Textual descriptions in mathematics

Change lemma--> Parsing textual descriptions in mathematics

Parsing textual description is a grammar based strategy for analyzing statements in order to understand them. Although mathematics makes extensive use of formal language, it also represent statements as textual descriptions. While experts do, students tend to not parse textual descriptions even if the can do so in other contexts.

Decoding work done

Step 1: Identification of bottleneck

Given mathematical statements in natural language students find it difficult to formalize such statements. This is especially the case for statements in predicate logic and set theory which involve more than one quantifier or if sets are nested, such as in

There is more than one bottleneck in mathematics which is difficult for all students in class.

The set P of all subsets of the set {1,…,10}, which are disjoint to the set {1,…,5}.

Step 2: Description of mental tasks needed to overcome the bottleneck

To illustrate the issue let's consider the following description of a set:

The set P of all subsets of the set {1,…,10}, which are disjoint to the set {1,…,5}.

In order to make sense of this statement one has to understand how all its entities relate to each other. In particular this involves the question to what nount the relative pronoun "which" refers to. This could be "set P", "all subsets" or "the set {1,...,10}" in the main clause. Students tyipcally find it hard to decide while experts use their grammatical knowledge about languages to decide:

The relative pronoun "which" needs to refer to a plural noun as indicated by the verb "are". This rules out all possibilities except "all subsets". Therefore the relative clause qualifies the subsets to be disjonint to the set {1,...,5}.

The setP of all subsets of the set {1,…,10}, which are disjoint to the set {1,…,5}.

The different approaches of novices and experts are well aligned with the findings of Good Enough Theory in Psycholinguistics which posits that human beings tend to use

Interview at DiZ

Step 3: Modelling the tasks

An activity suggested in asks students to answer the following question in a Peer Instruction setting:

Consider the following description of a set: The set P of all subsets of the set {1,…,10}, which are disjoint to the set {1,…5}. Which phrase does the relative pronoun “which” refer to?

(A) “The set P”

(B) “subsets”

(C) “the set {1,…,10}”

(D) to something else not mentioned in (A)-(C)

(E) This is no unambiguous answer to this question.

Step 4: Practice and Feedback

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Step 5: Anticipate and lessen resistance

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Step 6: Assessment of student mastery

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Step 7: Sharing

Decoding work on this issue has been published in Lost in Language Comprehension: Decoding putatively extra-disciplinary expertise.

Researchers involved

Peter Riegler

See also

• Sentence processing on Wikipedia