Engagement in Mathematics Dave Ebert ddeoregonsd org www

Engagement in Mathematics Dave Ebert dde@oregonsd. org www. oregonsd. org/webpages/debert

http: //www. keypress. com/x 25196. xml http: //www. youtube. com/watch? v=b. RJi. CEgf. Z 3 Q

A Mathematical Question… If you fold a sheet of paper 20 times, how thick will the paper be?

We Know What Makes a Difference in Student Learning n Quality teachers and teaching n Access to a challenging curriculum n Schools and classes that are organized so students are well known and supported (Linda Darling Hammond)

Different Opportunities for Different Students The learning opportunities provided for low-ability, average-ability, and high-ability grouped classrooms are different. Students in these different groups are offered different tasks, curriculum, and instruction. (Jo Boaler)

Different Cognitive Demands for Different Groups of Students in high-track classes typically have access to “high status” mathematics knowledge, ideas, and concepts, where students in low-track classes tend to repeat the same basic computational skills year after year. (William Tate)

Quality Teaching Matters Teaching has 6 to 10 times as much impact on achievement as all other factors combined. (Mike Schmoker)

Features of Effective Instruction n n Tasks – conceptual engagement and productive struggle Talk – mathematical discourse (James Hiebert)

Mathematical Tasks n n We use too many low-level tasks We need more high-level, cognitively demanding tasks

Cognitively Demanding Tasks Provide opportunities for students to explain, describe, justify, compare, assess, make decisions, plan, formulate questions, and exhibit creativity.

High Performing Countries In high-performing countries, teachers place greater cognitive demands on students by encouraging them to focus on concepts and connections among those concepts in their problem solving. (James Stigler and James Hiebert)

In Our Classrooms Teachers make mathematical tasks more explicit by breaking them down into smaller steps, specifying exact procedures to be followed, or even doing parts of the tasks. This robs students of the opportunity to develop meaningful mathematical understandings.

Productive Struggle is Necessary Struggle means that students expend effort to make sense of mathematics. Struggle does not mean being presented information to be memorized or being asked only to practice what has been demonstrated.

What Not to Do Don’t use word problems in which the premise is false. Don’t ask questions that only a math teacher would ask.

(Glencoe Mc. Graw-Hill)

U is for Undertow by Sue Grafton “In both junior high and high school, I had trouble staying focused in classes where I was doing poorly, math being my weakest subject. ”

“A train leaves Chicago for Boston traveling sixty miles an hour, while a second train leaves Boston, speeding toward Chicago at eighty miles an hour. A bird flies back and forth between the two… and that’s as far as I’d get. I’d start wondering why the bird was behaving so erratically, positing a virus affecting the bird’s internal gyroscope. I’d daydream about who was on the train and why they were going from Chicago to Boston. Then I’d fret about what was happening in Boston that residents had crowded into the fastest train out. I’d never been to Boston, and now I was forced to scratch it off my list. ”

Mathematical Talk Communication is an essential part of mathematics. It is a way of sharing ideas and clarifying understanding. Through communication, ideas become objects of reflection, refinement, discussion, and amendment. (NCTM)

Make Questioning Central to Teaching n Why? n How do you know? n Can you explain? (Steve Leinwand)

Asking Why: n n Emphasizes that it’s not about the answer, it’s about the process Creates a classroom culture of justifying answers Emphasizes interpretation, not calculation Fosters a culture that respects and encourages alternate solutions

Asking Why: n n n Lets students put forth their own thinking Empowers our students Helps our students make estimates Allows our students to make conjectures and draw conclusions Eliminates our students dependence on their teacher

Pre-Planned Questions Using purposely prepared questions helps orchestrate productive discussions and maintain high-level cognitive demands. (Jo Boaler and Megan Staples)

Enhanced Questions n n n Tell me more about what you were thinking How did you decide that? Can you justify that? What steps did you take? What made you think of that? (David Johnson)

Questions as Lesson Planning Use questions as a basis of planning a lesson

Let’s Return To Our Mathematical Task n n What makes this a quality mathematical task? What could make this a poor mathematical task? What questions can we ask our students? What questions would we expect our students to ask?

What are Some Elements of Good Mathematical Tasks?

In Conclusion n n Engage our students Quality teachers make a difference We need to use more high-level mathematical tasks for all our students We need to question our students more