Things I would do again:

Eric Mazur’s peer instruction research group has spun off, which uses laptops or even smart phones as glorified clickers. The range of questions it offers far surpass what ordinary clickers can do. When I tried it this semester, for instance, I could ask questions that allowed students to sketch graph shapes or even draw force diagrams (although they did look a little silly). It supports a dizzying array of question types but is pretty easy to use. On questions that it can’t immediately grade as right or wrong, it provides a good summary of student responses. I felt that it complemented Modeling well and allowed me to get at thorny misconceptions, such as those associated with Newton’s 3rd Law.


Pros: many question types, built-in functionality to form pairs of students for peer instruction, can provide direct feedback to students, allows instructors to tag and share questions and to clone and edit questions created by others

Cons: requires laptops (cost and distraction factors), yearly license cost

Ordinary Clickers

Pros: one-time cost

Cons: limited question types

Mathematical Notation: Summation

In response to David Wees’ post about summation notation, I’d like to suggest that the terseness of mathematical notation is a godsend when working on long calculations, but it should, perhaps, be collapsible for experts and expandable for beginners.  What I mean is that while

\sum\limits_{i=1}^{6} i^2 or

\sum\limits_{i\in1\ldots6} i^2

may suffice for the expert, beginners may prefer

\mathop{\Sigma\text{um}}\limits_{i\text{ from }1}^{\text{to }6} i^2 or

\mathop{\text{Sum}}\limits_{i\text{ from }1\text{ to }6} i^2 .

I hate the linear form that David Wees mentions,

Summation (i, 3, 6, i2) = 32 + 42 + 52 + 62 = 86,

for it loses the spatial memory aspect of the original summation convention.  The real problem seems to be that this last expression serializes for computers well, but the other mathematics is hard to type.  I would prefer “smaller bits” of mathematics, like Sum and Sequence:

Sum[Sequence[Lambda[i,i^2], 1..6]] or even Sum[(i->i^2)[1..6]]

It would be nice if computers would do f[A] if f[a] is defined for every a in A without some kind of function like Map or Apply.  Here’s a longer version of the last expression:


In fancy LaTeX form that might be:

\sum (i\mapsto i^2)[1..6] or

\sum \left[(i\mapsto i^2)[1..6]\right]

if we want to make the operator precedence completely clear.  Here, “1..6” is some Ruby-like syntactic sugar to mean the set (really: sequence) {1,2,3,4,5,6}.