Trig Ratios as Percents

My audience is mostly folks interested in math education in one way or another, so it’s no use starting this post off with “All you may know about trigonometry ratios is likely captured in the gibberish mnemonic SOHCAHTOA.” Your understanding of trigonometry ratios is no doubt more sophisticated than that.

But have you thought about trig ratios as percents? This will be enough for most of you:

sin θ = \(\mathtt{\frac{opposite}{hypotenuse} = \frac{?}{100}}\) = percent of hypotenuse length

It makes sense when you dredge up the 6th-grade math you remember and start making connections between it and the trigonometric ratios sine, cosine, and tangent (for example). After all, opposite : hypotenuse is the sine ratio, but it’s also just a ratio. If we think of it as a percent, we could say that if the sine of a reference angle is equal to 0.75, that means that the side opposite the angle in a right triangle is 75% the length of the hypotenuse. If the cosine were 0.75, that would mean that the side adjacent to the reference angle is 75% the length of the hypotenuse, since cosine is the ratio adjacent : hypotenuse. And a tangent of 0.75 means that the opposite side is 75% the length of the adjacent side, because tangent is simply the ratio opposite : adjacent.

The percent connection (or fraction; doesn’t have to be percent) strikes me as being immediately more useful for seeing meaning in values for trigonometric ratios. They usually go by students as just values which can’t be put into a sentence—a long list of changing decimals in a lookup table. Yet, the percent connection is right there, waiting for us to combine our middle school math knowledge with new material. We could model what this process of meaning-making actually looks like, rather than just ask them to go make meaning and hope for the best.

Of course, it also helps to be able to visualize what a sine of 0.75 looks like. Try, say, \(\mathtt{49^\circ}\) below on the unit circle and press Enter. That gives me something that looks pretty close to a sine of 0.75 (an opposite side that is \(\mathtt{\frac{3}{4}}\) the length of the hypotenuse, right?).

  θ = °

cos-sin-1
1-tan-sec
cot-1-csc

But the interactive tool, while helpful maybe, isn’t necessary, I don’t think. One can think about drawing a right triangle, say, with an adjacent side length about 80% of the hypotenuse length (a cosine of about 0.8). It will have to be longer than it is tall, relative to the reference angle, to make that work. The percent connection thus links a trigonometry ratio value to a simple and accessible visual.

An Example Problem: Testing Out the Percent Connection

41° 96 x

The basic mathematical (as opposed to contextual) trigonometry practice problem looks like this: Determine the length of \(\mathtt{x}\).

I can’t say the percent connection makes this a faster or more efficient process. What I would say is that knowing that the sine of 41° means the percent of the hypotenuse length represented by the opposite side length makes me feel like I know what I’m doing, other than moving numbers and symbols around. (Thinking about percents also gives us a way to estimate what my \(\mathtt{x}\) will be, if I know that the figure is drawn to scale.)

The sine of 41° is approximately 0.65605902899. With the percent connection, I know that this means that the opposite length is about 65.61% the length of the hypotenuse. It’s hard to overstate, I think, how useful it is to be able to wrap all of this number-and-variable work into one sentence like this: 96 is about 65.61% of x. I can climb the last few steps from there, by either dividing or setting up an equation—however the work happens, I at least have some background meaning to the numbers I’m playing with.

We can continue from there, of course (as we can without the percent connection, but so rarely do because the tedium of setting up and solving for the variable has overloaded us). The tangent of 41°, approximately 0.86928673781, tells us that the opposite side is about 86.93% the length of the adjacent side.

This guy gets it, and he seems to be the only one. It shouldn’t come as any surprise that he’s an experienced mathematics teacher a computer scientist who’s never taught. But, you know, it really should surprise us. Someday.


trigonometry

Searching the Solution Space

creativity

My reading in education has been a bit disappointing lately. This has everything to do with the relationship between what I’m currently thinking about and the specific material I’m looking into, rather than the books and articles by themselves. But Ohlsson’s 2011 book Deep Learning is so far a wonderful exception to the six or seven books collecting digital dust inside my Kindle, waiting for me to be interested in them again. The reason, I think, is that Ohlsson is looking to tackle topics that I am incredibly suspicious about, insight and creativity, in a smart and systematically theoretical way. The desire to provide technical, functional, connected explanations of concepts is evident on every page.

Prior Knowledge Constrains the Solution Space

Of particular interest to me is the idea that prior knowledge constrains a ‘problem space,’ or what Ohlsson wants to re-classify as a ‘solution space’:

A problem solution consists of a path through the solution space, a sequence of cognitive operations that transforms the initial situation into a situation in which the goal is satisfied. In a familiar task environment, the person already knows which step is the right one at each successive choice point. However, in unfamiliar environments, the person has to act tentatively and explore alternatives. Analytical problem solving is difficult because the size of a solution space is a function of the number of actions that are applicable in each situation—the branching factor—and the number of actions along the solution path—the path length. The number of problem states, \(\mathtt{S}\), is proportional to \(\mathtt{b^N}\), where \(\mathtt{b}\) is the branching factor and \(\mathtt{N}\) the path length. \(\mathtt{S}\) is astronomical for even modest values of \(\mathtt{b}\) and \(\mathtt{N}\), so solution spaces can only be traversed selectively. By projecting prior experience onto the current situation, both problem perception and memory retrieval help constrain the options to be considered to the most promising ones.

So, prior knowledge casts a finite amount of light on a select portion of the solution space, illuminating those elements which are consistent with representations in long-term memory and with a person’s current perception of the problem. It may even be the case that the length of the beam from the prior-knowledge flashlight corresponds to the limitations of working memory.

Crucially, this selectivity creates a dilemma. It is necessary to limit the solution space—otherwise, a person would be quickly overwhelmed by multiple, interacting elements of a problem situation—but, as is shown, prior knowledge (among other things) may restrict activation to those elements in the solution space which are unhelpful in reaching the goal.

It may be a goal, for example, for students to have a flexible sense of number, such that they can estimate with sums, products, differences, and quotients over a variety of numbers. Yet, students’ prior knowledge of working with mathematics can lead them to activate (and thus ‘see’) only ‘narrow’ procedural elements of solution spaces. The result can be that procedural mathematics is activated even when it serves no useful purpose at all.

This ‘tyranny’ of prior knowledge effects can be seen in the classic Einstellung experiments, a version of which is below—originally included in Dr Hausmann’s write-up on the topic, which I recommend highly. The goal below is to simply fill up one of the “jars” to the target level (the first target is 100). When you’re done, head over to Dr Bob’s site for the explanation. A similar obstacle to learning, described by S. Engelmann in his work, is called the problem of stipulation.

A: 21
B: 127
C: 3
Target: 100
0
0
0







But Creativity Theories Are Not Learning Theories

If one wanted to provide a slightly more serious intellectual justification for much of the popular folk-theorizing in education over the last decade—and then essentially replay its development, idea by idea—misinterpreting insight and creativity theories like Ohlsson’s would be an excellent strategy for doing so. (He never says, for example, that simply prior knowledge constrains the solution space, but that unhelpful prior knowledge does.)

It all seems to be there for the taking in these kinds of theories: the notion that solving problems is education’s raison d’etre, the idea that an unknowable future—rather than being just a fact that we must accept—can play a part as a premise in some chain of reasoning, the bizarre thought that removing instructional support can represent a game-changing way of restructuring the majority of learning time, a fluttering emphasis on collaboration and distributed cognition. All of this that has been humming in the background (and foreground) for a while in education fits comfortably and rationally inside creativity theories rather than learning theories.

From a learning-theory point of view, the problem of, say, thinking flexibly about number is primarily a problem of constructing better solution spaces—bring the goal within the flashlight’s view by instructing students (and thus making activation of ‘number sense’ more likely). Unfortunately, this requires a longer-term view, greater political will, and a bit of distance from everyday reality. Insight and creativity theories, on the other hand, assume that this number sense is already there but remains inert (students are only ever experiencing ‘unwarranted impasses’). The problem for insight theory becomes simply how to redirect the flashlight’s beam so that it uncovers the right knowledge. Along with the background assumptions listed above, these further assumptions of insight theories make them remarkably well tuned to the constraints of institutional teaching, both self-imposed and externally imposed. The practical work of teaching is still mostly a one-year-at-a-time affair, and sixth grade teachers, for example, do not have the time to remake solution spaces anew over the course of one year. What is within reach are redirection techniques suggested by insight theories. In this context, misinterpreting theories about insight for learning theories is practically inevitable.

Perhaps I’ll find out where Ohlsson makes learning theory and insight/creativity theory connect as I read further. But it’s worth noting that research Ohlsson himself conducted after the publication of this book has produced conclusions that run counter to certain predictions within it.

We’ll see!

Audio Postscript