Abstract:

In this paper the grammar and semantics of the specialized variety of English used for tutoring simple algebra problems is described and rendered into grammar rules for a software parser. When tutors and students discuss algebra problems they use a sublanguage, where words and syntactic constructions have meanings that are dictated by the subject domain. For example, when discussing algebra, “multiply exponents” means the exponents on several terms are to be multiplied together, while “multiply exponents by 2” means each individual exponent is doubled. In ordinary English, by contrast, “multiply weeds by 2” has a third meaning: make more weeds. The grammar of this sublanguage thus contains special parts of speech and verb-complement relations. When embedded in an ordinary English grammar, we show this grammar can isolate and describe the algebraic discussion embedded within a matrix of ordinary English utterances, for example a tutoring dialogue. This work is in service of the future Wooz-2 tutoring software, which will enable the deployment of partial intelligent tutoring system technology within the context of a human mathematics tutoring interaction.
From 48 transcripts of keyboard-to-keyboard algebra tutoring the vocabulary of words that have specialized meanings for algebra was identified and verb case frames were constructed. These case frames encode both the surface positions and syntagmatic roles of the complements. For example, one case frame for the verb “multiply” requires a plural noun phrase direct object (items to be multiplied together), while another case frame for “multiply” requires an additional “by” phrase indirect object (multiply each operand by something). We identified 18 different general patterns of case frames in the algebra sublanguage, meaning that even this small language has 18 different verb subcategorizations when analyzed closely. From these, we have written approximately 110 algebra sublanguage case frames, where each variety of each verb is described by a separate frame.
The algebra sublanguage vocabulary and grammar were encoded according to the Link grammar formalism and combined with the general-coverage English grammar that accompanies the Link Grammar Parser. The parse tree for “proceed to apply FOIL to multiply the two binomials above” treats “proceed to apply” as ordinary English while the remainder of the sentence is identified as a particular algebraic operation performed in a particular way on identified operands. To evaluate the coverage of the grammar, we classified parse results into complete, partial (meaning the algebra part was correct), and incorrect parses. The grammar was applied to 480 attested test sentences, resulting in 78% coverage with complete and partial parses.

Keywords:

Tutorial Dialogue, Algebra sublanguage, Intelligent Tutoring Systems, Case frame, Link Grammar Parser.