Posted at Oct 29/2018 08:03AM by Stu 8:
Moore seems to suggest that understanding the process by which an angle is measured in radians often (if not always) involves a student conceiving of angle magnitudes and arc lengths as quantities. More specifically, Moore explains how students develop the understanding that, given an angle whose vertex is at the center of a circle, a radian is the percent of a radius magnitude that fits within the magnitude of the arc subtended by the angle. In quantitative terms, a student can conceive of a radian’s magnitude through multiplicative comparison of the magnitude of a circle's radius with the magnitude of a subtended arc. Through this discussion, Moore seems to argue that students form radians by using the circle’s radius as a unit for measuring the arc subtended by an angle. I’m wondering whether anyone has found a way of conceiving angle measure as a relationship between quantities without using multiplicative comparison. Every example I’ve seen in Moore’s article involves either a multiplicative comparison between a subtended arc and a circle’s circumference or between a radius and a subtended arc.

Posted at Oct 29/2018 03:38PM by Stu 4:
@Stu 8

I have the same article as you, and agree with your analysis of Moore's article. The author definitely provides convincing evidence that students are better able to reason with trigonometry when they conceive of angle measure as a multiplicative comparison.

This is the DIRACC calculus TA in me, but are you envisioning an angle as a parameter when you are doing these basic examples? If so, how do you make the transition in thinking when the angle becomes a variable (e.g. trigonometric graphs, polar graphs, etc)? At that point I consider an angle as either a point on the circle or a radius varying along (i.e. around) the unit circle.

I may be wrong on this... does anyone have a better idea than me?

Posted at Oct 29/2018 04:46PM by Stu 8:
Now that I have an initial idea of how someone can conceive of a quantity as a parameter (thank you Pat!), my hypothesis would be that Moore intends for the students in his study to (eventually) develop a conception of an angle as a variable, but that working through the initial examples he uses (gips, quips, ect.) helps students to conceive of an angle as a parameter. Gips and quips (and degrees) discretize angle measure; a student using them might think of an angle as a specific fraction of a circle's circumference, but the fraction's denominator will be fixed and will depend on the fraction represented by one unit of the metric the student chooses (even if the end-point of an arc is somewhere in between two gip-marks, say, the student can round to the nearest gip or approximate where the point is relative to the gip-marks). Radians allow more room for conceptions of continuous angle measure since the length of an arc can change continuously and since someone can choose a circle with a radius of any real-valued magnitude for a protractor.

However, I think that even when working in radians, a student may not immediately conceive of an angle as a variable because the student can choose nice, integer values for arc and radius magnitudes with which he will measure angles (how many pre-calc or even calc students would we expect to choose numbers like e or sqrt(2) to represent the value for their circles' radii?). In order to help students conceive of an angle as a variable, a teacher can choose "ugly numbers" for the values of these quantities to help students recognize that they the quantities they compare to measure angles can have any values (well, presumably non-negative real values).

Just for clarification, a gip represents an angle which subtends an arc whose magnitude is valued at one-eighth of its circle's circumference. A quip has a similar definition, but for one-fifteenth. These are units Moore appears to have made up for investigating students' conceptions of angle measure.