According to The National Council of Teachers of Mathematics (NCTM) publication ‘Principles to Action: Ensuring Mathematical Success for All’ (2014) the 8 high leverage teaching practices that support meaningful learning are:
- Establish mathematics goals to focus learning
- Implement taks that promote reasoning and problem solving
- Use and connect mathematical representations
- Facilitate meaningful mathematical discourse
- Pose purposeful questions
- Build procedural fluency from conceptual understanding
- Support productive struggle in learning mathematics
- Elicit and use evidence of student thinking
Similarly the 2012 National Research Council report Education for Life and Work identifies the following essential features of instruction:
- Engaging learners in challenging tasks, with supportive guidance and feedback
- Using multiple and varied representations of concepts and tasks
- Encouraging elaboration, questioning, and self-explanation
- Teaching with examples and cases
- Priming student motivation
- Using formative assessment
Focus on rigor defined as a balance among conceptual understanding, procedural skills and fluency, and application with equal intensity.
Mathematics instruction should be intentionally designed and carefully orchestrated in the classroom, and should always focus on impacting student learning. Start with appropriately challenging learning intentions and success criteria. Teachers need to be clear about where their students are, where they need to go, and what achievement of learning milestones looks like. Good mathematics learning is embedded in discourse and collaboration – both with teachers and among peers – and is orchestrated around appropriately challenging tasks. Students should be doing more of the thinking and talking than the teacher. Must be partners in understanding with metacognition (thinking about their own thinking).
Surface, deep and transfer learning
Surface – initial development of conceptual understanding, procedural skills, and vocabulary of a new topic
Deep – begin to make connections among conceptual ideas, and practice and apply procedural skills with greater fluency. Plan, investigate, elaborate on their conceptual understandings and begin to make generalisations. Can facilitate transfer.
Transfer – ability to more independently apply deeply understood concepts and skills to new and novel situations