The article referenced at the end of this essay is excellent. Please read it. Most of the comments in this essay are rooted in that article.

In the opening paragraph the authors point out that throughout an individual’s mathematics education teaching the language of mathematics is an essential aspect of learning mathematics.

- Mathematical vocabulary affords access to concepts.
- Mathematical vocabulary is important to development of mathematical proficiency.
- Mathematical language is a pivotal component of mathematics success.
- A student’s knowledge of mathematical vocabulary can predict mathematical performance.
- Underdeveloped mathematical language skills impede overall mathematics learning.

These five benefits of learning definitions do not duplicate, nor do they refute the six roles I presented in the blog of 21 Feb 19 and repeat here.

Definitions in mathematics play several roles:

- They facilitate communication.
- They provide the foundation for new learning. All new learning requires a foundation of prior knowledge.
- They are essential to learning mathematics.
- They are the core element of every simple concept.
- They promote understanding.
- They facilitate applications.

Mathematical proficiency includes the ability to communicate and reason through written and spoken language.

In the 21^{st}
century teaching mathematics vocabulary is advocated by:

- The National Research Council,
- The National Council of Teachers of Mathematics,
- Common Core State Standards in Mathematics,
- Other organizations around the world.

Throughout my teaching career I emphasized overtly teaching the stipulative definitions found in mathematics. I was usually criticized for that position. It seems that research in the past dozen or so years establish that mathematical instruction in the area of language is imperative. Perhaps I am vindicated.

Cognitive scientists also provide solid justification for teaching and learning mathematics vocabulary.

“All new learning requires a foundation of prior knowledge.” Brown, Peter C. (2014-04-14). Make It Stick (p.5). Harvard University Press. Kindle Edition.

The learned definitions provide the foundation upon which additional new mathematics knowledge will rest.

“Research from cognitive science has shown that the sorts of skills that teachers want for students—such as the ability to analyze and to think critically—require extensive factual knowledge.” Willingham, Daniel T. (2009-06-10). Why Don’t Students Like School?: A Cognitive Scientist Answers Questions About How the Mind Works and What It Means for the Classroom (Kindle Locations 508-514). Wiley. Kindle Edition.

As learning progresses, a definition or family of definitions serve to hold together a large body of factual knowledge. These bundles are essential for critical thinking. They make it possible to engage in mathematical thinking. As a student learns more definitions, more bundles of high-quality factual knowledge become available from long-term memory. According to Willingham that contributes to successful thought.

“One of the factors that contributes to successful thought is the amount and quality of information in long-term memory.” Willingham, Daniel T. (2009-06-10). Why Don’t Students Like School?: A Cognitive Scientist Answers Questions About How the Mind Works and What It Means for the Classroom (Kindle Location 477). Wiley. Kindle Edition.

Please download and study the following paper.

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