January 9, 2003 (Addendum Feb. 3, 2002)

Invert and Multiply: the DIM Calculator

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.