Transcript 12748

How the ideas and language of
algebra K-5
set the stage for Algebra 8-12
MSRI, May 15, 2008
E. Paul Goldenberg
To save note-taking,
http://thinkmath.edc.org
Click download presentations link
(next week)
Language vs. computational tool
To us, expressions like (n – d)(n + d) can be manipulated
• to derive things we don’t yet know, or
• to prove things that we conjectured from experiment.
We can also use such notation as language (not manipulated)
• to describe a process or computation or pattern, or
• to express what we already know, e.g.,
(n – d)(n + d) = n2 – d2
Claim: While most elementary school children cannot use
algebraic notation the first two ways, as a computational tool,
most can use it the last two ways, as language.
Great built-in apparatus
Abstraction (categories, words, pictures)
 Syntax, structure, sensitivity to order
 Phenomenal language-learning ability
 Quantification (limited, but there)
 Logic (evolving, but there)
 Theory-making about the world irrelevance of orientation

In learning math, little differentiation
Some algebraic ideas precede arithmetic
Developmental

w/o rearrangeability 3 + 5 = 8 can’t make sense

Nourishment to extend/apply/refine built-ins
 breaking numbers and rearranging parts
(any-order-any-grouping, commutativity/associativity),
 breaking arrays; describing whole & parts
(linearity, distributive property)

But many of the basic intuitions are built in,
developmental, not “learned” in math class.
Algebraic language, like any language, is
Convention
Children are phenomenal language-learners
 Build it from language spoken around them
 Infer meaning and structure from use: not
explicit definitions and lessons, but from
language used in context
 Where “math is spoken at home” (not drill,
lessons, but conversation that makes salient
logical puzzle, quantity, etc.) kids learn it

Demand “does it work with kids?”
Algebraic language & algebraic thinking
Linguistics and mathematics
 Algebra as abbreviated speech
 A number trick
 “Pattern indicators”
 Difference of squares
 Systems of equations in kindergarten?
 Understanding two dimensional information

(Algebra as a Second Language)
Linguistics and mathematics
Michelle’s strategy for 24 – 8:
(breaking it up)
Algebraic ideas
A linguistic
idea (mostly)
Well, 24 – 4 is easy!
 Now, 20 minus another 4…
Arithmetic
 Well, I know 10 – 4 is 6,
and 20 is 10 + 10,
knowledge
so, 20 – 4 is 16.
 So, 24 – 8 = 16.

What is the “linguistic” idea?
28 – 8 on her fingers…
Fingers are counters,
good for grasping the
idea,
and
good (initially) for
finding or verifying
answers to problems
like 28 – 4, but…
Algebraic language & algebraic thinking
Linguistics and mathematics
 Algebra as abbreviated speech
 A number trick
 “Pattern indicators”
 Difference of squares
 Systems of equations in kindergarten?
 Understanding two dimensional information

(Algebra as a Second Language)
Algebra as abbreviated speech
 A number
(Algebra as a second Language)
trick
“Pattern indicators”
 Difference of squares

Surprise! You speak algebra!
5th grade
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
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.
 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!

Kids need to do it themselves…
Using notation: following 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 notation: undoing 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 14
8 7
3
20
Hard to undo using the words.
Much easier to undo using the notation.
Using notation: simplifying 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
16
8
7
3
20
Abbreviated speech: simplifying pictures
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
16
8
7
3
b
2b
2b + 6
20 b + 3
Notation is powerful!
Computational practice, but much more
 Notation helps them understand the trick.
 Notation helps them invent new tricks.
 Notation helps them undo the trick.
 Algebra is a favor, not just “another thing
to learn.”

Algebra as abbreviated speech

A number trick
 “Pattern

indicators”
Difference of squares
(Algebra as a second Language)
Children are language learners…
They are pattern-finders, abstracters…
 …natural sponges for language in context.

n
10
8
28 18 17
n–8
2
0
20
58 57
3
4
Go to index
Algebra as abbreviated speech
(Algebra as a second Language)
A number trick
 “Pattern indicators”

 Difference
of squares
Math could be fascinating!
 Is there anything less sexy than
memorizing multiplication facts?
 What helps people memorize?
Something memorable!

4th grade
Teaching without talking
Shhh… Students thinking!
35
36
15
16
2
3
4
5
6
80
81
7
8
9
10
11
12
13
Wow! Will it always work? Big numbers?
?
?
?
…
18
19
20
1600
21
22
…
38
39
40
41
42
Take it a step further
What about two steps out?
Teaching without talking
Shhh… Students thinking!
12
16
2
3
4
60
64
5
6
7
8
9
10
11
12
13
Again?! Always? Find some bigger examples.
?
?
?
?
…
28
29
30
31
32
…
58
59
60
61
62
Take it even further
What about three steps out?
What about four?
What about five?
75
100
4
5
6
7
8
9
10
11
12
13
14 15
Take it even further
What about three steps out?
What about four?
What about five?
1200
1225
29
30
31
32
33
34 35
36
37
38
39 40
Take it even further
What about two steps out?
1221
1225
29
30
31
32
33
34 35
36
37
38
39 40
“Mommy! Give me a 2-digit number!”
“OK, um, 53”
 “Hmm, well…

47

2500
about 50
48
49
50
51
52
53
…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 …)… 2500
Minus 3  3
–9
2491
But nobody cares if kids can
multiply 47  53 mentally!
What do we care about, then?
50  50 (well, 5  5 and place value)
 Keeping 2500 in mind while thinking 3  3
 Subtracting 2500 – 9
 Finding the pattern
 Describing the pattern

Algebraic language
Algebraic/arithmetic
Science
thinking
(7 – 3)  (7 + 3) = 7  7 – 9
(50 – 3)  (50 + 3) = 50  50 – 9
n
n–3
nQ?
+3
(n – 3)  (n + 3) = n  n – 9
Nicolina Malara, Italy: “algebraic babble”
Make a table; use pattern indicator.
Distance away What to subtract
1
1
2
4
3
9
4
16
5
25
d
dd
(7 – d)  (7 + d) = 7  7 – d  d
n
n–d
n+d
(n – d)  (n + d) = n  n – d  d
We also care about thinking!

Kids feel smart!
Why silent teaching?
Teachers feel smart!
 Practice.

Gives practice. Helps me memorize, because it’s memorable!

It matters!
Something new.
Foreshadows algebra. In fact, kids record it with algebraic language!

And something to wonder about:
How does it work?
One way to look at it
55
One way to look at it
Removing a
column leaves
54
Not “concrete vs. abstract”
semantic (spatial) vs. syntactic
Kids don’t derive/prove with algebra.
One way to look at it
Replacing as a
row leaves
64
with one left
over.
Not “concrete vs. abstract”
semantic (spatial) vs. syntactic
Kids don’t derive/prove with algebra.
One way to look at it
Removing the
leftover leaves
64
showing that it
is one less than
5 5.
Not “concrete vs. abstract”
semantic (spatial) vs. syntactic
Kids don’t derive/prove with algebra.
Algebraic language & algebraic thinking
Linguistics and mathematics
 Algebra as abbreviated speech
 A number trick
 “Pattern indicators”
 Difference of squares
 Systems of equations in kindergarten?
 Understanding two dimensional information

(Algebra as a Second Language)
Systems of equations
in Kindergarten?!
Challenge: can
you find some
that don’t work?
4+2=6
5x + 3y = 23
3+1=4
2x + 3y = 11
7 + 3 = 10
Math could be spark curiosity!
 Is there anything interesting about
addition and subtraction sentences?
 Start with 2nd grade
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
A Cross Number Puzzle
Don’t always start with the question!
7
6
13
5
3
8
12
9
21
Relating addition and subtraction
4
10
2
3
6
7
3
4
1
1
4
3
7
6
3
2
10
4
Ultimately, building the addition and subtraction algorithms
The algebra connection: adding
4
2
6
4+2=6
3
1
4
3+1=4
7
3
10
7 + 3 = 10
The algebra connection: subtracting
7
3
10
7 + 3 = 10
3
1
4
3+1= 4
4
2
6
4 +2 = 6
The eighth-grade look
5x
3y 23
5x + 3y = 23
2x
3y
11
2x + 3y = 11
3x
0
12
3x + 0 = 12
x=4
Algebraic language & algebraic thinking
Linguistics and mathematics
 Algebra as abbreviated speech
 A number trick
 “Pattern indicators”
 Difference of squares
 Systems of equations in kindergarten?
 Understanding two dimensional information

(Algebra as a Second Language)
Two-dimensional information
Words
Pictures Dana
Think of a number.
5
Double it.
10
Add 6.
16
Cory
4
8
14
Naming intersections, first grade
Put a red house at
the intersection of
A street
and N avenue.
Where is the
green house?
How do we go from
the green house to
the school?
Combinatorics, beginning of 2nd
a i

s n t
How many two-letter words can you make,
starting with a red letter and
ending with a purple letter?
Multiplication, coordinates, phonics?
a i
s n t
in
as
at
Multiplication, coordinates, phonics?
b pw s
il it in
l
k
br tr st
ic ac in
k k g
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)
With four different bottom blocks and
three different top blocks,
how many 2-block Lego towers
can you make?
Quick bail out!
Thank you!
E. Paul Goldenberg
 http://thinkmath.edc.org/

Quick recover
The idea of a word problem…

An attempt at reality
A situation rather than a “naked” calculation
The goal is the
problem, not the words
• Necessarily bizarre dialect: low redundancy or very wordy
The idea of a word problem…

An attempt at reality
A situation rather than a “naked” calculation
“Clothing the naked” with words makes it
linguistically hard without improving the
mathematics.
The
goalit isisthe
problem, not the words
In tests
discriminatory!
• Necessarily bizarre dialect: low redundancy or very wordy
• State ELA tests test ELA
• State Math tests test Math and ELA
Attempts to be efficient (spare)
Stereotyped wordingkey words
 Stereotyped structureautopilot strategies

Key words
We rail against key word strategies.
Ben and his sister were eating pretzels.
Ben left 7 of his pretzels.
His sister left 4 of hers.
How many pretzels were left?
So writers do cartwheels to subvert them.
But, frankly, it is smart to look for clues!
This is how language works!
Autopilot strategies
We make fun of thought-free “strategies.”
Many numbers: +
Two numbers close together: – or 
Two numbers, one large, one small: ÷
Writers create bizarre wordings with
irrelevant numbers, just to confuse kids.
But, if the goal is mathematics and
to teach children to think and
communicate clearly…
…deliberately perverting our wording to make
it unclear is not a good model!
So what can we do to help students learn to
read and interpret story-based problems
correctly?
“Headline Stories”
Less is more!
Ben and his sister were eating pretzels.
Ben left 7 of his pretzels.
His sister left 4 of hers.
What questions can we ask?
Children learn the anatomy of problems by
creating them. (Neonatal problem posing!)
“Headline Stories”

Do it yourself!
Use any word problem you like.

What can I do? What can I figure out?
Representing 22  17
22
17
Representing the algorithm
20
10
7
2
Representing the algorithm
20
2
10
200
20
7
140
14
Representing the algorithm
20
2
10
200
20 220
7
140
14 154
340
34 374
Representing the algorithm
20
2
1
10
200
20 220
7
140
14 154
340
34 374
22
x17
154
220
374
Representing the algorithm
20
2
1
10
200
20 220
7
140
14 154
340
34 374
17
x22
34
340
374
More generally, (d+2) (r+7) =
r
dr
2r
2r + dr
7
7d
14
7d + 14
2
dr + 7d
2r + 14
d
More generally, (d+2) (r+7) =
d
2
r
dr
2r
7
7d
14
37
x 25
600
140
150
35
925
dr + 2r + 7d + 14