tag:blogger.com,1999:blog-31085516.post3876576513743791394..comments2024-03-27T03:13:12.133-04:00Comments on Change of Basis: Philosophy 101DocTurtlehttp://www.blogger.com/profile/15154912977859107986noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-31085516.post-71835210804572352082010-10-06T16:48:28.602-04:002010-10-06T16:48:28.602-04:00Patrick, I miss our discussions (on philosophy, ma...Patrick, I miss our discussions (on philosophy, mathematics, politics, linguistics to name a few) a lot, too! <br />Just to defend myself to devil's comments that you are relaying...:<br /><br />1. The language we use in math is, I agree, very unnatural, as we are not used to that level of precision (and our natural language partly reflects that), or the associated abstraction, in our everyday life. Once this language is mastered, though, I don't know. It's the same with all kinds of puzzles and (say board) games. Maybe the 'rules of the game' are strange, most of the time artificial, and often hard to comprehend or explain to others, but once people get them, they spend endless hours playing those games, or solving those types of puzzles. Unfortunately, we (as math teachers) do a very bad job at even explaining the language, so I have to take devil's argument with grain of salt, here.<br /><br />2. Again I believe that very few math educators spend as much time and energy as you in making sure that the students get the 'rules of the game,' (and to make it clear that math is fun) let alone develop "critical thinking" or "abstraction." In other words, it may be the case that these studies usually assume that these skills exist at point A when studying their "portability" to point B.<br /><br /> Cheers.Nick Galatosnoreply@blogger.comtag:blogger.com,1999:blog-31085516.post-55406832826500158812010-09-29T20:46:22.717-04:002010-09-29T20:46:22.717-04:00Nick: Wow! I've never received this sort of e...Nick: Wow! I've never received this sort of extensive feedback on a blog post before!<br /><br />You've given a lot of food for thought...I'd like at this time only to play devil's advocate regarding a couple of things you've said:<br /><br />1. Stanislas Dehaene makes a very cogent argument (see my post from a few years back, under the tag "Dehaene," on his book <i>The number sense</i>) that mathematical thinking is actually very <i>unnatural</i>, and that is why many (most?) people have a very hard time doing it. This might stand in contradistinction to your assertion that people think mathematically without thinking it.<br /><br />2. There are a number of studies (none of which, sadly, I have handy at my fingertips) which show that there is very little "portability" to mathematical thinking, including the sort of abstraction that you and I do in our research every day. That is: when attempting to translate the skills (in "critical thinking" or "abstraction") students pick up in our courses to other domains of knowledge, generally some or all of the efficacy of those skills is lost. This fact makes the argument that "math teaches critical thinking skill which we can all use in daily life" an untenable one. I've shied away from this sort of argument for "what we do" in recent years, because of this reality of apparent "domain independence."<br /><br />That's not to say we shouldn't do what we do, but that we should be cautious about how we defend/explain it!<br /><br />I'm definitely going to have to think more deeply about your insightful comments. I'd forgotten how exciting our philosophical discussions in grad school always were! :)DocTurtlehttps://www.blogger.com/profile/15154912977859107986noreply@blogger.comtag:blogger.com,1999:blog-31085516.post-4989002357794512872010-09-29T14:40:30.853-04:002010-09-29T14:40:30.853-04:00(...Continued) I am often surprised when during a ...(...Continued) I am often surprised when during a course in which I failed to provide the above-mentioned “motivation” there are many, if not the majority of, students (not “math-oriented,” by the way) who -- I suppose because of the other parts, like active learning, the discovery method, etc -- get very much into the course and end up being motivated. Motivated not because I linked the course to their lives, or provided any hint of applicability, but because, I believe, they “motivated themselves.” Or rather, because no motivation was needed. Just the enjoyment of discovery -- of pure discovery, and often of abstraction -- with no relevance to usefulness, or to their lives or to applications, is enough to get them interested.<br /><br /> There is an article, I forget by whom, that claims that math (beyond basic arithmetic) is not needed in everyday life by the majority of people. Exceptions include mathematicians, and other researchers. Most people, will actually even use a calculator for basic arithmetic, and definitely no farmer will ever have really limited amount of fencing and want to enclose a maximum area; and even if he needs to and even if he had Calculus, most likely he will not differentiate any function to find the maximum. Although I do not quite agree with that article, I am convinced that we do not (or at least I do not) teach math because the facts will be useful to the students in their lives, or because they are applicable to so many areas related to our lives, which they indeed very much are. I teach math because of the training it provides (to the human brain, and also to other aspects of human development), because it teaches abstraction, and because it is fun! I think that this last thing is what my students (who end up being “self-motivated”) discover (via the discovery method etc.). It is, I believe, inevitable. [By the way, I believe that all human constructs (computer hardware and software included) are designed very mathematically, just because math is natural in abstract thinking, and in some sense innate to us.] In other words, I appeal to very primitive instincts of my students: it all amounts to just exposing them to and revealing to them the math and beauty behind all those symbols and behind that foreign language we use. (After all I am a hedonist, as we have discussed in the past, and I treat my students as such, as well.)<br /><br />I think next time, I will make a consious effort, an experiment of sorts, where I will try to completely abstain from “artificial” applications, or any reference to usefulness, and just try to capitalize on the hedonistic nature of my students, who I believe will love the subject, not because they will think they will make use of it in their busy lives, but just because they will discover that it is fun. (As I said I believe that it will also be beneficial to their thinking, but I won’t mention that either.) I will let you know how it goes. Thanks for the opportunity you gave me to do some thinking.Nick Galatosnoreply@blogger.comtag:blogger.com,1999:blog-31085516.post-55386868603867478042010-09-29T14:39:46.540-04:002010-09-29T14:39:46.540-04:00Patrick, my enjoyment while reading your statement...Patrick, my enjoyment while reading your statement was only surpassed by my amazement on how much I agree with what you wrote. The fact however that what I was reading was almost the same as my teaching philosophy put me into introspective mode. Indeed, I am also very practical in my approach -- and your philosophy is most practical, contradicting your early disclaimer -- and also very concious of my attempts to motivate my students. This is the part that I want to criticise; not the discovery method, the cooperative environment, the active learning or the concept driven approach.<br /><br />Why do I try so hard to motivate my students? Why do I struggle to provide the sense of security and feeling of confidence that you referred to? I say, it’s because I teach to the unmotivated, or hard to motivate student. Not that the easily-motivated student does not benefit from this approach; she does. But she has not been my main focus, I realize. Why? Well I suppose because students like her are fewer than students of the other kind. I want to make this clear, though: the distinction between easy-to-motivate and hard-to-motivate is very, very different than that of likes-math and doesn’t-like math, or is-diligent and not, and the list goes on.<br /><br />Then, while reading your philosophy, I got worried. Do I provide a good service to the hard-to-motivate students? I mean the easy-to-motivate students would benefit either way, and the hard-to-motivate ones get interested in the subject, they become willing to learn and they do learn. But, do I reinforce in this way their attitude to need well-motivated introductions in things?<br /><br />The problem is of course systemic to our society. In a world of video games and instant messages for teenagers (which by the way reinforce limited attention span) people learn to prioritize things in terms of relevance to their lives, applicability and usefulness (three terms used in your teaching philosophy). Subconciously knowing that these are their criteria, I rely on that to be able to approach them and motivate them, by providing examples, applications and situations that indeed relate to them, are applicable and can be perceived as useful. But, even though I succeed in teaching them the material (and also, and more importantly, the methodology and the tools and the way of thinking about the material), I still do so by “cheating” them (the term used in a very, very loose sense). And I do it by reinforcing their “addiction”. (Continued...)Nick Galatosnoreply@blogger.comtag:blogger.com,1999:blog-31085516.post-29769079429209487252010-09-28T19:45:15.494-04:002010-09-28T19:45:15.494-04:00Love it.Love it.Dirk Awesomehttps://www.blogger.com/profile/07639768642507941040noreply@blogger.comtag:blogger.com,1999:blog-31085516.post-55420578345164774432010-09-28T19:01:35.763-04:002010-09-28T19:01:35.763-04:00"How my students feel about what they do is a..."How my students feel about what they do is as important as what they do in the first place." - I cannot agree with this more.<br /><br />Overall, what I really like about your teaching philosophy is that it is deeply practical. Everything is done with the ultimate goal in mind: to foster learning. It is stupefying how many miss this point: that the goal of teaching is to teach. Sometimes, our perception of what teaching *is* gets in the way of what teaching is supposed to *do*.Unknownhttps://www.blogger.com/profile/14727409924841796103noreply@blogger.com