Transcript Math Workshop: Using Classroom Practice for Teacher

Mathematics (for all)
for Tomorrow:
How to start a job we can’t finish
Vail Symposium, 2006
Some ideas from Think Math!, a new NSF program
from
and Harcourt School Publishers
With support from
Curiosity, understanding, and
skill
No 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…
• … 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
tiny
part of Think Math!
Some 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 curious: Part
I
4+2=6
3+1=4
7 + 3 = 10
Just to get you curious: Part
II
Name a two-digit number…
A 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?
•
•
•
•
•
•
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
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
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
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
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
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
you first thought of.
• Aha! Your answer is 1!
Of course, they’ll need to
do it themselves!
Using the notation:
following the steps
Words
Think of a
number.
Double it.
Add 6.
Divide by 2.
What did you get?
Pictures Dan Cory Sand Chris
y
5a
10
16
8
7
3
20
Using the notation:
undoing the steps
Words
Think of a
number.
Double it.
Add 6.
Divide by 2.
What did you get?
Pictures Dan Cory Sand Chris
5a 4 y
10 8
16 14
8 7
3
20
Hard to undo using the words.
Much easier to undo using the notation.
Using the notation:
simplifying the steps
Words
Think of a
number.
Double it.
Add 6.
Divide by 2.
What did you get?
Pictures Dan Cory Sand Chris
5a 4 y
10 8
16 14
8 7
3
20
•
•
•
•
•
What are the kids
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…
soon becomes
A bit corny, but…
soon becomes
x
Kids are already
used to it!
• Focused arithmetic practice with a pattern
(function) and a “pattern indicator”
n
10
8
28 18 17
n–8
2
0
20
• Since 1st grade
58 57
3
4
A game, from 3rd grade
Puzzles, from 3rd grade
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.
h
t
u
1
4
4
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
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.
h t u
432
342
234
324
144
414
What’s My Number?
8
6
2
5
7
1
3
4
What’s My Number?
8
6
2
16
14
5
7
13
15
1
3
9
11
4
10
12
Combinatorics
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?
Early in
nd
2
grade
Combinatorics or Phonics?
a i
s n t
How 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?
a i
s n t
in
as
at
Or Multiplication in an
unfamiliar context?
b pw s
il it in
l
k
Or Multiplication in an
unfamiliar context?
b pw s
il it in
l
k
br tr st
ic ac in
k k g
Speaking of
multiplication…
…what could be
less sexy
than memorizing
4th grade
multiplication 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
memorize
multiplication facts?
Something memorable!
One way to look at it
55
One way to look at it
Removing a
column leaves
54
One way to look at it
Replacing as a
row leaves
64
with one left
over.
One way to look at it
Removing the
leftover leaves
64
showing that it
is one less than
5 5.
Where does this lead?
To do…
53
 47
Where does this lead?
To do…
53
 47
…I think…
3 more than 50
Where does this lead?
To do…
53
 47
…I think…
3 more than 50
3 less than 50
•50  50 (well, 5  5 and …) …
•Minus 3  3
2500
–9
Where does this lead?
To do…
53
 47
…I think…
3 more than 50
3 less than 50
•50  50 (well, 5  5 and …) …
•Minus 3  3
2500
–9
2491
Why does
it work?
50
53
47
3
Another view of multiplication:
from 2 x 3 = 6 to 22 x 17 =
374
a i
s n t
22
17
Another
view of
multiplicatio
n:
22 x 17
20
10
7
2
Another
view of
multiplicatio
n:
22 x 17
20
10
7
2
Another
view of
multiplicatio
n:
22 x 17
20
10
7
2
20200
x 10
2 x 10
20
20140
x7
2x7
14
20
10
7
2
200
20 220
140
14 154
340
34 374
20
2
1
10
7
200
20 220
140
14 154
340
34 374
22
x17
154
220
374
20
2
1
10
7
200
20 220
140
14 154
340
34 374
17
x22
34
340
374
22
17
374
22 x 17 =
374
22
17
374
22 x 17 =
374
22
17
374
374 ÷ 17 =
22
22
17 374
A kindergarten look at…
20
10
7
2
200
20 220
140
14 154
340
34 374
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.
Back to the very beginnings
blue
gray
6
small
Children can also summarize.
4
large
7
3
10
“Data” from the buttons.
Abstraction
If we substitute numbers for the original objects…
blue
gray
small
6
4
2
6
large
4
3
1
4
10
7
3
10
7
3
Puzzling
Don’t always start with the question!
7
6
13
5
3
8
12
9
21
Building the addition
algorithm
Only multiples of 10 in yellow. Only less than 10 in blue.
20
5
25
30
8
38
50
13 63
Relating addition and subtraction
4
2
6
7
3
10
3
1
4
3
1
4
7
3
10
4
2
6
…the subtraction algorithm
Only multiples of 10 in yellow. Only less than 10 in blue.
20
5
25
60
3
63
30
8
38
30
8
38
50 13
63
30
-5 25
25 + 38 = 63
63 – 38 = 25
…the subtraction algorithm
Only multiples of 10 in yellow. Only less than 10 in blue.
20
5
25
60 133
50
63
30
8
38
30
8
38
50 13
63
20
5
25
25 + 38 = 63
63 – 38 = 25
The algebra connection:
adding number sentences
4
2
6
4+2=6
3
1
4
3+1=4
7
3
10
7 + 3 = 10
The algebra connection:
subtracting number
sentences
7
3
10
7 + 3=10
3
1
4
3 + 1= 4
4
2
6
4 + 2= 6
The algebra connection:
subtracting “number”
sentences
5x
3y 23
5x + 3y = 23
2x
3y
11
2x + 3y = 11
3x
0
12
3x + 0 = 12
x=4
All from sorting buttons…
5x
3y 23
5x + 3y = 23
2x
3y
11
2x + 3y = 11
3x
0
12
3x + 0 = 12
x=4
Thanks!
Think Math !
www2.edc.org/thinkmath
Bye!
Thanks!
Think Math !
www2.edc.org/thinkmath
What is Think Math! ?
• NSF / EDC / Harcourt comprehensive K–5
• 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
6