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Algebraic Reasoning Institute

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(Learn about) To develop awareness of Curriculum Topic Study (CTS) as a tool you can use for connecting standards and research on learning to classroom practice. (Practice) To provide guided practice in using CTS. (Apply) To consider ways you might apply CTS to your work.

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NSF-funded TPC Professional Development Materials Project awarded to the Maine Mathematics and Science Alliance in partnership with West Ed 2 resources guides: Science and Mathematics Curriculum Topic Study books Facilitator’s Guide to Using Curriculum Topic Study Web Site: www.curriculumtopicstudy.orgwww.curriculumtopicstudy.org National Professional Development

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A process that incorporates systematic study of standards and research A set of tools and collective resources for improving curriculum, instruction, and assessment An intellectually engaging professional development experience where teachers focus on and discuss student learning

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CTS IS NOT: A remedy for weak content knowledge (CTS is used to enhance and support content learning) A collection of teaching activities (CTS describes considerations one must take into account when planning or selecting teaching activities) A description of “how to’s” (CTS helps you think through effective teaching based on knowledge of learning goals and how students learn) A quick fix (CTS takes time and dedication to use it effectively) The end-all for professional development (CTS helps you identify additional experiences that will help you grow as a teacher)

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Clarify and deepen knowledge of relevant curricular topics Develop a common knowledge base and language about standards and research Move beyond personal opinions and assumptions to consider key ideas and practices developed through consensus by the mathematics education community “Stand on the Shoulders of Giants”- Experts at your fingertips 24/7!

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Pre-service Teachers Beginning Teachers Experienced Teachers Teacher Leaders, Mentors, and Coaches Professional Developers Pre-Service and Graduate Mathematics Education Faculty Mathematicians Working with K-12 Teacher Education Curriculum Developers Curriculum, Instruction, and Assessment Committees

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What has been missing is a systematic, scholarly, deliberate process to help educators intellectually engage with standards and research on student learning so they can make effective use of them. CTS provides that “Missing Link.”

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Pair up at your tables. Open the CTS book at random. With your partner, do a quick scan of the page you opened to. What do you see that provides you with a “preview” of CTS? Mark the page with a sticky note. Repeat 2-3 times to get an initial sense of what is contained in the CTS book. Share an example with the group that particularly interests you and tell why.

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CTS Sections and Outcomes Selected Readings from CTS Resources Web Site- Supplementary Material www.curriculumtopicstudy.org www.curriculumtopicstudy.org See Chapter 2- The CTS Study Guide pp 21-23

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Each guide has 6 CTS sections (Left Column) Purposes of the sections I : Identify Adult Content Knowledge II : Consider Instructional Implications III : Identify Concepts and Specific Ideas IV : Examine Research on Student Learning V : Examine Coherency and Articulation VI : Clarify State Standards and District Curriculum Each section links to CTS sources and pre- vetted Readings (Right Column)

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Improve adult science literacy (I) Improve knowledge of content teachers teach (I) Examine instructional considerations (II) Identify alternative conceptions (IV) Identify “Big Ideas”, Concepts, Specific Ideas, and Skills (III) Consider developmental implications (II, IV) Examine scope and sequence (III) See connections within and across topics (V) Clarify state standards and district curriculum (VI)

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CTS Collective Resources- Experts at Your Fingertips 24/7 Indicates the resource is online

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Getting to Know the Resources Science Pages 24-26 Science for All Americans Science Matters Benchmarks for Science Literacy The National Science Standards Making Sense of Secondary Science Atlas of Science Literacy Mathematics Pages 27-30 Science for All Americans Beyond Numeracy Benchmarks for Science Literacy Principles and Standards for School Mathematics Research Companion Atlas of Science Literacy

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Divide the 6 resources among your table group. Choose one “expert” for each resource. Read the description of your resource from the CTS book, pp 27-30 Examine the resource, looking for notable features Describe the resource to your table group, pointing out notable features

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Scaffold: The structure and supports that a teacher or more knowledgeable helper provides to allow a learner to perform a task he or she cannot yet perform independently. (Vygotsky, 1978; Dixon-Krauss, 1996; Wertsch,1991.)

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STEP 1:Scan and select the CTS category. STEP 2: Scan the list of topics within the category that include the content you are examining. STEP 3:Select the CTS guide you will use. STEP 4: Determine which section(s) of the CTS guide will help you find the information you need. STEP 5:Select the resource(s) you will use, the grade span(s), and the readings. STEP 6: Examine the reading for information relevant to your topic and task. STEP 7: Record your findings. If you do not find what you need, go back to Step 2 and repeat with another topic.

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“What kinds of experiences should I provide my middle school students with when learning about percent?” Category? Numbers and Operations CTS Topic Guide? Percent Page Number of CTS Guide? Page 129

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“What kinds of experiences should I provide my middle school students when learning about percent?” Section? Section II Outcome? Consider Instructional Implications

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“What kinds of experiences should I provide my middle school students when learning about percent?” Which resource, grade level, and page numbers do I read? Benchmarks- Numbers, grade span essay, p 213 and/or NCTM, grades 6-8, Number and Operations, p 215 Understanding Numbers, pp 215-217 What part of the page do I focus on? Just the essay, not the bullets!

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“What specific ideas about statistics should I focus on at the 3-5 grade level?” Category? Data Analysis CTS Topic Guide? Statistical Reasoning or Summarizing Data Page Number of CTS Guide? Page 186 or 187

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“What specific ideas about statistics should I focus on at the 3-5 grade level?” Section? Section III (Could also include Section V and VI) Outcome? Identify Concepts and Specific Ideas

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Guide: Statistical Reasoning Which resource, grade level, and page numbers do I read? Benchmarks- 9D Uncertainty pages 227-230; 9E Reasoning pages 232-234; 12E Critical Response Skills pages 298-300 PSSM- 3-5 Data Analysis and Probability page 176 or 400 What part of the page do I focus on? Just the bulleted learning goals, not the essay.

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“What should adults know about mathematical equations? Category? Algebra CTS Topic Guide? Expressions and Equations Page Number of CTS Guide? Page 136

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“What should adults know about mathematical equations?” Section? Section I Outcome? Identify Adult Content Knowledge

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“What should adults know about mathematical equations?” Which resource, grade level, and page numbers do I read? Science for All Americans- Ch 9 pp 132-34 and/or Beyond Numeracy- Algebra, Some Basic Principles, pp 7-9

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“I’m curious to see how the concept and skills related to proportional reasoning develops from elementary grades through high school” Category? Integrated Topics CTS Topic Guide? Proportionality Page Number of CTS Guide? Page 198

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“I’m curious to see how the concept and skill related to proportional reasoning develops from elementary grades through high school” Section? Section V Outcome? Examine Coherency and Articulation

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“I’m curious to see how the concept and skill related to proportional reasoning develops from elementary grades through high school” What strand maps will you use? Ratios and Proportionality Is there a conceptual strand within a map you should focus on? Parts and Wholes, Description or Comparison, and Computation

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“I wonder what difficulties or common misconceptions I should anticipate when students create and interpret graphs?” Category? Integrated Topics CTS Topic Guide? Graphic Representation Page Number of CTS Guide? Page 196

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“I wonder what difficulties or common misconceptions I should anticipate when students interpret graphs?” Section? Section IV Outcome? Examine Research on Student Learning

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Students of all ages often interpret graphs of situations as literal pictures rather than as symbolic representations of the situations. Many students interpret distance/time graphs as the paths of actual journeys. In addition, students confound the slope of a graph with the maximum or the minimum value and do not know that the slope of a graph is a measure of rate. When constructing graphs, middle- school and high-school students have difficulties with the notions of interval scale and coordinates even after traditional instruction in algebra. For example, some students think it is legitimate to construct different scales for the positive and the negative parts of the axes. Alternatively, students think that the scales on the X and Y axes must be identical, even if that obscures the relationship. When interpreting graphs, middle-school students do not understand the effect that a scale change would have on the appearance of the graph. Finally, students read graphs point-by- point and ignore their global features. This has been attributed to algebra lessons where students are given questions that they could easily answer from a table of ordered pairs. They are rarely asked questions about maximum and minimum values; intervals over which a function increases, decreases or levels off; or rates of change.

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Students of all ages often interpret graphs of situations as literal pictures rather than as symbolic representations of the situations. Many students interpret distance/time graphs as the paths of actual journeys. In addition, students confound the slope of a graph with the maximum or the minimum value and do not know that the slope of a graph is a measure of rate. When constructing graphs, middle-school and high-school students have difficulties with the notions of interval scale and coordinates even after traditional instruction in algebra. For example, some students think it is legitimate to construct different scales for the positive and the negative parts of the axes. Alternatively, students think that the scales on the X and Y axes must be identical, even if that obscures the relationship. When interpreting graphs, middle-school students do not understand the effect that a scale change would have on the appearance of the graph. Finally, students read graphs point- by-point and ignore their global features. This has been attributed to algebra lessons where students are given questions that they could easily answer from a table of ordered pairs. They are rarely asked questions about maximum and minimum values; intervals over which a function increases, decreases or levels off; or rates of change.

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Involves listening to each other attentively; building on each others’ ideas and respectfully challenging each other; referring regularly to the texts and reading notes; provided evidence for opinions and theories; and asking questions and probing to clarify ideas.

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CTS is like learning any new skill. It takes practice, perseverance, focus, and a lot of effort!

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Choose 2-3 questions that interest you (Try to use different CTS sections and resources). Practice using the scaffold to answer the questions. Do not take shortcuts on round 1! Record notes from your reading that address the question (not the entire topic). Complete a reflection note. Post your reflection note on the reflection wall before you begin a new snapshot.

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What was the key learning for you? What surprised you about your CTS finding? How did CTS add to what you already knew? How will this information be useful to you in your work? What further questions does this raise for you?

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Turn to page 35 and examine the list of ways CTS can be applied. Turn to Chapter 4 on p 53 (or table of contents) to scan examples of suggestions and support materials for using CTS in different contexts. Which of these applications do you think you might use CTS for (or do you have a new one)? Do a “Quick Write” on your thoughts about using CTS

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Visit the CTS web site at www.curriculumtopicstudy.org www.curriculumtopicstudy.org Contact:

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This module was developed by Lynn Raith, Mathematics Curriculum Specialist K-12. Video courtesy of NYC District 2 and the Institute for Learning. These.

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