Mathematics and Language 3

Edited by Marsigit
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Doug Hainline:

Teachers who are themselves poor at mathematics will be poor teachers of mathematics. The best teachers of mathematics are confident about their mathematical abilities (with good reason), and enjoy mathematics.


They need not be mathematicians, but they should take pleasure in beautiful proofs and demonstrations, and in showing these to their students. They should be the kind of people who like to read popularized mathematics books, such as the kind the late Martin Gardner wrote, or that Ian Stewart or Keith Devlin write.

Of course, this is a counsel of perfection. We have to start where we are, and try to ensure, for the moment, that every child is exposed to at least one teacher like this in his or her first years of school, and that there is then some institutionalized path that children who become interested in mathematics can follow: Maths Clubs, for instance.

Of course, good teachers will know, and use, whatever mathematical abilities they discover latent in their students. But mathematics has to be taught. The idea that it is just lying, latent, in our pupils, is wrong (despite Socrates' famously, supposedly, demonstrating this the Meno).

Marsigit Dr MA :

@ Doug: Yes I agree that the teachers who are themselves poor at mathematics will be poor teachers of mathematics. Also I do agree that the best teachers of mathematics are confident about their mathematical abilities (with good reason), and enjoy mathematics.

By the way, I should be very careful with your term "beautiful proofs and demonstrations".

"Beautiful" proofs is your perception; but, your students may perceive that it is very "bad" proofs.

"Demonstrations" is your behave; but, what your students should behave?

"Showing the formula" is also your behave; and, it's a pity that your students just to look passively.

You seemed to force your life (math) to your students. From my perspective, it is dangerous for your students. So, again I have proved that the problems of math teaching are coming from the adults and not from the younger (students).

For me, the students need to learn mathematics; and not as your notion " But mathematics has to be taught". It is the students who NEED to learn math.

Your last notion indicate that your students are shadowed by your ambition to implant math to young generations.

For me, the younger students are free to learn; it is okay if they do not like math and they do not want to learn math. The problems, again, are not coming from the students but from the adults (teachers)

It is better if the students themselves (not the teacher) who claim beautiful math/formulas and demonstrate them to their mate or even to the teacher.

So, the problem is how the teacher are to facilitate their students in order to learn math happily.

David Reid:

Doug's point that school mathematics teachers need not be mathematicians is very realistic: most mathematics teachers never do any mathematics. Mathematicians rarely teach at the school level; most school mathematics teachers are only trained to apply mathematics, and after university they rarely do this beyond the level of school mathematics. However, loving mathematics is indeed a help, because a teacher who doesn't is likely to pass this dislike on to his/her students.

Marsigit has a point that beauty is in the eye of the beholder, and Doug's point that mathematical knowledge is not inborn is complementary to Marsigit's implicit point that the ability to find beauty in math is also not inborn. Alas, the world seems to be designing curricula which downplay or eliminate proof altogether, so that the student never gets much chance to judge whether he/she finds a proof beautiful. The reason is not only the tendency to teach what industry wants, but also that proof, when it was taught, has usually been taught in a way that led more to memorization that to thinking, especially with the school culture of learning for tests.

The dichotomy between teachers-teaching and students-learning is an artificial one: a student will not learn without being taught, and the teaching will be for naught if the students doesn't try to learn. There is effort needed on both sides. And it is an effort: we have to teach to the math-lovers, the math-indifferent, and the math-haters.

Marsigit is rather harsh on Doug, reading psychological tendencies between the lines that are not justified from the text: a rather strange practice for someone who is involved in mathematics, with its axiomatic method.

Marsigit accuses Doug of wishing to force mathematics on to students. This leads to the very thorny question as to how much mathematics, and which mathematics, is needed. Most of the students will not need more than a small percentage of the mathematics they need, whereas a small percentage of the students will need a large percentage of the mathematics. The problem is how to decide who will need what, and which mathematics to teach. Some school systems start splitting up the children according to ability and desire early in primary school, whereas most systems take the easy way out and teach everyone more or less the same mathematics. This is an unresolved problem, and I don't pretend to offer a solution here. But as long as the society requires the teaching/learning of more mathematics than most people need, then simply allowing the students to choose their own way of learning will not work, since most students will choose not to learn, due to a number of unfortunate factors. Therefore, although the word "facilitate" is now a very modish word in education, it is not clear what is really meant by it.

Doug Hainline :

 I'm not sure that I understand what Dr Marsigit is saying.

I have seen a fair amount of writing, among academics infected with the post-modernist disease, which seem at least congruent to some of the things Dr Marsigit seems to be advocating. I have to force myself to read these things.

What I believe is that there is an objective reality independent of human consciousness, that mathematical truths are absolute, that understanding them does not come easy to most people, but that it is worth understanding as much as you can. (Not so that your country's economy will grow faster, although that may be an outcome, but because expanding this sort of knowledge and ability is what it means to be human.)

I believe the truths of mathematics, and the skills involved in problem-solving and proofs, should be taught to all of our children, regardless of their wishes, until they become old enough to decide for themselves what they want to learn, which in our current culture is some time around 16 years of age.

How children learn the truths of mathematics and how they become, when they do, proficient mathematicians, is still an open question.
We as yet know little of how the brain works, and there are multiple ambiguities when it comes to discussing teaching and learning mathematics.

I favour Direct Instruction, including learning things by heart, as well as guided discovery and open-ended problems, collaborative learning, what have you. Whatever works, which may be different across cultures, and in any case has to be implemented by what are often very imperfect human agents and in social contexts which are not conducive to learning.

You can spend days reading about different approaches to teaching mathematics. I think many of these debates are sterile, and that the people who love to conduct them have little experience of actual classrooms.

The children we encounter there range from the highly intelligent, intellectually curious, across to those who are ... not. We have a duty to all of them.

I cannot see how any reasonable person would disagree with this.

David Reid :

I agree with the pedagogical side of Doug's last post. There are a couple of philosophical points about which his last line "I cannot see how any reasonable person would disagree with this " is unjustified. That is, he takes the side of Philosophical Materialism and the side of Classical Platonism. Whereas both of these sides have good arguments, so do their opposites, Philosophical Idealism and Neo-Formalism, respectively. (The "Neo-" is used to distinguish it from the Formalism of Hilbert's Program, which was discredited by the results of Kurt Gödel.)

Actually, most mathematicians and physicists take a golden mean between these extremes, since each extreme, while having their merits, also have their defaults. So, whatever your philosophical stance in these matters, it is simply not fair to say that your opponents are not reasonable. In any case, whichever your philosophical stance, it will not affect the pedagogical methods or aims below university level that Doug is discussing.

David Reid :

Dr. Marsigit, I am familiar with Prof. (emer.) Ernest's philosophy. On one side, any good logician (which Prof. Ernest studied for a while) or student of Model Theory clearly knows that mathematics rests upon the axioms chosen, that the axioms are chosen by human mathematicians, and that these mathematicians are a product of their genetic and social background. So far is obvious, and so I do not defend the straw man (Aunt Sally, if you are British) of naive mathematical absolutism that Prof. Ernest argues against.

On the other hand, the common misinterpretation of Prof. Ernest's social constructivism that mathematics depends entirely on social constructs is also flawed, since mathematics contains strong filters to restrict these choices. (Hence mathematics differs strongly from sociology.) In other words, given the infinite amount of mathematics possible, we choose part of it, but what is chosen would probably be found to be valid also by intelligent mathematicians from another planet. This is a far cry from the silliest social constructivism (not Prof. Ernest's, even though his theories easily give way to misinterpretation) which simply says that mathematics is invalid because it is a social construct.

In any case, the position that most practicing mathematicians adopt is somewhat in the middle between the naive absolutism and the extreme social constructivism which was so popular in the 1960's and 1970's. Unfortunately, it is not the more nuanced and realistic theories that are behind many of the theories of education which are practiced today because, as has been repeatedly pointed out, most teachers, as well as those who set the curricula or write the textbooks are not mathematicians. Nonetheless, most major theories -- social constructivism and others -- have yielded some good pedagogical practices (and, alas, a lot of rubbish as well), and one should take the best from each.

In general, the proof of the pudding is in the eating. Unfortunately, so far no one pedagogical theory has come up with a pudding worth eating, and so we will be depending on the ad hoc hybrids that characterize most education today -- including in those schools which give lip service to a singular concept but are then forced to alter these ideas in the face of the reality of the classroom.

Doug Hainline :

David: I agree absolutely with what you have said here, although I would be interested to learn more about the good pedagogical practices which have come from social constructivism. (I've recently been reading through back issues of Paul Ernest's journal, looking for useful work. I haven't found much yet, but I have not finished. I did come across an interesting piece on why we should teach logic along with mathematics, which, in the light of our discussion above, I thought made a lot of sense.)

I keep reading, from one side, that 'consructivism is a theory of learning, not of teaching'. I don't think it's even a theory, because it's not testable, but I have no objection to someone calling themselves a constructionist, if their kids learn maths.

I just want to see the evidence. I can well believe that some form of 'discovery-learning' by bright, motivated students under the guidance of a wise teacher, especially among older kids, can yield very good results. I think it's probably the best way to teach the last year or two of undergraduate maths (wasn't there a fellow named Moore at the University of Texas who did this -- his students were given proofs to do by themselves, as I recall, and became very good mathematicians as a result).

I doubt it works in inner-city classrooms among nine year olds. And I think there have been several meta-analyses of studies of 'constructivist' teaching vs 'direct instruction' which have shown that the former doesn't really get the results that the latter does.

Marsigit Dr MA :

@Reid: O yea great. I think Ernest's work is very great, because he produced something like a map of educational philosophy/ideology. I think you are very good reader of his works. Although, I still found that we have different interpretation of some of his points. I found that you still used a certain criteria from a certain dimension of life to judge the criteria from different dimension. It lead to a condition in which you seemed unfair in making some judgements.

In my point of view, your claimed that mathematics contains strong filters to restrict the choices based on the position of axiomatic mathematics. Again, in the case of younger learner, there are different world of life (learning math). There will be many mistakes if we judge them from the world of pure math. The young students do not learn the infinite amount of math possible. It is also not about the validity from other planet. The criteria for validity, truth, proof, construct, etc.are totally different. The children are the victims of naive absolutist who intervene much on the younger learner.

@Reid and Doug: I wish to argue the validity of the ground of pure math in developing school math curriculum and its textbooks. According to me, they are not accountable in doing so. I did not find any reason for pure math for their intervention in primary or secondary math teaching. It is clear that pure mathematicians will also teach pure mathematics to younger children. It is wrong. The younger learner must not learn pure mathematics. As Ebbutt and Straker (1995) suggested that the solution is that there should be School Mathematics i.e. mathematics that the student should learn.

David Reid :

For readability, I will make one post to answer Doug, and another to answer Marsigit. This one is addressed to Doug. You are referring to the late Prof. Robert Lee Moore, who developed a method which is still used in some universities in selected courses. A follower of the social constructivist school would have been proud to claim that Moore's method followed from social constructivism, but that would have been false; nor could it have, since there are important differences: the social constructivist would conclude that the students' final proofs would be valid proofs for that group, albeit perhaps not for another group.

That position, of course, would be rubbish, and this was not the position of Prof. Moore. However, there have been other spin-offs from social constructivism that are more worthwhile and not as naive. A good example is "Computer-supported collaborative learning" which is useful, for a couple of reasons. First, there is a trend in many Western secondary school classrooms for the students to think that the classroom is a chat room in which the teacher is just an annoying bit of noise that one can ignore most of the time.

However, these students are more willing to listen to one of their peers than the teacher. This is the "collaborative learning" part which is useful; in fact, I often use this trick: I get a student to say more or less the same that I would, but the students listen. In these cases I just insert necessary corrections or questions. Works like a charm. (Again, in secondary school. I haven't tried this in primary school.) Then, there is the computer part. Pupils above a certain age will more willingly listen to, and follow the instructions of, a computer than a human. (Some psychology student should do his doctoral thesis on this phenomenon.)

This is true even if the computer is just carrying a human's presentation. As well, then, an interactive computer lesson (once you figure out how to stop students from switching to Facebook) has other advantages, such as being able to have quick references. But apparently this method need not be reserved for undisciplined students: I have seen some research in South Korea (but I am afraid I cannot give you a reference at this moment) reporting some positive results from this method (although going under a different name, and practiced at university level). However, in South Korea the method was mixed with other methods, so the results were largely anecdotal. Nonetheless, I figure that this would have a place, not as the unique method of instruction, but as one element in a school mathematics program. That is, there is no reason not to mix this with the Direct Instruction method.

Finally, I fully agree that logic, when taught correctly (which it rarely is, in my view), should be taught at school level. Since we are in agreement, I will say no more in this post; however, I will comment a bit further on this in my post to Marsigit, since he does not agree.

David Reid :

To Dr. Marsigit: first, we must separate mathematics from mathematics education. As I pointed out in my post to Doug, Prof. Ernest's work has had some good spin-offs, even if indirect, for mathematics education. With respect to mathematics, his conclusions can be separated into two categories: the obvious ones about the limitations of mathematics, and the ones that are fuzzy enough to easily give rise to erroneous interpretations of the complex relationships between mathematics and society.

You refer to mathematics education, so I will say no more about his ideas on mathematics per se. Here, Prof. Ernest's main conclusion is that student's learning styles will depend on their respective social backgrounds, and that one must take this into consideration. I have taught in a large number of different cultures, and in each one I adapted my teaching style to the student's society's; even in a single school, my teaching styles for two different groups will usually differ. But this conclusion is not unique to the style of social constructivism espoused by the followers of Prof. Ernest.

Therefore Prof. Ernest's philosophy cannot always take credit for this conclusion. I understand what you mean when you say that the criteria for proof and truth are different in different environments. But you need to separate the ways of learning from mathematics per se. For mathematics per se, these criteria are not society-dependent, but the way a student to be convinced of something without formal proof is society-dependent.

The problem is that eventually the two concepts of truth and proof -- one subjective, one mathematical, need to eventually brought together. Students should eventually learn that their subjective ideas are not necessarily correct. I fully agree with you that one cannot start immediately in a topic in primary school with this synthesis, especially in primary school (and you seem to concentrate on primary education), as one needs to first appeal to their intuition. But even in primary school one can, little by little, impress upon the students that, whereas their intuitions are a useful starting point, and whereas one can develop this intuition quite a ways, one cannot always rely on intuition to take you where you want to go.

Mathematics partly uses intuition, but also partly uses methods which are no longer intuitive, nor which can be made to be so. It is especially important in secondary to get away from the pure subjective approach. This is also why pure mathematicians could be useful in organizing a curriculum -- both at primary and secondary level. An important aspect of mathematics which could be profitable for all students, regardless of their eventual professions, would be introducing concepts from logic, to help students think clearly. But not the formal notation of logic; rather the concepts. This can start in primary, and continue in secondary.

However, every time it has been introduced, it has been a fiasco, as most teachers below university level themselves do not have a clear idea of these concepts, and have made a mess of it. The solution has, alas, been to strip school mathematics of the very essence of mathematics, that of clear analysis. I am not sure what you consider pure mathematics, but at university level, pure mathematics includes Mathematical Logic, and some of the more useful and learnable aspects of Mathematical Logic would be more useful, if put into a form which students could understand and practice, than most of the formulas which are presently memorized by students.