The DIM 2003 Calculator is an interactive widget to help students learn
about dividing fractions. The intent is first to help them understand
why "invert and multiply" works (through a "missing factor"
approach), and then to provide opportunity for practice/consolidation.
A prerequisite for using this widget is a clear understanding of fraction
multiplication, including how to cancel factors before multiplication.

This is a "public beta" of the widget; the final version will
appear in the widgets section of MathDesign.

Design Notes. "Invert
and multiply" is one of the basic algorithms of grade school arithmetic.
I chose this one because, along with the long division algorithm, it has
the reputation of being one of the most difficult algorithms for students
to understand. I wanted to see if I could make it easier to see through
interactivity. I view the learning of algorithms as essentially an "understand,
abstract and consolidate" sequence, and tried to implement that here.
A side effect of designing this widget was the accumulation of a substantial
collection of notes about algorithms in general and how to present them
onscreen, which I'll try to enlarge upon in another post.

Interface. My general philosophy here was to make it difficult
or impossible for the student to make a mistake; hence, for example, terms
dragged to the wrong positions will not stay put. Standard computer interface
design recommends always allowing the user to undo the result of the immediately
preceding step. I've decided that, for teaching algorithms, it's probably
a bad idea to allow students to undo a correct step, so the widget "freezes"
any correct step. The widget also requires that any bad step be cleared
and corrected immediately, before proceeding. The original design had
additional buttons to be clicked to proceed to subsequent steps; I simplified
the interface by making the backgrounds of the windows clickable instead.
I chose to use an onscreen keypad for number inputs rather than the computer
keyboard to avoid problems with non-numerical entries and to avoid forcing
students to switch repeatedly between mouse and keyboard.

Feedback. The feedback for errors (e.g. trying to divide by
0) and other missteps is deliberately extensive in beginner mode and non-existent
(almost) in expert mode. The expectation is that students encountering
difficulties in expert mode will return to beginner mode until those difficulties
are sorted out. The feedback is also adaptive in that it reflects the
numbers and colours of the current example.

Colours. The colours are used to emphasize the pattern of the
numbers that results from the invert and multiply algorithm. To help maintain
focus on that pattern, most of the rest of the widget is black/gray/white
for contrast. The colours are shuffled for each example to avoid students'
abstracting a rule like "switch the red and blue terms" from
the examples. As per good usability practice, to accommodate red/green
and other forms of colour blindness, the colours convey only redundant
information (the numbers alone suffice).

Feedback anyone? I'd like to know:

Most importantly, do you think it would help students learn? Why or
why not?

Can you break it? I think I've got most of the bugs out, but there's
always another one, so ....

Does it behave as you think it should? or as you would expect it to?

Could the wording of the error messages be made clearer or otherwise
improved?

Any other comments or suggestions? How could I improve the calculator?

Thanks in advance for any input.

Note added Feb 3, 2002. Thanks to everyone who sent
feedback. The DIM Calculator now has its
own page. Any further feedback is still welcome.