Translating Research Findings into Classroom Practices that Give Students Agency, Competency, Commitment, and Authority Fifth Annual TEAM-Math Partnership Conference Tuskegee University Kellogg Conference Center September 5, 2008 Carol E. Malloy, Ph.D. University of North Carolina at Chapel Hill

Principal Investigators: Carol E. Malloy, Ph.D. Jill V. Hamm, Ph.D. Judith L. Meece, Ph.D. University of North Carolina-Chapel Hill – NSF Grant REC 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 MIDDLE Research Associate: Mark W. Ellis m

3 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 Population 31,980 African American54% Caucasian24.3% Hispanic15.7% Multiracial3.4% Asian2.4% Native American0.2% Students at or above grade level in math African American75% Caucasian 95% Hispanic77% Multiracial90% Asian >95% Native American81% Econ. Disadvantaged73% ELL68%

5 Framework for Looking at Reform Reform — Teacher's use of instructional practices and curricular materials that are aligned with NCTM’s Curriculum and Teaching Standards (1989, 1991) and the Principles and Standards for School Mathematics (2000). Reform — Teacher's use of instructional practices and curricular materials that are aligned with NCTM’s Curriculum and Teaching Standards (1989, 1991) and the Principles and Standards for School Mathematics (2000). Conceptual Understanding – Carpenter and Lehrer (1999) model to examine how students are given opportunities to develop conceptual understanding of mathematics. Conceptual Understanding – Carpenter and Lehrer (1999) model to examine how students are given opportunities to develop conceptual understanding of mathematics.

6 Looking at Instruction Pedagogy is seen in how a teacher's plans for and the resulting flow of the lesson including how students are given opportunities to learn. This includes the discourse that the teacher pursues in the lessons and the tools she uses. is seen in how a teacher's plans for and the resulting flow of the lesson including how students are given opportunities to learn. This includes the discourse that the teacher pursues in the lessons and the tools she uses.

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

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

9 Mathematical Interaction Mathematical Interaction I s the mathematical conversations or discourse that results from the instruction planned and modified by the teacher and initiated by students. I s the mathematical conversations or discourse that results from the instruction planned and modified by the teacher and initiated by students.

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

First, we investigated reform practice and student perception. 42 teachers 42 teachers 84 classrooms 84 classrooms 733 sixth graders in 53 classrooms 733 sixth graders in 53 classrooms 422 seventh graders in 31 classrooms. 422 seventh graders in 31 classrooms. –52.3 % African American –29.2 % Caucasian – 5.8% Hispanic

From Instruction Classroom instructional observations Classroom instructional observations Teacher interviews Teacher interviews Student perception surveys Student perception surveys

From Understanding Student conceptual understanding scores Student conceptual understanding scores End-Of-Course scores from State assessments End-Of-Course scores from State assessments

14 Looking at Conceptual Understanding A class has 28 students. The ratio of girls to boys is 4 to 3. How many girls are in the class? A class has 28 students. The ratio of girls to boys is 4 to 3. How many girls are in the class? Explain why you think your answer is correct. Concepts Assessed Understand and apply proportional reasoning used in scaling. Understand and apply proportional reasoning used in scaling. Understand that a fraction always represents a part-to-whole relationship. Understand that a fraction always represents a part-to-whole relationship. Understand that a ratio can represent part-to-part or part-to-whole relationships. Understand that a ratio can represent part-to-part or part-to-whole relationships.

15 Student Responses 1. There are 12 girls. I used the ratio and then added them up. (Shows columns of four 4s and three 3s adding up to 18 and 12, respectively.) I got lazy and actually counted out 4,3,4,3, etc.

I set up a ratio and proportion to find the answer. I think it is correct because there should be more than half the class girls. 4/7 = ?/28, 28 x 4 = 112, 112 /7 = 16 4/7 = ?/28, 28 x 4 = 112, 112 /7 = There are 16 girls. I figured this out because I knew that 16/12 was the same as 4/3 and gave me 28.

17 5. There are 16 girls. I used guess and check. Students wrote in space below not on the same line: 4/3 16/12 28/4 = 7 7 boys 12 4/3 16/12 28/4 = 7 7 boys I guess I divide 4 into 28 and the answer is the answer to the problem. 7 girls. 7. There are 16 girls (drawing below) 7. There are 16 girls (drawing below)

18 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? Questions

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 levels of reform scored higher on the conceptual understanding items. Students of teachers at the higher levels of reform scored higher on the conceptual understanding items.

20 Mid-High Reform High Reform Pedagogy Teacher directed Inquiry groups Teacher directed Inquiry groups Content Procedural & conceptual content Procedural Press Procedural & conceptual content Procedural and Conceptual Press Tasks Memorization, Procedural, and Conceptual Teacher provided solutions in groups Memorization, Procedural, and Conceptual Student justification dominated Assessment Questions, student work, questions, and responses Questions, student work, questions, responses, and peer and self assessments InteractionAuthority Teacher, behavioral Interaction teacher to student, student to student Authority teacher & student, mathematizing Interaction teacher to student, student to student

21 Perception Findings What is the student perception of their teachers’ instruction? Students who perceived a greater press for mastery and granting of authority by their teachers made more significant gains in conceptual understanding. Students who perceived a greater press for mastery and granting of authority by their teachers made more significant gains in conceptual understanding. These patterns did not differ by race or gender. These patterns did not differ by race or gender. Similar results emerged in predicting students’ change in EOG test scores between grades. Similar results emerged in predicting students’ change in EOG test scores between grades.

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

Framework The cognitive development of African American students is supported by use of instructional strategies grounded in learning preferences of African Americans. The cognitive development of African American students is supported by use of instructional strategies grounded in learning preferences of African Americans. Students’ opportunities to learn mathematics should be tied to their cultural experiences and social justice in their communities. Students’ opportunities to learn mathematics should be tied to their cultural experiences and social justice in their communities.

Cultural Experiences & Social Justice Culturally responsive and relevant pedagogies stress concern for the cultural experiences of students. Culturally responsive and relevant pedagogies stress concern for the cultural experiences of students. Social justice in mathematics education gives students tools and learning conditions to develop mathematical skills, knowledge as the gain understanding to become educated and effective citizens. Social justice in mathematics education gives students tools and learning conditions to develop mathematical skills, knowledge as the gain understanding to become educated and effective citizens.

Procedures  Selection of teachers based on student Conceptual Understanding growth  Analysis of teachers’ instructional methods and dispositions from observation and interview data and student perceptions of the selected teachers

The Students and Teachers 126 classrooms of 44 teachers with 946 students 126 classrooms of 44 teachers with 946 students non-gifted and non-inclusion non-gifted and non-inclusion 431 African American students: th, th, 82 8 th 431 African American students: th, th, 82 8 th

Data collected from MIDDLE relevant to this study Instructional observations Instructional observations Teacher interviews Teacher interviews Student conceptual understanding instruments Student conceptual understanding instruments Student perception surveys Student perception surveys

Selecting Teachers Greater than 15 African American students per teacher Greater than 15 African American students per teacher Greater than 10% growth on CU items or growth significant at the p<.05 level. Greater than 10% growth on CU items or growth significant at the p<.05 level. Four teachers who qualified on both criteria, taught 107 MIDDLE African American students who demonstrated an average percent increase of over 16% on conceptual understanding items with p values.000 < p <.04. Four teachers who qualified on both criteria, taught 107 MIDDLE African American students who demonstrated an average percent increase of over 16% on conceptual understanding items with p values.000 < p <.04.

Student increase in conceptual understanding by teacher TeacherStudentsClassrooms P Value % Increase JohnsonRescuer % SpearsFacilitator % CoddListener % WinstonInterrogator %

Results: The Teachers Rescuer “ 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 context Facilitator Facilitator “I’d rather they’d be working” Caucasian female, engineer for 5 yrs, lateral entry, 5 years experience. School had 700 students, 67% free and reduced lunch Students work for their knowledge Listener “Let the kids learn and generalize and see things.” Caucasian female, 15 years experience, in M.Ed. Program. School had 700 students 67% free and reduced lunch Caucasian female, 15 years experience, in M.Ed. Program. School had 700 students 67% free and reduced lunch Students learn to reason through listening to each other Interrogators “I try to do what is best for the students.” Caucasian female, 17 years experience, in M.Ed. Program. School had 600 students 39% free and reduced lunch –year round school Caucasian female, 17 years experience, in M.Ed. Program. School had 600 students 39% free and reduced lunch –year round school Using what students need—traditional and reform practice

Common Instructional Practices Reflecting on Practice Reflecting on Practice Building Communities of Learners Building Communities of Learners Giving Students Voice Giving Students Voice

Reflecting on Practice Blended memorization, procedural, and conceptual tasks Blended memorization, procedural, and conceptual tasks Involved fundamental concepts of the subject in lessons Involved fundamental concepts of the subject in lessons Respected students’ prior knowledge and preconceptions Respected students’ prior knowledge and preconceptions Were knowledgeable about content Were knowledgeable about content Listened and responded to students to anticipate their understanding and/or misunderstanding Listened and responded to students to anticipate their understanding and/or misunderstanding

Building Communities of Learners Encouraged students to generate conjectures, alternative solution strategies, and ways of interpreting evidence Encouraged students to generate conjectures, alternative solution strategies, and ways of interpreting evidence Created a climate of respect for what others had to say Created a climate of respect for what others had to say Valued intellectual rigor, constructive criticism, and the challenging of ideas Valued intellectual rigor, constructive criticism, and the challenging of ideas Encouraged elements of abstraction when important Encouraged elements of abstraction when important

Giving Students Voice Acted as a resource person, working to support and enhance student investigations Acted as a resource person, working to support and enhance student investigations Saw knowledge and authority in both teachers and students Saw knowledge and authority in both teachers and students Encouraged and valued active participation of students Encouraged and valued active participation of students Used learning communities that promoted student-teacher and student-student mathematical interaction Used learning communities that promoted student-teacher and student-student mathematical interaction

Common Dispositions Believed that all students could learn mathematics Believed that all students could learn mathematics Valued student motivation, involvement, effort, respectful behavior, and responsibility Valued student motivation, involvement, effort, respectful behavior, and responsibility Demonstrated concern to address the varied learning styles of their students and accommodated instruction based student learning preferences Demonstrated concern to address the varied learning styles of their students and accommodated instruction based student learning preferences

Demonstrated that knowledge their students brought into the classroom should be shared Demonstrated that knowledge their students brought into the classroom should be shared Helped their students feel safe in their classrooms and cared about their students and their learning Helped their students feel safe in their classrooms and cared about their students and their learning Were reflective about their practice Were reflective about their practice

Recognizing Cultural Experiences and Communities Was not seen in instruction or mentioned in interviews by two or more teachers Was not seen in instruction or mentioned in interviews by two or more teachers

What Did Students Experience? Agency Agency Competency and Commitment Competency and Commitment Authority Authority

Definitions a) a sense of agency over their own learning, through opportunities to define their own goals, and to submit and justify their own mathematical ideas, and to experience challenge with appropriate tasks; b) a sense of competency and commitment through support for persistent engagement in complex problems and opportunities for reflection on their mathematical thinking, and c) a sense of authority, through lessons that require students to take an active role in the creation and verification of mathematical ideas.

Agency Students –used different strategies to work out problems –felt safe asking questions about math when something does not make sense (2) –stated procedures make sense to them (2) Teachers encouraged us to –come up with new ways to solve problems (2) –figure out things for ourselves (2) –talk about why answer is not correct (2)

Sense of Competency and Commitment A class goal was to –understand how procedures work in math –improve our understanding of math –understand what we are doing to solve problems –think about how we solve problems Our teachers –asked questions that make us think –helped us understand what we were doing to solve problems – built on math we already knew (2) – gave rules to solve problems (2) We –learned to solve sets of similar problems (2)

Sense of Authority In our class –working in groups means that everyone shares our ideas –we used different strategies to work out problems –we work together to understand new math ideas (2) Teachers encouraged us to –try to understand why formulas work –invent ways to solve math problems (2) –figure out things for ourselves (2) –talk about why answer is not correct (2) Teacher’s comments help us to understand our errors (2)

Discussion During this time when little is mentioned about minority students who are achieving, it is clear that teachers can make a difference in what students learn. During this time when little is mentioned about minority students who are achieving, it is clear that teachers can make a difference in what students learn. We can be successful if we address the needs of students in mathematics classrooms as demonstrated through research. We can be successful if we address the needs of students in mathematics classrooms as demonstrated through research.

Teachers used high reform practices Pedagogy: Teacher directed and Inquiry groups Pedagogy: Teacher directed and Inquiry groups Content: Procedural & conceptual content and Procedural and Conceptual Press Content: Procedural & conceptual content and Procedural and Conceptual Press Tasks: Memorization, Procedural, and Conceptual and Student justification dominated Tasks: Memorization, Procedural, and Conceptual and Student justification dominated Assessment: Questions, student work, questions, responses, and peer and self assessments Assessment: Questions, student work, questions, responses, and peer and self assessments Interaction: Authority teacher & student, mathematizing and Interaction teacher to student, student to student Interaction: Authority teacher & student, mathematizing and Interaction teacher to student, student to student

Sense of Agency Teachers Believed that all students could learn mathematics Believed that all students could learn mathematics Valued student motivation, involvement, effort, respectful behavior, and responsibility Valued student motivation, involvement, effort, respectful behavior, and responsibility Demonstrated that knowledge students brought into the classroom should be shared Demonstrated that knowledge students brought into the classroom should be shared

Sense of Agency Teachers Presented memorization, procedural, and conceptual tasks Presented memorization, procedural, and conceptual tasks Involved fundamental concepts of the subject in lessons Involved fundamental concepts of the subject in lessons Encouraged to generate conjectures, alternative solution strategies, and ways of interpreting evidence Encouraged to generate conjectures, alternative solution strategies, and ways of interpreting evidence

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

Sense of Competency and Commitment Teachers Encouraged and valued active participation of students Encouraged and valued active participation of students Valued intellectual rigor, constructive criticism, and the challenging of ideas Valued intellectual rigor, constructive criticism, and the challenging of ideas Created a climate of respect for what others had to say Created a climate of respect for what others had to say Encouraged and valued active participation of students Encouraged and valued active participation of students Respected students’ prior knowledge and preconceptions Respected students’ prior knowledge and preconceptions

Sense of Authority Teachers Cared about their students and their learning Cared about their students and their learning Believed that all students could learn mathematics Believed that all students could learn mathematics Saw knowledge and authority in both teachers and students Saw knowledge and authority in both teachers and students Helped their students feel safe in their classrooms Helped their students feel safe in their classrooms

Sense of Authority Teachers Acted as a resource person, working to support and enhance student investigations Acted as a resource person, working to support and enhance student investigations Listened and responded to their students to anticipate their understanding and/or misunderstanding Listened and responded to their students to anticipate their understanding and/or misunderstanding Used learning communities Used learning communities Promoted student-teacher and student- student mathematical interaction Promoted student-teacher and student- student mathematical interaction

Falling Short Teachers’ instruction reflected the suggested strategies for African American student learning. Teachers’ instruction reflected the suggested strategies for African American student learning. However, the teachers fell short in the area of commitment to recognizing and using students cultural experiences and communities. However, the teachers fell short in the area of commitment to recognizing and using students cultural experiences and communities.

Questions What do you think we should do having knowledge from this research? What do you think we should do having knowledge from this research? What are large and small changes that can be made? What are large and small changes that can be made? How do we begin? How do we begin?

NCISLA. (2004).53 First “We” as Teachers and Teacher Educators Need A coherent vision of the structure of mathematical ideas and practice they are teaching structure of mathematical ideas and practice they are teaching conceptions, misconceptions, and problem-solving strategies that bring and their probable struggles conceptions, misconceptions, and problem-solving strategies that bring and their probable struggles learning trajectories students are likely to follow learning trajectories students are likely to follow tasks and tools that will provide knowledge about and support student learning tasks and tools that will provide knowledge about and support student learning scaffolding to support students to engage in sense making scaffolding to support students to engage in sense making class norms and activity structures that support learning class norms and activity structures that support learning

What is it “We” as Teachers and Teacher Educators Learn to Do? Value student discourse and verbal knowledge Value student discourse and verbal knowledge Develop activities and questioning to promote mathematical discourse among students and teacher Develop activities and questioning to promote mathematical discourse among students and teacher Encourage, support, and provide feedback to students as they learn Encourage, support, and provide feedback to students as they learn Understand that mathematics is in our students—not just in us and the books Understand that mathematics is in our students—not just in us and the books 54

55 We Must Create interdependent learning communities within the classroom Create interdependent learning communities within the classroom Expect that students can and will achieve conceptual and procedural understanding of the mathematics content Expect that students can and will achieve conceptual and procedural understanding of the mathematics content Create and use mathematical tasks that require students to “do mathematics” not just replicate procedures Create and use mathematical tasks that require students to “do mathematics” not just replicate procedures Understand that mathematics is in our students—not just in us and the books Understand that mathematics is in our students—not just in us and the books

Address the varied learning styles of our students and accommodate instruction based their learning preferences Address the varied learning styles of our students and accommodate instruction based their learning preferences Demonstrate that knowledge students bring into our classrooms should be shared Demonstrate that knowledge students bring into our classrooms should be shared Help our students feel safe in our classrooms and care about our students and their learning Help our students feel safe in our classrooms and care about our students and their learning Be reflective about our practice Be reflective about our practice

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

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