Supporting Rigorous Mathematics Teaching and Learning Deepening our Understanding of the CCSS Via a Constructed Response Assessment Tennessee Department of Education High School Mathematics Algebra 1 © 2013 UNIVERSITY OF PITTSBURGH Forms of Assessment Assessment as Learning Assessment of Learning © 2013 UNIVERSITY OF PITTSBURGH Assessment for Learning Session Goals Participants will: • deepen understanding of the Common Core State Standards (CCSS) for Mathematical Practice and Mathematical Content; • understand how Constructed Response Assessments (CRAs) assess the CCSS for both Mathematical Content and Practice; and • understand the ways in which CRAs assess students’ conceptual understanding. © 2013 UNIVERSITY OF PITTSBURGH Overview of Activities Participants will: • analyze Constructed Response (CRAs) in order to determine the way they are assessing the CCSS for Mathematics; • analyze and discuss the CCSS for Mathematical Content and CCSS for Mathematical Practice; • discuss what it means to develop and assess conceptual understanding; and • discuss the CCSS related to the tasks and the implications for instruction and learning. © 2013 UNIVERSITY OF PITTSBURGH The Common Core State Standards The standards consist of: The CCSS for Mathematical Content The CCSS for Mathematical Practice © 2013 UNIVERSITY OF PITTSBURGH Analyzing a Constructed Response Assessment © 2013 UNIVERSITY OF PITTSBURGH Tennessee Focus Clusters Algebra 1 Create equations that describe numbers or relationships. Solve equations and inequalities in one variable. Represent and solve equations and inequalities graphically. Interpret functions that arise in applications in terms of the context. © 2013 UNIVERSITY OF PITTSBURGH Analyzing Assessment Items (Private Think Time) Four assessment items have been provided: Buddy Bags Disc Jockey Decisions What’s the point? Paulie’s Pen For each assessment item: • solve the assessment item; and • make connections between the standard(s) and the assessment item. © 2013 UNIVERSITY OF PITTSBURGH 1. Buddy Bags For a student council fundraiser, Anna and Bobby have spent a total of $55.00 on supplies to create Buddy Bags. They plan to charge $2.00 per Buddy Bag sold. Anna created the graph below from an equation to represent the profit from the number of Buddy Bags sold. a. Determine the equation Anna used to create the graph if x represents the number of Buddy Bags sold and y represents the profit in dollars. Use mathematical reasoning to explain your equation. c. Anna says, “I connected the points to represent the equation, but by connecting the points I am not representing the context of the problem.” Use mathematical reasoning to explain why she is correct. © 2013 UNIVERSITY OF PITTSBURGH 60 50 40 30 Profit in Dollars b. Bobby claims that Anna’s graph is incorrect because it does not show that they plan to charge $2.00 per Buddy Bag. Do you agree or disagree with Bobby? Use mathematical reasoning to support your decision. 70 20 10 0 -10 0 10 20 30 40 50 -20 -30 -40 -50 -60 Number of Buddy Bags Sold 60 2. Disc Jockey Decisions The student council has asked Dion to be the disc jockey for the Fall Banquet. He has been asked to play instrumental music during the first hour while the students are eating dinner. During the last 15 minutes of the banquet the school choir will sing. For the remaining time, Dion will choose popular songs to play. a. Write an equation to determine the number of popular songs, p, that Dion can choose if the songs Dion chooses have an average run time of 3.5 minutes and the total time for the Fall Banquet is t minutes. Use mathematical reasoning to justify that your equation is correct. b. Use your equation from Part a to determine the number of popular songs that Dion can choose if the banquet will be held from 6:00 – 10:00pm. c. Dion decides to organize the music another way. He decides to play 50 popular songs. Write and solve an algebraic equation to determine the average run time, r, of the 50 popular songs Dion can choose if the average run time is represented in minutes by r. Use mathematical reasoning to justify that your equation is correct. © 2013 UNIVERSITY OF PITTSBURGH 3. What’s the point? Mr. Williams asks his Algebra 1 class to find the solutions to an equation in two variables with a domain in the set of real numbers. Colton correctly creates the table below using values from the domain of the equation. He then uses his table to create a graph. Colton’s Table x y 0 -4 2 -1.5 -2 -6.5 5 2.25 -8 -14 10 8.5 Colton's Graph 10 5 0 -10 -5 0 5 10 -5 -10 -15 a. Determine the equation Colton used to create the table. Use mathematical reasoning to justify that the equation is correct. b. Destiny sees Colton’s work and argues that any table contains just a subset of the solutions to Mr. Williams’ equation. Do you agree or disagree with Destiny? Explain why or why not. © 2013 UNIVERSITY OF PITTSBURGH 4. Paulie’s Pen Scarlet has to build a rectangular pen for her peacock, Paulie. Scarlet has P feet of fencing. a. Scarlet decides the length of the pen will be 12 feet. Write an equation describing the relationship between P and the width of the pen, w. Explain your thinking in the context of the problem. b. Solve the equation for w. Show your work. c. Scarlet changes her mind and decides that the width of Paulie’s pen will be half of the length of his pen. Write an equation describing the relationship between P and the width of the pen, w. Explain your thinking in the context of the problem. d. Solve the equation for w. Show your work. © 2013 UNIVERSITY OF PITTSBURGH Discussing Content Standards (Small Group Time) For each assessment item: With your small group, discuss the connections between the content standard(s) and the assessment item. © 2013 UNIVERSITY OF PITTSBURGH Deepening Understanding of the Content Standards via the Assessment Items (Whole Group) As a result of looking at the assessment items, what do you better understand about the specifics of the content standards? What are you still wondering about? © 2013 UNIVERSITY OF PITTSBURGH The CCSS for Mathematical Content CCSS Conceptual Category – Algebra Creating Equations★ (A –CED) Create equations that describe numbers or relationships A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. ★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard. Common Core State Standards, 2010 The CCSS for Mathematical Content CCSS Conceptual Category – Algebra Reasoning with Equations and Inequalities (A-REI) Solve equations and inequalities in one variable A.-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. A.-REI.B.4 Solve quadratic equations in one variable. A.-REI.B.4a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. A.-REI.B.4b Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Common Core State Standards, 2010 The CCSS for Mathematical Content CCSS Conceptual Category – Algebra Reasoning with Equations and Inequalities (A-REI) Represent and solve equations and inequalities graphically A.-REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). A.-REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★ A.-REI.D.12 Graph the solutions to a linear inequality in two variables as a halfplane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. ★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard. Common Core State Standards, 2010 The CCSS for Mathematical Content CCSS Conceptual Category – Functions Interpreting Functions (F-IF) Interpret functions that arise in applications in terms of the context F-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★. F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★ F-IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★ ★Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard. Common Core State Standards, 2010 Determining the Standards for Mathematical Practice Associated with the Constructed Response Assessment © 2013 UNIVERSITY OF PITTSBURGH Getting Familiar with the CCSS for Mathematical Practice (Private Think Time) • Count off by 8. Each person reads one of the CCSS for Mathematical Practice. • Read your assigned Mathematical Practice. Be prepared to share the “gist” of the Mathematical Practice. © 2013 UNIVERSITY OF PITTSBURGH 20 The CCSS for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Common Core State Standards for Mathematics, 2010, NGA Center/CCSSO 21 Discussing Practice Standards (Small Group Time) Each person has 2 minutes to share important information about his/her assigned Mathematical Practice. © 2013 UNIVERSITY OF PITTSBURGH 22 Discussing Practice Standards (Small Group Time) For each assessment item: With your small group, discuss the connections between the practice standards and the assessment item. © 2013 UNIVERSITY OF PITTSBURGH The CCSS for Mathematical Practice 1. 2. 3. 4. 5. 6. 7. 8. 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. Common Core State Standards, 2010 Deepening Understanding of the Practice Standards via the Assessment Items (Whole Group) Which standards for mathematical practice do you better understand? What are you still wondering about? © 2013 UNIVERSITY OF PITTSBURGH Assessing Conceptual Understanding © 2013 UNIVERSITY OF PITTSBURGH Rationale We have now examined assessment items and discussed their connection to the CCSS for Mathematical Content and Practice. A question that needs considering, however, is if and how these assessments will give us a good means of measuring the conceptual understandings our students have acquired. In this activity, you will have an opportunity to consider what it means to develop conceptual understanding, as described in the CCSS for Mathematics, and what it takes to assess for it. © 2013 UNIVERSITY OF PITTSBURGH Assessing for Conceptual Understanding The set of CRA items are designed to assess student understanding of expressions and equations. Look across the set of related items. What might a teacher learn about a student’s understanding by looking at the student’s performance across the set of items as a whole? What is varying from one item to the next? © 2013 UNIVERSITY OF PITTSBURGH Conceptual Understanding • What do the authors mean by conceptual understanding? • How might analyzing student performance on this set of assessments help us determine if students have a deep understanding of the assessed standards? © 2013 UNIVERSITY OF PITTSBURGH Developing Conceptual Understanding Knowledge that has been learned with understanding provides the basis of generating new knowledge and for solving new and unfamiliar problems. When students have acquired conceptual understanding in an area of mathematics, they see connections among concepts and procedures and can give arguments to explain why some facts are consequences of others. They gain confidence, which then provides a base from which they can move to another level of understanding. Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press The CCSS on Conceptual Understanding In this respect, those content standards which set an expectation of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics. Common Core State Standards for Mathematics, 2010, p. 8, NGA Center/CCSSO Assessing Concept Image Tall (1992) differentiates between the mathematical definition of a concept and the concept image, which is the entire cognitive structure that a person has formed related to the concept. This concept image is made up of pictures, examples and non-examples, processes, and properties. A strong concept image is a rich, integrated, mental representation that allows the student to flexibly move between multiple formulations and representations of an idea. A student who has connected mathematical ideas in this way can create and use a model to analyze a situation, uncover patterns and synthesize them to form an integrated picture. They can also use symbols meaningfully to describe generalizations which then provides a base from which they can move to another level of understanding. Brown, Seidelmann, & Zimmermann. In the trenches: Three teachers’ perspectives on moving beyond the math wars. http://mathematicallysane.com/analysis/trenches.asp Developing and Assessing Understanding Why is it important, when assessing a student’s conceptual understanding, to vary items in these ways? © 2013 UNIVERSITY OF PITTSBURGH Using the Assessment to Think About Instruction In order for students to perform well on the CRA, what are the implications for instruction? • What kinds of instructional tasks will need to be used in the classroom? • What will teaching and learning look like and sound like in the classroom? © 2013 UNIVERSITY OF PITTSBURGH Step Back • What have you learned about the CCSS for Mathematical Content that surprised you? • What is the difference between the CCSS for Mathematical Content and the CCSS for Mathematical Practice? • Why do we say that students must work on both the Standards for Mathematical Content and Standards for Mathematical Practice? © 2013 UNIVERSITY OF PITTSBURGH Functions and Modeling In modeling situations, knowledge of the context and statistics are sometimes used together to find algebraic expressions that best fit an observed relationship between quantities. Then the algebraic expressions can be used to interpolate (i.e., approximate or predict function values between and among the collected data values) and to extrapolate (i.e., to approximate or predict function values beyond the collected data values). One must always ask whether such approximations are reasonable in the context. http://commoncoretools.wordpress.com/2012/04/ccss_progression_functions_2012_12 04.bis.pdf, pg. 4 1. Buddy Bags Creating Equations★ (A-CED) Create equations that describe numbers or relationships A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Interpreting Functions (F-IF) Interpret functions that arise in applications in terms of the context F-IF.B.4 F-IF.B.5 ★ For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★ Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. ★ Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard. Common Core State Standards, 2010 2. Disc Jockey Decisions Creating Equations★ (A-CED) Create equations that describe numbers or relationships A.CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A.CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Reasoning with Equations and Inequalities (A-REI) Solve equations and inequalities in one variable A.-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. ★ Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard. Common Core State Standards, 2010 3. What’s the point? Creating Equations★ (A-CED) Create equations that describe numbers or relationships A.CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Reasoning with Equations and Inequalities (A-REI) Represent and solve equations and inequalities graphically A.-REI.D.10 ★ Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard. Common Core State Standards, NGA Center/CCSSO, 2010 4. Paulie’s Pen Creating Equations★ (A-CED) Create equations that describe numbers or relationships A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. ★ Mathematical Modeling is a Standard for Mathematical Practice (MP4) and a Conceptual Category, and specific modeling standards appear throughout the high school standards indicated with a star (★). Where an entire domain is marked with a star, each standard in that domain is a modeling standard. Common Core State Standards, 2010

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