Welcome to Math Designs

Improving Arithmetic and Algebraic Teaching and Learning

Math Designs Math Academy: Improving Arithmetic and Algebraic Teaching and Learning is designed to improve student’s facility and understanding of arithmetic, as well as, teaching arithmetic in a manner that lays a strong foundation for algebra. It is now widely accepted that we need to develop habits of mind that attend to the deeper underlying structure of mathematics (Kaput, 1999; Romberg & Kaput, 1998), as well as, a growing recognition that “algebraic reasoning can simultaneously emerge from and enhance elementary school mathematics” (National Council of Teachers of Mathematics, 2000). Algebraic reasoning includes: (a) the use of arithmetic as a domain for expressing and formalizing generalizations; (b) generalizing numerical patterns to describe functional relationships; (c) modeling as a domain for expressing and formalizing generalizations; and (d) generalizing about mathematical systems abstracted from computations and relations. (Blanton, Kaput 2000).

An example of an arithmetic idea that is foundational for algebra is equality. It is important for students to understand that equality is a relationship that expresses the idea that two mathematical expressions hold the same value, rather than just a signal to compute (Carpenter, Levi, Franke, Empson, Fennema, 1999). Nearly all manipulations on equations require understanding that the equal sign represents a relation. This definition of equality as a relationship allows students to perform operations on both sides of the equation while maintaining an equivalent relationship e.g., 5x + 32 = 97. If 90% of students who enter algebra believe that the equal sign is a signal to write an answer, then in algebra, they need to both rebuild their definition of equality and simultaneously make sense of new content. We can overcome this hurdle by elaborating the definition of equality in elementary school while learning operations, properties, inequalities, measurement conversions, etc. And then in algebra, students can focus just on new content. This is a win-win situation for both the elementary and algebra student. Success in algebra has enormous consequences for students and our economy.