Responding to Children’s Thinking and Diversity: A Reflection on 20 years of Research

Megan Loef FrankeUCLA

Cognitively Guided Instruction Thomas Carpenter (University

of Wisconsin) Elizabeth Fennema (University

of Wisconsin) Linda Levi (University of

Wisconsin) Susan Empson (University of

Texas) Ellen Ansell (University of

Pittsburgh) Vicki Jacobs (San Diego State

University) Elham Kazemi (University of

Washington) Dan Battey (Arizona State

Univ) Annie, Mazie, Sue, Barb,

Lilliam, Jo Ann, Kim, Janet, and many, many other teachers

Presentation Overview

Why focus on Children’s Mathematical Thinking Adding it Up Equity

Making use of the development of children’s mathematical thinking Research findings

Supporting the development of children’s mathematical thinking in classrooms

Considering Understanding: Adding it Up Recognizing that no term captures

completely all aspects of expertise, competence, knowledge, and facility in mathematics, we have chosen mathematical proficiency

five interwoven, interdependent strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, productive disposition.

Mathematical proficiency is not a one-dimensional trait, and it cannot be achieved by focusing on just one or two of these strands

About children’s thinking Children come to school

with mathematical knowledge

Children’s knowledge develops through well documented trajectories

Development of children’s thinking quite robust

Development of children’s thinking does not match the way adults solve problems

Development of Children’s Mathematical Thinking

Janelle has 7 trolls in her collection. How many more trolls will Janelle need to buy to have 11 trolls altogether?

How do you think children will solve this problem?Watch what children can do…

Direct modelingCounting strategyDerived FactRecall

How can a focus on children’s thinking help?

Notice what students’ can do

Make decisions based on what students’ know

Press for understanding

How can a focus on children’s thinking help?

Create multiple ways to participate

Support the development of mathematical identity—not just one way, sense making, question asking…

School Case Study

Longitudinal Study

CGI Research and Development

First Grade+/-

K- 3+/- x/÷ p.v.

K-5Algebraic thinking

All school+/- x/÷ p.v.

Experimental Study Teacher

Case StudiesFollow upTeacher/School

Experimental Study

Learning about the development of students’ mathematical thinking in classrooms

Development of tools to support learning in practice

Development of communities of inquiry

Evidence that Attending to Student Thinking Can Make a Difference CGI provides evidence that teachers’

classroom practice that

includes eliciting and making public student thinking,

involves eliciting multiple strategies, focuses on solving word problems and uses what is heard from students to make

instructional decisions

leads to the development of student understanding

Evidence that Attending to Student Thinking Can Make a Difference

Teachers who drew on

detailed knowledge of the development of students’ mathematical thinking within a domain

an organization of student thinking in relation to the mathematical content

notions that they could continue to learn from their practice …identity

supported the development of student understanding

Supporting teachers to make use of students’ mathematical thinking There is no single pattern or trajectory for

teachers as they come to make use of children’s thinking

Can get teachers to ask students how they solved problems

Challenging to support teachers to make use of what they hear from students, to engage students in comparing strategies, to move forward in their trajectories

Moving forward…learning more to support teacher learning and practice Pushing on the research

Learning through professional development

Moving towards understanding the details of practice through research

Listening to students talk makes it possible for the teachers (and other students) to monitor students’ mathematical thinking

The act of talking can itself help students develop improved understanding

Explaining to other students is positively related to achievement outcomes, even when controlling for prior achievement (Brown & Palincsar, 1989; Fuchs, Fuchs, Hamlett, Phillips, Karns, & Dutka, 1997; King, 1992;

Nattiv, 1994; Peterson, Janicki, & Swing, 1981; Saxe, Gearhart, Note, & Paduano, 1993; Slavin, 1987; Webb, 1991; Yackel, Cobb, Wood, Wheatley, & Merkel, 1990).

Less is known about teacher practices that are most effective for producing high-level discourse in the classroom

Details: Supporting the development of students’ mathematical thinking

In classrooms where:Students gave correct and complete explanations Students scored the highest on the assessments

Teachers: Used a fairly coherent set of problems Asked questions very specific to what students said Engaged students in thinking and talking about

important mathematical ideas arising out of their suggestions

All students participated in conversations about the mathematics

Learning through professional development

develop relationships: create a community where teachers can learn together about the teaching and learning of mathematics

where the activities of the community were embedded in teachers’ everyday work

make space for teachers to share their histories and make their practice public

focus on the details and structures around students’ mathematical thinking

Focus on what students can do (Counter-storytelling) attention to the artifacts and language that support the

development of students’ mathematical thinking in practice

Artifacts in our Professional Development work…

Framework for the development of student strategies within mathematical domains

• Problem types • Video of students and

classrooms • Language

How did you get that? Does that always work? Strategy and problem names

Number sentence index cards

Join Change Unknown

Avita has 7 rocks. How many more rocks does she need to collect to have 11 rocks altogether?

Join Result Unknown …

Artifact Travel

Ongoing use across settings

Attention to and unpacking of classroom use in PD

Trace where we are with ideas around artifact

Helps to see teacher use of artifact in both PD and classrooms – raise questions, note inconsistencies, conflict etc..

How artifacts support learning and practice

Focused on creating and negotiating meaning

Focused conversation across and within communities of practice

Supported the development of language and interaction that could be used to support the development of new relationships

Supported story telling across boundaries and be used to develop counter stories

Purposeful

Challenged the existing cultural practices

Development of Children’s Mathematical Thinking

Tom has 102 dog biscuits. His Dog Harmony eats 12 biscuits a day. How many days will it take Harmony to eat all of the dog biscuits?

Let’s see what children can do…

Attending to the Details of Children’s Mathematical Thinking Allows Teachers to:

Notice what students can do

Make decisions that build on what students know

Create openings for varied participation

Support the development of students who think of themselves as capable of making sense of mathematics

Development of Children’s Mathematical Thinking

Lucy had 38 dollars. One weekend she earned 25 making dollars raking leaves for her neighbors. How much money did Lucy have then?

Watch what children can do…What can you do?

Count by tens, solve problems using 20 and 30, take numbers apart and put them back together.

Mathematical Proficiency

conceptual understanding—comprehension of mathematical concepts, operations, and relations

procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and appropriately

strategic competence—ability to formulate, represent, and solve mathematical problems

adaptive reasoning—capacity for logical thought, reflection, explanation, and justification

productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

Development of Children’s Mathematical Thinking

8 + 4 = + 5

Can students solve without computing each side?

Let’s watch David…

Equal Sign Data (8+4= +5)

Student Responses1

Grade 7 12 17 12 & 17

1st & 2nd 5% 58 13 8

3rd & 4th 9 49 25 10

5th & 6th 2 76 21 2

1Falkner, K., Levi, L., & Carpenter, T. (1999). Children’s understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6, 232-6.

Evidence that Attending to Student Thinking Can Make a Difference

Students constantly surprise us…

Kindergarten data Fractions Algebraic thinking

Focusing on Making Student Thinking Explicit need to be able to use student strategies as

the center of the workgroup conversation, as one of the tools teachers interact with

expertise shared change in power structures, teacher as

expert provides an explicit trace of the group’s

thinking extends to other communities of practice centers the role of the professional developer

Moving Towards the Details of Practice

Need to know more about student participation in mathematics classrooms if we are to support teaching

Often large scale studies focus on what occurs in public discourse

Smaller scale studies document more specifically student participation and what that means for student learning Forman, et al, 1998; Lampert, 2001; Moschkovich, 2002; O’Connor & Michaels, 1996; Palincsar & Brown, 1984, 1989; Yackel, Cobb, & Wood, 1991

Want to look to the relationship between student participation, teaching, the mathematics and student outcomes

Development of Children’s Mathematical Thinking

19 Children are taking a mini-bus to the zoo. They will have to sit either 2 or 3 to a seat. The bus has 7 seats. How many children will have to sit 3 to a seat and how many can sit 2 to a seat?

How will children solve it?How about a kindergartener? 59% had a correct strategy 51% correct answer, 33% 1st graders, 26% 2nd

graders