Questioning and building procedural knowledge
- 1. PURPOSEFUL QUESTIONING AND BUILD PROCEDURAL FLUENCY FROM CONCEPTUAL UNDERSTANDING
- 2. Overview Review EffectiveTeaching and Learning Purposeful Questions Build Procedural Fluency from Conceptual Understanding
- 3. Review Match the effective math practice with the definition! Effective Practice Definition Meaningful discourse Solve and discuss task that promote reasoning and problem solving Uses goals to guide instructional decisions Establish math goals Using varied math representations High and low demand task Use and connect multiple representations Build shared understanding through exchange of ideas Give students opportunities to clarify understanding and construct via arguments. ImplementTask Student learn through situation goals within learning progressions. Visual, verbal, physical, symbolic, contextual
- 4. Pose Purposeful Questioning ■ Effective teaching of mathematics uses purposeful questions to assess and advance students’ reasoning and sense making about important mathematical ideas and relationships (NCTM, 2014)
- 5. TYPES OF QUESTIONS Question Type Description Example 1 Gathering Information Students recall facts, definitions, or procedures. When you write an equation, what does the equal sign tell you? What is the formula for finding the area of a triangle? 2 Probing thinking Students explain, elaborate, or clarify their thinking, including articulating the steps in solution methods or the completion of a task. As you drew that number line, what decisions did you make so that you could replace the 7 fourths on it? Can you show and explain more about how you used a table to find the answer to the Smartphone Plans task?
- 6. TYPES OF QUESTIONS Question Type Description 3 Making the mathematics visible Students discuss mathematical structures and make connections among mathematical ideas and relationships. What does you equation have to do with the band concert situation? How does that array relate to multiplication and division? 4 4 Encouraging reflection and justification Students reveal deeper understanding of their reasoning and actions, including making an argument for the validity of their work. How might you prove that 51 is the solution? How do you know that the sum of two odd numbers will always be even?
- 7. Funneling and Focusing Questions Funneling ■ Set of questions to lead students to a desired procedure or conclusions, while giving limited attention to students responses. Focusing ■ Teacher attending to what the students are thinking, processing them to communicate their thoughts clearly, and expecting them to reflect on their thoughts and those of classmates.
- 8. Teacher and Student actions Teacher ■ Advancing student understanding by asking questions that build on students thinking ■ Making certain to ask questions that go beyond gathering information ■ Ask intentional questions that make math more viable and accessible for students examination ■ Allow sufficient wait time so that more students can formulate and offer responses Students ■ Expecting to be asked to explain, clarify, and elaborate on their thinking ■ Thinking carefully about how to present their responses to questions clearly, without rushing to respond quickly ■ Reflecting on a justifying their reasoning, not simply provide answers ■ Listening to, commenting on, and questioning the contributions of their classmates.
- 9. Building Procedural Fluency from Conceptual Understanding ■ Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as the solve contextual and mathematical problems.
- 10. Fluency ■ Students are able to choose flexibly among methods and strategies to solve contextual and mathematical problems. ■ They understand and are able to explain their approaches ■ They are able to produce accurate answers efficiently. ■ Fluency builds on initial exploration and discussion of number concepts to using informal reasoning strategies based on meaning and prosperities of operations
- 11. Computational Fluency ■ More than memorizing facts or procedural steps. ■ Early works with reasoning strategies is related to algebraic reasoning ■ Composition (putting parts together) and decomposition (taking things apart)- leads to understanding properties of operations.
- 13. Learning procedures for multi digit computation needs to build from an understanding of their mathematical basis (Fuson and Beckmann 2012/2013; Russell 2000) Lets look-p. 43
- 14. 45 x 74 Array model OpenArray model Place value Conventional Algorithm