Active learning

Page about active student learning and conceptual questions.

Student-active learning and peer instruction (TNB) 

Student-Active Learning (SAL) pedagogies are a range of techniques centered around the ideas of promoting a very active engagement by students in their own learning - in thinking about and discussing the subject.   For a review of the literature demonstrating the success of various techniques in this area across science (but not, unfortunately, maths) see Evidence for the Efficacy of Student-active Learning Pedagogies by Froyd. 

Deslauriers et al's Improved Learning in a Large-Enrollment Physics Class  (summarised briefly in The Economist and also in the BPS Research Digest) reports striking success for SAL techniques as compared to conventional lecturing.  Peer Instruction is a major component in a collection of techniques designed to facilitate "deliberate practice".  Another  paper demonstrates similar improvements for a quantum mechanics class.  It is pointed out there that retention of conceptual knowledge is hugely better than that of factual knowledge.  (Is this part of the definition of "conceptual"?) 

The alternative to SAL is the "conventional lecture". The Lecture System in Teaching Science by Robert T. Morrison gives a rather devastating account of this in the context of Organic Chemistry. Some interesting quotes from this: 

• Nothing in the world is easier than giving a nice, smooth lecture,... 

• Of more active alternatives: You don't waste your time doing what Frank Lambert calls "presenting a board of elegantly organized material with beautiful answers to questions that the students have not asked." 

• Surely it must occur to every teacher at some time or other that lecturing is a suspiciously easy way to do what must be a tough job. 

Carl Wieman's excellent five-part article "Why Not Try A Scientific Approach To Science Education?" compares conventional lecturing with "Research-Based Instruction" (generally more interactive approaches based on research in cognitive psychology and supported by evidence of effectiveness).   His summary at the end of part 5: 

(An interesting paper "Comparing students’ and experts’ understanding of the content of a lecture" by Hrepic et al is referenced above. The main message seems to be that students perceptions of what is in a lecture can be quite different from lecturers' and that a lot of the changes in understanding are not in the direction the lecturer may have hoped!)  

See also "Optimizing Science Education..." for Wieman's view of how the 21s century University can teach in a resource-efficient way.  

Just in Time Teaching (JITT) is a process whereby students answer questions on a web-based system just before lectures (or other sorts of teaching sessions), and those then influence what the instructor does.  often amalgamated with Peer Instruction as part of the process of monitoring student reading.  

What does a "conceptual question" look like in Mathematics? 

First of all, what does a conceptual problem (ConcepTest or conceptual exam question) look like in Physics, as proposed by Mazur?  Inspection shows that in general: 

• They are, as far as possible, worded in everyday language without technical jargon or symbols. (The point being that we do not want to cue some application of a remembered but possibly not understood formula.) 

• They can be done with little or no calculation, but do involve understanding basic ideas. 

• They are not small variations on given examples: they force the application of the material in novel ways or scenarios. (Of course, whether this criterion is satisfied depends on what students have already been exposed to.) 

• They are questions that we would fear putting on an exam because it would be "giving away marks for nothing", but that we would find in practice that if the question is "unseen" students would find them challenging. 

All this is directly applicable to Mathematics.   Some questions we already ask are conceptual, but we are perhaps disinclined to set them in exams.  Here are one or two suggested examples: 

• Calculus:  The width and height of a rectangle are changing. Right now,  the width is 5 units and the height is 4 units.  Also right now,  the height  is increasing at a rate of 3 units per hour and the width is increasing at a rate of 7 units per hour.  What is the current rate of increase in the area of the rectangle? 

• On Eigenvectors:  A is a square matrix with an inverse and an e-value \lambda.  B is a square matrix with an e-value \mu.   Which are always true?  is an e-value of \(A^{-1} \)

• \[\lambda+\mu\)  is an e-value of \(A + B\) \(\lambda + \mu\]
• \(\lambda\mu\)  is an e-value of \(AB\)

• On Limits:  which of the following is a good characterisation of a sequence \(a_{n}\) converging to \(L\): 

• The \(a_{n}\) get closer and closer to \(L\)  without ever reaching it. 

• You can make \(a_{n}\) the  as close as you like to \(L\) by taking n large enough. 

• \(a_{n}\)Each  is closer to \(L\) than  \(a_{n-1}\) is.