Achieving the Impossible: Cultivating Creative Mathematical Thinkers

One of my goals as a math instructor is to cultivate creative thinkers out of my students.  I have taught at different levels and different settings: traditional and project-based learning.  I have tried using different innovative technological advances such as ALEKS to supplement my teaching.

However, I am going to stand by my “old-fashioned” belief: the best way to teach and learn mathematics is via face-to-face interaction.  Furthermore, project-based teaching cannot be effectively accomplished without classical pedagogy.  I am not opposed to technology; in fact, I love it. I am not opposed to project-based learning; I think that there are great things about PBL.  I will admit: I am a devoted classical academic!  I believe that students must have strong conceptual understanding of concepts so that the knowledge stays with them a long time, if not forever.

I am a lecturer.  I would like to think that I am a dynamic lecturer, though.  I usually prepare NOTES ahead of time and project on the board.  Most of the NOTES I create are blank, leaving space for students to fill in their own interpretation of what they hear and see.  I do not begin introducing a new concept with its definition, but rather ask students to “stumble” on the definition or theorem on their own by working in small groups and finally sharing their thoughts out loud as a whole class.  (To give PBL credit: I learned this technique during my brief tenure at High Tech High, a project-based learning charter school network in San Diego.)  I find that asking students provide their opinions/guesses/inputs on what they are about to explore is instrumental to students’ deep learning journey.

For example, instead of telling students the relationship between exponential and logarithmic, I project unlabeled graphs of both functions and asked to students to write down everything they notice about the two graphs.  (By this point, students have learned about inverse functions already.)  It is a really neat to hear what students have to say about the two graphs.  Some responses are in the outfield, while others are wildly creative.  At least one or two students will definitely say that the two graphs are inverse function.  If not, I guide students through “discovering” such relationship by labeling the coordinates and have them go from there.

I have introduced logarithmic and exponential functions with this simple activity and it seems to spark so much interests in students.  They feel inspired that they “discovered” the inverse relationship on their own.  It’s definitely better me standing up there and boringly write: exponential and logarithmic functions are inverse functions.

Besides creating activities that promote real thinking in my classes, I also encourage students to be “loose” with their presentation of their mathematical work.  I mean, the steps they show must be logical; nevertheless, how they arrive at the answer is irrelevant to me.  Math students in this day and age tend to be very stuck with a certain way to do math, as if mathematics is a dull, prescribed study.  There’s so much creativity that can go on in solving an equation.  One does not merely minus 5 both sides to keep the equation balanced.  Students must ponder: why are we subtracting 5 both sides?  Who gives us permission do such thing?  Subtracting 5 both sides without explanation seems likes a magical move!  I call it “black magic.”  Students laugh a lot when I say that but they often agree.  Once students understand why subtracting 5 both sides make sense, I do not care how and where they subtract the 5: in the same equation or underneath the equation.  (Side note: it makes absolutely no sense to subtract underneath the equation, by the way.)

I will post more examples of my work and students’ thinking processes in future posts.  For now, I am busy breastfeeding and giving ALEKS a chance by doing their Student Experience demo.

Happy math-ing!

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