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To get a deeper understanding of mathematics conceptual teaching and learning is supposed to be a much better approach than factual teaching and learning processes. Since the conceptual approach is much abstract it doesn't have a considerable impact on creating interest in the subject in average students. To fulfill this requirement we have the next level, that is contextual teaching and learning (CTL), where the priority is given to make sense of why we need to know about the concepts in mathematics.
Mathematics has a history that is rich in astonishing breakthroughs, false starts, misattributions, confusions, and dead ends. Therefore I think, historical developments over the generations play an important role in the contextual teaching and learning process, when it comes to discovering meaningful relationships between abstract knowledge of conceptual learning and real-world applications in a new environment outside the classroom.
As experienced educators what would you suggest regarding to use history of mathematics as a tool in CTL?

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    $\begingroup$ 1. Which level of schooling is this for? 2. The question would be improved by some paragraph breaks, and maybe some clarification of CTL and what it is to be contrasted against, for those of us who are not familiar with the concept from before. $\endgroup$
    – Tommi
    Oct 9, 2023 at 6:29
  • $\begingroup$ @Tommi this is for any level of mathematics education and I used abbreviation CTL for contextual teaching and learning. $\endgroup$ Oct 9, 2023 at 7:07
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    $\begingroup$ I would say the need to make sense of why mathematics is necessary is different in kindergarten, pure math PhD program, middle school and engineering mathematics course! $\endgroup$
    – Tommi
    Oct 10, 2023 at 5:01

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My advice is to minimize this sort of thing and do it more with strong track kids than with average or weak. What's really interesting and motivating is mastering the material. Your priority needs to be on training the kids, not diversions.

Of course as an instructor, you should have added knowledge of your topic and if a little drops now and then, can be ok. But it's a garnish. It's the sprig of parsley, not the schnitzel.

If I'm learning gymnastics or wrestling, the history is interesting. But it is a 4th order priority versus learning moves. Math class is the same.

What's intriguing to me is how weak many math instructors are on the general engineering curriculum and the use of math for the majority of students. Much bigger gap than math history. Just doing some orientation skimming of STEM texts in engineering, physics and chem would upgrade them. But even here, I will take a good trainer over someone who knows applications.

P.s. I don't like the supposed to be formulation. According to who? And are we sure? After all, a lot of silly stuff like learning styles, inquiry learning, whole language has been pushed because it sounded modern an liberal, both meanings, even though testing showed no benefit or even detriment. So beware the supposed to groupthink.

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    $\begingroup$ I'm sorry, you may have not understood what I meant there by contextual teaching and learning. It is mainly for the sake of giving sense of why particular concept is important and why you need to learn in a particular order of those concepts. That kind of sense is very much important for the students who believe mathematics is very difficult subject to follow and not relevant to real life. $\endgroup$ Oct 9, 2023 at 9:40
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    $\begingroup$ You asked for opinions on incorporating history into marh teaching. I responded. With a supporting rationale. It may not be what you wanted to hear. But when you ask a general question on a soft subject topic, you should at least read and consider different opinions. $\endgroup$ Oct 9, 2023 at 17:03
  • $\begingroup$ Also, the ps shows disagreement with your predicate. $\endgroup$ Oct 9, 2023 at 17:03
  • $\begingroup$ As mathematics teachers, we should respect alternatives and I appreciate sharing your view on my request. If you check you will see revised version of Blooms Taxonomy is much relevant to the contextual teaching and learning process than conceptual teaching and learning. $\endgroup$ Oct 10, 2023 at 3:17
  • $\begingroup$ "It's the sprig of parsley, not the schnitzel." Lol. $\endgroup$ Oct 19, 2023 at 18:36
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I am not a mathematics educator. I might have become one had I not discovered the world of computers in 1981. This put me on a computer science trajectory and ultimately resulted in a successful Silicon Valley career. I can therefore not talk to theories of learning and if or how they should inform classroom instruction. I have not had to face the challenges of fitting various teaching techniques into a prescribed time frame outside of acting as a private math tutor during my high school years and as a TA for computer science at university.

My mathematical educational consisted of the classical chalk and talk plus textbook approach, which worked well enough through high school (I had a STEM focus in high school and graduated in the top ten percent of my class), but I struggled terribly with the math-heavy undergraduate computer science studies at university. I eventually realized which learning style best suited me: Study of literature accompanied by experimentation 1, always with a view to practical applicability of whatever mathematical technique is being acquired. Being able to put mathematics into a real-life context has always been a motivating force for me. Conversely, I have had a certain visceral aversion to acquiring knowledge when I could not perceive its practical utility.

One aspect of the history of mathematics that has been a motivating factor for me personally are biographies of famous mathematicians. This started around age 13 when our math teacher told us the famous story of how Gauss' mathematical talents were first discovered. Shortly thereafter I found an old book in the school library relating lives of some important mathematicians. I specifically remember Cardano and Galois. As I recall it was a very Euro-centric and exclusively male selection. Not until a considerable time later did I learn about Ramanujan or Agnesi, for example. The fact that mathematics was advanced by people from all walks of life made it more relatable to me: Mersenne was a monk, Fermat was a lawyer, Cholesky a military officer. The humble origins of Gauss, Lambert, Bessel, and others particularly resonated with me.

I doubt that adding elements of the history of mathematics to a classical math curriculum will improve test scores. But based on my personal experience I think it can make math a bit more enjoyable, with the effect lasting beyond school and university. As a practical aspect, it may contribute to students not dropping math classes at the first opportunity. Because math was such a painful struggle during my undergraduate years (I almost flunked out of the CS program because of it), I elected not to take optional classes in differential equations, a decision I regretted twenty years later when that knowledge would have come in handy in my professional career.


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S. Breuer and G. Zwas, Numerical Mathematics: A Laboratory Approach. Cambridge University Press 1993
J. Borwein and D. Bailey, Mathematics by Experiment. A. K. Peters 2004
J. Borwein, D. Bailey, and R. Girgensohn, Experimentation in Mathematics. A. K. Peters 2004

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This is a narrow reply, limited to one aspect of math history that I have found effective in my teaching (college, and some high school, in the US). That aspect is showing that even professional mathematicians make mistakes, that various theorem proofs needed later revision. (And by implication, it is OK if a student makes a mistake.) A few examples:

  1. Kempe's incorrect proof of the $4$-color theorem. But his central contribution, Kempe chains, is essential for subsequent proofs.

  2. The discovery of the pseudorhombicuboctahedron. Lyusternik says: "It is remarkable that in the theory of semiregular polyhedra, which is over 2000 years old, there was a defect, which was recently discovered by the Soviet mathematician V. G. Ashkinuz. He discovered a 14th semiregular polyhedron..."

  3. The flaw in the proof of Cauchy's rigidity theorem, repaired a century later by Steinitz.

  4. Bricard's incorrect proof that eventually led to the Dehn invariant.

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IMHO, there are at least two "Histories of Mathematics". The first one is temporal. Usually it goes like that.

In year A, a not then known mathematician B wrote a letter to a famous specialist C in the field D containing the theorem E, to which the latter replied that it was "of little interest for the contemporary mathematics", so B switched to something else and was well on his way to develop the grounds of the theory F but at that moment the country G invaded the island H where both B and C lived and both were killed by the common soldiery during the siege of I. The theorem E was then rediscovered in year J by K, who, however left nothing but a cryptic note about it on the margins of the third volume of a twenty volume treatise by L (contemporary of A and B) of which only volume 3 and volume 13 survived the eagerness of the monks of the religious sect M to erase all traces of the heresy L professed in addition to his mathematical studies. Finally, E was rediscovered in year N by O, who accidentally happened to be the first representative of the race P to get the award Q for her work. The content of theory F was completely lost except for a few mentions in the works of R who lived S years after A and B and whose books were found by archeologists in some remote monastery T of the religious sect M where they were piled together with other literature of that time in preparation to be recycled for printing the booklets of prayers. Fortunately, the volcano U erupted just in time and buried the entire monastery T under the ashes before it happened. Some historians, however, maintain that R has never existed and his books were a fake created much later by a well-known alchemist, musician, and mathematician V to sell them as ancient rarities to the manuscript collector W to get out of debts. The most extravagant idea about what F might possibly be belongs to the well-known contemporary physicist X who maintains that it was a primordial version of Y, which was the mathematical foundation of the model Z of the Universe for which X got his Nobel prize.

If somebody wants to claim that the actual temporal history of mathematics is not as twisted, contorted, and random as what I wrote above, I will wholeheartedly agree. It is far crazier. Moreover, the more we learn about it, the crazier it becomes as far as I can tell. If one wants to memorize it, one immediately faces the problem of its Kolmogorov's complexity being proportional to its length. The only lesson one can derive from it in addition to what you can find in your childhood fairy tales is just the usual history mantra: "in addition to working hard, being persistent, having only moderate risk aversion, high power of concentration, and all other virtues from the fairy tales, you need to be in the right place at the right time doing the right thing to get an eternal fame".

The other history is logical. It is based on the idea that certain concepts are naturally related to certain other ones and the underlying ideas are the same across the results and even across the different fields. It ignores the names, the dates, and the biographies almost entirely (the mathematician names in it are just like the street names in towns with the idea that some person is behind every name, but most of them are just convenient identifiers of the locations used merely for the brevity, and the knowledge of the corresponding biographies is neither necessary, nor sufficient for navigating the map). It just takes all mathematics as a sum of knowledge and tries to build the simplest "Turing machine" (both the code and the tape content, but as we know they are indistinguishable once the universal machine is designed) that outputs all of it (or, at least, noticeable chunks of it). That is what we normally do when preparing to teach some ascending series of courses and that is what we store in our brains when we deal with the areas that we know really well. The Kolmogorov complexity here seems to be much smaller than the length of the output, so we can, say, go to the board and present the full course in measure theory in a non-stop way from the basic definitions to the most advanced topics covered in the graduate courses without a single look at any book or even a sheet of paper. The problem is that we have to communicate the whole protocol of the algorithm execution and the final output rather than the initial tape content to the students, so they have to build the initial tape from it themselves. The "universal Turing machine" here is usually called "mathematical maturity", which includes the ability to understand arbitrarily complicated mathematical constructs if they are presented from scratch and with full proofs (that is more or less equivalent to the ability of the universal Turing machine to read the tape content) but is not limited to it, and there is currently no way to build it directly in the brain either. They like to say that it "comes with experience", which doesn't explain much either. I haven't seen any good books on that kind of history and if my description of it seems even more twisted, incomprehensible, and sarcastic than my description of the temporal one, this is by no means surprising though I believe that I can explain what exactly I mean using an example of any topic of your choice I'm reasonably well familiar with.

So, what's the moral? For me it is that one can occasionally try to throw in some parts of any of these two (and, possibly, several more) possible "histories of mathematics" into the curriculum and see what effect it has on the students (I usually prefer to play with the second one myself) but it should be done with care and in small quantities. If you overdo the first, you'll not just overload their memory with unnecessary information, but may also interfere with their knowledge compression process by presenting things in the suboptimal for comprehension order and emphasizing random relations rather than natural ones. If you overdo the second, you'll soon discover that what you say is just incomprehensible to your students and the "historical" part of the message you want to deliver doesn't go through at all. Usually neither of them is included. However, a colleague of mine tried once to include elements of the temporal history into the analysis course telling the short stories about the lives of mathematicians whose names appeared in the course (mainly 17, 18, and 19 century ones). His account of the outcome was as follows (I cite his words nearly literally just translating from Russian)

I told my students the life story of X. When I received my teaching evaluations, one of them read "The teacher's accent is terrible and his lectures are hard, but he is a funny guy who told us many interesting stories. For example, when we learned about Y , he told us how during the Second World War he lived in a village with his grandfather and was sitting on his grandfather knee when the Germans went it and shot the grandfather"

If somebody can guess correctly what was the subject Y, who was X, and what was actually said of the events in the life of X that inspired this passage in the student evaluations, I'll send the first person with the right answer a small prize (the people who have heard that story before from me or from my colleague himself are, obviously, excepted from this offer).

Just my two cents.

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