Making the Common Core Standards for Mathematical Practice COME ALIVE in your classroom!

1. Making the Common Core Standards for Mathematical Practice COME ALIVE in your classroom! KATM Conference Presented by: Melisa Hancock melisa@ksu.edu

2. Implementing the Common Core State Standards • Requires: • Teacher knowledge of mathematics language and content. • Teacher knowledge of how to promote student involvement in mathematical practices. • Shifting students’ focus from “answer getting” to solving problems. • Establishing the classroom environment as a community of learners.

3. Get FIRED UP about Implementing the Common Core State Standards--- BEST PRACTICES ARE BACK!!!!!

4. “DOING” MATHEMATICS vs. MEMORIZING RULES Chinese Proverb I hear and I forget I see and I remember I do and I understand

5. Standards for Mathematical Practice • 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

6. Standards for Mathematical PracticesEnsure Your Kids are in the “Game”! The math practices teach kids EFFORT is what counts in mathematics achievement and struggling is OK!!! “Struggling in mathematics is no more the enemy than sweating is in basketball . . . it just means you are IN THE GAME!!!!!!!” Kimberly Sutton

7. Practice Standards Pocket Book Tear here

8. Practice Standard Pocket Book • Number each page 1-8. • Title on top of front page: Standards for Mathematical Practice • Write each Standard for Mathematical Practice on each page. • Paraphrase and/or take notes as we go along to help you understand what each standard means & “looks like” in the classroom.

9. Standards for Mathematical Practice • 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

10. Standards for Mathematical Practice -- PRE-TEST You Will Need: Practice Standards Sorting Mat and a set of Teacher Action Sorting Cards. • On Your Own -- Read each Action Card and match it with the correct Practice Standard. • When finished “matching”, turn to your elbow partner and compare.

11. How Did You Do? • Ask clarifying questions, such as “Which part do you agree or disagree with? • “What does it mean to have a remainder of 4 when you are dividing by 8?” • Design tasks that involve argumentation. • Represent their thinking with symbols. • Have a selection of tools readily available. • Describe quotient & remainder in words. • Focus thinking on relationships. • Explore other numbers.

12. Three Responses to a Mathematics Problem 1. Answer getting 2. Making sense of the problem situation 3. Making sense of the mathematics you can learn from working on the problem

13. Implementing the Common Core State Standards • Requires: • Teacher knowledge of mathematics language and content. • Teacher knowledge of how to promote student involvement in mathematical practices. The CORE of the CORE! • Shifting students’ focus from “answer getting” to solving problems. • Establishing the classroom environment as a community of learners and “patient problem-solvers”.

14. DEEP THINKING & TEACHING FOR UNDERSTANDING. . . . . Begins with expectations and the type of questions we ask of our students . . . That is when the Standards for Mathematical Practice COME ALIVE and your kids are “IN THE BALLGAME”!!!!!!!

15. PROBLEM SOLVING Problems we encounter in the “real-world”—our work life, family life, and personal health—don’t ask us what chapter we’ve just studied and don’t tell us which parts of our prior knowledge to recall and use. In fact, they rarely even tell us exactly what questions we need to answer, and they almost never tell us where to begin. They just happen. To survive and succeed, we must figure out the right question to be asking, what relevant experience we have, what additional information we might need, and where to start. And we must have enough stamina to continue even when progress is hard, but enough flexibility to try alternative approaches when progress seems too hard.

16. MP #1 asks students to develop that “puzzler’s disposition” in the context of mathematics. Teaching can certainly include focused instruction, but students must also get a chance to tackle problems that they have not been taught explicitly how to solve, as long as they have adequate background to figure out how to make progress. Young children need to build their own toolkit for solving problems, and need opportunities and encouragement to get a handle on hard problems by thinking about similar but simpler problems, perhaps using simpler numbers or a simpler situation. One way to help students make sense of all of the mathematics they learn is to put experience before formality throughout, letting students explore problems and derive methods from the exploration. For example, students learn the logic of multiplication and division—the distributive property that makes possible the algorithms we use—before the algorithms. The algorithms for each operation become, in effect, capstones rather than foundations. Another way is to provide, somewhat regularly, problems that ask only for the analysis and not for a numeric “answer.” CCSS Assessments: Problems deliberately designed to be different, to require students to “start by explaining themselves the meaning of a problem and looking for entry points to its solution.”

17. Make Sense of Problems & Persevere in Solving Them • In elementary grades, mathematically proficient students know that doing mathematics involves solving problems and discussing how they solved them. They APPLY their understanding to solve problems. • Students explain to themselves the meaning of a problem and look for ways to solve it. Your students may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, “Does this make sense?” They listen to the strategies of others and will try different approaches. They often will use another method to check their answers.

18. Example Task: Your math buddy comes to you and says, I worked the problem 124 divided by 8 on my paper and got 15r4. But when I did it on my calculator, it said 15.5. Which answer is right? How do you respond? • You want to help your students employ the Mathematical Practices. • This is the kind of problem we usually give an answer, move on and not think about it. • We want students to continue thinking about this type of problem, what they mean and continue thinking about what they can learn as they continue working on them. How do I interpret the SFM and embed them in ALL tasks I present to my students?

19. How do I encourage my students to make sense of problems and persevere in solving them? • Using the term “groups”, restate what 124 ÷ 8 means. • In the context of the problem, are the quotients 15 r 4 and 15.5 the same or are they different? • What do the quotients mean? • What does it mean to have a remainder of 4 when you are dividing by 8? • What does 0.5 mean? • What could you do to convince (show) your partner that the answers are the same or different?

20. 2. Reason Abstractly and Quantitatively Mathematically proficient K-5 students should recognize that a number represents a specific quantity. They connect the quantity to written symbols and create a logical representation of the problem at hand, considering both the appropriate units involved and the meaning of quantities.

21. Reason Abstractly & Quantitatively • Second graders who are learning how to write numerical expressions may be given the challenge of writing numerical expressions that describe the number of tiles in this figure in different ways. • Sudents might write 1+2+3+4+3+2+1 (the heights of the stairsteps from left to right) or 1+3+5+7 (the width of the layers from top to bottom) or 10+6 (the number of each color) or various other expressions that capture what they see. These are all decontextualizations—representations that preserve some of the original structure of the display, but just in number and not in shape or other features of the picture. Not any expression that totals 16 makes sense—for example, it would seem hard to justify 2+14—but a child who writes, for example, 8+8 and explains it as “a sandwich”—the number of blocks in the middle two layers plus the number of blocks in the top and bottom—has taken an abstract idea and added contextual meaning to it. • This Practice asks students to be able to translate a problem situation into a number sentence and be able to recognize the connection between all the elements of the sentence and the original problem. It involves making sure that the units (objects) in problems make sense. • Example, in decontextualizing a problem that asks how many busses are needed for 99 children if each bus seats 44, a child might write 99÷44. But after calculating 2r11 or 2¼ or 2.25, the student must recontextualize: the context requires a whole number answer, and not, in this case, just the nearest whole number. Successful recontextualization also means that the student knows that the answer is 3 busses, not 3 children or just 3.

22. Original Task: How do I encourage students to reason abstractly and quantitatively? Ask questions that: • focus the students thinking on the meaning of operations, such as “What does it mean to have a remainder of 4 when you are dividing by 8”? Give ample time for students to explore. (Focus is not on just getting the answer to this division problem—but making sense of this problem and numbers within the problem.) • encourage students to think about the remainder, relationships, why this remainder? • focus the students thinking on the meaning of numbers, such as “What does . 5 mean?” (Focus on 4 of 8 relationship). Is there a connection? Why? Explain? Show me your thinking.

23. 3. Construct Viable Arguments & Critique the Reasoning of Others In elementary, mathematically proficient students may construct arguments using concrete referents, such as objects, pictures, and drawings. They refine their mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?” Starting in 1st grade, they explain their thinking to others and respond to others’ thinking.

24. Defending & Justifying • We ALL know that children love to talk. But explanation—clear articulation of a sequence of steps or even the chronology of events in a story—is very difficult for children, often even into middle school. To “construct a viable argument,” let alone understand another’s argument well enough to formulate and articulate a logical and constructive “critique,” depends heavily on a shared context, especially in the early grades. Given an interesting task, they can show their method and “narrate” their demonstration. It does not make sense to have them try to describe, from their desks, an articulate train of thought, nor should we expect the other students in class to “follow” that description any better than—or even as well as—they’d follow the train of thought of a teacher who is just talking without illustrating. The standard recognizes this fact when it says “students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions.” The key is not the concreteness, but the ability to situate their words in context—to show as well as tell. • The inclination to “justify their conclusions” also depends on the nature of the task: certain tasks naturally pull children to explain; ones that are too simple or routine feel unexplainable. Depending on the context, “I added” can seem to a child hardly worth saying. And finally, skill at “communicating [a justification] to others” comes from having plentiful opportunities to do so. The way children learn language, including mathematical and academic language, is by producing it as well as by hearing it used. When students are given a suitably challenging task and allowed to work on it together, their natural drive to communicate helps develop the academic language they will need in order to “construct viable arguments and critique the reasoning of others.” One kind of task that naturally “pulls” children to explain is a “How many ways can you…” task.

25. Original Task: How do I encourage my students to construct viable arguments and critique the reasoning of others—at MY grade level? To develop the reasoning that this standard asks children to communicate, the mathematical tasks we give need depth. Problems that can be solved with only one fairly routine step give students no chance to assemble a mental sequence or argument, even non-verbally. Select tasks that depend on argumentation…rather than the goal being getting the answer to computation and/or following the correct steps. Ask questions that model how you want them to think……the desired thinking such as are you sure that the answers are different? • Can you convince me that they are different? • How would you convince me? • How can you explain to me what each answer means? • Can you analyze a peer’s solution and ask questions that result in them “catching” errors in their thinking, etc. etc.

26. 4. Model with Mathematics Mathematically proficient students experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, acting out, making a chart, list, or graph, creating equations, etc. Students need opportunities to connect the different representations and explain the connections. They should be able to use all of these representations as needed. Elementary students should evaluate their results in the context of the situation and reflect on whether the results make sense. They also evaluate the utility of models to determine which models are most useful and efficient to solve problems.

27. Model with Mathematics This Standard for Mathematical Practice can be easily misinterpreted or narrowly interpreted. Modeling with mathematics does NOT mean using concrete manipulatives. Although, the use of concrete manipulatives is necessary in developing conceptual understanding, this Practice Standard is referring to writing the mathematics equation that describes real-world situations. The intent is to ensure that the mathematics students engage in helps them see and interpret the world—the physical world, the mathematical world, and the world of their imagination—through a mathematical lens. One way, mentioned in the standard, is through the use of simplifying assumptions and approximations. Children typically find “estimation” pointless, and even confusing, when they can get exact answers, but many mathematical situations do not provide the information needed for an exact calculation.

28. Example Task: How do I encourage students to model with mathematics? • Providing a real-world problem and asking students to write the equation that the “situation” calls for. • Providing a mathematical model and asking students to write a “situation” that matches the equation. • For the original division task ask questions such as, How to represent their thinking about the “remainder” with symbols? Example:: 0.5 = 5 = 1 and 10 2 4 out of 8 is 4 = 1 8 2

29. 5. Use Appropriate Tools Strategically Mathematically proficient students consider the available tools (including estimation) when solving a mathematical problem and decide when certain tools might be helpful. For instance, they may use graph paper to find all the possible rectangles that have a given perimeter. They compile the possibilities into an organized list or a table, and determine whether they have all the possible rectangles. Or they may use unit cubes to fill a rectangular prism and then use a ruler to measure the dimensions.

30. Use Appropriate Tools Strategically • Base Ten Blocks, Ten-Frames, Cuisenaire Rods, Calculators are the “obvious” tools. This standard also includes “pencil and paper” as a tool, (such as: diagrams, two-way tables, graphs.” The number line and area model of multiplication are two more tools—both diagrammatic representations of mathematical structure—that the CCSS Content Standards explicitly require. So, in the context of elementary mathematics, “use appropriate tools strategically” must be interpreted broadly and sensibly to include many choice options for students. • Essential, and easily overlooked, is the call for students to develop the ability “to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations.” This certainly requires that students gain sufficient competence with the tools to recognize the differential power they offer; it also requires that their learning include opportunities to decide for themselves which tool serves them best. It also requires curricula and teaching to include the kinds of problems that genuinely favor different tools. It may also require that, from time to time, a particular tool is prescribed—or proscribed—until students develop a competency that would allow them to make “sound decisions” about which tool to use. • The number line is sometimes regarded just as a visual aid for children. It is, in fact, a sophisticated image used even by mathematicians. For young children, it helps develop early mental images of addition and subtraction that connect arithmetic with measurement. Rulers are just number lines built to spec! This number line image shows “the distance from 5 to 9” or “how much greater 9 is than 5.” Children who see subtraction that way can use this model to see “the distance between 28 and 63” as 35 , and to do so without crossing out digits and borrowing and following a rule they may only barely understand. In fact, many can learn to see this model in their heads, too, and do this subtraction mentally. This is essentially how clerks used to “count up” to make change. The number line model also extends naturally to decimals and fractions by “zooming in” to get a more detailed view of that line between the whole numbers. And it extends equally naturally to negative numbers. It thereby unifies arithmetic, making sense of what is otherwise often seen as a collection of independent and hard-to-remember rules. We can see that the distance from -2 to 5 is the number we must add to -2 to get 5: . And we can see why 42 – (-36) can also be written as 42 + 36: the “distance from -36 to 42 is . The number line remains useful as students study data, graphing, and algebra: two number lines, at right angles to each other, label the addresses of points on the coordinate plane.

31. Assumptions & Approximations as “Tools” Essential step of “reflecting” on whether the results make sense.”  What’s important here is not the context that’s used, but the kind of thinking it requires. Example, students in 3rd grade who are beginning to learn that there are many fractions equivalent to ½ can quickly become competent, and inventive, at contributing entries to a table, on the board, with three columns: fractions between 0 and ½, fractions equal to ½, and fractions between ½ and 1. They can then apply what they know to simplify, cleverly, such “naked arithmetic” problems such as “Arrange the fractions 4/9, 5/8, and 7/12 in order from least to greatest.” The comparison can be performed entirely without calculating by noting, first, that 4/9 is less than ½ (because 4 is less than half of 9) and it is the only one that is less than ½, so it is the smallest. The other two are both greater than ½, but 5/8 is one eighth greater than ½ whereas 7/12 is only one twelfth greater, so 5/8 is the largest. The point, of course, is not to replace one technique with another. The point is that mathematical thinking simplifies the work.

32. Original Task: How do I encourage students to use tools strategically? Pull out objects that help you SEE the decimal representation of the remainder, etc. (This is where manipulatives can be used) Indicate what’s happening in the mathematics situation. Asking questions such as: • Which of these things would help you understand what is going on with this problem: (Calculator, Blocks, Graph paper to show what is happening?) • Draw a model to show what you mean. How are your pictures alike or different? Have student create the use of tools by asking: 1)What is the calculator representing? 2)Which of these things will help you understand or think about this problem?

33. 6. Attend to Precision This Standard for Mathematical Practice is easily misinterpreted or narrowly interpreted. It is NOT talking about just getting the correct answer. It is much more than accuracy of computation. This Standard is referring to mathematical language used precisely (Mathematical terms and definitions embedded in our discussions, in words can we talk about what is happening in our representations, problems, and use them precisely so all understand what we are talking about ) Mathematically proficient elementary students develop their mathematical communication skills, they try to use clear and precise language in their discussions with others and in their own reasoning. They are careful about specifying units of measure and state the meaning of the symbols they choose. For instance, when figuring out the area of a rectangle they record their answers in square units, or when figuring the volume of a rectangular prism they record their answers in cubic units.

34. WE MUST “model” PrecisionAre these the same?

35. Are these the “same”?Concept of = (equal sign) Tilt or Balance the Equation • 3 +4 =2+ 5 • ?

36. Original Task: How do I encourage Students to attend to precision? Ask students to: • Describe the quotient and remainder in words. • Describe the decimal quotient in words. (What’s the connection between the two? • Explain the relationship between the divisor and the remainder. • Compare and contrast to different situations in which you might decide to use a decimal quotient instead of a remainder and justify your answer.

37. Structure & Repeated Reasoning The last 2 Standards for Mathematical Practice are more difficult to interpret or get a handle on because the language is very complex and the vocabulary used in not so familiar to us. But, they embody the nature of DOING MATHEMATICS!

38. 7. Look For and Make Use of Structure Mathematically proficient students look closely to discover a pattern or structure. For instance, students use properties of operations as strategies to multiply and divide (commutative and distributive properties). Intermediate grades might examine numerical patterns and relate them to a rule or a graphical representation.

39. Structure Mathematics has far more consistent structure than our language, but too often it is taught in ways that don’t make that structure easily apparent. If, for example, students’ first encounter with the addition of same-denominator fractions drew on their well-established spoken structure for adding the counts of things—two sheep plus three sheep makes five sheep, two hundred plus three hundred makes five hundred, and two wugs plus three wugs makes five wugs, no matter what a wug might be—then they would already be sure that two eighths plus three eighths makes five eighths. Instead, they often first encounter the addition of fractions in writing, as 2/8 + 3/8, and they therefore invoke a different pattern they’ve learned—add everything in sight—resulting in the incorrect and nonsensical 5/16.

40. Structure Structure is about what is going on in math that allows you to apply the equation, or symbolism, or to draw the diagram that you can draw. Structure is the skeleton of what is going on in the situation.

41. Original Task: How do I encourage my students to look for and make use of structure? Ask questions that focus students thinking on the important relationships within the problem. • So, with this problem focus questions that encourage looking at the relationships between the remainder & the divisor, i.e. 4 is half of 8. (That is the important part of the structure in this particular problem it’s all about relationships). • The important part of structure of this problem is NOT the dividend that matters. What matters is the divisor and the relationship of the remainder to the divisor. Looking at this relationship is the structure in this problem. • Decimal representation, not the dividend…..How does it relate to the previous representation with a remainder? Now the questions are related to why is this important to the problem and requires deductive reasoning.

42. 8. Look For and Express Regularity in Repeated Reasoning. In elementary grades mathematically proficient students look closely to discover a pattern or structure. For instance, students use properties of operations as strategies to multiply and divide (commutative and distributive properties). Intermediate students explore operations, use repeated reasoning to understand algorithms and make generalizations about patterns.

43. Regularity and Repeated Reasoning Although the description mentions “patterns”, be cautious about doing just what we are used to in terms of patterns. Some type of pattern is involved, but “regularity in repeated reasoning is much more than this. Again, this Standard for Mathematical Practice embodies the nature of DOING mathematics. It is not that simple and is more broad, than simply looking for patterns. It involves repetition in thinking that will lead you to ageneralization. It is inductive reasoning or looking at many examples that are somewhat the same and asking ourselves what can we learn from that sameness and how can we use that reasoning to take us farther and delve deeper into the structure and understanding the mathematics?

44. Original Task: How do I encourage my students to look for and express regularity in repeated reasoning? Division Task :1) Have student explore what happens with other numbers that when divided by 8 leave a remainder of 4. 12 ÷ 8 36 ÷ 8 804 ÷ 8 Why do I keep getting a remainder of 4? Have students explore remainders in other division problems that have a quotient that ends in .5. Example: 9 divided by 6 35 divided by 10 126 divided by 12.

45. CONCRETE – REPRESENTATIONAL-ABSTRACT (CRA) CRA instructional sequence consists of three stages: concrete, representation and abstract: • Concrete. In the concrete stage, the teacher begins instruction by modeling each mathematical concept with concrete materials (e.g., red and yellow chips, cubes, base-ten blocks, pattern blocks, fraction bars, and geometric figures). (The doing stage of mathematics) • Representational.In this stage, the teacher transforms the concrete mode l into a representational (semiconcrete) level, which may involve drawing g pictures; using circles, dots, and tallies; or using stamps to imprint pictures for counting. (The seeing stage of mathematics) • Abstract. At this stage, the teacher models the mathematics concept at a symbolic level, using only numbers, notation, and mathematical symbols to represent the number of circles or groups of circles. The teacher uses operation symbols (+, –, ) to indicate addition, multiplication, or division. (The symbolic stage of mathematics) CRA supports understanding underlying mathematical concepts before learning “rules,” that is, moving from a concrete model of chips or blocks for multiplication to an abstract representation such as 4 x 3 = 12.

46. Standards for Mathematical Practice “Understanding as points of intersection between expectations and practices” These Practice Standards are more complex and complete but will bring richness to our instruction. They are behaviors that can be observed in the classroom. Expectations UNDERSTANDING Math Practices

47. Three Responses to a Mathematics Problem 1.Answer getting 2.Making sense of the problem situation 3.Making sense of the mathematics you can learn from working on the problem

48. Standards for Mathematical Practice How can the CCSS for Mathematical Practice promote instructional change? When these standards are ALIVE in our classrooms, how will they change the focus of mathematics instruction from mechanical facility to conceptual understanding?

49. Practices: Developing Expertise “Students who engage in these practices, individually and with their classmates, discover ideas and gain insights that spur them to pursue mathematics beyond the classroom walls. They learn that effort counts in mathematical achievement. Encouraging these practices in students of all ages should be as much a goal of the mathematics curriculum as the learning of specific content.” Common Core State Standards, 2010

50. “Common Core State Standards are not intended to be new names for old ways of doing business. They are a call to take the next step. “ Excerpt from Common Core State Standards Document