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Extreme Math Makeover: State Assessment and Reporting in the Era of the Common Core. Mathematics Department Heads of the West Chicago Suburbs November 14, 2011 mathematical modeling Max McGee [email protected] 1 Extreme Math Makeover: State Assessment and Reporting in the Era of the Common Core. Mathematics Department Heads of the West Chicago Suburbs November 14, 2011 mathematical modeling Max McGee [email protected] 2 Extreme Math Makeover: State Assessment and Reporting in the Era of the Common Core.. Mathematics Department Heads of the West Chicago Suburbs November 14, 2011 mathematical modeling Max McGee [email protected] 3 Example from ACT A car averages 27 miles per gallon. If gas costs $4.04 per gallon, which of the following is closest to how much the gas would cost for this car to travel 2,727 typical miles? A. $ 44.44 B. $109.08 C. $118.80 D. $408.04 E. $444.40 Examples from OECD (PISA test) http://www.oecd.org/dataoecd/63/35/37464175.pdf Example from OECD (PISA test) For this question you need to draw together information from Figure 1 and Figure 2. The disappearance of the rhinoceros, hippopotamus, and auruchs from Saharan rock art happened: A. B. C. D. At the beginning of the most recent Ice Age. In the middle of the periods when Lake Chad was at its highest level After the level of Lake Chad had been falling for over a thousand years At the beginning of an uninterrupted dry period One-third of the PISA mathematics items are open-ended response Advance Organizers Common Core Standards and instructional practices Assessment Assessment Reporting What are your impressions of the Common Core Standards? Common Core Standards – Mathematical Practices 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 Common Core Standards – Mathematical Practices (cont.) Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning Mathematical Practices Describe ways in which students ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout their school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction. Mathematical Content … a balanced combination of procedure and understanding. Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Grade 6 Mathematics Multiple-Choice Sample Assessment Objective 10.6.05: Solve problems involving the probability of a simple event, including representing the probability as a fraction, decimal, or percent. 14 1. x = y 2. 2x = 2y 3. 2x + y = x + 2y 4. 2x + y - 2y = x + 2y - 2y 5. 2x + y - 2y = x 6. 2x + y - 2y - y = x - y 7. 2x - 2y = x - y 8. 2(x - y) = x - y 9. 2(x - y) (x - y) 10. 2(x - y) (x - y) 11. 2=1 (x - y) = = (x - y) (x - y) (x - y) Huh? Here is an algebraic equation that proves 2=1. Each step contains a new operation. Which step, if any, is incorrect? Advance Organizers Common Core Standards and instructional practices Assessment Assessment Reporting How about those assessments? “… in examining the secondary standards, curriculum, and assessments in Massachusetts, California, and Texas, we discovered that it was not the standards or even the curricula that drove teachers and schools nuts, it was the tests. The tests did NOT measure the standards, particularly the higher order ones, well or even at all. And they certainly did not do justice to the curriculum.” - David Pearson, 2010 Assessment “of” learning, “for” learning, and “to” learn! Summative Formative Performance ISBE, IBHE, and the P-20 Council should lead the nation in developing pioneering performance assessment items that are aligned with the revised Illinois Learning Standards and the 21stCentury Skills. With the assistance of educators, ISBE should provide exemplars of performance based assessment items, student work samples, rubrics, and resource materials (including videos and web based products) that educators can use to model and assess the revised standards and the 21stcentury knowledge and skills at the district, school, and classroom levels. The 24 PARCC states have committed to building a K-12 assessment system that: Builds a pathway to college and career readiness for all students, Creates high-quality assessments that measure the full range of the Common Core State Standards, Supports educators in the classroom, Makes better use of technology in assessments, and Advances accountability at all levels. The PARCC design will incorporate four features designed to significantly improve the quality and usefulness of large-scale assessments: Be anchored in college and career readiness, Enable deeper and richer assessments, Measure learning and provide information on student progress throughout the school year, and Provide timely results. PARCC will also leverage technology throughout the design and delivery of the assessment system. The overall assessment system design will include a mix of constructed response items, performance-based tasks, and computerenhanced, computer-scored items. The PARCC assessments will be administered via computer, and a combination of automated scoring and human scoring will be employed. Grade 6 Mathematics Multiple-Choice Sample Assessment Objective 10.6.05: Solve problems involving the probability of a simple event, including representing the probability as a fraction, decimal, or percent. 23 To prepare for the ISAT, one might make a typical test item like this one for fraction computation 2/3 of 1/4 is: A. B. C. D. 8/3 3/7 2/12 3/8 24 A Common Core Assessment item for fractions may look like: Use a blank sheet of paper to demonstrate how you figure out what 2/3 of 1/4 is, OR in the space below draw a model that shows how you figure out what 2/3 of 1/4 is. 25 What is Performance Assessment? An assessment that affords students the opportunity to demonstrate proficiency by creating, producing, presenting and/or performing a meaningful task. Common Core Standards – Third Grade Measurement and Data 1. Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram. Operations and Algebraic Thinking 8. Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.3 Third Grade Performance Based Assessment The Flat Stanley Elementary School day starts at 9:00 a.m. and ends at 3:15 p.m. Students have 30 minutes for lunch and 20 minutes for recess. Make a timeline to show how long (in hours and minutes) are they in class each day. At Jerome Horwitz Elementary, the day starts at 8:30 and ends at 3:30. Students have 40 minutes for lunch and 20 minutes for recess. Draw a picture to show long they are in class each day. (If the school year is 180 days long, write a number sentence and make a chart to show how much longer students at Jerome Horwitz spend in school? Common Core Standards – Eighth Grade Statistics and Probability Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. Common Core Standards – Eighth Grade Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.. Expressions and Equations 5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Eighth Grade Performance Based Assessment: Draw conclusions from this graph and explain your reasoning. Eighth Grade Performance Based Assessment: Draw a correlation line for the light blue dots and a correlation line for the dark blue dots (and triangle), calculate the slopes and explain why the two lines have similar or different slopes. Common Core Standards – Eleventh Grade Mathematic Modeling Modeling links classroom mathematics and statistics to everyday life, work, and decision-making. Modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions. Quantities and their relationships in physical, economic, public policy, social, and everyday situations can be modeled using mathematical and statistical methods. When making mathematical models, technology is valuable for varying assumptions, exploring consequences, and comparing predictions with data. Common Core Standards – Eleventh Grade Statistics and Probability Making Inferences and Justifying Conclusions 6. Evaluate reports based on data. Use probability to make decisions 7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). High School Performance Based Assessment A regional city has had lots of problems with gangs and violence over the years. The mayor, chief of police, and city council need your help. Review the available data, analyze and model these data to give the city a plan to reduce violence. After you complete your analysis and model, prepare a news release for the mayor briefly outlining your proposals that recommend a campaign strategy to curb the violence. Performance Based Assessment In the American League playoffs, Joe Girardi, the Yankees manager made a practice of intentionally walking a man when the first Texas hitter doubled. Was this smart to do? Would it be smart with one out? With two outs? Prepare a presentation explaining your conclusion. What model can you develop to assure that all schools spend an equitable amount on instruction per pupil? Enrolled % low EAV pp Tax Rate % Pr Tx Inst pp Evanston 2895 39% 867 K $2.10 84.3% $11,542 Thornton Fractional 1879 36% 285 K $3.00 60.8% $6,340 Quincy 1362 36% 106 K $4.20 50.6% $5,789 Example from ACT What is the value of x when 2x + 3 = 3x – 4 ? A. – 7 B. - 1/5 C. 1 D. 1/5 E. 7 Example from the U.K. Example from ACT What is the greatest common factor of 42, 126, and 210 ? F. G. H. J. K. 2 6 14 21 42 Example from Finland: Basic A solution of salt and water contains 25 per cent salt. Diluted solutions are obtained by adding water. How much water must be added to one kilogram of the original solution in order to obtain a 10 per cent solution? Work out a graphic representation which gives the amount of water to be added in order to get a solution with 2–25 % of salt. The amount of water (in kilograms) to be added to one kilogram of the original solution must be on the horizontal axis; the salt content of the new solution as a percentage must be on the vertical axis. Example from Finland: Advanced In a society the growth of the standard of living is inversely proportional to the standard of living already gained, i.e. the higher the standard of living is, the less willingness there is to raise it further. Form a differential-equation-based model describing the standard of living and solve it. Does the standard of living rise forever? Is the rate of change increasing or decreasing? Does the standard of living approach some constant level? Example from ACT The length, in inches, of a box is 3 inches less than twice its width, in inches. Which of the following gives the length, l inches, in terms of the width, w inches, of the box? F. l = w + 3 G. l = w + 3 H. l = w – 3 J. l = 2w + 3 K. l = 2w – 3 Example from Australia Your task is to design a space to store enough stackable chairs to seat all the staff and students in your school. You will: 1) 2) 3) Follow a series of steps to help you design a suitable space Use a research journal to record your ideas and rough working Write a report on the process and solutions Example from Australia (continued) Develop mathematical models for each dimension of a stack of chairs To help you think about the practicalities of storing chairs, use your model to find: 1) 2) The greatest number of chairs in one stack that can fit into a storage area with a 4m high ceiling The number of stacks that fit across a 3.2 m wide area if there are 10 chairs in each stack Grade 6 Mathematics Multiple-Choice Sample Assessment Objective 10.6.05: Solve problems involving the probability of a simple event, including representing the probability as a fraction, decimal, or percent. 47 48 49 La Jolla Spa Vacation or a Cool Convertible Mustang 50 Autographed Copy of NCLB 51 Another State Mandate 52 Create a model that illustrates or demonstrates why one of the following four answers is the correct one. What should the contestant do to increase his or her chance of winning the car and spa vacation? A. B. C. D. Stick with his or her first choice Switch, i.e. select the other door Either stick or switch--it doesn’t matter because the odds are 50-50 Go with his or her “gut” feeling 53 Still don’t believe it? Try the “Monty Hall Problem” yourself http://www.stat.sc.edu/~west/javahtml/LetsMakeaDeal.html http://math.ucsd.edu/~crypto/Monty/monty.html Or just Google Let’s Make a Deal or Monty Hall Problem 54 Let's Make a Deal Behind one of these doors is a car. Behind the other two is a goat. Click on the door that you think the car is behind. 55 Advance Organizers Common Core Standards and instructional practices Assessment Assessment Reporting Intel’s White Paper on Assessment The growing move to promote assessment of, for and as learning, rather than seeing testing as a separate disjointed element of the education enterprise, may provide opportunities for strengthening the teaching and learning of 21st century skills, as well as their assessment. The growing emphasis on school-based performance assessments in many countries appears to strengthen teaching where teachers learn more deeply about how to enact standards by participating in scoring and/or reviewing student work. Questions and comments welcome Max McGee 224.234.6129 (cell) [email protected]