Shifting Our Mindsets and Our
Actions from Remembering
HOW to Understanding WHY
Houston NCTM 11/20/14
Steve Leinwand
American Institutes for Research
sleinwand@air.org www.steveleinwand.com
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Decisions Decisions
LUNCH
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Decisions Decisions
STEVE
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But first:
A pre-session angry
overture
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The problem is universal!
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Ready?
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Get set. Go.
What is 8 + 9?
17 Bing Bang Done!
Vs.
Convince me that 9 + 8 = 17.
Hmmmm….
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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?
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4 + 29 =
How did you do it?
How did you do it?
Who did it differently?
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So…the problem is:
If we continue to do what we’ve
always done….
We’ll continue to get what we’ve
always gotten.
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Where is the opportunity to
learn?
Where is the sense-making?
Does anyone benefit from a
sheet like this?
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95
- 48 How did you do it?
or
Convince me that 95-48=47.
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In other words,
our questions make all the
difference.
(no pun intended)
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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
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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!
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Adding and Subtracting
Integers
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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”
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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
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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
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How nice if we wish to
continue using math to sort
our students!
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So what’s the alternative?
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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?
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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
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3 x 40
• 3 x 4 x 10 (properties)
• 40 + 40 + 40
• 12 with a 0 appended
•
3
120
40
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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
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Why bother?
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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?
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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
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Or using a model to support UndWhy:
www.thesingaporemaths.com
11 kg
Rahim
Siti
29 kg
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Wow – Look at HOW vs WHY?
7.5 ÷ 0.5
7.5 ÷ 0.25
Multiplying Decimals
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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.”
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But why?
How can this make sense?
How about a context?
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Understand Why
So? What do you see?
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Understand Why
gallons
Total
Where are we?
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Understand Why
4.2
gallons
$
Total
How many gallons? About how many?
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Understand Why
4.2
gallons
$ 4.39
Total
About how much? Maximum?? Minimum??
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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?
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Solving Simple Linear
Equations
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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
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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?
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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
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Let’s look at a silly problem
Sandra is interested in buying party favors for
the friends she is inviting to her birthday
party.
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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.
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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.
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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.
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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?
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So?
Your turn. How much?
How did you get your answer?
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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 384-5.
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Example 1:
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S.A.= 2πrh + 2 πr2
Find the surface area.
Homework: Page 385 1-19 odd
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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.
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• 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.
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• 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.
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The CCSSM Trojan Horse:
SMP 3: Construct viable
arguments and critique
the reasoning of others
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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.
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Thank You.
Go forth and take on the
world!
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