Regarding #25: my comment was nothing personal or to do with the topic, just a feeling that I did not feel the conversation was likely to move in a direction where I would be likely to be doing more than more or less repeating what I have already written.

I would just like to clarify that I am all for teaching teachers more advanced material: elementary number theory, for instance, can be an excellent way to develop the teachers’ competence in more basic matters, by applying their knowledge in a challenging and diverse context. But there is also no reason that a really good teacher could not introduce some of that back to their own classes, if the opportunity arose. One aspect of elementary number theory which makes it particularly suitable is precisely that one can avoid sets completely. I wrote a course for example along these lines, including a proof of quadratic reciprocity: you can the notes here, but they are in Norwegian.

It is certainly also true that the kind of approach that research in mathematical didactics suggests is effective is radically different from pretty much all teaching of mathematics at university level, and much teaching of mathematics elsewhere. And one of course has to work with the limitations imposed upon one, but, if one believes in something, it is necessary to try to change things if one sees an opportunity to better them.

If I have a strong opinion on these matters, it is because I feel that the status quo could be so much better, and I think that mathematicians could play a big role in this were they to become more directly involved.

Regarding programming: I think that this is a brilliant way to teach mathematics too, I think the two go hand in hand, and would advocate a curriculum in which they are much more closely intertwined.

]]>Zoran: By “I don’t see why they can’t also learn category theory” I didn’t mean it in the sense that the correct observation about lack of supportive educational infrastructure (for example, teachers who know cat theory) would falsify. I meant it in the sense of “they’re mature enough, so why not teach it to them”. (Teaching it to them would of course require changes in educational infrastructure).

Also, I totally agree that you need to have a high level of knowledge of other mathematics in order to properly appreciate/understand category theory. However, I also think you need to have a high (or atleast significantly above average) level of knowledge of just about everything in order to properly understand/appreciate philosophy, and I don’t think the optimal way to learn philosophy is to put it off until one is poised to *truly* understand it. Rather, I find it best to learn about general abstract concepts and dichotomies (platonism/aristotelianism, confucianism/daoism, relativism/substantialism, contextuality(or relativity)/absoluteness, systematicity/asystematicity, analytic/synthetic etc…) such that one has a hollow skeletal understanding of them, then descend to flesh out one’s understanding with examples of how those abstract concepts and dichotomies play out in particular* cases. once one has enough (& varied enough) particular examples, then one’s understanding converges again on the abstract purity of one’s initial novice level psuedo-understanding, only this time it is real understanding, has weight, is fleshed out**.

the tricky thing is finding the the proper level of generality that such that the amount & range of particular examples needed to ascend back to generality (with true understanding) is large enough to make the ascent back to generality interesting and profound, but not so large that in the descent to so many particular examples people get lost and forget about the general idea that motivated the descent.

*note: given how generic and abstract the meta-discipline that is philosophy is…. particular could mean for example the application of the platonism/aristotelianism divide to a platonic realist stance on mathematics vs an aristotelian realist stance (like james franklin’s). this “particular” application still resides at a very high level of generality.

**(you even have subprocesses nested within this process, where you are exposed to a skeletal abstract idea, you descend flesh it out in a variety of cases, then you ascend back to the level of abstract purity this time with proper understanding …. only this “abstract pure level” is on the concrete example level for a higher abstract idea).

]]>Of course, Urs, you know well that this is a statement in a completely different world, than the world of our school teachers.

]]>category theory … unlike programming

You are violating the holy trinity here. If the kids can code functionally, they are already making use of category theory.

]]>Thank you, Todd, now noted under mathematics education.

Trent: you can not expect that teachers will know category theory. Besides, category theory becomes useful only at a quite hi level of knowledge of other mathematics, unlike programming which has effects/can be applied in interesting way to a creator even after first few steps. Of course, you can have special groups of talents which may be motivated anyway, but this is not the question for mass scale education.

]]>Kids are learning computer programming at younger and younger ages. If junior highschoolers (7th-8th grade, ~ages12-13) can learn to code, I don’t see why they can’t also learn category theory.

]]>At the university level you have almost no time for students to have discovering things, at least in program I was given, unless I put a lot of homework for students and made this way my course more heavy on students time than other courses do.

There is of course the Moore method, which some swear by but others wouldn’t touch with a ten-foot-pole. This requires, I believe, a great deal of effort from the instructor to prepare carefully (and of course it requires a lot of student effort as well).

]]>Regarding #19: I was responding to the use of the word ’minor’.

There is no assertion that the methodology is minor but that I didnot intend to open the discussion to minor methodological units. Each methodological unit separately is a minor issue. To have say geometry taught or not is a major issue. The entire methodology for geometry is a major issue but needs much more space to discuss and different conversants.

I would argue that there should not be a rigid distinction between how to teach teachers mathematics and how to teach children mathematics

Well, the pedagogical faculty distinguish

course of mathematics as subject for teacher so that the teachers learn mathematics at a deeper level (and understand more than students so they can answer tricky questions of students by having bigger vision) and

the special courses of mathematical methodics/didactics.

Those pedagogical faculties which had just latter kind of courses had often in their output teachers with weak content knowledge. Of course, teachers should not learn advanced subjects but the same subject at deeper and more comprehensive level. Unfortunately I have to do more advanced subjects as well as the program of the courses (the program has been approved before my involvement) tells me, but in long run I will vote for change in that. So at the moment I am not doing my job so well.

what precisely one teaches is, from this point of view, of relatively little importance

This is a viewpoint which has been extremized in last couple of reforms in US and results are not satisfactory, my american colleagues interested in K8 and K12 education tell me. I am not sure how correct that is. Of course it is great if students discover themselves and if the teacher fosters that. Mostly effectively doing this is beyond the ability of present teacher population. At the university level you have almost no time for students to have discovering things, at least in program I was given, unless I put a lot of homework for students and made this way my course more heavy on students time than other courses do. Some teachers are praised to be great by some by making students work disproportionally much in their courses, but I think this is an abuse of teachers power. Of course, I will do more of that when i get stronger hold of the thing in future semesters, now I started in the midst of old program and can not invest that much time having to struggle with more basic issues.

I do not understand why people dealing with math education have so strong opinions in general on any side of any argument. Of course I can not waste more of your time but otherwise I do not understand these attitudes like “this is my last post on the topics” as if you have an unpleasant conversant (am I?) – or the topic is odious or unimportant. Thank you for the pointer to G.B.

]]>Regarding #19: I was responding to the use of the word ’minor’. In particular, I was responding to something which in fact is raised again in #19: I would argue that there should not be a rigid distinction between how to teach teachers mathematics and how to teach children mathematics; and that one cannot and should not separate ’what’ teachers should know from ’how’ they should impart that knowledge. The teachers, just as the children, should be learning mathematics as part of a rewarding, creative process of discovery - that is the point of view that I am discussing, which is the dominant one in research into mathematical didactics.

As I expressed in my very first sentence in #2, what precisely one teaches is, from this point of view, of relatively little importance, as long as the teachers arrive in the end at a deep level of understanding of that they will be teaching; *how* one teaches it is, on the other hand, of vital importance, because it is that rewarding, creative process of discovery, to use the same phrase again, that will lead to a deeper understanding.

I do not intend to contribute further to this discussion, I feel that I have said what I wished to say. If you are interested in learning more in this kind of point of view, you might wish to explore, for instance, the works of Guy Brousseau, which have been heavily influential. There are many other points of view, but most of them have something at least in common with that of Brousseau.

]]>- Ralph A. Raimi, Whatever Happened to the New Math?

As far as I understand, the principal objection of educators is not that set theory can not be made intuitive but that its relevance to the problems school students face is not crucial and they see sets usually separated from their experience and usages. So it kind of stays an exercise in abstraction. Even if example are given of plastic shape and vivid meaning those examples are perceived by many researchers and their test students as eventually little relevant, somewhat school-made. of course, other concepts like geometry etc. can be taught in terms of sets, but it appears that this is not necessary and at low level not beneficial. I am just trying to state what I heard and read. Another objection is that the educational reforms which emphasised on sets, primarily the, rather extreme, New Math failed. Note however that the courses for teachers and their textbooks in New Math program were done mainly by mathematics education experts without much input of mathematicians and that the teachers educated to teach that mathematics themselves understood the subject of proposed school mathematics rather insufficiently.

]]>“That is, I am not asking in this thread what do you think how to teach union of sets, but rather weather it should be thought at all, and if so, at which age.”

**IMO, If you are old enough to put your toys away, you are old enough to learn set theory.**

As soon as a kid can handle a bag with items in them (marbles in this example, though a poor choice due to size), that is when we start teaching set theory.

For example, you could first you have two bags and no marbles to explain membership and empty sets… then you have two bags and one marble to explain membership inclusion, you can put the marble in one bag but not both… then two bags and two marbles, three bags and one marble, etc, etc.

Then, IMO, you can talk how taking all marbles from one bag to the other is monic or epic, depending on how you label your bags, and you are a hop, skip, and a jump away from CT at a young age.

I can’t think of a single mathematical starting point that would work within the limitations of young children in terms of hand-eye coordination or analytical skills. CT is very visual, what with arrows and objects and all, but it is a good “abstract” starting point and IMO you need concrete foundations (sets) before you can ever hope to grok abstract ones (categories).

]]>I talked to a chemist at IRB minutes ago, he went to catholic kindergarten where the nuns were the teachers. And he learned the notion of sets from nuns at that kindergarten. They were teaching this concept routinely (his age is about 40 or so, so this is about 35 years ago).

Another question is not the set theory, but the question of much more of my immediate concern now and that is the right aprpaoch to the elementary geometry for K4 and K8 teachers. In 4th grade in Croatia they do learn the notion of Cartesian coordinate system and in later classes 5-8 they do have vectors in plane and a little bit in space at the end, but most geometry is via “synthetic methods”. The university teacher before me has therefore decided to develop the geometry from axioms in metric approach (the properties of metrics with values in real numbers is among the axioms). This is simpler approach then Hilbert’s but still very complicated and it is impossible for that level of students (future teachers of K1-K4 classes) to cover the geometry in one semester. On the other hand, to drop the building of the theory means giving rules and procedures. One needs something in between but I do not know a good approach and it is hard to improvise, the geometry synthetic way is hard. Even to prove simple things like a triangle with equal two sides have their two opposite angles equal as well. The textbooks on the internet are mainly terrible. The general notion of vector space is beyond the level of the course (and is taught in abstract algebra course which is a year after the geometry course!).

]]>I am not sure what you had in mind by ’minor methodological issues’, but if by this you intended a dismissal of the research in mathematical didactics mentioned in the first paragraph of #2

Where did I say anything even remotely similar to this ? I said

My primary concern is to find out what is the correct approach to school math education, in this thread primarily as far as choice of topics for math curriculum and approaches to the topic,rather then minor methodological issues.

That is, I am not asking in this thread what do you think how to teach union of sets, but rather weather it should be thought at all, and if so, at which age. So I know what teachers have to know in math, so that I teach them, then the others (didactics experts) will teach the how part (I need to know how part for teachers, not how part for students; in fact I am concerned with the latter as well, but this is not the principal goal of this thread). The details of this particular microlesson in school I leave to the experts, I am **not** teaching didactics, but I should know what subject teachers should know as I said
**within realistic boundary conditions** (of course I want to know what I should teach, not what some administrator tell me to teach, this boundary condition is the question of my owb fight, once I know what I should aim to).

it is something which relies on the teacher having a true understanding of the mathematics involved

Yes, so my objective in opening this thread is therefore the most relevant issue.

]]>One of the principal difficulties with teaching fractions is *exactly* that, by contrary with one’s experience with integers, the same notion can be expressed syntactically in many different ways. One of the principal first goals of teaching fractions is to find a way for the students to be comfortable with this. For this, *defining* a rational number as an equivalence class of integers is absolutely useless: it becomes a purely syntactic thing, and the whole point, with which I began, is that it is the *semantics* that the students (and teachers) need help with; anybody can mechanically apply syntactical rules, but it is quite another thing to have a deeper understanding of what one is doing.

I agree entirely that children, anybody, can grasp abstract ideas. But to do so, they have to be made *meaningful*. That is the challenge, and it is one that I do not think set theory helps with; one should be able to do it without any particular formal crutch.

I am not sure what you had in mind by ’minor methodological issues’, but if by this you intended a dismissal of the research in mathematical didactics mentioned in the first paragraph of #2, I would recommend that you think again! *How* mathematics is taught has everything to do with how well it will be learnt. This is mostly in fact *not* a question of pure pedagogy, it is something which relies on the teacher having a true understanding of the mathematics involved.

One of Askey’s articles about the problem I mention above is

]]>Thank you guys for joining (sad that Mike soon quit, hope he reverts back to the discussion).

My primary concern is to find out what is the correct approach to school math education, in this thread primarily as far as choice of topics for math curriculum and approaches to the topic, rather them minor methodological issues. Namely, I have to make correct choices when teaching school teachers (what I came now to be faced with) with realistic boundary conditions (motivation of future teachers to learn the subject, the standard in the society, the inclination of pedagogical community etc.). That is why I asked about majority of the countries rather than only in US (or Croatia where I live).

I noticed that one of the dominant issues in the thread above is the low standard in proficiency of the teachers. Academician Prof. Richard Askey has emphasised in many of his public and professional talks that one of the problems is that the pedagogical experts (that is the people influential in educational policy and university education for teachers) tend in recent decades to emphasise skill on how to teach but not on knowledge of the very subject (ineducational circles: **content knowledge**); they assume if a teachers know the theory how to teach well he will teach well even if he does not know the subject much better than his school students. And this gets into disasters as they may think that they do right while testing of teachers shown say large percentage of teachers who will add fractions by adding numerators and adding denominators.

As far as the above example that the fraction is a class of equivalence. Well, before doing any algebraic operations with numbers one needs to emphasise to students that if one multiplies the numerator and the denominator by the same number one is diving N times bigger thing to N times more people and gets the same result; so the two fractions are the same; so fractions are not what is written but indeed something what has many equivalent shapes. Now how to call this kind of sameness, and to treat this with equivalence is an abstracsy overkill when discussing this for the first time, but could be redone several years later in high school. So, saying it in formal words using words class of equivalence is counterproductive, but the idea has to be in the consciousness of the teacher and he has to teach it by having it implied somehow. There is a general view in education that kids can not grasp abstract ideas. But the relationships forming an abstract idea can be exhibited in examples and could be done in a way fostering child generalization. At the level of feeling, not at the level of formalized statement. If the educational experiments exhibited that a child can not learn to formally introduce the relations of equivalence and classes of equivalence that does not mean that the child can not grasp that for some purposes some things can be taken as the same, adn the differeces being abstracted to none. And this *is*, in my opinion, some level of understanding abstraction.

As far as knots and numbers, I definitely remember how late I learned to knot my shoes, about 3 years after counting to 10 (what was too early to remember myself). But this is off-topic.

]]>Regarding #14:

a) I have not suggested at any point the adoption of a ’knot-theoretic foundation’ for school mathematics curricula;

b) I have already made clear in #11 that to regard my proposal regarding knots as being ’opposed to’ anything is to completely misunderstand it.

Regarding #13: I will not reply directly. I just wish to clarify that I am perfectly aware of how rational numbers can be constructed and worked with category theoretically. I also wrote clearly that I am perfectly convinced that category theory can influence one’s teaching at all levels. It is quite another thing to suggest that one should base on a school curriculum on sets and categories directly.

If I am arguing that I don’t feel that this would be likely to address the fundamental challenges of teaching mathematics at school level, this is not some opinion that I am firing off on the spur of the moment; this is something that I have thought deeply about, and a conclusion that I have reached *despite* the fact that I might well have had the same point of view as Urs expresses, had I not become better acquainted, through direct experience of teaching mathematics to qualifying teachers, and through some study of the mathematical didatics literature, with how children learn mathematics, and with what the principal challenges in their so doing are.

When I put my question to Urs, this was in full respect for his views, having corresponded with him in one way or another for many years, and having met him in person.

]]>“I would contend that most children become acquainted with the tying of knots from their earliest days, through clothing, etc,”

At what age did your children learn to tie knots and how long was it after that they learned knot theory?

A practical implementation of your viewpoint would really help in understanding how a child is better served with a knot-theoretical foundation when they can’t tie knots as opposed to a set-theoretical one given that they can already count.

]]>“What insight does it give (in a direct way) into teaching multiplication of fractions, for instance?”

One is theory; the other is experiment.

HOW to do fractions is experiment; WHY fractions do what they do is theoretical/categorical.

As such, we need to be clear what we want to teach our kids, theory, experiment, or both.

Right now, for example, the USA is purely experimental, how to do something with no underlying framework to connect the disparate experiements…. there is no identification of the connection between “mult and addition” with “division and subtraction” for example.. or that the algebra that you use depends on the type of number you use (real, complex, hilbert, etc).

The tenor of this thread are the theorists saying that if people knew more theory right off the bat, they would do better experiments.

AH, the experimentalist retort, but if you know more experiments then you have a better grasp of theory.

And so the wheel keeps on turning as theorists and experimentalists keep working at cross purposes as to which appoach is best while the teachers are left blowing in the wind or worse, being told to go right (theory) one year and then left (experiment) the next and then something that is neither experiment nor theory but politics (common core) the next! :(

]]>Regarding #6: I would love category theory-influenced teaching to be dominant at university-level mathematics courses. Probably in the last years of school, it could begin to be introduced too, in a more or less direct way. And I certainly believe that one can do mathematics which is *influenced* by this point of view at all levels; I would consider my own teaching to be in this vein. However, I must say that I do not see a way that Sets for mathematics would be *directly* relevant for the majority of the school curriculum. What insight does it give (in a direct way) into teaching multiplication of fractions, for instance?

In reply to #8: I don’t really wish to discuss this here. I will just say that I chose the word ’acquainted’ deliberately: I would contend that most children become acquainted with the tying of knots from their earliest days, through clothing, etc, and almost all will become accustomed to the notion of a knot being tied long before they learn to count. But I am not trying to set up knots in opposition to counting, or anything else; I am simply arguing that they are as appropriate for the teaching of mathematics from the earliest years as more traditional topics.

]]>I completely agree with Mike’s comment in #5. Any reform of school mathematics must necessarily go hand in hand with reform of teacher education. I have worked in teacher education in Norway, and here the teachers go through again, in their education to become a teacher, the material that they will be teaching. This is the perfect opportunity to confront accumulated misunderstandings and misconceptions, and replace it with something that truly gives them the ability to teach mathematics well. In my experience the teachers, almost universally, are far from lacking in ability, i.e. they have the potential to be superb teachers of mathematics, but it can be challenging, in some cases, to break down the mentality they have inherited from their own schooling, and to get them to engage in understanding mathematics in a deeper way.

I have a lot of ideas about this, and have tried some of them out, mostly successfully, as far as I have been able, but to, for instance, introduce knot theory properly, one would need backing right the way through the framework of teacher education, and this is an enormous task, involving convincing politicians, schools, …. In Norway at least, there is no lack of political or societal will for improving mathematics standards, but it is still a gigantic task.

On a related but slightly different note, I would argue that whilst it is true that it is indispensable to improve teacher education in order to improve teaching of mathematics, one also has to improve the quality of the teachers teaching the teachers, and here mathematicians could take much more responsibility; as I wrote in #2, there is too much of a divide at present, in my opinion. Again, this is something which is a gigantic task.

]]>I am not interested in getting into an argument, but I will say that although there are undoubtedly many other problems with public education in the US, it is a fact that many of the elementary teachers that I have personally interacted with did not have the understanding of mathematics that I would regard as sufficient to teach it. This will be my last post on the subject.

]]>