Agenda

 
 MERI - University of Haifa - January 4, 2018

I. Questions & issues

A good challenge should

be easy to get started on, drawing on likely knowledge and past experience of the student 
arouse curiosity – successful challenges often build on apparent inconsistencies or start from unfamiliar or seemingly unintuitive premises
offer an engaging environment in which the student is invited (and indeed enticed) to make conjectures 
invite extensions to more general and/or analogous situations
not be perceived as a question to which there is a single correct answer

Why use interactive images in posing problems?

I believe that problems should elicit performance rather than recall
performance implies manipulating the givens of the problem in some way
what can be manipulated depends on the representation used
problems with interactive images that can be manipulated can provide insight & feedback & affords the possibility of self-assessment


For the sake of this discussion...
Operationally distinguishing between tasks that probe skill, and those that probe understanding
                      [both skill and understanding are necessary - neither is sufficient]

Tasks that probe Skill – call on the ability to answer problems using the tools [i.e., actions] of a single representation. Think of performing such actions as "executing procedures".
     whereas…

Tasks that probe Understanding – call on the ability to map the elements of mathematical objects and the actions one takes on them across at least two different complementary representations –

     Specifically…

this means interpreting how the tools of [i.e., actions in] one representation are related to the tools of the other representations & how each aspect of the mathematical object in one of the representations is represented in the others.


Number skill task - find a common denominators for adding or subtracting 4ths and 7ths.
   Number understanding task - explain slopes of lines & lattice intersections in {denominator, numerator} plane click here


Function skill task – write several expressions of the form x2 + px +q that all pass through any given point (X,Y)
•  Functions understanding task – What region in the {p,q} plane corresponds to quadratics with real roots? Complex roots?    What is the shape of the boundary of the region? click here

Shape skill task - Construct 5 examples of rectangles with a given perimeter
  Shape understanding task – What region in the {perimeter, area} plane corresponds to possible rectangles? click here



II.  Posing skill problems 
        [for each subject compare & contrast 2 or 3 examples with this question in mind] -

"What questions could/would you put to your students based on this applet?"
           arithmetic - the mathematics of number
                 geometry - the mathematics of shape
                       algebra - the mathematics of function          Click here



III.  Posing Understanding problems 
         [for each subject compare & contrast 2 or 3 examples with this question in mind] -

"What questions could/would you put to your students based on this applet?"
           arithmetic - the mathematics of number
                 geometry - the mathematics of shape
                        algebra - the mathematics of function          Click here


going further these materials were drawn from a website on selected strategies for posing problems
. Click here to see more.

going still further
- a two course sequence on "Problem Posing with Interactive Images" will be offered in 2018-19 
in the Mathematics for Teaching program at the Harvard Extension School. Click here for more information.