Speaking three different languages: The Contextual, the Mathematical and the Physical languages: “Forcing” students to understand ‘their’ Mathematics

Dvora Peretz, Michigan State University

Abstract: I present an alternative approach for non-major basic math courses in which the students are constantly engaged in verbally explaining each step of their doings and their motivation for doing it, while using three “different” languages.

Here, the Learning Space is viewed as consisting of the Contextual, the Abstract, and the Physical-Abstract sub-spaces. These three sub-spaces differ in the kinds of reasoning that are employed in each of them, in the language that is used in each of them, and in the kinds of activities that are performed in each of them. The transition among them is done by goal-driven means of re-phrasing into the language of that used in the ‘new’ sub-space.

In this approach, the students use DRAWINGS to show HOW and WORDS to explain WHY in many reasoning writing-tasks as well as in many investigations of other’s understandings and other’s solutions. Consequently, the students’ final project was to initiate “Mathematical-Doings” in others, by challenging them with an appropriate problem, to analyze twenty of these Doings-solutions and then to present their mini-research in class.

I describe a class engagement that focuses on a fractions situation and I bring examples of students’ work, as well as their thoughts about the course.

I propose to present an alternative approach for teaching basic Mathematics course to non-major students. In this approach the students are constantly engaged in verbally explaining each step of their doings and their motivation for doing it, while using three “different” languages. In addition, they are constantly required to relate various representations and doings across situations and across concepts.

Here, a holistic stand endorses a multiple-dimension view of the Learning Space: the Contextual, the Abstract, and an intertwined sub-space: the Physical-Abstract sub-space. These three sub-spaces differ in the kinds of reasoning that are employed in each of them, in the language that is used in each of them, and in the kinds of activities that are performed in each of them. The transition among the three sub-spaces is done by goal-driven means of re-phrasing into the language of that used in the ‘new’ sub-space. This perspective also provides the frame for relating the contextual concepts (for example, Altogether), the abstract-mathematical concepts (Addition), the Abstract-Set concepts (Union), and Physical-Abstract concepts (Join).

In addition, we used an abstract model, which conceptualizes all basic mathematical situations as the same, to lead, to channel and to mold the reasoning. Moreover, with the intention of developing the student’s awareness of their meta-cognitive processes we used the model to compare the same/different situations, reasoning and doings. Thus, by situating the student’s prior arithmetic knowledge in a meta-frame we help them to see part of the “Big Picture” and to better appreciate the kind of processes that makes Mathematics what it is.

In order to enhance the students reasoning skills I assigned many Reasoning writing-tasks as well as many investigations (trying to reason) of other’s understandings and other’s solutions. To illustrate, the students’ final project was to initiate “Mathematical-Doings” in others, young or adults, by challenging them with an appropriate contextual problem, and to analyze twenty of these Doings-solutions (using our triple-space approach). The students had to write a report for this as well as to present their mini-research in class.

So, I plan to present a class engagement that focuses on a Fractions situation in order to give the participant some understanding of what the proposed approach is all about, as well as to provoke a discussion about its significance. I’ll accompany my presentation with examples of students’ work, as well as their thoughts about the course.

Hence, I’ll show how we start the class discussion in the contextual space (story-problem, “real life” solution, estimating outcomes, etc.); model the situation (Abstract sets and elements); “translate” the goals (from the contextual to the abstract); analyze the abstract model (“fractions” of a set, “fraction” of an element, relations between sets, examining alternative models, etc.); discuss the mathematical description of the situation as well as the mathematical algorithm “to solve” it; “translate” the “Abstract” goals into “Physical-Abstract” goals (“make sets of a certain size”, “put elements equally into empty sets”, etc.); move “into” the Physical-Abstract space to “physically” do in order to achieve our “Physical-Abstract” goals; and then for the closure, we go back to the Contextual-Space to relate our “physical” outcomes to the situation (does it make sense? – fraction of a person?- a negative quantity?; Does it agree with our estimation? etc.); and then back again to the Abstract-Space to discuss the Mathematical Principles that underlines our “Physical” doings? (the commutative law that “helped” us or the “physical” meaning of “flipping” the Fraction, etc.).

It’s worth mentioning that the discussion in the contextual space takes place without any actual “doing” of writing, drawing or calculating anything, while all the doings in the Physical-Abstract Space are accompanied with DRAWINGS to show HOW and WORDS to explain WHY.

To conclude, I found this approach to be very efficient particularly in promoting students’ understanding of fraction’s situations. Actually, some of the students that first resisted the 'extremist' initiative eventually 'discovered' that ‘It’s so simple; I can't believe I did not understand it before’. However, this approach calls for persistence in implementation in order to be effective, as ‘It's too hard’ for the majority of the students, and for the most part they prefer ‘just tell me what to do and I'll do it…’.