Shifting Our Mindsets and Our Actions from Remembering HOW to Understanding WHY Steve Leinwand SUM

And what message do far too many of our students get? ( even those in Namibia !) 2

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Ready?? What is 8 + 9? 17 Bing Bang Done! Vs. Convince me that = 17. Hmmmm…. 4

8 + 9 = 17 – know it cold – add 1 to 9, subtract 1 from – decompose the 8 into 7 and 1 18 – 1 – add 10 and adjust – double plus 1 20 – 3 – round up and adjust Who’s right? Does it matter? 5

= How did you do it? Who did it differently? 6

So…the problem is: If we continue to do what we’ve always done…. We’ll continue to get what we’ve always gotten. 7

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Where is the opportunity to learn? Where is the sense-making? Does anyone benefit from a sheet like this? 10

How did you do it? or Convince me that 95-48=47. 11

In other words, our questions make all the difference. (no pun intended) 12

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 13

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! 14

Adding and Subtracting Integers 15

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” 16

Understand Why 5 + (-9) -Have $5, lost $9 -Gained 5 yards, lost 9 -5 degrees above zero, gets 9 degrees colder -Decompose 5 + ( ) -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 17

Let’s laugh at the absurdity of “the standard algorithm” and the one right way to multiply 58 x 47 18

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How nice if we wish to continue using math to sort our students! 20

So what’s the alternative? 21

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? 22

3 x 4 Convince me that 3 x 4 is Three threes are nine and three more for the fourth

3 x 40 3 x 4 x 10 (properties) with a 0 appended

3 x 47 3 (40 + 7) = 3 40s + 3 7s or

58 x x

Why bother? 27

28 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?

29 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

30 Or using a model to support UndWhy: kg Rahim Siti 29 kg

Multiplying Decimals 31

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.” 32

But why? How can this make sense? How about a context? 33

Understand Why So? What do you see? 34

Understand Why gallons Total Where are we? 35

Understand Why 4.2 gallons Total How many gallons? About how many? $ 36

Understand Why 4.2 gallons $ 4.39 Total About how much? Maximum?? Minimum?? 37

Understand Why 4.2 gallons $ 4.39 Total Context makes ridiculous obvious, and breeds sense-making. Actual cost? So how do we multiply decimals sensibly? 38

Solving Simple Linear Equations 39

3x + 7 = 22 How do we solve equations: Subtract 7 3 x + 7 = x = 15 Divide by Voila: x = 5 40

3x Tell me what you see: 3 x Suppose x = 0, 1, 2, 3….. 3.Let’s record that: x 3x How do we get 22? 41

3x + 7 = 22 Where did we start? What did we do? x 5 x 3 3x 15 ÷ x

3x + 7 = 22 X X X IIIIIII IIII IIII IIII IIII II X X X IIIII IIIII IIIII 43

Let’s look at a silly problem Sandra is interested in buying party favors for the friends she is inviting to her birthday party. 44

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. 45

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. 46

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. 47

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? 48

So? Your turn. How much? How did you get your answer? 49

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Putting it all together one way Good morning class. Today’s objective: Find the surface area of right circular cylinders. Open to page Example 1: 4 S.A.= 2πrh + 2 πr 2 Find the surface area. Homework: Page odd 58

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. 59

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. 60

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. 61

The CCSSM Trojan Horse: SMP 3: Construct viable arguments and critique the reasoning of others 62

` Questions? Comments? Thank You! 63