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Improving Student Problem Solving Through Reasoning & Revision

Jessica Murk, Windsor High School Math Teacher February 19, 2015

Sonoma County Leadership Network

+A Case for Revision

 The definition of mathematical proficiency has changed

+A Case for Revision   Previous definition:

CALIFORNIA HIGH SCHOOL EXIT EXAM

+A Case for Revision   New Definition:

SMARTER BALANCED ASSESSMENT CLAIMS

+A Case for Revision   The new expectations for students require new instructional

strategies.

Showing students exactly how to do a problem and then having them practice doing problems of the same type may work for Claim 1, but it will not work for Claims 3 and 4. Those require instructional strategies more like a writing class than a traditional math class.

+A Case for Revision

 The new expectations for students requires new instructional strategies.

 One becomes a better writer by writing, getting feedback, and revising.

 One does not improve by only writing first drafts.

 Math class often is about getting a certain percentage of problems “correct” and then moving on.

 Communicating reasoning and mathematical modeling are processes that can always be improved.

+The Target has Changed!

OLD DEFINITION OF PROFICENCY

NEW DEFINITION OF PROFICENCY

Tools for Reasoning and Revision

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+Overview of Our Time Together

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REASONING AND EXPLAINING 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others

MODELING AND USING TOOLS 4. Model with mathematics 5. Use appropriate tools strategically

SEEING STRUCTURE & GENERALIZING 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning

+Goals

 Big Picture of CCSS Shifts

 Some specific strategies to use in your classroom that can help with these shifts

 Give your students opportunities to engage in the SMPs using our current curriculum

+Tools for Reasoning & Revision

 Bongard Problems

 Developing Definitions

 Peer Feedback Template

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Reasoning Revision Tool #1: Bongard Problems

The Problem: Students struggle

with abstract reasoning

+Reasoning Revision Tool #1: Bongard Problems

 Students can test their rules on their own (Agency)

 The template allows them to write down their initial thoughts and then revise as they hear other ideas (Attend to precision)

 The “puzzle” element lowers fear and anxiety for students (Access/Equity)

+Bongard Problems

M.M. Bongard (1924 – 1971) was a Russion computer scientist interested in cognition and artificial intelligence. He introduced a set of problems (now called Bongard Problems) in his 1967 book Проблема Узнавания.

+ Let’s Try One

The goal is to identify a simple rule or property that applies to each of the figures on the left hand side AND none of the figures on the right hand side.

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I would like to add an option for their first draft and then a revised rule.

+Bongard Problem #94

+Student Responses

+Student Responses

+Student Responses

+Challenge

+Let’s Try Another One

+Revising as a Class: A written record of revision

+Reasoning Revision Tool #1: Bongard Problems

 Students can test their rules on their own (Agency)

 The template allows them to write down their initial thoughts and then revise as they hear other ideas (Attend to precision)

 The “puzzle” element lowers fear and anxiety for students (Access/Equity)

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Reasoning Revision Tool #2: Developing Definitions

The Problem: Students don’t understand the difference between a “mathematical” definition and a dictionary

definition

+Reasoning Revision Tool #2: Developing Definitions  Access student prior knowledge in an

authentic way (Use of assessment)

 They are not blank slates (Access/Identity)

 Make connections between what they already know and the more precise, mathematically rigorous definition (Cognitive Demand)

+Teaching Vocabulary

+Developing Definitions: Old School

 Look terms up in a dictionary

 Provide students the formal definition from the textbook through direct instruction

+Developing Definitions: New School

 Asking students what they already know

 Making connections between their prior knowledge and more precise, mathematically rigorous definitions

+Access Prior Knowledge

 We know it’s a good idea (research and credential programs tell us so)

 But HOW?!?

 How can we access a student’s prior knowledge in an authentic way?

+Geometry Vocabulary

+Geometry Vocabulary

+Geometry Vocabulary

+Geometry Vocabulary

Are these points

collinear? Can we think of the freeway as a segment? Or is it more of a line?

Which towns are

the endpoints

of the freeway?

+Reasoning Revision Tool #2: Developing Definitions  Access student prior knowledge in an

authentic way (Use of assessment)

 They are not blank slates (Access/Identity)

 Make connections between what they already know and the more precise, mathematically rigorous definition (Cognitive Demand)

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Reasoning Revision Tool #3: Peer Feedback Template

The Problem: Students lack experience in critiquing the

reasoning of others

+ Reasoning Revision Tool #3: Peer Feedback Template

 The authority shifts from teacher to student (Authority)

 Developing mathematical definitions and constructing arguments (Mathematics)

 Revising their definitions based on feedback (Cognitive demand)

+Peer Feedback with Revision

 Peer Feedback Template

 First Draft ≠ Final Draft

 Building on the ideas and conjectures that students have about reflections

 Goal is to help students move toward a more formal definition, which will ultimately be provided by the teacher

+ Giving Feedback… A Gentle Reminder  Unnatural in a math class

 Students have very little experience

 Rarely receive this type of feedback in a math class

 It takes practice!!!

+Giving Feedback

+A Little Help

+Let’s Try One

+Peer Feedback: Our Purpose

+Peer Feedback: Some Suggestions

•  Students write their argument individually

•  Each argument gets rotated to another table

•  Pairs work together on providing feedback to two arguments

•  This happens twice •  Arguments are returned to

original authors •  Students read through

feedback to consider where they might revise their original argument and write a second draft

+Peer Feedback

+Student Work – Lizbeth

+Student Work – Emily

+Student Work – Nicole

+ Reasoning Revision Tool #3: Peer Feedback Template

 The authority shifts from teacher to student (Authority)

 Developing mathematical definitions and constructing arguments (Mathematics)

 Revising their definitions based on feedback (Cognitive demand)

+Tools for Reasoning & Revision

 Bongard Problems

 Developing Definitions

 Peer Feedback Template

+Thank You!!

 Jessica Murk  Email: jmurk@wusd.org  Blog: themathymurk.blogspot.com  Twitter: @JessicaMurk13