**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

__p____erformance__ 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".

*w**hereas…*

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**