Constructivism in math
By Murray Bourne, 23 Sep 2009
Monica wrote to me recently:
In some of my math rules, it says that cos(-t)=cos(t) and sin(-t)=-sin(t)Why is the cos not changed to -cos(t) like the sin function is changed?
Grab your calculator and try the following.
sin 35° (write down the answer)
sin (-35°) (write down the answer - what can you conclude?)
Now, try it again, this time:
sin (-88°) Did the same thing happen?
Then try these 2, with cosine:
cos 50° (write the answer)
cos (-50°) (what can you conclude this time?)
To check your conclusion consider this pair:
These sort of "rules" are given to you in your text book as a summary of what someone discovered in the past. This is a good example where it is better that you discover it yourself - then the rule will make sense and you are more likely to remember it too.
Of course, we cannot get students to discover everything they need to know (you would never get through all the content).
But when math lessons are 100% rules- and formula-based, it is not surprising to get a question like this.
See the 3 Comments below.
23 Sep 2009 at 10:45 pm [Comment permalink]
Hmm, seems to me that a better way to understand this than by a few examples is by the definitions. If t is the rotational angle, cos t = x/r and sin t = y/r. The opposite angle, -t, will have the same x coordinate and the opposite y coordinate.
You can see that, both with examples, and by using your hands to represent the terminal side of the original angle and its opposite. I bet there are some good animations out there showing this.
24 Sep 2009 at 8:23 am [Comment permalink]
Thanks for your input, Sue. Actually, I'd normally approach this from 3 or 4 angles, depending on the response of the learner. Some students will understand the example approach, while some will respond better to visual cues.
4 Oct 2009 at 3:35 am [Comment permalink]
You are right that there are many applets showing this.
The one at http://qpr.ca/math/applets/trigfuncs/Applet.htm allows the user to interact directly with the reference point on the unit circle - and I think that swinging it back and forth across the x-axis gives pretty strong kinesthetic reinforcement of the symmetry properties of sin and cos.