Committing to the CoreRethinking Mathematics

for the 21st Century

Sara Good Heather Canzurlo

Welcome to Session 3!

REFLECTION TOOL REVISTED

• In Session 1 you assessed your level of implementation of the 8 Standards for Mathematical Practice.

• Today, we’d like you to complete the tool again to reflect on your growth during the past several months.

• After rating yourself, please indicate your choices by placing sticky dots on the graphs.

• Recognize how diagnostic teaching and common misconceptions can move student thinking forward.

• Distinguish between CGI problem types.

• Identify the key instructional shifts in transitioning to the Common Core.

• Commit to the PCSD Mathematics Instructional Framework.

THINKING ABOUT STUDENT THINKING

TASK• Examine and analyze the student thinking.

• Please respond to the prompts on your own.

• Place Xs on the line plot poster to show your choices for questions 1 and 2.

Ma & Pa Kettle

VIDEO

What does this have to do with traditional school mathematics?

What misconceptions did you notice in the video?

What would be your next teaching move? Why?

1. Which of the following do you think best represents Dylan’s thinking? a. Dylan seems to have subtracted upside down in the ones place. b. Dylan has used place value thinking to reason through the computation. c. The numbers happened to work out this time. d. Dylan does not understand the operation of subtraction.

2. How would you use Dylan’s response to further student thinking? a. Ask his classmates to explore the solution and determine if the approach is valid. b. Model how to borrow from the hundreds and the tens. c. I would have Dylan explain his thinking before making any instructional decisions. d. I would demonstrate that there is more than one way to obtain a solution.

• Share the reasoning for your choices.

• What can you infer from the group’s data?

• What are the implications for our teaching practice?

THINKING ABOUT STUDENT THINKING

How are these two questions different?

THINKING ABOUT STUDENT THINKING

Does she know it?

How does she

understand it?

Are we asking BOTH questions?

THINKING ABOUT STUDENT THINKING

How does she

understand it?

What does she know?

MATHEMATICAL UNDERSTANDING

We have to know where students are in order to get them where they’re going!

WHAT IS OUR BELIEF SYSTEM?

DEFICIT MODEL

ASSET MODEL

DIAGNOSTIC TEACHING

• http://math.serpmedia.org/tools_diagnostic.html

VIDEO

http://americaschoice.org/misconceptionsCorrecting Misconceptions

VS Remedial Learning

A study by Alan Bell and Malcolm Swan found that students whose teachers addressed and corrected misconceptions, rather than simply using remedial measures, achieved and maintained higher long-term learning results.

“Just add a zero”

TEACHING FROM STUDENT MISCONCEPTIONS

MISTAKES• Not intentional• Due to inattention or

carelessness• Made by a few• Occurs infrequently• Can usually identify error• Can usually self-correct

MISCONCEPTIONS• Consciously made• Students believe in

correctness• Made by many• Happens repeatedly• Unable to figure out;

student is committed• Often persist, even in the

face of counter-evidence

TEACHING FROM STUDENT MISCONCEPTIONS

• Teachers skillfully unearth misconceptions for scrutiny

• Misconceptions are the subject of rich discussion• Students experience disequilibrium and re-

construct existing schema to incorporate new knowledge

• Students develop robust conceptual understanding and are able to transfer it to novel situations

MATHEMATICAL UNDERSTANDING

Do we believe that we can positively impact student achievement?

IN PCSD WE BELIEVE:

ASSET MODEL

All students can think mathematically and can learn to do so in increasingly sophisticated ways

The best way to support mathematics learning is to know what and how students are thinking and how to move this thinking forward.

CGI: Cognitively Guided Instruction• Is not a curriculum, but rather an approach to teaching

mathematics – It offers a framework for thinking about student thinking

• Developed by education researchers Thomas Carpenter, Elizabeth Fennema, Penelope Peterson, Megan Loef Franke, and Linda Levi

• “Children do not always think about mathematics in the same ways that adults do. If we want to give children the opportunity to build their understanding from within, we need to understand how children think about mathematics.” (from the Introduction of Children’s Mathematics: Cognitively Guided Instruction, Carpenter & Fennema, et. Al)

CGICGI is rooted in two major principles:

1) Children bring informal, intuitive knowledge of mathematics to school with them which should serve as the basis for developing formal mathematics instruction

2) Math instruction should be based on the relationship between computational skills and problem solving, which leads to an emphasis on problem solving in the classroom instead of the repetition of number facts

CGIStages

– Creating a climate for communication– Posing problems– Problem solving– Reporting solution strategies

• OBSERVE• QUESTION• LISTEN

https://www.teachingchannel.org/videos/problem-solving-math?fd=1#

VIDEO

CGI and Problem Structures• CGI research has identified and classified word

problems by operation and type 14 types of structures based on children’s natural approaches

THE COMMON CORE STATE STANDARDS HELP US MOVE THINKING FORWARD

• they provide clear signposts along the way to the goal of college and career readiness for all students

• they empower students to understand what is expected of them and to become progressively more proficient in understanding and using mathematics

• convey a unified vision of the big ideas and supporting concepts within a discipline and reflect a progression of learning that is meaningful and appropriate

The companion Progressions Documents for each domain help inform our teaching!

• enable teachers to be better equipped to help students

Operations & Algebraic Thinking • The Common Core State Standards incorporate

CGI problem structures and provide detailed guidance for teachers page 9 of The Learning Progression

Document for Operations and Algebraic Thinking

Problem Structures Sorting Activity

K 1

K K

2

2

K

1

1

Problem Structures Sorting Activity

2 2

11

1

1

Problem Structures Sorting Activity

TASK• Read and discuss each word problem with your group.

• Place each problem card on the Problem Structure mat.

• Prepare to justify your choices.

Key

Everyday Mathematics & Problem Structures

SIX INSTRUCTIONAL SHIFTS

COMMON CORE STATE

STANDARDSFOR

MATHEMATICS

WORTHWHILE MATHEMATICAL

TASKS

THINKING ABOUT STUDENT THINKING

MATHEMATICS

MATHEMATICS INSTRUCTIONAL FRAMEWORKPLAN (Common Core State Standards)• Teachers design lesson plans aligned to the Common Core State Standards.• Teachers provide regular opportunities for students to engage in the Standards for

Mathematical Practice.• Teachers create a classroom environment which promotes equity, communication,

and collaboration.• Teachers use the Learning Progressions for each Domain to present mathematics as

a coherent system of ideas and concepts.• Teachers anticipate student misconceptions, potential strategies, and the range of

responses.• Teachers select multiple appropriate representations that express the same content.• Teachers draft essential, purposeful and thought-provoking questions prior to

instruction.• Teachers draw upon a wide variety of research- based, high impact instructional

strategies.

MATHEMATICS INSTRUCTIONAL FRAMEWORKASSESS (Thinking About Student Thinking)• Teachers administer formative and summative assessments to monitor student

progress toward the Common Core State Standards.• Teachers assess students’ growth in procedures, concepts, and problem solving.• Teachers look and listen for evidence of students engaging in the Standards for

Mathematical Practice, and offer constructive feedback.• Teachers use effective questioning techniques to uncover how students are thinking.• Teachers hold all learners accountable to high standards.• Teachers ensure that students are doing mathematics, not just hearing about it.• Teachers provide opportunities for students to self-assess and reflect on their

thinking.

MATHEMATICS INSTRUCTIONAL FRAMEWORKTEACH (Worthwhile Mathematical Tasks)• Teachers use the results of formative and summative assessment data to inform

instruction.• Teachers balance instructional time between procedures, concepts, and problem

solving.• Teachers capitalize on common misconceptions in order to help students construct

meaning.• Teachers help students to make connections across multiple representations.• Teachers strategically pose varied questions at all levels of Blooms Taxonomy.• Teachers effectively facilitate mathematical discussions among students.• Teachers incorporate rich, group-worthy mathematical tasks.• Teachers maximize student participation and interactions by using flexible, small-

group instruction, learning stations, and student interviews• Teachers challenge all students appropriately by differentiating instruction.

Are you ready?

We will PLAN, ASSESS, and TEACH with intention, and in full alignment with the Common Core.