SUPPORTING ALL STUDENTS’ PARTICIPATION IN ACADEMICALLY

RIGOROUS MATHEMATICS CLASSROOMS

KARA JACKSON PAUL COBB VANDERBILT UNIVERSITY

UNIVERSITY OF RENNES, NOVEMBER 2009

GOALS OF THE TALK

1) Develop a vision of academically rigorous mathematics instruction that is likely to support all students’ access to significant mathematical ideas– “good instruction plus”

GOALS OF THE TALK

2) Share conjectures and findings regarding aspects of institutional settings that support equitable opportunities to learn in middle-grades mathematics classroom

EDUCATIONAL ATTAINMENTIN THE U.S. CONTEXT

DISPARITIES IN EDUCATIONAL ATTAINMENT

IN THE U.S. CONTEXT• As of 2005, who had earned a post-secondary

education degree, ages 25-29?– About 34% of White youth– About 17% of African American youth– About 11% of Hispanic youth

(According to the U.S. Census Bureau, as cited in Darling-Hammond, 2007)

DISPARITIES IN MATHEMATICS ACHIEVEMENT IN THE U.S. CONTEXT

• Based on 2009 National Assessment of Educational Progress (NAEP) data, significant gaps remain between White students and their Black and Hispanic peers

(U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, Nations’ Report Card 2009)

DISPARITIES IN MATHEMATICS ACHIEVEMENT IN THE U.S. CONTEXT

• “Because all three racial/ethnic groups have made progress, neither the White – Black nor the White – Hispanic score gap in 2009 was significantly different from the corresponding gaps in 2007 or 1990.”

(U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, Nations’ Report Card 2009)

SINCE 1990…

• Achievement gaps in mathematics have remained constant or widened– Drops in per-pupil expenditures– Increase in enrollment of students– Increase in immigrant children attending school– Increase in concentrated poverty and homelessness– Increase in numbers of students requiring English

language services and special education services (Darling-Hammond, 2007)

WHY DO THE GAPS EXIST AND PERSIST?

• Schools that serve large numbers of students of color have fewer resources than schools that serve large number of White students– Qualified teachers– Class size– Curriculum offerings (including textbooks)– School facilities

• Unequal access to high-quality instruction(Darling-Hammond, 2007)

U.S. URBAN SCHOOL DISTRICTS

• Large numbers of poor, children of color• Large, bureaucratic systems• Under extreme pressure to show overall

improvement in achievement and to close achievement gaps – Discourse of high-stakes accountability

HOW DO URBAN DISTRICTS RESPOND?

• Most “teach to the test” and/or “game the system” (Elmore, 2000; Heilig & Darling-Hammond, 2008)

• Little guidance for districts to do otherwise (either from mathematics education or policy)

OUR ARGUMENT

• Mathematics education community needs to support schools and districts to improve the quality of mathematics learning opportunities in schools that serve large numbers of traditionally low-performing students

OUR ARGUMENT

• To do so, mathematics education community needs to develop a vision of “good instruction plus” that details concrete forms of practice likely to support traditionally low-performing students

OUR ARGUMENT

• Development of “good instruction plus” forms of practice is complex and demanding

• Therefore, teachers need school and district supports to develop these forms of practice

GOAL 1 OF TALK

• Development of a vision of academically rigorous mathematics instruction that is likely to support all students’ access to significant mathematical ideas– “good instruction plus”

VISION OF “GOOD INSTRUCTION”

Classroom as an Instructional System• Nature of the tasks (cognitively demanding)• Organization of classroom activities, including norms

for participation • Use of tools, including normative ways of using them• Nature of classroom discourse, including norms of

mathematical argumentation (e.g., Cobb, 2001; Hiebert et al., 1997)

TYPICAL ORGANIZATION OF PHASES OF REFORM-ORIENTED

LESSONS

• Teacher poses a cognitively-demanding task• Students work in small groups to solve the

task• Teacher orchestrates a whole class discussion

in which s/he builds on students’ (diverse) solutions to develop significant mathematical ideas

ASSUMPTION OF U.S. MATHEMATICS EDUCATION

REFORM PROPOSALS

• Good instruction for one is good instruction for all

EQUITY AND ACCESS

• Equity in terms of the opportunities that students have to learn mathematics– Equal versus equitable learning opportunities

• Access in terms of the mechanisms by which individual students or groups of students can participate substantially in classroom mathematical activities

PROVISIONAL VISION OF “GOOD INSTRUCTION PLUS”

Three related aspects of instruction• Explicit negotiation of the social and

sociomathematical norms of the classroom• Purposeful posing of the task• Cultivation of students’ mathematical interests

EXPLICIT NEGOTIATION OF NORMS

• Social norms– “characteristics of the classroom community that …

are jointly established by the teacher and students” (Cobb et al., 2001, p. 122)

– Not specific to any subject matter– E.g., explaining solutions, justifying solutions,

attempting to make sense of others’ solutions, expressing (dis)agreement with others, questioning others’ solutions

(Cobb, Stephan, McClain, and Gravemeijer,

2001)

EXPLICIT NEGOTIATION OF NORMS

• Sociomathematical norms – Specific to the discipline of mathematics– E.g., what counts as a different mathematical

solution, what counts as an efficient mathematical solution, what counts as an acceptable solution

(Cobb, Stephan, McClain, and Gravemeijer, 2001)

EXPLICIT NEGOTIATION OF NORMS

• All students need opportunities to learn what is expected mathematically and how to participate in all phases of the lesson.

EXPLICIT NEGOTIATION OF NORMS

• Norms that support students’ access to one another’s reasoning:– Teacher should press students to explain and

justify not merely their solution methods but also the reasons for using particular methods rather than others

– Teacher should support students’ understandings of how varied solution strategies are related to one another

PURPOSEFUL POSING OF THE TASK

• Problem-solving tasks often provide students a situation in which to ground their mathematical thinking

PROBLEM-SOLVING TASK

PROBLEM-SOLVING TASK

PURPOSEFUL POSING OF THE TASK

• Build a shared understanding of

– important aspects of the non-mathematical context

– situation-specific images of the key mathematical ideas embedded in the task

so that all students can engage productively in solving the task.

CULTIVATION OF STUDENTS’ MATHEMATICAL INTERESTS

• Characteristics of a task– Prior familiarity with the phenomenon to be

investigated; students have developed an awareness of the phenomenon either in school or out of school

– Prior awareness of the specific question to be investigated and initial familiarity with the processes or mechanisms involved

– Resolution of the “problem” is of value to students or a broader audience

(Cobb, Hodge, Visnovska,& Zhao , 2007)

VIDEO OF TEACHER ENACTING FORMS OF “GOOD INSTRUCTION

PLUS” INSTRUCTIONAL PRACTICES

• Productive posing of the task

GOAL 2 OF TALK

• Share conjectures and findings regarding aspects of institutional settings that support equitable opportunities to learn in middle-grades mathematics classroom

CONJECTURED INSTITUTIONAL SUPPORTS FOR TEACHERS’ DEVELOPMENT OF EQUITABLE FORMS OF

AMBITIOUS INSTRUCTIONAL PRACTICES

• Access to rigorous mathematics curriculum – (e.g., Schoenfeld, 2002)

• Provision of high-quality professional development focused on equity-specific instructional practices in mathematics

CONJECTURED INSTITUTIONAL SUPPORTS FOR TEACHERS’ DEVELOPMENT OF EQUITABLE FORMS OF

AMBITIOUS INSTRUCTIONAL PRACTICES

• Un-tracked instructional program – (e.g., Boaler, 1997; Gamoran, Nystrand, Berends,

& LePore, 1997; Oakes, 1985)

• Positive category systems for describing students in relation to views of mathematics – (e.g., Horn, 2007; Jackson, 2009; Martin, 2000;

Moschkovich, 2007)

WHAT DO WE MEAN BY CATEGORIES?

• Distinguish types of phenomena, objects, and people

• Categories render some aspects as visible and some as invisible (Bowker & Star, 1999)

• Formal (e.g., NCLB categories, academic tracks) and informal (e.g., “smart”); circulate locally and more widely

• Always an empirical question as to what people mean by the categories they use

• “Frames problems of practice” (Horn, 2007)

WHAT DO WE MEAN BY CATEGORY SYSTEMS?

• Shared by majority of participants in a community

• Emergent phenomena

• Category systems are naturalized/normalized over time (Bowker & Star, 1999; Foucault, 1995/1977)

WHAT DO WE MEAN BY “POSITIVE” CATEGORY SYSTEMS?

• Teachers did not tend to describe students as having innate or fixed abilities or characteristics

• When teachers described groups of students, they tended to describe the instructional actions they took to support the groups of students

• Mathematics teachers tended to frame student motivation as a relation between the individual student and classroom instruction

ANALYSIS

• Cross-case analysis of 2 schools in the same district (A) that had “positive” category systems and sophisticated visions of HQMI–One school (A4) had notably better

opportunities to learn and student value-added achievement results for sub-populations than the other school (A5)

FOCUS OF ANALYSIS

• Explain why a positive category system was “productive” in A4 and not in A5 through an analysis of 3 related aspects of the institutional setting• Quality of professional development• Teachers’ access to expertise• Accountability relations between instructional

leaders and teachers

PRE- CASE SELECTION

• Coded Round 1 District A interview data for the following:– Categories participants used to describe groups of students

and the characteristics they ascribed to those categories– Pedagogical actions teachers described taking to meet the

perceived needs of different groups– Instructional leaders’ instructional expectations, particularly

for differentiation– Extent to which participants took responsibility for student

learning– Supports specific to issues of equity (e.g., ELLs)– Stances toward curriculum and mathematics

CRITERIA FOR CASE SELECTION

• Schools in District A with more than 1 participating teacher (n = 8)

• Majority of teachers in a school expressed positive categories the majority of the time

• Majority of teachers had sophisticated visions of high-quality mathematics instruction

SELECTED A4 & A5

SIMILARITIES BETWEEN A4 & A5• Positive category system• Teachers’ visions of high-quality mathematics

instruction• PreK-8 Schools, large % of economically

disadvantaged students• Size of schools, 3 middle-grades math teachers• Did not track in 6th or 7th grade• Offered one advanced course in 8th grade

(Algebra)• Used Connected Mathematics Program

DIFFERENCES

A4 A5PREDOMINANTLY LATINO POPULATION PREDOMINANTLY AFRICAN AMERICAN

POPULATION

46% of STUDENTS ARE ELLs, SINGLE LANGUAGE

30% of STUDENTS ARE ELLs, MULTIPLE LANGUAGES

EVERY STUDENT TAKES 1 MATH CLASS EVERY STUDENT TAKES 2 MATH CLASSES

PRINICPAL’S VISION OF HQMI IS CONTENT-FREE

PRINCIPALS’ VISION OF HQMI IS SOPHISTICATED

RECONSTITUTED IN 2007

DIFFERENCES: OPPORTUNITIES TO LEARN

Task Potential

Mean1 or 2 (-)3 or 4 (+)Median

Task Implementation

Mean1 or 2 (-)3 or 4 (+)Median

Discussion

Mean1 or 2 (-)3 or 4 (+)Median

A4(n = 3)

+3.5

+3.5

+3

A5*Deleted one teacher b/c no discussions (n=2)

+3

+3

-2

ANALYSIS OF INSTITUTIONAL STRUCTURES, RESOURCES, AND

SOCIAL RELATIONSHIPSQuality of Professional Development Specific to Equity

Teachers’ Access to Expertise Specific to Equity

Accountability Relations between Ts and P

A4 Ongoing support of the ELL Department

2 Ts in Designing Group Work

Culturally Responsive training (one-off)

No time to meet

Ts and P turn to one of the teachers, whose vision is most sophisticated and has equity- and math-specific expertise

Principal communicated instructional expectations particular to supporting all students’ learning

A5 Highly-mobile population (one-off)

2 Math Consultants, but not focused on supporting teacher learning

One isolated T

Principal did not communicate instructional expectations particular to supporting all students’ learning

COMMUNICATION OF INSTRUCTIONAL EXPECTATIONS AT A4

Well basically it starts with the lesson plan of expecting that I’m going to look at my student’s test data, get to know my students well just within the classroom of being able to have more individual ideas about what’s going on with each student and plan a good lesson that takes into account where each students is at and what they need. There’s the expectation that as I’m planning that lesson that I’m thinking about what activities am I going to do, how is that going to motivate the students, how is it going to teach the standards are the expected to be taught. How am I going to [get] students actively involved in that lesson? It’s basically looking at all those good quality teaching things and thinking about how is that going to play out within that lesson. And … the expectation is that while I’m delivering that lesson that I am differentiating from my students. That I have some way of being able to figure out at the end of the lesson did they get it? What do I need to do tomorrow? What happened that I didn’t expect and what am I going to do to able to deal with that? You know did it go better than I thought and I need to move on? Did it not go so well and I need to bring something else in and present it a different way? He’s expecting me to be reflective about that.

COMMUNICATION OF INSTRUCTIONAL EXPECTATIONS AT A5

T: He expects us to run a classroom and to operate in the building. He’s very clear on that and that, that has, that has been great.

I: And what does he say?T: It’s just making sure that … as far as clear expectations, the

kids …should expect to know what…work is to be completed, how it’s to be completed, when it’s to be completed by….[H]e expects us to deliver lessons as far as inquiring, questioning and those kind of things, …behavior management, you know, are we going to run morning meetings, is that part of our management plan, are we gonna use infractions and referrals.

IMPLICATIONS

• Importance of principal communicating clear instructional expectations regarding how to support all students’ learning

• Nature of the instructional expectations that the principal needs to communicate is related to the nature of teachers’ expertise (and access to expertise)

REVISITING OUR EQUITY-SPECIFIC CONJECTURES

• Rigorous curriculum, un-tracked instructional program, category systems and sophisticated visions of mathematics might be necessary but are not sufficient for increasing opportunities to learn (and hence, student achievement) for low-performing groups of students.

REVISITING OUR EQUITY-SPECIFIC CONJECTURES

• What else?– Teachers’ access to expertise in equity-specific

ambitious forms of teaching mathematics– Principal presses teachers to support all students

learning

EMPIRICALLY OPEN (AND IMPORTANT) QUESTION

• What do principals need to know and be able to do to support teachers’ development of ambitious and equitable instructional practices?

GOALS FOR OUR WORK

• Short-term goal: Provide actionable recommendations to districts

• Long-term goal: Revise our conjectures about institutional structures, resources, and social relationships for supporting teachers’ development of instructional practices that reflect “good instruction” plus

EQUITY AND ACCESS TEAMVanderbilt UniversityKara Jackson, PIPaul Cobb, Co-PIRichard Milner & Robert Jiménez, Senior ResearchersGlenn Colby, Annie Garrison, Lynsey Gibbons, Jonee

Wilson, Graduate Assistants

Collaborators from Other UniversitiesMelissa Boston (Duquesne University)Lindsay Clare Matsumura (University of Pittsburgh)

STUDENTS AS INNATELY ONE WAY OR ANOTHER

Because I figure, you know the good, the kids who want to learn are going to learn and the smart kids will always be smart. It’s the bottom dwellers that you want to see how they handle … instruction.

SUPPORTING STRUGGLING STUDENTS

[W]henever I can, … I’ll take some of the kids that are not at grade level and I’ll pre-teach them something that when they get into those groups they have something that the other kids don’t know. So it raises their status. Like I had a whole group of kids that were like, like if they were doing 3 times 1 ½ or something, they were multiplying the numerator and the denominator by 3. And so, taking those kids and teaching them how you can write whole number as a fraction…. something real simple. And then they went back and they felt great cause they shared that with their whole group.

MOTIVATION AS A RELATIONSHIP

All of my students are motivated at different times. I think all of them want to learn…they’re motivated by different things. Every single one of those kids at some point has shown me that they want to learn and has put effort at some point in that class. So I have to believe that all of them want to learn within that classroom.