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!

Before you scramble to take notes

http://thinkmath.edc.org

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

What could mathematics be like?

Is there anything interesting about addition and subtraction sentences?

It could be spark curiosity!

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

What could mathematics be like?

Is there anything less sexy than memorizing multiplication facts?

What helps people memorize? Something memorable!

It could be fascinating!

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

Take it a step further

What about two steps out?

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… …

???

Take it even further

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

“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

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!

One way to look at it

5 5


5 4

Removing a column leaves


6 4

Replacing as a row leaves

with one left over.


6 4

Removing the leftover leaves

showing that it is one less than

5 5.

How does it work?

47 3

5053

47

350 50– 3 3

= 53 47

An important propaganda break…

“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

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!

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
















Kids need to do it themselves…

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

Using notation: undoing steps


510168 7 3 20

Dana

Cory

Sandy

Chris

Words48

14

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

Pictures

Using notation: simplifying steps


510168 7 3 20

Dana

Cory

Sandy

Chris

Words Pictures4

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!

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

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

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

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.

Representing processes

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

ideas — multiplicationprocesses — the multiplication algorithm

Representing multiplication, itself

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?

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

Multiplication, coordinates, phonics?

a i s n t

as in

at

Multiplication, coordinates, phonics?

w s ill

it

ink

b p

st

ick

ack

ing

br

tr

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?

Representing 22 1722

17

Representing the algorithm20

10

2

7


10

2

7

200

140

20

14


10

2

7

200

140

20

14

220

154

37434340


10

2

7

200

140

20

14

220

154

37434340

2217

154220374

x1


10

2

7

200

140

20

14

220

154

37434340

172234

340374

x1

22

17 374

22 17 = 374

22

17 374

22 17 = 374

Representing division (not the algorithm)

“Oh! Division is just unmultipli-cation!”

22

17 374

374 ÷ 17 = 222217 374

A kindergarten look at20

10

2

7

200

140

20

14

220

154

37434340

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.

6

4

7 3 10

Back to the very beginnings

Children can also summarize.

“Data” from the buttons.

blue gray

large

small

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

Puzzling

5

Don’t always start with the question!

21

8

13

912

7 6

3

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

63

38

25

1350

20 5

830

Relating addition and subtraction6

4

7 3 10

4 2

3 16

4

7 3 10

4 2

3 1

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

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

The algebra connection: adding

4 2

3 1

10

4

6

37

4 + 2 = 6

3 + 1 = 4

10+ =7 3

The algebra connection: subtracting

7 3

3 1

6

4

10

24

7 + 3 = 10

3 + 1 = 4

6+ =4 2

The algebra connection: algebra!

5x 3y

2x 3y 11

23 5x + 3y = 23

2x + 3y = 11

12+ =3x 0x = 4

3x 0 12

All from sorting buttons

5x 3y

2x 3y 11

23 5x + 3y = 23

2x + 3y = 11

12+ =3x 0x = 4

3x 0 12

“Skill practice” in a second grade

VideoVideo

Thank you!

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

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)

Keeping things in one’s head

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2

3

4

8

75

6

Documents

From Buttons to Algebra: