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Ingredients for Successful Lessons: Challenging Tasks & Questions that Count Gail Burrill Michigan State University

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The urn Calculus Nspired, 2010

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Increasing at a decreasing rate?

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Overview Challenging tasks –Examples –What makes worthwhile tasks Questions –Why questions –Examples of “good” questions The role of technology

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Opportunities for discussion Tasks have to be justified in terms of the learning aims they serve and can work well only if opportunities for pupils to communicate their evolving understanding are built into the planning. (Black & Wiliam, 1998)

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The Mast A sailboat has two masts. One is 5m, the other 12m, and they are 24m apart. They must be secured to the same location using one length of rigging. What is the least amount of rigging that can be used?

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The mast P Locate P so AP + PB is a minimum A B

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Measuring/arithmetic P Locate P so AP + PB is a minimum A B AP+BP

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Using algebra P Locate P so AP + PB is a minimum A B

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Using geometry P Locate P so AP + PB is a minimum A B C Reflect B to C over the deck line

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Is P ’ the solution? Why or why not? P Locate P so AP + PB is a minimum A B Find D, the intersection of the diagonals, and construct perpendicular from D to the deck at P’. D P’P’

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Worthwhile Tasks Focused on important mathematics; clear mathematical goal (intent & justification) Provide opportunities for discussion Provoke thinking and reasoning about the mathematics; high level of cognitive demand Engage students in the CCSS mathematical practices Create a space in which students “wonder, notice, are curious” SSTP, 2013

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Triangles Draw a triangle ABC Construct the perpendicular bisector of side AB Construct the perpendicular bisector of side BC Make a conjecture about the perpendicular bisector of side AC. Move point A What do you observe?

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A small park is enclosed by four streets, two of which are parallel. The park is in the shape of a trapezoid. The perpendicular distance between the parallel streets is the height of the trapezoid. The portions of the parallel streets that border the park are the bases of the trapezoid. The height of the trapezoid is equal to the length of one of the bases and 20 feet longer than the other base. The area of the park is 9,000 square feet. a. Write an equation that can be used to find the height of the trapezoid. b. What is the perpendicular distance between the two parallel streets? The Task?

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The task? 10. Mrs. Dorn operates a farm in Nebraska. To keep her operating costs down, she buys many products in bulk and transfers them to smaller containers for use on the farm. Often the bulk products are not the correct concentration and need to be custom mixed before Mrs. Dorn can use them. One day she wants to apply fertilizer to a large field. A solution of 55% fertilizer is to be mixed with a solution of 44% fertilizer to form 22 liters of a 47% solution. How much of the 55% solution must she use? 6 L11L21L19L

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A rubric for inquiry math tasks Harper & Edwards, 2011

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Worthwhile tasks involve Multiple representations Multiple strategies for solutions Multiple solutions Multiple entry points Models to develop concepts Critical thinking Opportunity for reflection Making connections among strands, concepts

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Characteristics of tasks- Urn & the Mast Multiple representations (urn) Multiple strategies for solutions (mast) Multiple solutions Multiple entry points (urn, mast) Models to develop concepts (urn) Critical thinking (urn, mast) Opportunity for reflection Making connections among strands, concepts (mast)

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Characteristics of tasks- Urn & the Mast Multiple representations (urn) Multiple strategies for solutions (mast) Multiple solutions Multiple entry points (urn, mast) Models to develop concepts (urn) Critical thinking (urn, mast) OPPORTUNITY FOR REFLECTION Connections among strands, concepts (mast)

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To PROBE or uncover students’ thinking. understand how students are thinking about the problem. discover misconceptions. use students’ understanding to guide instruction. To PUSH or advance students’ thinking. make connections notice something significant. justify or prove their thinking. The only reasons to ask questions are: (Black et al., 2004)

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00:04:25 T The thing we're gonna learn about …is exponential growth. 00:04:29 T …we have 2 cubes. This would be like 2 to the 1st power. 00:04:34 T So if we made it 2 squared, which would be 2 times 2, we would see that it grows to 2 squared. That's two times two, right? 00:04:44 T Two cubed is 2, times 2, times 2. 2 to the 3 rd power… 00:04:53 T Then if we go two to the fourth, you're looking.. 00:05:05 T Now two to the fourth is how much? 00:05:08 SN Sixteen. 00:05:14 T Okay. So two to the fifth would be how much? 00:05:17 SN Twenty-five. 00:05:18 SN Twenty-five? 00:05:19 SN No. 00:05:20 SN Twenty. 00:05:21 SN Thirty-two. 00:05:24 T Two to the fourth is 16…. 00:05:26 T And we take that and multiply it by two and we get? NCES TIMSS US Video 1999

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Lesson on linear function T: Zach, what did your group find out? What did you discuss? Z: If you slide the B, it changes the location on the x and y-axis. T: When you slide B? Z: Yeah. And the A, just rotates. It keeps, I think, yeah, the y-axis on the same point. But changes the x-axis. T: What do you mean? Show us what you’re talking about. Z: Here. So this is a. T: And what’s happening?..... … T: Interesting. But how do you know that? I can’t see the y- intercept up there. How do you know it’s rotating around the y- intercept? S: ‘Cause of the sliders. T: What? …. Go back. Go back. How do you KNOW that it’s rotating around the y-intercept without even seeing it? Functions & sliders, 2012

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Inquiry Questions Explain what something means; what is …. Choose and evaluate strategies: What advantages does….have? Compare and contrast: How are they alike? How different? Given an action, predict forward: “What would happen if.. ?” Given a consequence, predict backward: “What do I do if I want... to happen?” “Is it possible to... ?” Require analyzing a connection/relationship: “When will... be (larger, smaller, equal to, exactly twice, etc.) compared to...?” “When will... be as large (small) as possible?” “When will... be as large (small) as possible?” Generalize/make conjectures: “When does... work?” “Describe how to find...?” “Is this always true?” Justify/prove mathematically: “Why does... work?” Change assumptions inherent in the problem Interpret information, make and justify conclusion: “The data support… ; “This… will make ….happen because…” Dick & Burrill, 2009

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Unpredictability and Predetermination Deliberate: Clear intent and justification- about what we do in teaching, not just about what we expect but also about what we do as teachers in organizing and implementing a lesson –Lessons that enable students to learn are not “accidents” or “good” days; careful and intentional planning goes a long way Practice the role of teaching Take risks -

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*Active 2013

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NFL Quarterback Passing Ranking Burrill & Hopfenberger, 1998

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Step 1: Complete passes divided by pass attempts. Subtract 0.3, then divide by 0.2 Step 2: Passing yards divided by pass attempts. Subtract 3, then divide by 4. Step 3: Touchdown passes divided by pass attempts, then divide by.05. Step 4: Start with.095, and subtract interceptions divided by attempts. Divide the product by.04. The sum of each step cannot be greater than or less than zero. Add the sum of Steps 1 through 4, multiply by 100, and divide by 6. An Alternate Formula?

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Responding to questions ‘In composing a useful response, the teacher has to interpret the thinking and the motivation that led the pupil to express the answer. It helps if the teacher first asks the pupil to explain how he or she arrived at that answer, then accepts any explanation without comment and asks others what they think. This gives value to the first answer, and draws the class into a shared exploration of the issue. In doing this the teacher changes role, from being an interviewer of pupils on a one-to-one basis to being a conductor of dialogue in which all may be involved.’( Black, 2009 ).

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Procedures as worthwhile tasks Jeopardy A solution is 3+2i. Concave up for x>2 and x 2 and x<-1 and an asymptote at x=-1. Has an axis of symmetry at x=3 and passes through (2,1) Solution is π/4 + 2nπ

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Procedures as worthwhile tasks Sort quadratic equations (by form, by number of solutions, by common x-intercept, …) trig equations (by form, number of solutions, …) Analyze “student” work for correct solutions

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Good questions engage students in the 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 Use appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning CCSS, 2010

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Use appropriate tools strategically make sound decisions about using tools, recognizing both the insight to be gained and their limitations use technology to visualize the results of varying assumptions, explore consequences, and compare predictions with data use technological tools to explore and deepen understanding of concepts. identify relevant external mathematical resources and use them to pose or solve problems CCSS 2010

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As a tool for doing mathematics - a servant role to perform computations, make graphs, … As a tool for developing or deepening understanding of important mathematical concepts The role of technology Dick & Burrill, 2009

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Functions/area AP Calculus AB 2003 It e m N o. Co rre ct An sw er Percentage Correct by Grade Tota l Per cen t Cor rect 54321

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From characteristics of f ’ to f to f ” Calculus AB 2003

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4.Assume that y = log 2 (8x) for each positive real number x. Which of the following is true? A) If x doubles, then y increases by 3. B) If x doubles, then y increases by 2. C) If x doubles, then y increases by 1. D) If x doubles, then y doubles. E) If x doubles, then y triples. Algebra and Precalculus Concept Readiness Alternate Test (APCR alternate) – August 2013

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2. Suppose =. If B changes from x to 2, how does A change? A. A changes to 2. B. A changes to 2A. C. A changes to (1/2)A. D. A changes to A/(log 2). E. A changes to A + log 2. Algebra and Precalculus Concept Readiness Alternate Test (APCRalternate) – August 2013

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25. Which of the following defines f −1 for f (t ) = ln(t + 2) ? A) f −1 (t) = e t − 2 B) f −1 (t) = e t+2 C) f −1 (t) = e t−2 D) f −1 (t) = e t /2 E) f −1 (t) = e t + 2 Algebra and Precalculus Concept Readiness Alternate Test (APCR alternate) – August 2013

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Life time vs income Gapminder

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Logarithms Logarithmic scaleLinear scale Gapminder

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A special sequence

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Handshakes How many handshakes are possible with 3 people? With 5? Find a general rule for the number of handshakes for n people and verify your rule.

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How many handshakes? People Handshakes People Handshakes

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How many handshakes? n(n-1) n = 1 n = 2 n = 3 n(n+1) 2 H = or 2 H = ?????

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A task Choose two whole numbers a and b (not too large) Compute a 2 +b 2 = a 2 +b 2 = a 2 - b 2 = 2ab =

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a,b to produce a 2 -b 2, 2ab, a 2 +b 2 Geometry Nspired, 2009

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Pythagorean Triples

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Chips & Probability 1. You have a bag with 6 chips in two different colors, red and blue. You draw two chips from the bag without replacement. a. What is the probability the chips are the same color? b. What is the probability you have one of each color? 2. You have a bag with two different colors of chips, red and blue. If you draw two chips from the bag without replacement, how many of each color chip do you need to have in the bag for the probability of getting two chips of the same color to equal the probability of getting two chips, one of each color

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Tasks we give and questions we ask should ensure students are actively involved in choosing and evaluating strategies, considering assumptions, and receiving feedback. are actively involved in choosing and evaluating strategies, considering assumptions, and receiving feedback. encounter contrasting cases- notice new features and identify important ones. encounter contrasting cases- notice new features and identify important ones. struggle with a concept before they are given a lecture struggle with a concept before they are given a lecture develop both conceptual understandings and procedural skills develop both conceptual understandings and procedural skills National Research Council, 1999; 2001

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“ taking mathematics is not enough ” Students should acquire the habit of puzzling over mathematical relationships - why is a formula true; why was a definition made that way? It is the habit of questioning that will lead to understanding of mathematics rather than merely to remember it, and it is this understanding that college courses require. The ability to wrestle with difficult problems is far more important than the knowledge of many formulae or relationships. More important than the knowledge of a specific mathematical topic is the willingness to tackle new problems. Harvard University Harvard University

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“ Teaching ” Practices Think deeply about simple things. Ross Never say anything a kid can say. Reinhart If the class ends after the students have explained their work, there is no need for a teacher. Takahashi When students don’t seem to understand something, my instinct is to consider how I can explain more clearly. A better way is to think “They can figure this out. I just need the right question.” Kennedy I know what they have learned when I observe them in a place where they have never been. Cuoco

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Mathematics Association of America. Algebra and Precalculus Concept Readiness (APCR). (2013). Using Research to Shape Instruction and Placement in Algebra and Precalculus MAA Director of Placement Testing, Bernard Madison. Mathematics Association of America. Black, P. (2009). Looking again at formative assessment. Learning and Teaching Update, 30. Black, P. & Wiliam, D. (1998). “Inside the Black Box: Raising Standards Through Classroom Assessment”. Phi Delta Kappan. Oct. pp Black, P. Harrison, C., Lee, C., Marshall, E., & Wiliam, D. (2004). “Working Inside the Black Box: Assessment for Learning in the Classroom,” Phi Delta Kappan, 86 (1), Burrill, G. & Hopfensperger, P. (1997). Exploring Linear Relations. Palo Alto CA: Dale Seymour Publications. Calculus Nspired. (2010). Math Nspired. Texas Instruments Education Technology. References

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College Board (2003) AP Calculus AB Free-Response Questions. apcentral.collegeboard.com/apc/members/exam/exam_questions/2003.html Common Core Standards. College and Career Standards for Mathematics 2010). Council of Chief State School Officers (CCSSO) and (National Governor’s Association (NGA) Cuoco, A. (2003). Personal correspondence Dick, T., & Burrill, G. (2009). Dick, T., & Burrill, G. (2009). Presentation at Annual Meeting of National Council of Teachers of Mathematics. Washington DC Functions & Sliders. (2012). Functions & Sliders. (2012). Video Clip from T-Cubed Common Core State Standards Professional Development Workshop. Brennan, B., Olson J. & the Janus Group. Curriculum Research & Development Group. University of Hawaii at Manoa, Honolulu HI (2011). Gapminder World. (2009). Math Nspired. Texas Instruments Education Technology. Geometry Nspired. (2009). Math Nspired. Texas Instruments Education Technology. Harper, S., & Edwards, T. (2011). A new recipe: No more cookbook lessons. The Mathematics Teacher. 105(3). Pp Harvard University. cdn=education&tm=54&f=10&su=p ip_&tt=2&bt=1&bts=1&z u=http%3A//www.admissions.college.harvard.edu/apply/preparing/index.ht ml%23math cdn=education&tm=54&f=10&su=p ip_&tt=2&bt=1&bts=1&z u=http%3A//www.admissions.college.harvard.edu/apply/preparing/index.ht ml%23math cdn=education&tm=54&f=10&su=p ip_&tt=2&bt=1&bts=1&z u=http%3A//www.admissions.college.harvard.edu/apply/preparing/index.ht ml%23math

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Kennedy, D. (2002). Talk at National Council of Teachers of Mathematics Annual Meeting. Boston MA. National Center for Education Statistics (NCES). (2003).Third International Mathematics and Science Study (TIMSS), Video Study. U.S. Department of Education. National Research Council. (1999). How People Learn: Brain, mind, experience, and school. Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). Washington, DC: National Academy Press. National Research Council (2001). Adding It Up. Kilpatrick, J., Swafford, J., & Findell, B. (Eds.) Washington DC: National Academy Press. Also available on the web at Reinhart, Steven C., (2000). Never say anything a kid can say! Mathematics Teaching in the Middle School. Apr. 2000, 478–83. Ross, A. In interview with Jackson, A. (2001). Interview with Arnold Ross, Notices of the American Mathematical Society, pp Summer School Teachers Program. (2013). Reflecting on Practice Course. Park City Mathematics Institute. Institute for Advanced Study Takahashi, A. (2008). Presentation at Park City Mathematics Institute Secondary School Teachers Program

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