1

Shifting Our Mindsets and Our Actions from Remembering

HOW to Understanding WHY

Houston NCTM 11/20/14Steve Leinwand

American Institutes for Researchsleinwand@air.org www.steveleinwand.com

2

Decisions Decisions

LUNCH

3

Decisions Decisions

STEVE

4

But first:

A pre-session angry overture

5

The problem is universal!

6

7

Ready?

8

Get set. Go.

What is 8 + 9?17 Bing Bang Done!

Vs.Convince me that 9 + 8 = 17.

Hmmmm….

9

8 + 9 =17 – know it cold 10 + 7 – add 1 to 9, subtract 1 from 87 + 1 + 9 – decompose the 8 into 7 and 118 – 1 – add 10 and adjust16 + 1 – double plus 120 – 3 – round up and adjust

Who’s right? Does it matter?

10

4 + 29 =How did you do it?How did you do it?

Who did it differently?

11

So…the problem is:

If we continue to do what we’ve always done….

We’ll continue to get what we’ve always gotten.

12

13

Where is the opportunity to learn?

Where is the sense-making?Does anyone benefit from a

sheet like this?14

15

95 - 48 How did you do it? orConvince me that 95-48=47.

16

In other words, our questions make all the

difference.(no pun intended)

17

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

18

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!

19

Adding and Subtracting Integers

20

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”

21

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

22

Let’s laugh at the absurdity of “the standard algorithm” and the one

right way to multiply

58x 47

23

3 5

58x 47 406 232_2726

24

How nice if we wish to continue using math to sort

our students!

25

So what’s the alternative?

26

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?

27

3 x 4Convince 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 4

12

28

3 x 40

• 3 x 4 x 10 (properties)• 40 + 40 + 40• 12 with a 0 appended

• 3

40

120

29

3 x 47

• 3 (40 + 7) = 3 40s + 3 7s• 47 + 47 + 47 or 120 + 21

• 3

40 7

120 21

30

58 x 47

40 7

50

8

58 x 47

56 350 320 2000

2726

31

Why bother?

32

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?

33

The old (only) way or RemHow:

Let S = the weight of Siti’s clothesLet R = the weight of Rahim’s clothesLet 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

34

Or using a model to support UndWhy:www.thesingaporemaths.com

11 kg

Rahim

Siti

29 kg

Wow – Look at HOW vs WHY?

7.5 ÷ 0.5

7.5 ÷ 0.25

36

Multiplying Decimals

37

Remember How 4.39x 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.”

38

But why?How can this make sense?

How about a context?

39

Understand Why

So? What do you see?

40

Understand Why

gallons

Total

Where are we?

41

Understand Why

4.2 gallons

Total

How many gallons? About how many?

$

42

Understand Why

4.2 gallons

$ 4.39

Total

About how much? Maximum?? Minimum??

43

Understand Why

4.2 gallons

$ 4.39

Total184.38

Context makes ridiculous obvious, and breeds sense-making. Actual cost? So how do we multiply decimals sensibly?

44

Solving Simple Linear Equations

45

3x + 7 = 22

How do we solve equations:

Subtract 7 3 x + 7 = 22 - 7 - 7 3 x = 15

Divide by 3 3 3

Voila: x = 5

46

3x + 7

1. Tell me what you see: 3 x + 72. 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?

47

3x + 7 = 22

Where did we start? What did we do?

x 5 x 3 3x 15 ÷ 3

+ 7 3x + 7 22 - 7

48

3x + 7 = 22

X X X IIIIIII IIII IIII IIII IIII II

X X X IIIII IIIII IIIII

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.

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.

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.

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.

53

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?

54

So?

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

55

56

57

58

59

60

61

62

63

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.

3Example 1:

4 S.A.= 2πrh + 2 πr2

Find the surface area. Homework: Page 385 1-19 odd

64

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.

65

• 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.

66

• 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.

67

The CCSSM Trojan Horse:

SMP 3: Construct viable arguments and critique the reasoning of others

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,

andWork without collaboration is not sustaining.

Ergo: Our job, as professionals, at its core, is to help people envision, understand, practice, receive feedback and collaborate.

In Conclusion

68

69

Thank You.

Go forth and take on the world!