To learn more about the development of multiplication and division understanding in elementary and middle school, please view the following short videos:
Secondary math courses build upon and support the elementary focus of teaching for deep understanding before teaching procedurally. Many of the basic models introduced in elementary are re-visited and expanded upon in middle school and in high school allowing students to make connections with prior learning.
To understand the meaning, the use and connections between addition, multiplication, subtraction, and division;
To understand the "why" and "how" of operations and not just memorizing steps (what most of us learned in school);
To select from a repertoire of computational strategies (including mental computation) for each operation;
To be proficient mathematicians;
To promote students' capacity to think flexibly of numbers as sums and differences of other numbers with all mental and written calculations.
The procedure, also known as an algorithm, is sometimes a more efficient strategy for students when in written form.
As students make connections and show understanding of the operation conceptually, they begin using an algorithm.
Differentiated instruction in small group provides the flexibility for teachers to assess children's readiness to move to the procedure.
Nix the Tricks: Why we do not teach math tricks without teaching meaning.
NCTM.org: National Council of Teachers of Mathematic
National Academies: National Research Council, Adding it Up: Helping Children Learn Mathematics
References: Department of Education and Training of Western Australia (2007). First Steps in Mathematics.