4 Overview Challenging tasks Questions The role of technology Examples What makes worthwhile tasksQuestionsWhy questionsExamples of “good” questionsThe role of technology
5 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)
6 The MastA 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?
7 The mastAB12Locate P so AP + PB is a minimum5P24
8 Measuring/arithmetic Locate P so AP + PB is a minimumAB12AP+BP1230111330.229.8141029.616829.435P24
9 Using algebraABLocate P so AP + PB is a minimum125P24
10 Using geometry Locate P so AP + PB is a minimum 125Reflect B to C over the deck linePC24
11 Is P’ the solution? Why or why not? ABLocate P so AP + PB is a minimum12D5Find D, the intersection of the diagonals, and construct perpendicular from D to the deck at P’.PP’24
12 Worthwhile TasksFocused on important mathematics; clear mathematical goal (intent & justification)Provide opportunities for discussionProvoke thinking and reasoning about the mathematics; high level of cognitive demandEngage students in the CCSS mathematical practicesCreate a space in which students “wonder, notice, are curious”SSTP, 2013
13 Triangles Draw a triangle ABC Construct the perpendicular bisector of side ABConstruct the perpendicular bisector of side BCMake a conjecture about the perpendicular bisector of side AC.Move point AWhat do you observe?
14 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?
15 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
16 A rubric for inquiry math tasks Harper & Edwards, 2011
17 Worthwhile tasks involve Multiple representationsMultiple strategies for solutionsMultiple solutionsMultiple entry pointsModels to develop conceptsCritical thinkingOpportunity for reflectionMaking connections among strands, concepts
18 Characteristics of tasks- Urn & the Mast Multiple representations (urn)Multiple strategies for solutions (mast)Multiple solutionsMultiple entry points (urn, mast)Models to develop concepts (urn)Critical thinking (urn, mast)Opportunity for reflectionMaking connections among strands, concepts (mast)
19 Characteristics of tasks- Urn & the Mast Multiple representations (urn)Multiple strategies for solutions (mast)Multiple solutionsMultiple entry points (urn, mast)Models to develop concepts (urn)Critical thinking (urn, mast)Opportunity for reflectionConnections among strands, concepts (mast)
20 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 connectionsnotice something significant.justify or prove their thinking.
21 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
22 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
23 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 problemInterpret information, make and justify conclusion: “The data support… ; “This… will make ….happen because…”Dick & Burrill, 2009
24 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 lessonLessons that enable students to learn are not “accidents” or “good” days; careful and intentional planning goes a long wayPractice the role of teachingTake risks -
27 An Alternate Formula?Step 1: Complete passes divided by pass attempts.Subtract 0.3, then divide by 0.2Step 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 interceptionsdivided 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.
28 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).
29 Procedures as worthwhile tasks JeopardyA 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
30 Procedures as worthwhile tasks Sortquadratic equations (by form, by number of solutions, by common x-intercept, …)trig equations (by form, number of solutions, …)Analyze “student” work for correct solutions
31 Good questions engage students in the mathematical practices Make sense of problems and persevere in solving themReason abstractly and quantitativelyConstruct viable arguments and critique the reasoning of othersModel with mathematicsUse appropriate tools strategicallyAttend to precisionLook for and make use of structureLook for and express regularity in repeated reasoningThe 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
32 Use appropriate tools strategically make sound decisions about using tools, recognizing both the insight to be gained and their limitationsuse technology to visualize the results of varying assumptions, explore consequences, and compare predictions with datause technological tools to explore and deepen understanding of concepts.identify relevant external mathematical resources and use them to pose or solve problemsCCSS 2010
33 The role of technologyAs a tool for doing mathematics - a servant role to perform computations, make graphs, …As a tool for developing or deepening understanding of important mathematical conceptsDick & Burrill, 200933
34 Percentage Correct by Grade Functions/areaItem No.Correct AnswerPercentage Correct by GradeTotal Percent Correct54321AP Calculus AB 2003
35 From characteristics of f ’to f to f ” Calculus AB 2003
36 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
37 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
38 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 + 2Algebra and Precalculus Concept Readiness Alternate Test (APCR alternate) – August 2013
48 Chips & Probability1. 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
49 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 lecturedevelop both conceptual understandings and procedural skillsNational Research Council, 1999; 2001
50 “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
51 “Teaching” Practices Think deeply about simple things. Ross Never say anything a kid can say. ReinhartIf the class ends after the students have explained their work, there is no need for a teacher. TakahashiWhen 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.” KennedyI know what they have learned when I observe them in a place where they have never been. Cuoco
52 ReferencesAlgebra 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. ppBlack, 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.
53 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.htmlCommon 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 correspondenceDick, T., & Burrill, G. (2009). Presentation at Annual Meeting of National Council of Teachers of Mathematics. Washington DCFunctions & 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.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). PpHarvard University.
54 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 atReinhart, 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, ppSummer School Teachers Program. (2013). Reflecting on Practice Course. Park City Mathematics Institute. Institute for Advanced StudyTakahashi, A. (2008). Presentation at Park City Mathematics Institute Secondary School Teachers Program