Teaching+for+learning

Basic ideas about teaching for learning
Students have to learn the concepts behind the computations, not just the algorithms. If they don't learn the concepts along with the procedures, many (or most) won't remember the procedures into the next year.
 * First big idea:**

Important mathematical concepts cut across many procedures, so learning them in one context makes it easier to learn them in another context.

Important early mathematical concepts include: - What multiplication and division mean - What the distributive property is - Why the area of a rectangle is a good representation for multiplication - What a fraction means (there are several meanings) - What place value is and how to use it

Students need to learn mathematics in context, so they know why it is important, what it helps us do, and when to use it.
 * Second big idea:**

Real world contexts in which students solve problems not only deepen their understanding of why mathematics is important, but strengthen their problem solving abililties.

Also, contexts can give clues to students about how to solve problems. Early problem solving (grade K-3) is not about "key words" but about the action or relationships in problems. See Teaching Children Mathematics: Cognitively Guided Instruction by

The NCTM Process Standards are central to all kinds of mathematics teaching and learning. They include: - Reasoning and proof - Problem solving - Connections (making connections among ideas and representations) - Communication (communicating one's ideas clearly, explaining one's reasoning) - Representations (using different kinds of representations -- drawings, models, graphs, etc.)
 * Third big idea:**