WHAT DIFFERENCE WILL THEY MAKE?

1. WHAT DIFFERENCE WILL THEY MAKE? Michael Savoy, Ph.D. Math Specialist

2. What are standards? Standards define what students should understand and be able to do. Standards must be a promise to students of the mathematics they can take with them. We haven’t kept our old promise and now we make a new one. What difference will it make?

3. Creating Better Standards in Mathematics (Phil Daro) Phil Daro Chair, Mathematics College and Career Readiness Standards Work Group; Writing Team, K–12 Mathematics Standards Committee; Senior Fellow, America’s Choice How do you create better standards in mathematics? http://commoncoreanswers.com

4. Lessons Learned After two decades of standards based accountability: • Too many standards • Lack of student motivation • “Cover” at “pace” is a failure • Tells teachers to ignore students • Turn the page regardless • Shrug your shoulders and do what “they” say • Mathematics is not a list of topics to cover • Singapore: “Teach less, learn more”

5. Lessons Learned • TIMSS: math performance in the US is being compromised by a lack of focus and coherence in the “mile wide, inch deep” curriculum • Hong Kong students outscore U.S. students on the grade 4 TIMSS, even though Hong Kong only teaches about half of the tested topics. U.S. covers over 80% of the tested topics. • High-performing countries spend more time on mathematically central concepts: greater depth and coherence.

6. Answer Getting vs. Learning Mathematics United States How can I teach my kids to get the answer to this problem? Use mathematics they already know. Easy, reliable, works with bottom half, good for classroom management. Japan How can I use this problem to teach mathematics they don’t already know?

7. Overview State-led and developed common core standards for K-12 in English/language arts and mathematics Focus on learning expectations for students, not how students get there. States allowed to add additional state-specific standards up to 15%.

8. YES, the CCSSM: define what students should understand and be able to do are based on evidence and research provide clear signposts (alignment) along the way to the goal of college and career readiness for all students are clear and consistent are a call to take the next step stress conceptual understanding of key ideas are informed by other top-performing countries set grade-specific standards (Common Core State Standards Initiative 2010, 3–5)

9. NO, the CCSSM do not: define the full range of supports appropriate for English language learners dictate curriculum or teaching methods define the intervention methods or materials necessary to support students who are well below or well above grade-level expectations define the full range of supports appropriate for students with special needs (Common Core State Standards Initiative 2010, 3–5)

10. Standards of Mathematical Content

11. Math Content Standards • Mathematical Performance: what kids should be able to do • Mathematical Understanding: standards for what kids need to understand

12. Performance • Performance: what kids should be able to do • multiply and divide within 100 • 3rd grade sample

13. Understanding • Understanding: what kids should understand about mathematics • 3rd grade sample • Understand properties of multiplication and the relationship between multiplication and division. • Table Talk: • Why is that our kids do not perform as well as students in other countries do?

14. Standards for Mathematical Content Organization by Grade Bands and Domains (Common Core State Standards Initiative 2010)

15. How to Read the Standards • introduction (see page 13)

16. How to Read the Standards • Overview (see page 14)

17. Organization of CCSS Domain Standard Cluster (Common Core State Standards Initiative 2010, 16)

18. The high school mathematics standards: Call on students to practice applying mathematical ways of thinking to real world issues and challenges Require students to develop a depth of understanding and ability to apply mathematics to novel situations, as college students and employees regularly are called to do Emphasize mathematical modeling, the use of mathematics and statistics to analyze empirical situations, understand them better, and improve decisions Identify the mathematics that all students should study in order to be college and career ready. Overview of High School Mathematics Standards

19. How to Read the Standards: High School • Additional mathematics that students should learn in order to take advanced courses such as calculus, advanced statistics, or discrete mathematics is indicated by (+), as in this example: • (+)Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers). • All standards without a (+) symbol should be in the common mathematics curriculum for all college and career ready students. • The high school standards are listed in conceptual categories: • Number and Quantity • Algebra • Functions • Modeling • Geometry • Statistics and Probability

20. How to Read the Standards: High School • Introduction (p. 62)

21. How to Read the Standards: High School • Overview (p. 63)

22. Focus and Progression Grade Bands K–5 6–8 High School Operations andAlgebraic Thinking Expressions and Equations Algebra Number andOperations inBase Ten The NumberSystem ConceptDomains Number andOperations–Fractions (Common Core State Standards Initiative 2010)

23. Standards of Mathematical Practice

24. Essential Questions What are the Standards for Mathematical Practice and why are they important for mathematical proficiency? What do we mean by “understanding” in mathematics and how do the Standards for Mathematical Practice lead to understanding?

25. A ProblemDO NOT SOLVE Make as many rectangles as you can with an area of 24 square units. Use only whole numbers for the length and width. Sketch the rectangles, and write the dimensions on the diagrams. Write the perimeter of each one next to the sketch. What if the perimeter is 24 units? What questions do you ask yourself as you encounter this problem? How do these questions help you to develop a solution approach?

26. Meta-Cognition • Thinking about thinking. • The unconscious process of cognition. • Meaning making It is hard to articulate how you think about thinking. It is even harder to model

27. Meta-cognition • Modeling the thinking strategies • Using multiple representations • Talking about thinking- what are you thinking? Why? • Could you approach this a different way? • Accountable Talk

28. Meta-cognition implications for lessons. • Make thinking public • Use multiple representations • Offer different approaches to solution • Ask questions about the problem posed. • Set a context, define the why of the problem • Focus students on their thinking not the solution • Solve problems with partners • Prepare to present strategies

29. Introduction to the Standards for Mathematical Practice Mathematical proficiency is more than “getting the answer”—it includes the process of using mathematical concepts effectively as identified in the Standards for Mathematical Practice. The mathematical practices are consistent for all the grade levels even though they manifest themselves differently as students grow in mathematical maturity.

30. Background of Mathematical Practices National Council of Teachers of Mathematics Principles and Standards (2000): Process Standards National Research Council’s Report Adding It Up (2001): Mathematical Proficiencies

31. Standards for Mathematical Practice (Phil Daro)

32. Text Rendering Read your assigned mathematical practice description from the Common Core State Standards. Describe the standard in your own words. Find sentences, phrases, and words that are particularly significant. Discuss with your selections with your group. What does the Practice mean for classroom practice and student understanding. Think about and describe what it may look like, sound like and/or feel like in the classroom.

33. Standards for Mathematical Practice • Make sense of problems and persevere in solving them. • Reason abstractlyand quantitatively. • Construct viable argumentsand critiquethe reasoningof others. • Model with mathematics. • Use appropriate toolsstrategically. • Attend to precision. • Look for and make use of structure. • Look for and express regularityin repeated reasoning.

34. Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. • Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. • They analyze givens, constraints, relationships, and goals. • They plan a solution pathway rather than simply jumping into a solution attempt. • They monitor and evaluate their progress and change course if necessary. • Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?”

35. Standards for Mathematical Practice 2. Reason abstractly and quantitatively. • Mathematically proficient students make sense of quantities and their relationships in problem situations. • They bring two complementary abilities to bear on problems involving quantitative relationships: • the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents • and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. • Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities; and knowing and using different properties of operations and objects.

36. Standards for 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 counter examples. • They justify their conclusions, communicate them to others, and respond to the arguments of others. • 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.

37. Standards for Mathematical Practice 4. Model with mathematics. • Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. • They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. • They can analyze those relationships mathematically to draw conclusions. • They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

38. Standards for Mathematical Practice 5. Use appropriate tools strategically. • Mathematically proficient students consider the available tools when solving a mathematical problem. • These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. • Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. • They detect possible errors by strategically using estimation and other mathematical knowledge. • They are able to use technological tools to explore and deepen their understanding of concepts.

39. Standards for Mathematical Practice 6. Attend to precision. • Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. • They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. • They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. • They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.

40. Standards for Mathematical Practice 7. Look for and make use of structure. • Mathematically proficient students look closely to discern a pattern or structure. • Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. • They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. • They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects.

41. Standards for Mathematical Practice 8. Look for and express regularity in repeated reasoning. • Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. • As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details.

42. Fair Share Problem There is a delicious cookie that 4 children at a party want to share. Show how the 4 children might share the cookie. How much does each child get? Before the cookie is handed out, one child leaves without getting her piece. How could the 3 children left share the cookie fairly? How much does each child get? (Philipp 2005)

43. Video Example of Fair Share Problem Looking for evidence of mathematical thinking. (Philipp 2005)

44. Thank you for your participation! Have a great day !!!!

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