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From Buttons to Algebra: Learning the ideas and language of algebra, K-12 from and Harcourt School Publishers Rice University, Houston, Sept 2007 Paul Goldenberg http://thinkmath.edc.org Some ideas from the newest NSF program, Think Math!

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From Buttons to Algebra:. Paul Goldenberg http://thinkmath.edc.org Some ideas from the newest NSF program, Think Math!. Learning the ideas and language of algebra, K-12. from and Harcourt School Publishers Rice University, Houston, Sept 2007. http://thinkmath.edc.org - PowerPoint PPT Presentation

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Page 1: From Buttons to Algebra:

From Buttons to Algebra: Learning the ideas and language of algebra, K-12

from and Harcourt School PublishersRice University, Houston, Sept 2007

Paul Goldenberg http://thinkmath.edc.org

Some ideas from the newest NSF program, Think Math!

Page 2: From Buttons to Algebra:

Before you scramble to take notes

http://thinkmath.edc.org

With downloadable PowerPointat http://www.edc.org/thinkmath/

Page 3: From Buttons to Algebra:

What could mathematics be like?

Is there anything interesting about addition and subtraction sentences?

It could be spark curiosity!

Page 4: From Buttons to Algebra:

Write two number sentences…

To 2nd graders: see if you can find some that don’t work!

4 + 2 = 6

3 + 1 = 4

10+ =7 3

Page 5: From Buttons to Algebra:

What could mathematics be like?

Is there anything less sexy than memorizing multiplication facts?

What helps people memorize? Something memorable!

It could be fascinating!

Page 6: From Buttons to Algebra:

Teaching without talkingShhh… Students thinking!

Wow! Will it always work? Big numbers??

38 39 40 41 42

3536

6 7 8 9 105432 11 12 13

8081

18 19 20 21 22… …

??

1600

1516

Page 7: From Buttons to Algebra:

Take it a step further

What about two steps out?

Page 8: From Buttons to Algebra:

Shhh… Students thinking!

Again?! Always? Find some bigger examples.

Teaching without talking

1216

6 7 8 9 105432 11 12 13

6064

?

58 59 60 61 6228 29 30 31 32… …

???

Page 9: From Buttons to Algebra:

Take it even further

What about three steps out?What about four?What about five?

Page 10: From Buttons to Algebra:

“OK, um, 53” “Hmm, well…

…OK, I’ll pick 47, and I can multiply those numbers faster than you can!”

To do… 53

47

I think… 50 50 (well, 5 5 and …)… 2500Minus 3 3 – 9

2491

“Mommy! Give me a 2-digit number!”2500

47 48 49 50 51 52 53

about 50

Page 11: From Buttons to Algebra:

Why bother? Kids feel smart! Teachers feel smart! Practice.

Gives practice. Helps me memorize, because it’s memorable!

Something new. Foreshadows algebra. In fact, kids record it with algebraic language!

And something to wonder about: How does it

work?

It matters!

Page 12: From Buttons to Algebra:

One way to look at it

5 5

Page 13: From Buttons to Algebra:

One way to look at it

5 4

Removing a column leaves

Page 14: From Buttons to Algebra:

One way to look at it

6 4

Replacing as a row leaves

with one left over.

Page 15: From Buttons to Algebra:

One way to look at it

6 4

Removing the leftover leaves

showing that it is one less than

5 5.

Page 16: From Buttons to Algebra:

How does it work?

47 3

5053

47

350 50– 3 3

= 53 47

Page 17: From Buttons to Algebra:

An important propaganda break…

Page 18: From Buttons to Algebra:

“Math talent” is made, not found

We all “know” that some people have…musical ears,mathematical minds,a natural aptitude for languages….

We gotta stop believing it’s all in the genes! And we are equally endowed with much of it

Page 19: From Buttons to Algebra:

A number trick Think of a number. Add 3. Double the result. Subtract 4. Divide the result by 2. Subtract the number

you first thought of. Your answer is 1!

Page 20: From Buttons to Algebra:

How did it work? Think of a number. Add 3. Double the result. Subtract 4. Divide the result by 2. Subtract the number

you first thought of. Your answer is 1!

Page 21: From Buttons to Algebra:

How did it work? Think of a number. Add 3. Double the result. Subtract 4. Divide the result by 2. Subtract the number

you first thought of. Your answer is 1!

Page 22: From Buttons to Algebra:

How did it work? Think of a number. Add 3. Double the result. Subtract 4. Divide the result by 2. Subtract the number

you first thought of. Your answer is 1!

Page 23: From Buttons to Algebra:

How did it work? Think of a number. Add 3. Double the result. Subtract 4. Divide the result by 2. Subtract the number

you first thought of. Your answer is 1!

Page 24: From Buttons to Algebra:

How did it work? Think of a number. Add 3. Double the result. Subtract 4. Divide the result by 2. Subtract the number

you first thought of. Your answer is 1!

Page 25: From Buttons to Algebra:

How did it work? Think of a number. Add 3. Double the result. Subtract 4. Divide the result by 2. Subtract the number

you first thought of. Your answer is 1!

Page 26: From Buttons to Algebra:

How did it work? Think of a number. Add 3. Double the result. Subtract 4. Divide the result by 2. Subtract the number

you first thought of. Your answer is 1!

Page 27: From Buttons to Algebra:

How did it work? Think of a number. Add 3. Double the result. Subtract 4. Divide the result by 2. Subtract the number

you first thought of. Your answer is 1!

Page 28: From Buttons to Algebra:

Kids need to do it themselves…

Page 29: From Buttons to Algebra:

Using notation: following steps

Think of a number.Double it.Add 6.Divide by 2. What did you get?

510168 7 3 20

Dana

Cory

Sandy

Chris

Words Pictures

Page 30: From Buttons to Algebra:

Using notation: undoing steps

Think of a number.Double it.Add 6.Divide by 2. What did you get?

510168 7 3 20

Dana

Cory

Sandy

Chris

Words48

14

Hard to undo using the words.Much easier to undo using the notation.

Pictures

Page 31: From Buttons to Algebra:

Using notation: simplifying steps

Think of a number.Double it.Add 6.Divide by 2. What did you get?

510168 7 3 20

Dana

Cory

Sandy

Chris

Words Pictures4

Page 32: From Buttons to Algebra:

Why a number trick? Why bags?

Computational practice, but much more Notation helps them understand the trick. Notation helps them invent new tricks. Notation helps them undo the trick. But most important, the idea that notation/representation is powerful!

Page 33: From Buttons to Algebra:

Children are language learners…

They are pattern-finders, abstracters… …natural sponges for language in context.

n 10n – 8 2

80

2820

18 173 4

58 57

Page 34: From Buttons to Algebra:

hundreds digit > 6 tens digit is

7, 8, or 9

the number isa multiple of 5

the tens digit isgreater than thehundreds digit

ones digit < 5

the number

is even

tens digit < ones digit

the ones digit istwice the tens digit

the number isdivisible by 3

A game in grade 3

Page 35: From Buttons to Algebra:

3rd grade detectives!

I. I am even.

h t u

0 01 1 12 2 23 3 34 4 45 5 56 6 67 7 78 8 89 9 9

II. All of my digits < 5III. h + t + u = 9

IV. I am less than 400.

V. Exactly two of my digits are the same.

432342234324144414

1 4 4

Page 36: From Buttons to Algebra:

Is it all puzzles and tricks?

No. (And that’s too bad, by the way!) Curiosity. How to start what we can’t finish. Cats play/practice pouncing; sharpen claws. We play/practice, too. We’ve evolved fancy

brains.

Page 37: From Buttons to Algebra:

Representing processes

Bags and letters can represent numbers. We need also to represent…

ideas — multiplicationprocesses — the multiplication algorithm

Page 38: From Buttons to Algebra:

Representing multiplication, itself

Page 39: From Buttons to Algebra:

Naming intersections, first gradePut a red house at the intersection of A street and N avenue.

Where is the green house?

How do we go fromthe green house tothe school?

Page 40: From Buttons to Algebra:

Combinatorics, beginning of 2nd

How many two-letter words can you make, starting with a red letter and ending with a purple letter?

a i s n t

Page 41: From Buttons to Algebra:

Multiplication, coordinates, phonics?

a i s n t

as in

at

Page 42: From Buttons to Algebra:

Multiplication, coordinates, phonics?

w s ill

it

ink

b p

st

ick

ack

ing

br

tr

Page 43: From Buttons to Algebra:

Similar questions, similar image

Four skirts and three shirts: how many outfits?

Five flavors of ice cream and four toppings: how many sundaes? (one scoop, one topping)

How many 2-block towers can you make from four differently-colored Lego blocks?

Page 44: From Buttons to Algebra:

Representing 22 1722

17

Page 45: From Buttons to Algebra:

Representing the algorithm20

10

2

7

Page 46: From Buttons to Algebra:

Representing the algorithm20

10

2

7

200

140

20

14

Page 47: From Buttons to Algebra:

Representing the algorithm20

10

2

7

200

140

20

14

220

154

37434340

Page 48: From Buttons to Algebra:

Representing the algorithm20

10

2

7

200

140

20

14

220

154

37434340

2217

154220374

x1

Page 49: From Buttons to Algebra:

Representing the algorithm20

10

2

7

200

140

20

14

220

154

37434340

172234

340374

x1

Page 50: From Buttons to Algebra:

22

17 374

22 17 = 374

Page 51: From Buttons to Algebra:

22

17 374

22 17 = 374

Page 52: From Buttons to Algebra:

Representing division (not the algorithm)

“Oh! Division is just unmultipli-cation!”

22

17 374

374 ÷ 17 = 222217 374

Page 53: From Buttons to Algebra:

A kindergarten look at20

10

2

7

200

140

20

14

220

154

37434340

Page 54: From Buttons to Algebra:

Back to the very beginnings

Picture a young child with a small pile of buttons.

Natural to sort.

We help children refine and extend what is already natural.

Page 55: From Buttons to Algebra:

6

4

7 3 10

Back to the very beginnings

Children can also summarize.

“Data” from the buttons.

blue gray

large

small

Page 56: From Buttons to Algebra:

large

small

blue gray

If we substitute numbers for the original objects…Abstraction

6

4

7 3 10

6

4

7 3 10

4 2

3 1

Page 57: From Buttons to Algebra:

Puzzling

5

Don’t always start with the question!

21

8

13

912

7 6

3

Page 58: From Buttons to Algebra:

Building the addition algorithmOnly multiples of 10 in yellow. Only less than 10 in blue.

63

38

25

1350

20 5

830

Page 59: From Buttons to Algebra:

Relating addition and subtraction6

4

7 3 10

4 2

3 16

4

7 3 10

4 2

3 1

Page 60: From Buttons to Algebra:

The subtraction algorithmOnly multiples of 10 in yellow. Only less than 10 in blue.

63

38

25

1350

20 5

830

25

38

63

-530

60 3

830

25 + 38 = 63 63 – 38 = 25

Page 61: From Buttons to Algebra:

The subtraction algorithmOnly multiples of 10 in yellow. Only less than 10 in blue.

63

38

25

1350

20 5

830

25

38

63

520

60 3

830

25 + 38 = 63 63 – 38 = 25

50 13

Page 62: From Buttons to Algebra:

The algebra connection: adding

4 2

3 1

10

4

6

37

4 + 2 = 6

3 + 1 = 4

10+ =7 3

Page 63: From Buttons to Algebra:

The algebra connection: subtracting

7 3

3 1

6

4

10

24

7 + 3 = 10

3 + 1 = 4

6+ =4 2

Page 64: From Buttons to Algebra:

The algebra connection: algebra!

5x 3y

2x 3y 11

23 5x + 3y = 23

2x + 3y = 11

12+ =3x 0x = 4

3x 0 12

Page 65: From Buttons to Algebra:

All from sorting buttons

5x 3y

2x 3y 11

23 5x + 3y = 23

2x + 3y = 11

12+ =3x 0x = 4

3x 0 12

Page 66: From Buttons to Algebra:

“Skill practice” in a second grade

VideoVideo

Page 67: From Buttons to Algebra:

Thank you!

E. Paul Goldenberg http://thinkmath.edc.org/

Page 68: From Buttons to Algebra:

Learning by doing, for teachers

Professional development of 1.6M teachers To take advantage of time they already have,

a curriculum must be…Easy to start (well, as easy as it can ge)

Appealing to adult minds (obviously to kids, too!)

Comforting (covering the bases, the tests)

Solid math, solid pedagogy (brain science, Montessori, Singapore, language)

Page 69: From Buttons to Algebra:

Keeping things in one’s head

1

2

3

4

8

75

6