*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