Math Education Wars

From IAE-Pedia

Contents 1 Introduction 2 What is Math? 3 Information Overload—The Totality of Accumulated Knowledge 4 More About the Discipline of Mathematics 5 History of the Math Wars 5.1 Mathematically Correct 5.2 Mathematically Sane 5.3 Compromise: A Middle Road 6 National Mathematics Advisory Panel 6.1 The NMAC Report 6.2 Reaction to the NMAC Report 7 Final Remarks 8 References 9 Author or Authors

During the 1990s, the teaching of mathematics became the subject of heated controversies known as the math wars. The immediate origins of the conflicts can be traced to the “reform” stimulated by the National Council of Teachers of Mathematics ’Curriculum and Evaluation Standards for School Mathematics. Traditionalists fear that reform-oriented, “standards-based” curricula are superficial and undermine classical mathematical values; reformers claim that such curricula reflect a deeper, richer view of mathematics than the traditional curriculum. An historical perspective reveals that the underlying issues being contested—Is mathematics for the elite or for the masses? Are there tensions between “excellence” and “equity”? Should mathematics be seen as a democratizing force or as a vehicle for maintaining the status quo?—are more than a century old. This article describes the context and history, provides details on the current state, and offers suggestions regarding ways to find a productive middle ground. (Schoenfeld, 2004)

Introduction

Reading, writing, and arithmetic have been part of the “core” educational curriculum since writing was first developed about 5,200 years ago. Thus, from a formal education point of view, math is an old discipline. In addition, informal math education certainly existed as long as we have had language that included counting words.

Even if we only go back to the first development of written language, this means we have had over 5,000 years to experiment with different ways of teaching and learning math, and the challenge of what content to put into the math curriculum.

Over the centuries, math has become more and more important to more and more people. Math research has led to substantial and continuing increases in the totality of collected math knowledge. This totality is so large that nowadays a person earning a doctorate in math has studied only a very small fraction of this accumulated math knowledge. As the totality of accumulated math knowledge has continued to grow, the math education system has continually adjusted the topics it wants students to learn about, and the breadth and depth of knowledge that students are to achieve. Math education research has led to the development of better ways to teach and learn math.

There are many different aspects of this challenge of dealing with potential changes in the content and teaching of math. For example, what math is so important that it should be part of the required core curriculum for all students? Is algebra so important that every student should be required to pass an algebra course—or, a high stakes algebra exam—in order to graduate from high school? What about geometry, probability, statistics (including the collection and analysis of data)?

The development of electronic digital calculators ad computers has added considerably to the challenges facing our math education system. These aids to doing math have become increasingly powerful and readily available over the past half-century. If a calculator or computer can solve a particular type of math problem, what do we what students to learn about that type of math problem? Should we have a math curriculum that emphasizes students routinely using calculator and computer aids to representing and solving math problems? Computers have made possible quite sophisticated Computer-assisted Learning (CAL). There are some components of the math curriculum in which CAL is more effective than the typical whole class instruction provided by a typical classroom teacher.

The possible roles of calculators and computers in the math curriculum have been an ongoing source of disagreement among various math education stakeholder groups. However, that is not the only reason for the current math “wars.” The remainder of this document explores various aspects of the current math education wars.

What is Math?

To understand and effectively participate in the math wars, it helps to have good insights into what mathematics is and what the commonly accepted goals of math education are. Many people have addressed the question, “What is mathematics?” This section contains some examples of what they have said.

“Mathematics is the queen of the sciences.” (Carl Friedrich Gauss, German mathematician, physicist, & prodigy; 1777 – 1855.)

Gauss is one of the greatest mathematicians of all time, and this is an often-quoted statement. It says that from a mathematician’s point of view, math is very important. However, it does not provide any information about what math is. Here is a quote from Alan Schoenfeld that provides us with more detailed information.

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—on gaining mathematical understanding. 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.

George Polya was one of the leading mathematicians of the 20th century, and he wrote extensively about problem solving. The Goals of Mathematical Education (Polya, 1969) is a talk that he gave to a group of elementary school teachers.

To understand mathematics means to be able to do mathematics. And what does it mean doing mathematics? In the first place it means to be able to solve mathematical problems. For the higher aims about which I am now talking are some general tactics of problems—to have the right attitude for problems and to be able to attack all kinds of problems, not only very simple problems, which can be solved with the skills of the primary school, but more complicated problems of engineering, physics and so on, which will be further developed in the high school. But the foundations should be started in the primary school. And so I think an essential point in the primary school is to introduce the children to the tactics of problem solving. Not to solve this or that kind of problem, not to make just long divisions or some such thing, but to develop a general attitude for the solution of problems. [Bold added for emphasis.]

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

As the following quotation from the same talk indicates, Polya was particularly concerned with helping students learn to think mathematically when working on problems. Notice the emphasis on representing problems in the abstract words and symbols of math.

We wish to develop all the resources of the growing child. And the part that mathematics plays is mostly about thinking. Mathematics is a good school of thinking. But what is thinking? The thinking that you can learn in mathematics is, for instance, to handle abstractions. Mathematics is about numbers. Numbers are an abstraction. When we solve a practical problem, then from this practical problem we must first make an abstract problem. Mathematics applies directly to abstractions. Some mathematics should enable a child at least to handle abstractions, to handle abstract structures.

Moursund (2006) discusses math cognitive development and the development of math maturity. Quoting from that reference:

The term math maturity is widely used by mathematicians and math educators. For example, a middle school teacher may say, “I don’t think Pat has the necessary math maturity to take an algebra course right now.” It is clear that the teacher is not talking about Pat’s math content knowledge. Probably Pat has completed the prerequisite coursework. Perhaps Pat is weak in math reasoning and thinking, tends to learn math by rote memorization, has little interest in math, and shows little persistence in working on challenging math problems.

Perhaps the dominant component in the literature of math maturity is “proof” and the logical, critical, creative reasoning and thinking involved in understanding and doing proofs. The following list contains this and some additional components of math maturity. An increasing level of math maturity is demonstrated by:

1. An increasing capacity in the logical, critical, creative reasoning and thinking involved in understanding and doing proofs.

2. An increasing capacity to move beyond rote memorization in recognizing, posing, representing, and solving math problems. This includes transfer of learning of one’s math knowledge and skills to problems in many different disciplines.

3. An increasing capability to communicate effectively in the language and ideas of mathematics. This includes:

A. Mathematical speaking and listening fluency.

B. Mathematical reading and writing fluency.

C. Thinking and reasoning in the language and images of mathematics.

4. An increasing capacity to learn mathematics—to build upon one’s current mathematical knowledge and to take increasing personal responsibility for this learning.

5. Improvements in other factors affecting math maturity such as attitude, interest, intrinsic motivation, focused attention, perseverance, having math-oriented habits of mind, and acceptance of and fitting into the “culture” of the discipline of mathematics. The term math maturity is widely used by mathematicians and math educators. For example, a middle school teacher may say, “I don’t think Pat has the necessary math maturity to take an algebra course right now.” It is clear that the teacher is not talking about Pat’s math content knowledge. Probably Pat has completed the prerequisite coursework. Perhaps Pat is weak in math reasoning and thinking, tends to learn math by rote memorization, has little interest in math, and shows little persistence in working on challenging math problems.

The dominant component in the literature of math maturity is “proof” and the logical, critical, creative reasoning and thinking involved in understanding and doing proofs.

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.

The responses to the “What is Math” question given above come from people with a deep knowledge and understanding of math. The following description of our math education system reflects good insight into math education from a layperson point of view.

For most students, school mathematics is an endless sequence of memorizing and forgetting facts and procedures that make little sense to them. Though the same topics are taught and retaught year after year, the students do not learn them. Numerous scientific studies have shown that traditional methods of teaching mathematics not only are ineffective but also seriously stunt the growth of students' mathematical reasoning and problem-solving skills. Traditional methods ignore recommendations by professional organizations in mathematics education, and they ignore modern scientific research on how children learn mathematics (Battista, 1999).

Many adults remember their math education experience as “an endless sequence of memorizing and forgetting facts and procedures that make little sense.” Thus, there are major differences in perspective of “What is Math” and what constitutes a good math education system.

Information Overload—The Totality of Accumulated Knowledge

The totality of accumulated knowledge is both very large and is growing quite rapidly. Depending on the definitions used and the ways this totality is measured, it is often estimated to be doubling every five to ten years, or perhaps more rapidly. Such a rapid pace of increase overwhelms the traditional hard copy library approach to collecting and storing information, and making it available to large numbers of people. It overwhelms any approach of schools teaching students the knowledge and skills that will suffice for the rest of their lives. It has led to the situation where many people feel that they are burdened by an information overload.

Computer technology has provided substantial help in addressing this problem. We now have virtual libraries such as the Web that are far larger than the largest of hardcopy libraries. Moreover, virtual libraries are far easier to access than physical libraries. Even elementary school students can learn to retrieve information from virtual libraries. The information growth problem can also being attacked by placing greater emphasis on learning to learn and “just in time” learning. Help students increase their self-reliant learning skills. Give them an education that provides the background that facilitates just in time learning—learning when one has specific need for the learning.

Computer technology provides another aid to addressing this information growth problem. A computer can solve many of the problems and accomplish many of the tasks that people have previously done “by hand and brain.” For example, a graphing calculator can quickly draw a graph of a function. An equation-solving calculator can solve many different kinds of equations. Tax preparation software can provide a great deal of help in preparing an income tax return. A global positioning system (GPS) can help solve the problem of where you are located and how to get to a specific other location.

Even with all of these aids, people tend to feel we have an information overload and that they need still more help in dealing with this information overload. Each academic discipline taught in our schools faces the challenge of dealing with a limited amount of time to help students learn the discipline, and a steadily growing amount of accumulated data, information, knowledge, and wisdom in the discipline.

More About the Discipline of Mathematics

So far, we have used the term discipline rather loosely. In a typical school, the curriculum is divided into pieces (blocks of time, courses), each devoted to study within a particular discipline. Over the years, this approach to developing the overall curriculum in a school has become relatively standard, although other approaches are certainly possible. For example, we know that reading and writing are part of the language arts curriculum, but their use cuts across many different curricula. Thus, some schools stress the teaching of reading and writing within specific time slots, but also stress “reading and writing across the curriculum.”

The same holds true for math. Some schools integrate the teaching of math and science. A few schools have both math courses and an emphasis on roles of math in the other courses being taught.

While formal schooling is typically divided into discipline-specific chunks, the world outside of school does not tend to be that way. Read world problems tend to be interdisciplinary, cutting across many disciplines. Thus, students are educated in discipline-specific chunks but expected to solve problems and accomplish tasks in an outside of school environment that is not divided into such chunks.

Math is an academic discipline. It has some of the general characteristics of other academic disciplines, and some specific differences from each other academic discipline. Here is a relatively long bulleted list that can help one define a specific discipline and distinguish it from other disciplines. Each academic discipline can be defined by a combination of:

• The types of problems, tasks, and activities it addresses. How people work individually and/or in teams, face to face and by other modes of communication, to solve the problems, accomplish the tasks, and to ”do” the discipline.
• Its accumulated accomplishments, such as its results, achievements, products, performances, scope, power, uses, impacts on the societies of the world, and so on.
• Its history, culture, methods of communication, and language (including notation and special vocabulary).
• Its methods of teaching, learning, assessment, and thinking, and what it does to preserve and sustain its work and pass it on to future generations. Its preparation of teachers, coaches, tutors, apprentice masters, practitioners, scholars, researchers, and so on.
• Its tools, methodologies, and types of evidence and arguments used in solving problems, accomplishing tasks, and recording and sharing accumulated results.
• How it accommodates and/or assimilates changes within its discipline and within other possibly related disciplines. (With respect to the discipline of math, for example, other related areas or disciplines include science, technology, employment, people, and societies.)
• The knowledge and skills that separate and distinguish among (a) a novice, (b) a person who has a personally useful level of competence, (c) a reasonably competent person, (d) an expert, and (e) a world-class expert. Each discipline has its own ideas as to what constitutes a high level of expertise within the discipline and its sub disciplines.

You might find that this list is somewhat overwhelming. This particular IAE-pedia article is about math. You began to learn math at the time you began to learn oral language. You studied math for many years during your precollege education. You may well have taken some math courses in college. Thus, you know a great deal about math.

Go through the bulleted list. Test your insights into math by thinking briefly about each of the bulleted items from a math point of view.

Next, select a second discipline that you have extensively studied. Spend a few minutes analyzing your knowledge and understanding of math and a second discipline. Pay special attention to what—in your mind—is the same and is different between the two disciplines. When you get down to the last bulleted item, make a determination of your current level of expertise in the two disciplines. Does your current level of expertise in each of the disciplines meet your current needs? Does it meet needs that you expect to have in the future? What are you doing to maintain or improve your level of expertise in each of the disciplines?

It is easy to see why the various math education stakeholder groups have disagreements as to appropriate content, teaching processes, and assessment for precollege math education. It is easy to find fault with some aspects of the overall system. Moreover, it is easy to pick some particular measurement of some particular aspect of the system, and find evidence that this aspect of math education is not doing as well as one would like. The comparison might be between schools, (including private versus public) school districts, states, regions of the country, or countries.

For example, much is made of the data from the various international comparisons. Many countries get higher scores than the United States on the Trends in International Mathematics and Science Study. This leads many people to think that there must be something wrong with the math education system in the United States.

Even without the push from specific politicians, the past fifty years has seen major efforts to improve science and math education in the United States. The National Council of Teachers of Mathematics took a lead role in math education reform in its 1989 Standards. The National Science Foundation has been one source of leadership and funding. Many different private foundations and many different professional societies have added their efforts.

History of the Math Wars

Alan Schoenfeld is a highly respected math educator. His 2004 article on the Math Wars provides a history of changes going on is the world that led to agreements that our math education system needed improvement. He chronicles the processes used in developing and implementing changes, and then the reactions of various groups of people to the changes being implemented.

The National Council of Teachers of Math developed a set of math education Standards that it published in 1989. Quoting from Schoenfeld (2004):

On the radical side, the Standards challenged (or was seen as challenging) many of the assumptions underlying the traditional curriculum. As noted above, the traditional curriculum bore the recognizable traces of its elitist ancestry: The high school curriculum was designed for those who intended to pursue higher education. Yes, it is true that half the students dropped out of the mathematics pipeline each year after Grade 9—but as some see it, this is because honest-to-goodness mathematics is hard. The 50% annual attrition rate was taken by some as confirmation of the difficulty of mathematics. For them, there was the suspicion that the curriculum would have to be dumbed down in order for more students to succeed. That is, only a bastardized curriculum (a lowering of real standards) could result in greater success rates. These fears were exacerbated by, among other things, the “increased attention/decreased attention” charts in the Standards. Topics to receive decreased attention included the following: complex paper-and-pencil computations, long division, rote practice, rote memorization of rules, teaching by telling, relying on outside authority (teacher or an answer key), memorizing rules and algorithms, manipulating symbols, memorizing facts and relationships, the use of factoring to solve equations, geometry from a synthetic viewpoint, two column proofs, the verification of complex trigonometric identities, and the graphing of functions by hand using tables of values. These can be seen as the meat and potatoes of the traditional curriculum.

…

Epistemologically, with its focus on process, the Standards could be seen as a challenge to the “content-oriented” view of mathematics that predominated for more than a century. Each of the three grade bands began with the following four standards: mathematics as problem solving, mathematics as communication, mathematics as reasoning, and mathematical connections. Only after these four process standards were described did the Standards turn to what has traditionally been called mathematical content.

In short, the seeds for battle were sown—not that anyone at the time could predict that the Standards would have much impact or that the battle would rage.

The next two subsections are brief summaries from the two major stakeholder groups in the Math Wars.

Mathematically Correct

The two sides in the Math Wars agree that math education needs to be improved and that it can be improved. The Mathematically Correct side focuses on a need to place more emphasis on the basics and less emphasis on the various reforms such as New Math, New-New Math, and Whole Math.

Quoting from the Mathematically Correct Website (Mathematically Correct, n.d.):

Mathematics achievement in America is far below what we would like it to be. Recent "reform" efforts only aggravate the problem. As a result, our children have less and less exposure to rigorous, content-rich mathematics.

The advocates of the new, fuzzy math have practiced their rhetoric well. They speak of higher-order thinking, conceptual understanding and solving problems, but they neglect the systematic mastery of the fundamental building blocks necessary for success in any of these areas. Their focus is on things like calculators, blocks, guesswork, and group activities and they shun things like algorithms and repeated practice. The new programs are shy on fundamentals and they also lack the mathematical depth and rigor that promotes greater achievement.

Quoting from Mathematically Correct (1996):

The impending changes in mathematics education are not based on any change in the mathematics that has been developed over thousands of years. Rather, they are based on a cluster of notions from teaching philosophy and a desire to implement them all at once. The driving force behind these changes is dissatisfaction with the continued declines in the achievement of American students, coupled with the idea that a set of goals should be developed that all students can attain. The position taken is that poor math achievement is the result of the traditional curriculum and the way it has been implemented by teachers. The fact that math education in countries with high levels of achievement does not look like these new programs, but rather like intensified versions of our own traditional programs, is never addressed.

…

Another philosophical notion is the idea of Complete Math, which is a replacement term for Whole Math. Just as Whole Language attempted to skip the basics of phonics and go directly to reading literature, Whole Math attempts to cast aside computational basics and go to final productions that rely on math at some level. This view holds that math is not used only in equations, but in writing and discussion as well. The implication is that students should write essays and have group discussions about math. The major problem with this method is that students end up spending hours working on essays, again detracting from their chance to practice basic skills. This has the substantial risk that potentially controversial moral lessons make their way into assignments. Another problem with the emphasis on language skills, which many feel is misplaced in math class, is the disadvantage to students with speech and hearing problems or non-native English speakers.

Mathematically Sane

The Mathematically Sane Website was developed to help counter the Mathematically Correct Website. Quoting from the Mathematically Sane Website:

There are at least two sides to every issue, including the so-called “Math Wars.”

• For too long, however, the public has heard primarily from the side of the traditionalists. MathematicallySane.com has been developed to balance the equation.

• For too long the case for reform has been unfairly characterized as “Fuzzy Math.” Mathematically Sane has been created to provide an alternative—and more accurate—view of reform by making a compelling case that changes in our nation’s mathematics programs are imperative for our students’ future success and for the economic health of our nation.

• For too long there has been an insidious—and often unanswered—campaign to return mathematics instruction to the failed practices of the past. Mathematically Sane has been developed to provide a broad array of evidence that reform initiatives have been successful and have raised student achievement in school districts across the country.

Accordingly, Mathematically Sane's mission is to advocate—broadly and persuasively—for the rational reform of school mathematics.

A number of leading math educators have spoken in favor of the math reform movement. The following statements from Robert Reys are representative:

The National Council of Teachers of Mathematics, a nonprofit organization of mathematics teachers, has published a set of content standards in math called Principles and Standards for School Mathematics. Consistent with these standards, some textbooks are now integrated—topics from arithmetic, algebra, geometry, statistics, and probability are naturally connected. Integration is commonplace in countries, such as Japan, whose students excel on international mathematics tests. But most US schools are still mired in a 19th-century course sequence of Algebra I, Geometry, and Algebra II.

Throughout most of the 20th century, statistics and probability were not taught in school. Yet today, one cannot read and understand Newsweek, USA Today, or countless other news sources without being able to interpret statistical information. As a student, I used a slide rule to do some computations; today, I use a calculator. I also spent endless hours doing computations and rarely learning to estimate. Now I rarely do any tedious computations but regularly call upon estimation to decide if a calculator result is reasonable.

True reform would allow calculators, graphing calculators, and other readily available technological tools to help students solve equations in an instant, replacing pages of written procedures and hours of work. A mathematics curriculum should reflect what is important for the future and include advances in technology. (Reys, 2002)

Notice the emphasis on seeking a change from a 19th century curriculum and on incorporating Information and Communication Technology.

Compromise: A Middle Road

Both sides in the math wars agree that our math education system needs to be improved. One side tends to stress that we can and should do a better job of teaching the traditional basics. The other side stresses the need for research-based reforms that better address the math needs of people living in the 21st century.

A number of people argue that the math reform movement has tried to move too rapidly. They note that the reform curricula tend to crate a schism between what parents think math is and what math education ought to be, and what their children are experiencing.

Phil Daro (n.d.) is playing a major role in trying to facilitate discussions between the two groups and to mediate a compromise “middle ground” end to the war. He suggests that one way to reclaim the middle ground, is to define it clearly—to specify a set of propositions that will call for some degree of compromise from reformers and traditionalists alike. That middle ground would be broadly encompassing, containing propositions that most people would find reasonable (or at least livable). Daro offered a draft “Math Wars Peace Treaty” (or perhaps “Math Wars Disarmament Treaty”) that includes the following stipulations:

Whose fault is the status quo? How should we improve the status quo? What are the highest priorities for change? What are the best strategies for change? We have among us agreements and disagreements about these things. But about these things we agree:

• The status quo is unacceptable. Its defenders are wrong, mathematics instruction must improve;

• Teachers, especially K–8 teachers, should learn more mathematics throughout their careers;

• No students should be denied a fair chance to learn mathematics because they have been assigned unqualified mathematics teachers;

• All students should have a copy of their mathematics books to take home;

• Research and evidence should be used whenever it is available to inform decisions.

We also agree that students should learn to:

• add, subtract, and multiply single digit numbers automatically and accurately.

• add, subtract, multiply and divide integers, decimals and fractions accurately, efficiently, and flexibly without calculators.

• understand the mathematics they study and use.

• use the mathematics they know to solve problems with calculators and computers.

• be fluent with the symbolic language of algebra and understand how to use the basic laws of algebra when working mathematics problems.

• explain and justify their reasoning and understand the reasoning of others.

• reason with increasing rigor and mathematical maturity as they advance through the curriculum.

• formulate, represent and solve mathematical problems.

• apply their mathematical knowledge and know-how to analyze and solve unfamiliar problems.

• approach learning and using mathematics with a sense of efficacy: “I can learn it and use it; mathematics makes sense.”

• approach mathematics with diligence and curiosity, systematically and inventively; with the concentration to execute a procedure accurately and the courage to use initiative and imagination.

National Mathematics Advisory Panel

In 2006, President Bush appointed the National Mathematics Advisory Panel NMAP). It issued its final report on March 13, 2008.

Some people expected that the work of the NMAP might provide information and advice that would help to reconcile differences between the two extremes in the Math Wars. That has not proven to be the case. Indeed, a new battlefront has emerged between those who are supportive of the NMAC report and those who are against it.

Here are some sources of informaiotn that readers may find useful.

The NMAC Report

The Final Report, reports of three subcommittees and five Task Groups, together with two appendices containing the Presidential Executive Order setting up the panel, and details of the panel membership and other personnel, can be downloaded from www.ed.gov/about/bdscomm/list/mathpanel/index.html.

Reaction to the NMAC Report

Greer, Brian (2008). Guest Editorial: Reaction to the Final Report of the National Mathematics Advisory Panel. TMME, vol5, nos.2&3, p.365, Retrieved 4/24/09: http://www.math.umt.edu/tmme/vol5no2and3/TMME_vol5nos2and3_a17_pp.365_370.pdf.

Quoting from this document:

The subcommittees dealt with Standards of Evidence, Instructional Materials, and a National Survey of Algebra Teachers and the task groups with Conceptual Knowledge and Skills, Learning Processes, Teachers and Teacher Education, Instructional Practices, and Assessment. Henceforth "the report" refers to these documents collectively, and "Final Report" to the summary document.

Practitioners, scholars, and researchers within the field of mathematics education were underrepresented on the panel, and accorded surprisingly little input to the style and content of the report. In this collection of papers, some of us raise a number of issues that we find troublesome in the report. Many others issues, notably practices of assessment in school mathematics, are equally deserving of scrutiny.

…

In the NMAP Final Report, the main aim is clearly stated. Mathematics (and science) education is seen as key to economic competitiveness, with implications, moreover, for national security. Thus, it is declared (p. xi) that "the safety of the nation and the quality of life – not just the prosperity of the nation – are at issue."

Nationalistic motivations of this kind are by no means confined to the United States, but are most strongly expressed in this country. Yet, there is an alternative worldview in which mathematics and science are seen as having a central role in solving the problems of humankind in general.

D'Ambrosio (2003) has written passionately about the ethical responsibilities of mathematicians and mathematics educators:

It is clear that Mathematics is well integrated into the technological, industrial, military, economic and political systems and that Mathematics has been relying on these systems for the material bases of its continuing progress. It is important to look into the role of mathematicians and mathematics educators in the evolution of mankind. … It is appropriate to ask what the most universal mode of thought – Mathematics – has to do with the most universal problem – survival with dignity.

I believe that to find the relation between these two universals is an inescapable result of the claim of the universality of Mathematics. Consequently, as mathematicians and mathematics educators, we have to reflect upon our personal role in reversing the situation. (Emphasis in original).

The quoted material given above provides a start in identifying the source of major disagreements about the report. These quotes point out disagreements about the makeup of the committee and the orientation of the committee.

(An aside.) As I (David Moursund) was writing this section, I was holding two young kittens (not quite four weeks old) that my wife and I are providing foster care for. We started with three kittens that were about a week old. There eyes were still closed and, of course, they needed bottle feeding.

All three received the same loving care. One died yesterday, perhaps from congenital reasons. As I petted the remaining two, one purred loudly and the other meowed loudly. Three kittens—different even at birth, and more different now. This kitten situation makes me think about the types of challenges teachers face in implementing our math education system.

Final Remarks

Neither side in the Math Wars has presented a careful analysis of current and potential roles of Information and Communication Technology in the content, instruction, and assessment of a desirable math curriculum. Let’s suppose that the Middle Road Compromise becomes widely accepted. With the continuing progress is ICT, this meas we will have a widening gap between the math education curriculum and what is possible if one more fully accepts and makes use of ICT.

A good piece of this difference is seen in the capabilities of Computer Algebra Systems (CAS, n.d.). Quoting from the Wikipedia, CAS systems typically provide:

• simplification to the smallest possible expression or some standard form, including automatic simplification with assumptions and simplification with constraints

• substitution of symbolic, functors or numeric values for expressions

• change of form of expressions: expanding products and powers, rewriting as partial fractions, constraint satisfaction, rewriting trigonometric functions as exponentials, etc.

• partial and total differentiation

• symbolic constrained and unconstrained global optimization

• partial and full factorization

• solution of linear and some non-linear equations over various domains

• solution of some differential and difference equations

• taking some limits

• some indefinite and definite integration, including multidimensional integrals

• integral transforms

• arbitrary precision numeric operations

• Series operations such as expansion, summation and products

• matrix operations including products, inverses, etc.

• display of mathematical expressions in two-dimensional mathematical form, often using typesetting systems similar to TeX (see also Prettyprint)

In short, a huge amount of the manipulative and computational aspects of math that are currently being taught in the K-14 curriculum can be carried out be a CAS. Think of the disagreements between what simple arithmetic computations students should learn to do mentally or using paper and pencil procedures. Then carry this thought to essentially all of the computational and symbol manipulation procedures students learn in K-14 math. You can see that this provides a serious challenge to our math education system.

[Note dated 7/5/09, and not yet properly integrated in this document. Wolfram Alpha is software that can do many of the CAS operations and is available free on the Web. See http://www.wolframalpha.com/. This software is built upon the Mathematica system, a very powerful CAS system.]

References

Battista, Michael T. (February 1999). Ignoring research and scientific study in education. Phi Delta Kappan. Retrieved 4/26/06 http://www.pdkintl.org/kappan/kbat9902.htm.

CAS (n.d.). Computer Algebra System. Wikipedia. Retrieved 11/11/07: http://en.wikipedia.org/wiki/Computer_algebra_system.

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Author or Authors

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