Performance Based Assessment in Mathematics Hawaii March 5th, 2012 Brad Osborn

Objectives Participants will develop an understanding of the Common Core State Standards through performance tasks Participants will identify the implications of the CCSS assessments to instruction, assessment, leadership and professional development.

Along the Way to Common Standards What are the characteristics of students who are college and career ready? College and Career Readiness College and career readiness can be defined as the level of preparation a student needs to enroll and succeed—without remediation—in a credit-bearing general education course at a postsecondary institution that offers a baccalaureate degree or transfer to a baccalaureate program, or in a high- quality certificate program that enables students to enter a career pathway with potential future advancement. (Educational Policy Improvement Center 2011)

Along the Way to Common Standards College and Career Readiness Students (Common Core State Standards Initiative 2010c, 7) demonstrate independence build strong content knowledge respond to the various demands of audience, task, purpose, and discipline comprehend as well as critique value evidence use technology and digital media strategically and capably come to understand other perspectives and cultures

Teach Less, Learn More Overview of the Common Core State Standards for Mathematics The Common Core State Standards are aligned with college and work expectations are clear, understandable, and consistent include rigorous content and application of knowledge through high order skills build upon strengths and lessons of current state standards are informed by other top performing countries so that all students are evidence-based (Common Core State Standards Initiative 2010a) are prepared to success in our global economy and society

Innovative Items Items in which students manipulate graphic elements to provide a response Delivered by a computer Scored electronically May have multiple correct answers Cannot be easily translated to paper

Benefits of Innovative Items Innovative Items Assess content in different more engaging ways Measure a broader range of skills Allow for more authentic alignment with learning situations Include process skills and higher-order thinking skills Improve presentation of complex and dynamic information Reduce reading load through visual supports Reduce effect of successful guessing Allow for rapid electronic scoring of open-ended items 7

Innovative Math Sample

Innovative Math Sample (continued) 9 Sample Innovative Item

Salary problem After receiving 4.3% salary increase, a teacher’s aide is now earning $34,000 per year. What did she earn last year? Show your strategy and make a model Explain how you know your solution process is correct.

The Cube What is the volume of a cube whose edges each measure 3 centimeters? What is the surface area of a cube whose edges each measure 3 centimeters? A student named Eddie says, “No matter what size the cube is, the number you get when you calculate its surface area is always twice as big as the number you get when you calculate its volume.” Is Eddie right? Show how you know.

Performance Tasks What did you notice about this task? What are students being asked to demonstrate? How does this align with what you are currently doing? 12

What do we want students to perform Standards for understanding- the content standards are very specific about what students need to understand, and even differentiate the learning continuum between developing understanding and understanding Standards for performance- what students should be able to do with their standard for understanding, how they demonstrate their understanding Standards for Mathematical Practice: varieties of expertise that math educators should seek to develop in their students.

Standards 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, pp.6-7

Performance tasks typically... Are open-ended Have more than one way to answer correctly Require active performance rather than passive reaction Need a scoring guide or rubric

Good Performance Tasks Are clear and unambiguous Set parameters for what the answer should look like Measure something important Are not a simple substitute for a multiple-choice question Require reasoning, synthesis, evaluation, higher- order thinking

Very Complex Tasks What’s the demand on the students? What knowledge and skills are tapped into? What is the cognitive challenge or depth of knowledge demand? Is this task different from the typical tasks given to students in your class/school/district now? How do you think students in our schools today would react to these kinds of tasks? How would you approach teaching this? Discuss at your table and report out.

Sample Performance Item- What grade level?

Sample High School Task LUCKY DIP 25¢ a chance 3 balls the same color Win $5 Ann is in charge of a Lucky Dip to raise money for charities. There are 3 barrels, and each contains an equal number of red, green, white, and black balls. The balls are buried in sawdust so that you cannot see them before you pick one out. To play the game, you give Ann your 25¢, then you pick one ball from each barrel. You win $5 if all three balls are the same color. a) Calculate the probability that you will win the $5 if you play once. b) Do you think that the Lucky Dip will raise money for the local charities. Show your calculations. c) Ann wants to change the game increase the amount of money it makes for the charities. Describe two different kinds of changes that she could make to the Lucky Dip and find how much is likely to be raised for the charities after each change.

Sample High School Task PRICE REDUCTION A. Two for the price of one B. Buy one and get 25% off the second C. Buy two, get 50% off the second one D. Three for the price of two 1. Which of these four different offers gives the biggest price reduction? _____________________________________________________________________ Explain your reasoning clearly: _____________________________________________________________________ 2. Which of these four different offers gives the smallest price reduction? _____________________________________________________________________ Explain your reasoning clearly: _____________________________________________________________________

Sample High School Task You have the job of organizing a table tennis league. 7 players will take part. All matches are singles. Every player has to play each of the other players once. There are four tables at the club. Games will take up to half an hour. The first match will start at 1:00 p.m. Plan how to organize the league so that the tournament will take the shortest possible time. Put all the information on a poster so that the players can easily understand what to do.

Sample High School Task The Fresha Drink Company is marketing a new soft drink. The drink will be sold in a can that holds 200 cm 3. In order to keep costs low, the company wants to use the smallest amount of aluminum. Find the radius and height of a cylindrical can that holds 200 cm 3 and uses the smallest amount of aluminum.

Implications Discuss in your groups the implications for teaching and learning? Be prepared to share out.

There is no decision that teachers make that has a greater impact on students’ opportunities to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages students in studying mathematics. Lappan & Briars, 1995 Mathematical Tasks: A Critical Starting Point for Instruction

Going Deeper Session A: Evidence of Mathematical Practices in Student Work Session B: Developing an Instructional Model to Support the Mathematical Practice Standards Session C: Complex Math Tasks Lead to Accountable Talk- Evidence of the Mathematical Practice Standards