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Lucy West

Agenda Teaching with a Big Ideas Focus What is a Big Idea? What’s the difference in terms of teaching? Connecting Big Ideas and CCSS Place Value Designing a Lesson From a Big Ideas Perspective Examining Student Work Providing Actionable Feedback

What is a Big Idea in Math? A Big Idea is a statement of an idea that is central to the learning of mathematics, one that links numerous mathematical understandings into a coherent whole?

From Answers to Ideas On a farm there were 10 cows and 5 horses. How many animals in all? Make up a problem for which the following equation could be used: = 15 For the same problem, use a different equation.

Operations Meanings and Relationships The same number sentence (e.g. 12-4=8) can be associated with different concrete or real-world situations, AND different number sentences can be associated with the same concrete or real-world situation.

Big Idea # 5 Read the subset of ideas under Big Idea # 5—select one or more and either write or find a problem that could be used to bring out the idea(s). If you choose “bare number” problems, consider how they might be presented in ways that bring out the concepts rather than explain the concept.

Rich Tasks Combine Big Ideas There were 15 animals on the farm. There were cows and horses. How many of each could there be?

From Closed to Open A rectangle is 20 yards long and 10 yards wide. What are the perimeter and the area of the rectangle? What do students need to know and do. Use the information above to write an “Open” questions with a much higher cognitive demand.

From Closed to Open I want to make a garden in the shape of a rectangle. I have 60 yards of fence for my garden. What might the area of the garden be? What will students need to know and do?

Variation of Garden Problem Given a diagram of a fenced in rectangular garden plot with dimensions six meters by eight meters, find its area and perimeter. 3 Design a second garden plot using less fencing, but providing greater area. Design a third garden plot using more fencing but providing less area. Of all possible rectangular designs using the original amount of fencing which provides the greatest area? 8

How Do We Increase Cognitive Demand of Typical Math Tasks? Typical QuestionOpen Question A polar bear is about 20 times as heavy as Ali. If Ali weighs 125 lbs., what is the approximate weight of the polar bear? A polar bear weighs about 2500 lbs. How many children do you need to have the same weight? What might students do to solve the open question?

What are the characteristics of the open question? No fixed “key word”—can be division, multiplication, repeated addition, or ratios Arouses natural curiosity—a real meaningful problem Not all data is given—have to make assumptions; estimate (mathematize) Make connections to themselves—use own weight as a benchmark Activates reasoning—sense-making is central

How do you construct rich questions? Just take the answer to a closed question and make it the question. Try it with the following: $2.00-$1.66 = $.44 Round off $1.29 to the nearest tenth Which fraction is less, 3/4 or 4/5?

Were your questions something like: The difference between two amounts of money is $.44. What might the two amounts of money be? OR $____ - $_____ =$.44 What numbers might be rounded off to 1.3? Name as many fractions as you can that are less than 3/4. Explain how you know they are less than 3/4.

Subtract What comes next and why? Why does this work? Write down a reason for why this work, share it with a partner and refine it.

Analyzing Student Work Purpose—to decide what to focus on and/or give feedback Describe what you see—no inference Understand—what does the student know? Wonder—what are your questions? Focus—what is one mathematical idea or quality of work expectation that you want the child to revisit? Feedback—what question might you ask this child to prompt action toward learning or improvement?

The Process Students solve the problem Teacher (and coach) examine the student work from a strategies and big idea perspective Teacher (and coach) select 2-3 pieces of work to share in the whole group conversation the next day based on strategies and mathematical ideas A student’s work is projected for the class to see Children read and describe the work to understand the student’s thinking

The Process The student who created the work then talks about his work and addresses questions and contradictions Then the class discusses what needs to be on a finished piece of work Then the student goes back with the same piece of work and revises it Next day the teacher poses a new problem and the process is repeated. Day two they look at different strategies and compare and contrast

Analyzing Student Work Purpose—to decide what to focus on and/or give feedback Describe what you see—no inference Understand—what does the student know? Wonder—what are your questions? Focus—what is one mathematical idea or quality of work expectation that you want the child to revisit? Feedback—what question might you ask this child to prompt action toward learning or improvement?

Results Students begin to consider the readers point of view and write more explicitly about their thinking Label their work Their ability to communicate their ideas and understand other people’s strategies visibly improves They realize that what is in their heads needs to be made evident on paper