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<?xml version="1.0" standalone="yes"?> <Paper uid="C04-1196"> <Title>Understanding Students' Explanations in Geometry Tutoring</Title> <Section position="4" start_page="0" end_page="0" type="metho"> <SectionTitle> TUTOR SYNTACTIC PROCESSING SEMANTIC PROCESSING UPPER MODEL KNOWLEDGE </SectionTitle> <Paragraph position="0"> for geometry theorems. The results of the classification are passed back to the tutor.</Paragraph> <Section position="1" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 1.1 Example of Compositional Build of the Semantic Representation </SectionTitle> <Paragraph position="0"> To see how the compositional building of semantic representations works, let's consider the last step in parsing the sentence: The measure of a right angle is 90 degrees.</Paragraph> <Paragraph position="1"> A set of simplified knowledge base definitions necessary for building its representation, in Loom's definitional language, is: Based on these definitions and the rules of the grammar, the system builds the representation below for the subject of the example above, expressed in Loom's assertional language: (tell (:about measure-1 (:create Angle-Measure))) (tell (:about angle-1 (:create Right-Angle) (measure measure-1))) The system also builds this structure for the verb phrase: (tell (:about measure-2 (:create Angle-Measure) (unit 'degree) (value 90))) (tell (:about being&having-1 (:create Being&Having) (attribute measure-2))) The two structures are illustrated in Figure 2. Then the parser applies the grammar rule for clauses, given below in simplified form. Connectsemantics will assert an attribuend relation between instances being&having-1 and measure-1, relation specified in the lexicon as the semantic role of the verb's subject.</Paragraph> <Paragraph position="3"> Loom then classifies being&having-1 as an instance of the more specific concept Ascription, and this classification triggers production ascription-production. The production will combine the two measure instances, measure-1 and measure-2, into a single instance, resulting in the structure below: This structure is then classified against a hierarchy of concept definitions representing classes of possible explanations. A few of them are shown in Figure 4.</Paragraph> </Section> </Section> <Section position="5" start_page="0" end_page="0" type="metho"> <SectionTitle> 3 Specific Problems in Students' </SectionTitle> <Paragraph position="0"/> <Section position="1" start_page="0" end_page="0" type="sub_section"> <SectionTitle> Explanations </SectionTitle> <Paragraph position="0"> The basic approach we take to the content equivalence problem is to provide the right inference rules to make the logic system derive the same semantic representation for all sentences that are semantically equivalent. Below we present how this is done in some specific cases.</Paragraph> </Section> <Section position="2" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 1.2 Variation of Syntactic Structure </SectionTitle> <Paragraph position="0"> Even when the choice of content words is the same, the same meaning can be conveyed through a variety of syntactic structures. Some cases, like that of passive versus active constructs, can be taken care of in the grammar. Other cases require specific knowledge about the domain of discourse. One such situation is that of prepositional phrases attached in different places in the sentence, without changing the meaning, like in these examples: In a triangle angles opposite to congruent sides are congruent.</Paragraph> <Paragraph position="1"> Angles opposite to congruent sides in a triangle are congruent.</Paragraph> <Paragraph position="2"> Angles in a triangle opposite to congruent sides are congruent.</Paragraph> <Paragraph position="3"> Angles opposite to congruent sides are congruent in a triangle.</Paragraph> <Paragraph position="4"> The solution in our system comes as a concept definition that identifies the container relation at the assertion level, and percolates it down to involved objects.</Paragraph> <Paragraph position="5"> A similar case is that of using constructs specific to the domain of discourse.</Paragraph> <Paragraph position="6"> The measures of these two angles are equal. These two angles are equal in measure.</Paragraph> <Paragraph position="7"> Knowledge about the semantics of &quot;equal&quot; and &quot;measure&quot; is involved in determining that &quot;equal in measure&quot; means the same thing as &quot;measures ... are equal&quot;. We can model this knowledge by defining a rule that will identify cases of &quot;equal in some measurable quantity&quot; and will generate a structure with the meaning of &quot;equal quantity&quot;. (defconcept Equal-in :is (:and Equal (:some belongs-to Measure)))</Paragraph> <Paragraph position="9"> The use of relative and subordinate clauses can also lead to a large variety of syntactic structures without a significant change in meaning: The sum of the measures of complementary angles is 90 degrees.</Paragraph> <Paragraph position="10"> If angles are complementary, then the sum of their measures is 90 degrees.</Paragraph> <Paragraph position="11"> The measures of the angles sum to 90 degrees, because they are complementary angles.</Paragraph> <Paragraph position="12"> Complementary angles are angles whose measures sum to 90 degrees.</Paragraph> <Paragraph position="13"> These sentences all express the same theorem about complementary angles using respectively a single clause sentence, a conditional clause, a subordinate clause, or a relative clause. Because the semantic representation we build does not keep any trace of the original syntactic structure, such variations are automatically ignored. For example, the structure built for the first sentence is: All that is needed to achieve semantic equivalence is a reference resolution mechanism that identifies referents at the semantic level with their antecedents. In the example above the system would solve thing-1 to angle-1.</Paragraph> </Section> <Section position="3" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 1.3 Variation of Content Words </SectionTitle> <Paragraph position="0"> Many times differences in the content words used in the sentence do not make any difference at the meaning level. An obvious case is that of synonyms. However there are cases when different words are used as synonyms only in certain contexts. For instance: Angles ABC and BAC are equal.</Paragraph> <Paragraph position="1"> Angles ABC and BAC are congruent.</Paragraph> <Paragraph position="2"> Versus: The measures of angles ABC and BAC are equal. *The measures of angles ABC and BAC are congruent.</Paragraph> <Paragraph position="3"> Here the synonymy holds only when the objects involved in the relation are geometry objects, and it is not allowed when they are measures. We can make this distinction by defining &quot;congruent&quot; as a specialized case of &quot;equal&quot;: Moreover, we can add a production rule that will perform the inference that if the measures of some objects are equal, then the objects themselves are congruent. This rule will make the third sentence above be recognized as equivalent to the first two sentences.</Paragraph> <Paragraph position="4"> A related phenomenon is that of using very generic functional words in usual language to denote specific relations among the concepts of the domain.</Paragraph> <Paragraph position="5"> The angles of a linear pair sum to 180.</Paragraph> <Paragraph position="6"> The angles that form a linear pair sum to 180. The angles that are elements of a linear pair sum to 180.</Paragraph> <Paragraph position="7"> In these examples the angles are actually the elements of the linear pair. However in the first two sentences the relation is expressed either through a preposition, or through a generic verb like &quot;form&quot;. Recovering the explicit relation and thus being able to determine that the three examples above are semantically equivalent requires once again a model of the domain of discourse. We can model this first by defining the element-of relation as a more specific version of the generic relation belongs-to expressed by &quot;of&quot;. This definition will make the system build the same representation for the first sentence as for the third one.</Paragraph> <Paragraph position="8"> (defconcept Set :is-primitive Thing) (defrelation element-of :is (:and belongs-to (:domain Thing) (:range Set))) Second, we can define a production rule that recognizes a &quot;form&quot; configuration and asserts a &quot;belongs-to&quot; relation between the arguments, thus generating for the second sentence the same representation as for the first one: Another similar situation is that when students use the definition of a concept expressed in terms of more generic concepts, instead of its name. Adjacent angles on a line sum to 180 degrees. Linear angles sum to 180 degrees.</Paragraph> <Paragraph position="9"> The ability to recognize such examples as being semantically equivalent, with the right degree of generality, is conditioned by the possibility to model the definitions of those specific concepts within the framework of the system. This case can be dealt with by defining &quot;linear angles&quot; as &quot;adjacent angles on a line&quot;:</Paragraph> </Section> <Section position="4" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 1.4 Syntactic Ambiguity </SectionTitle> <Paragraph position="0"> Syntactic ambiguity in many cases does not reflect semantic ambiguity. One such possibility is prepositional phrase attachment. That is, following only the grammar rules, many times a prepositional phrase could be an adjunct/ argument of several preceding components. A deeper look at those alternative attachments reveals that most of them can be discarded because they do not result in a meaningful sentence. However, in absence of detailed knowledge about the meaning of the words in the sentence and their possible interactions, an NLU approach would not be able to disambiguate among them.</Paragraph> <Paragraph position="1"> The sum of the measures of the three interior angles in a triangle is equal to 180 degrees.</Paragraph> <Paragraph position="2"> The subject in this example contains three prepositional phrases: &quot;of the measures&quot;, &quot;of the three interior angles&quot;, and &quot;in a triangle&quot;. While the first one can only be attached to one place: the noun &quot;sum&quot;, the second one already can be attached to two places: &quot;sum&quot; or &quot;measures&quot;, and the third one can be attached to three places: &quot;sum&quot;, &quot;measures&quot;, or &quot;angles&quot;, resulting in a total of 6 different valid parses. By adding appropriate restrictions to the definitions of the concepts involved, our approach can make some of these combinations invalid during the parsing process. In our example we can restrict sums to only apply to elements that are measures, and thus eliminate the attachment of prepositional phrase &quot;of the three interior angles&quot; to &quot;the sum&quot;. And then we can restrict the containment relation to have geometry objects on both sides, and thus eliminate the attachment of &quot;in a triangle&quot; to either &quot;the sum&quot; or &quot;the measures&quot;.</Paragraph> <Paragraph position="3"> The presence of anaphora in students' explanations results in cases where sentences with different sets of words are semantically equivalent. Recognizing the semantic equivalence of such cases leads to the necessity to have an accurate reference resolution mechanism, which allows us to build the right semantic representation for the sentence.</Paragraph> <Paragraph position="4"> The resolution of referents to antecedents is done in our system at the semantic level. That is we simply try to merge the semantic representation of the referent with that of the antecedent. This mechanism has the advantage that the logic system will make sure that all semantic constraints associated with the two discourse referents are enforced, so that elements that are incompatible will fail the merge. This takes care both of the number restrictions, as well as all other semantic features, like taxonomic compatibility between the concepts involved.</Paragraph> <Paragraph position="5"> Finding the right referent for an anaphor is not always easy. Syntactic criteria can help with disambiguation among candidates, but there are cases where they cannot lead to a unique antecedent. Adding semantic constraints to the solution can increase the accuracy considerably.</Paragraph> <Paragraph position="6"> If the lengths of two sides of triangles are equal, then the measures of the angles opposite them will also be equal.</Paragraph> <Paragraph position="7"> In this example there are five possible candidates as antecedent for the pronoun &quot;them&quot;: &quot;the lengths&quot;, &quot;two sides&quot;, &quot;a triangle&quot;, &quot;the measures&quot;, and &quot;the angles&quot;. Constraints of the Binding Theory implemented in our system eliminate &quot;the angles&quot;, since &quot;them&quot; is a personal pronoun that has to be free within its local domain. Constraints on number eliminate &quot;a triangle&quot;, as being singular, while &quot;them&quot; is plural. Then semantic constraints attached to the definition of relation &quot;opposite&quot; can eliminate both &quot;the lengths&quot; and &quot;the measures&quot;, by asking that geometry objects can oppose only other geometry objects:</Paragraph> </Section> </Section> class="xml-element"></Paper>