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

The urn Calculus Nspired, 2010

Increasing at a decreasing rate?

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

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)

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?

The mast A B 12 Locate P so AP + PB is a minimum 5 P 24

Measuring/arithmetic Locate P so AP + PB is a minimum A B 12 AP+BP 12 30 11 13 30.2 29.8 14 10 29.6 16 8 29.43 5 P 24

Using algebra A B Locate P so AP + PB is a minimum 12 5 P 24

Using geometry Locate P so AP + PB is a minimum 12 5 Reflect B to C over the deck line P C 24

Is P’ the solution? Why or why not? A B Locate P so AP + PB is a minimum 12 D 5 Find D, the intersection of the diagonals, and construct perpendicular from D to the deck at P’. P P’ 24

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

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?

The Task? 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? www.iroquoiscsd.org/cms/lib/NY19000365/Centricity/Domain/105/CN__097_LESSON_13.1.PDF

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 L 11L 21L 19L http://answers.yahoo.com/question/index?qid=20100224163459AASjkx7

A rubric for inquiry math tasks Harper & Edwards, 2011

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

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)

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)

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.

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

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? The slides on questions and teacher moves should take about 10 minutes. Here is a transcript of part of the video we just saw. Note the questions Brandon asks. How often do we ask questions like this in either our pd work or our classrooms? Just a minute or two for any thoughts or can go right to the next slide. Note Brandon is “relentless” in asking how do you know – connects to the teacher moves which will be raised in the slide after the next (slide 13). Functions & sliders , 2012

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?” 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

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 -

*Active 2013

NFL Quarterback Passing Ranking Burrill & Hopfenberger, 1998

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.

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).

Procedures as worthwhile tasks Jeopardy A solution is 3+2i. Concave up for 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

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

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 The mathematical practices should be woven through every lesson as appropriate - these are ways of thinking about and doing mathematics that should permeate all of the work students do- and should not be taught in an isolated - today we will model with mathematics. The examples that follow suggest how these practices might be incorporated using some Nspired activities. CCSS, 2010

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

The role of technology 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 Dick & Burrill, 2009 33

Percentage Correct by Grade Functions/area Item No. Correct Answer Percentage Correct by Grade Total Percent Correct 5 4 3 2 1 AP Calculus AB 2003

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

4. Assume that y = log2 (8x) for each positive real number x 4.Assume that y = log2 (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

2. Suppose 𝐴 = 𝑙𝑜𝑔𝐵. If B changes from x to 𝑥2, how does A change. A 2. Suppose 𝐴 = 𝑙𝑜𝑔𝐵. If B changes from x to 𝑥2, how does A change? A. A changes to 𝐴2. 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

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

Life time vs income Gapminder

Logarithms Logarithmic scale Linear scale Gapminder

A special sequence

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.

How many handshakes? People Handshakes 1 0 2 1 3 3 4 6

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

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

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

Pythagorean Triples 3 4 5 12 13 7 24 25 9 40 41 11 60 61 84 85 15 112 113 17 144 145 19 180 181 21 220 221

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

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

“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

“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

References 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. 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. Burrill, G. & Hopfensperger, P. (1997). Exploring Linear Relations. Palo Alto CA: Dale Seymour Publications. Calculus Nspired. (2010). Math Nspired. Texas Instruments Education Technology. www.ti-mathnspired.com/login/?next=/

College Board (2003) AP Calculus AB Free-Response Questions. apcentral 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). Presentation at Annual Meeting of National Council of Teachers of Mathematics. Washington DC 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. http://www.gapminder.org/world/ Geometry Nspired. (2009). Math Nspired. Texas Instruments Education Technology. www.ti-mathnspired.com/ 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. 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. Ross, A. In interview with Jackson, A. (2001). Interview with Arnold Ross, Notices of the American Mathematical Society, pp. 691-698. 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