What could the teacher do differently to earn a higher score on one or more of the rubrics? Using the 3Cs model described by Gutstein (2016), discuss the instruction in the video in

ICL 7502/4007 Fall 2021

Equity Project

The Equity Project is intended to allow you to apply some of the ideas we have explored through the

creation of a lesson that could be taught in your classroom. There are two parts to the Equity Project: (1)

a lesson plan; and (2) a reflection.

Lesson Plan

Design a lesson that connects mathematics content to issues of social justice or community concerns in

your school context. (See Gutstein (2016) reading and the resource “SJMathGuide” for ideas of what

this might look like).

Your lesson plan must meet the following criteria:

1. It must target a well-defined mathematical concept – what Gutstein refers to as classical

knowledge.

2. It must also integrate at least one of the other 3Cs: community knowledge OR critical

knowledge. (I realize that Gutstein advocates for curriculum and instruction that includes all

three Cs. However, for the purpose of this assignment, the connections can involve either

community knowledge or critical knowledge).

3. There is not a required format for the lesson plan. However, it should include the following:

• Learning objectives • Lesson procedures (including what you and the students will be doing) • Learning tasks/activities

• Questions that you will ask the students – You must include the questions that you would ask to ensure that students CONNECT classical mathematical knowledge with community or critical knowledge.

• Instructional resources and materials used

Reflection In addition to the Lesson Plan itself, you should include responses to the following questions:

1. How will the lesson plan help students build connections among classical, community, and/or cultural

knowledge? 2. How does the lesson support equity and/or social justice? 3. As you think about incorporating lesson such as this one into your mathematics instruction, what is

one “take-away” from this assignment? In other words, what have you learned from creating this lesson plan?

NOTE: You can use an existing lesson plan as a foundation for your lesson. However, you need to be sure

that your lesson plan is connected to the students and community in which you teach (or will teach).

For example, the SJMathGuide includes a description of an activity involving “The Toxic Circle.” The

mathematics is transferrable (many communities have restrictions regarding how close certain

businesses or facilities can be to schools). But the setting of this particular scenario is a Chicago

neighborhood. You could use a similar premise, but you would need to tailor it to a situation in your

school’s community.

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Instruction Rubric 6: Learning Environment

MTH6: How does the candidate demonstrate a respectful learning environment that supports students' engagement in learning?

The Guiding Question

The Guiding Question addresses the type of learning environment that the candidate establishes and the degree to which it fosters respectful interactions between the candidate and students, and among students.

Key Concepts of Rubric:

 Respect—A positive feeling of esteem or deference for a person and specific actions and conduct representative of that esteem. Respect can be a specific feeling of regard for the actual qualities of the one respected. It can also be conduct in accord with a specific ethic of respect. Rude conduct is usually considered to indicate a lack of respect, disrespect, whereas actions that honor somebody or something indicate respect. Note that respectful actions and conduct are culturally defined and may be context dependent. Scorers are cautioned to avoid bias related to their own culturally constructed meanings of respect.

 Rapport—A close and harmonious relationship in which the people or groups understand each other's feelings or ideas and communicate well.

For the following term from the rubric, see the handbook glossary:

 Learning environment

Primary Sources of Evidence:

Video Clip(s)

Instruction Commentary Prompt 2

Note that for the Instruction Task, the commentary is intended to provide context for interpreting what is shown in the video. Candidates sometimes describe events that do not appear in the video or conflict with scenes from the video—such statements should not override evidence depicted in the video.

Scoring Decision Rules

► Multiple Criteria  N/A

► AUTOMATIC 1  None

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Unpacking Rubric Levels

Level 3

Evidence that demonstrates performance at Level 3: In the clip(s):

 The candidate's interactions with students are respectful, demonstrate rapport (evidence of relationship between candidate and students and/or ease of interaction that goes back and forth based on relevance or engaged conversation), and students communicate easily with the candidate.

 There is evidence that the candidate facilitates a positive learning environment wherein students are willing to answer questions and work together without the candidate or other students criticizing their responses.

 There is evidence of mutual respect among students. Examples include attentive listening while other students speak, respectful attention to another student's idea (even if disagreeing), working together with a partner or group to accomplish tasks.

Below 3

Evidence that demonstrates performance below 3: The clip(s):

 Do not exhibit evidence of positive relationships and interactions between candidate and students.

 Reveal a focus on classroom management and maintaining student behavior and routines rather than engaging students in learning.

What distinguishes a Level 2 from a Level 3: At Level 2,

 Although clip(s) reveal the candidate's respectful interactions with students, there is an emphasis on candidate's rigid control of student behaviors, discussions, and other activities in ways that limit and do not support learning. There is a general lack of rapport with students, as the candidate is interacting with students mainly to control behavior.

What distinguishes a Level 1 from a Level 2: At Level 1, there are two different ways that evidence is scored:

1. The clip(s) reveal evidence of candidate-student or student-student interactions that discourage student contributions, disparage the student(s), or take away from learning.

2. The classroom management is so weak that the candidate is not able to, or does not successfully, redirect students, or the students themselves find it difficult to engage in learning tasks because of disruptive behavior.

Note: Classroom management styles vary. Video clips that show classroom environments where students are productively engaged in the learning task should not be labeled as disruptive. Examples of this may include students engaging in discussion with peers, speaking without raising their hands, or being out of their seats.

Above 3

Evidence that demonstrates performance above 3: The clip(s)

 Reveal a positive learning environment that includes tasks/discussions that challenge student thinking and encourage respectful student-student interaction.

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What distinguishes a Level 4 from a Level 3: At Level 4,

 The learning environment supports learning tasks that appropriately challenge students by promoting higher-order thinking or application to develop new learning. There must be evidence that the environment is challenging for students. Examples include: students cannot answer immediately, but need to think to respond; the candidate asks higher- order thinking questions; students are trying to apply their initial learning to another context.

 The learning environment encourages and supports mutual respect among students, e.g., candidate reminds students to discuss ideas respectfully with each other.

What distinguishes a Level 5 from a Level 4: At Level 5,

 The learning environment encourages students to express, debate, and evaluate differing perspectives about content with each other. Perspectives could be from curricular sources, students' ideas, and/or lived experiences.

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Instruction Rubric 7: Engaging Students in Learning

MTH7: How does the candidate actively engage students in developing conceptual understanding, procedural fluency, AND/OR mathematical reasoning and/or problem- solving skills?

The Guiding Question

The Guiding Question addresses how the candidate provides video evidence of engaging students in meaningful tasks and discussions to develop their understanding of mathematical concepts, procedures, and/or reasoning/problem-solving skills.

Key Concepts of Rubric:

For the following terms from the rubric, see the handbook glossary:

 Engaging students in learning

 Assets (personal, cultural, community)

Primary Sources of Evidence:

Video Clip(s)

Instruction Commentary Prompt 3

Note that for the Instruction Task, the commentary is intended to provide context for interpreting what is shown in the video. Candidates sometimes describe events that do not appear in the

video or conflict with scenes from the video—such statements should not override evidence depicted in the video.

Scoring Decision Rules

► Multiple Criteria  Criterion 1 (primary): Engagement in learning tasks

 Criterion 2: Connections between students' academic learning AND/OR assets (personal, cultural, community) and new learning

 Place greater weight or consideration on the criterion 1 (engagement in learning tasks).

► AUTOMATIC 1  None

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Unpacking Rubric Levels

Level 3

Evidence that demonstrates performance at Level 3:

 Primary Criterion: The clip(s) show that the students are engaged in learning tasks that provide opportunities for students to focus on conceptual understanding, procedural understandings, and reasoning and/or problem solving. Although these content understandings are evident in conversations, they are addressed at a cursory level. For example, the candidate has a student summarize how he found the value of an angle formed by two parallel lines cut by a transversal, using the concept of supplementary angles and the procedure (angle relationships) used. The candidate relates this to the congruent supplement theorem, but moves on immediately, leaving the instruction at a cursory level.

 Secondary Criterion: The clips show the candidate making connections to students' prior academic learning to help them develop the new content or skills.

Below 3

Evidence that demonstrates performance below 3:

 Students are participating in tasks that provide little opportunity to develop conceptual understanding or mathematical reasoning and/or problem-solving skills.

What distinguishes a Level 2 from a Level 3: At Level 2,

 Students are participating in rote tasks that primarily focus on following step-by-step procedures AND/OR concepts and provide little opportunity to develop mathematical reasoning and/or problem-solving skills.

 The structure of the learning task or the way in which it is implemented constrains student development of content and skills.

 In addition, the candidate may refer to students' learning from prior units, but the references are indirect or unclear and do not facilitate new learning.

What distinguishes a Level 1 from a Level 2: At Level 1,

 The learning tasks seen in the video clip(s) have little relation to the central focus identified.

 In addition, the candidate is not using either students' prior academic learning or assets (personal, cultural, community) to build new learning.

Above 3

Evidence that demonstrates performance above 3:

 The learning tasks as seen in the clip(s) are structured to engage students to develop understandings of concepts and procedures through mathematical reasoning and problem-solving skills.

 Connections between students' prior academic learning and assets (personal, cultural, community) are made to support the new learning.

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What distinguishes a Level 4 from a Level 3: At Level 4,

 The learning tasks in the clip(s) include structures or scaffolding that promote the learning of concepts, procedures AND mathematical reasoning and/or problem-solving skills. Students must interact with the content in ways that are likely to either extend initial understandings or surface misunderstandings that the candidate can then address.

 In addition, the candidate draws upon not only prior academic learning, but also students' assets (personal, cultural, community) to develop new learning. Examples of experience outside of school may include. .. calculating how many hours in an after school job at a reasonable rate for teenage employees would be needed to save enough for some object with current popularity, using baseball batting averages to illustrate the comparison of decimals or measures of center, any metaphor based on familiar objects or situations for that particular group of students.

What distinguishes a Level 5 from a Level 4: At Level 5,

 The learning tasks as seen in the clip(s) are structured or scaffolded so that students will develop and use concepts, procedures, AND reasoning and/or problem solving in ways that are appropriately challenging directly related to new learning.

 In addition, the candidate encourages students to connect and use their prior knowledge and assets (personal, cultural, community) to support new learning.

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Instruction Rubric 8: Deepening Student Learning

MTH8: How does the candidate elicit responses to promote thinking and to develop conceptual understanding, procedural fluency, AND mathematical reasoning and/or problem-solving skills?

The Guiding Question

The Guiding Question addresses how, in the video clip, the candidate brings forth and builds on student responses to guide learning; this can occur during whole class discussions, small group discussions, or consultations with individual students.

Key Concepts of Rubric:

 Significant content inaccuracies—Content flaws within processes or examples used during the lesson will lead to student misunderstandings and the need for reteaching.

Primary Sources of Evidence:

Video Clip(s)

Instruction Commentary Prompt 4a

Note that for the Instruction Task, the commentary is intended to provide context for interpreting what is shown in the video. Candidates sometimes describe events that do not appear in the video or conflict with scenes from the video—such statements should not override evidence depicted in the video.

Scoring Decision Rules

► Multiple Criteria  N/A for this rubric

► AUTOMATIC 1  Pattern of significant content inaccuracies that are core to the central focus or a key learning objective for the learning segment

Unpacking Rubric Levels

Level 3

Evidence that demonstrates performance at Level 3:

 The candidate prompts students to offer responses that require thinking related to either concepts or procedures OR to mathematical reasoning and problem solving, e.g., by using "how" and "why" questions. Some instruction may be characterized by initial questions focusing on facts to lay a basis for later higher-order questions in the clip(s). A pattern of evidence supporting a particular score level has a heavier weight than isolated evidence in another score level.

Below 3

Evidence that demonstrates performance below 3:

 In the clip(s), classroom interactions provide students with limited or no opportunities to think and learn.

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What distinguishes a Level 2 from a Level 3: At Level 2,

 The candidate asks questions that primarily elicit right/wrong or yes/no answers and do little to encourage students to think about the content being taught.

What distinguishes a Level 1 from a Level 2: At Level 1,

 There are almost no opportunities shown in the clip(s) that students were able to express ideas. (For example, the candidate does most of the talking during the video)

Automatic Score of 1 is given when:

 There is a pattern of significant content inaccuracies that will lead to student misunderstandings.

 The candidate makes a significant error in content (e.g., introducing an inaccurate definition of a central concept before students work independently) that is core to the central focus or a key standard for the learning segment.

Above 3

Evidence that demonstrates performance above 3:

 In the clip(s), the candidate uses student ideas and thinking to develop students' mathematical learning or their abilities to evaluate their own learning.

What distinguishes a Level 4 from a Level 3: At Level 4,

 The candidate follows up on student responses to encourage the student or his/her peers to explore or build on the ideas expressed.

 The candidate uses this strategy to develop students' understanding of mathematics concepts, procedures, AND reasoning and/or problem-solving skills.

 Examples of "building on student responses" include: referring to a previous student response in developing a point or an argument; calling on the student to elaborate on what s/he said; posing questions to guide a student discussion; soliciting student examples and asking another student to identify what they have in common; asking a student to summarize a lengthy discussion or rambling explanation; and asking another student to respond to a student comment or answer a question posed by a student to move instruction forward.

What distinguishes a Level 5 from a Level 4: At Level 5, the candidate meets all of Level 4 AND

 There is evidence in the clip(s) that the candidate structures and supports student- student conversations and interactions that facilitate students' ability to evaluate and self- monitor their learning.

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Instruction Rubric 9: Subject-Specific Pedagogy: Using Representations

MTH9: How does the candidate use representations to develop students' understanding of mathematical concepts and procedures?

The Guiding Question

The Guiding Question addresses how the candidate uses representations (e.g., charts, graphs, metaphors, equations) in the clip(s) to build students' understanding of mathematical content.

Key Concepts of Rubric:

 Conceptual understanding—Students demonstrate conceptual understanding in mathematics when they recognize, label, and generate examples of concepts; use and interrelate models, diagrams, manipulatives, and varied representations of concepts; identify and apply principles; know and apply facts and definitions; compare, contrast, and integrate related concepts and principles; recognize, interpret, and apply the signs, symbols, and terms used to represent concepts (NAEP, 2003; What Does the NAEP Mathematics Assessment Measure?).

 Procedural Fluency—"…refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skills in performing them flexibly, accurately, and efficiently. … These methods include, in addition to written procedures, mental methods for finding certain sums, differences, products, or quotients, as well as methods that use calculators, computers, or manipulative materials such as blocks, counters, or beads." (National Research Council, (2001). Adding it Up: Helping Children Learn Mathematics, p. 121).

 Representation—the act of capturing a mathematical concept or relationship in some visual, symbolic or physical form OR the form itself (adapted from NCTM Principles and Standards for School Mathematics, p. 67)

Primary Sources of Evidence:

Video Clip(s)

Instruction Commentary Prompt 4b

Note that for the Instruction Task, the commentary is intended to provide context for interpreting what is shown in the video. Candidates sometimes describe events that do not appear in the

video or conflict with scenes from the video—such statements should not override evidence depicted in the video.

Scoring Decision Rules

► Multiple Criteria  N/A for this rubric

► AUTOMATIC 1  Representations that are not appropriate or used inappropriately for the content being taught

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Unpacking Rubric Levels

Level 3

Evidence that demonstrates performance at Level 3:

 In the clip(s), the candidate guides conversation and/or structures explorations using representations that help students develop understanding of mathematical concepts and procedures.

Below 3

Evidence that demonstrates performance below 3:

 In the clip(s), the candidate is not using representations effectively to guide student learning.

What distinguishes a Level 2 from a Level 3: At Level 2,

 The candidate attempts to use representations to facilitate understanding of mathematical concepts and procedures, but the connections between them are not strong enough or clear enough to be effective.

What distinguishes a Level 1 from a Level 2: At Level 1,

 The candidate stays focused on facts or procedures and fails to make connections to concepts.

Automatic Score of 1 is given when:

 The representations are significantly inappropriate for the intended learning.

 The use of the representations will lead to significant student misunderstandings.

Above 3

Evidence that demonstrates performance above 3:

 In the clip(s), the candidate is making strategic choice or use of representations to develop students' mathematical learning.

What distinguishes a Level 4 from a Level 3: At Level 4,

 In the clip(s), the candidate provides opportunity for students to use problems or visual representations to guide and encourage students to think more deeply about mathematical concepts and procedures.

What distinguishes a Level 5 from a Level 4: At Level 5, in the clip(s), the candidate meets Level 4 AND

 Structures and supports student-student conversations to help them evaluate their own use of representations to explore concepts and conjectures and to solve problems.

“Our Issues, Our People—Math as Our Weapon”: Critical Mathematics in a Chicago Neighborhood High School

Eric “Rico” Gutstein University of Illinois at Chicago

This article provides an example of, and lessons from, teaching and learning critical mathematics in a Chicago public neighborhood high school with a social justice focus. It is based on a qualitative study of my untracked, 12th-grade mathematics class, a full-year enactment of mathematics for social and racial justice. Students were Black and Latin@ from a low-income, working-class community with a tradition of resis- tance. Any neighborhood student could enroll without selection criteria. The class goal was for students to cocreate a classroom in which they would learn and use college- preparatory, conceptually based mathematics to study and understand social reality to prepare themselves to change it. Through analyzing my practice, I address possibilities and challenges of curriculum development and teaching, examine student learning, and pose questions and directions for further research and practice.

Key words: Critical mathematics; Equity; Freire; Mathematics for social justice; Race; Racism

I wrote the numbers on the board as class wound down. Students silently and soberly stared at 150,000 – 291,000 = 92,000 (an “equation” derived from a recursive function, modeling a mortgage). I talked as I wrote: “You’ve paid two hundred and ninety-one thousand dollars on a one-hundred-and-fifty-thousand-dollar mortgage, and you still owe ninety-two thousand dollars. Check that math out. One hundred and fifty thousand minus two hundred and ninety-one thousand equals ninety-two thousand.” I paused as students looked and mumbled to themselves and neighbors. “Think about that. Hey! You started with a hundred and fifty—you paid two ninety-one—and you still owe ninety-two thousand dollars. What’s going on here?”

Antoine: “They’re taking your money.”1

Daphne: “The bank is taking advantage of you.”

This research was supported in part by a fellowship from the Great Cities Institute, University of Illinois at Chicago. Acknowledgments to student “Crew” members (Vero González, Rut Rodríguez, Amparo Ramos, George C

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