Algebraic Reasoning Institute

1. Algebraic Reasoning Institute July, 2011 Math & Science Collaborative at the Allegheny Intermediate Unit

2. Overview of the Institute • Goals Overview • Conceptual Flow Graphic • Reasoning Algebraically about Operations from the DMI series • Common Core State Standards for Mathematics (CCSSM) • Thinking Mathematically (CGI) • The work of Dr Margaret Smith from UPitt Math & Science Collaborative at the Allegheny Intermediate Unit

3. Overview of the Institute • Content Assessment • Will serve as a pre and post-test to the course • Not a timed assessment Math & Science Collaborative at the Allegheny Intermediate Unit

4. Overview • Insert conceptual flow and goals overview. Math & Science Collaborative at the Allegheny Intermediate Unit

5. Math & Science Collaborative at the Allegheny Intermediate Unit

6. Math & Science Collaborative at the Allegheny Intermediate Unit

7. Common Core State Standards Design • Building on the strength of current state standards, the CCSS are designed to be: • Focused, coherent, clear and rigorous • Internationally benchmarked • Anchored in college and career readiness* • Evidence and research based Math & Science Collaborative at the Allegheny Intermediate Unit

8. Common Core Standards for Mathematics • Grade-Level Standards • K-8 grade-by-grade standards organized by domain • 9-12 high school standards organized by conceptual categories • The Common Core proposes a set of Mathematical Practices that all teachers should develop in their students. Math & Science Collaborative at the Allegheny Intermediate Unit

9. Standards for Mathematical Practice Math & Science Collaborative at the Allegheny Intermediate Unit • Carry across all grade levels • Describe habits of mind of a mathematically proficient student

10. CCSS Mathematical Practices Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Math & Science Collaborative at the Allegheny Intermediate Unit

11. Underlying Frameworks • 5 Process Standards • Problem Solving • Reasoning and Proof • Communication • Connections • Representations Math & Science Collaborative at the Allegheny Intermediate Unit

12. Underlying Frameworks Math & Science Collaborative at the Allegheny Intermediate Unit

13. Underlying Frameworks Math & Science Collaborative at the Allegheny Intermediate Unit

14. Strands of Mathematical Proficiency • Conceptual Understanding – comprehension of mathematical concepts, operations, and relations • Procedural Fluency – skill in carrying out procedures flexibly, accurately, efficiently and appropriately • Strategic Competence – ability to formulate, represent,, and solve mathematical problems • Adaptive Reasoning – capacity for logical thought, reflection, explanation, and justification • Productive Disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy Math & Science Collaborative at the Allegheny Intermediate Unit

15. The Standards for Mathematical Practice Math & Science Collaborative at the Allegheny Intermediate Unit

16. The Standards for Mathematical Practice Math & Science Collaborative at the Allegheny Intermediate Unit

17. The Standards for Mathematical Practice Math & Science Collaborative at the Allegheny Intermediate Unit

18. CCSS Mathematical Practices Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Math & Science Collaborative at the Allegheny Intermediate Unit

19. The Standards for Mathematical Practice Math & Science Collaborative at the Allegheny Intermediate Unit • Read the practice assigned to your group. • Think about what this practice might look like in action • What would this practice sound like in action • Make a poster of the main ideas. • The poster may be a bulleted list, picture, diagram, or any other method that conveys the main ideas of the practice.

20. The Standards for Mathematical Practice • Take a moment to examine the first three words of each of the 8 Mathematical Practices.What do you notice? Mathematically Proficient Students…. Math & Science Collaborative at the Allegheny Intermediate Unit

21. The Standards for Mathematical Practice • Consider the verbs that illustrate the student actions each practice. • For example, examine Practice #3: Construct viable arguments and critique the reasoning of others. • Highlight each of the verbs in the description of that standard. • Discuss with a partner: What jumps out at you? Math & Science Collaborative at the Allegheny Intermediate Unit

22. Mathematical Practice #3: Construct viable arguments and critique the reasoning of others Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Math & Science Collaborative at the Allegheny Intermediate Unit

23. Mathematical Practice #3: Construct viable arguments and critique the reasoning of others Mathematically proficient students: • understand and use stated assumptions, definitions, and previously established results in constructing arguments. • make conjectures and build a logical progression of statements to explore the truth of their conjectures. • analyze situations by breaking them into cases, and can recognize and use counterexamples. • justify their conclusions, communicate them to others, and respond to the arguments of others. • reason inductively about data, making plausible arguments that take into account the context from which the data arose. • compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is. • construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. • determine domains to which an argument applies. • listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Math & Science Collaborative at the Allegheny Intermediate Unit

24. Mathematical Practice #3: Construct viable arguments and critique the reasoning of others Mathematically proficient students: • understand and use stated assumptions, definitions, and previously established results in constructing arguments. • make conjectures and build a logical progression of statements to explore the truth of their conjectures. • analyze situations by breaking them into cases, and can recognize and use counterexamples. • justify their conclusions, communicate them to others, and respond to the arguments of others. • reason inductively about data, making plausible arguments that take into account the context from which the data arose. • compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is. • construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. • determine domains to which an argument applies. • listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Math & Science Collaborative at the Allegheny Intermediate Unit

25. Mathematical Practice #3: Construct viable arguments and critique the reasoning of others • On a scale of 1 (low) to 6 (high), to what extent is your school/district promoting students’ proficiency in Practice 3? • Evidence for your rating? Math & Science Collaborative at the Allegheny Intermediate Unit

26. Standards for Mathematical Practice • SMP1: Explain and make conjectures… • SMP2: Make sense of… • SMP3: Understand and use… • SMP4: Apply and interpret… • SMP5: Consider and detect… • SMP6: Communicate precisely to others… • SMP7: Discern and recognize… • SMP8: Notice and pay attention to… Math & Science Collaborative at the Allegheny Intermediate Unit

27. Implementation Issue Math & Science Collaborative at the Allegheny Intermediate Unit

28. Math & Science Collaborative at the Allegheny Intermediate Unit

29. React to the Following Statements in Writing • When you hear the word algebra what kinds of mathematical ideas come to mind? • What, if anything, does algebra have to do with the content you teach? • What might it mean to engage with children on algebraic ideas? Math & Science Collaborative at the Allegheny Intermediate Unit

30. Mathematical Themes • Are two different definitions of even numbers equivalent? • What comprises an argument that a statement is always true when you cannot check every number? • What are generalizations about adding and multiplying odd and even numbers and how can they be proved? Math & Science Collaborative at the Allegheny Intermediate Unit

31. Hue and Julio scenario In a second-grade classroom the teacher commented there were an even number of children in class that day. • Hue said, “I knew it was even because when we lined up for lunch everyone had a partner.” • Julio said, “I knew it was even because when we split into two groups, the two groups were equal and no one was left over.” Math & Science Collaborative at the Allegheny Intermediate Unit

32. How can you show that these definitions describe the same set of numbers; how can you show these definitions are equivalent? Math & Science Collaborative at the Allegheny Intermediate Unit

33. Conjecture: The sum of two odd numbers is even. • Work with a partner to make an argument for how you know the conjecture is true. Math & Science Collaborative at the Allegheny Intermediate Unit

34. Math & Science Collaborative at the Allegheny Intermediate Unit

35. Math Practices in The Classroom Math & Science Collaborative at the Allegheny Intermediate Unit Which mathematical practice standards are evident ? How does the teacher support students with the Mathematical Practice Standards?

36. Acknowledgement This material is based on work supported by the SW PA MSP 2010 funds administered through the USDOE under Grant No. Project #: RA-075-10-0603. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the granting agency. Math & Science Collaborative at the Allegheny Intermediate Unit