Teaching for Mastery in Math

We completed a whole unit on prisms. Students worked diligently to explore surface area, perimeter of the net, and volume. We talked about cubes and rectangular prisms. Yet when some were asked to choose the definition of a prism on a test, they were unable to do it. Why?

As I think about this, I recognize the need to have students utilize the specific mathematical language in written explanations as part of math explorations. Explaining what they did in writing using key mathematical words will solidify vocabulary knowledge and understanding.

This reminder makes me think about the essential ingredients to math study. What is it that we have to include in order to help students master concepts? As I think of this, I can think of the following components:

• Introduction of the big idea of the concept. This invites students into the learning in thought provoking ways.
• Introduction to the concept with similarity and difference. For example teach what the topic connects with and also teach what is in opposition to. This helps to provide context for the learning.
• Explicitly teach the concept using models, real world stories, numbers, and words.
• Give students the opportunity to integrate the concept into engaging, meaningful, and explicit practice.
• Provide students with the opportunity to apply the learning into a real world context giving students a chance to construct knowledge.
• Let students teach the concept to others. This demands deep understanding.
• Have students and reflect and write about the concept using specific language/vocabulary. 
• Review students' work, provide feedback, and give them a chance to correct their errors, ask questions, an deepen understanding.
• Teach the concept with many modalities including music, words, images, video, story, building, manipulatives, number problems, online programs, games, and more.

The challenge to deep learning is often time. As educators we sometimes cram learning into the year in order to meet state and systemwide expectations. This cramming often does not result in deep learning or mastery. The challenge is to find ways to teach and embed the learning in rich, multi-modal, broad ways that result in mastery, engagement, and success.

This leads to the challenge of learning design--how do we work together to design rich, blended, and broad units of study that encourage students to reach mastery?

What would you add to this discussion as I want to continue to consult this list and grow this practice in order to teach children well. Thanks in advance for your ideas.