Mathematical Blasphemies [MBs]

(a + b)² = a² + b²
I call mistakes such as the one above Mathematical Blasphemies or MB for short. The inertial appeal of such mistakes for a typical math student is incredible. As instructors we must address these MBs with a variety of tools.  One such tool that I use is to put together a set of multiple choice questions that I call Clickerisms, whereby students use clickers to select the most accurate answer amongst a set of MB choices. A few of these Clickerisms that address various MBs will be presented in future blog entries (ISA!)

Meanwhile share your own thoughts on how you address MBs. Thank you

Response to Jonathan's Post

Jonathan at http://infinitesums.com/ wrote: "... There were no in class examples like it. Though I did mention that the largest exponent has control of behavior throughout the course of the material. ... Now the conundrum: is this a good test question because it showcases who internalized the concept completely? Or a bad test question because the answer is so obvious to me, the seasoned veteran, and hahaha can you believe these inexperienced teenagers are no match for me?"

0) Thank you Jonathan for sharing this information. 1) There need not be an example for every eventuality of any given math concept for we must leave room for students' brains to remain active and at edge.
2) Mentioning something does not necessarily mean students grasped it but it would be better if the students be expected to pull the leading term out and use limits over and over again to internalize the inevitable  rule. I noticed that the student did not show the limits approach in the sample shown, which means she/he just memorized the rule rather than understood its essence.
3) I think the question is good question for it helps us as well as the student to reexamine the essence of the concept at hand.
4) A former student of mine discovered a cool method to deal with these kind of problems while taking one my tests on the same subject. He figured that given any problem of this nature, he would have to always write both numerator and denominator Polynomials in such a way that they would always be in descending order and he would supply zero coefficients if necessary to match the upper and lower Polynomials degree-wise. This enabled him to always have only one rule to work with, namely that of equal degrees. He presented this to the class and everyone adopted it and started calling it Gordy's method. Pass this along to your students and let me know how it plays out.

Cool Free Trig Apps for You & Your Students

I have been covering trig as of late and there are three free apps that I recommend for everyone, students and teachers alike. Following are links to these three apps in order of Coolism rating.

Coolism: FunTrig

Cooler: Trig Wizard

Cool: iUnit Circle

3 Trig Functions Animated

Today in class students were treated to an animation that illustrates how the motion of a point on the unit circle that generates the usual graphs students see in their calculators when they graph the sine, cosine, and tangent functions. The artistic beauty of math shows up when the animation is left to proceed beyond a given set of cycles.

Cool Pi Article

The Huffington Post Science section has an article on Pi titled Are the Digits of Pi Random?

Share your thoughts about the article, its content, and how it may be used to enhance our students' learning and understanding of the underlying math concepts.

Think-Pair-Share becomes Nadji's Think-Pair-Teach-Present

I have always used think-pair-share or peer instruction as Eric Mazur would call it but back in April 26 of this year as I was about to have my PreCalculus students embark on yet another Trig challenge  when a variation on think-pair-share suddenly sprung up in my head. Here are the steps of what I like to call Think-Pair-Teach-Present.

A) Instead of assigning the same set of problems to all students to later discuss with their classmates, why not split the class in two (by having students call one and two) and assign each of them half of the problems in the set. 

B) Each student works individually on his/her assigned set of problems based on her/his number pick (1 or 2.)

C) Once the individual work is completed, all students with a given assigned problem set would meet with students who are working on the same set to discuss their respective solution processes, critique them, and arrive at an agreed upon set of solution steps.

D) Now, each student who has problem set 1 (and became "expert" on her/his set) gets to teach a counterpart student who has problem set 2 and then the roles get reversed.

E) Finally, one of these newly formed pairs of "teachers" volunteer to present the to everyone in the class what they were taught by their partner, not their own solution set, to ensure that the peer teaching process did indeed work.  

Please, give this method a try in your classroom and post comments on how it worked for you.

Math Abound

I attended my son's graduation from the University of Michigan and was fascinated by the abundance of math during the ceremony inside Hill auditorium. I posted one of the pictures from the ceremony to http://www.101qs.com/2176-curves-abound (check it out!)