Mathematics (for all) for Tomorrow: How to start a job we can’t finish from and Harcourt School Publishers With support from Vail Symposium, 2006 Some

Mathematics (for all) Mathematics (for all) for Tomorrow: for Tomorrow:

How to start a job we can’t How to start a job we can’t finishfinish

from and Harcourt School Publishers

With support from

Vail Symposium, 2006Vail Symposium, 2006

Some ideas from Think Math!, a new NSF program

Curiosity, Curiosity, understanding, and understanding, and

skillskillNo trade-offNo trade-off

• We can’t afford a trade-off.• We can’t afford to think there is one.

• Fortunately, there doesn’t have to be one.

My personal goal…My personal goal…

• … as a teacher, is to get kids to notice how smart they really are.

• … as a curriculum developer, is to get teachers to notice how smart they are…

• …and to notice how smart their kids are!

One One tinytiny part of part of Think Think Math!Math!

Some ways to play algebraicallySome ways to play algebraically • There’s more to the program, of course, including fractions, geometry, statistics, logic and reasoning, word problems…

• Building algebra: the ideas and the language

Just to get you Just to get you curious: Part Icurious: Part I

4 + 2 = 6

3 + 1 = 4

10+ =7 3

Just to get you Just to get you curious: Part IIcurious: Part II

Name a two-digit number…

A number trickA number trick, near the end of 4th

grade • 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?How did it work?

How did it work?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.


How did it work?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.











• Aha! Your answer is 1!

Of course, they’ll Of course, they’ll need to do it need to do it themselves!themselves!

Using the Using the notation: notation:

followingfollowing the the stepssteps

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 the Using the notation: notation: undoingundoing

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

510168

48147 3 20

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

Dana

Cory

Sandy

Chris

Words Pictures

Using the Using the notation: notation:

simplifyingsimplifying the the stepssteps

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

510168

48147 3 20

Dana

Cory

Sandy

Chris

Words Pictures

What are the kids What are the kids getting?getting?

• Computational practice and more.• Notation helps them understand the trick.

• Notation lets them invent new tricks.• Notation helps them undo the trick.• And a most important idea… Notation/representation is powerful!

A bit corny, but…A bit corny, but…

soon becomes

A bit corny, but…A bit corny, but…

soon becomesx

Kids are already Kids are already used to it!used to it!

• Focused arithmetic practice with a pattern (function) and a “pattern indicator”

• Since 1st grade

n 10n – 8 2

8

0

28

20

18 17

3 4

58 57

A game, from 3rd A game, from 3rd gradegrade

hundre

ds dig

it > 6tens di

git is

7, 8, or 9

the number is

a multiple of 5

the tens digit isgreater than thehundreds digit

ones digit < 5

the nu

mber

is eve

n

tens digit < ones digi

t

the ones digit istwice the tens digit

Puzzles, from 3rd Puzzles, from 3rd gradegrade Who Am I? I. I am even II. All of my digits < 5 III. h + t + u = 9 IV. I am less than 400 V. Exactly two of my digits are the same.htu

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 < 5

III. h + t + u = 9

IV. I am less than 400.

V. Exactly two of my digits are the same.

432342234324144414

1 4 4

What’s My Number?What’s My Number?

1

2

3

4

8

75

6

What’s My Number?What’s My Number?

1

2

3

4

8

75

6

9

10

11

12

161

513

14

CombinatoricsCombinatoricsFour 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?

Early in 2Early in 2ndnd grade grade

Combinatorics or Combinatorics or Phonics?Phonics?

a i s n tHow many two-letter combinations can you make starting with one of these two vowels, and ending with one of these three consonants?as in at …

Or coordinates?Or coordinates?

a i s n t

asin

at

Or Multiplication in Or Multiplication in an unfamiliar an unfamiliar

context?context?w s i

ll

it

ink

b p

Or Multiplication in Or Multiplication in an unfamiliar an unfamiliar

context?context?w s i

ll

it

ink

b p

st

ick

ack

ing

br

tr

Speaking of Speaking of multiplication…multiplication…

……what could be what could be less sexy less sexy

than memorizing than memorizing 4th grade 4th grade

multiplication multiplication facts?facts?

Just the facts

• Kids already know 4 4, 5 5, 6 6, 7 7, …

• Have most others and easily work out what they don’t have memorized.

• Goal now is to consolidate!

What helps kids What helps kids memorize memorize

multiplication multiplication facts?facts?

Something Something memorable!memorable!

One way to look at it

a a

5 5


a a

a

5 4

Removing a column leaves


a

a a

a

6 4

Replacing as a row leaves

with one left over.


a

a a

a

6 4

Removing the leftover leavesshowing that it is one less than

5 5.

Where does this Where does this lead?lead?

To do… 53 47


To do… …I think… 53 3 more than 50 47


To do… …I think… 53 3 more than 50 47 3 less than 50

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


To do… …I think… 53 3 more than 50 47 3 less than 50

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

2491

Why Why does it does it work?work?

47

3

5053

a i s n t

Another view of Another view of multiplication:multiplication:

from 2 x 3 = 6 to from 2 x 3 = 6 to 22 x 17 = 37422 x 17 = 374

Another Another view of view of multiplicmultiplication:ation:22 x 1722 x 17

22

17


20

10

2

7


20

10

2

7

20

10

2

7

20 x 10

20 x 7

2 x 10

2 x 7

200

140

20

14

20

10

2

7

200 20

14140

220

154

37434340

20

10

2

7

200 20

14140

220

154

37434340

2217154220374

x1

20

10

2

7

200 20

14140

220

154

37434340

172234340374

x1

22 x 17 = 22 x 17 = 374374

22

17 374

22 x 17 = 22 x 17 = 374374

22

17 374

374 ÷ 17 374 ÷ 17 = 22= 22

22

17 374

2217374

20

10

2

7

200 20

14140

220

154

37434340

A kindergarten A kindergarten look at…look at…

Back to the very Back to the very beginningsbeginnings

Picture a young child with a small pile of buttons.

Natural to sort.

We help children refine and extend what is already natural.

Back to the very Back to the very beginningsbeginnings

Children can also summarize.

“Data” from the buttons.

large

small

blue gray

6

4

7 3 10

large

small

blue gray

6

4

7 3 10

4 2

3 1

If we substitute numbers for the original objects…

10

4

6

37

AbstractionAbstraction

PuzzlingPuzzling

5

Don’t always start with the question!

21

8

13

912

7 6

3

Building the addition Building the addition algorithmalgorithm

Only multiples of 10 in yellow. Only less than 10 in blue.

63

38

25

1350

20 5

830

Relating addition and Relating addition and subtractionsubtraction

4 2

3 1

10

4

6

37

7 3

3 1

6

4

10

24

……the subtraction the subtraction algorithmalgorithm


63

38

25

1350

20 5

830

25

38

63

-530

60 3

830

25 + 38 = 63 63 – 38 = 25

……the subtraction the subtraction algorithmalgorithm


63

38

25

1350

20 5

830

25

38

63

520

60 3

830

50 13

25 + 38 = 63 63 – 38 = 25

The algebra The algebra connection: connection: addingadding number number

sentencessentences4 2

3 1

10

4

6

37

4 + 2 = 6

3 + 1 = 4

10+ =7 3

The algebra The algebra connection: connection:

subtractingsubtracting number number sentencessentences

7 3

3 1

6

4

10

24

7 + 3 = 10

3 + 1 = 4

6+ =4 2

The algebra The algebra connection: connection:

subtractingsubtracting “number” “number” sentencessentences

5x 3y

2x 3y 11

23 5x + 3y = 23

2x + 3y = 11

12+ =3x 0x = 4

3x 0 12

All from sorting All from sorting buttons…buttons…

5x 3y

2x 3y

23

12+ =3x 0x = 4

5x + 3y = 23

2x + 3y = 1111

3x 0 12

Think MathThink Math ! !

www2.edc.org/thinkmath

Thanks!

www2.edc.org/thinkmath

Bye!

Thanks!

Think MathThink Math ! !

What is Think Think

Math! Math! ?• NSF / EDC / Harcourt comprehensive K–56

• Not a supplement to other programs;no need to supplement it.

• Built to conform with NCTM Standards, and to meet and exceed the high-stakes state tests

• Fully researched in classrooms with children and teachers

• Easy to teach from; easy to learn from • Skills, understanding, problem solving, algebraic focus from K

Documents

Mathematics (for all) for Tomorrow: How to start a job we can’t finish from and Harcourt School Publishers With support from Vail Symposium, 2006 Some