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Shifting Our Mindsets and Our Actions from Remembering HOW to Understanding WHY Houston NCTM 11/20/14 Steve Leinwand American Institutes for Research [email protected] www.steveleinwand.com 1 Decisions Decisions LUNCH 2 Decisions Decisions STEVE 3 But first: A pre-session angry overture 4 The problem is universal! 5 6 Ready? 7 Get set. Go. What is 8 + 9? 17 Bing Bang Done! Vs. Convince me that 9 + 8 = 17. Hmmmm…. 8 8+9= 17 – know it cold 10 + 7 – add 1 to 9, subtract 1 from 8 7 + 1 + 9 – decompose the 8 into 7 and 1 18 – 1 – add 10 and adjust 16 + 1 – double plus 1 20 – 3 – round up and adjust Who’s right? Does it matter? 9 4 + 29 = How did you do it? How did you do it? Who did it differently? 10 So…the problem is: If we continue to do what we’ve always done…. We’ll continue to get what we’ve always gotten. 11 12 13 Where is the opportunity to learn? Where is the sense-making? Does anyone benefit from a sheet like this? 14 95 - 48 How did you do it? or Convince me that 95-48=47. 15 In other words, our questions make all the difference. (no pun intended) 16 Mathematics • A set of rules to be learned and memorized to find answers to exercises that have limited real world value OR • A set of competencies and understanding driven by sense-making and used to get solutions to problems that have real world value 17 And Alt apps and mult reps emerge from this why/convince me • Effective teachers of mathematics elicit, value, and celebrate alternative approaches to solving mathematics problems so that students are taught that mathematics is a sense-making process for understanding why and not memorizing the right procedure to get the one right answer. • Effective teachers of mathematics provide multiple representations – for example, models, diagrams, number lines, tables and graphs, as well as symbols – of all mathematical work to support the visualization of skills and concepts. Also know as rational, doable DIFFERENTIATION! 18 Adding and Subtracting Integers 19 Remember How 5 + (-9) “To find the difference of two integers, subtract the absolute value of the two integers and then assign the sign of the integer with the greatest absolute value” 20 Understand Why 5 + (-9) - Have $5, lost $9 - Gained 5 yards, lost 9 - 5 degrees above zero, gets 9 degrees colder - Decompose 5 + (-5 + -4) - Zero pairs: x x x x x O O O O O O O O O - On number line, start at 5 and move 9 to the left 21 Let’s laugh at the absurdity of “the standard algorithm” and the one right way to multiply 58 x 47 22 3 5 58 x 47 406 232_ 2726 23 How nice if we wish to continue using math to sort our students! 24 So what’s the alternative? 25 Multiplication • • • • • • What is 3 x 4? How do you know? What is 3 x 40? How do you know? What is 3 x 47? How do you know? What is 13 x 40? How do you know? What is 13 x 47? How do you know? What is 58 x 47? How do you know? 26 3x4 Convince me that 3 x 4 is 12. • 4+4+4 • 3+3+3+3 • Three threes are nine and three more for the fourth • 3 12 4 27 3 x 40 • 3 x 4 x 10 (properties) • 40 + 40 + 40 • 12 with a 0 appended • 3 120 40 28 3 x 47 • 3 (40 + 7) = 3 40s + 3 7s • 47 + 47 + 47 or 120 + 21 • 3 120 21 40 7 29 58 x 47 40 50 8 7 58 x 47 56 350 320 2000 2726 30 Why bother? 31 Just do it: Siti packs her clothes into a suitcase and it weighs 29 kg. Rahim packs his clothes into an identical suitcase and it weighs 11 kg. Siti’s clothes are three times as heavy as Rahims. What is the mass of Rahim’s clothes? What is the mass of the suitcase? 32 The old (only) way or RemHow: Let S = the weight of Siti’s clothes Let R = the weight of Rahim’s clothes Let X = the weight of the suitcase S = 3R S + X = 29 R + X = 11 so by substitution: 3R + X = 29 and by subtraction: 2R = 18 so R = 9 and X = 2 33 Or using a model to support UndWhy: www.thesingaporemaths.com 11 kg Rahim Siti 29 kg 34 Wow – Look at HOW vs WHY? 7.5 ÷ 0.5 7.5 ÷ 0.25 Multiplying Decimals 36 Remember How 4.39 x 4.2 “We don’t line them up here.” “We count decimals.” “Remember, I told you that you’re not allowed to that that – like girls can’t go into boys bathrooms.” “Let me say it again: The rule is count the decimal places.” 37 But why? How can this make sense? How about a context? 38 Understand Why So? What do you see? 39 Understand Why gallons Total Where are we? 40 Understand Why 4.2 gallons $ Total How many gallons? About how many? 41 Understand Why 4.2 gallons $ 4.39 Total About how much? Maximum?? Minimum?? 42 Understand Why 4.2 gallons $ 4.39 184.38 Total Context makes ridiculous obvious, and breeds sense-making. Actual cost? So how do we multiply decimals sensibly? 43 Solving Simple Linear Equations 44 3x + 7 = 22 How do we solve equations: Subtract 7 Divide by 3 Voila: 3 x + 7 = 22 -7 -7 3 x = 15 3 3 x = 5 45 3x + 7 1. Tell me what you see: 3 x + 7 2. Suppose x = 0, 1, 2, 3….. 3. Let’s record that: x 3x + 7 0 7 1 10 2 13 4. How do we get 22? 46 3x + 7 = 22 Where did we start? What did we do? x 5 x3 3x 15 ÷3 +7 3x + 7 22 -7 47 3x + 7 = 22 X X X IIIIIII IIII IIII IIII IIII II XXX IIIII IIIII IIIII 48 Let’s look at a silly problem Sandra is interested in buying party favors for the friends she is inviting to her birthday party. 49 Let’s look at a silly problem Sandra is interested in buying party favors for the friends she is inviting to her birthday party. The price of the fancy straws she wants is 12 cents for 20 straws. 50 Let’s look at a silly problem Sandra is interested in buying party favors for the friends she is inviting to her birthday party. The price of the fancy straws she wants is 12 cents for 20 straws. The storekeeper is willing to split a bundle of straws for her. 51 Let’s look at a silly problem Sandra is interested in buying party favors for the friends she is inviting to her birthday party. The price of the fancy straws she wants is 12 cents for 20 straws. The storekeeper is willing to split a bundle of straws for her. She wants 35 straws. 52 Let’s look at a silly problem Sandra is interested in buying party favors for the friends she is inviting to her birthday party. The price of the fancy straws she wants is 12 cents for 20 straws. The storekeeper is willing to split a bundle of straws for her. She wants 35 straws. How much will they cost? 53 So? Your turn. How much? How did you get your answer? 54 55 56 57 58 59 60 61 62 Putting it all together one way Good morning class. Today’s objective: Find the surface area of right circular cylinders. Open to page 384-5. 3 Example 1: 4 S.A.= 2πrh + 2 πr2 Find the surface area. Homework: Page 385 1-19 odd 63 Putting it all together another way Overheard in the ER as the sirens blare: “Oh my, look at this next one. He’s completely burned from head to toe.” “Not a problem, just order up 1000 square inches of skin from the graft bank.” You have two possible responses: - Oh good – that will be enough. OR - Oh god – we’re in trouble. 64 • Which response, “oh good” or “oh my” is more appropriate? • Explain your thinking. • Assuming you are the patient, how much skin would you hope they ordered up? • Show how you arrived at your answer and be prepared to defend it to the class. 65 • Exit slip: Sketch an object and it’s dimensions that has a surface area of about 100 square inches? • Homework: How many square cm of skin do you have and be prepared to show how you arrived at your answer. 66 The CCSSM Trojan Horse: SMP 3: Construct viable arguments and critique the reasoning of others 67 In Conclusion People won’t do what they can’t envision, People can’t do what they don’t understand, People can’t do well what isn’t practiced, But practice without feedback results in little change, and Work without collaboration is not sustaining. Ergo: Our job, as professionals, at its core, is to help people envision, understand, practice, receive feedback and collaborate. 68 Thank You. Go forth and take on the world! 69