How to Learn Stipulative Definitions in Mathematics

Share Cognitive Science Implications for Teaching Mathematics
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Math is not learned like a language. Children can learn vocabulary and more complex syntax of common language by mere exposure. They can’t learn math that way. http://www.danielwillingham.com/daniel-willingham-science-and-education-blog/what-the-ny-times-doesnt-know-about-math-instruction.

The primary reason for this difference is the nature of definitions of the words in the two languages. Definitions in common language are lexical (they report common usage) while definitions in mathematics are stipulative (they stipulate the meaning).

Riccomini et. al. make the assertion that to improve students’ overall mathematical performance, educators need to recognize the importance of, and use research-validated instructional methods to teach, important mathematical vocabulary. The purpose of their article is to provide teachers with an overall understanding of the impact of mathematical vocabulary on proficiency and specific evidence-based instructional strategies to promote the learning of essential vocabulary in mathematics.

It is clear that learning the language of mathematics cannot be left to chance or mere exposure, it must be addressed directly and that begins with learning the definitions of important words.  Learning the stipulative definitions in mathematics is a multi-step and prolonged process.  I will try to describe one such process.

Step 1:  The teacher presents the absolutely correct stipulative definition.  The student must memorize this definition.  The presented definition will be the result of centuries of refinement by experts.  The idea of “stating in your own words” is foolish because experts have refined this definition to the point that even the slightest modification will make it incorrect. The student must memorize this definition word for word, even if it does not “make sense” to the student.  Understanding will come later.

Step 2: The student must put this definition into his/her implementation of Leitner Boxes to use of flashcards. See my blog posts for July 17, 19, and 24 of 2018.  Instead of this manual method, the student might implement a similar method using one of the online programs discussed in a few posts to be presented late in April 2019.

Step 3:  The teacher must present examples of the creature being defined.  Presentation of examples must occur at the same time as when the wording of the definition is presented and must occur again and again over a period of days, weeks, and months.  These definitions should be as varied as possible. Don’t use only integers, use all types of numbers, use simple and complex examples.  The teacher must constantly be alert to any opportunity to relate a creature to its definition.

Step 4:  The teacher must present non-examples of the creature being defined.  Repeat the discussion for Step 2 here.

Step 5:  The teacher must provide links to previously learned concepts.   Mathematics is a coherent well-structured collection of facts.  Neither the coherence nor the structure will be evident to the student.  The teacher must make these connections.

Step 6:  The teacher must provide and discuss all appropriate classifications.   First and foremost, the teacher must make it clear whether this new creature is a relation, an operation, or an object.  (See my blog post for 19 Feb 19).  Beyond this initial classification, the teacher should point out and use Venn Diagrams (or other diagrams) to show any other classification.  For example, linear polynomial is a type of polynomial which is a mathematical thing.  These classifications and the structure they provide are important to an understanding of the definition and its role in mathematics. Remember some creatures lend themselves to several classifications.

Step 7:  The teacher must provide and discuss any related previously learned knowledge.   For example, when introducing polynomials, the teacher should relate them to the integers.  This should continue as new information is gained about polynomials.  For example, The Division Algorithm for polynomials should be related to The Division Algorithm for the Integers. This is a bit difficult for the teacher, but it really is essential if your students are to learn the definition and its role in mathematics.

Step 8:  The teacher must provide and discuss a variety of exercises related to the definition.  This must occur again and again over a period of days, weeks, and months. 
As a means of interleaved retrieval these exercises should be mixed into discussions of future material.  Generally, the teacher must create these exercises “on the fly”.  For example, when a complex number appears in some class work, ask your students to identify the complex component, or compute the norm, plot the complex number in the complex plane, or any number of other questions that require retrieving information from long-term memory.

Final Step:  The teacher must recognize that all elaboration of a definition expands and deepens the student’s understanding.  Therefore, the teacher must continuously present elaboration at every opportunity. After some time the student will understand the concept so well that its precise stipulative definition will not seem to be memorized, rather it will simply be the only reasonable statement defining the concept.  By this time the student will also be able to provide elaboration of his own beyond those originally provided by the instructor or instructional material (See blog of 12 Apr 18).

It should be clear from the above that memorization is only the first step and that understanding is achieved only after other learning strategies are employed over an extended period of time.  It is important to recognize that all related study and learning is based on the exact precise stipulative definition.

It is also important that the teacher interleave these steps into other instruction which might happen in the normal course of events.  Nobody said teaching was easy.

Paul J. Riccomini, Gregory W. Smith, Elizabeth M. Hughes & Karen M. Fries (2015) The Language of Mathematics: The Importance of Teaching and Learning Mathematical Vocabulary, Reading & Writing Quarterly, 31:3, 235-252, DOI: 10.1080/10573569.2015.1030995 To link to this article:  http://dx.doi.org/10.1080/10573569.2015.1030995

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