teachmathdemo.files.wordpress.com · Web viewClassroom Practices Module Table of Contents Overview of Classroom Practices Module 2 Activity 1: Analyzing Video Cases Goals and Description

Classroom Practices ModuleTable of Contents

Overview of Classroom Practices Module 2

Activity 1: Analyzing Video CasesGoals and Description 4Lesson Outline 5Instructor Reflections 7Ideas from TEACH Math 9Additional Resources 9Tools and Handouts 12

Activity 2: Analyzing Curriculum SpacesGoals and Description 31Lesson Outline 31Instructor Reflections 34Ideas from TEACH Math 34Additional Resources 35Tools and Handouts 37

Activity 3: Analyzing Mathematics Lesson PlansGoals and Description 40Lesson Outline 40Instructor Reflection 41Ideas from TEACH Math 41Additional Resources 42

Activity 4: Analyzing Mathematics LessonsDescription 43Tools and Handouts 44

References 53

Module developed by the TEACH Math Project, supported by the National Science Foundation (DRL # 1228034). Any opinions, findings, and conclusions or recommendations are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Classroom Practices Module

Overview of Classroom Practices Module

Summary of module:This module consists of four activities, each of which can be enacted multiple times throughout a mathematics methods course. The four activities are: 1) analyzing video cases, 2) analyzing curriculum spaces, 3) analyzing lesson plans, and 4) analyzing an observed lesson or one’s own teaching practice. In these activities, prospective teachers (PSTs) analyze classroom practices through four lenses: teaching, learning, task, and power & participation. These activities can be completed in any order (though we typically start with Activity 1) and are designed to be repeated multiple times over the semester to support PST learning. Goals for PSTs engaged in this module include the following:

PSTs will identify and analyze aspects of mathematics teaching and learning that support teachers eliciting and building on children’s multiple mathematical knowledge bases.

PSTs will see mathematics classrooms in more detail, suspend everyday interpretation, and learn to look for patterns that otherwise go unnoticed at a conscious level.

PSTs will identify and analyze aspects of mathematics curriculum materials with respect to eliciting and building on children’s multiple mathematical knowledge bases.

The module supports PST learning in part by introducing PSTs to two tools – Video Lenses and the Curriculum Spaces Analysis Table. The Video Lenses are used in Activities 1 and 4 to analyze enacted instruction (on video or in person) and the Curriculum Spaces Analysis Table is used in Activities 2 and 3 to analyze written representations of teaching (in textbooks or in lesson plans). Both tools focus PSTs’ attention on tasks, teaching, learning, and power and participation, as well as children’s multiple mathematical knowledge bases (MMKB).

Module Activities Overview:Activity 1: Analyzing Video CasesPre-service teachers analyze video cases using a quadruple “lens” approach (e.g. task, learning, teaching, power & participation) to analyze each case. This activity takes place over time such that PSTs analyze multiple videos using the various lenses in a rotating pattern. This activity supports PSTs to examine children’s mathematical ideas (e.g. concepts, skills, problem-solving strategies); knowledge resources (e.g. mathematical, family, community, cultural, linguistic, personal); participation and status issues; and instructional strategies that facilitate mathematical thinking and reasoning of students with varied cultural and linguistic backgrounds, math experiences and confidences.

Activity 2: Analyzing Curriculum SpacesPSTs use the Curriculum Spaces Analysis Table to analyze and adapt one or more lessons from commonly used curriculum materials to open “spaces” for eliciting, building on, and integrating children’s multiple mathematical knowledge bases.

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Activity 3: Analyzing Mathematics Lesson Plans PSTs evaluate a lesson plan they have written for its potential to open spaces for eliciting and building on children’s MMKB. PSTs use the Curriculum Spaces Analysis Table from Activity 2 for this analysis of a written lesson plan. The analysis is intended to inform instructional planning and lesson development.

Activity 4: Analyzing Mathematics LessonsPSTs use the video lenses to analyze enacted teaching. This teaching could be another teacher’s practice observed by the PST or video recordings of the PSTs’ own teaching.

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ACTIVITY 1: ANALYZING VIDEO CASES

Goals and Description of Analyzing Video Cases

GOALS for Analyzing Video Cases: The goal of this activity, ideally repeated multiple times across the semester, is for PSTs to begin to see mathematics classrooms in more detail, suspend everyday interpretation, and learn to look for patterns that otherwise may go unnoticed at a conscious level. A more specific goal is for PSTs to begin to identify and analyze aspects of mathematics teaching and learning reflective of eliciting and building on children’s MMKB.

DESCRIPTION of Analyzing Video Cases:PSTs analyze multiple video cases attending to specific foci – the teaching, student learning, the task, and power and participation patterns. Typically, our first video case has been the Marshmallow video case (Annenberg Foundation, 2013). This first enactment of the video case activity happens relatively early in the semester – often during the first or second class of the course. Other video cases are selected by the methods instructors based on course needs. One recommendation is for the final video case to be an analysis of the Questioning Data video (Annenberg Foundation, 2013). Another nice option is to revisit the Marshmallow video again at the end of the semester. Analyzing video cases from different lenses allows PSTs to begin to see what it might mean to elicit and build on children’s MMKB in instruction and supports PSTs in moving away from only looking at teaching to consider mathematics teaching and learning issues such as the role of the task, student learning, and power and participation patterns.

Possible Video Cases The following list of video cases, arranged by grade level and then alphabetically within grade levels by the title of the video, are described in more detail in Video Case Descriptions handout (Handout CP 8).

People Patterns (Annenberg) Lady Bugs (Annenberg) Amazing Equations (Annenberg) Get to 100 (Storeygard, 2009) Marshmallows (Annenberg) Buying and Cooking a Turkey (Fosnot & Dolk, 2001) Multiplication (Schifter, Bastable, & Russell, 2009) Developing Children’s Conceptions of Equality (Carpenter, Franke & Levi, 2003) Pencil Box Staining (Annenberg) Valentine Exchange (Annenberg) Sharing a Submarine Sandwich (Fosnot & Dolk, 2002) Fraction Tracks (Annenberg) Questioning Data (Annenberg)

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Lesson Outline for Analyzing Video Cases

LAUNCH (for the first enactment of the activity – Introduction to Video Lenses)The launch for this activity consists of two components: (a) a brief overview by the instructor about the video case analysis that will be occurring over time in the course and (b) a quick examination of a central question from each of the four lenses.

Instructor OverviewPSTs are informed that they will be examining mathematics teaching and learning from four perspectives – teaching, learning, task and power & participation – over time in the course. Introduction to Video Case Analysis Ask PSTs in class to spend time reviewing each of the four questions on the

Introducing the Lenses handout (Handout CP 1). Note: This introductory activity generally occurs early in the semester. As such, we suggest you direct PSTs’ attention to the four questions from each lens, rather than the center question, which you might note to PSTs you’ll return to later or you might omit completely for this introductory activity. We also encourage reviewing general guidelines for viewing and discussing video cases with the Guidelines for Discussing Video Cases handout (Handout CP 2).

Provide PSTs a specific focus question from the Introducing the Lenses handout for the viewing of the first video case. Below are some suggestions for ways to distribute the focus questions across and within small groups:Option A: Assign each small group a separate lens question.Option B: Assign each small group all four lens questions, but have individual students in each group responsible for a separate lens question.

EXPLORE (for the first enactment of the activity)The explore phase of Activity 1’s Introductory Activity consists of PSTs viewing Annenberg’s Marshmallow video using their assigned focus question(s), followed by small and whole class discussion.

Analysis of Marshmallow VideoWatch the Marshmallow Video with PSTs viewing from the perspective of their assigned lens(es). After viewing the video, give PSTs a moment to jot down notes about what they noticed in the video based on their lens(es). Have PSTs engage in discussion in their small groups about their responses to their prompts, and support PSTs in grounding their claims using evidence from the video.

SUMMARIZE (for the first enactment of the activity)To summarize the lesson, engage in a whole group discussion. One goal of this discussion is to help PSTs begin to understand the nature of the four lenses and distinctions between them. Discussion questions might include:

What is the evidence in the video you have for that claim? In what ways are our claims about teaching (teaching lens asks about ways

teacher elicits and responds to student thinking) different from claims from the learning perspective (what math understandings or confusions are indicated in

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students’ work)? What might our evidence from the video consist of for each of these areas?

LAUNCH (for subsequent enactments of the activity)The launch for this activity consists of PSTs taking time to review the lenses that will be the focus of the enactment. (Handouts CP 3 - CP 6). This might be initially done outside of class with a follow-up review in class, or entirely in class.

Assigning Lenses: For Video #2, have your class focus on two of the lenses (e.g., learning and teaching). For Video #3, have your class focus on the other two lenses (e.g., task and power & participation). Focusing on two lenses at a time allows for a discussion and understanding of how the lenses focus attention on different aspects of instruction, as well as how the lenses interact. For Video #4 (recommended: Questioning Data or Marshmallow video for 2nd time) consider repeating two lenses you most want to focus with in your class, or consider using all four lenses. If using all four lenses, instructors could again choose to use the introductory activity handout that has just one prompt from each lens. Below are some suggestions for ways to distribute the focus questions across and within small groups:

Option A: Assign each table a lens. Next, assign each student at each table a particular prompt within the lens. Once lens prompts are assigned, ask students to think about their question, review the other questions, and think about how their specific prompt is similar or different from the others. One question on each lens goes to the “resources/ knowledge bases” component and is the same for each lens. Instructors might consider assigning one student to this question or assigning all students to this question (in addition to their individual prompt). Option B: Each small group focuses on one lens in its entirety. Assign each table a lens. Ask students to read through all of the questions for their lens.

EXPLORE (for subsequent enactments of the activity)The explore phase of this lesson consists of PSTs watching the video from the perspective of their assigned lens/prompt and discussing their analyses. Recall that for Video #2 and Video #3 we recommend focusing on only two lenses at a time as a class. Another option to consider is having students watch a video multiple times on their own before class so that they are familiar with the context before being assigned a particular lens.

Small group discussion: After watching the video, have each student take a moment to jot down notes for their lens/prompt. Then, have each table spend some time discussing their thoughts for their particular lens/prompt in their small group. Encourage students to focus on grounding their comments in evidence from the video. Each small group will create a poster or set of notes to summarize their findings. They should be careful to ground their claims using evidence from the video. They should include on their poster or notes a short list of questions or unresolved issues their group faced. They should prepare to share their findings with the whole class. Note: Often the posters are bulleted lists identical to what PSTs have already written on their notes pages. Some suggested options that might help avoid this repetition are suggested below (this is not an exhaustive list):

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Encourage PSTs to create a poster that consists of pictures and/or words.

Explicitly tell PSTs that you do NOT want the poster to be identical to the lists on their notes pages. Encourage them to choose one major point to highlight and explain why that was important. Encourage students to include with this major point a question they have or an unresolved issue their group faced.

SUMMARIZE (for subsequent enactments of the activity)To summarize, discuss the two lenses as a whole group in depth. Focus on how the lenses allow you to notice different things. After some discussion, facilitate a summary discussion using some or all of the following prompts:

Reflecting across these two (or four) lenses, what resources and knowledge (e.g., mathematical, cultural, community, family, linguistic, students’ interests, peers) did students, the teacher, or the task use to support learning?

What are the similarities and differences in the ways the two (or four) lenses focused your attention on specific aspects of instruction?

A goal of this course is for you to consider ways and outcomes for integrating these multiple knowledge bases in mathematics instruction. What are the ways that the teacher and/or students integrated these knowledge bases and resources in instruction? (The phrase “integrating multiple knowledge bases” is used for the first time here- in this summary. This prompt serves to pull out and summarize key ideas that may have already been expressed in the discussion of each of the lenses.)

How did the use of these resources influence students’ learning? (Students are likely to mention engagement and motivation first. Work to connect issues of engagement and motivation with learning of important mathematics (i.e., go beyond just making learning fun). Encourage them to think about how using these resources might also influence issues of identity, competence, using mathematics for social justice/ power (critical mathematics), and changing views on the nature and value of mathematics.

What are other opportunities or ways that the teacher and/or students might have integrated students’ knowledge bases and resources in this lesson? Why might you choose to pursue these alternatives (or why not)? (This question is not intended to generate negative criticism about what the teacher did, but instead is aimed at considering alternative approaches and what they might afford for learning, challenges in implementing, etc.)

INSTRUCTOR REFLECTIONSUse your reflections to guide how you support PSTs in future activities and discussions.

What kinds of things do PSTs tend to notice about students’ strategies and mathematical thinking? How do they attend to / respond to strategies and/or algorithms that may be unfamiliar and/or unexpected? What do they notice about the nuances of students’ strategies? (In other words, what do they attend to? And in what detail do they attend to students’ mathematical thinking?)

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What kinds of things are absent in PSTs’ comments and/or written reports? (In other words, what are they not attending to or what details are they missing?)

In general, what kinds of things do PSTs tend to notice about students? What kinds of competencies did they notice? What kinds of participation did they notice? What did they notice about students’ interactions with their peers? (In other words, what do they attend to?)

What kinds of things are absent in PSTs’ comments and/or written reports? (In other words, what are they not attending to?)

What meaning or value do they seem to assign to the ways in which students participate, the nature of their social interactions, the range of skills and competencies they demonstrate (including those that may not typically be seen as relevant to mathematics?

How are PSTs positioning students’ out of school knowledge and experiences? What meaning or value do they seem to assign to mathematical activity in students’ homes or communities? (In other words, what kinds of awareness do you notice?)

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Ideas from TEACH Math Incorporating Additional Instructional Goals: In an examination of our own practice

with respect to launching the video analysis in our mathematics methods courses (Roth McDuffie et al., 2014), we have found that we often used the Launch of a particular video as a place to incorporate other instructional goals – goals that complement the project goals planned for the video analysis activity. In this way, we could incorporate the video analysis activity while attending to existing instructional goals, rather than simply adding one more activity to an already packed methods course. For example, at times we have engaged PSTs in the mathematics of the Marshmallow video by giving them marshmallows and having them do the mathematics task, attending to the mathematics content and processes entailed in the problem. Doing this early in the semester can support the goal of disrupting PSTs’ views about mathematics consisting only of procedures and may serve as a place to connect to the content and practice standards in the Common Core. As another example, with the Developing Children’s Conceptions of Equality video, we have used that video to attend to both the multiple lenses and toward a goal of PSTs exploring the mathematics related to equations and the idea of relational thinking in particular, which was then focused on in more depth within the lesson. Or, we provide the task (8 + 4 = ___ + 5) with the goal of supporting PSTs in anticipating students’ thinking, asking PSTs to predict student responses and confusions prior to watching the video.

Importance of Repeated Enactments: As noted in the overview, we have found that it is very important to enact this activity multiple times over the course of the semester, providing PSTs with opportunities to view instructional through different lenses over time and to improve their noticing skills.

One Lens: While we most often use two or four lenses as a class to view video, we have also at times had the whole class use just one lens to view and analyze video. This allows for an in-depth discussion of the lens and can be particularly useful for lenses that PSTs are struggling to use.

Written Cases: At some sites, these lenses have also been productively used for PSTs to analyze written cases of instruction.

Handout CP 7: In several of our courses, we have found this handout useful for focusing PSTs’ attention on particular discourse and questioning moves. It can also be useful for Activity 4, later in the module.

Additional Resources Aguirre, J., Turner, E., Bartell, T. G., Drake, C., Foote, M. Q., & Roth McDuffie, A.

(2012). Analyzing effective mathematics lessons for English learners: A multiple mathematical lens approach. In S. Celedón-Pattichis & N. Ramirez (Eds.), Beyond good teaching: Advancing mathematics education for ELLs (pp. 207–222). Reston, VA: National Council of Teachers of Mathematics.

Roth McDuffie, A., Foote, M.Q., Bolson, C., Turner, E.E., Aguirre, J.M., Bartell, T.G., Drake, C., & Land, T. (2014). Using video analysis to support prospective K-8 teachers’ noticing of students’ multiple mathematical knowledge bases. Journal of Mathematics Teacher Education 17(3), 245-258.

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Roth McDuffie, A., Foote, M. Q., Drake, C., Turner, E., Aguirre, J., Bartell, T. G. & Bolson, C. (2014). Use of video analysis to support prospective K-8 teachers’ noticing of equitable practices. Mathematics Teacher Educator, 2(2), 108-140.

Sherin, M. G., & Han, S. Y. (2004). Teacher learning in the context of a video club. Teaching and Teacher Education, 20, 163-183.

VanZoest, L. & Stockero, S., (2008). Using a video-case curriculum to develop preservice teachers’ knowledge and skills. In M. Smith, S. Friel, & D. Thompson (Eds.) Cases in mathematics teacher education: Tools for developing knowledge needed for teaching (Association of Mathematics Teacher Educators Monograph No. 4., pp. 117-132). San Diego, CA : Association of Mathematics Teacher Educators.

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Handouts for Activity 1 begin on the next page.

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Classroom Practices ModuleHandout CP 1 – Introducing the Lenses

What is/are the central mathematics ideas in this task? (i.e., identify specific concepts, processes, skills, problem solving strategies).

1. TASK: What makes this a good and/or problematic task? How could it be improved?

3. TEACHING: How does the teacher elicit students’ thinking and respond? (e.g., moves, questions, responses to students’ correct answers/ mistakes/ partial solutions, decisions).

4. POWER & PARTICIPATION: Who participates? Does the classroom culture value and encourage most students to speak, only a few, or only the teacher?

RESOURCES & KNOWLEDGE BASES STUDENTS USE

(e.g., mathematical, cultural, community, family, linguistic, students’ interests, peers)

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Handout CP 2:Video Case Discussion Guidelines

Looking at video: Assume responsibility for making sense of what students are

saying. Assume responsibility for making sense of what the teacher is

saying. Assume responsibility for making sense of the task/activity. Take a few minutes to think about what each student/teacher

says, how she/he says it, to whom, how it’s taken up by others, etc.

You can make some notes during and after the video about questions you have, points you want to make, segments of the video that you find interesting.

Discussing and exploring the video: Keep the conversation grounded in what you see in the video,

talk about the children and teacher in the video rather than children or teachers in general.

Use examples from the video to ground your comments, questions, or claims.

Stay focused on the group’s questions. Assume that students are making sense and that there is

knowledge and expertise in what they are saying.

Creating Summary of Findings• Create a poster that summarizes your findings• Ground your claims using evidence from the video/transcript• Be sure to include any remaining questions or unresolved issues• Be prepared to share your findings with the whole groupThese guidelines are borrowed and adapted from the Chèche Konnen Center, TERC, Cambridge, MA and J. Moschkovich’s math video case work.

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Handout CP 3: Teaching Lens

1. How does the teacher elicit and respond to students’ thinking? (e.g., moves, questions, responses to students’ correct answers/ mistakes/ confusions/ partial solutions/ approaches).

2. What opportunities does the teacher create for diverse learners to communicate their mathematical understandings - show what they know? (Consider: English learners, different genders, various cultures, varied math backgrounds and confidence levels, etc.) To what extent and how are various forms of communication are supported, encouraged, and valued (e.g., verbal and non-verbal expressions, gestures, written work (symbols, pictures, diagrams), work with manipulative/models/physical objects, etc.)? How does the teacher help to build (or diminish) students’ sense of competence and self-confidence.

3. How does the teacher implement the task in a way that maintains or changes the cognitive demand? Does the teacher’s implementation of the task maintain high cognitive demand? Alternatively, does the teacher’s implementation of the task diminish or lower the level of cognitive demand? In what ways did the teacher modify the task upon implementation that changed the cognitive demand?

4. What resources and knowledge does the teacher use/draw upon to support students’ math understanding (e.g., mathematical, cultural, community, family, linguistic, students’ interests, peers)? What does the teacher do to help students make connections within mathematics (e.g., between various mathematical concepts, processes, representations, strategies, etc.)? What does the teacher do to help students connect mathematics with relevant/authentic issues or situations in their lives? What does the teacher do to encourage students to use their own prior knowledge and/or peers as resources for learning?

5. How do the teacher’s actions support student learning? Interpret the results of your analysis above with respect to student learning.

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Handout CP 4: Learning Lens

1. What specific math understandings and/or confusions are indicated in students’ work, talk, and/or behavior? Describe concepts, processes, skills, and/or problem solving strategies that students appear to know/understand as they engage in the lesson. Describe areas of confusion and/or misunderstanding. What did the students do, say, or show that indicated these understandings or confusions?

2. How do students communicate their understandings and sense making of others’ thinking (e.g., provide justifications, explanations, respond to questions, written work, representations, manipulatives, models, gestures)? How do students share their thinking? Consider evidence of understanding and/or learning in the form of: verbal and non-verbal expressions, gestures, written work (symbols, pictures, diagrams), work with physical objects (manipulatives, models), behavior, etc.

3. In what ways does student engagement reflect conceptual and/or procedural learning?

4. What resources or knowledge do students draw upon to understand and solve the math task? (e.g., mathematical, cultural, community, family, linguistic, students’ interests, peers) What connections are students making between mathematics and their experiences and/or prior knowledge? Consider how these forms of resources/ knowledge influence learning.

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Handout CP 5: Task Lens

1. Describe the main mathematical task that students are asked to do:

2. What makes this a good and/or problematic task (e.g., What is the cognitive demand of the task? How does the task allow for multiple entry points? How does the task allow for multiple solution strategies? How does the task encourage use of multiple representations? How are students expected to show how they thought about the problem/solution? How does the task help you and students to understand and assess students’ mathematical thinking (as students work, and/or after they have a solution))?

3. What is/are the central math ideas in this task (i.e., identify specific concepts, processes, skills targeted with task)? Consider national and state standards. Do the central ideas align with students’ grade/age, prior knowledge, and/or background? Describe instances when a central idea is evident and when alignment is/ is not observed.

4. What resources and knowledge (e.g., mathematical (M), cultural (Cul), community (Comm), family (Fam), linguistic (Lang), students’ interests (SI), peers (Peer)) does this task activate and/or connect to? Code notes for forms of resources and knowledge used/activated by the task (M, Cul, Comm, Fam, Lang, SI, Peer). Consider how these connections or lack of connections influence learning and what opportunities might exist to better connect the task to students.

5. How does the task provide (or not) learning opportunities for students? Interpret the results of your analysis above with respect to student learning.

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Handout CP 6: Power and Participation

1. Who participates? Does the classroom culture value and encourage most students to speak, only a few, or only the teacher? Where does the majority of the math “work” take place in the classroom (e.g., front of room, small group, individual desks) and how does this contribute to participation? In classes characterized by high levels of mathematical discourse, there is considerable teacher-student and student-student interaction about mathematical ideas; the interaction is reciprocal, and it promotes shared understandings. Record who the teacher involves/calls on and whether students of various subgroups are proportionally represented (e.g., females and males, racial/ethnic subgroups, etc.).

2. Who holds authority for knowing mathematics? Do some students hold more status than others? Does the teacher have the final word on correct answers/solutions? Are students encouraged to convince themselves or others? Are some students higher/ lower status than others? Are some students assumed to “know” or “be correct” or “be better” at math? Are some students sought out or avoided as partners or group members? Who assigns high/low status (teacher, other students) and on what basis (academic performance, social standing, athletic performance)?

3. What evidence indicates that differences in approaches and perspectives are/ are not respected and valued? Is a sense of respect for all members evident? Is there an effort to understand one another’s perspectives? Is there evidence that members are aware of and value different educational experiences and backgrounds (e.g., mathematical, cultural, community, family, linguistic, students’ interests, peers)

4. How do issues of participation, authority, and status (above) influence learning? Interpret the results of your analysis above with respect to student learning.

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Handout CP 7: Types of Teacher Moves, Prompts, Questions, and Responses

(Adapted from Neumann, M. (2007) and Jacobs & Ambrose (2008))Type and Description Examples

Academic Praise “Interesting strategy.”“I like your thinking in solving that problem.”

Nonacademic Praise “You’re being nice and quiet today.”“I like how you put your name at the top of your test.”

Facilitating HelpTeacher facilitates learning by providing students with suggestions, hints and cues that enable them to complete the work themselves.

“How does solving 60 times 50 help you solve 62 times 51?”“Looking at the hundreds chart, what do you notice about the numbers that follow numbers that end in 9?”

Short-circuitsTeacher (in attempting to provide help) prevents or short-circuits student’s success by taking over the learning process.

“Give me your pencil. When multiplying, you first…”“You’ve got this part wrong – 60 times 50 is 3000, not 300.”

Academic CriticismTeacher directs critical remarks at the lack of intellectual quality OR teacher attributes academic failure to lack of effort.

“I don’t think you’re good at mathematics.”“This is a simple problem that you got wrong.” “You could do the math if you just put your mind to it and worked harder.”

Nonacademic Criticism(Mild or Harsh) Teacher makes negative comments about violations of conduct, rules, and forms; behaviors; and other nonacademic areas.

“Megan, you need to raise your hand.”“Tom, stay in line.”“Tom, I told you to get in line! I don’t want to talk to you again about this. The next time I say something, no recess!”

Academic InformationTeacher gives information related to the lesson content.

“The sum of the interior angles for any triangle is 180 degrees.”

Nonacademic InformationTeacher gives information that is procedural or related to classroom management.

“I need you to put your desk there, into that group of 4, for today’s lesson.”

Repeating or RevoicingTeacher repeats a student’s answer or comment verbatim, rather than asking the student to repeat it OR teacher restates a student’s response and adds new language or advances the response for a specific purpose.

S: “I got 36”T: “Okay, Sarah said 36 is the answer.”

S: “The distance around is 24.”T: “Tom found a perimeter of 24 inches.”

Supporting and Extending Mathematical Thinking Before a Correct Answer is Given (with subcategories):

“What do you know about the problem?”“I know you like baseball. Can you tell me about batting averages? [Teacher listens].

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1. Teacher ensures that a student understands the problem by asking what the student knows about it, rephrasing, and/or relating it to a familiar context.

2. Teacher changes the mathematics in the problem to match student’s level of understanding by using easier/harder numbers or by changing the structure.

3. Explore student’s work to see what he/she has done so far.

4. Remind the student to use other strategies.

Think about how calculating batting averages might connect to the mean in this problem?”

“Tell me about what you have tried so far... How did you decide to start with 5 for this part (referencing specific parts of the solution)?”

“Last week you tried making a table to see the patterns in how the numbers are changing. Would that strategy help you with this problem?”

Supporting and Extending Mathematical Thinking After a Correct Answer is Given (with subcategories):

1. Teacher promotes reflection on strategies used by asking for explanations and posing clarifying questions about details (connecting strategies, quantities, and relationships in the problem).

2. Teacher encourages students to explore multiple strategies and connections between strategies.

3. Teacher connects students’ thinking to symbolic notation by asking for a number sentence to go with a problem and by asking students to record strategies.

4. Teacher generates follow up problems that are linked to a completed problem by using more challenging numbers and encouraging more sophisticated strategies.

“What made you think to use this strategy for this problem?”“When do you think this strategy works best?”“How did making a table help you predict what would happen next?”

T: “When you made your bar graph for students’ favorite foods, what did you notice?”S: “The bar for pizza was tallest.”T: “Why was it tallest?”S: “From our survey, pizza has the most tally marks on our chart. So more children chose pizza than any other food. “

“Can you write a number sentence to show what you did?”“Can you label your equation to show what the numbers represent from the problem?”

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Handout CP 8: Description of Videos Used in TEACH MATH Video Analysis Activity

Note: Videos are arranged first by grade level and then alphabetically within grade levels by title of video.

Clip Title People PatternsBrief Summary The teacher first reviews the meaning of patterns. She then

uses some students to illustrate patterns (i.e., boy-girl-boy-girl) and asks the remaining students to describe and identify the pattern they see. Then students work in small groups to make their own pattern, and to represent their pattern on paper using drawings, words, or symbols. The lesson ends with a group discussion of the patterns students generated.

Reason for Inclusion in Set of Videos

Early childhood example. Racially/ethnically diverse urban classroom (predominantly African American students along with some White and Asian students). Students are engaging in significant mathematical analysis with multiple representations.

Grade Level / School Type

Kindergarten/urban elementary

School Location Milwaukee, WisconsinMath Content Focus Identify, Making and Extending PatternsLength of Clip 13 minutesPublic Availability Available as part of Annenberg Media’s “Teaching Math: A

Video Library, K-4.” Entire set of 11 DVDs is available for purchase at www.learner.org.Also available on-line as a “video on demand” file at: http://www.learner.org/resources/series32.html (Clip #38)

Additional Discussion Prompts

What is the value of asking children to represent patterns on paper? What is value of asking children to represent patterns using their peers?

How did the task design create opportunities for students to draw on home or community-based knowledge?

What else might the teacher have done to privilege multiple knowledge bases in this lesson?

Related Readings Economopoulos, K. (1998). The mathematics of pattern in kindergarten. Teaching Children Mathematics (5) 4, 230-233.

Clip Title LadybugsBrief Summary Before this lesson, students were asked what they wanted to

learn about, and they chose ladybugs. In this lesson, after observing ladybugs with a magnifying glass, students make bar graphs and a class chart to record the number of heads, wings, feet and antennas ladybugs have. At the end of the

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http://www.learner.org/resources/series32.html

http://www.learner.org/


lesson, the teachers facilitate a whole class discussion that helps students make connections among real objects, diagrams, and numerals.


Racially/ethnically/linguistically diverse classroom (predominantly Latino students, many English learners, some Native American students, some White). Students’ ideas and interests are solicited to motivate learning. Mathematical connections to the world and among representations are emphasized.


1st grade/urban elementary

School Location Tucson, ArizonaMath Content Focus Collecting, organizing, representing and analyzing dataLength of Clip 14 minutesPublic Availability Available as part of Annenberg Media’s “Teaching Math: A



What are the pros and cons of having children choose their own topics for investigation? What tensions might arise?

How did students use home and community-based knowledge to support their participation in the lesson?

How did the teachers create an environment that supported students with varied linguistic backgrounds?

Related Readings -Torres-Velasquez, D. (2005). Culturally responsive mathematics teaching and English Language Learners. Teaching Children Mathematics (11)5, 249-255.-Kahn, L. & Civil, M. (2001). Unearthing the mathematics of a classroom garden. In E. McIntyre, A. Rosebery, & N. González (Eds.), Classroom diversity: Connecting school to students’ lives (pp. 37-50). Portsmouth, NH: Heinemann.

Clip Title Amazing EquationsBrief Summary Students investigate the concepts of addition and subtraction

as they generate story problems for the number of the day. Students use everyday language and home/community experiences to pose problems. At the end of the lesson, students share their problems, including a pictorial and a symbolic representation of their problem, with their peers.


Racially/ethnically diverse urban classroom (predominantly African-American, also Latino, Asian, and White students). Teacher elicits story problems based on students’ experiences, interests, and mathematical knowledge.

Grade Level / 1st and 2nd grades/urban elementary

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School TypeSchool Location Milwaukee, WisconsinMath Content Focus Addition/subtraction problem solvingLength of Clip 14 minutesPublic Availability Available as part of Annenberg Media’s “Teaching Math: A



What does Ms. Pearson do to support collaboration among students?

What evidence do you see that students are drawing on home-based or community knowledge as they pose problems?

What strategies does Ms. Pearson use to support all students’ participation in whole class discussion?

Related Readings -Lo Cicero, A. M., De La Cruz, Y., & Fuson, K. (1999). Teaching and Learning Creatively: Using Children’s Narratives. Teaching Children Mathematics (5)9. 544-547. -Lo Cicero, A. M., Fuson, K. & Allexsaht-Snider, M. (1999). Mathematizing children’s stories, helping children solve word problems, and supporting parental involvement. In L. Ortiz-Franco, N. Gonzalez and Y. De La Cruz (Eds.) Changing the faces of mathematics: Perspectives on Latinos (pp. 59-70). Reston, VA: NCTM.

Clip Title Get to 100Brief Summary A second-grade teacher introduces the game “Get to 100”

from Investigations in Number, Data, and Space curriculum materials (Russell et al., 2008). The game involves students taking turns rolling two dice (the dice are labeled with multiples of 5: 5, 10, 15, and 20). The students move their game pieces on a 100 chart, advancing their pieces based on their rolls. As they play, students record their rolls in a continuous number string, working between the symbolic representation of their rolls and the cumulative sum as represented by the location of the piece on the 100 chart. The object of the game is to get to 100 on the chart. The video shows the teacher working with a pair who are struggling to navigate the 100 chart, keep track of their moves, and add on to a previous sum. The teacher uses focused questioning and elicits students’ thinking and explanations.


The interaction features a teacher starting from and building on students’ mathematical thinking. Teacher moves and the use of the 100 chart representation support students in

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working through confusions.Grade Level 2nd gradeSchool Location Information not provided.Math Content Focus Addition with multiples of 5 and 10, counting by groups,

structure of the base ten number system.Length of Clip 14 minutesCommercialAvailability

Available on the CD that accompanies the following text:Storefygard, J. (2009). My kids can. Portsmouth, NH: Heinemann.


How did the teacher elicit students’ understandings and confusions?

How did the teacher incorporate this information in deciding what to do next?

How did the teacher encourage the students to listen to and support each other’s learning?

How did the students use number strings and the 100 chart together to understand addition and patterns in the base ten number system?

Related Readings Original lesson can be found at: Russell et al. (2008). “How many tens? How many ones?” Investigations in number, data, and space (2nd Grade). Glenview, IL: Pearson, Scott Foresman.

Clip Title MarshmallowsBrief Summary The video begins with a second grade class in Tucson

graphing the number of marshmallows they think they can eat on an upcoming class camping trip. After this discussion and figuring out the average number of marshmallows students will eat, they then move to figuring out how many students can be fed from one bag of marshmallows and ultimately the number of bags of marshmallows that the class will need. Spanish and English (although mainly English) are used in the classroom.


Racially/ethnically/linguistically diverse classroom (predominantly Latino students, many English learners, some Native American and White students). Both English and Spanish are used in instruction. Lesson links to knowledge gained at home through discussion with parents/guardians. Students use and explain multiple solution methods developed through group work.

Grade Level 2nd gradeSchool Location Tucson, ArizonaMath Content Focus Graphing, whole number operations, statistics and

probability, reasoning, problem solvingLength of Clip 17 minutesAvailability Available as part of Annenberg Media’s “Teaching Math: A

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How was the task posed and developed as the lesson progressed?

How did the learning environment encourage student reasoning?

Discuss student discourse regarding leftover marshmallows.

Clip Title Buying and Cooking a TurkeyBrief Summary This unit presents a series of investigations that includes

determining the cost of a turkey and the fruits and vegetables included in a Big Dinner. One investigation examines the cost for purchasing a 24-pound turkey at $1.25 per pound. In another investigation, students determine how long this 24-pound turkey should cook, at 15 minutes per pound. The teacher introduces the context for learning, supports an investigation, and facilitates a math discussion. The investigation concludes with a teacher interview, during which the teacher discusses her teaching moves and language.


Racially/ethnically diverse urban classroom (mix of White, Asian, African American and Latino students). Problem is posed as an authentic problem for the teacher. Students and prospective teachers can solve problem in a variety of ways: repeated addition, algorithm for multiplication, doubling, doubling and halving, a ratio table, and so forth.


3rd grade/urban classroom

School Location New York City (Manhattan)Math Content Focus Multiplicative thinkingLength of Clip Multiple clips availableCommercialAvailability

Video accompanies the following text: Fosnot, C. T., & Dolk, M. (2001). Young mathematicians at work: Constructing multiplication and division. Portsmouth, NH: Heinemann.


What is the mathematical learning goal for this lesson? Why is the choice of 24 and $1.25 significant? Why do you think Dana (the teacher) has chosen the

follow-up (Cooking a Turkey) investigation? How is the scenario different from and yet similar to the earlier investigation?

Clip Title MultiplicationBrief Summary Four clips of student interactions with peers and the teacher

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displaying multiple strategies to solve multiplication problems (word and numeric).The main tasks are the following:• Third grade (How many legs are on 29 elephants) 2 students present strategies, and teacher uses of clarifying questions and questions that elicit student thinking.• Fifth Grade: 29 x 12. Brief clip shows one fifth grade girl sharing her strategy with her partner. She worked with 30 x 12 instead of 29. Second brief clip shows teacher asking another student to explain this solution. Teacher uses questions to prompt student discourse and push reasoning & justification.• Fifth Grade 36 x 17. A fifth grade boy shares his strategy for solving 36 x 17 . He first envisions the problem in a familiar context: I have 36 bowls and 17 cotton balls in each one. Then, he breaks 36 apart (10 x 17, 10 x 17, 10 x 17, 6 x 17) and does a series of easier multiplication calculations (partial products) to find his answer.


Racially/ethnically/linguistically diverse classrooms (highlighting the work of African American and Latino students in an urban school). Teacher moves that help students get “unstuck,” push student thinking and refine math arguments. Features short examples of students communicating their mathematical ideas and justifying their reasoning.

Grade Level/School Type

3rd and 5th grades/urban elementary

School Location New YorkMath Content Focus Multiplication with two digits quantities.Length of Clip 10 minutes - total time to see all clips. Individual clips are

short.CommercialAvailability

Video that accompanies the following text: Schifter, D. Bastable, V., and Russell, S.J. (2009). Developing Mathematical Ideas (2nd Edition), Number and Operations, Part 1: Building systems of Tens: Calculating whole numbers and decimals. Boston, MA: Pearson.


• What teacher moves did you find powerful? Why?• What was a powerful example of student learning in these

clips? What made it powerful?Related Readings • Russell, S. J. (2000). Developing computational fluency with

whole numbers. Teaching Children Mathematics, 7(3), 154-158.

Clip Title Developing Children’s Conceptions of EqualityBrief Summary A fourth grade bilingual (English/Spanish) teacher has her

first discussion about the equal sign with her students. The

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teacher uses a variety of open number sentences and true/false number sentences to elicit students’ understandings about the meaning of the equal sign. When children express problematic conceptions or confusions, the teacher poses problems to challenge those conceptions/confusions. The clip includes excerpts from a 25-minute whole class discussion.


Racially/ethnically/linguistically diverse classroom (predominantly Latino students, many English learners). Use of both Spanish and English in instruction/discussion. Teacher facilitation supports students in arriving at an understanding of the equal sign.


4th grade/urban elementary

School Location Los AngelesMath Content Focus Algebraic Reasoning; Understanding the equal sign as a

statement of equalityLength of Clip 10.5 minutesCommercialAvailability

Available on the CD that accompanies the text: Carpenter, T. P., Franke, M., & Levi, L. (2003). Thinking Mathematically: Integrating Arithmetic and Algebra in the Elementary School. Portsmouth, NH: Heinemann.


How did the teacher respond to children’s problematic conceptions or confusions? Why do you think she posed the series of tasks that she posed?

What kinds of support did the teacher provide for students with varied levels of proficiency in English?

What might the teacher have done to privilege multiple knowledge bases (cultural, familial, community) in this lesson?

Related Readings -Falkner, K., Levi, L., & Carpenter, T. (1999). Children’s understanding of equality: A foundation for algebra. Teaching Children Mathematics(6)4, 232-236.

Clip Title Pencil Box StainingBrief Summary Fourth-graders are faced with the task of finding out how

much stain to buy from the hardware store and encounter problems as they work with many mathematical ideas in the context of a real application.


Racially/ethnically diverse classroom (primarily White students, some African American students). Problem is a real-world problem designed to connect with students’ experiences and interests. Students draw on multiple content areas within mathematics to solve the problem.

Grade Level 4th grade

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School Location Lexington, MassachusettsMath Content Focus Measurement, fractions and decimals, problem solving,

reasoningLength of Clip 16-17 minutesPublicAvailability

Available as part of Annenberg Media’s “Teaching Math: A Video Library, K-4.” Entire set of 11 DVDs is available for purchase at www.learner.org.Also available on-line as a “video on demand” file at: http://www.learner.org/resources/series32.html (Clip #27)


How did students differ in their approaches to solving the problem?

In what ways is mathematics connected to the outside world?

How did Mr. Levy respond to the fact that students came up with different solutions?”

Clip Title Valentine ExchangeBrief Summary A bilingual (English-Spanish) fourth-grade class uses a

Valentine’s Day card exchange problem to explore mathematical relationships and share problem solving strategies. Students start with a smaller version of the task – calculating the number of exchanges among 3 students – and then calculate the number of cards needed if all students in the class (24) exchange valentines. During their problem solving, students discover patterns and are asked to explain their strategies and reasoning.


Racially/ethnically/linguistically diverse classroom. Shows all parts of a problem-based lesson (Launch, Explore, Summary) with students explaining their thinking throughout. Latina student elaborates correct solution.

Grade Level/School Type

4th Grade/urban elementary

School Location Tucson, ArizonaMath Content Focus Patterns, Problem SolvingLength of Clip 14 minutesPublic Availability Available as part of Annenberg Media’s “Teaching Math: A



How did the teacher elicit student thinking without imposing her own ideas?

How appropriate was this problem for the students? Explain your reasoning.

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In what ways is language (Mathematics, Spanish) used as a resource for learning and teaching math?

How did the task, materials, and environment all facilitate problem solving in this lesson?

Related Readings -Moschkovich, J.N. (1999) Learning mathematics in two languages. In Walter Secada (Ed) Changing the faces of mathematics: Perspectives on Multiculturalism and gender equity. (pp. 85-93). Reston, VA: NCTM.-Garrison, L. (1997). Making the NCTM Standards work for emergent English speakers. Teaching Children Mathematics (4)3, 133-138.

Clip Title Sharing a Submarine SandwichBrief Summary Students engage in a math workshop; the teacher introduces the

context for learning, supports an investigation, and facilitates a math discussion. In the investigation, students work in small groups to determine if the distribution of submarine sandwiches on a field trip was fair and to figure out what portion of a sandwich each person received. The investigation concludes with an interview with the teacher, in which the teacher discuss her teaching moves and language.


Racially/ethnically diverse urban classroom (mix of Latino, White, and African American with a small percentage of American Indian, Alaskan, Asian American or Pacific Islander). Provides a variety of direct modeling solutions by upper grade students who are exploring comparisons of fractions non-algorithmically.

Grade Level 4th-5th grade classroomSchool Location New Rochelle, New YorkMath Content Focus FractionsLength of Clip Multiple clips availableCommercialAvailability

Video accompanies the following text: Fosnot, C. T., & Dolk, M. (2002). Young mathematicians at work: Constructing fractions, decimals, and percents. Portsmouth, NH: Heinemann.


What are the key features of the teacher’s launch of this task? In what ways are the number choices important?

Focus on teacher questioning: What questions is the teacher asking? What talk moves is the teacher employing?

Clip Title Fraction TracksBrief Summary Using a board game, a fifth-grade class studies and practices

fractions and equivalent fractionsReason for Inclusion in Set of Videos

Racially/ethnically diverse classroom. Features a game situation as a way to connect to students’ interests and experience. The game is set up as cooperative, rather than

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competitive, as a way to promote students’ competence and minimize status issues.

Grade Level 5th gradeSchool Location Brookline, MassachusettsMath Content Focus Fractions, problem-solvingLength of Clip 15 minutesPublicAvailability



What are the pros and cons of using cooperative games? How do materials, such as fraction pieces, enhance student

understanding?

Clip Title Questioning DataBrief Summary The lesson begins with a whole group discussion of a local

newspaper article, and a poll that students conducted related to the article. (The article described local mall owners’ desire to institute a dress code for shoppers.) Next, students work in small groups to interpret and write about graphs from various newspapers OR to organize data from surveys they designed about topics of personal interest. The lesson ends with a whole group discussion of students’ interpretations and questions.


Racially/Ethnically Diverse classroom (predominantly Latino students, some White students). Lesson connects to a current issue in the students’ community and draws on students’ interests as they choose graphs from newspapers to explore data displays. The teacher positions a student who struggled with a representation as an expert in appropriate use of that representation for another group.

Grade Level 4th-6th gradeSchool Location Tucson, ArizonaMath Content Focus Collecting, organizing, representing and analyzing dataLength of Clip 16 minutesPublicAvailability



• How did the students draw upon critical knowledge?• How did (or how might have) students’ mathematical activity

enhance their critical understanding of important local, national, or global issues?

• How did (or how might have) students’ critical understandings

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about a topic impact the way they made sense of the mathematics?

Related Readings(see critical case #7 on the previous page)

-Turner, E. & Font Strawhun, B. (2007). Posing problems that matter: Investigating school overcrowding. Teaching Children Mathematics (13)9, 457-463.-Turner, E. & Font Strawhun, B. (2004, Winter). “With Math it’s like you have more defense” Rethinking Schools online, http://www.rethinkingschools.org/archive/19_02/math192.shtml

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ACTIVITY 2: ANALYZING CURRICULUM SPACES

Goals and Description of Analyzing Curriculum Spaces

GOALS for Pre-Service Teachers (PSTs):PSTs will integrate understandings and practices related to children’s multiple mathematical knowledge bases and the use of mathematics curriculum materials through this activity. Specifically, PSTs will identify and enact adaptations to curriculum materials that create “spaces” for eliciting and building on children’s multiple mathematical knowledge bases.

DESCRIPTION of Analyzing Curriculum Spaces: PSTs will complete analyses of curriculum materials in groups of 2-4 students during class. PSTs will analyze one or more curriculum lessons using a Curriculum Spaces Analysis Table. There are two versions of this table (Handouts CP 9 and CP 10). The longer version supports PSTs on identifying spaces within the materials and designing adaptations for leveraging and expanding those spaces. The shorter version focuses PSTs on specific strategies that can be used to open spaces in curriculum materials. A lesson we have frequently used for this activity is the Everyday Mathematics lesson (Grade 4, 6-2 Strategies for Division) featured in our practitioner article, Three Strategies for Opening Curriculum Spaces (Drake et al., in press). PSTs might then complete the curriculum analysis activity and table for other lessons.

Lesson Outline for Analyzing Curriculum SpacesLAUNCH:Pass out the Curriculum Spaces Analysis Table (Handout CP 9 or CP 10) for PSTs to record their thoughts.

Introduction to Curriculum Spaces TableIntroduce the notion of a “space” in curriculum (from our article Three Strategies for Opening Curriculum Spaces (Drake et al., in press)) and consider a couple of example spaces for initial discussion (e.g., space to make a real-world connection or space for students to share their mathematical thinking) and PSTs’ initial thoughts about the nature of the space (at this point, any comments from PSTs are welcome). To start the discussion, you might consider having PSTs read the Drake et al. (in press) article either before class or as a jigsaw activity during class. You may prompt PSTs to create a list of possible curriculum adaption strategies and questions they would ask themselves when evaluating or modifying a textbook lesson. Some prompts that we have found helpful for this discussion include the following:

What factors need to be considered when selecting and/or adapting a mathematical task?

Identify and describe at least 5 different ways to adjust curriculum to meet students' needs.

Describe three different ways to minimize status issues in a lesson.

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Introduce and discuss one or both versions of the Curriculum Spaces Analysis Table as a class to see if students have questions before beginning.

EXPLORE:PSTs analyze the mathematics lesson Strategies for Division from Grade 4 Everyday Mathematics (UCSMP, 2007, p. 406-411) – or any other textbook lesson chosen by the instructor - in small groups. They complete the Curriculum Spaces Analysis Table as they work. They might also explore connections between the lesson and state standards, as well as connections to what they have learned about children’s strategies for solving the type(s) of problem(s) in the lesson.

OR

As a whole group discussion, analyze the selected textbook lesson together using the Curriculum Spaces Analysis Table. Guide PSTs to notice places where connections to mathematical thinking and children’s funds of knowledge takes place (or is limited), various strategies introduced, potential mathematical pitfalls or errors in the text, as well as consistency between lesson objectives, task(s), and learning outcomes. Then discuss ways to open up the space by adapting the task. Following the whole group discussion, have PSTs select another textbook lesson and conduct the same analysis in pairs or small groups.

SUMMARIZE:The summary part of this activity consists of a debriefing discussion in class, either in small or whole groups. Present the following prompts to guide discussion around PSTs’ analyses of mathematics curriculum materials. PSTs might also address these questions in individual written reflections. To support instructor facilitation of the discussion, it is helpful for instructors to be familiar with the adaptations proposed in the Teaching Children Mathematics article (Drake et al., in press) and to suggest any adaptations that are not raised by PSTS.

Looking over your analysis of the curriculum materials, what stands out for you as especially important for making the lesson:

o effective in promoting students’ learning? Please explain why and what possible outcomes you envision.

o ineffective or limited for promoting students learning? Please explain why and what possible outcomes you envision.

For your above comments, as you consider specific students or sub-groups of students (e.g., think about classes in which you have spent time), do you have any additional reflections?

What kinds of adaptations did you propose for this lesson and why? What are the strengths and limitations of using this tool to analyze curriculum in

ways that are responsive to children’s math thinking and multiple resources/funds of knowledge?

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After completing this activity, do you think it is possible for teachers to use curriculum materials in ways that are responsive to children’s mathematical thinking and children’s many resources and knowledge bases? If so, how? If not, why not?

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Instructor Reflections on Analysis of Curriculum Spaces. Use your reflections to guide how you support PSTs in future activities and discussions.

1) What kinds of things do PSTs tend to notice about the various curricula (In other words, what do they attend to?).

2) What kinds of things are absent in PSTs’ comments and/or written reports? (In other words, what are they not attending to?)

3) How are PSTs making sense of what they learn and know about students and their communities to consider the ways of adapting curriculum materials? How are they positioning students’ out of school knowledge and experiences? What meaning or value do they seem to assign to mathematical activity in students’ homes or communities? (in other words, what kinds of awareness do you notice?)

Ideas from TEACH MathThe descriptions below are suggested preparation activities for PSTs that you might engage in if you have time in your course.

Exploration of Curriculum MaterialsIt would be helpful if PSTs spent some time before this module exploring curriculum materials – the features of different curriculum materials, comparing and contrasting different curriculum series, etc.

Discussion of Features of Good Mathematics Tasks (and discussions of related readings)We have found it helpful for PSTs to consider, read about, and discuss important features of good mathematics tasks, including tasks having multiple entry points, meaningful contexts, and requiring high levels of cognitive demand. We have also found it helpful to have this discussion while simultaneously referring to and introducing the Curriculum Spaces Analysis Table to the PSTs.

Cognitive Demand Task Sort [highly recommended]A useful preparation activity would be the cognitive demand task sort (Smith, Stein, Arbaugh, Brown & Mossgrove, 2004).

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Additional ResourcesArbaugh, F. & Brown, C.A. (2004). What makes a mathematical task worthwhile? Designing

a learning tool for high school mathematics teachers. In Rheta N. Rubenstein & George W. Bright (Eds)., Perspectives on the Teaching of Mathematics, Sixty-sixth yearbook (pp. 27-41). Reston, VA: NCTM.

Boston, M.D. & Smith, M.S. (2009). Transforming Secondary Mathematics Teaching: Increasing the Cognitive Demands of Instructional Tasks Used in Teachers’ Classrooms. Journal for Research in Mathematics Education, 40(2), 119-156.

Breyfogle, M.L. & Williams, L.E. (December, 2008). From the classroom: Designing and implementing worthwhile tasks. Teaching Children Mathematics, 15(5), p. 276.

Drake, C., Land, T., Bartell, T.G., Aguirre, J., Foote, M.Q., Roth McDuffie, A. & Turner, E. E. (in press). Three strategies for opening curriculum spaces: building on children’s multiple mathematical knowledge bases while using curriculum materials. Teaching Children Mathematics.

Featherstone et al. (2011) Chapters 5-6. Addressing status issues through lesson design & addressing status issues during the lesson. In Smarter Together!: Collaboration and Equity in the elementary math classroom. Reston, VA: NCTM.

Jackson et al (2012). Launching complex tasks. Mathematics Teaching in the Middle School 18(1) 24-29.

Karp, K. & Howell, P. (2004). Building responsibility for learning in students with special needs. Teaching Children Mathematics. 11(3) 118-126.

Roth McDuffie et al, (2011). Tailoring tasks to meet student needs. Mathematics Teaching in the Middle School. 16(9), 550-555.

Smith, M. (2004). Beyond presenting good problems: how a Japanese teacher implements a mathematics task. In Rheta N. Rubenstein & George W. Bright (Eds)., Perspectives on the Teaching of Mathematics, Sixty-sixth yearbook (pp. 96-106). Reston, VA: NCTM.

Smith, M.S., Bill, V. & Hughes, E.K. (2008). Thinking through a Lesson: Successfully Implementing High-Level Tasks. Mathematics Teaching in the Middle School, 14(3), October 2008.

Smith, M.S., Hughes, E.K., Engle, R.A., & Stein, M.K. (May 2009). Orchestrating Discussions. Mathematics Teaching in the Middle School, 14(9), p. 548.

Smith, M. S., Stein, M. K., Arbaugh, F., Brown, C. A., & Mossgrove, J. (2004). Characterizing the cognitive demands of mathematical tasks: a task-sorting activity. In G. W. Bright & R. N. Rubenstein (Eds.), Professional Development Guidebook for Perspectives on the Teaching of Mathematics: Companion to the Sixty-sixth Yearbook (pp. 45 – 72). Reston, VA: NCTM.

Stallings, L.L. (2007). See a different mathematics. Mathematics Teaching in the Middle School, 13(4), p. 212.

Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2000). Implementing Standards-based Mathematics Instruction: A casebook for professional development. Reston, VA: NCTM.

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Handouts for Activity 2 follow this page.

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Classroom Practices ModuleHandout CP 9: Curriculum Spaces Analysis Table

Title and Source of Lesson:

QUESTIONS

OVERALL LESSONWhat are the central mathematical goals or ideas of this lesson?

QUESTIONS for LESSON PHASES LAUNCH EXPLORE SUMMARY LESSON PERIPHERIES1

1) What makes the task(s) in each phase of the lesson good and/or problematic? Consider: Multiple entry points Representations used Level of cognitive demand Language supports Alignment with lesson goal(s)

2) Where are opportunities for activating or connecting to family/cultural/community knowledge in each phase of the lesson?

3) How does each phase of the lesson open spaces for making real-world connections?

Do students have opportunities to make

1 Lesson peripheries are anything in the textbook lesson that is not part of the “main” launch/explore/summarize lesson. The peripheries are often ideas for differentiation or extension, typically found in the margins of the teacher’s guide and/or at the top or bottom of the page and/or set apart from the rest of the text by a box, shading, or other formatting.

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Classroom Practices Moduletheir own connections?

4) Where are opportunities for students to make sense of the mathematics and develop/use their own solution strategies and approaches?

5) What kinds of spaces exist for children to share and discuss their mathematical thinking with the teacher and the class?

How does the lesson create opportunities for students with varied mathematical and linguistic backgrounds and confidences to communicate their math understanding?

6) Where does the math authority reside in the lesson (e.g. only with teacher, only with textbook, only a few students, shared among teacher and students)?

POSSIBLE LESSON ADAPTATIONSGiven your responses above, what kinds of adaptations might you make to each of the phases of the lesson?What kinds of adaptations might you make to the overall lesson structure or order?

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Classroom Practices ModuleHandout CP 10: Curriculum Spaces Analysis Table (Short)

Strategies for Opening Curriculum Spaces forChildren’s Mathematical Thinking and Children’s Funds of Knowledge

#1 Rearrange Lesson ComponentsFind lesson components that focus on 1) having students make connections between the task and their prior knowledge and experiences, 2) providing support for students to develop their own strategies, and 3) encouraging students to share and explain their strategies.

Once you identify these components, think about ways to rearrange the lesson.- Frontload Problem Solving. Make problem solving tasks the “main tasks” of the lesson (rather than at the end, or for homework)- Consider Omitting Lesson Components that “tell,” “direct” or “show.” Use extra time to elicit children’s experiences and strategies.

#2 Adapt Tasks-Adjust numbers or offer multiple number choices (students can select numbers that are “just right” for them)-Encourage multiple strategies and/or multiple representations-Adjust task contexts so that they reflect students’ experiences (consider shared school experiences, home and community experiences)-Ask students to help generate task contexts (Ask students to identify mathematical ideas/practices at home or in the community; ask them to write about these examples of mathematics, or take pictures. Use students writing or photos to generate contexts for future tasks.)-Create openings in tasks where students can contribute their own ideas, experiences and information (ask students to collect information from home or from the community that they will need to work on the task)

#3 Ideas for Opening Spaces Across Lesson PhasesLAUNCH EXPLORE SUMMARY

-Motivate tasks via multiple representations such as photos, videos or stories-Encourage students to share what they notice about these photos or stories, as a way of inviting students into the task context-Ask students to brainstorm possible strategies or approaches to the task-Elicit students’ experiences, stories and practices related to the task, or the central mathematical idea of the lesson-Leave portions of the task open; complete task details using students ideas and experiences

-Invite students to solve tasks in multiple ways, and/or to represent their thinking about the task using multiple representations-Invite students to adjust and adapt tasks based on their experiences with the context or mathematical ideas-Probe students’ thinking, including their reasoning about the task and its solution-Include scaffolds that support the participation and learning of students with varied mathematical and linguistic backgrounds (access to tools, bilingual peers, etc)

-Ask students to compare, contrast and connect different strategies for solving the task-Talk with students about how the strategies they used to solve the task compare to how they (or their families) would approach the task at home, or in the community-Discuss the solution, including how real-world considerations might impact whether the solution is reasonable, possible, or realistic

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ACTIVITY 3: ANALYZING MATHEMATICS LESSON PLAN(S)

Goals and Description of Analyzing Mathematics Lessons Plan(s)

GOALS for Analyzing Mathematics Lesson Plan(s): PSTs will identify and analyze aspects of a mathematics lesson plan that support teachers eliciting and building on children’s multiple mathematical knowledge bases.

DESCRIPTION of individual activity: PSTs use the Curriculum Spaces Analysis Table (see Handout CP 9 and/or CP 10 from Activity 2) to evaluate their own mathematics lesson’s potential to support mathematics learning and elicit and build on children’s multiple mathematical knowledge bases. The analysis is designed to inform PSTs’ instructional planning and lesson development. Be explicit with PSTs that it is not necessary to address every aspect of this table in a lesson plan, but rather to be clear on what their lesson plan does and does not address so that they can speak to those issues and the potential related implications. In addition to a completed Curriculum Spaces Analysis Table, PSTs should also reflect in writing on the suggested prompts detailed below. This activity could be completed multiple times during the course.

Lesson Outline for Analyzing Mathematics Lesson Plan(s)

LAUNCH:The launch for this activity is a brief overview given by the instructor of the assignment.

EXPLORE:The explore part of this activity consists of PSTs beginning an analysis of a mathematics lesson plan in class and continuing their analysis outside of class. After PSTs have been introduced to the idea of using the Curriculum Spaces Analysis Table, they analyze a lesson plan that they have drafted. The goal is to start the analysis in class where students can ask questions and get support; they then complete the analysis and turn it in later. PSTs’ analysis results in a) a written assignment that is turned in to the methods instructor or b) an in-class quick-write about the activity.

SUMMARIZE:The summary part of this activity consists of both PSTs’ written reflection as well as a debriefing discussion in class on the day the written assignment is submitted. The following prompts should guide PSTs’ written reflections (write-ups or quick writes):

Summarize what you learned from your analysis of your lesson plan. What are the strengths and limitations of your lesson plan?

Next, think about how you might use what you have learned to inform your lesson planning. What strategies or areas would you like to strengthen as a result of this analysis? Given an example of how you might strengthen one area (this can be in this lesson or in subsequent lessons). How does this analysis help, if at all, your math lesson planning process to meet the math learning needs of your students?

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In the debriefing discussion held in class on the day the assignment is submitted, the focus is on what PSTs learned from this activity. Some guiding questions for this debriefing session might include:

What did you discover about analyzing your mathematics lesson in relation to making thinking visible and integrating children’s multiple mathematical knowledge bases?

Did anything surprise you? Did anything challenge your thinking?

Instructor Reflections on Analyzing Mathematics Lesson Plan(s) Use your reflections to guide how you support PSTs in future activities and

discussions. What evidence do PSTs use to rate their mathematics lessons (activities, focus questions, math tasks, participation structures)?

How do assignment reflections illustrate developing attention and awareness to multiple mathematical knowledge bases?

In what ways do PSTs envision changing their math lessons? What trends/dimensions do PSTs discuss the most? How do they discuss these changes (provide detailed examples? Vague statements such as “I’ll use mixed ability groups” and not say why).

Ideas from TEACH Math about Analyzing Mathematics Lesson Plan(s)Across the group, we have tended to use this activity later in the semester, as a synthesizing activity. This activity also provides an opportunity to connect to the research and practitioner literature related to culturally relevant mathematics teaching.

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Additional ResourcesHiebert, J., Morris, A. K., Berk, D., & Jansen, A. (2007). Preparing teachers to learn from

teaching. Journal of Teacher Education, 58, 47-61Jansen, A., & Spitzer, S. M. (2009). Prospective middle school mathematics teachers’

reflective thinking skills: Descriptions of their students’ thinking and interpretations of their teaching. Journal of Mathematics Teacher Education, 12(2), 133-151.

Morris, A. (2006). Assessing pre-service teachers' skills for analyzing teaching. Journal of Mathematics Teacher Education, 9(5), 471-505.

Morris, A. K. (2007). Factors affecting pre-service teachers' evaluations of the validity of students' mathematical arguments in classroom contexts. Cognition and Instruction, 25(4), 479-522.

Pre-service teachers examine gender equity in teaching math. http://www.nctm.org/profdev/content.aspx?id=15621

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http://www.nctm.org/profdev/content.aspx?id=15621


ACTIVITY FOUR: ANALYZING MATHEMATICS LESSONS

Description of Analyzing Mathematics Lessons

In some of our courses, we have also supported PSTs in analyzing instances of teaching using one or more lenses. Below, we provide two sample assignment descriptions that we have used with PSTs for this activity (Handouts CP 11 and CP 12). Handout CP 11 was used by PSTs observing a lesson in their field placement, while Handout CP 12 was used for PSTs to analyze videos of their own teaching in their field placement.

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Handout CP 11: Classroom Observation with Lenses Assignment

Observe ONE mathematics lesson in your practicum classroom. Consider LEARNING during your observation. This is NOT to be an assessment of the general way mathematics classes evolve in you setting, but a clear and specific description and analysis of what transpired in ONE LESSON.

DURING AND AFTER THE OBSERVATION: Take extensive notes during the lesson you observe. Immediately after the observation, read through your notes, and write notes using the observation prompts below (C: From the lens of Learning). Respond directly to each focus area to make sure that you have documented as much as possible relative to the observation prompts. Keep all of these notes to submit with your observation analysis and report. (Points will be deducted if notes are not turned in.) These notes will serve as the basis for your paper.

In a paper of 2-4 pages, please provide (a) the general information about the students and mathematics taught that day, (b) a summary description of what happened in class that day, (c) an analysis of the learning that you observed, and (d) a short conclusion. It would be helpful to structure your paper by responding to the questions posed below. A. General Information: In a short paragraph (or in a list) provide the following information.

Date of Observation Grade Level Total number of students and demographic information about students as far as you

know it (e.g. number of students of each gender, race/ethnicity, ELL, Special Ed, and/or any other pertinent background information)

Mathematics Topic for the Lesson Curricular Materials Used (Textbook, activity sheets, etc.) (Note: If lesson is taken

from a published curriculum, specify particular curriculum used, for example, Everyday Math.)

Start and ending times of observation

B. Description/Summary of Lesson: Describe/summarize what happened in class using your observation notes. This section should include pertinent detail related to the focus of the observation and describe an overall context for “what went on” (key math problems/activities, teacher’s and students’ questions, and responses from the lesson, etc). The description of context and classroom activity should provide the reader with a clear image of what happened. Imagine you are writing this report for someone that has not been in the classroom (which I will not have been). Thus, you need to describe the classroom activity enough to provide a context for me. You may want to consider asking someone outside of the class to read this paper for you – can they follow it? What additional information did they need to make sense of what you are saying?

C. Analysis of Lesson from the lens of LEARNING

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What specific mathematical understandings and/or confusions were indicated by students’ work, talk, and/or behavior?

How did students communicate their own understandings and their sense making of the thinking of others (e.g. did they provide justifications, explanations; did they respond to questions, written work, representations, manipulatives, models, gestures, etc.)

What resources of knowledge (e.g. mathematical, cultural, community, family, linguistic, students’ interests, peers, the teacher) did students draw upon to understand and solve the mathematics task?

How do you gauge the learning of your case study student in consideration of the above questions?

D. Short Conclusion: In a paragraph, describe how effective/successful the lesson was at promoting participation and learning. What strategies and/or approaches you would like to emulate or avoid with respect to promoting participation and equalizing power within your classroom? Link your ideas to one or more of our class readings.

NOTE: Please submit all notes made during the observation and during analysis after the observation. These notes can be hand written. You can either (a) scan hand-written notes or type your notes and upload them to Blackboard, or (b) turn in hand-written notes in class on the day the assignment is due. If you type your notes, type them exactly as they were written. Save changes for the paper.

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Handout CP 12: Whole Class Lesson ProjectLesson Plan, Write Up, and Presentation

PART ONE: Lesson PlanYour lesson should be a whole group problem-solving lesson that uses Van de Walle’s Before, During, After structure. The lesson must:

Relate to the math content that your class is working on at the time you teach

Draw from existing curriculum materials (Investigations, etc) Use the lesson plan template Address state standards Involve at least 7-10 minutes of whole group discussion at the

end of the lesson Involve at least 10-15 minutes of individual, partner or small

group work

FIRST DRAFT of Whole Class Lesson Plan Due (Bring to Class): DATEIn Class on DATE, you will meet with peers to review one another’s lesson plans and provide feedback. You will attend in particular to the mathematical task, students’ opportunities to participate, the teacher moves you outlined for supporting diverse groups of learners, and the ways that the lesson connects to and draw upon children’s multiple mathematical funds of knowledge (their mathematical thinking, linguistic knowledge, cultural knowledge, community or family experiences etc).

REVISED Lesson Plan Due: Before you Teach You will revise your lesson plan based on your in-class analysis session and your

peer’s feedback. Your revised plan much include:o A brief description (1 paragraph) of the changes you made in your lesson

plan as a result of your in-class analysis and peer feedback. How did you use what you learned to revise your lesson plan? What part(s) of your lesson do you think was strengthend as a result of this analysis? Given an example of how you strengthened one area (this can be in this lesson or in subsequent lessons). How did this analysis help, if at all, your math lesson plan to meet the math learning needs of your students?

o Your revised lesson plan, using the lesson plan template.

You will upload the descriptions noted above, and the revised plan to D2L for instructor review. Before you teach the lesson, your instructor MUST provide you with feedback, and let you know that the lesson is ready for you to teach. The revised version of the lesson plan will assigned a grade.

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PART TWO: Teach and VIDEO TAPE the Lesson Teach the Lesson on DATES

Lesson should be taught to entire class or a significant portion of the class (unless special arrangements have been made with the instructor)

Lesson should be the length of a standard math period in your APPS classroom: for example, 25-30 minutes (kindergarten) – 1 hour or more (upper elementary)

Collect the student work produced during the lesson. If appropriate, you may use this student work in your final class presentation AND/OR submit it to go with your write-up.

VIDEOTAPE YOUR TEACHING: This video is a key part of your assignment. Please arrange ahead of time to have a peer, or your mentor teacher video tape your lesson. Have the person operating the camera follow YOU, and YOUR INTERACTIONS WITH STUDENTS at all times. During the whole group discussion part of the lesson, try to capture the entire flow of the discussion (your comments, students’ comments, etc..) as much as possible. You can’t videotape your own lesson (because you need to be focused on teaching), nor can you set up a camera in the back of the room and let it run while you teach. The flip cameras will be available to use for video taping.

PART THREE: View and Analyze the VideotapeOne way to reflect upon the effectiveness of your teaching with respect to building on children’s multiple mathematical knowledge bases and supporting all children in learning mathematics is to evaluate your teaching from the perspective of four lenses (teaching, learning, task, power and participation). No one lesson can (or should) successfully address each component of each of the four lenses. Do not expect your lesson, as planned or as implemented, to do this. However, your lesson (as planned and implemented) should address some aspects of the four lenses, and what your lesson addresses, what it does not, what you planned for it to accomplish and how it did or did not do that are questions that serve to inform and develop your teaching practice.

For this activity, you will watch and re-watch your video tape using THE STUDENT LEARNING LENS AND EITHER THE TEACHING LENS OR THE POWER AND PARTICIPATION LENS to inform your reflection (the student learning lens is required). That is, you will watch the video once to get a general sense of the lesson, again with a focus on the Student Learning Lens, and again with a focus on either the teaching lens or the power and participation lens.

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LENSES AND ASSOCIATED PROMPTS

STUDENT LEARNING LENS (REQUIRED)5. What specific math understandings and/or confusions are indicated in students’ work, talk, and/or behavior? Describe concepts, processes, skills, and/or problem solving strategies that students appear to know/understand as they engage in the lesson. Describe areas of confusion and/or misunderstanding. What did the students do, say, or show that indicated these understandings or confusions? 6. How do students communicate their understandings and sense making of others’ thinking? (e.g., provide justifications, explanations, respond to questions, written work, representations, manipulatives, models, gestures)How do students share their thinking? Consider evidence of understanding and/or learning in the form of: verbal and non-verbal expressions, gestures, written work (symbols, pictures, diagrams), work with physical objects (manipulatives, models), behavior, etc. 7. In what ways does student engagement reflect conceptual and/or procedural

learning? 8. What resources or knowledge do students draw upon to understand and solve the math task? (e.g., mathematical, cultural, community, family, linguistic, students’ interests, peers) What connections are students making between mathematics and their experiences and/or prior knowledge? Consider how these forms of resources/ knowledge influence learning.

TEACHING LENS (CHOICE FOR SECOND LENS)6. How does the teacher elicit and respond to students’ thinking? (e.g., moves,

questions, responses to students’ correct answers/ mistakes/ confusions/ partial solutions/ approaches).

7. What opportunities does the teacher create for diverse learners to communicate their mathematical understandings - show what they know? (Consider: English learners, different genders, various cultures, varied math backgrounds and confidence levels, etc.) To what extent and how are various forms of communication are supported, encouraged, and valued (e.g., verbal and non-verbal expressions, gestures, written work (symbols, pictures, diagrams), work with manipulative/models/physical objects, etc.)? How does the teacher help to build (or diminish) students’ sense of competence and self-confidence.

8. How does the teacher implement the task in a way that maintains or changes the cognitive demand? Does the teacher’s implementation of the task maintain high cognitive demand? Alternatively, does the teacher’s implementation of the task diminish or lower the level of cognitive demand? In what ways did the teacher modify the task upon implementation that changed the cognitive demand?

9. What resources and knowledge does the teacher use/draw upon to support students’ math understanding? What does the teacher do to help students make connections within mathematics (e.g., between various mathematical concepts, processes, representations, strategies, etc.)? What does the teacher do to help students connect mathematics with relevant/authentic issues or situations in their lives? What

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does the teacher do to encourage students to use their own prior knowledge and/or peers as resources for learning?

POWER AND PARTICIPATION LENS (CHOICE FOR SECOND LENS)3. Who participates? Does the classroom culture value and encourage most students to speak, only a few, or only the teacher? Where does the majority of the math “work” take place in the classroom? (e.g., front of room, small group, individual desks) and how does this contribute to participation? In classes characterized by high levels of mathematical discourse, there is considerable teacher-student (T-S) and student-student (S-S) interaction about mathematical ideas; the interaction is reciprocal, and it promotes shared understandings. Record who the teacher involves/calls on and whether students of various subgroups are proportionally represented (e.g., females and males, racial/ethnic subgroups, etc.).4. Who holds authority for knowing mathematics? Do some students hold more status than others? Does the teacher have the final word on correct answers/solutions? Are students encouraged to convince themselves or others? Are some students higher/ lower status than others? Are some students assumed to “know” or “be correct” or “be better” at math? Are some students sought out or avoided as partners or group members? Who assigns high/low status (teacher, other students) and on what basis (academic performance, social standing, athletic performance)?3. What evidence indicates that differences in approaches and perspectives are/ are not respected and valued? Is a sense of respect for all members evident? Is there an effort to understand one another’s perspectives? Is there evidence that members are aware of and value different educational experiences and backgrounds (e.g., mathematical, cultural, community, family, linguistic, students’ interests, peers) 5. How do issues of participation, authority, and status (above) influence learning? Interpret the results of your analysis above with respect to student learning.

Use the questions above to guide your analysis. Collect specific examples to use in your write-up, including direct quotations from the lesson. Stopping the tape and viewing each segment multiple times is absolutely essential. You just won’t catch everything the first, or even the second time around.

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PART FOUR: Lesson Analysis Write Up (Due DATE) Prepare a write-up on your analysis of your teaching of your lesson (around 1500 words). Be as SPECIFIC as possible in your writing. Anytime you make a claim (i.e., “students really seemed to understand the task” or “I noticed that students were drawing on one another as resources) you need to support your claim with evidence and specific examples (including quotes) from your lesson. Your write-up must include the following components.

1) In general (Summary of Lesson and Lesson Effectiveness): Briefly describe the main task of the lesson, and what happened during the lesson, and in the group discussion in particular. How effective was the lesson at promoting participation from all students? How effective was the lesson at prompting or developing student learning? In what ways did you elicit and build on children’s multiple mathematical knowledge bases?

2) Analysis of Lesson According to First Lens – STUDENT LEARNING LENS: Explain how you conducted your analysis for this lens (what did you keep track of, what did you focus on, etc..) Include a detailed description of what you learned about student understanding, ideas, confusions and/or misunderstandings. Include examples, with evidence, of student comments, actions, work that indicate procedural understanding, and conceptual understanding. Include examples of how students communicated their understandings, and, if relevant, how they made sense of the ideas of others. This analysis should include specific statements about what happened during the lesson, and each statement should be supported by specific evidence and examples from the lesson.

3) Analysis of Lesson According to Second Lens – TEACHING or POWER AND PARTICIPATION. State the second lens that you chose, and why you selected that lens. Explain how you conducted your analysis for this lens (what did you keep track of, what did you focus on, etc..) Include a detailed description of the results of your analysis for this lens. This analysis should include specific statements about what happened during the lesson, and each statement should be supported by specific evidence and examples from the lesson.

4) Extending Your Thinking: Next, think about how you might use what you have learned from your analysis to inform your future mathematics instruction. What strategies or areas would you like to strengthen as a result of this analysis? If you were to teach this lesson again, or a similar lesson, what would you do differently next time, and why? Include a specific example of changes you might make to strengthen one area.

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PART FIVE: Final Class Presentation (DATE) Select a 3-4 minute excerpt to show in class. Your clip should contain a moment (or moments) in your lesson that are you wondering about, and about which you would like input from your peers. It might be a moment in which you have questions about what students understood or were thinking, about a teaching move you made, about student status/participation issues or about what else you could have done. TRANSCRIBE this excerpt and make 15 copies to bring to class on the day you present.

You will have a 10 minute block of time for your presentation. You are responsible for filling this time but not exceeding it! In this time, you should

1) Provide a quick overview of the lesson: What grade level did you teach and what was the task? How did you consider children’s multiple mathematical funds of knowledge as you were designing and implementing the lesson?

2) Introduce your 2-3 minute video excerpt: Why did you choose this excerpt? What specific questions would you like to explore with the help of the group? Where does this excerpt occur in the lesson? What happens just before the excerpt starts?

3) Ask for clarifying questions: Your group might have questions about what students said or what happened. Invite these questions first.

4) Lead a discussion of the excerpt. You might consider different ways of facilitating this. For example, do you want to give your group a minute to write down their thoughts? Do you want to use one of the round robin protocols to make sure that everyone has a opportunity to tell you what they think?

5) Summarize what you have learned from the group. You might share any of the insights you included in your final paper.

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ReferencesAnnenberg Foundation (Producer). (2013). Teaching math: A video library K–4

[Video clips]. Retrieved from http://www.learner.org/resources/series32.htmlCarpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically:

Integrating arithmetic and algebra in elementary school. Portsmouth, NH:Heinemann.

Fosnot, C. T., & Dolk, M. (2001). Young mathematicians at work: Constructing multiplication and division. Portsmouth, NH: Heinemann.

Fosnot, C. T., & Dolk, M. (2002). Young mathematicians at work: Constructing fractions, decimals, and percents. Portsmouth, NH: Heinemann.

Jacobs, V.R. & Ambrose, R.C. (2008). Making the most of story problems. Teaching Children Mathematics 15(5), 260-266.

Neumann, M.D. (2007). Preservice teachers examine gender equity in teaching mathematics. Teaching Children Mathematics 13(7), 388-395.

Schifter, D., Bastable, V., and Russell, S. J. (2009). Developing mathematical ideas (2nd ed.), Number and operations, part 1: Building systems of tens: Calculating whole numbers and decimals. Boston, MA: Pearson.

Smith, M. S., Stein, M. K., Arbaugh, F., Brown, C. A., & Mossgrove, J. (2004). Characterizing the cognitive demands of mathematical tasks: a task-sorting activity. In G. W. Bright & R. N. Rubenstein (Eds.), Professional Development Guidebook for Perspectives on the Teaching of Mathematics: Companion to the Sixty-sixth Yearbook (pp. 45 – 72). Reston, VA: NCTM.

Storeygard, J. (2009). My kids can. Portsmouth, NH: Heinemann.UCSMP. Everyday Mathematics: The University of Chicago School Mathematics

Project. Chicago: McGraw Hill, 2007.

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Documents

teachmathdemo.files.wordpress.com · Web viewClassroom Practices Module Table of Contents Overview of Classroom Practices Module 2 Activity 1: Analyzing Video Cases Goals and Description