Farmer Jack Farmer Jack harvested 30,000 bushels of corn over a ten-year period. He wanted to make a table showing that he was a good farmer and that his harvest had increased by the same amount each year. Create Farmer Jack’s table for the ten year period.

Research Does not provide answers Does provide information we can use to make informed decisions about the things we do in our classrooms Goal: Evidence based decisions

Novices and Experts You are a mathematician – an expert. How might the way you did this problem be different from the ways students would do it?

Teaching Preparing students to become experts in mathematics

Experts Recognize meaningful patterns of information Organize knowledge around core concepts or big ideas Have conditionalized knowledge Can easily retrieve knowledge relevant to particular task Understand what is easy and what is difficult for Are able to monitor their understanding and decide when it is not adequate

Organize knowledge around core concepts or big ideas What are the big ideas here? The basic principles? How are they organized or chunked?

Recognize meaningful patterns of information What patterns of information are here? How would you help students “see” those patterns? What questions might you ask?

Have conditionalized knowledge What prerequisite knowledge do students need to understand and do the task? What subset of ideas are relevant to this task?

Can easily retrieve knowledge relevant to particular task What “sorting” strategies might students use to help surface the particular knowledge required for this problem? What problem domain might include this problem and how would students know?

Understand what is easy and what is difficult for novices What difficulties might students encounter in doing the problem? What might you do to help students use their prior knowledge to make this one meaningful and to help them find the solution?

Are able to monitor their understanding and decide when it is not adequate How can you put different solutions together so that students will see where they are in their level of understanding? And so they can see where they might go in their way of thinking about the solution?

What do we know about teaching and learning? What does it mean to learn? What is important in thinking about teaching that will enable learning to take place?

The Teaching Principle Effective teaching requires understanding what students know and need to learn and challenging and supporting them to learn it well. (NCTM, 2000)

The Learning Principle Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge. (NCTM, 2000)

What does it mean to learn with understanding? Make connections Communication Reflection (Hiebert et al, 1992)

What evidence do we look for in our students? Flexible retrieval of important information Recognize the big ideas in contexts other than the original Make connections to prior knowledge /see relationships Describe how a concept works and what it means Recognize what they don’t understand and look for some resolution

Understanding vs memorization What does it mean to understand? What do we know about understanding? What kind of experiences have you had related to memorization and learning for your students? What do you try to do with them? What is the role of definitions in understanding?

Active vs Passive What are the kinds of things you want to happen in your classroom? Who is the center of what happens in the classroom?

Monitor learning and providing feedback What is feedback? Why is it important? What ways do you use to provide feedback? During class discussion? On tests and quizzes? On homework?