Concepts and Procedures of Teaching

On this page, we list a partial list of teaching procedures and concepts for an inquiry-based classroom. We are convinced that there are still pieces missing, as we keep discovering more and more layers of our teaching.

  • Procedure: Exploration before Explanation.
    Concept: Students need to build their sense making on prior knowledge. (Note: This means learning happens differently for different students as they all have different prior knowledge. The teaching and learning has to be differentiated)
  • Procedure: Individual students present solution to the whole class. Type A: The expectation is that students present a “perfect” solution. Type B: Or, the expectation is to present the student’s current understanding, including misunderstandings, ideas, partial solutions, possible mistakes, or where they are stuck.
    Concept: For the audience–understanding and critiquing other people’s thinking, learning about alternate ways of solving a problem, For the presenter–confidence-building, taking and making constructive use of audience critiques, deepen understanding of mathematical ideas;
    Type A: Develop skill in delivering coherent, formally correct, professional presentations,
    Type B: Develop a growth mindset from working with productive mistakes.
  • Procedure: Facilitating whole class discussions: not explaining mathematical ideas, instead facilitating the students’ talking to each other.
    Concept: Mathematical authority and agency lies with the students. Whatever is happening must make sense to the students.
  • Procedure: Teacher arrives in class early and sits to chat with students.
    Concept: Develop rapport with students which creates the energy in the classroom with helps students be comfortable with taking risks and sharing their thinking. Teacher gains information to better support the individual students.
  • Procedure: Choose questions that have a low floor and a high ceiling, as much as possible.
    Concept: We need access for all students. If we choose a “low floor”, then everybody can get started. At the same time, some students have higher prior knowledge and need to be challenged so that they can learn at their learning edge. We want problems to be challenging since learning includes being confused and then making sense of the concepts. Also, all students need to be exposed to difficult concepts even though they learn at their own pace.
  • Procedure: Create early successes (by choosing problems that are “just right”).
    Concept: Students need to be motivated and feeling successful allows them to dive into more difficult problems later on. Early successes build students’ confidence and perseverance.
  • Procedure: Students are sitting in groups, often chosen to be homogeneous according to prior knowledge, working speed, curiosity, work ethic, verbal participation and personality.
    Concept: Students learn better when community support is available. They also work on many non-content skills here, like team work, listening to and critiquing other students’ thinking, standing up for their own ideas, asking questions, making sense of different ways of thinking about the problems, etc.
  • Procedure: Facilitator chooses and sequences presentations.
    Concept: Make connections between mathematical ideas, or making a big idea visible to the class.
  • Procedure: Individual students present their polished solution to the whole class.
    Concept: Audience–understanding and critiquing other people’s thinking, learning about alternate ways of solving a problem, Presenter–develop skill in delivering coherent, formally correct, professional presentations, confidence-building, taking and making constructive use of audience critiques.
  • Procedure: The facilitator is “just” asking questions. Subtle hints are only given so that students do not feel frustrated enough to give up. There is little evaluation of what’s “right”.
    Concept: One of the goals of inquiry is to let students be the mathematical authority in the classroom. They decide what counts as a correct solution. This helps students feel ownership of the concepts and solutions, to fully make sense of the concepts, and to feel confident in their ability to be mathematicians.
  • Procedure: Ideas/definitions/theorems/explanations/proofs are generated by students.
    Concept: As before, the mathematical authority lies with the students. The concepts make sense to students, and can be more easily remembered without just rote memorization.
  • Procedure: Students define classroom norms. The facilitator needs to establish a positive relationship with the students (depends very much on facilitator personality).
    Concept: there needs to be good rapport between facilitator and students for inquiry to happen. Students need to feel safe and respected to take the risk of being wrong in front of their peers and the facilitator.
  • Procedure: Facilitator provides a sequence of tasks, a developmental progression through content.
    Concept: Students learn mathematical content best in progressions that connect big ideas. The tasks need to be meaningful, rich, group worthy, and manageable (see early successes).
  • Procedure: Using journals, classroom conversations, and video clips to discuss growth mindset and productive mistakes.
    Concept: We believe that there is also a developmental progression through meta goals. For example, students need to feel successful to start believing in themselves. Growth mindset ideas will help keep them motivated, and productive mistakes will help them develop more perseverance. All these meta goals are necessary for learning to happen.