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 • About 60% of U.S. youth graduate from high schools • About 60% of U.S. high school graduates attend post-secondary education; about 50% of them graduate college with a degree • In sum, less than 30% of any U.S. age group receives a post-secondary degree (Darling-Hammond, 2007)
DISPARITIES IN EDUCATIONAL ATTAINMENTIN 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 classroommathematical 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
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
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?