what does " one over cos square theta" look like?
what I "see" ...
$\displaystyle \frac{1}{\cos^2{\theta}} = \sec^2{\theta}$
probably because of the expression's use in calculus and its relatively easy antiderivative ...
$\displaystyle \int \sec^2{\theta} \, d\theta = \tan{\theta} + C$
you don't see any thing else? the statement " the yellow tulip " creates a vivid mental picture, I see a vivid mental picture when I think of this expression, just wondered if anyone experiences this
when you use certain types of mathematical expressions over long periods of time they can take on a geometrical meaning, particularly when they are geometric, this happens with allsorts of expressions, like A^2 or A^3, I did post this in math philosophy for that reason
I think I understand what you are getting at but in my experience math does not work that way. For example when you go through a chain of equivalent statements in solving an equation you cannot really expect to visualize each next statement and what it "looks like". It is exemplary if you can recognize the nature of the problem, whether you have sufficient information to solve it and the direction towards which to approach a solution ... that and avoiding dumb computational errors.
I think that gifted mathematicians do "see things" but more in the sense of recognizing architecture, or some heretofore unseen commonality between things apparently different. This is not to say that such things as the definitions of basic functions cannot be pictured in effective ways, and pointed out as being present all around us ... periodic motion (buzzing bee wings, flowers swaying back and forth, the earth turning, etc, linear, logarithmic, exponential relations, triangles in structure (look around for anything that leans or slants) ... but pictures replacing mathematical expressions, that would be tantamount to inventing a new symbolic language I think.
What does " one over cos square theta" look like? My first notion it's ... its a number, second, its a rule for assigning corresponding values along the number line, third ... well, there you go into its properties, its domain, range, the fact that its periodic, but really, unless it is the end product, has some physical correspondence that I am interested in it is just like some isolated word without context.
I know you are interested in pedagogical considerations. Here is a question for you ... if I give you a cosine, what have you got ... many answers but really I think you can find one in every point of view as well as slothfully standing about everywhere. (laughing with you only).
thank you so much for your input ray, I know this is only a tentative start, and I'm really looking forward to some more input, I want to give an example of this, suppose a large auger for drilling rock is being continually worn down in a tapering fashion, and requires the helical part to be cut away and replaced regularly, the information required to manufacture the replacement helix is, radius, radius/cos theta, radius/cos square theta,tortion/degree and pitch, I have described how a template can be made that contains this information, now this thin sheet metal template get's a hole drilled in it and hung on a nail on the wall, there to be gotten down and used at anytime in the future, now this template is four sided with one right angle, it clearly looks nothing like the section of helicoidal plane that is created from it, however this template can be held up to other augers to see if it is the correct one or not, ok so far nothing particularly interesting in anything I have described,
r/cos square theta happens to also be the answer to the question " I cut a cylinder at angle theta and an ellipse is created, what is the radius of curvature of the short semi-axis? so on one hand we have a question that perplexed the minds of Newton, Kepler and possibly Taylor or Mclaron, and on the other we have a rather rudimentary uninteresting four sided plain template, containing the blood sweat and tears of some of the greatest minds of the last five hundred years, so when I look at that template I "see" r/cos square theta, when I see the expression r/cos square theta it conjures the image of the template,
An interesting questions that goes further back. What do you see for 'one'? I have been thinking about whether it would help some students if they could see numbers as 'concrete' objects. There's a couple of ways I have been trying to realise this one of which is through this blog. Don't know whether this helps or not.
the template I describe seems to me to be a very strange thing, I don't know whether it is a true representation of what one over cos square theta actually looks like, I didn't know that it could look like anything, I always took it to be just a trigonometric function, I'd seen it graphed obviously, but I never thought of it as a geometric shape, and of course I can't help but think if that idea can be extended a little?
What is the level of the students you teaching/encouraging?
Your example is illuminating but still I think your question is like saying, "What is black?". The problem is that r/cos^{2}Theta is an answer to many, many questions. How long would you need to wait before anyone else presented it as a solution to the problem that you described?
Obviously, with your experience, you know a lot of math but now you are trying to solve a different kind of problem ... how to get others interested and motivated in a subject that keeps getting more interesting the further you get into it ... but ... how to get them started. I think that is what you are considering(?).
It seems to me that math with concrete applications is motivational. Seeing the world as a collection of phenomena many aspects of which are "of a pattern" that lend themselves to, what? ... descriptions of correspondences to elements on the number line rendered by operations within operations? Realizing THAT ... discovering how to match the pattern of a phenomena to a family of functions properly constrained to create a model, and then learning how to manipulate that model using identity transformations to present a view yielding the information that was previously invisible, THAT, is a valuable skill.
BUT ... and I am just arguing a point, not claiming the standing to spout the gospel … it seems that modern mathematics argues itself as a grand game that feeds from and nourishes the sciences and the engineering applications but is itself a separate enterprise. For example, a scientist locked in a metal tube in deep space wouldn’t have much to do, but a mathematician with a pencil/ paper and a comfortable place to sit could become engaged for eternity. (Not just saying rah rah mathematicians, there is limit to how long sitting is good for the human animal and we do not live in a metal tube.)
That mathematics is distinct from it’s application is not a pure point, one would could also write poetry in the metal tube describing ones inner life but no matter how deep, the circumscription would necessarily be severe.
Since this is a blab space and we do have as a focus, pedagogy, the question remains how does one reconcile math as method of exploring the world at one pole with math as a vast abstract game played by embroidering rules and studying consequences as the other pole?
Trying to lessen or remove the abstraction by finding ways to visualize expressions would be venerated if it were possible but, to repeat a point, isn’t that effort an attempt to find a new set of picture symbols for describing and manipulating ideas … whether generated by application or fancy. And actually isn’t that what modern math is all about, the reduction of symbol sets to the barest essence in order to provide a unambiguous lexicon? It is just that pictures have not been found efficacious.
Games have rule sets and strategies, one can discuss those things. Maybe math classes should have more discussion about rules and strategies, examining sets of problems and not just tunneling through the selected odd numbered problems at the end of the chapter. Sure one MUST solve many, many problems but how many math classes featured a review to discuss which problems were just repeats with mild variation and those which really added a new twist. How many math classes ask one to outline problem solving procedures as though one had a little pal to do the grunt work … work these problems to completion, supply an actionable outline for solving these problems?
How about challenging students to find ways to classify problems, getting them to engage not just in the search for an answer but the recognition that there is an underlying simplicity between certain problems that are not as different as they might first seem … isn’t that a mathematical pursuit too?
I respect your knowledge and your intent but perhaps you could illustrate the teaching problem you are encountering … is it something like, given a simple equation like:
ln(x) = 4,
the student just freezes?
How does one break down the elements of such a statement? Does one provide a real life application, a Cartesian graph, a set of transformation rules? What connects? How does one go from counting things on ones fingers to explaining the meaning of that statement?
For me (meager knowledge that I have) I would say the most important thing to recognize and that will get you out of your brain freeze is:
1) THERE EXISTS AN X. I think that is the single most important thing, we are talking about a number x.
2) X could have been anything but alas, it’s potential is constrained by being part of an equation. X’s mother had dreamed her son might grow up and join a function and generate a sequence of success but no, wah-wah, x is caught in an equation, and what is the result of that? Poor x is likely to have only one role in life, perhaps a little more. ok, ok … melodrama, hey that is what sticks.
3) Still not staring the equation in the eye, what next? X is hidden, we cannot see its value, basically we need to liberate it, bring x into the light. We need to pick the lock of its incarceration by applying transformation rules. (Wait, what is transformation rule? Time to step into a digression, a brief lecture on the existence of a hierarchy of operations and identities arising from Peano’s axioms, just the thrust of the idea is powerful, perhaps more powerful then becoming an expert.)
4) Ok, so then one plays the game, applies the transformation rule by composing each side of the equation into e^(_) and liberating x = e^4.
It seems to me that this sort of ‘at a distance’ abstraction of the essence of problem is perhaps more powerful then peering into the meaning of
ln(x) = 4 itself.
What I mean is the method of analysis I described is even more powerful then saying:
ln(x) = 4 is a statement that says x can be expressed as a product string of e’s four in length, i.e. x = (e)(e)(e)(e).
I find this second analysis (sic?) powerful too (yes, I am a simple guy). Note that this last insight leads to a solution in one step. What it does not do however is to leave me prepared to deal with other statements which might otherwise put me in panic if I did not have special insight. For example 1/(cos(x)^2) = 4.
Aaaaah, what is this strange thing, brain freeze! (not now, but before), Wait … X EXISTS, onward, incarcerated damsel in distress, transformation rules at the ready … unless Freddy ( the magician) has special insight, always appreciate a shortcut, a get out of jail free card.
Well, enough. But again, is there a particular teaching experience that might illuminate your desire to picture expressions?
Further note. You later wrote
“I don't know whether it is a true representation of what one over cos square theta actually looks like …”
I don’t think it looks like anything, it’s just a rule, a recipe, what it outputs, metal shavings, birthday cakes, a decoded message, just depends on what you are using it to model. As an application math is usually asked to provide some correspondence, some output for a given input. Math education hopefully provides a box of tools for making the connection … and insights into shortcuts.
Mathsedge:
I am not a teacher and it’s hard for me to focus on those blog examples, they seem to obscure the basic ideas in too much pretty detail.
I sort of favor the idea of a balance scale as the primitive embodiment of the equal sign, how you group the contents of each pan, pairing apples and oranges or just “like with like” really doesn’t matter, neither does factoring/partitioning the contents into special collections (boxes of fruits).
Regarding one. “one” implies separation, discrimination, the possibility of recognizing a pair as two “ones”, the possibility of counting by ones, the notion of equivalency in number regardless of kind … 3 horses, 3 ducks, 3 is 3.
Once numbers are abstracted from the natural world they can be regarded in their own right, their own inherent properties studied … isn’t it amazing that all counting numbers can be reduced to the product of those few “special” numbers named “prime”. Isn’t it amazing that one can count not just by ones, or two’s or three’s but by powers of tens or any other number. Isn’t it surprising that mirroring the natural world there exists both rational and irrational elements … those things understood by decomposition and those things always one step beyond “rational” comprehension? Isn’t it unfortunate that like a perverse invasion of entropy one can easily create a snarl, that might be impossible to disentangle … like solving higher degree polynomials, or encoded messages. Isn’t it amazing the range of problems that can be solved?
Well, back to the salt mines for now, there are many things I have only the faintest grasp of, fortunately I like the taste of salt.
I really don't know whether this imagery is effective. Hope you didn't take my previous comment as dissing your work. I am much in favor of illustrations, crutches, shortcuts, mnemonic devices while at the same time recognizing that in practice the manipulation of mathematical symbols is a lawyers game. Ultimately it seems to me we try and see the symbols patterns as pictures and come to recognize opportunities for their step wise transformation by the application of rules.
That being said there is still the question of motivation and the seeding of innovation and experimentation. Attempts such as yours are at the forefront of that effort. It may be that your blog is opening eyes, it may be that you will continue to become increasing efficient at what you are attempting, it may be that others will open up a fruitful dialog with you ... certainly efforts such as yours are the only means of advancing the educational cause. Salute.
The truth is neither do I yet, though now I am retired I have only tried it with my Grandson who seemed to cotton on OK. It seems, if only I could think of how to start it off, a discussion of the effectiveness of what I am attempting should be in a new thread! Thank you for the encouragement.