IBL Contributed Paper Session at MathFest 2013

Quick FYI -- If you are going to MAA MathFest (LINK), consider submitting an abstract and/or attending the IBL BEST Practices session.  MAA IBL MathFest Contributed Paper Session, Summer 2013

-SY

Inverse Pale Blue Dot

A departure from IBL topics for this post... 

I spent part of last week with my family taking a much needed break in Yosemite.  As we made our way back to civilization yesterday, we were rudely confronted with news of the massacre at an elementary school in CT.  My thoughts go out to those who have been affected by this terrible event.  

While in Yosemite, I made an image that I think is relevant.  I have always lived in urban areas so I rarely get to see the stars.  On this trip, one of the evenings was moonless and clear, so I took my camera and tripod to make a "wide field" night photo from Tunnel View.  At the time I was out there taking pictures, I thought of "A Pale Blue Dot" by Carl Sagan, and how this experience was the inverse of the Pale Blue Dot image.  Instead of the Voyager capturing an image of the earth as a single pixel, I was on the pale blue dot capturing a mass of stars and galaxies that make even a great national park infinitesimally small by comparison.

"There is perhaps no better demonstration of the folly of human conceits than this distant image of our tiny world. To me, it underscores our responsibility to deal more kindly with one another, and to preserve and cherish the pale blue dot, the only home we've ever known." -- Carl Sagan

Indeed it is all our responsibility to deal more kindly with one another.

Learning is Nonlinear

Learning is nonlinear.  

When a college student is learning to prove a sequence converges in Real Analysis or when a first grader is constructing the notion of grouping powers of ten, there is a period of mental construction that goes on where the ideas, language, problem solving, all that have to be put together coherently.  This process is highly nonlinear.  Certainly it isn't a matter of just transferring information from us to them.  Learning to do math is much more than say the simple act of memorizing your friend's phone number.  Learning a new idea takes time, energy, experimenting, mulling, messing around, all that.  Learning to creatively solve nonroutine problems requires even more of this kind of time.

What does knowing "learning is nonlinear" tell us about teaching?  One thing we can take away is that traditional courses tend to be set up linearly.  Section 1 on Monday, section 2 on Tuesday,... The factory model or industrial revolution genetics of our system become apparent here.  We line them up in rows, bolt on the parts one-by-one, and off they go.  Job well done, right?   

Linear teaching models are dangerously attractive, because they are easier to plan, follow the linearity of books, and institutions can be manufactured along these lines for less cost.  It's efficient on a cost basis, from the standpoint of human capital, because it requires little investment in faculty development and associated resources for teaching (e.g. technology, tools, materials, training, support).

Linear teaching is however inefficient in time allocation within class with regards to learning.  Students struggle on different things at different times.  Basic stuff is easily learned (even outside of class), and linear teaching may inadvertently allocate too much class time to basic material.

Further linear teaching may not allocate enough class time on harder material, which then conflicts with nonlinear learning.  When students need time and support to make it over the next tough section, the class moves on, ideas are half-baked, and misconceptions are formed or allowed to linger.  Students' foundation of thinking is not set upon bedrock, but rather a shifting, insecure base.

What is perhaps the most difficult part for a teacher regarding this linear vs. nonlinear issue (especially for traditional instructors) is the blind spot.  The blind spot I'm talking about is the not having a detailed understanding of student thinking.  If you don't know your students, you don't know where they are likely to struggle, other than through "inverse scattering theory," where in the past you've given quizzes or tests on a particular subject and students did poorly on certain items.  Of course it was too late for that batch of students, but at least the next batch will be slightly better off... Such issues are classic symptoms of traditional assessments, where your only learn about your students from tests (i.e. summative assessment).  Before you give an exam, you should be able to rank order your students (assuming a small enough class size), and then give a test that more or less confirms this.  If you can't do this, then you don't know your students well enough.

The ability to see the nonlinearity of learning is closely connected to student-centered teaching.  Giving students a chance to play and explore, gives us insight into their thinking.  (Just talk to them and ask them what they have tried on a particular problem.)   We use insights into their thinking to guide our instruction and do what is more likely to be useful as the next task.  If things are going well, students will just tell you, "I'm stuck on this..."  Great!  Now you know what the next move should be centered on.  

Linear assessment also takes the form of weighted averages used for grades.  If you rely solely on a weighted average of the hw, exams, final, and do not account for nonlinear learning, then this may be harmful to students.  If a student doesn't understand a topic in week 4, but figures it out by the end of the term, then how should the low score on first test be interpreted and used?  This is a topic of debate, and I am not going to say there is one right way to handle this.  But the point here is that you need to think about this issue and do something consistent with your values regarding education and what the point of education is.  Many faculty use a system where the final exam can make up for earlier lower scores at least to some degree.  This kind of policy is then an acknowledgement of nonlinear learning.  Let us now complete the circle by embracing nonlinear models for (student-centered) teaching!

Lecturing is linear.
Learning is nonlinear.
Teaching is nonlinear.