Advanced Summer Camp for students age 11-14
who show high promise and love mathematics

An Interview with Prof. George R. Thomas

The following is an essay written by George R. Thomas in an "interview" format. The purpose of the essay is solely to inform the reader the school education attitude of the founder of the camp just as he has tried to inform about the camp in the Why MathPath essay.

QUESTION: You stated in your essay "Why MathPath?" that remedies for the malais in education must be sought. In your opinion, what are these 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.

What is taught
Mathematics content should not be chosen like people choose clothes according to what today is in style. The school curriculum content should be influenced by what mathematical disciplines have prevailed in the foundational core and by how much the young human brain has advanced. The first of these two tells me that 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. Noting that algebra skills and a confident familarity 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 repeates a good portion of it should be removed; that is, a streamlined and exhaustive algebra curriculum is needed - I believe the NCTM Standards address this. If the Algebra I - Algebra II - Pre-calculus sequence addresses the NCTM Standards and the repetition occasionally present is of ideas, not formulas, algorithms and computations, then it would suffice. However, the NCTM standards are wrong in not stressing the extreme importance of 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. ). 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 Student's 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. 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 the grade.

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, say in junior high or high school especially, 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% [4] 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."[4] The islands can compete with each other - for the very intelligent, this might be more fun than TV football. (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." [2] 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 bee hive. 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 lot of black board writing space and placement of projection screen so the teacher's work on the board is not blocked.
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 [3]: 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 class room 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/xn 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 2013. 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 highschool 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. Are you also against the 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 2012, 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 in to the human brain!)

QUESTION: What is your opinion of our teachers?
ANSWER: Adequately Qualified Teachers
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 [1]. 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.
Teacher Training
The responsibility for the state of bad teachers partly rests with the teacher education colleges; and that is another whole issue. What we can do now with the teachers we have is to implement proper teacher in-service and out-service training (sending them, for example, to at least the regional meetings of the NCTM). This allows the teachers who need to improve to interact 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 of the individual and the level of the job. By level of job I mean in this context that high school teachers should be paid more than middle school teachers who should be paid more than elementary school teachers. Further, salaries should be proportional to not only academic and professional qualifications, but also performance. 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 work-load of the math teacher?
ANSWER: The school teacher and the university professor are two categories of over-worked people. At least the university teacher work-load 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 class and class activity preparation, counseling, and committee work and the remaining time spent either on 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.

[1] Jerald, C.D, "All Talk, No Action", The Education Trust, 2002
[2] Merseth, K.K., "How Old Is the Shepherd? - An Essay About Mathematics Education, " Phi Delta Kappan, vol. 74 (March 1993), pp. 548-554
[3] Perelman, A., "Amusing algebra. Edited and supplied by V. G. Boltyansky. “Nauka”, Moscow, 1976. (In Russian.)
[4] Stein, S.K., "Mathematics for the captured student," The American Mathematical Monthly, November 1972, pp. 1023-1032

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