Course 1:

Course 1: Mathematics as a Second Language
Target Audience: This course was designed for teachers of middle level mathematics,
however the majority of the course content is accessible and applicable to teachers of
grades K-8. Thus, training in elementary education is the only necessary prerequisite.
Graduate Credit Hours: 3
Usual Delivery Format: One-week, on-site, summer course
Text: Gross, H. & Gross, K.. Mathematics as a Second Language.
Description: This course lays the foundation for developing the “habits of mind of a
mathematical thinker,” a theme that is further developed in subsequent M2 courses. The
approach is to understand arithmetic (number) and (introductory) algebra as a means of
communicating mathematical ideas (i.e. as a language). The course will stress a deep
understanding of the basic operations of arithmetic, as well as the interconnected nature
of arithmetic, algebra and geometry. Attention is given to connections with other areas of
mathematics and emphasizes the development of an appreciation for the importance of
careful reasoning, problem solving and communicating mathematics both orally and in
One distinctive characteristic of the text (and therefore the course) is its use of what is
referred to as the “adjective-noun theme”. The authors of the text argue that “numbers
are adjectives that modify nouns (or other adjectives); and that an adjective in isolation
(without reference to a noun) leaves an incomplete picture. A second distinctive
characteristic of the text is its premise that arithmetic, algebra and geometry are
interconnected and that the study of mathematic should reflect their interconnected
Coarse Goals: The primary objectives of the course are (1) to provide a solid conceptual
understanding of the operations of arithmetic, as well as the interrelationships among
arithmetic, algebra and geometry; and (2) to develop the “habits of mind of a
mathematical thinker”.
Topics: include arithmetic vs. algebra; solving equations; place value and the history of
counting; inverse processes; the geometry of multiplication; the many faces of division;
rational vs. irrational numbers; and the one-dimensional geometry of numbers.
Instructional Style: As sections in the text are covered, participants work
collaboratively on problem sets that reinforce the material studied in that section. In
addition, participants work special “Habits of Mind” problems that challenge them to
develop their problem solving and adaptive reasoning ability.