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Achieving Instructional Equity Zenaida Aguirre-Muñoz, Ph. D. West Texas Middle School Math Science Partnership Texas Tech University June – July, 2010

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Expectations Design Instruction Around Big Ideas Maximize Growth Potential Plan to Scaffold & Differentiate Help Students Reason Mathematically Draw on Students’ Language & Culture ◦ If time permits

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Self-Monitoring Activities Form Submissions Blogging Conference Presentations

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‘‘opportunities to learn do not exist for learners who cannot take advantage of them’’ (Haertel et al., 2008, p. 6).

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6 th grade student explanation of the relationship between area, volume, and distance. I think they are a chain, so if you know your volume, you will be able to find your area, so like a chain if you know one you’ll know the other. So I think the relationship is that if you know one you’ll know the other. If you know your calculations of volume, you’d be able to find your area. If you use what volume is which is length and width, area, and perimeter. With volume you’ll be able to find area, and with area you’ll be able to find out your distance.

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Unit: u Unit: u Unit: u 2 Unit: u 2 Unit: u 3 Unit: u 3 Unit: Name Unit: Name Area formulas SIZE Measure 0- dimension Number 0- dimension Number 1-dimension Length 1-dimension Length 2-dimensions Area 2-dimensions Area 3-dimensions Volume 3-dimensions Volume Counting Distance formula Volume formulas Unit Analysis

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Defining & Identifying Big Ideas

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Teaching for ‘exposure’ Teaching without objectives, with ‘fun’ activities Neither empowers students to solve complex problems

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Emphasize Big Ideas ◦ Highly selective concepts and principles ◦ Clarify connections between smaller concepts ◦ Facilitate links to new concepts and problem solving situations Build students’ understanding and use of conceptual knowledge

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Reveals how different natural phenomena follow the same flow of matter/energy that represents a rectangular figure. Convection: a specific pattern of cause-and-effect relations involving phenomena that range from a pot of boiling water to ocean currents to earthquakes. Links several smaller ideas (Density, heating and cooling, force, and pressure) and strategies together: to demonstrate how they operate in similar ways.

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Unit: u Unit: u Unit: u 2 Unit: u 2 Unit: u 3 Unit: u 3 Unit: Name Unit: Name Area formulas SIZE Measure 0- dimension Number 0- dimension Number 1-dimension Length 1-dimension Length 2-dimensions Area 2-dimensions Area 3-dimensions Volume 3-dimensions Volume Counting Distance formula Volume formulas Unit Analysis Size: measurement of an object which is based on its dimensionality. Links smaller ideas (dimen- sion, distance, area, volume) and strategies (formulas & unit analysis) to demonstrate how they are related. Reveals how the process of translation is similar across objects of different dimen- sions.

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Used to teach a variety of math content and strategies Provide referential starting points for new math concepts and strategies ◦ Include, size, proportion, estimation, etc Explicitly described and modeled by the teacher

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Unwrap Standards 1.Underline content nouns Represent concepts (what students need to know) 2.Circle verbs Represent skills (what students need to be able to do) 3.Examine verbs to determine the intended level of thinking/reasoning Correspond to Bloom’s Taxonomy 4.Determine organizing/’power’ concepts (big ideas)

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StandardExpectations (5.10) Measurement. The student applies measurement concepts involving length (including perimeter), area, capacity/volume, and weight/mass to solve problems. The student is expected to: (A) perform simple conversions within the same measurement system (SI (metric) or customary); (B) connect models for perimeter, area, and volume with their respective formulas; and (C) select and use appropriate units and formulas to measure length, perimeter, area, and volume.

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Concepts (nouns)Skills (verbs)Potential Big Ideas Measurement Length Area Weight/mass Capacity/volume Perimeter Units Formulas Models Applies Solve Perform Connect Select Use Measure Size/measurement, dimensionality

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SkillKCApAnSEType of “Thinking Question” Applies Solve Perform Connect Select Use Measure X X X X X X X X

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Elements of Conceptually- Based Instruction

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C-Scope identifies concepts that can be used as starting points Teacher should identify “power” concepts and develop students understanding of the relationships between concepts Should be foundational to the lesson Should be applicable across lessons (e.g., size)

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Should Focus Attention on Big Ideas Should Generate higher-level thinking Instruction focused on deep conceptual knowledge results in higher achievement Instruction focused on lower level skills leads to smaller gains over time

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Strategy: ◦ Examine verbs in question prompt (instruction) ◦ Think about the steps involved in the expected solution strategies Review page 9 in the instructional guide and discuss why each task is categorized the way it is. Share your findings with the class.

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Review the case study you received on Wednesday. As you review think about the following: ◦ Why do you think Kevin and Fran selected the tasks they did? Where the tasks capable of bringing out the ideas they thought were important? Explain. ◦ How was Kevin’s approach to students different than Fran’s approach? Examine the tasks presented to students and determine the level of reasoning involved. ◦ Was the task selection related to Kevin and Fran’s success? Explain.

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Strategy refers to a routine that leads to both the acquisition and use of knowledge The ultimate purpose of a strategy is meaningful application, HOWEVER For diverse learners, acquisition is most reliable when instruction focuses on stretegy first The purpose of strategy instruction is to illuminate expert cognitive processes (mathematical reasoning) so that they are visible to the novice learner

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Volume Strategy Instruction 1.Link to prior knowledge 2.Introduce new strategy 3.Compute area 4.Compute volume 5.Write complete answer

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How would you modify the instruction to reflect what you have learned this week about size? Compare the volume strategy with that which is described for proportion. How can strategy instruction be implemented for proportion?

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Use visual maps, models to present big ideas Visual aids should make obvious the connections that are important Refer back to links during instruction and in feedback to students Feedback to students should draw attention to big idea and links among concepts Use and emphasize words to call attention to big ideas

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Diverse learners benefit from good strategy instruction if and only if the strategies are designed to result in transferable knowledge of their application.

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Use Handout 5 to brainstorm and outline how strategy instruction could be done on a unit focused on the measurement of geometric shapes. Be prepared to share your outline.

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Instruction should introduce and combine information in ways that result in new or more complex knowledge. ◦ What concepts need to be integrated for size? ◦ In what sequence should these concepts be taught?

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Provide multiple meaningful practice opportunities using big idea with new strategy Apply big idea to the math strategy using a variety of problem solving situations Pair a visual cue with each math big idea Post visual cue along with one sentence describing why the big idea is important

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After a recent review of your math TAKS scores, you notice that 30% of your students scored significantly lower on the measurement items of the test. Design a higher-level reasoning task involving the big idea of size. Include the following information in the description of the design: ◦ What visual aids would you provide? ◦ What strategy would you introduce? ◦ How would you model the strategy and its connection to the big idea?

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Theoretical Foundations

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Less than 3% growth of K-12 US population 56% growth of ELLs between1995 and 2005 Greatest increases in areas with traditionally little to no ELL populations Providing equal opportunity to learn content and skills continues to be a critical issue Teacher training and curricular materials in short supply

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‘‘opportunities to learn do not exist for learners who cannot take advantage of them’’ (Haertel et al., 2008, p. 6).

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Students develop higher-order functions through language use. ◦ Mental processes involved in higher-order thinking (e.g., math reasoning) From the socio-cultural/situative perspective, language mediates the development of higher mental processes (synthesis, evaluation) Thus, the basic argument in education is that language plays a critical role in the development of conceptual understanding.

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Language is the main vehicle of thought and all language use is based on social interaction Language supports thinking and is evident when inner speech is overt: ◦ “Oops, that can’t be right…Maybe I should start by making a function table…Ah, good! I see why that relationship is off.”

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Language develops almost exclusively from interaction. Thought is essentially internalized speech (age 2+), and speech emerged in social interaction. Learning occurs first thru social interaction-on the inter-psychological plane, then is internalized in the intra-psychological plane. Cognitive Development Thought Language Age 2

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The child does not merely “copy and paste” what they see and hear. Internalization is a process of transformation involving appropriation and reconstruction. ◦ All learning is co-constructed ◦ Learner transforms the social learning into individual learning over time ◦ Takes place in the zone of proximal development (ZPD) Can occur between peers ◦ Joint construction of knowledge ◦ Must foster active involvement, initiative, and autonomy--AGENCY

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Many students do not exercise their agency. Participation moves from apprenticeship (marginal participation) to appropriation (doing math) ◦ Qualitative changes in participation Over time, students appropriate the ways of thinking, acting, and interacting that is valued in school. “It is more revealing to observe students’ participation in academic activity over time, to see how their potential is gradually realized” (Walqui & van Lier, 2010, pp. 12)

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Interaction that fosters appropriate support and leads to higher level functioning (not too much and not too little) Requires explicit planning and incorporating supports or scaffolds to enable learners to take advantage of learning opportunities It is NOT simply helping students complete tasks they cannot do independently. ◦ The teacher would be doing all the (talking and) thinking Scaffolds allow students to interact in their ZPD Every ZPD is unique AND constantly changing

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In order for teachers to maximize a child’s growth potential, scaffolding entails routinely differentiating the scaffolds provided to individual students across topics and tasks and to continue to do so over time.

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‘‘opportunities to learn do not exist for learners who cannot take advantage of them’’ (Haertel et al., 2008, p. 6).

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Structure & Processes

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Scaffolding is used imprecisely. is often conceived of a structure, ignoring the process. enables differentiation to occur. Is a structural instructional element AND an instructional process Is how the ZPD is established and learning takes place.

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Involves both the predictable unpredictable aspects of the instructional context Predictable ◦ The structure of instruction (task design) ◦ Planning and nature of task/activity Unpredictable ◦ Process of carrying out instructional events/activities ◦ Moment-by-moment words and actions ◦ Teacher’s responsiveness to students unexpected actions (feedback to students)

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Allows teachers to identify signs of an emerging skill, such as a word, behavior, or expression, and use it to engage the student in higher level functioning Allows the student to take increasing control of the thinking Control of thinking is shared Entices the student to take as much initiative as possible

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Allow for learner autonomy and initiative Neither stifling of development nor lead to chaos Facilitate the process (lead to the identification of signs emerging skill) Consider the following description: ◦ The builders put a scaffold around a building that needs to be renovated, but the scaffold itself is only useful to the extent that it facilitates the work to be done. The scaffold is constantly changed, dismantled, extended, and adapted in accordance with the needs of the workers. In itself, it has no value.

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Read dialogues found on Table 2, page 19 of the instructional guide ◦ Identify instances of scaffolding. ◦ Identify who has the control of the direction of the interaction? Compare the interactions captured on page 20 of the instructional guide. ◦ Who has control of the direction of the interaction? ◦ What are the instances of scaffolding?

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More Planned More Improvised Continuity and Coherence task repetition with variation; connecting tasks and activities; project- based or action-based learning Supportive Environment environment of safety and trust; experiential links and bridges Intersubjectivity mutual engagement; being ‘in tune’ with each other Flow student skills and learning challenges in balance; students fully engaged Contingency task procedures and task progress dependent on actions of learners Emergence, or Handover/Takeover increasing importance of learner agency

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Directly describe and model the skill. Perform the skill/task while thinking aloud (asking and answering questions aloud). Provide immediate and specific feedback. ◦ Incorrect response: praise the student for effort while also describing and modeling the correct process/response; ASK QUESTIONS! ◦ Correct response: provide positive reinforcement by specifically stating what it is they did correctly; ASK QUESTIONS! As students demonstrate success, ask for an increased number of student responses or ask more complex questions. Continue to fade your direction, prompting students to complete more and more of the problem solving process: Relinquish CONTROL When students understand the problem-solving process, invite them to actively problem-solve with you ◦ Let STUDENTS ‘TAKE OVER’ ◦ students direct problem-solving, students ask questions Let student accuracy of responses guide your decisions about when to continue fading your direction.

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Listen carefully to the scaffolding demonstration video. What are the instances of scaffolding? Who is in control of the interaction? What would you do differently? Why?

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Thinking Questions

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Focusing instruction on big ideas is necessary but insufficient Requires ongoing monitoring of student understanding of those big ideas The prevailing form of questioning is low- level fill in the blank questions Instruction is focused on getting students to say the right things

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Dialogue 1 (IRE)*Dialogue 2 (IRF)** Teacher: Excuse me. Student: Yes? Teacher: Can you tell me how I can get to Highway 1 from here? Student: No problem! You go straight that way and see traffic light. When traffic light, you…left, then go, eh, go more….straight and then the Highway 1, you will see it. Teacher: Okay. Listen. Go straight TO the traffic light, turn left, and go straight ahead UNTIL you see the sign for Highway 1. Student: Ehm…go straight TO traffic light…(etc) Teacher: Excuse me. Student: Yes? Teacher:Can you tell me how I can get to Highway 1 from here? Student: No problem! You go straight that way and see traffic light. When traffic light, you…left, then go, eh, go more….straight and then the Highway 1, you will see it. Teacher: Thanks! Student: You welcome!

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‘‘opportunities to learn do not exist for learners who cannot take advantage of them’’ (Haertel et al., 2008, p. 6).

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Using different kinds of questions for different purposes can help differentiate instruction Help monitoring conceptual understanding Increase the proportion of students who remain engaged in conversations about important math ideas ◦ Engaging questions ◦ Refocusing questions ◦ Clarifying questions

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Open-ended with multiple acceptable answers Invite students into discussion Keep them engaged in conversation Re-engage students who have “tuned-out.” Students with low math self-efficacy benefit most by being invited into discussions that reward multiple solutions based on alternative, accurate math reasoning.

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Get students back on track or to move away from a dead-end strategy ◦ Used instead of simply telling students what to do differently Remind students about some important aspect of a problem they may be overlooking

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Help students explain their thinking or help the teacher understand their thinking Can be used when: A.A student understands an idea but the language used to explain that thinking is not clear or precise a.“What does ‘it’ refer to?” B.More needs to be revealed about a student’s thinking to make sense of it a.“How did you get that answer?”

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Two Similar Rectangles Engaging Question: How can we decide what value the question mark stands for? Refocusing Question: What does it mean for two rectangles to be similar? Clarifying Question:(In response to a student who says that the answer is 5) How did you get 5? 43 6?

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Suppose the learning target for a lesson is to distinguish area from perimeter. In a class discussion a student says the area is 50 centimeters. If the teacher wants to refocus the student to the general math idea (unit analysis), what question can be posed? If the teacher wants to call attention to (clarify) the student’s response what question can be posed?

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The conversation in Handout 6 occurs when the teacher stops to talk with two students who have been playing a game (based on Fraction Tracks). Considering how the questions are phrased, what do you think the purposes of the following questions are? How could you move your pieces across to the other side if your card was ? Could you go and then ? Can I move the whole now?

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Suppose a teacher asks students what number goes in the box (below) to make a true number sentence = A student replies “382.” The teacher then asks a clarifying question “How did you get 382?” To which the student replies “I added 152 and 230.” As the student replies, she notices that another student wrote “= 622” after 240. What fundamental misconception do these responses represent? Turn to page 27 in guide.

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Go to Page 25 in Instructional Guide for process instructions.

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Use ‘Teacher Talk’ to Model Ways of Thinking about Mathematics

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There is value in helping students conceptualize math as more than a set of procedures. Students need to understand that math is a thinking and reasoning process rather than a set of steps to go through to get the right answer. A focus on language use enables teachers to develop and reinforce norms for talking mathematics in valued ways which, in turn, affects students’ math beliefs and self- efficacy

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Stepping out More explicit language moves that include reflection on math actions; talking about math. That’s a great example of the kind of explanation I’m looking for. It’s important that you not only give your answer but that you also explain what you did and why you did it. I want you to explain the process you went through, not just give an answer.

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Revoicing Less explicit language moves that allow the teacher to reformulate a student’s response by clarifying or extending what a student has said in an effort to help other students understand the math significance of the contribution Recast student’s verbal contributions in more technical terminology with slight changes so as to move the discussion forward, leading to more conceptually-based explanations that originated with the student’s contribution.

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Revoicing is used to clarify statements, make connections, or fill in missing elements of an explanation. “By helping students articulate their understanding, teachers provide opportunities for students to agree or disagree with the reformulated version, teaching them to explain their reasoning.” (emphasis added, pp. 25)

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Refer to page 31 in teacher guide.

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Class Norms Students positioned to be thinkers and explainers within math community… Transmitted Message Math: Is flexible Is about meaning Makes sense Has reasons for its procedure Requires particular ways of reasoning and explaining Language Moves Using language moves, the teacher is able to: Request multiple solutions Amplify solutions Revoice to make math processes clearer and more precise Make students aware of math thinking and relationships Help students develop ways of thinking and talking about math

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Planning for classroom interaction is a way to offer all students opportunities to observe math reasoning in action and to develop their own abilities with math reasoning. Attending to language moves in the classroom that both reveals your own thinking processes and clarifies those of your students is a step toward constructing more meaningful math learning for all students,

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Developing reasoning about math takes time Teachers at all levels can help students begin to do so by modelling ways to: ◦ talk about math, ◦ reason about the activities they are engaged in. Refer to page 32 for assignment.

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‘‘opportunities to learn do not exist for learners who cannot take advantage of them’’ (Haertel et al., 2008, p. 6).

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