The Math Wars

 

by Jerome Dancis

 

 

An abridged version of this article appeared in the Faculty Voice March 2000 (The faculty newsletter at the Univ. of Maryland).

 

 

"Now, for at least the fourth time in a hundred years, school boards and university professors and PTA leaders are engaged in a bitter debate over how to teach arithmetic". ( Jay Mathews in his article on the Math Wars in the Washington Post Magazine, Feb. 2000) Secretary of Education Riley has called for an end to the Math Wars and urged Math educators to work together in improving mathematics education; never mind that the Department of Education continues to take the side of the "Reformers". The NSF has awarded grants for Teacher Enhancement Programs and Teacher Preparation programs for "Reform". When President Clinton called for a national voluntary Grade 8 math test, the committee to write the specs, was stacked with "Reformers".

 

In December,1997, California's state Board of Education was about to vote on math standards for public school students by injecting more math into math -- actually expecting kids to memorize multiplication tables in the third grade and master long division in the fourth[1]. On , the day before the final vote, Luther S. Williams, (then) assistant director of the National Science Foundation, fired off a letter to board president Yvonne Larson. He reminded Larsen that his group "cannot support individual school systems that embark on a course that substitutes computational proficiencies for a commitment to deep, balanced, mathematical learning.''

 

In September, 1999, the new California Standards were published, standards which are very different from those of the Reformers. In October, 1999, the Department of Education announced a list of exemplary mathematics textbooks. (Riley had repeated postpond making the announcement -- possibly because a mathematician on his Expert Panel had written strongly against the panel's recommendations.) In reaction, a cryptic public letter[2] to the Secretary of Education was published in the Nov. 18, 1999 Washington Post calling for the Secretary to "withdraw your premature recommendations" and try again since several books on the list contained "serious mathematical shortcomings". This letter was signed by about 200 professors, mostly of mathematics, including Bill Adams, our Mathematics Department's Vice-chairman for Undergraduate Educ and yours truly, and four Nobel laureates.

 

The failure of traditional mathematics instruction and how it hinders college education.

 

Knowledge and understanding of arithmetic and algebra is crucial for understanding the simple mathematics and other quantitative concepts that arise in a wide variety of college freshmen courses and lack of such student knowledge puts an unfair burden on both instructors and students. (For an example, see Problems 1 and 2 below.) It mucks up many a student's education.

 

 

Some high schools required that a student score only 30 points (on a 100 point test) to pass and only 65 points for an A on the 1999 Montgomery County Public Schools' Algebra 1. (All the middle schools required 60 points for passing). A passing score of 30 means students will need remediation in college.

 

Elementary Algebra (Math 001) was the largest single mathematics course on our campus in the 1990s. Half the students therein, had received a grade of B or A in high school Algebra II[3]. Having to repeat two years of mathematics, already studied in high school, is not limited to the weak students, it extends to many strong students also. Dr. Frances Gulick[4] , noted that about one third of the students in her precalculus class (Math 115) had already "completed" calculus in high school.

 

Problem 1. (from an elementary nutrition course on our campus.) It is a fact that fat has 9 calories per gram and protein has 4 calories per gram. If a piece of meat consists of 90 grams of protein and 15 grams of fat, how many calories does it have? (495)

 

Class time, used for this instruction, reduces time available for teaching nutrition.

 

In elementary sociology classes on our campus, students struggle with "percents" in problems like:

 

Problem 2. A cohort has 1000 males and 160 females. Suppose that 25% of the males and 15% of the females have blue eyes. How many in this cohort have blue eyes?[5] (Ans. 274)

 

The lack of fluency in Arithmetic and Algebra I among students pressures colleges and high schools to "dumb-down" (course deflation) a variety of social science and mathematics courses.

 

In the 1970s, Jack Goldhaber, then Chairman of the Mathematics Department, assigned Professors David Schneider (now emeritus) and Larry Goldstein to write a textbook for Math 110 (Finite Math). Their text was used for many years on our campus. Later, when the students arriving in this class were not sufficiently fluent in high school algebra to handle this text, it was replaced by a dumbed down text.

 

"There are other hidden, but measurable, costs. Laurence Steinberg, a psychology professor at Temple University, noted last year that his institution's requirement for two semesters of psychological statistics for majors is not a cause to celebrate high standards. Rather, it is an admission that it now takes two semesters to learn what used to be done in one".[6]

 

Large numbers of college students change majors under the duress of difficulty with the mathematics in a required course. Often, a major reason for their difficulties/failure is lack of fluency in high school mathematics.

 

"When Grant Scott, a biology teacher, had to teach his chemistry students at Howard High School [in Howard County] how to change centimeters to meters, he just told them to move the decimal two places -- rather than illustrating the concept. ... 'Forty-five minutes later, only three of them got it.' ", [7] (Not so hard since 100 centimeters make a meter, just like 100 cents make a dollar.) The new California Standards require that students learn this in Grade 4.

 

At the request of a local public school system (1980's), Prof. Jim Greenberg (College of Education and Director of UMCP's Center for Teaching Excellence) organized a series of discussions on the topic: What contributes to the failure of college freshmen? The participants in the initial discussions were faculty members (who have much connection with freshmen) from 10 departments on our campus, together with high school teachers and UMCP freshmen). The main conclusion was that: "The overemphasis on testing, skill development, and fact level content, etc. [in high school] seems to have inhibited [student] interest in learning, motivation, ability to work with and enjoy ideas, use creativity and attain satisfaction from an educational experience." In a later discussion among college faculty members (mostly from departments of speech and communication including UMCP's Andrew Wolvin, professor of Communication Arts and theatre and Barbara Williams, then with our Institute for Urban studies), it was noted that: "Entering college freshmen appear severely limited in their ability to read critically, synthesize information, interact effectively with both peers and instructors in academic settings, and participate actively in discussions."[8] This is a natural consequence of these activities not being included in the curriculum of most school systems.

 

High school courses are largely determined by the textbooks. Now we discuss textbooks:

 

That there is little value in reading school textbooks and that textbooks were going from bad to worse was documented in the 1980's by Harriet Tyson-Bernstein's[9] in her book: The Textbook Fiasco; A Conspiracy of Good Intentions. Prof. Davis of Worcester Polytechnic Institute wrote[10] : "The [high school precalculus textbook] ... is no more mathematics than the noise made by trained seals is music. But the trained seal approach abounds in textbooks and in classrooms. It never provides a foundation of fundamental ideas ... . It never offers intellectual challenges, or chances to build confidence and problem-solving skills." Mathematics textbooks basically teach skills and calculation procedures without teaching understanding, without teaching when and how to use the skills, and without teaching how to think through problems. No wonder that students do not read their calculus textbooks; only foreign students read the textbooks, and they do so to improve their English! (Observed at U.C. Berkeley by Uri Treisman.)

 

What is the most basic rule of traditional American mathematics textbook publishers? It is the "two-page spread"; the entire day's lesson must fit on two pages. In sharp contrast are the Singapore mathematics textbooks which have a single day's lesson going on for pages. The Singapore mathematics textbooks are traditional mathematics taught properly. They will be on sale in the U.S. as soon as the names of people and vegetables are changed to American ones.

 

Many traditional texts/classes "educational beat-up" the students by placing them in a non-viable pedagogical situation where chance of success is small. This is why our otherwise great country is largely populated by people who say (mostly without shame) "I was never good at mathematics".

 

End of deductive reasoning. Until the 1950's, students studied Euclidian geometry in high school. Starting with a small number of axioms, they proved and watched the teacher prove many theorems. In this way they were provided with extensive training in deductive reasoning. Students learned that a statement in plane geometry was true because they had seen a proof of its validity. Since then, 100 theorems have been renamed axioms and their proofs (now being redundant) have disappeared from the textbooks. Now a statement is true in plane geometry because the book/teacher says it is so (It is an axiom.). Deductive proofs have been exiled to the last quarter of the textbook; not enough for students to learn this topic. The teaching of deductive proofs in plane geometry is banned in the Montgomery County school system. Students now arrive in college with little or no training in deductive reasoning, a serious educational handicap.

 

Prof. Barry Simon, Chairman of the Mathematics Department at California Institute of Technology in, "A Plea in Defense of Euclidian Geometry "[11] , "mourned this loss of what was a core part of education for centuries." as he noted "what is really important is the exposure to clear and rigorous arguments. ... "They can more readily see through the faulty reasoning so often presented in the media and by politicians". Also, they would have less difficulty adjusting to and understanding college courses.

 

Math-Education Prof. Guershon Harel [12] wrote "It is imperative to reinstate Euclidian geometry [based on deductive proofs], in the high school curricula ... [it] is a concrete system where students can learn the concept of a deductive system." I strongly agree, at least in those school systems where the geometry teachers are fluent in deductive geometry. One could obtain a State of MD high school mathematics teacher certification (before 1990) without having taken any course in geometry or in deductive reasoning.

 

The Second International Mathematics Study (SIMS) concluded that the major reasons for the low achievement in mathematics in U.S. schools, are that the mathematics curriculum is underachieving, very repetitive, ineffective and inefficient.[13] My children (in the fast academic track) were taught one third less mathematics in high school than I was (in the standard academic track) in the 1950s.[14] Serious training in deductive proofs and word problems have disappeared. Prof. Barry Simon noted: "The dumbing down of high school education in the United States, especially in mathematics and science, is a crime that must be laid at the doorstep of the educational establishment".

 

Summarized as Traditional: pedantically rigorous drill and kill on skills, no thinking, or conjecturing. Unreadable outline books pretending to be textbooks. Excessive memorizing, absence of reasoning things through.

 

All this cries out for dramatic solutions for improving mathematics instruction. To the rescue or semi-rescue or pseudo-rescue (depending on one's perspective) came:

 

The self-styled "Reform Movement".

 

The Reform movement was largely organized by professors of mathematics education. It advocates much use of hand calculators, no drill, emphasizes on concepts, group learning, students discovering mathematics for themselves which includes much conjecturing, students inventing their own algorithms for arithmetic, equity, mathematics education for all.

 

In order to earn Department of Education approval, the Reform Movement textbooks had to document that they were more effective somewhere than the traditional textbooks/programs. Not hard to do considering that the traditional textbooks/programs are so ineffective (as just noted).

 

The Reform Movement is a package of many good ideas, many half-baked ones and many counterproductive ones ; also many half-developed good ideas.

 

I am a proponent of and have used discovery learning and group learning along with lecturing in my teaching for decades. I accomplish this in college mathematics courses , without reducing the quantity or quality or rigor of the mathematics in the course.

 

The Reform elementary school curriculum is far more interesting than the traditional one. Having students work together in groups is far better than each student working alone doing the traditional busywork. Having students making conjectures and asking questions are valuable things that should be part of any program. But conjectures should be educated guesses not distracting or frivolous (time-wasting) ones. They should be checked and followed by a rigorous justification. Checking that a conjecture works on a few examples is good, but not enough. It is a pseudo-justification; it does not prove that it is valid in all situations/problems. This type of mis-teaching will lead students to believe all sorts of incorrect things and not just in mathematics.

 

Many Reformers believe that a teacher should be the guide on the side instead of the sage on the stage. But, just as with sports' coaches, it is useful for a teacher to be _both_ a guide on the side and a sage on the stage.

 

Patricia F. Campbell, Professor of Mathematics Education at UMCP, has developed a reform style of pedagogy, which includes training in computation. She provided training in her effective methods for teaching mathematics to the teachers at 3 of the 4 poor Montgomery County elementary schools cited in a May 16,1999 Post article as doing exceptionally well. She organized Project IMPACT. Her reward was to be assigned to do the same for the City of Baltimore public schools.

 

While there is variety in reform, we can still talk about the Reform movement as the movement exemplified by the NCTM standards and some of its spokesmen and textbooks. This broad brush approach will misrepresent some of its participants.

 

Steven Leinwand is the co-chairman of the U. S. Dept. of Education's Expert Panel (on textbooks) and the top mathematics adviser at Connecticut's Department of Education. In his article "It's Finally Time to Abandon Computational Algorithms"[15], he began:

 

"It's time to confront those nagging doubts about continuing to teach our students computational algorithms for addition, subtraction, multiplication, and division [like 23 x 37]. It's time to acknowledge that teaching these skills to our students is not only unnecessary, but counterproductive and downright dangerous! And it's time to proclaim that, for many students, real mathematical power, on the one hand, and facility with multi-digit pencil and paper computational algorithms, on the other hand, may be mutually exclusive." ...

 

"Today, real people in real situations regularly put finger to button and make critical decisions about which buttons to press, not where and how to carry threes into hundreds columns."

 

"No longer simply perpetuators of the bell curve, where only some survive and even fewer truly thrive, schools and their mathematics programs must instill understanding and confidence in all". ... Most compelling to Leinwand is the "sense of failure and the pain unnecessarily imposed on hundreds of thousand of students in the name of mastering these obsolete procedures".

 

I strongly agree with Leinwand's statements of parts of the problem in the last paragraph. But I strongly disagree with his solution, namely, not to teach children how to multiply 23x37. It is like throwing out the baby with the dirty bath water.

 

Students given this type of Reform math training will be able to Problems 1 and 2 above, if they have their hand calculators handy. Without their hand calculators, they will know which numbers to multiply, but will have difficulty with the multiplication.

 

Similarly, in their much quoted 1998 article, "The Harmful Effects of Algorithms in Grades 1-4", Constance Kamii and Ann Dominick wrote: "Algorithms not only are not helpful in learning arithmetic, but also hinder children's development of numerical reasoning ..."

 

Invoking Piaget's constructivism, Kamii and Dominick wrote: "Children in the Primary grades should be able to invent their own arithmetic without the instruction they are receiving from textbooks and workbooks."

 

As Wu[16] has noted: "Why not consider the alternate approach of teaching these algorithms properly before advocating their banishment from classrooms?"

 

Wu also wrote: "What is left unsaid is that when a child makes up an algorithm, the act raises two immediate concerns: One is whether the algorithm is correct, and the other is whether it is applicable under all circumstances." To carefully check and correct many new student algorithms periodically is a sizable task. Also as Liping Ma has documented[17], many teachers do not have sufficient knowledge of the mathematics they are teaching, to be qualified algorithm checkers. Teachers currently insist that students present fractional answers in simplest terms so that there will be a unique right answer and they will not have to check whether each student's unsimplified fraction is equivalent to the answer in the teachers manual.

 

The Reform movement also advocates an "integrated curriculum" which mixes algebra and geometry. This makes as much sense as the teaching of European history and American History in alternate months.

 

Dr. Jerome Epstein gave the following (pre-algebra) problem to such a second-year "integrated" algebra and geometry class. It was solved by none of the (mostly Grade 10) students.

 

Problem 3. Solve x/2 = (3/4)x +1.

 

The reform movement professors of mathematics education largely organized and wrote The National Council of Teachers of Mathematics (NCTM) Standards in the late 1980s. The NCTM is the professional society of school mathematics teachers. Their standards and the AAAS criteria are for all students; there is no separate higher standards for students going on to college. When adopted by a school system, Reform methods and textbooks are used for all students even though this is a dumbing down for college bound students.

 

The reform movement professors of mathematics education largely organized and wrote The National Council of Teachers of Mathematics (NCTM) Standards[18] in the late 1980s. The NCTM is the professional society of school mathematics teachers. Their standards and the AAAS criteria are for all students; there is no separate higher standards for students going on to college. When adopted by a school system, Reform methods and textbooks are used for all students even though this is a dumbing down for college bound students.

 

The NCTM response, to the low level of students skill at using fractions, has been to prescribe decreased attention to fractions in algebra. The NCTM standards state: "This is not to suggest that valuable time should be devoted to exercises like (17/24) + (5/18) or 5 3/4 x 4 1/4". ... "Division of fractions should be approached conceptually".

 

This is what the Grade 6 reform book "Mathland" does: "Rather than relying on algorithms, where memorization of rules is the focus, the Mathland approach relies heavily on active thinking. To solve problems such as 1/4 — 1/2 , students need to be able to visualize the question: How many halves go into one-fourth? This kind of fluency enables students to use their own logical and visual thinking skills to really know what the solution (1/2) means". Good, but I do not know how to visualize 19/74 — 17/23?

 

In Oct. 1998, the NCTM released its Proposed Principles and Standards (for the next decade). This revision is less revolutionary then its earlier Standards. It has invited much feedback from a wide variety of organizations and individuals. The final version will be released at the NCTM convention this spring.

 

The verbose, 700 page NCTM proposed standards do not even consider the question of raising the content of the mathematics curriculum back to the levels of the 1950s.

 

Several reform textbooks have been written mostly by professors of mathematics educ, some with support from the NSF or Dept. of Educ.

 

Many mathematics teachers are not fluent in the sterile math curriculum they are currently teaching, never mind an enriched curriculum. They will be overwhelmed by a program, which has new topics or treats topics in a non-traditional manner, especially if thinking is involved, even if the text is a big improvement. A short staff development session is not likely to be sufficient.

 

The NCTM proposed Principles and Standards do strongly argue for reasoning and proof in all grades. But the level being proposed is so low as to be embarrassing. Details below.

 

The NCTM Proposed Standards do not advocate the reinstatement of Euclidian geometry [based on deductive proofs], in the high school curricula. In fact, the phrase "deductive proof" is not even mentioned in the proposed Standard 7: Reasoning and Proof for high school -- pages 316-8.

 

The phrase "Formal proof" appears on line 31, Page 316 followed by a pictorial example of a useless triviality: The sum of two consecutive 'triangular numbers' is a square. This so called "formal proof" consists largely of looking at the diagram on page 317. This is an example of an elementary school, not high school level of proof. This demonstrates that the proposed standards are highly underachieving. The concept of 'triangular numbers' is useless and distracting. It provides fodder to those who say school mathematics will be useless to me after I escape high school.

 

Page 317 top paragraph. The only example of a 'sophisticated' reasoning situation mentioned is to explain that rational numbers may be converted to repeating decimals and vice versa, like 1/3 = .33333.. . To me this not so 'sophisticated' reasoning situation belongs in middle school not high school; at least in those school systems where the Grade 6-8 teachers are fluent in fractions and decimals. The main use of these conversions is to generate busywork exercises.

 

Page 246 Line 24 Problem solving for middle school. Asks the useless question: If we build semicircles, etc. on the the sides of a right triangle, then the sum of the areas [of the semicircles] on the legs equals the area [of the semicircle] on the hypotenuse. This encourages students to spend time on useless, frivolous, trivial and time-wasting conjectures and to provide fodder to those who say school mathematics will be useless to me after I escape high school.

 

The proposed standards Page 219 [Arithmetic] Operations and their Properties for Grades 6-8, includes the arithmetic of decimals and fractions like: 1.4 + .67 and 2/3 + 3/4. The new California Standards require this teaching in Grades 4 and 5, resp. I agree, at least for those school systems where the Grades 4-5 teachers are fluent in fractions and decimals -- many are not..

 

The proposed NCTM Principles are very verbose about emphasizing the importance of equity. During the last 3 decades much equity has been achieved in mathematics education as good consequences of the civil rights and feminist movements. In addition, much equity has been achieved by the easy, cheap method of dumbing down the mathematics curriculum. If the proposed NCTM standards are implemented, more equity will be achieved by simply dumbing down the mathematics curriculum. Programs like Pat Campbells should be the rule, the reality is that they are the exception.

 

Some extremists believe that discovery learning means no lecturing and all group work. They also believe lecturing and group work are incompatible. A friend (knowledgeable in the Math Wars) stated her amazement that discovery learning and rigor could be part of the same teaching method. The polarization results in words carrying much (not always valid) baggage. The terms "Traditional" and "Reform" are too often interpreted (or misinterpreted) to the bulk of the things I listed.

 

Faculty members and other parents in Princeton NJ were so aghast at how the Reform program was destroying their children's education that they organized a charter school (Read "Why Charter Schools? -- The Princeton Story" by research scientist and ex-school board member, Dr. Chiara Nappi, available on the web at http://www.edexcellence.net/library/wcs/wcs.html).

 

Often the choice is presented as restricted to either understanding or skills. Wu makes the point " ... as if thinking of any sort -- high or low -- could exist outside of content knowledge. ... in mathematics, skills and understanding are completely intertwined. ... "Our children need courses which teach both skills and understanding.

 

"They really don't know anything about mathematics," said David Klein, a mathematics professor at California State University, North Ridge, and co-author of the letter to Riley, as he noted that the number of freshmen needing remedial help in the California state university system has doubled over the past 10 years. Two years ago, California stepped back from the reforms, mandating more pencil-and-paper calculating and traditional drill and practice mathematics.

 

To the rescue or semi-rescue or pseudo-rescue from the Reformers, came:

 

The new Calif. Standards (a third way).

 

In opposition to the reform movement arose Mathematically Correct, a politically-incorrect parents group and several professors of mathematics, especially Dick Askey[19] of the Univ. of Wisc., H. Wu of U.C. Berkeley, and Jim Milgram of Stanford and Henry Alder.

 

The Fall 1999 issue of the American Educator, (the quarterly professional journal of the American Federation of Teachers) has four good articles. The one by Wu, "Basic skills versus conceptual understanding: a bogus dichotomy in mathematics education", starts roughly as follows:

 

"Education seems to be plagued by false dichotomies. 'Facts vs.~higher order thinking' is an example of a false choice that we often encounter these days, as if thinking of any sort --- high or low --- could exist outside of content knowledge. In mathematics education, this debate takes the form of 'basic skills or conceptual understanding.' This bogus dichotomy would seem to arise from a common misconception of mathematics held by a segment of the public and the education community: that the demand for precision and fluency in the execution of basic skills in school mathematics runs counter to the acquisition of conceptual understanding. The truth is that in mathematics, skills and understanding are completely intertwined. In most cases, the precision and fluency in the execution of the skills are the requisite vehicles to convey the conceptual understanding. There is not 'conceptual understanding' and 'problem solving skill' on the one hand and 'basic skills' on the other. Nor can one acquire the former without the latter."

 

Mathematically Correct won the latest battle in California. The new Mathematics Framework for California Public Schools (K-12) reflects their views. Available on the Web at http://www.cde.ca.gov/cdepress/math.pdf. None of the current textbooks, traditional or reform meet these new California Standards.

 

The new California Standards bring back two-column deductive proofs as the basis of high school Geometry. The number of axioms is reduced to a reasonable 16. The new California Mathematics Framework emphasizes a balanced mathematics curriculum. It stresses the critical interrelationship among computational proficiency, problem-solving ability and conceptual understanding of all aspects of mathematics.

 

Having lessons end with a wide variety of problems on material that the students were taught months and years ago is an important feature of the popular traditional Saxon textbooks. These Saxon-reviews change the subject just when the students are starting to internalize the day's new material; this disrupts the learning process. The California Framework does not permit this type of busywork; it states: "review must be developing automaticity or preparation for further learning" (Page 231).

 

Testing. The teacher allocated half of the time of my child's mathematics lessons in Grades 6 and 7 just to testing. (just the teacher's tests, not counting stansardized tests.) The California Framework does not permit this squandered of class time; it states: "Minimize loss of instructional time [due to testing]" (Page 197).

 

All states would be wise to adopt the new Mathematics Framework for California Public Schools (K-12).

 

These Mathematics Framework for California Public Schools are the opposite of the message delivered at a 1999 staff development session for Algebra I teachers in Montgomery County, a rich suburb of Washington, D.C; it was summarized to me as "Do not worry about the students understanding algebra -- Just be sure they can put anything on their Hand calculators".

 

 

 

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Partial Update on "The Math Wars"

 

 

 

The NCTM had solicited and received much feedback from a wide variety of organizations and individuals on its 1998 Proposed Principles and Standards (PPS). Many improvements were made as these standards were moved somewhat in my direction. It is still on a considerably lower level than the new Calif. Framework and Standards (1999). That the NCTM standards have been changed does not mean that the Reformers have changed their views or that the number of Reformers has been reduced. The final version, Principles and Standards for School Mathematics (PSSM)was released after my article, "The Math Wars" appeared.

 

 

I had noted that the only example of a formal proof for high school [in the PPS] really belonged in elementary school. This demonstrated that the NCTM writers held highly underachieving standards for students. Fortunately, the feedback persuaded the writers to move this example down to its appropriate Grade 3-5 level (P. 159).

 

 

I had criticized the only example of a 'sophisticated' reasoning situation for high school students mentioned in the PPS; fortunately, this example appears to have disappeared during the revision.

 

 

The phrase "deductive proof" was not even mentioned in the PPS Standard 7: Reasoning and Proof for high school. This phrase has been added in the PSSM; but there is still no mention of deductive proofs for a geometric theorem in the PSSM.

 

 

Much of the low standards and foolishness remained. The arithmetic of decimals and fractions like: 1.4 + .67 and 2/3 + 3/4 is still postponed to Grades 6-8.

 

 

In the "Problem Solving for Middle School" section (Page 261 main paragraph), the following useless and frivolous question is still asked: What if we build semicircles, etc. on the sides of a right triangle, will the sum of the areas of the semicircles on the legs equals the area of the semicircle on the hypotenuse. (adapted from Stephen Brown and Marion Walters's book The Art of Problem Solving [1983]).

 

 

The following is an appropriate problem for Grade 3-5 students; but PSSM (Page 258 bottom) postpones it until Grades 6-8:

 

"[Teachers] can challenge students with problems that have more than one answer, such as the following (adapted from Gelfand and Shen's Algebra book [1993], p. 3): Make a sum of 1000, using some eights (8s) with some plus signs (+s) inserted".

 

Answer: 1000=888+88+8+8+8.

 

That a problem that has more than one answer does not a challenging problem make. The thought involved in solving this problem has nothing to do with the fact there are several answers.

 

 

 

 


[1] Debra Saunders San Francisco Chronicle essay of Dec. 19, 1997, called Man of Science Has a Problem With Real Math.

[2] The letter may be found at http//www.mathmaticallycorrect.com/riley.htm, complete with links to web sites containing (not easily found) referenced documents.

[3] Survey in the late 1980's by Math Lecturer Lynn Cleary.

[4] a lecturer in mathematics at UMCP and wife of recent Campus Senate Chairman Prof. Denny Gulick.

[5] I am collecting examples like these of simple problems involving simple mathematics from non-hard-science disciplines. Donations from readers will be appreciated.

[6] From Lois Cronholn's wonderful article "Why One College Jettisoned All Its Remedial Courses" in Chronicles of Higher Education (1999).

[7] Linda Perlstein's February 15, 1999, Washington Post, front page article "Right Teacher, Wrong Class".

[8] John L. Brown, Curriculum Dialogues, Prince Georges County Public School System, (1987) Page 6

[9] She was a member of the Board of Education for the Montgomery County Public Schools.

[10] Paul Davis, Teaching mathematics and Training seals, SIAM News (the newsletter of the [professional] Society for Industrial and Applied Math.), (March 1987) page 7.

[11] in his February 6, 1998 Los Angeles Times article.

[12] of Purdue Univ. in his "Preprint "Greek vs Modern Mathematics Thought ...".

[13] The SIMS report was the booklet The Underachieving Curriculum (1987) by Curtis C. McKnight et al. It was sponsored by the International Association for the Evaluation of Educational Achievement. It was one of the most thorough analysis of mathematics education in American schools in the 1980's. The conclusions of SIMS were not accepted by school administrators, thereby justifying a decade of inaction on mathematics education reform. A major purpose of the massive and expensive Third International Mathematics Study (TIMS) was to double check the SIMS results.

[14] This can be documented by comparing the New York State Regents high school mathematics exams from the 1950s and 1990s.

[15] reprinted in Wisconsin Teacher of Mathematics, Winter 1995

[16] Dr. H. Wu's great article, Basic Skills Versus Conceptual Understanding -- -- A Bogus Dichotomy in Math Education" in The American Educator, American Federation of Teachers, Fall 1999.

[17] Liping Ma Knowing and Teaching elementary Math, Lawrence Erlbaum Associates, Mahwah, N.J. (1999), partially summarized by Dick Askey in The American Educator, American Federation of Teachers, Fall 1999.

[18] According to the NCTM's Curriculum and Evaluation Standards for School Mathematics (the Standards) (1989). On the web at http://www.enc.org/reform/journals/ENC2280/nf_280dtoc1.htm

[19] member of the National Academy of Sciences