7/15/16 Sustaining High CogniCve Demand in Intermediate Algebra Jason Slowbe Great Oak High School San Diego, CA @TheSlowbe jason.slowbe@gmail.com #NCTMinst • Slides and handout will be on Ins1tute website: h3p://nctm.org/hs16 • Everything will be on my own website: h3p://theslowbe.weebly.com • Please tweet throughout today’s session @theslowbe #NCTMinst 2 1
7/15/16 Goals for this workshop • Avoiding focus on “answer-geFng” • What is Cogni1ve Demand? • Modifying exis1ng problems and tasks, not “reinven1ng the wheel” • Work 1me crea1ng a shared resource to take back to our classrooms and colleagues 4 @theslowbe #NCTMinst Toward BeJer PD… Ar1cle on my website, h3p://theslowbe.weebly.com Johnson, Karen. “5 Things Teachers Want from PD, and How Coaching and Collabora1on Can Deliver Them – If Implementa1on Improves”. EdSurge.com, June 28, 2016 h3ps://www.edsurge.com/news/2016-06-28-5-things-teachers-want-from- pd-and-how-coaching-and-collabora1on-can-deliver-them-if-implementa1on- improves 5 @theslowbe #NCTMinst 6 @theslowbe #NCTMinst 2
7/15/16 7 @theslowbe #NCTMinst 8 @theslowbe #NCTMinst Our Google Doc… • Go to h3p://theslowbe.weebly.com, go to this workshop’s page, open the Google Doc • Populate the document with some “ordinary” Algebra 2 problems from each unit 9 @theslowbe #NCTMinst 3
7/15/16 If there was one “big” thing you could change about Algebra 2, what would it be? Phil Daro on “answer-geSng” • h3ps://www.youtube.com/watch? v=DgTnmRyV9bc • Call to ac1on: Delay answer-ge2ng 11 @theslowbe #NCTMinst Swooping in • h3p://www.1mssvideo.com/97 12 @theslowbe #NCTMinst 4
7/15/16 Goals for this workshop • Avoiding focus on “answer-geFng” • What is Cogni1ve Demand? • Modifying exis1ng problems and tasks, not “reinven1ng the wheel” • Work 1me crea1ng shared resource to take back to our classrooms 13 @theslowbe #NCTMinst What is CogniCve Demand? Peg Smith, Juli Dixon, others: • Task selec1on and implementa1on are important… • …but what about Algebra 2 content specifically? 14 @theslowbe #NCTMinst What is CogniCve Demand? This is a bridge of length three. Determine the number of beams in a bridge of any length. Dixon Nolan Adams Mathema1cs 2016 15 @theslowbe #NCTMinst 5
7/15/16 High CogniCve Demand Task This is a bridge of length three. Describe how the following generaliza1ons were visualized: ( ) ( ) ( ) 3 + 4 n − 1 n + 2 n + n − 1 3 n + n − 1 Dixon Nolan Adams Mathema1cs 2016 16 @theslowbe #NCTMinst Levels of CogniCve Demand Lower-Level • Memoriza1on Demands • Procedures without connec1ons Higher-Level • Procedures with connec1ons Demands • Doing mathema1cs Smith & Stein, “Selec1ng and Crea1ng Mathema1cal Tasks: From Research to Prac1ce”, 2012 17 @theslowbe #NCTMinst What “High CogniCve Demand” Looks Like • Messy! • Students “grapple with complexity” • Students respond directly to other students • Students do the sense-making • Leverage Standards for Mathema9cal Prac9ce 18 @theslowbe #NCTMinst 6
7/15/16 Rethinking “I-We-You” • Not “I-we-you” = gradual release • Instead “You-we-I” = produc1ve struggle 19 @theslowbe #NCTMinst What “High CogniCve Demand” Looks Like – Teacher moves • Facilita1ng discourse among students – “What did she say?” – “Is that what she said?” – “You seem to disagree. Why?” – Summarize different responses, then say “It sounds like you have more to talk about” and walk away – S: “I don’t get it” T: “Then you have a ques1on to ask (other student)” – Teacher rarely gives authorita1ve answer to ques1on 20 @theslowbe #NCTMinst What “High CogniCve Demand” Looks Like – Teacher moves • Ques1on students’ correct answers! Give no indica1on of correctness • “I saw a student do ______. What do you think?” • Have [student 2] use [student 1]’s approach to solve the problem • Wait 1me 21 @theslowbe #NCTMinst 7
7/15/16 CogniCve Demand in Algebra 2 • Promp1ng students to understand each others’ thinking keeps students engaged longer and at a higher cogni1ve level • Leveraging the Mathema9cs Teaching Prac9ces to focus on processes, not just answers • Give students more credit: give them the opportunity then let them pursue it 22 @theslowbe #NCTMinst Goals for this workshop • Avoiding focus on “answer-geFng” • What is Cogni1ve Demand? • Modifying exis1ng problems and tasks, not “reinven1ng the wheel” • Work 1me crea1ng shared resource to take back to our classrooms 24 @theslowbe #NCTMinst 8
7/15/16 Pose Purposeful QuesCons From NCTM’s Principles to Ac9ons (pp35-36): “Purposeful ques9ons allow teachers to discern what students know and adapt lessons to meet varied levels of understanding, help students make important mathema9cal connec9ons, and support students in posing their own ques9ons.” 25 @theslowbe #NCTMinst Modifying Problems to Increase CogniCve Demand Generalize Apply a Context Reversibility Multiple Representations Uniqueness Open-Middle Flexibility Defend/Dispute a Claim 26 @theslowbe #NCTMinst Generalize Then : Write an equa1on for an ellipse that contains the point (6,0). Now : Write 2 more equa1ons of ellipses that all contain the point (6,0). Write a general rule for any ellipse containing (6,0) Write a general rule for any ellipse containing ( a ,0) 27 @theslowbe #NCTMinst 9
7/15/16 Generalize Then : Determine the end behavior of y = 2 x – 3 Now : Write a rule to determine the end behavior of any exponen1al func1on: y = a b (x – c ) + d 28 @theslowbe #NCTMinst Reversibility Then : Find the first four iterates of f ( n ) = n 2 + 2 where n 0 = -1. Now : Given -1, 3, 11, 123, … Explain the pa3ern and determine a (recursive) formula 29 @theslowbe #NCTMinst Reversibility Then : Solve this system of equa1ons: x + 2 y + z = 4 y − z = − 1 − 2 x + z = − 5 Now : Create a system of equa1ons whose solu1on is (3,0,1) 30 @theslowbe #NCTMinst 10
7/15/16 Reversibility Then : Find csc(A): E 10 8 Y A 6 Now : 5/4 is the cosecant of which angle? 31 @theslowbe #NCTMinst Uniqueness “Is there more than one solu1on?” “How do you know?” Notes: • Ask this ques9on regularly, even when there is only 1 unique solu9on • Use interes9ng ques9ons regularly that have non-unique solu9ons 32 @theslowbe #NCTMinst Flexibility NCTM on “procedural fluency”: • Apply procedures accurately, efficiently, and flexibly • Transfer procedures to different problems and contexts • Build or modify procedures from other procedures • Recognize when one strategy/procedure is more appropriate to apply than another 33 @theslowbe #NCTMinst 11
7/15/16 Flexibility Then : Factor completely: y = x 3 – 5 x 2 + 3 x – 15 Now : Factor completely: y = ( x 3 – 6 x 2 + 5 x ) + ( x 2 – 2 x – 15) Now factor again using a different approach. Which do you prefer? Write an expression for which you prefer the other approach. 34 @theslowbe #NCTMinst Flexibility Then : Write an equa1on for: Now : Write two equa1ons using two different func1ons. Which do you prefer? Why? Sketch a graph that would make you prefer the other func1on 35 @theslowbe #NCTMinst Apply a Context • Recall: • Create your own context for: B = 4n + (n-2) 36 @theslowbe #NCTMinst 12
7/15/16 Apply a Context How is “root multiplicity” represented in this graph? What do they tell us about the game? http://www.nba.com/gametracker/#/20160508/CLEATL/lp/analysis 37 @theslowbe #NCTMinst MulCple RepresentaCons www.nctm.org 38 @theslowbe #NCTMinst MulCple RepresentaCons Then : The height of a hot air balloon can be modeled by h ( t ) = –0.08 t 4 + 4.7 t 3 – 84.3 t 2 + 539 t . Find all local extrema. Now : Create a be3er representa1on for the company to display in their adver1sement that gives more clear details about the flight. 39 @theslowbe #NCTMinst 13
7/15/16 Open-Middle 40 @theslowbe #NCTMinst Open-Middle 41 @theslowbe #NCTMinst Open-Middle 42 14
7/15/16 Defend/Dispute a Claim y = a sin( b ( x – c )) + d Esme: “There is only one value for b that makes this func1on have period 3π.” Do you agree or disagree with Esme? Write a paragraph to explain your reasoning. 43 @theslowbe #NCTMinst Defend/Dispute a Claim Esme: “To find the complex roots of a quadra1c, I can reflect the parabola ver1cally over its vertex then use the real x -intercepts.” Do you agree or disagree with Esme? Support your answer mathema1cally. 44 @theslowbe #NCTMinst Defend/Dispute a Claim Esme: “To find the complex roots of a quadra1c, I can reflect the parabola ver1cally over its vertex then use the real x -intercepts.” 45 @theslowbe #NCTMinst 15
7/15/16 Defend/Dispute a Claim Esme: “Every polynomial has a unique factoriza1on.” Do you agree or disagree with Esme? Support your answer mathema1cally. 46 @theslowbe #NCTMinst Phillips Exeter Academy h3p://www.businessinsider.com/what-its-like-to-a3end-phillips-exeter-academy-2014-11?op=1 47 @theslowbe #NCTMinst 48 @theslowbe #NCTMinst 16
7/15/16 Modifying Problems to Increase CogniCve Demand Generalize Apply a Context Reversibility Multiple Representations Uniqueness Open-Middle Flexibility Defend/Dispute a Claim 49 @theslowbe #NCTMinst ModificaCon and ReflecCon Designate 1 person from your group to share: 1) one of your favorite modifica1ons 2) one reflec1on on the strategies 50 @theslowbe #NCTMinst 17
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