What is Mathematics
- Mathematics is not a way of hanging numbers on things so that quantitative answers to ordinary questions can be obtained. It is a language that allows one to think about extraordinary questions ... getting the picture does not mean writing out the formula or crunching numbers, it means grasping the mathematical metaphor. (James Bullock, 1994.) Quoted from the book Metaphors & Analogiesby Rick Wormeli available at http://www.stenhouse.com/shop/pc/viewprd.asp?idProduct=9178&r=eu10113?pos=sponstop2&adv=stenhouse.
What is Mathematics? Article #1, Dave Moursund
This short article is written for adults interested in helping children learn about math. Begin by reading and thinking about the three quotes to the right of Spiderman. These quotes capture the essence of empowering the learner, responsibilities that empowerment brings, and math as problem solving. The third quote provide a good answer to question of what is mathematics.
“With great power comes great responsibility." (Peter Parker’s uncle Ben in a 2002 Spiderman movie.)
“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.” (George Polya, a leading mathematician and math educator of the 20th century.)
Accumulated Human Knowledge
Math is part of the accumulated knowledge of the human race. Math knowledge and skills empower a person. A healthy human brain is naturally curious and has a great capacity to learn. A newborn baby’s brain has a built-in capacity to learn natural languages. It already knows a little math, and it has the capacity to learn a great deal of math.
Today’s children are born into a complex and changing world. The totality of human knowledge is huge and is growing very rapidly. This knowledge is so vast that it is divided into disciplines or specializations. Mathematics is one of these broad, deep disciplines with a huge and ever growing accumulation of knowledge. Tens of thousands of math research articles are published each year. There is a huge accumulation of how to teach math, how to learn math, the history of math, and uses of math. Mathematics is one of the great achievements of humankind, and it is useful in many different disciplines of study.
In Your Mind, What Is Math?
As an adult, you know a lot about math and you use your knowledge every day. Thus, if I ask you “What is mathematics?” you can give me an answer that fits with how you view and use math. You are empowered by your ability to use math in dealing with money, time, distance, area, weight, and other problem areas. Your insights into math and your uses of math help to guide you as you help children to learn math through their informal and formal education.
However, the chances are that your answer to the "what is math" question is different than that of many other people. In terms of educating our children, it is helpful if we can have some agreement on what math is and what it is important for children to learn about math.
Mathematicians like to quote the famous mathematician Leopold Kronecker (1823–1891) who said: "God created the natural numbers, all else is the work of man." Long before we had written language and schools, people learned to count and make other simple uses of math.
From their early childhood caregivers, young children learn to count and how to determine the number of objects in a small set. Above that level of math understanding, knowledge, and skill, our formal math education begins to click in. For example, we have various numeral systems such as Roman numerals (I, II, III, IV, V), Hindu-Arabic numerals (1, 2, 3, 4, 5), Chinese numerals (一, 二, 三, 四, 五), and Arabic numerals (٠, ١, ٢, ٣, ٤). We have specially defined words and symbols for addition, subtraction, multiplication, division, and other operations on numbers. We have fractions and decimals. We have sub discipline of math such as algebra, geometry, statistics, probability, and calculus. We have various systems for measuring distance area, and quantity, such as the metric system and the English system.
We have developed aids to solving math problems. For example, you probably own a calculator, wristwatch, and ruler. You routinely use these as aids to doing arithmetic calculations, to determine time of day, to measure length, and so on. A car contains a speedometer and an odometer. Nowadays, it may contain a global positioning system (GPS) that can display a map and give you oral instructions as to when and where to turn to get to a specified destination.
Thus, one way to answer the question “What is mathematics?” is to name some of its sub disciplines, name some of its tools, and name some of its achievements. However, this is of limited value in talking to young children about math and helping them to develop personally useful understanding of the field. What mathematically empowers children?
A Language for Thinking About, Representing, and Solving Problems
A different way to think about math is that it empowers people who seek to represent and solve a wide range of problems in different disciplines. Math is both a special language and a special approach to representing, thinking and reasoning about, and solving certain kinds of problems. Because there has been such a large amount of research in math over the years, there is a huge accumulation of how to solve a wide variety of math problems. If a “real world” problem can be represented mathematically, this may be quite useful in solving the problem.
Here are two ideas that help to define goals of math education and empowering learners.
- Math fluency is being able to read, write, speak, listen, think, and understand communication in the language of mathematics. This is somewhat akin to developing fluency in a foreign language.
- Math maturity is being able to make effective use of the math that one has studied. It is the ability to recognize, represent, clarify, and solve math-related problems using the math one has studied. Thus, a fifth grade student can have a high or low level of math maturity relative to math content that one expects a typical fifth grader to have learned.
Suggestion to Teachers, Parents, and Exhibit Guides
As an adult, you can view the Fun 2, 3, 4 math exhibit through your adult eyes and select out some ideas and information that seems important from your point of view. Then you can view the exhibit through a child’s eyes. What might the child learn that will empower the child to move toward your level of insights and math maturity? Interact with children to help make these ideas explicit and meaningful.
What is Mathematics? Article #2, Dave Moursund
This answer to the "What is math" question is quoted from the book:
Moursund, D.G. (June 2006). Computational thinking and math maturity: Improving math education in K-8 schools. Access at http://uoregon.edu/~moursund/Books/ElMath/ElMath.html.
Many people have addressed the question, “What is mathematics?” See, for example, (Lewis, n.d.) and the many publications of the National Council of Teachers of Mathematics. Here are two good examples of answers to the question, “What is mathematics?”
- Mathematics is an inherently social activity, in which a community of trained practitioners (mathematical scientists) engages in the science of patterns—systematic attempts, based on observation, study, and experimentation, to determine the nature or principles of regularities in systems … The tools of mathematics are abstraction, symbolic representation, and symbolic manipulation. However, being trained in the use of these tools no more means that one thinks mathematically than knowing how to use shop tools makes one a craftsman. Learning to think mathematically means (a) developing a mathematical point of view—valuing the processes of mathematization and abstraction and having the predilection to apply them, and (b) developing competence with the tools of the trade, and using those tools in the service of the goal of understanding structure—mathematical sense-making (Schoenfeld, 1992).
Notice the emphasis on thinking mathematically. One gains increased expertise in math by both learning more math and by getting better at thinking and problem solving using one’s knowledge of math.
- Mathematics is built on a foundation which includes axiomatics, intuitionism, formalism, logic, application, and principles. Proof is pivotal to mathematics as reasoning whether it be applied, computational, statistical, or theoretical mathematics. The many branches of mathematics are not mutually exclusive. Oft times applied projects raise questions that form the basis for theory and result in a need for proof. Other times theory develops and later applications are formed or discovered for the theory. Hence, mathematical education should be centered on encouraging students to think for themselves: to conjecture, to analyze, to argue, to critique, to prove or disprove, and to know when an argument is valid or invalid. Perhaps the unique component of mathematics which sets it apart from other disciplines in the academy is proof—the demand for succinct argument that from a logical foundation for the veracity of a claim (McLoughlin, 2002).
Notice the emphasis on proof or disproof.
The word proof comes up in most attempts to define mathematics. Of course, the idea of proof or proving something is not restricted just to mathematics. A trial lawyer attempts to prove his or her case. A person attempts to prove that another person is wrong in a particular situation. Researchers in science attempt to prove scientific theories.
Each discipline has its own ideas and standards about what constitutes a proof. Math proofs are designed to answer, once and for all, the correctness or incorrectness of a “mathematical” assertion. Suppose, for example, that I am exploring the sum of three consecutive integers. I see that 6 + 7 + 8 = 21, and 11 + 12 + 13 = 26. After looking at a lot of examples, I conjecture that if the first of the three consecutive integers is odd, then the sum is an even integer; if the first integer is even, then the sum is an odd integer. Looking at lots of example, and not finding any counter examples, may increase my confidence that my conjectures are correct. However, my failure to find a counter example does not constitute a proof. Think about definitions of odd and even integers. See if you can construct a convincing proof that my conjectures are correct.
Then think about whether K-8 students, once they have encountered definitions of odd and even integers, might be able to develop convincing proofs. If the conjecture given above is too difficult for students at a particular age, how about considering the simpler conjecture that the sum of two consecutive integers is odd. A young child attacking this task might make use of small cubes, physically lining up rows of cubes to represent integers, and then arguing from the patterns that result.
Finally, be aware that there are lots of simple proof-type situations that can be constructed for use in the K-8 school setting. To give one more example, suppose that students have learned the mathematical word mean. You might then have them compute the mean of various sets of three consecutive integers, looking for a pattern. Quite likely some of the students will note that the answers they obtain are always the middle one of the three consecutive integers. Can they construct a convincing argument that this is always the case? What if one wants to find the mean of five consecutive integers?
When you present young students with such problems, you want to think carefully about what they may be learning. The examples given above might lead some students to think that the mean of a set of consecutive integers is an integer. With a little encouragement, some of your students might conjecture and then attempt to prove that “The mean of an odd number of consecutive integers is an integer, and the mean of an even number of consecutive integers is not an integer.”
The terms fluency and proficiency are often used in talking about goals and expertise in mathematics. The following definition of math proficiency is quoted from Kilpatrick et al. (2001), a report written for the National Academy of Sciences. Mathematical proficiency, as we see it, has five components, or strands:
- conceptual understanding—comprehension of mathematical concepts, operations, and relations
- procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
- strategic competence—ability to formulate, represent, and solve mathematical problems
- adaptive reasoning—capacity for logical thought, reflection, explanation, and justification
- productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.
Warning! The mathematical proficiency bulleted list reflects many hundreds of hours of thinking by some of the world’s leading math educators. Did you read it in a reflective manner? Did you work to construct your own meaning? What aspects of the presented ideas will you remember five minutes from now, a day from now, or a year from now?
For the most part, answers to the “what is math” question do not depend on specific areas of math content. The question and answers are part of math maturity. As you think about the mathematical proficiency bulleted list, you are working to increase an aspect of your math maturity that is very important to being a good teacher of math.
As you construct and/or make use of a math lesson plan, you can think about how it fits in with and contributes to increasing your students’ mathematical proficiency. For example, compare having students work on a drill and practice page of arithmetic computations, versus students solving word problems, versus students creating word problems, versus students reading a science book and identifying math usage in science.
What is Mathematics? Article #3, Gene Maier
The ideas in this section come from:
- Maier, Eugene (June, 2000). Problem Solving. Gene's Corner, Math Learning Center. Retrieved 2/27/2011 from http://www.mathlearningcenter.org/node/2471.
Here are several quotations form the article:
- … what mathematics is about. Making conjectures, seeking relationships, validating theories, searching for solutions, verifying results, communicating findings--in short, problem solving. To do mathematics is to solve problems.
- Problem solving--at least the phrase--has always been part of school math. Every math textbook series claims to emphasize it and every list of standards gives it special attention. Here in Oregon, the Department of Education's performance assessment in mathematics is a "problem-solving" test. Passing this test is a requirement for the Certificate of Initial Mastery--concocted as part of the Oregon Educational Act for the 21st Century in hopes of convincing the world that a tenth-grade Oregon education really means something.
- However, in contrast to the professional world of mathematics, school mathematics isn't synonymous with problem solving. In school, problem solving is likely to be considered just another topic, along with adding fractions, multiplying decimals, or finding perimeters. And so teachers attempt to teach it like any other topic …
- I suspect our students would be better problems solvers if we would quit treating problem solving as just another mathematics topic to be taught, but rather regard all mathematics as problem solving, and teach it accordingly. Most school math is ancient history--the mathematics that occurs in the curriculum are answers to mathematical questions that were posed years, or even centuries, ago. But these questions are new to our students.
- Mathematics doesn't have to be taught as a cut-and-dried, here's-how-you-do-it subject. It can be taught in a reflective, inquisitive mode. No matter what the topic, students' perceptions and suggestions can be explored, tested and refined. If every topic were introduced as a problem to be investigated rather than a process to be mastered, there would be no need to treat problem solving as a separate topic with its own set of rules and procedures. Students' ability to solve problems would evolve naturally, hand in hand with their mathematical knowledge and sophistication.
What is Mathematics—Additional Resources
There are many excellent expository papers about math and math education. Here are some that I have enjoyed reading:
Thurston, William (1990).Mathematics education. This article originally appeared in the Notices of the AMS 37 (1990), 844– 850. Retrieved 10/15/2012 from http://arxiv.org/abs/math/0503081. (See the Download button on the right side of that page.)
Quoting from the beginning of the article:
- Mathematics education is in an unacceptable state. Despite much popular attention to this fact, real change is slow.
- Policymakers often do not comprehend the nature of mathematics or of mathematics education. The ‘reforms’ being implemented in different school systems are often in opposite directions. This phenomenon is a sign that what we need is a better understanding of the problems, not just the recognition that they exist and that they are important.
- I am optimistic that our nation will find solutions to these problems. Problems rising from failure of understanding are curable. We do not lack for dedication, resources, or intelligence: we lack direction.
Steen,Lynn Arthur (2012). Reflections on mathematics and democracy. Mathematical Association of America. Retrieved 10/15/2012 from http://www.maa.org/pubs/FOCUSoct-nov12_Steen.html. This is the Revised text of a paper presented at a joint AMS-MAA Special Session at MathFest 2012 in Madison, Wisconsin. The session was organized by David Mumford and Solomon Garfunkel; other speakers were William McCallum, Hyman Bass, and Joseph Malkevitch.
Quoting from the paper:
- Today we are at the cusp of a 25-year effort to develop a common set of standards for school mathematics (and other subjects). Not surprisingly, these proposed “Common Core” standards3 improve on, rather than overthrow, orthodoxy. They generally adhere to the traditional curriculum—especially in their emphasis on algebra—while also encouraging limited innovation.
- Ten years ago I addressed the first question posed to this panel in Mathematics and Democracy—a collection of essays from a variety of professionals both inside and outside mathematics.4 (These essays are available for free downloading on the MAA website.) The chief message of this volume is that the mathematics taught in school bears little relationship to the mathematics needed for active citizenship. That mathematics we called quantitative literacy (QL) to contrast it with traditional school mathematics which, historically, is the mathematics students needed to prepare for calculus.
- Mathematics and quantitative literacy are distinct but overlapping domains. Whereas mathematics’ power derives from its generality and abstraction, QL is anchored in specific contexts and real world data. An alternative framing of the challenge for this panel is to ask whether perhaps QL might be a more effective approach to high school mathematics for all.
What is Mathematics For?
Dudley, Underwood (2010). What is mathematics for. Notices of the AMS. Retrieved 5/2/2010 from http://www.ams.org/notices/201005/rtx100500608p.pdf.
Underwood Dudley is a mathematician, retired from DePauw University. This 6-page article provides a little bit of the history of the teaching of math in schools and a discussion of uses of this math education. Here is an interesting historical tidbit quoted from the article:
- Of course, you may think, those were the ancients; in modern times we have learned better, and arithmetic at least has always been part of everyone’s schooling. Not so. It may come as a surprise to you, as it did to me, that arithmetic was not part of elementary education in the United States in the colonial period.
The article discusses the trend toward requiring all students to take algebra. He says:
- A few years ago I was at a meeting that had on its program a talk on the mathematics used by the Florida Department of Transportation. There is quite a bit. For example, the Florida DoT uses Riemann sums to determine the area of irregular plots of land, though it does not call the sums that. After the talk I asked the speaker what mathematical preparation the DoT expects in its new hires. The answer was, none at all. The DoT has determined that it is best for all concerned to assume that the background of its employees includes nothing beyond elementary arithmetic. What employees need, they can learn on the job.
The article ends with the statement:
- What mathematics education is for is not for jobs. It is to teach the race to reason. It does not, heaven knows, always succeed, but it is the best method that we have. It is not the only road to the goal, but there is none better. Furthermore, it is worth teaching. Were I given to hyperbole I would say that mathematics is the most glorious creation of the human intellect, but I am not given to hyperbole so I will not say that. However, when I am before a bar of judgment, heavenly or otherwise, and asked to justify my life, I will draw myself up proudly and say, “I was one of the stewards of mathematics, and it came to no harm in my care.” I will not say, “I helped people get jobs.”
What is Algebra?
As the discipline of mathematics has grown over thousands of years, various sub disciplines have been developed. Examples include arithmetic, algebra, geometry, and statistics.
Algebra is a very important component of mathematics. It is often called the "language of mathematics." See Communicating in the Language of Mathematics. Also see:
- Steen, Lynn Arthur (Fall 1999). Algebra for All in Eighth Grade: What's the Rush? Middle Matters, the newsletter of the National Association of Elementary School Principals, Vol 8, No. 1, Fall 1999, pp. 1, 6-7. Retrieved 8/7/09: http://www.stolaf.edu/people/steen/Papers/algebra.htm.
Quoting from the National Council of Teachers of Mathematics:
- Algebra is a way of thinking and a set of concepts and skills that enable students to generalize, model, and analyze mathematical situations. Algebra provides a systematic way to investigate relationships, helping to describe, organize, and understand the world. Although learning to use algebra makes students powerful problem solvers, these important concepts and skills take time to develop. Its development begins early and should be a focus of mathematics instruction from pre-K through grade 12. Knowing algebra opens doors and expands opportunities, instilling a broad range of mathematical ideas that are useful in many professions and careers. All students should have access to algebra and support for learning it.
- Algebra provides a way to explore, analyze, and represent mathematical concepts and ideas. It can describe relationships that are purely mathematical or ones that arise in real-world phenomena and are modeled by algebraic expressions. Learning algebra helps students make connections in varied mathematical representations, mathematics topics, and disciplines that rely on mathematical relationships. Algebra offers a way to generalize mathematical ideas and relationships, which can then be applied in a wide variety of mathematical and nonmathematical settings.
Here is one more important piece of NCTM insight into algebra education. Quoting from the same source:
- The development of algebraic concepts and skills does not occur within a single course or academic year. An understanding of algebra as a topic, a course of study, and a collection of mathematical understandings develops over time, and students must encounter algebraic ideas across the pre-K-12 curriculum. At the elementary level, teachers should help students develop fluency with numbers, identify relationships, and use a variety of representations to describe and generalize patterns and solve equations. Secondary school teachers should help students move from verbal descriptions of relationships to proficiency in the language of functions and skill in generalizing numerical relationships expressed by symbolic representations. Teachers should also help students develop skills in the strategic use of a range of technological tools, including graphing calculators, spreadsheets, statistical software, and computer algebra systems. Because knowing algebra is essential in a wide variety of careers and professions, students should have the guidance of highly qualified teachers as they learn algebra.
The quoted material given above is profound, deep, and not easily understood by many of the teachers who provide students with their first introduction to algebra. Consider, for example, a primary school teacher providing students with problems such as:
Here the "box" represents an "unknown number" that is to be filled in to form a correct number sentence. Or, more abstractly, the box is a variable.
Later in elementary school students will encounter the use of the symbol x or other symbols in place of the box. Thus, a fourth grade teacher may proudly proclaim, "I am teaching the students algebra," when having students solve problems such as:
- 6 + x + 5 = 25
- 5 + 2x = 21
- 3 + 5x = 7 + 3x
In these equations, x is a variable.
A variable is a very important and "deep" concept in math. Quoting from the James & James Mathematical Dictionary:
- A variable is "A symbol used to represent an unspecified member of some set. A variable is "a place holder" or a "blank" for the name of some member of the set.
This definition represents several thousand years of "struggle" by mathematicians to come up with a good definition. For some discussion of this history, see Steven Wolfram.
The primary school teacher faces the problem of helping students gain an initial foundation that can be built upon in the future. Notice the use of the word "set" in the James & James definition. In 1874 Georg Cantor published a research article that marks the birth of set theory.
That is, "set" is a relatively recent mathematical concept. Quoting from the Georg Cantor reference given above:
- Georg Cantor (1845–1918) was a German mathematician, born in Russia. He is best known as the creator of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic.
In summary, many elementary school teachers are "in way over their heads" as they introduce their students to algebraic concepts such as variable and set. The students, who are at the Piagetian concrete operations level of cognitive development are facing abstract ideas that stretched the minds of mathematical researchers who first developed the ideas.
For most students, the learning approach used is to memorize some rules, with relatively little understanding of the rules.
Perhaps you have heard the statement: "The road to hell is paved with good intentions." That may well be a statement that applies to having many elementary school teachers teaching initial concepts of algebra that are above their own heads and well beyond the cognitive developmental levels of most of their students.
Some Additional Dave Moursund Thoughts 10/22/08
Math is an old, broad, and deep discipline. From an Information Age Education point of view, a good answer to the "what is math" question needs to:
- Help differentiate math from other disciplines. What is there about math that separates it from economics, history, literature, or the hundreds of other disciplines and sub disciplines that are taught in precollege and higher education?
- Help students and our education system understand what math knowledge, skills, and understanding is necessary for people of different ages and different walks in life. In place of the word "necessary" one might also discuss "desirable."
- Address the issue of roles of Information and Communication Technology as part of math and as an aid to making use of math in many different disciplines.
All three questions need to deal with the fact that math is woven into the fabric of many other disciplines. For example, every discipline deals with problems solving and task accomplishing that is specific to the discipline. Often this involves use of some math.
The Information Age Education Wiki contains a number of math education pages. Most are aimed at preservice and inservice teachers of math, and to teachers of teachers of math.
The Math Education Digital Filing Cabinet contains links to the IAE math pages and to some other valuable math education resources available on the Web.
ACM News Release (10/12/2011). Experimental mathematics: Computing power leads to insights. American Mathematics Society. Retrieved 10/17 2001 from http://www.ams.org/news/ams-news-releases/ams-news-releases. See http://www.ams.org/notices/201110/rtx111001410p.pdf for a copy of the journal article that is being referred to. Quoting from the news release:
- Providence, RI—In his 1989 book The Emperor's New Mind, Roger Penrose commented on the limitations on human knowledge with a striking example: He conjectured that we would most likely never know whether a string of 10 consecutive 7s appears in the digital expansion of the number pi. Just 8 years later, Yasumasa Kanada used a computer to find exactly that string, starting at the 22869046249th digit of pi. Penrose was certainly not alone in his inability to foresee the tremendous power that computers would soon possess. Many mathematical phenomena that not so long ago seemed shrouded and unknowable, can now be brought into the light, with tremendous precision.
Here is a list of math computational uses given in the research article:
- gaining insight and intuition;
- visualizing math principles;
- discovering new relationships;
- testing and especially falsifying conjectures;
- exploring a possible result to see if it merit formal proof;
- suggesting approaches for formal proof;
- computing replacing lengthy hand derivations;
- confirming analytically derived results.
Kilpatrick, J.., Swafford, J., & Findell, B. (2002). Adding it up: Helping children learn mathematics. The National Academies Press. Available free on the Web. Retrieved 4/26/06: http://www.nap.edu/books/0309069955/html/index.html.
Lewis, Robert H. (n.d.). Mathematics: The most misunderstood subject. Retrieved 5/4/06: http://www.fordham.edu/mathematics/whatmath.html.
Notices of the ACM December 2008). A Special Issue on Formal Proof. Retrieved 11/12/08: http://www.ams.org/notices/200811/.
- Using computers in proofs both extends mathematics with new results and creates new mathematical questions about the nature and technique of such proofs. This special issue features a collection of articles by practitioners and theorists of such formal proofs which explore both aspects.
McLoughlin, M.M. (3 August 2002). The central role of proof in the mathematics canon: The efficacy of teaching students to do proofs using a fusion of modified, more traditional, and reform methods. Retrieved 7/22/04: http://facstaff.morehouse.edu/~pmclough/MAA. [I was not able to find this article online on 4/27/06 or on 11/7/2011.]
Schoenfeld, Alan (1992.) Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. [Chapter 15, pp. 334-370, of the Handbook for Research on Mathematics Teaching and Learning (D. Grouws, Ed.). New York: MacMillan, 1992.] Retrieved 4/27/06: http://www-gse.berkeley.edu/faculty/AHSchoenfeld/LearningToThink/Learning_to_think_Math.html.
Author or Authors
The initial version of this article was written by David Moursund.