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## Teaching Math to Diverse Adolescent Learners:

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**Teaching Math to Diverse Adolescent Learners:**Achieving Instructional Equity June – July, 2010 Zenaida Aguirre-Muñoz, Ph. D. West Texas Middle School Math Science Partnership Texas Tech University**Workshop Overview**• 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**Expectations**• Self-Monitoring Activities • Form Submissions • Blogging • Conference Presentations**Instructional Equity Premise**‘‘opportunities to learn do not exist for learners who cannot take advantage of them’’ (Haertel et al., 2008, p. 6).**Getting Started**6th 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.**SIZE**Measure 0-dimension Number 1-dimension Length 2-dimensions Area 3-dimensions Volume Counting Distance formula Volume formulas Area formulas Unit Analysis Unit: Name Unit: u Unit: u2 Unit: u3 Measurement of Geometric Shapes**Design Instruction Around Big Ideas**Defining & Identifying Big Ideas**The Challenge**• Teaching for ‘exposure’ • Teaching without objectives, with ‘fun’ activities • Neither empowers students to solve complex problems**Development of Conceptual Knowledge**• 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**Earth Science Example**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. Reveals how different natural phenomena follow the same flow of matter/energy that represents a rectangular figure.**SIZE**Measure 0-dimension Number 1-dimension Length 2-dimensions Area 3-dimensions Volume Counting Distance formula Volume formulas Area formulas Unit Analysis Unit: Name Unit: u Unit: u2 Unit: u3 Middle School Math Example 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.**Summary of Big Ideas**• 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**Identifying Big Ideas**• Unwrap Standards • Underline content nouns • Represent concepts (what students need to know) • Circle verbs • Represent skills (what students need to be able to do) • Examine verbs to determine the intended level of thinking/reasoning • Correspond to Bloom’s Taxonomy • Determine organizing/’power’ concepts (big ideas)**Organizing Content –Handout 1**Size/measurement, dimensionality**Organizing Content –Handouts 2&3**X X X X X X X X**Design Instruction Around Big Ideas**Elements of Conceptually-Based Instruction**Identify Math Key Concepts**• 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)**Design and/or Selection of Tasks**• 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**Try it!**• 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.**Task Design Case Study**• 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.**Teach Strategies Before Introducing Authentic Tasks**• 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**Strategy Example Case (Handout 4)**• Volume Strategy Instruction • Link to prior knowledge • Introduce new strategy • Compute area • Compute volume • Write complete answer**Apply It!**• 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?**Demonstrate Links between Big Ideas, Prior Knowledge, &**Strategies • 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**Remember…**Diverse learners benefit from good strategy instruction if and only if the strategies are designed to result in transferable knowledgeof their application.**Try It!**• 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.**Apply Conceptual Understanding to New Content**• 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?**Groundwork for Conceptual Understanding**• 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**Apply It!—Classroom Scenario 1**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?**Maximize Growth Potential**Theoretical Foundations**The Challenge**• 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**Instructional Equity Premise**‘‘opportunities to learn do not exist for learners who cannot take advantage of them’’ (Haertel et al., 2008, p. 6).**Use Language to “Think Together”**• 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.**Thought and Language**• 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.”**Interaction, Language, & Thought—Vygotsky**• 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. Age 2 Cognitive Development Thought Language**Learning as Transformation**• 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**Learning as Change in Participation Over Time**• 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)**Getting in the Zone**• 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**Summary of Growth Potential**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.**Instructional Equity Premise**‘‘opportunities to learn do not exist for learners who cannot take advantage of them’’ (Haertel et al., 2008, p. 6).**Plan to Scaffold & Differentiate**Structure & Processes**The Challenge**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.**Scaffolding De-Mystified**• 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)**Successful Scaffolding**• 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**Tasks that Promote Autonomy**• 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.**Teacher Interactions that Promote Autonomy**• 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?**Implementation**• 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 whilealso 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.**View and Analyze**• 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?