Overview 

QUESTION: You had stated that “remedies for the malaise in education must be sought.” In your opinion, what are those remedies?

ANSWER: Observing, as I did, that remedies must be sought is easy. Prescribing the remedies is not. Now, even before considering remedies, we need to know what the problem is — note that we are pointing the finger at ourselves rather than at other countries.  Let us simplify the problem as this: The student in junior high or high school in general is not getting the concepts. Let us also simplify and say that the fault does not lie with students, parents, the surrounding society, or school funding, or even with the shortcomings of a particular teacher. In short, let us say the problem is with content and pedagogy. What is taught and how it is taught is then our question.

Note that these comments constitute only my opinion. It is offered so that parents and teachers know my understanding of mathematics education — an understanding as imperfect as it must be — which has guided the various camps I founded. In the passage of time, this opinion will become more and more outdated but would serve as a time capsule from yonder years!

Caution!

The goal of math education is not just skill fluency but also conceptual understanding. Skill fluency is easily instilled. My comments on content and pedagogy are about conceptual understanding as the end result. So that my comments reflect my philosophy on mathematics instruction which guided me in the founding of the various mathematics camps, these are being made without looking at the Common Core State Standards for Mathematics (CSSM).

I believe that the importance of pedagogy is inversely proportional to student age.  As I am not an expert in pedagogy particularly for the very young, my comments below concern the instruction of the older students — say those older than age 9,

Content - What Should Be Taught?

By content, I mean topic areas.

Euclid's geometry should not be removed from being a course in high school  — as well as middle school — but presented in a way that beautifully exhibits the deductive method it contributed to mathematics as a whole. This means the two-column proof format in the geometry classes should remain. 

A second geometry course based on sets and the mappings known as isometries and similarities should be offered so that students not only see the beauty and power of geometry but also realize what geometry really is and get an idea of why there are other kinds of geometries than Euclidean geometry.

Noting that algebra skills and a confident familiarity with functions and trigonometry are what are needed to take the freshman college math courses, any algebra course that does not exactly build on the previous algebra course but repeats a good portion of it should be removed; that is, a streamlined and exhaustive algebra curriculum is needed. Again, I believe the CSSM address this. If the Algebra I — Algebra II — Pre-calculus sequence addresses the CSSM and the repetition occasionally present is of ideas, not formulas, algorithms and computations, then it would suffice.

In algebra courses, word problems must be given where a topic is amenable to word problems. Word problems are the breeding ground for the serious study of the sciences. Word problems illustrate the transition from the concrete to the abstract, from arithmetic to algebra. Word problems are the 'manipulatives' we need, and the texts and the homework sheets should have enough well-chosen word problems. 

Include also word problems that do not give real-world answers. (Example: If the father is 35 and the son is 7, how many years later will the father's age be eight times the son's age?  The answer is -3. [6]). This informs the abstractness and generality of mathematics and that math is as much about things beyond this world! 

A solid foundation in sets, functions and trigonometry should be preferred over taking a course in high school Calculus.  Then the students will not be required to take algebra in college, and they can delve deep into the college calculus course and the student would get to experience "the greatest technical advance in exact thinking" that the great John von Neumann said Calculus is. Instead of trying to teach fractals, let us ensure that ALL students know how to add fractions and why the addition works. Instead of talking about Students and chi-square distributions, let us ensure that ALL are conversant enough with the normal distribution to the extent that they would understand the pre-election polls. 

Other core courses could include probability & statistics, combinatorics, and foundational concepts of computer science.

Testing student understanding

Look for ways — perhaps, through nation-wide or at least state-wide exit tests (not multiple-choice format) for each grade — to test student understanding rather than skill; this will reduce the mechanical drilling and the teaching of tricks to pass tests. A nationwide or statewide exit test can have greater quality due to the central funding and can even be designed to emphasize the learning outcomes of critical as well as analytical thinking as they relate to a national or statewide curriculum, for example CSSM, for the grade.

Pedagogy (How Math is Taught)

In considering how math is taught, we are led to considering the common practice today — the traditional lecture method. The efficacy of the method is an ongoing debate. My feeling is that in a given course in junior high or high school, lecture is sometimes appropriate while at other times a more interactive format is. Even during a lecture, there can be interactive intervals.

Student Islands in school classes

I believe that a class should be split up into groups of size between three and six — this size is an interesting topic for investigation. The groups should form islands of desks and chairs facing the front of the class, with room between islands for the teacher to walk about. There are several advantages to forming islands. Let me mention four: 

(a) Whereas only about 30% [7] of the students follow the lecture in a traditional class, the percentage would be much more with the islands. 

(b) The islands are suitable for having the class do problems. They can discuss problems right there. "The main audio-visual tool, the most dependable and responsive component in any coordinated instructional system, is the classmate of the student."[7] The islands can compete with each other — for the very intelligent, this might be more fun than TV football; for all, this should be at least a more supportive and interactive arrangement than sitting alone and doodling often.

(c) An island system enables large class sizes, say 50 as opposed to 20. This would result in immense cost savings in education. What we need is not small class but larger classroom. 

(d) Islands reduce the "strong dependence on the teacher as source of learning relative to other sources ... and the student's belief that they cannot learn math on their own" and "need some knowledgeable authority to provide assistance." [5] My opinion is that a third of class time should be left to students to do their thing on the topic the teacher discussed. This has the greater chance of the student self-organizing around the topic, which is perhaps the effective mode of learning. Islands promote self-organized learning. 

*****  A criticism of Islands could be that it would increase the noise level of the classroom. True, but hold your ear close to a beehive. The best noise in the world is the noise of work. And this informs that portable classrooms are not suitable for math classes. 

Learning-space Design

The foregoing, student islands, actually comes under the design of learning space. There is more to learning space. One problem today is the design of lecture halls as multipurpose multimedia spaces. Projection screens that block the board are not suitable for a math class. Math classes need a lot of blackboard writing space and projection screens must be placed so the teacher's work on the board is not blocked.

Homework

Homework given to students must be carefully selected and coordinated regarding time and quality. Both time and quality are important for homework or you harm the student. Too much time spent on homework will not only not improve grades but also adversely affect the student's social development. A rule of thumb is 10 minutes total for each grade. Thus a grade 3 student would be required to do about 30 minutes of homework a day, whereas a grade 12 student would do about two hours. And that is not all math but all subjects combined. This involves coordination among teachers. Quality of the work assigned is equally important. The photocopied math sheets are the easy way for the teacher. But the standard "math sheet" is boring. No wonder, most students don't like math! Doing the homework in any subject should ideally make for an excited, happy, and engaged time for the child. The drill sheets in math do not evoke this.  So, go for assignments that involve not only practice, for some amount of practice is useful and even necessary, but also calls for a brain response different from practice.  For example, in a sheet of algebra word problems, include a bonus problem that is a problem-posing problem. That is, the student poses her/his own problem that involves the same algebraic manipulation as in one of the problems on the sheet but a different answer. The father-son age problem I gave above is an example of this; I made it up from a problem of A. Perelman [6]: If the father is 32 and the son is 5, how many years later will the father's age be ten times the son's age? The answer is -2.

Calculators in class

Calculators should be banished from the school math classrooms, for these implementers of algorithms prevent the student from becoming fluent with numbers. The devil called "product sales" has gotten hold of the unsuspecting school teacher or the bribed calculator-pushing mathematician (speaking at NCTM and other large teacher meetings). The only justifiable need for a calculator in the mathematics classroom is to check a time-consuming graph, but the advantage is small compared to the harm in the student's over-reliance for simpler chores. Let every math teacher acquire a confident familiarity with the graphs of the six trigonometric functions, the conics, the exponential function, the logarithm function, and the 1/x^n function, and let every student memorize the multiplication table, instead. A calculator is indeed an aid to computation in statistics classes; but statistics is not mathematics; it is applied mathematics.  

(I wrote the preceding in 2006. Now it is 2022. I see that calculators have evolved so much that the line blurs between them and computers. So I must revise my thinking. Instead of banishing calculators, let us banish our use of it for simple work like multiplication of two integers where one of the multiplicands is a single-digit, and additions and subtractions of any numbers unless involving a bunch of numbers in a step. However, I still hold the view that a pure math class does not need calculators of any kind. In fact, they can be an impediment in seeing the limiting process of calculus. Calculators give the false impression of infinite precision even when an answer is supposed to be an infinite decimal. The value of the square root of 2 is square root of 2. Calculators give the false impression that the answer is the important aspect as opposed to the process or idea.) 

This is not to say that a student should not have a calculator. An elementary school student could be given simple calculators so they can use it to check basic algebraic operations they have done. Middle school and high school students could be given scientific calculators so they can use them in statistics and science classes to do tedious computations. 

QUESTION: You seem to be against calculators in school classrooms. What do you think of student use of computers in schools?

ANSWER: I am against the over-use of computers to drill students. Certainly, a well-designed interactive math session explaining a math concept can be good on a computer since it may do a better job than a particular teacher in a particular school. Moreover, the students, being used to computer games, may take to it easily. The Epsilon-Delta definition of continuity or limits, for instance, would be ideal to explain on a computer!  

(I wrote the preceding in 2006. Today, in 2022, I can actually see a role for computers not only in education but also in the very activity of mathematics. There will be several core uses, one being the checking of the correctness of a mathematics proof; in the very distant future, the computer might even replace the mathematician in generating proof from an input of hypotheses. Another use even for the present age will be as a facilitator of conjectures. How so? Computer algebra systems, an online dictionary of integer sequences, algorithms, and the like comprise a telescope to the firmament of mathematics; they can show the patterns to make the guesses. I am more bullish now about computers in schools. I now believe that in the distant future computers would have a bigger role than the human teacher or the school; a whole math course might be downloaded in seconds into the human brain!)

QUESTION: What is your opinion of our teachers?

ANSWER: Adequately qualified teachers are a necessity.

First ensure that high school and middle school mathematics teachers hold degrees in mathematics. The minimum qualification for high school and middle school math teachers should be a Bachelor of Science degree in math — this is not the situation now, 2006 A.D. — when 35% of high school mathematics classes are taught by teachers without even a minor in mathematics [3]. This is not surprising; less than 1 in 100 ninth graders go on to earn a math degree as of this writing. The only option we now have is to attempt to certify teachers using national criteria and re-certify them periodically. While a math degree is necessary, it is not sufficient to be a mathematics teacher.

Teacher Training and continued learning

A mathematics teacher must have a “profound understanding of fundamental mathematics” [4} and content knowledge specific to teaching (“mathematical knowledge for teaching” [1]).

The responsibility for the state of bad teachers rests on several factors including the teacher-education colleges, the structure of the workday, and the teacher learning support system  By structure of the workday, I mean the hours allotted for teaching, one-on-one mentoring for students, mentoring of colleagues, and free time. By teacher learning support system I mean interaction with the wonderful math teachers in one's own school and elsewhere. Attending regional/national meetings will help teachers learn more effective new ways of teaching the new ideas. Corporations do this kind of thing all the time; they call it staff development.

Teacher pay

Pay them — being in the most important profession in the world — salaries that are attractive relative to other professions. Where is the money? Take it from the savings from increasing class sizes by forming student islands in class. I am not advocating the same salary for all teachers.

Lowering teaching load

Distribute the heavy work load so that each teacher has adequate daily in-school preparation time to be more effectively involved in "INSTRUCT! I'm ON" — instruction.

QUESTION: You say you are not advocating the same salary for all teachers. Can you elaborate? 

ANSWER: Salary should be proportional to the qualification as well as the quality of the individual. Qualification is generally easy to obtain, namely, for example, obtaining a more advanced degree. But quality is the greater issue. Clearly, not all teachers can be of the same quality, but they can strive to attain it. One way could be what is practiced in all universities by having three kinds of professorial ranks, based on teaching or research achievement. China has long demonstrated that this works in schools too; the school teaching career there moves through three ranks: second rank, first rank, and master teacher [2]. The salary would then depend not just on the number of years served but more on the ranking achieved. Finally, the lowest salary in all schools should be attractive enough so that teachers do not leave the profession due to low salaries.

QUESTION: Can you throw more light on distributing the workload of the math teacher? 

ANSWER: The school teacher and the university professor are two categories of over-worked people. At least the university teacher workload is distributed in a manner that they can happily bear. That is, they get release-time for research, which is their happy activity. The teacher's work can become boring even if they are conscientious. So, I would suggest that instead of teaching five periods, they teach two, but never more than three! Part of the released time could be spent on lesson and class activity preparation, mentoring students and junior colleagues, committee work and self-improvement such as reading, taking a course, attending weekly, bi-weekly, or monthly math discussion groups attended by other teachers, and getting involved in academic projects. Occasionally they could take on an extra load for a teacher who has to go to a conference. Concerning projects, you do not have to be teaching in a great school to do an academic project. There are always projects that are relevant to the type of school you are teaching in. 

References

[1] D.L. Ball, S.T. Lubienski, and D.S. Mewborn, Research on teaching mathematics: the unsolved problem of teachers’ mathematical knowledge,” in V. Richardson (ed.) Handbook of research on teaching (4th ed.), Macmillan, New York, 2001, pp. 433-456.

[2]  Roger Howe, “Knowing and teaching elementary mathematics – How are we doing?”, Notices of the American Mathematical Society, June/July 2020

[3]  C.D. Jerald "All Talk, No Action", The Education Trust, 2002

[4] Liping Ma, “ Knowing and teaching elementary mathematics,” Lawrence Erlbaum Assoc. Inc., Mahwah, New Jersey 1999.

[5] K. K. Merseth, "How Old Is the Shepherd? - An Essay About Mathematics Education," Phi Delta Kappan, vol. 74 (March 1993), pp. 548-554

[6] A. Perelman,,  " Amusing algebra,” edited and supplied by V. G. Boltyansky. “Nauka”, Moscow, 1976. (In Russian.)

[7] S.K. Stein, "Mathematics for the captured student," The American Mathematical Monthly, November 1972, pp. 1023-1032