Algebra and thinking:
what babies and little kids tell us about algebra
Kaput Center, February 9, 2011 — E. Paul Goldenberg
Education Development Center
© 2011 E. Paul Goldenberg
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Brain science’s influence on
standards
curriculum
assessment
teaching
Brain science’s influence on
standards
curriculum
assessment
teaching
Babies know things at birth!
How can we know that?
 Well, we ask them!
 Same or different? Which do you prefer?


HOW DO THEY
ANSWER?!
What do they know?
Mommy’s voice
 Facial expressions
 Which expression fits a voice
 How to match a face
 Foreign languages
 Perfect pitch
 Math… (to be continued)

And they think about things!
And they remember!
 And they communicate.
 And they perform experiments!

The miracle of communication
We start out not speaking at all.
 By about 1, some words.
 By about 5 or 6, a full half of
our adult
vocabulary!
And virtually
all of our
grammar!
Math and musical ears
Our use of pitch.
We all have perfectly working equipment.
 Only the training is different!
 Equally adept at learning our language.
 No reason to believe we aren’t
equally adept at
learning mathematics.

Babies already know some math!
They recognize symmetry
 They know quantity
 What about probability?
 We all have the “math gene”!

What’s this have to do w/ algebra?
Knowing what children already know, how
they learn, and how much they can learn lets
us be appropriately ambitious!
 We can design to take advantage of the way
children think and learn…
 …and to enhance, extend, refine what they
do naturally.

(distributed learning, use of language)
Enough generalities! Examples!
Distributive property…
…of multiplication over addition.
 But built into cognition
before children multiply!

A number trick--a multi-step arithmetic process







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!
4th grade
How did it work?

Think of a number.
How did it work? -- building a symbolic system

Think of a number.
How did it work?
Think of a number.
 Add 3.

How did it work?
Think of a number.
 Add 3.
 Double the result.

How did it work?
Think of a number.
 Add 3.
 Double the result.
 Subtract 4.

How did it work?
Think of a number.
 Add 3.
 Double the result.
 Subtract 4.
 Divide the result by 2.

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?
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 Imani Cory Amy Chris
5
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 Imani Cory Amy Chris
5
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 Imani Cory Amy Chris
5
10
16
8
4
7
3
20
Abbreviated speech: simplifying pictures
Words
Think of a number.
Double it.
Add 6.
Divide by 2.
What did you get?
Pictures Imani Cory Amy Chris
5
10
16
8
4
7
3
b
2b
2b + 6
20 b + 3
Why a number trick? Why bags?
Computational practice, but much more
 Notation helps them understand the trick.

invent new tricks.

undo the trick.
 But most important, the idea that
notation/representation is powerful!
Algebra Is Downright Useful

Children are language learners…
a story from 2nd grade
They are pattern-finders, abstracters…
 …natural sponges for language in context.

n
10
8
n–8
2
0

28 18 17 11 12 58 57
20 10 9 3 4 50 49
Important algebraic idea: 10 – 8, vs. n – 8
Language in context
A game in grade 3
3rd grade detectives!
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
432
342
234
324
144
414
Distilling algebra: three fractions

Describing what we know — learnable early

Notation is never “obvious” or discoverable —
needs “native speaker” and use in context
Deducing what we don’t know — comes later
 No need to be taught as a package deal.


Properties: Distributive property already
there.
Addition and Subtraction:
From buttons to algebra
Challenge: can
you find some
that don’t work?
4+5=9
5x + 3y = 23
3+1=4
2x + 3y = 11
5
17 + 46 = 13
Math could be spark curiosity!
 Is there anything interesting about
addition and subtraction sentences?
 Start with 2nd grade
Children classify things

Words are a kind of sorting! And babies
learn these words from the most chaotic data.
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
Detour: How’d they learn the facts?
One hand, plus
 All about 10
 Making it (How far from 10?)
 Adding, subtracting it
 Remembering to use it!
 Using it
 Adjusting with it (+ 8)
 Four-hand addition

A video from
nd
2
grade: mental math
The addition algorithm
Only multiples of 10 in yellow. Only less than 10 in blue.
20
5
25
30
8
38
50
13 63
25 + 38 = 63
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 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
All from buttons!
Mental math vs. visual vs. print
Different!
 The pairs to 10, 20, 30
were mental (probably visual)
 The “negative a million”
was visual (partly also linguistic)
 “two fifths + three fifths”
is linguistic
 2/5 + 3/5 is print

Vision…

Can babies focus? Yes and no… Especially yes.
Vision…
Can they focus? Yes and no… Especially yes.
 Seeing is built in, but
they also “learn” about
seeing.
 Totally different image
on the retina, but baby
quickly learns to
recognize objects in
any position.

Vision…
So these are all the same object in different
positions:
 We don’t start out knowing that, but learn
it.

Vision and reading…
So these are all the same object in different
positions:
p d qb
 We don’t start out knowing that, but learn
it.
 Then, in order to read, we must unlearn it!
 It’s not a defect that children reverse letters!
was ≠ saw

Reading requires special ways of
seeing
Orientation matters: d ≠ b
 Order matters: was ≠ saw
 (In English) We read left to right.
Dog bites boy ≠ Boy bites dog.
 We need to monitor our seeing!
EXECUTIVE FUNCTION

Math-reading uses different visual
conventions
Words
Pictures Dana
Think of a number.
5
Double it.
10
Add 6.
16
Cory
4
8
14
32 ≠ 32
3+4×2
9 + 6 = __ + 5
3(1/3 + 1/6) 1 + g (x + 1)
Actually even regular text reading
isn’t strictly left to right.
Aoccdrnig to a rscheearch at Cmabrigde
Uinervtisy, it deosn't mttaer in waht oredr the
ltteers in a wrod are, the olny iprmoetnt tihng
is taht the frist and lsat ltteer be at the rghit
pclae. The rset can be a toatl mses and you
can sitll raed it wouthit porbelm. Tihs is
bcuseae the huamn mnid deos not raed ervey
lteter by istlef, but the wrod as a wlohe.
Conjecture…

Because we don’t need to attend to the
insides of words in order to be able to read,
some of kids’ spelling difficulties arises
from not having activities
in which they do need to attend to the
insides of words.
Reading different, language same
Language and mathematics
Michelle’s strategy for 24 – 7:
(breaking it up)
Algebraic ideas
A linguistic
idea (mostly)
Well, 24 – 4 is easy!
 Now, 20 minus another 3…
Arithmetic
 Well, I know 10 – 3 is 7,
and 20 is 10 + 10,
knowledge
so, 20 – 3 is 17.
 So, 24 – 7 = 17.

“Twenty four” as a name
What are the “linguistic” ideas?
24 – 7 on her fingers…
Fingers are counters,
for grasping the idea,
and
(initially) for finding
or verifying answers
to problems like 24 – 7…
but then let language take over!
“Five hundred” and “five sevenths” as names
Five hundred plus two hundred
Our language makes
this “obvious” to
children.
 Ask first graders
about five ninths plus
two ninths.

“Conservation” of number

Commutative and associative properties:
What WW Sawyer called the “any order
any grouping” property
“Conservation” of number

A 4-year-old may see “more” here

than here,
even after counting correctly.
 While that’s true, 2 + 3 = 5 can’t make sense.
When quantity is stable…

They already know that
=
but they also know that print is weird!
2+5 ≠ 5+2
 was ≠ saw so maybe
 They have to unlearn again! But only about print! Not math.

Why puzzles?
Preschoolers’ play — the lever toy
 Because we’re not cats!
 Puzzles grab
attention!
 Puzzles give
permission to think!

“Executive function”
Monitoring one’s thinking
 Extending one’s short term memory
 Picturing and manipulating ideas and objects
in one’s head
 Counting what we can’t see!

(how many fingers don’t you see, “spilled ink,” four hand addition)

What can I do? vs. Whatamispoztado?!
Distributive property, half of 48
Monitoring one’s thinking
 Extending one’s short term memory

Associative property, 5 × 42
Monitoring one’s thinking
 Extending one’s short term memory

Go forth and teach! 
What’s that got to do w/ algebra?

Left as an exercise for the reader.
Or research! 
Attention and silence


“
If it’s
quiet, it
can sneak
up on me.
I’d better
watch
closely!
When we say preschoolers can’t pay attention, we really mean that
They can’t not pay attention.”
Putting things in one’s head
8
6
2
5
7
1
3
4
2nd grade