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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.
• Your answer is 1!
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.
• Your answer is 1!
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.
• Your answer is 1!
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.
• Your answer is 1!
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.
• Your answer is 1!
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.
• Your answer is 1!
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
One way to look at it
a a
a
5 4
Removing a column leaves
One way to look at it
a
a a
a
6 4
Replacing as a row leaves
with one left over.
One way to look at it
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
Where does this Where does this lead?lead?
To do… …I think… 53 3 more than 50 47
Where does this Where does this lead?lead?
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
Where does this Where does this lead?lead?
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
Another Another view of view of multiplicmultiplication:ation:22 x 1722 x 17
20
10
2
7
Another Another view of view of multiplicmultiplication:ation:22 x 1722 x 17
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
Only 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 the subtraction algorithmalgorithm
Only 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
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
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