Ingredients for Successful Lessons: Challenging Tasks & Questions that Matter

1. Ingredients for Successful Lessons: Challenging Tasks & Questions that Matter Gail Burrill Michigan State University burrill@msu.edu

2. Triangles, perimeter & area Dick & Hollebrands, 2011

3. 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)

4. Looking at an Angle Looking at an Angle. Feijs et al, 2006.

5. Gliders • Two different gliders start from the same height. Which glider goes farther: one with a glide ratio of 25/185 or one with a glide ratio of 20/155? • Explain your reasoning.

6. Characteristics of tasks • Multiple representations • Multiple strategies for solutions • Multiple solutions • Multiple entry points • Models to develop concepts • Critical thinking • Opportunity for reflection • Connections among strands, concepts Different contexts for same concept Different concepts from same context

7. Characteristics of tasks- Glide ratios, area & perimeter • Multiple representations • Multiple strategies for solutions • Multiple solutions • Multiple entry points • Models to develop concepts • Critical thinking • Opportunity for reflection • Connections among strands, concepts Different contexts for same concept Different concepts from same context

8. More 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?

9. Characteristics of tasks- which were present in last task as given? • Multiple representations • Multiple strategies for solutions • Multiple solutions • Multiple entry points • Models to develop concepts • Cognitive demand - require critical thinking • Connections among strands, concepts Different contexts for same concept Different concepts from same context

10. A rubric for inquiry math tasks Harper & Edwards, 2011

11. Equivalent Expressions • Which expression is equivalent to 3(8x-2y+7)? • 24x-2y+7 • 24x-6y+21 • 8x-6y+21 • 11x-5y+10 Albert wants to simplify the expression: 8(3–y) + 5(3–y). Which of the following is equivalent to the expression above? A. 39 – y B. 13(3 – y) C. 40(3 – y) D. 13(6 – 2y)

12. Equivalent Expressions Which expression is equivalent to 3(8x-2y+7)? 23%* A. 24x-2y+7 33% B. 24x-6y+21 26% C. 8x-6y+21 18% D. 11x-5y+10 Albert wants to simplify the expression: 8(3–y) + 5(3–y) Which of the following is equivalent to the expression above? 29% A. 39 – y 40%* B. 13(3 – y) 7% C. 40(3 – y) 24% D. 13(6 – 2y) (Michigan 2007, Grade 8) (Florida 2006, grade 9)

13. Formative Assessment

14. Now what? 13(3-y) 39-13y 13(3-x) 39-y (24-8y)+(15-5y) 13(3+y) 24+15 39(y) 39-3y 5(3-y)+8(3-y) 7(3-y)+5(3-y) 8(3+y)+3(5+y)

15. The only reasons to ask questions are:(Black et al., 2004) • 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.

16. 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 3rd 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

17. Making It Happen - Facilitating Discourse • What would you predict would happen, you are not actually going to do it, what would you predict would happen? • Group 3, you guys think… think that, you guys said that when you slant the sides the area stays the same and so do the sides but…the angles change. Why do you think that is true. • Can you show us up there? • Try that, I don't know? • So, what does that mean? • How did you measure the area? • So, so what she is saying is that…what's the formula again? • ..so you're saying the length times the width….lets soak that in for a second. Bringing it all together video clip, 2012

18. Reasoning/Sense Making Questions • Compare and contrast • Predict forward • Predict backward • Analyze a connection/relationship • Generalize/make conjectures • Justify/prove mathematically • Consider assumptions inherent in the problem and what would happen if they were changed • Interpret information, make/ justify conclusions Burrill & Dick, 2009

19. NFL Passing Leaders *Active 2012

20. NFL Quarterback Passing Ranking Burrill et al, 1998

21. An Alternate Formula? 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 2.375 or less than zero. Add the sum of Steps 1 through 4, multiply by 100, and divide by 6.

22. What we do with tasks • Setting up Adaptation/modification • Implementation Respond to student questions Prompts Monitor student work • Discussion Choose solutions to share Sequence solutions to meet mathematical goal Manage solution strategies Ask questions Consolidate the math using student work (Stein & Smith, 2011)

23. A special sequence

24. 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.

25. How many handshakes? • People handshakes 1 0 2 1 3 3 4 6

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

27. A task Choose two whole numbers a and b (not too large) Compute a2 +b2 = a2 - b2 = 2ab =

28. a,b to produce a2-b2, 2ab, a2+b2 Geometry Nspired, 2009

29. Pythagorean Triples

30. 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? • 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

31. Mathematical Practices: How students should work • 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

32. 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 anduse them to pose or solve problems CCSS 2010

33. 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

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

35. “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

36. “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

37. References • Black, P. & Wiliam, D. (1998). “Inside the Black Box: Raising Standards Through Classroom Assessment”. Phi Delta Kappan. Oct. pp. 139-148. • 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), 9-21. • Bringing it all together. (2012). Video clip shown at the 2012 annual National Council of Supervisors of Mathematics annual meeting, Philadelphia PA. film by Brennan, B., Olson J. & the Janus Group. Curriculum Research & Development Group. University of Hawaii at Manoa, Honolulu HI (2009). • Burrill, G. & Dick, T. (2009). Presentation at Annual Meeting of National Council of Teachers of Mathematics. Washington DC • Burrill, G. & Hopfensperger, P. (1997). Exploring Linear Relations. Palo Alto CA: Dale Seymour Publications. • 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

38. Dick, T., & Hollebrands, K. (2011). Focus in high school mathematics: Technology to support reasoning and sense. Reston VA: National Council of Teachers of Mathematics • Florida Department of Education (2006). FCAT Mathematics Released Items, Grade 9. • Harper, S., & Edwards, T. (2011). A new recipe: No more cookbook lessons. The Mathematics Teacher. 105(3). Pp 180-188. • Harvard University. http://collegeapps.about.com/gi/o.htm?zi=1/XJ&zTi=1&sdn=collegeapps&cdn=education&tm=54&f=10&su=p897.11.336.ip_&tt=2&bt=1&bts=1&zu=http%3A//www.admissions.college.harvard.edu/apply/preparing/index.html%23math • Kennedy, D. (2002). Talk at National Council of Teachers of Mathematics Annual Meeting. Boston MA. • Looking at an angle. (2003). Mathematics in Context Project. Directed by Tom Romberg & Jan deLange.ChicagoIL: Encyclopedia Britannica. • Michigan Department of Education. (2007). Released item mathematics grades 8 fall. www.michigan.gov/mde/0,1607,7-140-22709_31168_31355-95470--,00.html

39. National Center for Education Statistics (NCES). (2003).Third International Mathematics and Science Study (TIMSS), Video Study. U.S. Department of Education. http://nces.ed.gov/timss/video.asp • 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 www.nap.edu. • Reinhart, Steven C., (2000). Never say anything a kid can say! Mathematics Teaching in the Middle School. Apr. 2000, 478–83. • Smith, M. S., & Stein, M.K., (2011). The five practices for organizing productive mathematical discussions. Reston, VA: National Council of Teachers of Mathematics. • Ross, A. In interview with Jackson, A. (2001). Interview with Arnold Ross, Notices of the American Mathematical Society, pp. 691-698. • Takahashi, A. (2008). Presentation at Park City Mathematics Institute Secondary School Teachers Program