one of today’s emails is from a math teacher i had at university long ago, with whom i didn’t talk since i was 19. seems that back then i was already complaining about the way maths are taught, and that at some point i did a very cool presentation on my ideas and approach. i long ago lost count on the amount of little battles i’ve fought in form of talks, presentations and lectures , so of course i don’t really remember that particular one he’s referring to. however, from what he tells me in his email, seems that it had quote some effect in him.
now that i think about it, my main point has always been the same: that during school, maths are formalized way too early. or perhaps, alternatively, that we never let kids use maths, play with them, and discover things by themselves before things get formalized. therefore they never get the chance to build intuitions. and trying to do maths without intuitions is like trying to find your way in a new city without a map.
my motivation to believe so comes from the observation that language learning, which usually is much smoother than math learning, happens very differently. basically, by the time kids are introduced to the formalization of language, to grammar and its syntax and structure, they already know how to speak. furthermore, they can perfectly read and write. not surprisingly, telling them that “a verb is an action”, “a noun is an object” and “an adjective is a qualifier” suffices to teach them the terms we collectively agreed on for speaking of language, because they already know intuitively what a noun or verb is. indeed, in language learning formalization happens to be trivial, painless, smooth, and natural, for kids are already familiar with language. hence the last step of organizing the knowledge within a framework is actually the easy part.
with maths, however, that’s no the case. that one terrible day the teacher arrives and announces “kids, we are going to learn trigonometry today” and introduces the notion of a cosinus and sinus, it is also the very first day kids ever get to examine angles and circles for the first time in their lives. crazy. no wonder this super-premature formalization of cosinus and sinus functions and their interrelations (described by weird formulas) alienates students to death. how wouldn’t it.
to make an analogy, the situation is pretty much like that of pretending to teach musical theory and what a major 5th or a minor 3rd sound like the very first day in their lives that the students see and listen to a piano. of course, it makes a lot more sense to let them first play music, often for many years, and only afterwards teach them its structure formally. any other approach to music teaching would be condemned for making no sense at all. obviously! yet, that’s exactly what we are doing with mathematics, throwing the whole thing at once to the students.
so, instead, what if kids had a chance to play with sinus waves, angles, circles, projections and shadows during 4 or 5 years? perhaps, that would allow them build their own intuitions and naturally understand these things. this could happen by means of visual problem solving, sound exercising, programming, drawing or any other method of direct manipulation of these objects. and what if after all of this playing, and only after the playing, we’d go for the formalization of the subject. my bet is that maths wouldn’t be any more difficult to kids than it is to learn language.
i believe that there’s a fundamental flaw in the current teaching method, and that it is one of the main reasons that prevents most people from grasping what maths actually are. and this is not a random believe, i’d say i can prove it. from my own studying experience during my engineering masters, which was heavily based in an “intuition construction” approach (including programming, osmosis, sleeping and relating things to other easier subjects), to all the people i ever met who i saw explaining the most abstract topics by using of very simple analogies and examples (topics that otherwise would have been considered difficult). such close examples have repeatedly proved to me the power of intuition and its role as the basis on which formal knowledge should be built upon.
so, taking into account that of course every person is different, and acknowledging the fact that every student has a different way to built intuitions – as some people are musical, others visual, others need spatial movement, others systematization, others narrative, others discovery – i think that a period of “playing with maths” is something that would benefit the grand majority if not all of the students. so, my suggestion would be to allow the students play for a few years indeed, and delay the formalization to later.
to make things even sadder, lets point to the fact that we are living in a world of computers, digital images and sounds, where not only we finally have the medium to make this new “intuitively playing with maths” paradigm possible and almost unavoidable. yet, we are avoiding it. seems to me digital devices are perceived only as a recreational gadgets. which is a pity, cause, perhaps, all those kids texting each other during the long math class are in fact holding the answer to the problem we are dealing with here right in their hands. this probably doesn’t need more explanation, for we all know that if there’s anything we humans have ever invented that is visual, dynamic, touchable, audible, interactive, intuitive, and well, a great platform for playing, that’s the smartphone (/computer). we do have the best intuition-making tool in our hands and pockets.
a second but not less attention worthy issue is that of aesthetical teaching. i usually refer to this as “sexiness in teaching”. i have the feeling that no matter how good the content of your message or story is, the format you present it in matters just as much in terms of effectivity.
these days we all expect to see beauty everywhere. not only we all want to look good or even be sexy (to the point people can get disappointed if they think they are not perceived as following the current standard of beauty), but we actually project the beauty as a value into the places we visit, items we consume and objects we own. indeed, the best selling devices and gadgets are sold for their design and beautiful lines not for their easy of use or technical features. cars are designed with very specific lines too which certainly don’t follow efficiency or any other technical criteria. every single movie and magazine undergoes heavy composition and color modifications to optimize their visual attractiveness. at more individual level, we no longer upload a picture to our online photo albums without passing it through color filters in an attempt to make it look “cool”. we as a society and as individuals, expect beauty everywhere, regardless of the quality of the object, subject, content or idea we are dealing with. a famous architect was putting it this way when talking about sustainability: the brutal truth to make sure an architectural piece is sustainable, in that it stays there for longs periods of time, is not to make it better solar powered or more heat efficient only – instead, the answer lays in making it beautiful, to be eye catching so it never gets aestehtically obsolete, that will ensure it stays there forever.
others forms of beauty are engagement and humor. of course no good speech can be expected to not start with a joke, then show strong passion and heart, and do explicit use of verbal repetition and reiteration of the key ideas in the speech. that, we probably could indeed call a beautiful speech.
so, coming back to the topic of teaching maths, my second idea for today is that the realizations of the importance of beauty in the format of whatever content you are dealing with regardless of its quality, probably should apply to teaching too. now, if these principles of beauty should be applied just as a direct and simple design issue in the material used for teaching, or if there are actually more profound ways to understand beauty in the teaching process is something i cannot foresee yet. what i do know is that a set of exercises in a blackboard or an online website are certainly not the sexiest way to present the otherwise beautiful ideas that math teachers aspire to teach.