# Thread: what do you " see" when you think about an expression

1. ## Re: what do you " see" when you think about an expression

ray I have been absorbing your thoughts for a few days now and have read that many times, i have always associated maths with pictures, sketches, drawings, graphs, algebra, chalk marks on the floor, scribbles on the backs of cigarette boxes, full-scale drawings, weird rabattements that no one ever understood or even believed were real, perhaps it's more to do with the impact that maths has on our personal lives, i recall getting up late and needing to be at an airport 60 miles away in one hour, the realisation that i would need to start getting beyond 92 miles per hour quickly set in, the mean value theorem very nearly cost me my life that day, i always associate it with being late, or motorway carnage, i could draw it i guess but it also represents a multitude of other things, it's funny how we " see" things, objects, situations every moment of every day yet find it hard to represent them in a true way, be it personal or otherwise, i am still none the wiser as to mathematical expressions, in the new year i will be attempting to show how to represent simple structures in plan, elevation and auxiliary plain, i want to show how to abstract from the drawing completely with the idea of cosines, i have attempted to show how to do this many times with absolutely no success whatsoever, there doesn't seem to be a graspable stepping stone between a drawing and the concept of a cosine, the concept of rise over run is never an issue, i don't refer to it as a tangent as this word is too confusing and sin's don't need to be mentioned either, ray i do appreciate that this sounds like a simple problem but it is not, measuring from auxiliary planes is simply not acceptable as we will be constructing real parts to a real building, i need some kind of way of " seeing" them on the drawing, i have learnt that people understand right angled triangle but fail to understand an auxiliary plain, then of course there is the concept of" angle who's cosine is" which seems to fly in the face of the first understanding, it may be simple to you and i ray but these two concepts are completely irreconcilable with each other in the mind of someone who has never met these ideas, the students have learning difficulties and very low levels of concentration, there is no way i can start waffling on about some circle such that, ultimately we will be drawing wood working joints with solid angles on them for cutting, i read somewhere that maths is the art of giving the same name to completely different things, cosine is a ridiculous word, it sounds puzzling to begin with, i need a new word for it, preferably which begins with cos and i need a word for it's argument, they are going to have to press that button on the calculator at some stage and it's argument has to have some relevance to the appearance of the index notation that will appear on the calculator, it just simply has to make sense ray, i must apologise for my word rabattement, apparently it isn't a word, perhaps i could use cos as a pneumonic? or the beginning of another word that describes a pneumonic that says what it does, remembering that the rule needs to explain that multiplying by it and dividing by it is a back and forth conversion, i know I'm asking a lot but i think this is the right place to be asking this question

2. ## Re: what do you " see" when you think about an expression

First, you are asking about pedagogy, not mathematics per se. Moreover, you are asking about pedagogy about a specific branch of mathematics for a specific category of people. I suspect that the more specific you can make your question, the better the answers you get will be. If the question is how best to visualize a cosine, the answers, when not facetious, are likely to be all over the lot. If the question is explaining to remedial students how and when to use the trig functions on a calculator when drawing practical conclusions from technical drawings, you may get no answers, but any you do get are likely to be relevant.

Second, Ray has mentioned the distinction between math as a technical tool and math as an intellectual inquiry. In my experience, few people can be motivated to make any intellectual inquiry at all, let alone sustained inquiry. You have to make the practical use of some aspect of math apparent to motivate most people. You seem to have decided that a visualization of a function will do the trick. I am not sure that any visualization other than a graph is possible, and I am not sure that, for the students you are concerned with, a graph will be helpful.

Third, it may be useful to think about two different issues. One is what do the three trig functions tell us generically. The other is how to use them in the specific field that you are trying to teach. I for one cannot help on the latter at all: I had to look up what an auxiliary plane is. You may decide that the first issue can be avoided if what you are teaching is how to set up a lathe from reading a drawing.

Fourth, if you do need to provide some general understanding of trig functions, I would try two things. I would treat them as constant ratios of the lengths of right triangles with equal angles. Why is that important? Because they are constant ratios, they scale up from drawing to thing. I do not disagree that the unit circle on a Cartesian plane is a simpler approach to sine and cosine as periodic functions over real numbers, but cartesian planes, real numbers, and periodic functions all seem to be concepts that will have no corellates in the minds of your students. The second thing I would do is to show them an old-fashioned table giving sine, cosine, and tangent for different angles, and tell them their calculators have the table built in. Show them how to get the cosine of an angle from the table and from the calculator. Show them how to get the angle corresponding to a value of the tangent from the table and frm the calculator. In short, make things concrete and pertinent to what they MUST learn.

3. ## Re: what do you " see" when you think about an expression

Follow up thought. I am entirely opposed to the idea that in order to learn arithmetic, you must first learn a little bit about groups and rings and fields, which historically came from generalizing arithmetic. It turns people off from learning any math: the stuff seems either trivial or pointless and has no obvious practical use. We would be far better off if people said, "I only know a little bit of math" rather than "I don't get math." It would be great if every machinist was an aficionado of number theory, but every machinist has to know how to interpret drawings.

4. ## Re: what do you " see" when you think about an expression

I completely agree with JeffM.

I got my MS in Math in the 60's. .After I had mastered Calculus III
and Differential Equations, there were courses in "abstract algebra"
which included set theory, Boolean algebra, groups, rings, fields, etc.

My first job was at a college. .Imagine my surprise when Math 101
turned out to be a "Theory of Arithmetic" course in which I had to
teach these concepts to incoming Freshmen.

This was, of course, the beginning of the craze for the New Math,
in which children would be taught to multiply in base-eight as well
as base-ten. .(See "New Math" by Tom Lehrer.)

My students (mostly Elementary Ed majors) balked at the material.
Of course, they asked, "Why do we have to learn this stuff? .What's
it good for?" .I had no good answers, but I did my best to justify it.

teaching traditional mathematics: PreCalculus, Calculus, etc.

5. ## Re: what do you " see" when you think about an expression

WARNING: Beer soaked rambling/opinion/observation ahead. Read at your own risk. Not to be taken seriously. In no event shall Sir jonah in his inebriated state be liable to anyone for special, collateral, incidental, or consequential damages in connection with or arising out of the use of his beer (and tequila) powered views.
Originally Posted by Soroban
This was, of course, the beginning of the craze for the New Math,
in which children would be taught to multiply in base-eight as well
as base-ten. .(See "New Math" by Tom Lehrer.)
Walter Warwick Sawyer "criticized the New Math movement, which included the people who had hired him."
Makes me wonder if fans of this movement were on lsd.

6. ## Re: what do you " see" when you think about an expression

i like many others inherited a form of drawing called the tangent method, I'm not sure whether it is " a geometry" in the mathematical sense or not, it does however have six definitions of tangent, none of which refer to it as a trigonometric function, followed by two statements about tangents which again have no correspondence with trig, I learnt this method before I knew anything at all about maths, A tangent is described as a straight line that is "tangent to" a curve at the point where the curve joins the straight part, " tangent to" is defined as being in line with, and tangent angle means the angle between two tangents, Plan drawings contain tangents, sometimes very many and they are referred to as tan m or tan kl ect, the true length of a tangent at theta in the plan would be tan m divided by cos theta, the height gained by a tangent in the plan becomes tan m multiplied by tan theta, I think this type of drawing may be where the term tangent was taken from, is it better to refer to one thing using two different words than it is to refer to two different things using the same word? you clearly see why I intend to use the term rise over run as opposed to tangent, this way of drawing was pretty much abandoned by myself and many others as there seemed no possible place for it in the work place, mathematics simply made it a redundant waste of time, in fact I'm horrified at the amount of time wasted on it in a commercial sense, having said that it is still used in modelling rising curved lines and mocking up shapes for investigation and has on more than a few occasions given me an insight as to how to tackle a problem, in the past it was used full scale but now we pretty much fit it onto a single piece of A4 paper, it folds up into a 3D model and is used for isometric projection from the plan drawing, it is the only possible way I can think of to " see" the plan in it's true way, this is at the heart of the matter, I want to somehow introduce the idea that cosines are a way of "seeing", cosines are very different from the other trig functions as they are the key in plan drawings, despite the fact that it is called the tangent method, tangents are useless here, if I hold a sphere at arms length, say a football and I cast my eye from left to right across the surface I am "seeing" lots and lots of little measured lengths, each one being divided by the cosine of the particular angle at that part, so seeing cosines is simply seeing shape, jeff I am definitely going to print some cos tables up, but rather than simply being a list of numbers I want to show how the eye is seeing these lengths so maybe with added pictures of the eye looking at an oblique length as a hypotenuse and the adjacent obscured below it so this way of seeing is completely obvious, I really appreciate this opportunity to get some dialogue going on this, jeff and soraban, i am self taught but i do remember spending time trying to grasp other branches of maths thinking that i maybe missing out on something, i very quickly dismissed set theory as being of little or no use, calculus has been my obsession, and ray, your post really inspired me to look again at a problem that has been bugging me for years, so thank you,

7. ## Re: what do you " see" when you think about an expression

I almost flunked mechanical drawing (and penmanship) when I was a kid so I am pretty lost in what you are talking about.

However, the slope of the tangent line to a curve is in fact the limit of the tangent of an angle in a specific triangle. So in one case "tangent" is being used as an adjective, and in the other as a noun. Then, however, "line" got dropped from "tangent line", and the adjective turned back into a noun, giving one noun with two different senses.

8. ## Re: what do you " see" when you think about an expression

Originally Posted by jonah
WARNING: Beer soaked rambling/opinion/observation ahead. Read at your own risk. Not to be taken seriously. In no event shall Sir jonah in his inebriated state be liable to anyone for special, collateral, incidental, or consequential damages in connection with or arising out of the use of his beer (and tequila) powered views.

Walter Warwick Sawyer "criticized the New Math movement, which included the people who had hired him."
Makes me wonder if fans of this movement were on lsd.
I wonder if such fans would have been taken in by what was ultimately found to be a cart before the horse approach?!

9. ## Re: what do you " see" when you think about an expression

I guess we are referring to modern maths, I think at that time we had just entered the computer age so I imagine it was supposed to prepare the curriculum for what was to become computer programing, what happened to it? is it still around? or were they simply bombed out of their minds on acid and watching far too much star-trek

10. ## Re: what do you " see" when you think about an expression

Yes, you are probably right... dawning of the computer age. Drugs had nothing to do with it I am sure, I mean really, what, an exotic dream put out in blueprint form?

Possibly if it was just a failure in pedagogy, an error in preparation and presentation, then perhaps it might have been a dream come true. How well illustrated were the texts? Did they connect to either the common notions of math as a presence in the world, or the root of the idea, about the counting game. What did finally happen? Was it tested and after four years a decision about curriculum and development swung away from the New Math? ... or did it make even the teachers, parents, administrators grumpy and so fizzle shortly after launch?

I think the one thing I have not heard yet, was the vision of its proponents, wouldn't some argue for the program, I mean if you are going to put out a new line of development then perhaps you have some accomplishment in mind?

Anyway, it might be interesting to look into it. Hey, perhaps the internet might assist.

Am still considering the thrust of your vision about ways to see math that engage a comprehension on some plane, other then the symbolic and ultimately the interplay between the two, that is sort of what I am getting is your aim, but of course very specific in application.

Perhaps the other "plane" is an application plane, food for some intuitive notion of the meaning behind an equation. But if that is the case, then a handy place to look is in math texts, screening for pictures, graphs, flags of importance ... for ideas which in the least might better define the enterprise in question. Good New Year to you MW.

11. ## Re: what do you " see" when you think about an expression

what does " one over cos square theta" look like?

Like a billboard saying

$one \ over \ cos \ \square \ theta$

12. ## Re: what do you " see" when you think about an expression

Originally Posted by Rebesques
Like a billboard saying

$.....one$
$cos \ \square \ theta$
....

13. ## Re: what do you " see" when you think about an expression

One answer might be: the complex exponential function, in all it's Eulerian glory. But such a reply does little to invoke the imagination of someone who has never heard of the complex numbers.

Here is how I think people "actually learn things": they observe, or perhaps are even told, some factual information. Over time, they see a "pattern" to the sea of facts. This lets them "forget the details", and "concentrate on the pattern".

At first, you may just know (from desperately difficult computation using your fingers) that 3 + 4 = 4 + 3. After doing this various times, with various numbers, you may suspect that a + b = b + a is *always* true. It may a relief, when a teacher (or a book) informs you that this is the case. You might wonder (or you might not), "when is this true...for what kinds of things?". Even being able to pose this question (which one might do at a very young age), the MACHINERY for demonstration is still some time off. In point of fact, "field axioms" (the above is just one example of them) are gradually introduced as one meets numbers of more intricacy (fractions, say, or the more-dreaded "square roots").

By the time high school has finished, one may know most of the field axioms, without ever having been told the concept of "field". This is fine, the "default fields" (the rational numbers, and their big brother, the reals), are the most well-known, and often used. Having some "concrete examples" is necessary for understanding the "motivation" for the "abstraction" of their structure. One learns about the appearance of the body, and the various parts of it, before undertaking a study of medicine.

However, once one is "mature" enough to do so, learning "abstractions" allows one to forget a great deal of "particulars", rules of computation are far more powerful than "doing by rote". So it's a 3-part process:

Exposure->Abstraction->Application.

"New Math" undoubtedly failed for trying to do "too much, too fast". It is unrealistic to expect young children to be capable of the kind of abstract thought required for abstract algebra. It is *not* however, unrealistic to expose said children to the concept of "permutation": (indeed, most high-school curricula have realized that combinatoric basics are part-and-parcel of making students "computer-savvy"), or to do simple things with small finite sets of objects (a pre-cursor to logic). "Pin the tail on the donkey" is an excellent illustration (exposure) of "correspondence", and certainly poses little intellectual challenge. There is an INFANT's toy that requires a small child to place geometric objects in matching holes, another example. Not ALL ways of teaching the underpinnings of "advanced subjects" have to be beyond the ken of mere mortals.

Unfortunately, the bitter after-taste of "New Math" has left us with "standardized test scores", something that encourages robotic behavior, unsuitable as preparation for anything requiring "real thought". Every child should be able to add "evens and odds" with at least as much facility as adding 3 and 4. They do NOT need the formal apparatus of an equivalence relation (that can be done later), but there's nothing "hard" about it. MANY "hard problems" actually are EASIER just by thinking in terms of "evens and odds" it's a near-perfect way to show "an easy abstraction" without actually going into the fussy details.

What was wrong (in part) about "new math" was the *way* it was taught, incompetently and incoherently. There should be more PLAY in learning. There was a video I saw, from an award-winning calculus teacher. His method-he would pose a problem. The students would try to figure out a way to solve it. Anything was allowed, if they wanted to find the volume of a cylinder by filling it with beans, so be it (personally, I would use pancakes-because, tasty!). Imagine the difference in our "standardized test scores" if math class was the class everyone looked FORWARD to, because it was lots cooler than reading Shakespeare.

Anyway, I digress. I see lots of things when I "visualize math"-sometimes the "written code", sometimes shapes/graphs, sometimes ideas which I form mental pictures of that there are no words for. Information can be encoded in a variety of different ways-one woman became famous for encoding algebraic formulae in the quilts and purses she wove.

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