A Strategy that Fosters Translating Between Multiple Representations

Mathematical communication (Schoenfeld et. al., 1992, Koedinger & Nathan, in press) is increasingly seen as a goal for students, and in particular, translating between algebraic (e.g. "m/s+h"), English (e.g. "the total trip time") and concrete (e.g. "7 hours") representations. So far we have discussed students translating from the English to algebraic or concrete representations. The next two strategies explore asking students to translate in different ways including: 1) translating from the algebraic to the English and 2) within the English representation.

Subgoal Reification Strategy

The subgoal reification strategy is inspired by Corbett & Anderson’s (1995) lisp tutor that showed learning gains when the students we asked to post (or reify) their subgoals before attempting to write lisp code. We have added to our tutor the ability to choose a similar strategy. At T2 below, the student is asked for the name of the subgoal. At T3 the student is asked to explain in English how to achieve that subgoal. At T4 the tutor asks for the subgoal in algebra. T5 and T6 repeat the pattern from T3 and T4.

T1: Hello. [Again, doing the "bike-trip" problem] Please write an expression for the total time the trip took.

S1: help!

T2: In order to find the total time, what do you first have to find out? [The student is presented with pull-down menu to select a quantity-noun phrase]

S2: [menu:] The time actually on the bikes

T3: Correct. Can you explain how you would find the time actually on the bikes)? [Presented with the following sentence to fill in the blanks: "The time actually on the bikes is equal to (select quantity) (select operator) (select quantity)." The student can select the 2 quantities and the operator from menus. ]

S4: The time actually on the bikes is equal to [1st menu:] the distance biked [2nd menu:] divided by [3rd menu:] the speed of the bikes.

T4: Good. Now say what that [note the simple pronoun substitution] in symbols.

S5: m/s

T5: Correct, Can you explain how you would find the total time of the trip using the time on the bikes?

S6: The total time of the trip is equal to [1st menu:] the amount of time for the break [2nd menu:] added to [3rd menu:] the time actually on the bikes.

T6: Can you say that in symbols?

S6: b+m/s

T7: Correct.

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