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Supporting Rigorous Mathematics Teaching and Learning Deepening Our Understanding of CCSS Via A Constructed Response Assessment Tennessee Department of Education High School Mathematics Geometry © 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 Assessments (CRAs) in order to determine the way the assessments are assessing the CCSSM; • analyze and discuss the CCSS for Mathematical Content and 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 Geometry Understand congruence in terms of rigid motions. Prove geometric theorems. Define trigonometric ratios and solve problems involving right triangles. Use coordinates to prove simple geometric theorems algebraically. © 2013 UNIVERSITY OF PITTSBURGH Analyzing Assessment Items (Private Think Time) Four assessment items have been provided: Park City Task Getting in Shape Task Lucio’s Ride Task Congruent Triangles Task For each assessment item: • solve the assessment item; and • make connections between the standard(s) and the assessment item. © 2013 UNIVERSITY OF PITTSBURGH 1. Park City Task Park City is laid out on a grid like the one below, where each line represents a street in the city, and each unit on the grid represents one mile. Four other streets in the city are represented by . a. Dionne claims that the figure formed by , , , and is a parallelogram. Do you agree or disagree with Dionne? Use mathematical reasoning to explain why or why not. b. Triangle AFE encloses a park located in the city. Describe, in words, two methods that use information in the diagram to determine the area of the park. c. Find the exact area of the park. © 2013 UNIVERSITY OF PITTSBURGH 2. Getting in Shape Task Points A (12, 10), J (16, 18), and Q (28, 12) are plotted on . coordinate plane below. the a. What are the coordinates of a point M such that the quadrilateral with vertices M, A, J, and Q is a parallelogram, but not a rectangle? b. Prove that the quadrilateral with vertices M, A, J and Q is a parallelogram. c. Prove that the quadrilateral with vertices M, A, J and Q is not a rectangle. d. Determine the perimeter of your parallelogram. © 2013 UNIVERSITY OF PITTSBURGH 3. Lucio’s Ride When placed on a grid where each unit represents one mile, State Highway 3 111 runs along the line = x + 3, and State Highway 213 runs along the line = 3 x 4 - 13 . 4 4 The following locations are represented by points on the grid: • Lucio’s house is located at (–3, 1). • His school is located at (–1, –4). • A grocery store is located at (–4, 0). • His friend’s house is located at (0, 3). a. Is the quadrilateral formed by connecting the four locations a square? Explain why or why not. Use slopes as part of the explanation. b. Lucio is planning to ride his bike tomorrow. In the morning, he plans to ride his bike from his house to school. After school, he will ride to the grocery store and then to his friend’s house. Next, he will ride his bike home. The four locations are connected by roads. How far is Lucio planning to ride his bike tomorrow if he plans to take the shortest route? Support your response by showing the calculations used to determine your answer. © 2013 UNIVERSITY OF PITTSBURGH 4. Congruent Triangles a. Locate and label point M on 2 such that it is of the 5 distance from point S to point U. Locate and label point T on such that it is 2 of the distance from point 5 S to point N. Locate and label point Q on such 2 that it is of the distance 5 from point N to point U. b. Prove triangles TNQ and QMT are congruent. © 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 – Geometry Congruence (G-CO) Understand congruence in terms of rigid motions G-CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. G-CO.B.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. G-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Common Core State Standards, 2010 The CCSS for Mathematical Content CCSS Conceptual Category – Geometry Congruence (G-CO) Prove geometric theorems G-CO.C.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. G-CO.C.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. G-CO.C.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. Common Core State Standards, 2010 The CCSS for Mathematical Content CCSS Conceptual Category – Geometry Similarity, Right Triangles, and Trigonometry (G-SRT) Define trigonometric ratios and solve problems involving right triangles G-SRT.C.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G-SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles. G-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★ ★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 – Geometry Expressing Geometric Properties with Equations (G-GPE) Use coordinates to prove simple geometric theorems algebraically G-GPE.B.4 G-GPE.B.5 G-GPE.B.6 G-GPE.B.7 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★ ★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. 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 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 geometric properties. 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 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 the Standards for Mathematical Practice? © 2013 UNIVERSITY OF PITTSBURGH The CCSS for Mathematical Content K-8 Distribution of Domains K – 8 Domain K 1 2 3 4 5 6 7 8 x x x Ratios and Proportional Relationships x x The Number System x x x Expressions and Equations x x x Counting and Cardinality x Operations and Algebraic Thinking x x x x x x Number and Operations in Base Ten x x x x x x x x x Number and Operations - Fractions Measurement and Data x x x x x x Geometry x x x x x x Functions Statistics and Probability © 2013 UNIVERSITY OF PITTSBURGH x x x x The CCSS for Mathematical Content − High School Geometry is one of 6 Conceptual Categories which specify the mathematics that all students should study in order to be college and career ready. • Number and Quantity • Algebra • Functions • Modeling • Geometry • Statistics and Probability © 2013 UNIVERSITY OF PITTSBURGH Examples of Key Advances from Previous Grades or Courses – Geometry • Because concepts such as rotation, reflection, and translation were treated in the grade 8 standards mostly in the context of hands-on activities, and with an emphasis on geometric intuition, high school Geometry will put equal weight on precise definitions. • In grades K–8, students worked with a variety of geometric measures (length, area, volume, angle, surface area, and circumference). In high school Geometry, students apply these component skills in tandem with others in the course of modeling tasks and other substantial applications (Mathematical Practice #4). © 2013 UNIVERSITY OF PITTSBURGH Examples of Key Advances from Previous Grades or Courses – Geometry (continued) The algebraic techniques developed in Algebra I can be applied to study analytic geometry. Geometric objects can be analyzed by the algebraic equations that give rise to them. Some basic geometric theorems in the Cartesian plane can be proven using algebra. PARCC Model Content Frameworks for Mathematics, October 2011, pp. 53-54 1. Park City Expressing Geometric Properties with Equations (G-GPE) Use coordinates to prove simple geometric theorems algebraically G-GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★ ★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. Moved By This Situation Expressing Geometric Properties with Equations (G-GPE) Use coordinates to prove simple geometric theorems algebraically G-GPE.B.4 G-GPE.B.7 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. ★ ★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. Lucio’s Ride Expressing Geometric Properties with Equations (G-GPE) Use coordinates to prove simple geometric theorems algebraically G-GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). G-GPE.B.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★ ★ 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 4. Congruent Triangles Expressing Geometric Properties with Equations (G-GPE) Use coordinates to prove simple geometric theorems algebraically G-GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). G-GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Common Core State Standards, 2010