Translating Research Translating Research Findings into Classroom Findings into Classroom

Practices that Give Practices that Give Students Agency, Students Agency,

Competency, Competency, Commitment, Commitment, and Authorityand Authority

Fifth Annual TEAM-Math Partnership ConferenceFifth Annual TEAM-Math Partnership ConferenceTuskegee UniversityTuskegee University

Kellogg Conference CenterKellogg Conference CenterSeptember 5, 2008September 5, 2008

Carol E. Malloy, Ph.D.Carol E. Malloy, Ph.D.University of North Carolina at Chapel HillUniversity of North Carolina at Chapel Hillcmalloy@email.unc.edu

Principal Investigators:

Carol E. Malloy, Ph.D.

cmalloy@email.unc.edu

Jill V. Hamm, Ph.D.jhamm@email.unc.edu

Judith L. Meece, Ph.D.meece@email.unc.edu

        

          

      

University of North Carolina-Chapel Hill – NSF Grant REC 0125868

l l l l l l l l l l l l l l l l l l l l l l l l l

MIDDLEResearch Associate:

Mark W. Ellismellis@exchange.fullerton.edu

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Purpose

To better understand how mathematics reform affects students’ development as mathematics knowers and learners

To identify the processes that explain changes in students’ mathematical learning and self-conceptions

District Demographics

PopulationPopulation31,980 31,980

African AmericanAfrican American 54%54%

CaucasianCaucasian 24.3%24.3%

HispanicHispanic 15.7%15.7%

MultiracialMultiracial 3.4%3.4%

AsianAsian 2.4%2.4%

Native AmericanNative American 0.2%0.2%

Students at or above Students at or above

grade level in mathgrade level in math

African AmericanAfrican American 75%75%

Caucasian Caucasian 95%95%

HispanicHispanic 77%77%

MultiracialMultiracial 90%90%

AsianAsian >95% >95%

Native AmericanNative American 81%81%

Econ. DisadvantagedEcon. Disadvantaged 73%73%

ELLELL 68%68%

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Framework for Looking at Framework for Looking at ReformReform

Reform — Teacher's use of instructional Reform — Teacher's use of instructional

practices and curricular materials that are practices and curricular materials that are aligned with NCTM’s aligned with NCTM’s Curriculum and Teaching Curriculum and Teaching Standards Standards (1989, 1991) and the (1989, 1991) and the Principles Principles and Standards for School Mathematics and Standards for School Mathematics (2000).(2000).

Conceptual Understanding – Carpenter and Conceptual Understanding – Carpenter and Lehrer (1999) model to examine how Lehrer (1999) model to examine how students are given opportunities to develop students are given opportunities to develop conceptual understanding of mathematics.conceptual understanding of mathematics.

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Looking at InstructionLooking at Instruction

PedagogyPedagogy is seen in how a teacher's plans for and is seen in how a teacher's plans for and

the resulting flow of the lesson the resulting flow of the lesson including how students are given including how students are given opportunities to learn. This includes opportunities to learn. This includes the discourse that the teacher pursues the discourse that the teacher pursues in the lessons and the tools she uses.in the lessons and the tools she uses.

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Content

Includes the objectives of lesson Includes the objectives of lesson including where the student is being led including where the student is being led and allowed to advance and the subject and allowed to advance and the subject matter, both procedural and conceptual, matter, both procedural and conceptual, that students will gain.that students will gain.

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TasksTasks

Represent the mathematical work that Represent the mathematical work that students are engaged in during class students are engaged in during class and the opportunity students have to and the opportunity students have to internalize the work they do. Of internalize the work they do. Of particular interest are characteristics particular interest are characteristics of classrooms and instruction that of classrooms and instruction that maintain high-level cognitive maintain high-level cognitive demands or produce a decline of high-demands or produce a decline of high-level cognitive demands. level cognitive demands.

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Mathematical InteractionMathematical Interaction

IIs the mathematical s the mathematical conversations or discourse that conversations or discourse that results from the instruction results from the instruction planned and modified by the planned and modified by the teacher and initiated by students.teacher and initiated by students.

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AssessmentAssessment

Includes the ways that the teacher Includes the ways that the teacher determined what students had determined what students had learned, specifically, evidence of learned, specifically, evidence of student performance, the relation of student performance, the relation of student understanding to content student understanding to content being taught, feedback to students, being taught, feedback to students, and student involvement in critique.and student involvement in critique.

First, we investigated reform First, we investigated reform practice and student practice and student perception.perception. 42 teachers42 teachers 84 classrooms84 classrooms 733 sixth graders in 53 classrooms733 sixth graders in 53 classrooms 422 seventh graders in 31 classrooms422 seventh graders in 31 classrooms..

– 52.3 % African American52.3 % African American– 29.2 % Caucasian29.2 % Caucasian– 5.8% Hispanic 5.8% Hispanic

From InstructionFrom Instruction

Classroom instructional observationsClassroom instructional observations

Teacher interviewsTeacher interviews

Student perception surveysStudent perception surveys

From Understanding

Student conceptual understanding Student conceptual understanding scores scores

End-Of-Course scores from State End-Of-Course scores from State assessmentsassessments

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Looking at Conceptual Looking at Conceptual UnderstandingUnderstanding A class has 28 students. The ratio of A class has 28 students. The ratio of

girls to boys is 4 to 3. How many girls girls to boys is 4 to 3. How many girls are in the class?are in the class?

Explain Explain whywhy you think your answer is correct. you think your answer is correct.

Concepts AssessedConcepts Assessed• Understand and apply proportional reasoning Understand and apply proportional reasoning

used in scaling.used in scaling.• Understand that a fraction always represents Understand that a fraction always represents

a part-to-whole relationship.a part-to-whole relationship.• Understand that a ratio can represent part-Understand that a ratio can represent part-

to-part or part-to-whole relationships.to-part or part-to-whole relationships.

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Student ResponsesStudent Responses

1.1. There are 12 girls. I used the ratio There are 12 girls. I used the ratio and then added them up. (Shows and then added them up. (Shows columns of four 4s and three 3s columns of four 4s and three 3s adding up to 18 and 12, adding up to 18 and 12, respectively.)respectively.)

2.2. 16. I got lazy and actually counted 16. I got lazy and actually counted out 4,3,4,3, etc.out 4,3,4,3, etc.

1616

3. 3. 16. I set up a ratio and proportion to 16. I set up a ratio and proportion to find the answer. I think it is correct find the answer. I think it is correct because there should be more than because there should be more than half the class girls. half the class girls.

4/7 = ?/28, 28 x 4 = 112, 112 /7 = 4/7 = ?/28, 28 x 4 = 112, 112 /7 =

1616 4. There are 16 girls. I figured this out 4. There are 16 girls. I figured this out

because I knew that 16/12 was the because I knew that 16/12 was the same as 4/3 and 16 + 12 gave me 28.same as 4/3 and 16 + 12 gave me 28.

1717

5. 5. There are 16 girls. I used guess and There are 16 girls. I used guess and check. Students wrote in space below check. Students wrote in space below not on the same line:not on the same line:

4/3 16/12 28/4 = 7 7 boys 12 4/3 16/12 28/4 = 7 7 boys 12

6. I guess I divide 4 into 28 and the 6. I guess I divide 4 into 28 and the answer is the answer to the problem. 7 answer is the answer to the problem. 7 girls.girls.

7. There are 16 girls (drawing below)7. There are 16 girls (drawing below)

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Does teacher instruction based on levels of reform practice influence students’ conceptual understanding of mathematics in the middle grades?

What are students’ perceptions of their teachers’ instruction?

What do you think we found?

QuestionsQuestions

Reform Level Findings

Teacher reform levels explained 10% of the between-classroom variation in conceptual understanding scores, with student control variables in the model.

Students of teachers at the higher Students of teachers at the higher

levels of reform scored higher on levels of reform scored higher on the conceptual understanding items.the conceptual understanding items.

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Mid-High Mid-High ReformReform

High ReformHigh Reform

PedagogyPedagogy Teacher directed Teacher directed

Inquiry groupsInquiry groupsTeacher directed Teacher directed

Inquiry groupsInquiry groups

ContentContent Procedural & conceptual Procedural & conceptual contentcontent

Procedural PressProcedural Press

Procedural & conceptual Procedural & conceptual contentcontent

Procedural and Conceptual Procedural and Conceptual PressPress

TasksTasks Memorization, Procedural, Memorization, Procedural, and Conceptual and Conceptual

Teacher provided solutions Teacher provided solutions in groupsin groups

Memorization, Procedural, Memorization, Procedural, and Conceptual and Conceptual

Student justification Student justification dominateddominated

AssessmeAssessmentnt

Questions, student work, Questions, student work, questions, and responsesquestions, and responses

Questions, student work, Questions, student work, questions, responses, and questions, responses, and peer and self assessmentspeer and self assessments

InteractioInteractionn

AuthorityAuthority

Teacher, behavioralTeacher, behavioral

Interaction Interaction

teacher to student, student teacher to student, student to studentto student

AuthorityAuthority

teacher & student, teacher & student, mathematizingmathematizing

Interaction Interaction

teacher to student, student to teacher to student, student to studentstudent

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Perception FindingsPerception FindingsWhat is the student perception of their

teachers’ instruction? Students who perceived a greater press for Students who perceived a greater press for

masterymastery and and granting of authority granting of authority by their by their teachers made more significant gains in teachers made more significant gains in conceptual understanding. conceptual understanding.

These patterns did not differ by race or These patterns did not differ by race or gender. gender.

Similar results emerged in predicting Similar results emerged in predicting students’ change in EOG test scores students’ change in EOG test scores between grades. between grades.

Another InvestigationAnother Investigation

What are the instructional strategies and dispositions of teachers who help African American middle grades students gain conceptual understanding in mathematics?

What are successful students’ What are successful students’ perceptions of their teachers’ perceptions of their teachers’ instruction?instruction?

FrameworkFramework

The cognitive development of African The cognitive development of African American students is supported by use of American students is supported by use of instructional strategies grounded in instructional strategies grounded in learning preferences of African learning preferences of African Americans. Americans.

Students’ opportunities to learn Students’ opportunities to learn mathematics should be tied to their mathematics should be tied to their cultural experiences and social justice in cultural experiences and social justice in their communities. their communities.

Cultural Experiences & Social Cultural Experiences & Social JusticeJustice

Culturally responsive and relevant Culturally responsive and relevant pedagogies stress concern for the pedagogies stress concern for the cultural experiences of students.cultural experiences of students.

Social justice in mathematics Social justice in mathematics education gives students tools and education gives students tools and learning conditions to develop learning conditions to develop mathematical skills, knowledge as the mathematical skills, knowledge as the gain understanding to become gain understanding to become educated and effective citizens. educated and effective citizens.

Procedures

Selection of teachers based on student Selection of teachers based on student Conceptual Understanding growthConceptual Understanding growth

Analysis of teachers’ instructional Analysis of teachers’ instructional methods and dispositions from methods and dispositions from observation and interview data and observation and interview data and student perceptions of the selected student perceptions of the selected teachersteachers

The Students and TeachersThe Students and Teachers

126 classrooms of 44 teachers with 126 classrooms of 44 teachers with 946 students 946 students

non-gifted and non-inclusionnon-gifted and non-inclusion 431 African American students: 159 431 African American students: 159

66thth, 190 7, 190 7thth, 82 8, 82 8thth

Data collected from MIDDLE Data collected from MIDDLE relevant to this studyrelevant to this study

Instructional observations Instructional observations Teacher interviewsTeacher interviews Student conceptual understanding Student conceptual understanding

instruments instruments Student perception surveysStudent perception surveys

Selecting Teachers

Greater than 15 African American students Greater than 15 African American students per teacher per teacher

Greater than 10% growth on CU items or Greater than 10% growth on CU items or growth significant at the p< .05 level. growth significant at the p< .05 level.

Four teachers who qualified on both criteria, Four teachers who qualified on both criteria, taught 107 MIDDLE African American taught 107 MIDDLE African American students who demonstrated an average students who demonstrated an average percent increase of over 16% on conceptual percent increase of over 16% on conceptual understanding items with p values .000 < p understanding items with p values .000 < p < .04.< .04.

Student increase in conceptual understanding by teacher

TeacherTeacher StudentsStudents ClassrooClassroomsms

P ValueP Value % % IncreaseIncrease

JohnsonJohnson

RescuerRescuer3030 44 .000.000 15.8%15.8%

SpearsSpears

FacilitatorFacilitator3737 22 .040.040 11.2%11.2%

CoddCodd

ListenerListener2121 22 .014.014 19%19%

WinstonWinston

InterrogatInterrogatoror

1919 33 .001.001 19.4%19.4%

Results: The TeachersRescuer “Rescuer “They don’t realize they know before I teach them “

African American male, lateral entry, 8 years experience, coach. School had 450 students, 56% free and reduced lunch.

Discipline and saving students through contextDiscipline and saving students through context

Facilitator Facilitator “I’d rather they’d be working”Caucasian female, engineer for 5 yrs, lateral entry, 5 years Caucasian female, engineer for 5 yrs, lateral entry, 5 years

experience. School had 700 students, 67% free and reduced experience. School had 700 students, 67% free and reduced lunch lunch

Students work for their knowledgeStudents work for their knowledge

Listener Listener “Let the kids learn and generalize and see things.”“Let the kids learn and generalize and see things.” Caucasian female, 15 years experience, in M.Ed. Program. Caucasian female, 15 years experience, in M.Ed. Program.

School had 700 students 67% free and reduced lunch School had 700 students 67% free and reduced lunch Students learn to reason through listening to each other Students learn to reason through listening to each other

Interrogators Interrogators “I try to do what is best for the students.”“I try to do what is best for the students.” Caucasian female, 17 years experience, in M.Ed. Program. Caucasian female, 17 years experience, in M.Ed. Program.

School had 600 students 39% free and reduced lunch –year School had 600 students 39% free and reduced lunch –year round schoolround school

Using what students need—traditional and reform practiceUsing what students need—traditional and reform practice

Common Instructional PracticesCommon Instructional Practices

Reflecting on PracticeReflecting on Practice Building Communities of Building Communities of

Learners Learners Giving Students Voice Giving Students Voice

Reflecting on PracticeReflecting on Practice

Blended memorization, procedural, Blended memorization, procedural, and conceptual tasks and conceptual tasks

Involved fundamental concepts of the Involved fundamental concepts of the subject in lessons subject in lessons

Respected students’ prior knowledge Respected students’ prior knowledge and preconceptionsand preconceptions

Were knowledgeable about contentWere knowledgeable about content Listened and responded to students to Listened and responded to students to

anticipate their understanding and/or anticipate their understanding and/or misunderstanding misunderstanding

Building Communities of LearnersBuilding Communities of Learners Encouraged students to generate Encouraged students to generate

conjectures, alternative solution conjectures, alternative solution strategies, and ways of interpreting strategies, and ways of interpreting evidenceevidence

Created a climate of respect for what Created a climate of respect for what others had to sayothers had to say

Valued intellectual rigor, constructive Valued intellectual rigor, constructive criticism, and the challenging of ideascriticism, and the challenging of ideas

Encouraged elements of abstraction Encouraged elements of abstraction when important when important

Giving Students VoiceGiving Students Voice

Acted as a resource person, working to Acted as a resource person, working to support and enhance student support and enhance student investigationsinvestigations

Saw knowledge and authority in both Saw knowledge and authority in both teachers and studentsteachers and students

Encouraged and valued active Encouraged and valued active participation of studentsparticipation of students

Used learning communities that Used learning communities that promoted student-teacher and student-promoted student-teacher and student-student mathematical interactionstudent mathematical interaction

Common Dispositions

Believed that all students could learn Believed that all students could learn mathematics mathematics

Valued student motivation, Valued student motivation, involvement, effort, respectful involvement, effort, respectful behavior, and responsibility behavior, and responsibility

Demonstrated concern to address the Demonstrated concern to address the varied learning styles of their students varied learning styles of their students and accommodated instruction based and accommodated instruction based student learning preferences student learning preferences

Demonstrated that knowledge their Demonstrated that knowledge their students brought into the classroom students brought into the classroom should be sharedshould be shared

Helped their students feel safe in their Helped their students feel safe in their classrooms and cared about their classrooms and cared about their students and their learningstudents and their learning

Were reflective about their practiceWere reflective about their practice

Recognizing Cultural Recognizing Cultural Experiences and Experiences and CommunitiesCommunities

Was not seen in instruction or Was not seen in instruction or mentioned in interviews by two or mentioned in interviews by two or more teachers more teachers

What Did Students What Did Students Experience?Experience?

AgencyAgency Competency and Commitment Competency and Commitment AuthorityAuthority

Definitions

a)a)a a sense of agencysense of agency over their own learning, through over their own learning, through opportunities to define their own goals, and to opportunities to define their own goals, and to submit and justify their own mathematical ideas, submit and justify their own mathematical ideas, and to experience challenge with appropriate and to experience challenge with appropriate tasks; tasks;

b)b)a a sense of competencysense of competency and and commitmentcommitment through through support for persistent engagement in complex support for persistent engagement in complex problems and opportunities for reflection on their problems and opportunities for reflection on their mathematical thinking, and mathematical thinking, and

c)c) a a sense of authority, sense of authority, through lessons that require through lessons that require students to take an active role in the creation and students to take an active role in the creation and verification of mathematical ideas. verification of mathematical ideas.

AgencyAgency

Students Students – used different strategies to work out used different strategies to work out

problems problems – felt safe asking questions about math when felt safe asking questions about math when

something does not make sense (2)something does not make sense (2)– stated procedures make sense to them (2) stated procedures make sense to them (2)

Teachers encouraged us to Teachers encouraged us to – come up with new ways to solve problems (2) come up with new ways to solve problems (2) – figure out things for ourselves (2)figure out things for ourselves (2)– talk about why answer is not correct (2)talk about why answer is not correct (2)

Sense of Competency and Commitment

A class goal was toA class goal was to– understand how procedures work in mathunderstand how procedures work in math– improve our understanding of math improve our understanding of math – understand what we are doing to solve understand what we are doing to solve

problemsproblems– think about how we solve problemsthink about how we solve problems

Our teachers Our teachers – asked questions that make us think asked questions that make us think – helped us understand what we were doing to helped us understand what we were doing to

solve problemssolve problems– built on math we already knew (2)built on math we already knew (2)– gave rules to solve problems (2)gave rules to solve problems (2)

WeWe– learned to solve sets of similar problems (2)learned to solve sets of similar problems (2)

Sense of AuthoritySense of Authority

In our class In our class – working in groups means that everyone shares our working in groups means that everyone shares our

ideas ideas – we used different strategies to work out problems we used different strategies to work out problems – we work together to understand new math ideaswe work together to understand new math ideas(2)(2)

Teachers encouraged us toTeachers encouraged us to– try to understand why formulas work try to understand why formulas work – invent ways to solve math problems (2)invent ways to solve math problems (2)– figure out things for ourselves (2)figure out things for ourselves (2)– talk about why answer is not correct (2)talk about why answer is not correct (2)

Teacher’s comments help us to understand our errors Teacher’s comments help us to understand our errors (2)(2)

Discussion

During this time when little is mentioned During this time when little is mentioned about minority students who are about minority students who are achieving, it is clear that teachers can achieving, it is clear that teachers can make a difference in what students learn. make a difference in what students learn.

We can be successful if we address the We can be successful if we address the

needs of students in mathematics needs of students in mathematics classrooms as demonstrated through classrooms as demonstrated through research.research.

Teachers used high reform practices Pedagogy: Teacher directed and Inquiry groupsPedagogy: Teacher directed and Inquiry groups Content: Procedural & conceptual content and Content: Procedural & conceptual content and

Procedural and Conceptual PressProcedural and Conceptual Press Tasks: Memorization, Procedural, and Tasks: Memorization, Procedural, and

Conceptual and Student justification dominatedConceptual and Student justification dominated Assessment: Questions, student work, Assessment: Questions, student work,

questions, responses, and peer and self questions, responses, and peer and self assessmentsassessments

Interaction: Authority teacher & student, Interaction: Authority teacher & student, mathematizing and Interaction teacher to mathematizing and Interaction teacher to student, student to studentstudent, student to student

Sense of AgencySense of Agency

TeachersTeachers Believed that all students could learn Believed that all students could learn

mathematics mathematics Valued student motivation, Valued student motivation,

involvement, effort, respectful involvement, effort, respectful behavior, and responsibility behavior, and responsibility

Demonstrated that knowledge Demonstrated that knowledge students brought into the classroom students brought into the classroom should be shared should be shared

Sense of AgencySense of Agency

TeachersTeachers Presented memorization, procedural, Presented memorization, procedural,

and conceptual tasksand conceptual tasks Involved fundamental concepts of the Involved fundamental concepts of the

subject in lessonssubject in lessons Encouraged to generate conjectures, Encouraged to generate conjectures,

alternative solution strategies, and alternative solution strategies, and

ways of interpreting evidenceways of interpreting evidence

Sense of CompetencySense of Competency and and CCommitmentommitment TeachersTeachers

Demonstrated concern to address the Demonstrated concern to address the varied learning styles of their students varied learning styles of their students and accommodated instruction based and accommodated instruction based student learning preferencesstudent learning preferences

Sense of CompetencySense of Competency and and CCommitmentommitmentTeachersTeachers Encouraged and valued active participation Encouraged and valued active participation

of students of students Valued intellectual rigor, constructive Valued intellectual rigor, constructive

criticism, and the challenging of ideascriticism, and the challenging of ideas Created a climate of respect for what Created a climate of respect for what

others had to sayothers had to say Encouraged and valued active participation Encouraged and valued active participation

of students of students Respected students’ prior knowledge and Respected students’ prior knowledge and

preconceptionspreconceptions

Sense of AuthoritySense of Authority

TeachersTeachers Cared about their students and their Cared about their students and their

learninglearning Believed that all students could learn Believed that all students could learn

mathematics mathematics Saw knowledge and authority in both Saw knowledge and authority in both

teachers and studentsteachers and students Helped their students feel safe in their Helped their students feel safe in their

classrooms classrooms

Sense of AuthoritySense of Authority

TeachersTeachers Acted as a resource person, working to Acted as a resource person, working to

support and enhance student support and enhance student investigationsinvestigations

Listened and responded to their Listened and responded to their students to anticipate their students to anticipate their understanding and/or misunderstanding understanding and/or misunderstanding

Used learning communities Used learning communities Promoted student-teacher and student-Promoted student-teacher and student-

student mathematical interaction student mathematical interaction

Falling Short

Teachers’ instruction reflected the Teachers’ instruction reflected the suggested strategies for African suggested strategies for African American student learning.American student learning.

However, the teachers fell short in However, the teachers fell short in the area of commitment to the area of commitment to recognizing and using students recognizing and using students cultural experiences and cultural experiences and communities. communities.

QuestionsQuestions

• What do you think we should do What do you think we should do having knowledge from this having knowledge from this research?research?

• What are large and small changes What are large and small changes that can be made?that can be made?

• How do we begin?How do we begin?

What is it “We” as Teachers What is it “We” as Teachers and Teacher Educators Learn and Teacher Educators Learn to Do?to Do?• Value student discourse and verbal Value student discourse and verbal

knowledge knowledge • Develop activities and questioning to Develop activities and questioning to

promote mathematical discourse promote mathematical discourse among students and teacheramong students and teacher

• Encourage, support, and provide Encourage, support, and provide feedback to students as they learnfeedback to students as they learn

• Understand that mathematics is in our Understand that mathematics is in our students—not just in us and the booksstudents—not just in us and the books

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Address the varied learning styles of our Address the varied learning styles of our students and accommodate instruction students and accommodate instruction based their learning preferences based their learning preferences

Demonstrate that knowledge students Demonstrate that knowledge students bring into our classrooms should be bring into our classrooms should be sharedshared

Help our students feel safe in our Help our students feel safe in our classrooms and care about our students classrooms and care about our students and their learningand their learning

Be reflective about our practiceBe reflective about our practice

If we do these things, then our Students will LEARN Mathematics

Thank You

ReferencesReferencesBeyer, L. E. (Ed.). (1996). Beyer, L. E. (Ed.). (1996). Creating democratic classrooms: The struggle to integrate theory and Creating democratic classrooms: The struggle to integrate theory and

practicepractice. New York: Teachers College Press.. New York: Teachers College Press.

Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for Journal for Research in Mathematics Education, 29Research in Mathematics Education, 29(1), 41-62.(1), 41-62.

Bransford, J.D., Brown, A.L. & Cocking, R. R. (Eds.). (1999). Bransford, J.D., Brown, A.L. & Cocking, R. R. (Eds.). (1999). How people learn: Brain, mind, experience, How people learn: Brain, mind, experience, and school.and school. Washington, D.C.: National Academy Press.Washington, D.C.: National Academy Press.

Briars, D. J. & Resnick, L. B. (2000). Standards, assessments – and what else? The essential elements of Briars, D. J. & Resnick, L. B. (2000). Standards, assessments – and what else? The essential elements of standards-based school improvement. National Center on Education, Standards, and Student standards-based school improvement. National Center on Education, Standards, and Student Assessment (CRESST).Assessment (CRESST).

Carpenter, T. P., & Lehrer, R. (1999). Teaching and learning mathematics with understanding. In E. Carpenter, T. P., & Lehrer, R. (1999). Teaching and learning mathematics with understanding. In E. Fennema & T. A. Romberg (Eds.), Fennema & T. A. Romberg (Eds.), Mathematics classrooms that promote understanding Mathematics classrooms that promote understanding (pp. 19-32). (pp. 19-32). Mahwah, NJ: LEA.Mahwah, NJ: LEA.

Cousins-Cooper, K. M. (2000). Teacher expectations and their effects on African American students’ Cousins-Cooper, K. M. (2000). Teacher expectations and their effects on African American students’ success in mathematics. In W. G. Secada, M. E. Structhens, M. L. Johnson, & W. F. Tate (Eds.), success in mathematics. In W. G. Secada, M. E. Structhens, M. L. Johnson, & W. F. Tate (Eds.), Changing the faces of mathematics: Perspectives on African AmericansChanging the faces of mathematics: Perspectives on African Americans (pp. 15-20). Reston, VA: (pp. 15-20). Reston, VA: NCTM.NCTM.

Delpit, L. (1995). Delpit, L. (1995). Other people’s children: Cultural conflict in the classroomOther people’s children: Cultural conflict in the classroom . New York: W. W. Norton & . New York: W. W. Norton & Co., Inc.Co., Inc.

Ellis, M. E., Malloy, C. E., Meece, J. L., & Sylvester, P. R. (2007). Convergence of observer ratings and Ellis, M. E., Malloy, C. E., Meece, J. L., & Sylvester, P. R. (2007). Convergence of observer ratings and student perceptions of reform practices in sixth-grade Mathematics Classrooms. student perceptions of reform practices in sixth-grade Mathematics Classrooms. Learning Learning Environments ResearchEnvironments Research..1010(1). (1).

Gay, (2000). Gay, (2000). Culturally responsive teaching: Theory, research, and practiceCulturally responsive teaching: Theory, research, and practice . New York: Teachers . New York: Teachers College Press.College Press.

Glaser,Glaser, B. G. (Spring, 1965). The constant comparative method of qualitative analysisB. G. (Spring, 1965). The constant comparative method of qualitative analysis.. Social ProblemsSocial Problems, , 1212(4), pp. 436-445.(4), pp. 436-445.

Grouws, D. A., & Lembke, L. O. (1996). Influential factors in students motivation to Grouws, D. A., & Lembke, L. O. (1996). Influential factors in students motivation to learn mathematics: The teacher and classroom culture. In M. Carr (Ed.), learn mathematics: The teacher and classroom culture. In M. Carr (Ed.), Motivation in mathematicsMotivation in mathematics (pp. 39-59). Cresskill, NJ: Hampton Press. (pp. 39-59). Cresskill, NJ: Hampton Press.

Gutiérrez, R. (2000). Advancing African-American, urban youth in mathematics: Gutiérrez, R. (2000). Advancing African-American, urban youth in mathematics: Unpacking the success of one math department. Unpacking the success of one math department. American Journal of Education, American Journal of Education, 109109(1), 63-111.(1), 63-111.

Hale-Benson, Janice. Hale-Benson, Janice. Black Children: Their Roots, Culture, and Learning StylesBlack Children: Their Roots, Culture, and Learning Styles. . Baltimore, MD: Johns Hopkins Press, 1986.Baltimore, MD: Johns Hopkins Press, 1986.

Hamilton, L. S., McCaffrey, D., F., Stecher, B. M., Klein, S. P., Robyn, A., & Buglairi, D. Hamilton, L. S., McCaffrey, D., F., Stecher, B. M., Klein, S. P., Robyn, A., & Buglairi, D. (2003). (2003). Educational Evaluation and Policy Analysis, 25, Educational Evaluation and Policy Analysis, 25, 1-29.1-29.

Harris, Olita D. “Equity in Classroom Assessment.” In H. Roberts, J. C. Gonzales, O. D. Harris, Olita D. “Equity in Classroom Assessment.” In H. Roberts, J. C. Gonzales, O. D. Harris, D. J. Huff, A.M. Johns, R. Lou, & O. L. Scott (Eds.), Harris, D. J. Huff, A.M. Johns, R. Lou, & O. L. Scott (Eds.), Teaching from a Teaching from a Multicultural PerspectiveMulticultural Perspective, (pp. 77-90). Thousand Oaks, CA: Sage1994., (pp. 77-90). Thousand Oaks, CA: Sage1994.

Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers' mathematicalHill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers' mathematical knowledgeknowledge for teaching on student achievement. for teaching on student achievement. American Educational Research American Educational Research Journal, 42Journal, 42(2), 371-406. (2), 371-406.

Hilliard, A. G. Alternatives to IQ Testing: an Approach to the Identification of Gifted Hilliard, A. G. Alternatives to IQ Testing: an Approach to the Identification of Gifted Minority Children. Final Report to California State Department of Education, Minority Children. Final Report to California State Department of Education, (1976). (1976).

hooks, b. (1994). hooks, b. (1994). Teaching to transgress: Education as the practice of freedomTeaching to transgress: Education as the practice of freedom. New . New York: Routledge.York: Routledge.

Howard, T. C. (2002). Hearing footsteps in the Dark: African American students’ Howard, T. C. (2002). Hearing footsteps in the Dark: African American students’ descriptions of effective teachers. descriptions of effective teachers. Journal of Education for Students Placed at Risk, Journal of Education for Students Placed at Risk, 77(4), 425-444.(4), 425-444.

Irvine, J, J. (1992). Making teacher education culturally responsive. In M. E. Dilworth. Irvine, J, J. (1992). Making teacher education culturally responsive. In M. E. Dilworth. (ed.) (ed.) Diversity in Teacher EducationDiversity in Teacher Education, pp. 79-92. San Francisco, CA: Jossey-Bass , pp. 79-92. San Francisco, CA: Jossey-Bass Publishers.Publishers.

King, Joyce, E. (1994). The purpose of schooling for African American children. In E. R. King, Joyce, E. (1994). The purpose of schooling for African American children. In E. R. Hollins, J. E. King, & W. C. Hayman (Eds.), (pp. ). Hollins, J. E. King, & W. C. Hayman (Eds.), (pp. ). Teaching Diverse Populations: Teaching Diverse Populations: Formulating a Knowledge BaseFormulating a Knowledge Base. Albany, NY: State University of New York Press.. Albany, NY: State University of New York Press.

Ladson,-Billings, G. (1994). Ladson,-Billings, G. (1994). The Dreamkeepers: Successful Teachers of African The Dreamkeepers: Successful Teachers of African American ChildrenAmerican Children. San Francisco, CA: Jossey-Bass.. San Francisco, CA: Jossey-Bass.

Ladson-Billings, G. (1995). Toward a theory of culturally relevant pedagogy. American Ladson-Billings, G. (1995). Toward a theory of culturally relevant pedagogy. American Educational Research Journal, 32(3), 465-491.Educational Research Journal, 32(3), 465-491.

Malloy, C. E. (2008). Looking throughout the world for democratic access to Malloy, C. E. (2008). Looking throughout the world for democratic access to mathematics. In L. D. English, (Ed.), mathematics. In L. D. English, (Ed.), Handbook of International Research in Handbook of International Research in Mathematics Education, Second Edition, (pp. 20-31),Mathematics Education, Second Edition, (pp. 20-31), Taylor and Francis/Routledge.Taylor and Francis/Routledge.

Malloy, C. E. (2004). Equity in mathematics education is about access. In R. Rubenstein Malloy, C. E. (2004). Equity in mathematics education is about access. In R. Rubenstein & G. Bright, Eds., & G. Bright, Eds., 2004 NCTM Yearbook: Effective mathematics teaching (pp.1-14).2004 NCTM Yearbook: Effective mathematics teaching (pp.1-14). Reston, VA, NCTM.Reston, VA, NCTM.

Malloy, C. E. (1997). Including African American students in the mathematics Malloy, C. E. (1997). Including African American students in the mathematics community. In J. Trentacosta, (Ed.), community. In J. Trentacosta, (Ed.), Multicultural and gender equity in the Multicultural and gender equity in the mathematics classroom: The gift of diversitymathematics classroom: The gift of diversity, (pp.23-33). Reston, VA: National , (pp.23-33). Reston, VA: National Council of Teachers of Mathematics, Council of Teachers of Mathematics,

Malloy, C., & Malloy, W. (1998). Issues of culture in mathematics teaching and learning. Malloy, C., & Malloy, W. (1998). Issues of culture in mathematics teaching and learning. The Urban Review,The Urban Review, 3030(3), 245-257. (3), 245-257.

Malloy, C. E., & . Meece, (under review) J. L. The Relations between Reform Instruction Malloy, C. E., & . Meece, (under review) J. L. The Relations between Reform Instruction and Student Learning in Middle Grades Mathematics Classrooms. and Student Learning in Middle Grades Mathematics Classrooms.

Miller, S. P., & Hudson, P. J. (2007). Using evidence-based practices to build Miller, S. P., & Hudson, P. J. (2007). Using evidence-based practices to build mathematics competence mathematics competence related to conceptual, procedural, and declarative related to conceptual, procedural, and declarative knowledge. Learning Disabilities Research & Practice,knowledge. Learning Disabilities Research & Practice, 2222(1), 47-57.(1), 47-57.

Moses, R. P., & Cobb, Jr., C. E. (2001). Moses, R. P., & Cobb, Jr., C. E. (2001). Radical equations: Math literacy and civil rights.Radical equations: Math literacy and civil rights. Boston; Beacon Press.Boston; Beacon Press.

NCTM. (1989). NCTM. (1989). Curriculum and evaluation standards for school mathematics. Curriculum and evaluation standards for school mathematics. Reston, Reston, VA: NCTM.VA: NCTM.

NCTM. (1991). NCTM. (1991). Professional standards for teaching mathematics. Professional standards for teaching mathematics. Reston, VA: NCTM.Reston, VA: NCTM.NCTM. (2000). NCTM. (2000). Principles and standards for school mathematicsPrinciples and standards for school mathematics. Reston, VA: NCTM.. Reston, VA: NCTM.Sawada, D., Piburn, M., Falconer, K., Turley, J., Benford, R., & Bloom, I. (2000). Sawada, D., Piburn, M., Falconer, K., Turley, J., Benford, R., & Bloom, I. (2000).

Reformed teaching observation protocol (RTOP) Reformed teaching observation protocol (RTOP) (ACEPT Technical Report No. IN00-(ACEPT Technical Report No. IN00-1). Tempe, AZ: Arizona Collaborative for Excellence in the Preparation of Teachers.1). Tempe, AZ: Arizona Collaborative for Excellence in the Preparation of Teachers.

Shade, B. (1989). The influence of perceptual development on cognitive style: Cross Shade, B. (1989). The influence of perceptual development on cognitive style: Cross ethnic comparisons. ethnic comparisons. Early Child Development and CareEarly Child Development and Care, 15, 137-155., 15, 137-155.

Shafer, M. C. (2001, April). Instructional quality in the context of reform. Paper Shafer, M. C. (2001, April). Instructional quality in the context of reform. Paper presented at the Research Pre-session of the annual meeting of National Council of presented at the Research Pre-session of the annual meeting of National Council of Teachers of Mathematics Teachers, Orlando, FL. Teachers of Mathematics Teachers, Orlando, FL.

Silver, E. A., & Stein, M. K. (1996) The QUASAR Project: The “revolution of the possible” Silver, E. A., & Stein, M. K. (1996) The QUASAR Project: The “revolution of the possible” in mathematics instructional reform in urban middle schools. in mathematics instructional reform in urban middle schools. Urban Review, 30Urban Review, 30(4), (4), 476-521.476-521.

Skemp, R.R. (1986). Skemp, R.R. (1986). The psychology of leaning mathematics. The psychology of leaning mathematics. Hillsdale, NJ: LEA.Hillsdale, NJ: LEA.

Smith, L. B., Stiff, L. V., & Petree, M. R. (2000). Teaching mathematics to the least Smith, L. B., Stiff, L. V., & Petree, M. R. (2000). Teaching mathematics to the least academically prepared African American students. In W. G. Secada, M. E. Structhens, academically prepared African American students. In W. G. Secada, M. E. Structhens, M. L. Johnson, & W. F. Tate (Eds.), M. L. Johnson, & W. F. Tate (Eds.), Changing the faces of mathematics: Perspectives Changing the faces of mathematics: Perspectives on African Americanson African Americans (pp. 89-96). Reston, VA: NCTM. (pp. 89-96). Reston, VA: NCTM.

Sowder, J. T., & Phillipp, R. A. (1999). The value of interaction in promoting teaching Sowder, J. T., & Phillipp, R. A. (1999). The value of interaction in promoting teaching growth. In J. T. Sowder & B. P. Schappelle, (Eds.), growth. In J. T. Sowder & B. P. Schappelle, (Eds.), Providing a foundation for teaching Providing a foundation for teaching mathematics in the middle grades, mathematics in the middle grades, (pp. 223-250). Albany, NY: SUNY.(pp. 223-250). Albany, NY: SUNY.

Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing Implementing standards-based mathematics instruction: A casebook for professional development.standards-based mathematics instruction: A casebook for professional development. New York, NY: Teachers College Press.New York, NY: Teachers College Press.

Stiff, L. (1990) African-American students and the promise of the curriculum and Stiff, L. (1990) African-American students and the promise of the curriculum and evaluation standards. In T. J. Cooney and C. R. Hirsh, (Eds.)evaluation standards. In T. J. Cooney and C. R. Hirsh, (Eds.)In Teaching and Learning In Teaching and Learning Mathematics in the 1990s,Mathematics in the 1990s, (pp. 152-158). Reston: VA: National Council of Teachers of (pp. 152-158). Reston: VA: National Council of Teachers of Mathematics.Mathematics.

Supovitz, J. A., Mayer, D. P., & Kahle, J. B. (2000). Promoting inquiry-based instructional Supovitz, J. A., Mayer, D. P., & Kahle, J. B. (2000). Promoting inquiry-based instructional practice: The longitudinal impact of professional development in the context of practice: The longitudinal impact of professional development in the context of systemic reform. systemic reform. Educational Policy 14Educational Policy 14(3), pp. 331-356.(3), pp. 331-356.

Van Haneghan, J. P., Pruet, S. A., & Bamberger, H. J. (2004). Mathematics reform in a Van Haneghan, J. P., Pruet, S. A., & Bamberger, H. J. (2004). Mathematics reform in a minority community: Student outcomes. minority community: Student outcomes. Journal of Education for Students Placed at Journal of Education for Students Placed at RiskRisk, , 99(2), 189–211.(2), 189–211.

Vithal, R. (1999). Democracy and authority: A complementarity in mathematics Vithal, R. (1999). Democracy and authority: A complementarity in mathematics education? education? Zentralblatt für Didaktik der MathematikZentralblatt für Didaktik der Mathematik, , 9898(6), 27-36. (6), 27-36.

Walker, C. M. (1999). The effect of different pedagogical approaches on mathematics Walker, C. M. (1999). The effect of different pedagogical approaches on mathematics students’ achievement. Paper presented at the annual meeting of the American students’ achievement. Paper presented at the annual meeting of the American Educational Research Association, Montreal. (ERIC Document Reproduction Service Educational Research Association, Montreal. (ERIC Document Reproduction Service No. ED 431011).No. ED 431011).

Wenglinsky, H. (2004). The link between instructional practice and the racial gap in Wenglinsky, H. (2004). The link between instructional practice and the racial gap in middle schools. middle schools. Research in Middle Level Education Online, 28Research in Middle Level Education Online, 28(1), 1-18,(1), 1-18,

West, C. (2005). West, C. (2005). Democracy mattersDemocracy matters. New York: Penquin. . New York: Penquin. Willis, M. G. (1992). Learning styles of African American children: Review of the literature Willis, M. G. (1992). Learning styles of African American children: Review of the literature

and interventions. In A. K. H. Burlew, W. C. Banks, H. P. McAdoo, and D. A. Azibo and interventions. In A. K. H. Burlew, W. C. Banks, H. P. McAdoo, and D. A. Azibo (Eds.), (Eds.), African-American Psychology,African-American Psychology, (pp. 260-278). Newbury Park, CA: Sage. (pp. 260-278). Newbury Park, CA: Sage.