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 they are able to 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. As is often the case, here this can be done by only using grammatical (rather than semantic) information. In particular this involves the question to which noun the relative pronoun "which" refers to. This could be the phrases

• "set P",
• "all subsets" or
• "the set {1,...,10}"

in the main clause. Students typically find it hard to decide while experts use their grammatical knowledge about languages to decide (as revealed in a Decoding interview):

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

It is worth noting that in this particular example experts exlusively use grammatical information. No semantic knowledge of mathematics is needed. In constrast to that students seem to rely on semantic experience when relating parts of a sentence to each other.

These different approaches of novices and experts are well aligned with the findings of Good Enough (GE) Theory in Psycholinguistics. This theory is in contrast to traditional theories of language comprehension which assume that sentence processing is algorithmic and based on grammar. GE posits that human beings do not necessarily make use of syntactic structure, in particular when it comes to more complex sentences. They tend to use a semantic heuristic rather than syntactic algorithms such as parsing in order to obtain the meaning of more difficult sentences. This approach, however, is futile for learners as they cannot yet have access to semantic heuristics.

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.

At a certain point during Peer Instruction students will explain their reasoning to their neighbors. Most likely, some students will bring up the idea of parsing, possibly not named such. The modelling will then be done by one of theses students. Of course instructors can complement this by modelling how they parse.

Step 4: Practice and Feedback

The activity described in Step 3 above as well as the activities described in Step 5 below allow students to practice parsing and to receive feedback.

Step 5: Anticipate and lessen resistance

Parsing textual description of mathematics involves skills which students are more likely to attribute to language education than mathematics. In particular in STEM disciplines students might have uneasy memories about their language education. Hence, simply asking students to use skills taught earlier to them in secondary school might trigger emotional blockage. Note that the Modelling task described above avoids this as there is no need for the instructor to do so. Students able to parse will explain to their fellow students how to do it. Doing so, they typically do not refer to language education.

Possible resistance can be further lessened by group activities like the following:

What or who is lukized, i.e. what is “lukized” referring to?

1. The montillation of quasselties which are lukized has been bractered.

2. The montillation of quasselties which is lukized has been bractered.

3. The montillations of quasselties which are lukized have been bractered.

They can be used to show students that they are able to parse in situation where they do not have any semantic information.

Step 6: Assessment of student mastery

Assessment of student mastery can be easily done by using items similar to that shown in Step 3 above.

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