Mathematically Correct

A PROGRAM for RAISING the LEVEL of STUDENT
ACHIEVEMENT in SECONDARY SCHOOL MATHEMATICS

Submitted by Frank Allen
All students should be given the opportunity to master
academic subject matter calibrated against world norms.
National Alliance for Business

Many of the serious problems we face in secondary school mathematics today are due to deteriorating social conditions in the home and community over which our teachers and school administrators have no control, and for which they should not be faulted. In some school districts, "students" attend school sporadically, and are often in no condition to learn when they do attend. Problems arising from this situation should not be attributed to deficiencies in curriculum, teaching or assessment. We believe that much can be done to raise the level of student achievement in secondary school mathematics without making radical changes in any of these areas. We present our recommendations under four headings.

I. Objectives.

A clear statement of objectives in terms of subject matter requires that there be provided, for each course, an accurate description of the mathematics we expect students to learn. These descriptions could take the form of course syllabi of the kind utilized in most of the other industrialized countries in the world. The successful completion of each course should contribute to the student's understanding of the symbolic language of mathematics, and to his ability to employ this language in the construction of valid mathematical arguments (proofs) and in the solution of problems.

This is not the place for a detailed description of course syllabi. We believe that they should be consistent with our traditional curriculum in secondary school mathematics, with which we are reasonably content. We insist that these course syllabi be comparable with those employed in other countries where students have reached high levels of achievement. To insure this we specify that these syllabi shall be prepared by committees of mature, well-established mathematicians appointed on a regional or preferably national basis, with some input from experienced teachers of secondary level mathematics.

II. Description of a Learning Situation in Which Our Objectives Can Be Achieved.

1. Differences in learning rates must be recognized and provided for. In most secondary schools the student population presents wide variations in student interest, aptitude and training with respect to mathematics and there is no point in pretending otherwise. Schools are to be commended for making curricular provisions for these variations. It is not unusual for our larger high schools to provide as many as six levels of instruction for grade nine. These are not established by faculties bent on making invidious comparisons. It is simply impossible to meet the needs of the entire school population without them.

2. We reassert the principle that the secondary school mathematics curriculum must be organized around its own internal structure, and not around problem solving as the NCTM s "Agenda for Action" requires. There are, of course, many such structures, but all of them are logically consistent in the sense that the prerequisites for each new concept are available when it is introduced. This logical structure is the very essence of mathematics. The idea that mathematics is a hierarchy of propositions forged by logic on a postulational base should begin to form in the student's mind about grade nine and be thoroughly established by grade twelve.

3. The teaching of mathematics should be regarded as an extension of the teaching of language, in which facility in two-way translation between language and mathematical symbolism is emphasized. Efforts to develop an awareness of the intimate relationship which exists between grammar, mathematics and logic should begin with games of "How do we know" in the early grades, continue with the introduction of formal proof not later than grade nine and culminate in the ability to read and write valid essay proofs not later than grade twelve.

This linguistic approach to secondary mathematics has several advantages. It helps the student to develop the gradually formalized natural language so essential for mathematical discourse, and indeed for the expression of rational thought in any field.

It tends to build bridges between mathematics and other subjects. It fosters the idea that reasoning in mathematics has much in common with reasoning elsewhere. This will help to counteract the idea that there is something esoteric and arcane about mathematics. It will help to dispel the notion, already too widely held, that one must have a mathematical mind in order to deal with the peculiar thinking required in mathematics. This idea has served to isolate our subject from the main stream of public consciousness to an extent that we cannot possibly counteract by our current over-emphasis on real world applications.

The two-way translation between language and mathematical symbolism is important for problem solving. First, we express the problem situation in mathematical language (L to M) thus bringing it under the sway of the powerful laws that govern mathematical calculations. Second, we apply these laws to perform these calculations. Here we can use a computer or calculator, but we should understand the nature of the operations being performed. Finally, we must interpret our mathematically expressed result in terms of the original problem (M to L).

4. We must take a balanced view of the role of problem solving in school mathematics, lest our preoccupation with it causes us to fragment and distort the very mathematics that makes problem solving possible.

Problems are the life blood of mathematics. But we must not fail to convey to our students that the body of mathematics is given structure and coherence by the bones and sinews supplied by definitions, postulates and proof. Make no mistake, a person's problem solving ability depends on how much mathematics he understands. Moreover, one of the principal objectives of problem solving in high school is to inculcate a better understanding of the basic mathematical theory. It is this understanding that will enable the student to deal with problems that are today unforeseen and unforseeable. A student who solves a problem has devised a key that will open a specific lock. A student who understands the mathematical theory underlying his solution has a master key that will open many locks. Each problem should be placed in its proper mathematical context by citing the principles used in its solution.

5. We must also try to develop a little more confidence in the idea that mathematics is interesting for its own sake, and that a problem can be interesting, challenging and instructive without being obviously attached to some real world application. These ideas open the door to an appreciation of the recently much neglected cultural and aesthetic values of mathematics. As Davis and Hersch put it, Blindness to the aesthetic values is widespread and can account for the feeling that mathematics is dry as dust, as exciting as a telephone book. Contrariwise, appreciation of this element makes the subject live in a wonderful manner and burn as no other creation of the human mind seems to do. Teachers who have this appreciation should try to transmit it to their students.

6. We must promote the idea that mathematics, properly taught, makes unique and indispensable contributions to the development of the student's ability to think and communicate in a logical manner. For many students the value of this ability, which is by no means confined to the field of mathematics, far transcends the value of the mathematical facts learned. The eminent mathematician Jean A. E. Dieudonne has eloquently expressed the concept that enhancement of the student's ability to think effectively as a major goal of mathematical study. For what good do we seek? Certainly it is not to introduce them (students) to collections of more or less ingenious theorems about the bisectors of the angles of a triangle or the sequence of prime numbers, but rather to teach them to order and link their thoughts according to the methods mathematicians use, because we recognize in this exercise a way to develop a clear mind and excellent judgment. It is the mathematical method that ought to be the object of our teaching, the subject matter being only well-chosen illustrations of it.

7. We need to take a more balanced attitude regarding the building of basic mathematical skills. Jeremy Kilpatrick describes our problem very well in the July 1988 issue of the NCTM's research journal, One of the most venerable and vexing issues in mathematics education today concerns the trade-off between proficiency and comprehension, between promoting the smooth performance of a mathematical procedure and understanding how and why the procedure works. He continues Researchers -- should not dismiss too lightly the question of how and where skill development fits into the school mathematics curriculum. Recent research in cognitive science suggests that a strong knowledge base is needed for problem solving, and surely some of this base should be composed of procedural knowledge. Furthermore, conceptual knowledge both supports and is supported by what Brownell termed ‘meaningful habituation,’ the almost automatic performance of a routine that is based on understanding.

We want our teachers to feel free to seek this balance on their own, uninhibited by strictures against A tight focus on manipulative facility with which the NCTM Standards are replete, or the notion that technology has supplanted the skills students need.

8. We urge caution in the use of calculators and dynamic geometries.

III. The grading system.

First we consider the internal grading system which has long been used by teachers to determine course grades and to provide feed-back for improving instruction.

At mid-century the following principles were widely accepted as valid by high school teachers. They are still valid today.

Now we turn to the external grading system.

Every student must have the opportunity to take the externally set and externally graded examinations covering the standard syllabus that has been adopted for each course on a regional/national basis. There will, of course, be variations in the amount of time that students need to prepare for these examinations. For example, some students may need two years of study to prepare for the regionally administered examination in first year algebra (roughly algebra through quadratics).

These externally set examinations provide a basis for comparing the student's performance with regional, national or, perhaps even world norms. They invoke what John Bishop calls The Power of External Standards. In his article by that title in the Fall 1995 issue of American Educator, the AFT magazine, Bishop says: Such standards are established when the student's mastery of a common curriculum taught in high school is assessed by examinations that are set and graded on the national or regional level. He continues: The standards reflected in these exams are visible and public as well as demanding. In France and the Netherlands, for example, questions and answers are published in the newspapers and available in video texts. And the grades a student gets are extremely important because they signal the student's achievement to colleges and employers. They influence the jobs that graduates get and the universities and programs to which they are admitted. How well graduating seniors do on these exams also affects the reputation of a school and, in some cases, the number of students applying for admission to the school.

This describes the situation where high school seniors take batteries of examinations covering the basic courses, for which they have prepared over a four or five year period.

We would begin the establishment of externally set and graded examinations on a course-by-course basis. There would be one such examination covering the prescribed syllabus in algebra, another for geometry, etc. Exam results would be publicized on a regional/national basis. Such exams do much more than compare classmates with each other. They set performance levels for each course, measure the extent to which each student has mastered the prescribed syllabus and compare his performance with regional norms. These examinations will be prepared by committees of mature, well-established mathematicians appointed on a regional/national basis.

IV. We Reestablish the Teacher's Role as an Expositor and the Student's Role as a Learner.

The dictionary defines a teacher as ... a director, one who imparts knowledge, an instructor. The teacher in secondary school has always been regarded as one who explains. We believe that the teacher of secondary level mathematics should be confirmed in his centuries-old role as an expositor and director of learning.

There are, of course, many ways to direct learning and each teacher should use those that work best for him. Cooperative learning can be used occasionally as it has been in the past. But teachers should not be under pressure to use it because, used excessively, it tends to relegate the teacher to the role of facilitator. To facilitate means to make easy. This sends the wrong message. For most people, the learning of mathematics is not easy. It requires hard work, sustained effort, intense concentration and diligent attention to homework. The student needs the kind of experience that the individual study of mathematics provides in order to learn how to learn. Without it, he may graduate, steeped in self esteem, but totally unprepared to meet the ever-changing demands of an intensely competitive world, where there are no facilitators to make things easy.

We want our teachers of secondary mathematics to have at least an undergraduate major in mathematics. They should be encouraged to continue their study of mathematics at the graduate level and in graduate math education courses that emphasize mathematical content, if such can be found.

We would give them the wide discretion that well-trained professionals deserve. We would set performance goals for students and would rely on the ingenuity of our mathematically competent classroom teacher to find many ways to achieve them.

As Bishop observes, with the establishment of external standards, the teacher's situation improves dramatically. The teacher with high standards is no longer regarded as a taskmaster whose demands are to be evaded, but rather as ... a coach or mentor whose advice and expertise help students to achieve a goal they care about. Moreover, they are released from the pressure now exerted by students, administrators and parents to grade on the curve, lower standards and inflate grades. This pressure is fast becoming intolerable and it is destroying education in America.

The learning situation for students is also vastly improved. No longer rated against each other, they would be striving to reach a standard that has been externally set. Diligent students would no longer be resented and derided as "nerds" who are lifting the curve and thus making it more difficult for their classmates. As Bishop notes: ... students will work very hard and achieve at a high level if you make clear what you want them to learn and if there are serious consequences attached to their achievement.

We believe that it is time for the gradual introduction of external standards in secondary school mathematics, on a voluntary course-by-course basis. We believe that the public will support this. We cite the first recommendation in the report, The Challenge of Change: Standards to Make Education Work for All Children, recently produced by the National Alliance for Business. All students should be given the opportunity to master academic subject matter calibrated against world-class norms.

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It is the duty of teachers and parents to call upon the best advice obtainable (in this case, from mathematicians) to set standards of student performance and then demand that students meet these standards. When standards are held firm and the student is required to adjust to them, we have a process that can be accurately described as education. In recent years we have seen a distressing reversal of this process. Students don't listen very well? Adjust by downgrading oral exposition by the teacher, and, perhaps resort to cooperative learning. Students don't like the curriculum? Change the curriculum, perhaps by emphasis on practical applications in an effort to recapture student interest. Students don't do well on standardized tests? Try to discredit these tests by proclaiming that they are not and cannot be valid measures of student achievement. This stultifying process where changes take place in the system rather than in the student is education turned on its head. It is destroying education in America, and it must be stopped.

We have described a way to stop it, and to put math education in America on a par with that in other industrialized countries. We realize that the establishment of external standards will encounter powerful opposition. But it is long overdue. With the support of concerned parents throughout the nation, we believe that it can be achieved.

Submitted by
Professor Frank B. Allen
Advising Member, Mathematically Correct
Professor Emeritus of Mathematics, Elmhurst College
Former President, National Council of Teachers of Mathematics
Former Chair, Division of Natural Sciences and Mathematics, Elmhurst College

April 1996


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