteries in Europe for Gregorian chant, using symbols known as neumes; the earliest surviving musical notation of this type is in the Musica disciplina of Aurelian of Réôme, from about 850. There are scattered survivals from the Iberian peninsula [sic] before this time of a type of notation known as Visigothic neumes, but its few surviving fragments have not yet been deciphered.

The ancestry of modern symbolic music notation originated in the Roman Catholic Church, as monks developed methods to put plainchant (sacred songs) to paper. The earliest of these ancestral systems, dating from the 8th century, did not originally utilize a staff, and used neum (or neuma or pneuma), a system of dots and strokes that were placed above the text. Although capable of expressing considerable musical complexity, the dots and strokes could not exactly express pitch or time. This system served mainly as a reminder to one who already knew the tune, rather than a means by which one who had never heard the tune could sing it exactly at sight.

We are all used to the term "musician" to describe a person who has a high level of knowledge and skill in some parts of the discipline of making music. Music is such a large discipline that no one can gain a high level of expertise over the entire discipline. A world class opera singer need not know how to play a violin, compose music, or direct an orchestra. However, world class singers, violinists, composers, and conductors are all immersed in the culture of music and each can communicate effectively in the language of music.

Many musicians have a high level of expertise as music teachers. Indeed, perhaps because music instruction tends to be performed by good music educators, many musicians learn a great deal about how to teach music as they learn music. In any case, many musicians learn to teach music and come to depend on this knowledge and skill as a source of income.

Our educational system accepts the idea that a person teaching music should be a native language speaker of music (that is, a musician) and a music educator. In telling contrast, American schools have a large percentage of people teaching math who are ill-prepared in the content and the teaching of math.

Most children grow up in a home environment is which there is singing and other music. They may learn jingles and other music from radio, television, or Internet ads, from computerized games, from music storage and playback devices, and so on. If parents and other caregivers have a reasonably high level of music interest and fluency, their children will learn a lot of music by being immersed in such an environment.

However, learning to play a musical instrument, learning to read music, learning to compose music, learning to sing well individually and in a group, and so on, all require many years of informal and formal instruction plus lots of practice.

Three important characteristics of informal and formal music education are:

  1. The music learner is frequently given the opportunity to observe (listen to) the performance of others who have a relatively high level of expertise. Math has this characteristic for very few children—or adults.
  2. Music is a human endeavor that is often performed in a group setting. This setting promotes and facilitates communication and sharing. This sharing—making music together—is an important aspect of music. There are large intrinsic and extrinsic rewards in such musical sharing. Again, this is seldom true for math.
  3. Music provides a learning environment in which the learner can readily tell that he or she is gaining in expertise and can demonstrate to others this increasing expertise. Math exhibits some of this characteristic, but not nearly as strongly as music.

Each of these characteristics of music education offers us some insights into math education.

The first item suggests that we can improve math education by improving the general math environment. For example, we currently teach math during a specific math period in school, and math may receive very little attention during the rest of the day. A child's math-related environment at home may be quite limited. Our educational system has accepted the idea of reading and writing across (throughout) the curriculum. How about making this into "the three R's across the curriculum"?

The second item points to a major challenge in math education. While math, like music, is a human endeavor, it mainly lacks the group learning and performance aspects of music. The intrinsic and extrinsic reward structure in math is quite different from that in music.

The third item suggests we can improve math education by creating better feedback mechanisms and by helping students gain expertise in sensing and assessing their own progress. Research in math eduction supports these ideas.

For reflection and discussion. Think about the three ideas listed above in terms of your personal math education experiences. What does your personal analysis suggest to you in terms of things you can do to improve the math education experiences of students?

Doing and Consuming Music and Math

From an educational point of view, it seems important to distinguish between those who learn to "do" music (perform at some level) and those who merely consume (listen to) music. Both categories of people existed before the development of electronic technology. We can educate for each of these endeavors, and most people have some level of interest in and expertise in each of them. Participation in the second category has been greatly influenced by the development of technology for the broadcasting, recording, and playback of sound, and new developments in electronic music composition and performance.

We all "do" math. When my stomach rumbles and I glance at my watch, I do a mental calculation of how long it has been since my last meal and how long before my next meal. When I fill my car's gas tank, I estimate the cost and the miles per gallon since the last fill up. In a store, I pay some money and get some change, then I check the correctness of the change.

We all consume math, but not in the same way that we consume music. For example, I use my cell phone to make a call. The cell phone system makes quite sophisticated use of computers and a variety of other equipment. People designing the equipment and the overall phone system made extensive use of math. So, when I use my cell phone, I am making extensive use of math. However, that does not give me the same sense of feeling contentment that I get by "making use of music" when I am listening to a live or high-quality recording of an expert musical performance.

ICT and Music

The overall discipline of music has been strongly impacted by Information and Communication Technology (ICT). The history of this impact certainly dates back to the development of the telephone (so music could be transmitted over a wire) and recording devices (so music could be stored, edited, and widely distributed).

We now have relatively inexpensive electronic hardware and software for creating, editing, storing, playback, sharing, and performing music. With such technology, a grade school student can learn to compose and edit music, and can use a computer as a performance instrument. This is an important idea. When grade school students compose music and use a computer in the editing and playback process,they can hear what they produce, improve it by editing, and share their music with others. This is in marked contrast to what typically happens in math instruction.

Computer technology allows people to build personal libraries of music. Computer technology can "notate" music—that is produce a written musical score from a live or recorded music performance. Artificial intelligence music generation systems have generated music "in the style of" various world class composers, music deemed comparable in quality to human-created compositions. These electronic musical instruments and a wide range of storage and editing tools have greatly changed the music recording and performance industries.

Relating back to math education, one of the key ideas given above is that ICT has brought us a number of new electronic musical instruments to "do" music. In math, ICT has brought us 6-function, scientific, graphing, and equation-solving calculators, and computers with vastly greater powers. These new math tools are instruments that can be used to "do" math. Thus, math education is faced by the problem of determining the extent to which it wants to facilitate students learning to use these electronic tools (instruments) and pay less or no attention to other more traditional tools. We can ask, "What is so special about paper and pencil computational algorithms?"

Here is an idea that might interest you. Solving a complex math problem or producing a proof in math is somewhat akin to composing in music. A musical composer does not have to have a high level of expertise in playing each of the instruments needed to perform a composition. That is, composing and performing are two different skills. Of course, the composer needs to know the capabilities and limitations of the various musical instruments and the human limitations in playing the instruments.

In math, for example, a person working to understand and solve a math problem can imagine having a three-dimensional picture of a particular three-dimensional geometric figure, and being able to readily view this picture from different directions. We have long had computer graphics software that can produce such three-dimensional representations.

Somewhat similarly, a person working to solve a math problem may decide that it would be helpful to fit a mathematical function (perhaps a quadratic, cubic, or higher order polynomial function) to some data, and then find the places where this function crosses the x axis. Computer programs have long existed that readily accomplish such tasks.

To continue this example, the math problem solver may decide that it would be helpful to perform various statistical computations on some data. Computer programs that can carry out such computations incorporate a huge amount of collected knowledge from a great many mathematical statisticians.

Perhaps you see a pattern emerging here. Think of a person attempting to solve a complex math problem as a composer, developing a set of instructions that can be carried out to solve the problem. The math problem-solving composer can draw upon the performance capabilities of calculators and computers. The math problem-solving composer does not need to have a high level of expertise in areas in which calculators and computers can readily produce very high levels of performance.

Here is another math education idea coming from music. One of the reasons math education is part of the core curriculum is that math is a powerful aid to solving problems in many different disciplines. What computers and other aspects of ICT have done is make it possible to automate finding the solution to many of these problems. Consider an analogy between having an automated tool that solves a particular problem, and having a music storage and playback device. A music consumer gets to listen to and have the benefits of recorded music. A person using a computerized tool that incorporates many mathematical functions gets the advantage of a mathematical performance.

In summary, ICT has strongly affected doing and consuming music. It has equally strong potentials in math. However, our math educational system has made only modest progress in realizing (making use of) these potentials.

Compare and Contrast with Chess

This section examines the game and language of chess. Extensive research on good chess players has given us good insight into the role of chunking (see above) in learning to be a good chess player. There is considerable similarity between this chunking used in playing chess (solving the problem of making a good move) and chunking used in solving problems in math.

Chess (just like checkers)is played on an 8-square by 8-square board. The game has a long history and is played by millions of people. At the higher levels of play there are masters grandmasters, international grandmasters, and world champions. At lower levels, grade school children can learn the game and there are tournaments for players at all levels. Many websites explain the rules and some of the details of the game.

Chess is a game of skill. This is in marked contrast to card games such as bridge and poker, where the "luck of the draw" makes a big difference in the short run. In a chess game, two players compete against each other. The game ends in a win for one of the players, or a draw. There is some advantage to being the first player to make a move, but in a chess match between two players, they take turns in going first. The "luck" that chess players talk about is when their opponent overlooks a possible good move or fails to see that a move s/he is about to make is a bad move.

Chess has an oral and written language. However, chess is not a significant part of the everyday life of most people. Nor is chess one of the nine areas of multiple intelligence identified by Howard Gardner. See Children who grow up in a "chess-playing" family may learn the game when they are quite young and gain from the collected oral history of chess that is part of the family conversation.

Chess has been extensively studied to help understand problem solving and how humans can improve their problem-solving skills. It has also been extensively studied by people working in the field of artificial intelligence. How does one go about developing a computer program that is good at playing chess? What can we learn about human intelligence and the education of children through the study of how humans and computers learn and become better at playing chess?

You don't have to be a skilled chess player to understand the chess section in this Communication in the Language of Mathematics document. Here is enough background to get you going.

Chess Notation

The columns (files) of the 8 x 8 board are lettered a, b, … h, and the rows (ranks) are numbered 1, 2, … 8. In chess, the person playing the White pieces always moves first. The lettering and numbering notation used to identify the spaces on the board is convenient and natural from the point of view of the person playing the White pieces.

The names of the pieces (in English) are abbreviated as follow: K=King, Q=Queen, R=Rook, B=Bishop, N=Knight, and P=Pawn. This board coordinate system and the piece name abbreviations make it quite easy to record all of the moves in a game.

For example, here are the first few moves of a game. White always moves first, and White's moves are in the left column. The sequence of moves given below indicates that White’s Bishop captures Black’s Knight on White’s fourth move.

  1. Pe2 to e4 — Pe7 to e5
  2. Ng1 to f3 — Nb8 to c6
  3. Bf1 to b5 — Pa7 to a6
  4. Bb5 x Nc6 —

This notation can be tightened up considerably. Here is a tighter notation that conveys the same information. The notation assumes that the reader knows the legal moves. Thus, the first move of Pe4 means that White's pawn that is at e2 is moved to the e4 location. It is the only pawn that can legally move to that location at this point in the game.

  1. Pe4 Pe5
  2. Nf3 Nc6
  3. Bb5 Pa6
  4. BxN

Notice how easy it is to make an exact record of a chess game and to learn to read such a record. Contrast this with musical notation and learning to read music, or the notations used in math. From a notational point of view, music and math are far more complex than chess. It takes only s few minutes to learn the written language used to store a record of a game of chess.

Communication is More than Just Notation

This section discusses some of the learning that takes place as one develops into a good chess player. If you have not played chess, or not played it much, the ideas will probably pass you by. But, for each idea, think about whether it might be applicable in learning to being better at math problem solving. Keep asking yourself, "What 'big ideas' in math problem solving do I know that I use regularly and that I can help my students learn to use?"

Here is a piece of information useful for understanding the example in the following paragraph. In chess, a Knight's move must be either two horizontal and one vertical square or one horizontal and two vertical squares. This allows a Knight to attack various pieces that cannot, in turn, be attacking back.

Like any well-developed discipline, chess has an extensive vocabulary. Also, as in any discipline-specific vocabulary, many chess terms are adapted from natural language vocabularies. For example, you might think of a fork as an eating utensil. Of course, you have heard of a tuning fork used in music. You have heard of a fork in a road. In the diagram, White has just moved the Knight to d7, actually forming a triple fork. This particular fork of King, Queen, and Rook is also known in chess as a “family fork.”

You might be able to guess meanings of terms such as open file and Queenside. Other terms such as check, gambit, castle, and fianchetto (Italian for "on the flank") are more challenging.

This special vocabulary and notation are important for communicating about and thinking about chess. However, there is more to such endeavors than just vocabulary and notation. Your brain stores images that represent emotions, sounds, smells, pictures, and so on. Your brain draws upon these mental images as it works to solve problems and accomplish tasks. Good chess players have stored many thousands of chess patterns (chess chunks) in their brains. For them, a short look at a chess game in progress provides information needed to retrieve mental chunks of information related to possible future outcomes of the game in progress.

In this aspect of communication with one's self, there are clear similarities among chess, math, and music. In each discipline, one learns chunks, stores them, and learns to think in terms of these chunks. Some of the chunks have names, while others are mental patterns that one accesses through other means such as mental pictures, sounds, "gut-level feelings," and so on.

Learning a discipline-specific chunk and how to make effective use of it is a step toward increased expertise in a discipline. However, accumulating a large number of chunks in and by itself does not make one into a high-level expert in a discipline. It is learning to "see," "sense," "hear," "feel," "recognize," etc., the relationships among chunks, and to make use of appropriate combinations of these chunks, that is key to having a high level of expertise in a discipline.

The discipline-specificity issue is worth repeating. Substantial research supports the need for discipline-specific knowledge and skill (discipline-specific chunks) in order to have a high level of expertise in that discipline. That is why, for most people, it takes so many years of effort in order to become a high-level expert in a discipline. But it takes more than just rote memorization of chunks to achieve this high level of expertise.

Chess Strategy

There are many chess websites available. On the Web one can read about chess, see the rankings of the best players, follow tournaments in progress, play against human opponents, play against a computer, or try your hand at solving challenging chess problems. These websites can be used just for fun and can also be used to gain increased expertise as a chess player.

Here is a personal story. I learned to play chess when I was relatively young. That is, I learned the legal moves and to play well enough so that it was fun to play with other kids my age.
A number of years later I became interested enough in the game to read a couple of chess books. One was a "how to" book that explained some of the strategies that good players find useful. Another was a chess history book, looking at some of the great players and games from the past. I found both types of book enjoyable. The "how to" book substantially increased my level of playing ability.
This is an "aside." Notice that I read these chess books for fun and to learn to be a better chess player. At about the same time I read some books such as The World of Mathematics that had little to do with the math I was being taught in school. Our educational system places a lot of emphasis on students learning to read well enough so they can learn by reading. In math education, however, we do not take much advantage of a student's steadily improving expertise in reading. Think back over your own math education. Did you ever read a math book for fun or to further your math knowledge and skills beyond what was being taught in school?

Here is a simple example. In chess, one of the key ideas is to maintain the mobility of your pieces. That is, to keep as many move options available for your pieces as possible. Another strategy is to gain control of the center of the board. Among other things, control of the center tends to increase your own level of mobility and decrease your opponent's level of mobility.

Suppose that you had studied a book discussing these two strategies, and that it contained some examples of how to make use of the strategies. You then play a game against an opponent who (up until now) was your equal, but who had not received formal instruction (from a teacher, book, or opponent) on these two strategies. The chances are quite good that you will now be the superior player. This little bit of formal instruction gives you a large advantage over an "unschooled" opponent.

This is an important idea. It might well be that, as you play chess, you will discover some of these strategies for yourself. However, there are many strategies that are useful at various points in a typical game. Many have been discovered and carefully analyzed by world class chess players. The accumulated knowledge in this area is far more than one person could discover (unaided by the previous work of others) in a lifetime.

Now, consider my opponent who is consistently losing to me because I have been making use of these two strategic concepts. My opponent may carefully analyze these (losing) games and eventually discover the concepts of mobility and center control. Alternatively, I might mention the two ideas and illustrate them in a game that we have recently completed. In both cases, I am assuming that we have written down the moves from the games, so that we have a written record that allows us to analyze games we have played in the past.

Chess is both a fun game for children and a discipline of fierce, ego-involved competition. With few exceptions, it takes ten thousand or more hours of study and practice to become an international grandmaster, assuming inherent talent. Much of this time and effort is spent studying games that have been played by exceptionally good players in the past, and games one has played in the past.

These insights into learning chess strategies provide some useful insights into learning math. There are many different strategies for attempting to solve math problems. Many of these are designed to aid in communicating with one's self, such as by drawing a diagram, making a table, creating a mental model or image, and so on.

Determining a Chess Player's Strength

Chess is a competitive game. If two players of approximately equal chess-playing strength play against each other a number of times, they will each win about half of the games. If one player is much stronger than the other, this player will win almost all the time.

Over the years, the discipline of chess has developed a relatively accurate means for determining a player's strength. The method is somewhat like that used in rating teams in competitive sports. Careful records are kept of how well players do against each other in different tournaments. Even if two players have never played against each other, they will have played against players whose strength or rank has been determined through tournament play.

As in competitive sports, there are chess tournaments pitting the top players in the world against each other, and there are world championship matches in which two players compete against each other, with the winner designated as the world champion.

While there are competitions in both math and music, there is essentially nothing like the level of competition one finds in chess. There are many world class mathematicians, and there are many world class musicians. However, there is no world champion mathematician determined by head-to-head competition.

Computers have brought an additional approach to chess rankings. One can compare (and rank) humans in how well they do playing against various computer chess programs. People throughout the world can compare themselves in terms of how well they do against a particular chess program, set at a particular level of difficulty.

Without a competition and ranking system, math students have no easy way to compare their own math strengths against each other. Let me share a personal example. For students in college, there is a national math competition called the Putnam competition. Throughout the United States, on one specific day, entrants spend the day working on 12 problems. This is done on their own campuses, and the test is carefully proctored.

I was the best math undergraduate at my university. I knew this because of having taken course with the other top students—in some sense, competing with these students in math classes and the tests given in the classes. I thought of myself as being quite good at math.

I competed in the Putnum contest in both my junior and senior years. In both years, a quarter to a third of all entrants scored better than me! Objectively, I was quite good. Only the better students would enter the competition, and I was better than about seven of ten other such students. Even so, my ego's feeling of “being quite good” suffered significantly.

One of the goals in the No Child Left Behind Act is to move toward a ranking system in math education that can be used to measure the relative strengths of schools. The people supporting this type of "competition" believe that it will help to improve the precollege math educational system in this country.

It is not at all obvious that making math into a competitive "sport" will lead to improved math learning and performance for students as a whole. Indeed, it might well do just the opposite. Those who are not highly talented and highly motivated in math (as well as those who are not basically competitive in what they do) may well choose not to compete. One might well see widespread implementation of the sentiment: "I’m not very good at math. Why should I compete, when I will always come out in the bottom half?"

Contrast this with a person learning that the knowledge and skills they are gaining in math empower them to do various things they need or want to do. Through study and practice, they get better at doing those things. This suggests that math education can become more successful through helping students, individually as necessary, to grasp the personal advantages (empowerment) they accrue through their math studies.

For reflection and discussion: Suppose we had a computer program that could "play the game" of math at different difficulty levels. A student studying math could play against this game to determine his or her current math ranking. Here, we are assuming that this "game of math" is good enough to be used throughout the world to determine a student's math level of learning, understanding, and overall "math strength." How do you think this would affect math education? To help your thinking on this question, you might want to read David Moursund's short article Chesslandia: A Parable.

Rote Memory

It may feel to you that a discussion of rote memory is a far reach from a discussion of communication in math and math education. Here is the way I see it. Much of what a person does when attempting to solve problems and accomplish tasks in any discipline involves communicating with her or him self. One consciously communicates with data, information, knowledge, and wisdom stored in one's brain. One carries on a mental conversation. Indeed, you probably know people who verbalize—talk out loud to themselves—during this thinking.

One can memorize with little or no understanding of what is being memorized. One way to think about this is in a stimulus/response setting. A person's brain (or, some other animal's brain) is trained to respond in a specified manner to a specified stimulus. The stimulus elicits the response, and the responder does not need to have an “understanding” of the meaning of what is stored in the brain and produces the response.

Of course, we can also have stimulus/response learning in which the response has meaning to the learner. You may be able to respond quite rapidly to the stimulus 8 x 7 =, and produce a response of 56. Upon further reflection, you realize that you have done a "multiplication fact" problem-solving task, producing an answer of 56.You may realize that this answer is a little bigger than 50 and a lot less than 100. You may realize that a score of 56 on a hundred point test may not lead correspond to a good grade on the test.

Rote memory, with or without understanding, can be used in the storage and retrieval of part of the collected knowledge with a particular area. This can be quite helpful in solving some of the frequently occurring problems within that area.

Here is an illustration from the game of chess. A chess game begins with White and Black each having 16 playing pieces. It is possible to carefully analyze the board situations that result after all possible sequences of one move by each player, two moves by each player, and so on. Of course, the number of possible sequences grows exponentially, and soon becomes so huge that no person (indeed, even all the past and present chess players in the world) can analyze all of them.

However, what has been done is that many of the interesting and potentially good opening sequences of moves have been carefully analyzed by high level chess experts. A huge amount of this collected chess knowledge is available in books and in other media. Any person who has learned to read chess notation can access this collected chess knowledge.

A chess player gains a considerable advantage by studying these well-analyzed sequences of opening moves and by memorizing a large number of them. Rote memory of the results of work done by others is a good substitute for "reinventing the wheel." In a game between reasonably highly ranked chess players, the first half dozen or more moves by each player tend to be made quite quickly, using rote memory.

After that, the thinking and chess problem-solving begins. Each player soon encounters a position (a chess problem) that he or she has never encountered before. However, even here having a large repertoire of memorized chunks is very important. In essence, such chunks correspond to parts of a game position. The good chess player recognizes parts of the problem as being similar to or even exactly the same as parts of board positions that he or she has carefully analyzed in the past.

Math education can be approached via rote memory. We can have a student memorize facts, definitions, and algorithms. Rote memory is useful in solving frequently occurring problems. Moreover, math problem solving makes use of chunks much in the same way as chess playing does.

It turns out, however, that math is played on a much larger playing board (many more playing pieces) than is chess. In dealing with the math people encounter in their everyday lives, they quickly move beyond the point where rote memory suffices. In novel problems, problem solvers quickly move beyond being able to succeed from rote memory and enter the mode of attempting to make effective use of chunks of information stored in their long-term memories. A large repertoire of such chunks and lots of experience in drawing on such chunks is essential in dealing with challenging math problems.

Artificial Intelligence and Chess Playing

Consider the following type of competition. Well before the competition begins, competitors are given a copy of a very comprehensive dictionary. During the contest, competitors are given a definition from the dictionary. Their task is to say and spell the word that has been defined.

By dint of considerable study and practice, a person can get very good at this rote memory game. But a computer can become letter perfect in a very short period of time. Computers are very good at rote memory, and computers can store the spelling and pronunciation of all of the words in a dictionary.

When researchers in artificial intelligence went looking for a game to study in which (at the time) humans were much better than computers, many decided on chess. One can become better at chess by rote memory, but chess is far more than a rote memory game.

In chess, rote memory is quite helpful at the beginning of a game. It can also be quite helpful near the end of a game in which each player has lost a number of pieces. In that situation, there are relatively few pieces left on the board. Many such end games have been carefully analyzed by chess experts and computers. The results are available in books and databases.

It is in the mid game—after use of the memorized openings and before use of memorized end games—that intelligence is needed. How does a human chess player analyze possible moves in order to select the one that improves their situation and/or damages the opponents’ position? One way to gain insight into this is through working with skilled chess players. Get them to "think out loud" as they analyze chess problems. Of course, there are also many books full of the written analyses of games played by good chess players.

This interaction with expert problem solvers has been used in many different disciplines. It has led to the development of expert systems (computer programs that are good at solving challenging problems) in many different disciplines.

This is part of the challenge for educators in the information age. Thousands of researchers are working on developing computer programs that make use of computer capabilities (machine intelligence, artificial intelligence) to solve or help solve problems in various disciplines. Sometimes the artificial intelligence methods parallel human intelligence methods. Often they don't. Rather, they make use of methods that take advantage of the large memory and great speed of computer systems.

The first chess-playing computer programs were very weak compared to humans. However, over the years, computers got much more capable, and chess-playing programs got much better. By 1997, an IBM computer named Deep Blue beat Garry Kasparov, the world's human chess champion! A dedicated chess machine called Hydra was programmed to take advantage of a combination of rote memory and a type of artificial intelligence relevant to chess playing—and became as good at the world's best human chess players. The success of chess-playing computers did not lead to the game of chess gradually going away. Nowadays there are chess matches that pit computer against computer. There are chess matches that pit human plus computer against human plus computer. Many chess players practice their skills against computer programs. With all of this, chess remains a game that many people learn to play and enjoy playing against human and computer opponents.

Artificial intelligence has been used in other game-playing computer programs. IBM undertook the challenge of developing a computer program that could play the popular TV game show Jeopardy. In this game, opponents are given an answer to a quite specific question, and compete to see who can first correctly state the question. In February of 2011, an IBM computer system named Watson defeated two human champions in this game. See

Artificial Intelligence and Solving Math Problems

Artificial intelligence, rote memory, and steadily increasing computer memory size and speed have been applied in mathematics. For many years, there have been high quality Algebra Systems. Such computer programs can solve a wide range of math problems.

You know that an inexpensive 6-function calculator can add, subtract, multiply, divide, and compute square roots. That is, it can "do" some of the things we are teaching grade school students to do through rote memory and through use of memorized algorithms. A modern Computer Algebra System (CAS) has this same level of capability up through calculus and linear algebra. That is, in every part of the math curriculum where rote memory and use of memorized algorithms is useful, artificially intelligent CAS systems can do (typically, faster and more accurately) what we are teaching students to do by hand.

The math education community needs to think carefully about the steadily growing "intelligence" of computer systems. If a computer system can solve a certain category of math problems, what "by hand and by brain" knowledge and skills in this specific area do we want students to acquire?

Our math educational system has been struggling with this situation for years. To the extent that math resembles a competitive game, computers are far better than humans at many aspects of it. It seems evident that this math education quandary will continue to exist for the foreseeable future.

There are no widely agreed on solutions to the computers versus humans in math issue. The Common Core State Standards Math Initiative is down-playing the importance of computers. See and

For reflection and discussion. In chess, the development of computer programs that can outplay even the best of human players has not resulted in the demise of the game. Chess players enjoy the head-to-head competition with each other and the social aspects of being part of the chess community. Within in certain areas, computers are far better at math than humans. How is this affecting what we are doing in math education? Are there aspects of math that are very large numbers of people want to learn because they are fun—personally and socially rewarding—independently of whether computers can do them better than humans? For example, to what extent is Sudoku a math game?
For reflection and discussion. Suppose that our education system decided that all math education above the 8th grade was elective. Any course requiring a higher level of math knowledge and skill could clearly specify the higher level of math prerequisite that was required. However, students could well graduate from high school and college without taking math courses above the 8th grade level—or, by only taking such higher level math courses when they had a clear personal need to do so. What are your thoughts on how this would affect our overall educational system?

Final Remarks

For me, the ideas that I have discovered and explored while writing this article are quite thought-provoking. While my initial focus was on communication and math education, many of the ideas apply to learning in every academic discipline. For example, brain research on chunking is applicable in any discipline, including disciplines as diverse as carpentry, dancing, and Texas hold ‘em. Skill in creating and using personal chunks is an essential component of self-talking and planning in solving challenging problems.

Information and Communication Technology (ICT) brings a new dimension to communication. One way to think about this is that a computer is an artificially intelligent machine that one can communicate with and make use of in solving problems, accomplishing tasks, and learning. Thus, within each discipline in our educational system, educators are now faced by the challenge of helping their students communicate effectively with computers and other artificially intelligent aids to solving problems and accomplishing tasks.

This challenge is especially large in math and in other disciplines where computers are especially useful (powerful, capable) in solving or helping to solve problems and accomplish tasks. That is, many chunks include procedures that one can learn to carry out "by hand" but that computers can carry out faster and (often) more accurately. For each of these, a student could have an option of learning all details of the chunk, or of learning that such a chunk exists and that a computer can accurately and rapidly carry out the details of the procedure(s) associated with the chunk.

For reflection and discussion: What are your thoughts on education, including having a student learn about (including how to retrieve and how to make use of) a number of computerized chunks?

In my opinion, our math educational system spends far too much time helping students to learn (memorize, often with little understanding) to do things that computers can do faster and more accurately. This uses up so much of the math education time, that relatively little time is spent on understanding, creative thinking, problem posing, and other activities in which human intelligence far exceeds computer intelligence.

Some of this memory work is important. Often a person is called upon to make real time decisions (quick decisions) based on using math knowledge and skills. As computerized processing and information retrieval systems get better, and as more computer intelligence (including math-related computer intelligence) is built into machines, we will need to continually reexamine those aspects of math that need to be stored in one's head.

For reflection and discussion: Think back over this article. Identify one or two ideas that you found particularly interesting and that you tend to agree with. Find one or two that you fond uninteresting and/or that you strongly disagree with. Do a compare and contrast, working to increase your insight into communication aspects of improving math education in our information age world.


Annenberg Media (n.d.). Mathematics illuminated. Retrieved 5/31/08 from

This is a set of 13 free half-hour videos and accompanying instructional materials. The first of these talks about mathematics as a language and explores prime numbers.

Ball, D.L., Hill, H.C., & Bass, H. (Fall 2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator. Retrieved 7/13/2010 from

Quoting the first part of the article:

With the release of every new international mathematics assessment, concern about U.S. students’ mathematics achievement has grown. Each mediocre showing by American students makes it plain that the teaching and learning of mathematics needs improvement. Thus, the country, once more, has begun to turn its worried attention to mathematics education. Unfortunately, past reform movements have consisted more of effort than effect. We are not likely to succeed this time, either, without accounting for the disappointing outcomes of past efforts and examining the factors that contribute to success in other countries. Consider what research and experience consistently reveal: Although the typical methods of improving U.S. instructional quality have been to develop curriculum, and—especially in the last decade—to articulate standards for what students should learn, little improvement is possible without direct attention to the practice of teaching. Strong standards and quality curriculum are important. But no curriculum teaches itself, and standards do not operate independently of professionals’ use of them. To implement standards and curriculum effectively, school systems depend upon the work of skilled teachers who understand the subject matter. How well teachers know mathematics is central to their capacity to use instructional materials wisely, to assess students’ progress, and to make sound judgments about presentation, emphasis, and sequencing.
That the quality of mathematics teaching depends on teachers’ knowledge of the content should not be a surprise. Equally unsurprising is that many U.S. teachers lack sound mathematical understanding and skill. This is to be expected because most teachers—like most other adults in this country—are graduates of the very system that we seek to improve. Their own opportunities to learn mathematics have been uneven, and often inadequate, just like those of their non-teaching peers. Studies over the past 15 years consistently reveal that the mathematical knowledge of many teachers is dismayingly thin.1 Invisible in this research, however, is the fact that the mathematical knowledge of most adult Americans is as weak, and often weaker. We are simply failing to reach reasonable standards of mathematical proficiency with most of our students, and those students become the next generation of adults, some of them teachers. This is a big problem, and a challenge to our desire to improve.

Dynarski, et al. (March 2007). Effectiveness of reading and mathematics software products: Findings from the first student cohort. Report to Congress. Retrieved 2/19/08: Quoting from the report's Executive Summary:

The main findings of the study are: ,
  1. Test Scores Were Not Significantly Higher in Classrooms Using Selected Reading and Mathematics Software Products. Test scores in treatment classrooms that were randomly assigned to use products did not differ from test scores in control classrooms by statistically significant margins.
  2. Effects Were Correlated With Some Classroom and School Characteristics. For reading products, effects on overall test scores were correlated with the student-teacher ratio in first grade classrooms and with the amount of time that products were used in fourth grade classrooms. For math products, effects were uncorrelated with classroom and school characteristics.

Ericcson, K.A. (n.d.). Superior memory of experts and long-term working memory (LTWM): An updated and extracted version of Ericsson (in press). Retrieved 3/4/08:

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

The Mathematics Course : 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 analyze 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.

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:

Logan, R.K. (2000). The sixth language: Learning and living in the Internet age. Toronto, Canada: Stoddard. (See also.)

Mazur, E. (n.d.). Interview by Marty Abrahamson. Retrieved 2/24/08: p Mazur is a physics professor and highly acclaimed teacher at Harvard. Quoting from the interview:

  1. Mazur assigns reading and expects his students to email him questions about what they do not understand.
  2. Mazur assigns reading and gives an online quiz to see what they do not understand.
  3. Mazur make sues of "clickers" (hand held student response units) in class to get feedback from students.


MA: … Despite the fact that it is possible to accomplish all of these objectives and many more with a single question, do you think that it is useful to have a specific primary objective when designing and planning the delivery of a question?
EM: Oh, yes ! I often actually use students' questions. I actually use this now with a teaching technique called "Just-in-Time Teaching" …. Basically, the students read before class and then they tell me in an e-mail what they find difficult or confusing. I use that to prepare my lecture. In other words rather than lecture on what I find difficult, I will take some of their confusion and bounce it straight back at them.

Miller, G.A. (1956). The magical number seven, plus or minus two: Some limits on our capacity for processing information. Originally published in The Psychological Review, 1956, vol. 63, pp. 81-97. Retrieved 2/29/08:

Moursund, D.G. (June 2006). Computational thinking and math maturity: Improving math education in K-8 schools. Retrieved 3/4/08:

Polya, G. (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. vol. 16—no. 4. Retrieved 5/19/08: Quoting the Abstract:

Because people forget much of what they learn, students could benefit from learning strategies that provide 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. [Bold added for emphasis.]

Sylwester, R. (March 2008). How children learn a language: Part 2 – Knowing what to say and how to sayit. Brain Connection. Retrieved 3/19/08: Quoting from the article:

Last month’s column described how mirror neurons provide us with a mental template of the active motor neurons of someone who is speaking. The person’s comments create an analogous template of the content of the speaker’s thoughts. So if a person says cat, it activates the mirror neurons our brain uses to say cat, but also the neurons that process our memories and images of cat as a concept.
When I begin to write an article, I have a general sense but no set outline of what I hope to write. I explore the concept on my keyboard, and the article gradually begins to emerge. As in conversation, the focus may shift from the original idea. At one point, though, everything becomes clearer, and then considerable rewriting sharpens the text. This often also occurs in a conversation or meeting, when a consensus suddenly occurs, and the issue is then quickly resolved.
What’s odd is that when things are most confusing, I’ll often suddenly wake up from sleep with the mental clarity that had eluded me while writing during the day. I have no explanation for this, except that my thoughts about current tasks seem to continue at a subconscious level, whether awake or asleep. We’ve all experienced this when we can’t recall a familiar name. We go on with other thoughts, and then hours later the name suddenly pops up in our mind.
This suggests that while thought and language are perhaps two sides of a single coin, thought can occur without language—and alas, a lot of language occurs without thought.

Umeå Mathematics Education Research Centre (n.d.) Retrieved 4/23/2014 from Quoting from this website:

In UMERC, several members are active in research that focuses on aspects of language and communication. This page describes the research agenda of this group.…
One goal of mathematics education is for students to develop the ability to communicate using the forms of language and forms of representations that are typical for mathematics. Aspects of language are thus important parts of mathematics. However, sometimes there is a wish to separate language from mathematics, e.g. when constructing mathematical test items and wanting them not to measure reading ability but mathematical ability. Therefore, there is a need to investigate more thoroughly properties and functions of language in relation to mathematical content and to knowledge and learning of mathematics.

Waters, R. (3/4/08). World-wise web? Financial Times. Retrieved 4/5/08:

This article looks at possible futures of the Web. It focuses specifically on increasing linguistic "intelligence" of the Web. Web 3.0 will have a much better ability to "read" the content of websites, extract meaning, and link this meaning to that stored in other Websites.

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

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The original version of this document was written by David Moursund. Editing and a number of revisions were provided by Dick Ricketts.