A Panel Discussion on the Mathematics

That Should be Taught in K - 12

Richard Askey

Prof. of Mathematics, Emeritus

University of Wisconsin

R. James Milgram

Prof. of Mathematics

Stanford University

H.-H. Wu

Prof. of Mathematics

U. of California, Berkeley

INTRODUCTORY AND FOUNDATIONAL ISSUES

A Brief History of NCTM and its relationship with the AMS and MAA In 1915 the Math Association of America (MAA) was spun off by the American Mathematics Society (AMS) and in 1920 NCTM was created with the major purpose of helping preserve mathematics in the public schools.

The MAA formed a committee to help defend high school mathematics from attacks from, among others, social efficiency educators who demoted subject matter including mathematics to focus on what students would need for work. Algebra was viewed as useless for most students. This MAA committee was supported by regional mathematics associations, but some members in these groups felt that there should be a national group of school mathematics teachers to help defend the profession. The MAA sent repesentatives to the meeting which formed NCTM and welcomed the new association. Thus, from the start there was close cooperation between NCTM and the professional mathematics organizations.

``In early 1919, C.M. Austin led an effort initiated by the Men's Mathematic Club of Chicago to determine if other groups were interested in forming a national organization of mathematics teachers that could defend the profession against these external attacks. Austin's committee called for a meeting to coincide with an NEA conference in Cleveland. There, 127 mathematics teachers from twenty states formed the National council of Teachers on Mathematics (NCTM) on 24 February 1920 (Rappaport 1965). MAA, which had authorized NCMR officials to attend the organizational meeting, welcomed the new association ("Notes and News" 1920).^0.1

The problems with school instruction in mathematics that led to the creation of the NCTM haven't changed that much if at all. In 1923 the important report The Reorganization of Mathematics in Secondary Education^0.2 was published. It's authors were leading American mathematicians together with leading high school math teachers and it summarizes over six years of work by this distinguished group. Virtually the same book could have been written today. For example, to match up to the fact that the leading foreign countries did algebra in the eighth grade we have the following suggested outlines of the mathematics that should be taught in grades 7 - 9.

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These three societies worked closely together and shared common goals for fully 50 years, and even today they have similar overriding goals. Today, after a thirty year period where there has been relatively little interaction between the members of the AMS and NCTM this situation seems to be changing, and there is hope that in the near future these societies will once more cooperate. Certainly, widespread mathematical competence has become even more critical to our well-being than was the case 50 years back.

The Need for Renewed Cooperation between NCTM, MAA, and AMS. There are two basic issues here. The first is that the long isolation of NCTM from the professional mathematics community has resulted in many errors and questionable practices creeping into the accepted mathematics that is taught in K - 12. The second is the dramatically more important role that mathematics plays in modern society.

THE STRUCTURE OF MATHEMATICS AND ITS CORRECT PRESENTATION

IN SCHOOL INSTRUCTION

A major purpose of this handout is to summarize the five main characteristics of mathematics.

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1. Precision. All statements in mathematics are unambiguous.

   For example, consider the question

   $$\vbox{ \noindent \sl Two third graders in different classes want to know which classroom is bigger. How can they tell?}$$

   The difficulty here is that the term ``bigger" is ambiguous. Does it refer to volume, floor area, perimeter, or perhaps the number of desks? Without a precise understanding of what is to be meant by bigger in the context of this question, the problem is not a question in mathematics.

2. What you see is what you get. There are no hidden assumptions in mathematics.

   This can be subtle. We tend to make unarticulated assumptions all the time. For example consider the problem

   $$\hbox{\sl What is the fifth term in the sequence that starts 1, 2, 4, 8?}$$

   Most people in this country would say 16. However, there are an infinite number of rules that can be given which produce $1,2,4,8$ as their outcomes at the first four positions, but produce different results at the fifth position. There are hidden assumptions being made in problems like this - among them that the rules for sequences like this are linear or quadratic polynomials in $n$ or are given by simple exponential functions like $k^n$ . By contrast, here is a correct pattern problem from a Hungarian second grade text.

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3. Definitions are the cornerstone of mathematics.

   Precise definitions are not just to be memorized but understood and used. Without precise definitions there cannot be mathematics. Even in the earliest grades it is possible to give students precise definitions that they can use and understand. Indeed, when looking at the programs from countries which have a history of successful mathematics education, one of the most persistent differences from our programs is that they provide precise definitions from the beginning and we don't. Here is an example from page 7 of a Russian second grade text:

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4. Logical reasoning is the glue that holds mathematics together.

   Logical reasoning is behind every step in mathematics. Logical reasoning in mathematics can only be carried out with precisely defined concepts, and this reinforces the need for precise definitions. Another name for logical reasoning is proof.

5. Problem solving is what mathematicians do.

   Mathematical problem solving is the application of logical reasoning to precisely defined concepts and previously proven statements in order to get to desired conclusions. Because all of mathematics is about problem solving, problem solving cannot be separated out from the rest of mathematics as a separate subject. Good examples of questions that emphasize problem solving skills demand original ideas, but should use and help deepen students understanding of the material they are currently learning. For example, here is a problem for students learning about place-value:

   $$\vbox{\hsize =4in \noindent\sl If you have to add three 14-digit ... ...ws the leading 8 digits in a number, how would you make use of the calculator?}$$

THE CRITICAL ROLE OF MATHEMATICS IN TODAYS SOCIETY AND THE NEED

FOR TOP-FLIGHT MATH INSTRUCTION IN THE EARLY GRADES

The critical need for bringing the professional mathematicians and the K - 12 teachers of mathematics together once again is indicated by

• the dramatically increased importance of high level mathematics knowledge for the best jobs,
• the very confused notions of what mathematics is that are reflected in the K - 8 or K - 12 curricula in this country
• the unrealistic depreciation of the value of mathematics in K - 12 education that seems to be commonly held by educators in the U.S.A.

In February, 2004, Alan Greenspan told the Senate Banking Committee that the threat to the standard of living in the U.S. isn't from jobs leaving for cheaper Asian countries. Much more important is the drop in U.S. educational standards and outcomes.

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Current estimates are that over the next 15 years at least 3.3 million jobs and 136 billion dollars in wages will move to East Asia.^0.3

$$\epsfysize 1.5in\epsfbox{Robot018.ps}... ... of all high school graduates in this country have taken some college courses.}$$

This is highly relevant to the issue of mathematics in K - 12 since it turns out that the clearest predictor of success in college is success in high school mathematics. Consider this data from the U.S. Department of Education that shows success in Algebra II is the strongest indicator of success in college, and the more mathematics beyond algebra II that students take in high school the better their chances of success in college.^0.4

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Given that college is the gateway to the best jobs, it is ironic that the situation for mathematics instruction in the public schools is still precarious in this country. We are graduating fewer and fewer students in technical areas like engineering and the sciences at the same time that demand for workers in these areas is growing dramatically.

Due, in large part, to the failure of this country to educate a sufficient number of students in mathematics and science, the United States has, as predicted, already begun to experience serious losses of high level jobs to other countries that do a much better job in this area.

SEATTLE Sept 14, 2004: The U.S. information tech sector lost 403,300 jobs between March 2001 and this past April, and the market for tech workers remains bleak, according to a new report.

Perhaps more surprising, just over half of those jobs 206,300 were lost after experts declared the recession over in November 2001, say the researchers from the University of Illinois-Chicago.

In all, the researchers said, the job market for high-tech workers shrank by 18.8 percent, to 1,743,500 over the period studied.

Our situation has been similar to that of the United Kingdom in recent years, and recently the situation there has become catastrophic. Here are some quotes from a recent article in The Guardian^0.5

The closure of Hull University's mathematics department announced this week has fuelled fears that parts of the country will become a "wasteland" for the subject.

Four other maths departments in England have closed since 1999 and the number of maths students has plummeted by more than 2,200 over the same period. The recent closure of Exeter University's chemistry department provoked widespread concern about the future of science departments.

``The effect is creating mathematical wastelands in parts of the country at a time when the government is saying we need more students to study maths and that we need to encourage people into maths teaching," she [VP of the London Mathematical Society Prof. Amanda Chetwynd], told the Times Higher Education Supplement.

Robert Reich, U.S. Secretary of Labor in the Clinton Administration, described the situation very clearly in a recent article.^0.6

``The problem isn't the number of jobs in America; it's the quality of jobs. Look closely at the economy today and you find two growing categories of work - but only the first is commanding better pay and benefits. This category involves identifying and solving new problems. Here, workers do R&D, design and engineering. Or they're responsible for high-level sales, marketing and advertising. They're composers, writers and producers. They're lawyers, bankers, financiers, journalists, doctors and management consultants. I call this `symbolic analytic' work because most of it has to do with analyzing, manipulating and communicating through numbers, shapes, words, ideas. This kind of work usually requires a college degree. ...

``The second growing category of work in America involves personal services. Computers and robots can't do these jobs because they require care or attentiveness. Workers in other nations can't do them because they must be done in person. Some personal-service workers need education beyond high school - nurses, physical therapists and medical technicians, for example. But most don't, such as restaurant workers, cabbies, retail workers, security guards and hospital attendants. In contrast to that of symbolic analysts, the pay of most personal-service workers in the U.S. is stagnant or declining. That's because the supply of personal-service workers is growing quickly, as more and more people who'd otherwise have factory or routine service jobs join their ranks.''