What do Attention, Memory, and Language Learning have...

What do Attention, Memory, andLanguage Learning have to do

with Problem Solving?MassMATEMassMATE,, June 11, 2008June 11, 2008

E. Paul GoldenbergE. Paul Goldenberg

To save note-taking,http://thinkmath.edc.orgClick download presentations link

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(next week)

Ideas built into

a comprehensive K-5 curriculum from

First, what is problem solving? If I had a 10' pole and you had a 20' pole… Problems vs. Exercises Real problems don’t ask what chapter you’re in. Real problems matter. What’s real to a child? Curiosity Alejandro

Puzzles Learning! Word problems?

Keeping things in one’s head

1

2

3

4

875

6

Andrew

Algebra for little ones

IdeasIdeas LanguageLanguage

Surabhi and Devon

Language vs. computational toolTo us, expressions like (n – 1)(n + 1) can be manipulated• to derive things we don’t yet know, or• to prove things that we conjectured from experiment.

Claim: While most elementary school children cannot usealgebraic notation the first two ways, as a computational tool,most most cancan use it the last two ways, as language use it the last two ways, as language..

(n – 1)(n + 1) = n2 – 1

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.,

Great built-in apparatus

Abstraction (categories, words, pictures)Abstraction (categories, words, pictures) Syntax, structure, sensitivity to orderSyntax, structure, sensitivity to order Phenomenal language-learning abilityPhenomenal language-learning ability Quantification (limited, but there)Quantification (limited, but there) Logic (evolving, but there)Logic (evolving, but there) Theory-making about the world Theory-making about the world irrelevance of orientationirrelevance of orientation

In learning math, little differentiationIn learning math, little differentiation

Some algebraic ideas precede arithmetic

w/o w/o rearrangeabilityrearrangeability 3 + 5 = 8 3 + 5 = 8 cancan’’t make senset make sense NourishmentNourishment to to extend/extend/apply/refineapply/refine built-ins built-ins

breaking numbers and rearranging parts breaking numbers and rearranging parts(any-order-any-grouping, (any-order-any-grouping, commutativity/associativitycommutativity/associativity)),,

breaking arrays; describing whole & partsbreaking arrays; describing whole & parts(linearity, distributive property)(linearity, distributive property)

But many of the basic intuitions are built in,But many of the basic intuitions are built in,developmental, not developmental, not ““learnedlearned”” in in math class.math class.

Developmental

Algebraic language, like any language, is

Children are phenomenal language-learners Build it from language spoken around them Infer meaning and structure from use: not

explicit definitions and lessons, but fromlanguage used in context

Where “math is spoken at home” (not drill,lessons, but conversation that makes salientlogical puzzle, quantity, etc.) kids learn it

Convention

Algebraic language & algebraic thinking

Linguistics and mathematics Algebra as abbreviated speech (Algebra as a Second Language)

A number trick“Pattern indicators”Difference of squares

Systems of equations in kindergarten? Understanding two dimensional information

Linguistics and mathematicsMichelle’s strategy for 24 – 8:

Well, 24 Well, 24 –– 44 is easy! is easy! Now, 20 minus Now, 20 minus anotheranother 4 4…… Well, I know Well, I know 1010 –– 4 4 is 6,is 6,

and 20 is 10 + 10,and 20 is 10 + 10,so, so, 2020 –– 4 is 16. 4 is 16.

So, 24 So, 24 –– 8 = 16 8 = 16..

A linguisticidea (mostly)

Algebraic ideas(breaking it up)

Arithmeticknowledge

What is the “linguistic” idea?

28 – 8 on her fingers…Fingers are counters,

good for grasping the idea, andgood (initially) for finding or verifying answers to problems like 28 – 4, but…

Thi

Algebraic language & algebraic thinking

Linguistics and mathematics Algebra as abbreviated speech (Algebra as a Second Language)

A number trick“Pattern indicators”Difference of squares

Systems of equations in kindergarten? Understanding two dimensional information

Algebra as abbreviated speech (Algebra as a second Language)

A number trick “Pattern indicators” Difference of squares

Surprise! You speak algebra!Surprise! You speak algebra!

5th grade5th grade

A number trick

ThinkThink of a number. of a number. Add 3.Add 3. Double the result.Double the result. Subtract 4.Subtract 4. Divide the result by 2.Divide the result by 2. Subtract the numberSubtract the number

you first thought of.you first thought of. Your answer is 1!Your answer is 1!

http:http://thinkmath//thinkmath..edcedc.org/.org/

How did it work?

ThinkThink of a number. of a number. Add 3.Add 3. Double the result.Double the result. Subtract 4.Subtract 4. Divide the result by 2.Divide the result by 2. Subtract the numberSubtract the number

you first thought of.you first thought of. Your answer is 1!Your answer is 1!

How did it work?

ThinkThink of a number. of a number. Add 3.Add 3. Double the result.Double the result. Subtract 4.Subtract 4. Divide the result by 2.Divide the result by 2. Subtract the numberSubtract the number

you first thought of.you first thought of. Your answer is 1!Your answer is 1!

How did it work?

ThinkThink of a number. of a number. Add 3.Add 3. Double the result.Double the result. Subtract 4.Subtract 4. Divide the result by 2.Divide the result by 2. Subtract the numberSubtract the number

you first thought of.you first thought of. Your answer is 1!Your answer is 1!

How did it work?

ThinkThink of a number. of a number. Add 3.Add 3. Double the result.Double the result. Subtract 4.Subtract 4. Divide the result by 2.Divide the result by 2. Subtract the numberSubtract the number

you first thought of.you first thought of. Your answer is 1!Your answer is 1!

How did it work?

ThinkThink of a number. of a number. Add 3.Add 3. Double the result.Double the result. Subtract 4.Subtract 4. Divide the result by 2.Divide the result by 2. Subtract the numberSubtract the number

you first thought of.you first thought of. Your answer is 1!Your answer is 1!

How did it work?

ThinkThink of a number. of a number. Add 3.Add 3. Double the result.Double the result. Subtract 4.Subtract 4. Divide the result by 2.Divide the result by 2. Subtract the numberSubtract the number

you first thought of.you first thought of. Your answer is 1!Your answer is 1!

How did it work?

ThinkThink of a number. of a number. Add 3.Add 3. Double the result.Double the result. Subtract 4.Subtract 4. Divide the result by 2.Divide the result by 2. Subtract the numberSubtract the number

you first thought of.you first thought of. Your answer is 1!Your answer is 1!

How did it work?

ThinkThink of a number. of a number. Add 3.Add 3. Double the result.Double the result. Subtract 4.Subtract 4. Divide the result by 2.Divide the result by 2. Subtract the numberSubtract the number

you first thought of.you first thought of. Your answer is 1!Your answer is 1!

Kids need to do it themselves…

Using notation: following steps

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

510168 7 3 20

Dana

Cory Sandy

ChrisWords Pictures

Using notation: undoing steps

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

510168 7 3 20

Dana

Cory Sandy

ChrisWords

14

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

Pictures

Using notation: simplifying steps

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

510168 7 3 20

Dana

Cory Sandy

ChrisWords Pictures4

Abbreviated speech: simplifying pictures

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

510168 7 3 20

Dana

Cory Sandy

ChrisWords Pictures4 b

2b2b + 6b + 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 (Algebra as a second Language)

A number trick “Pattern indicators” Difference of squares

Children are language learners…

They are pattern-finders, abstracters… …natural sponges for language in context.

n 10n – 8 2

80

2820

18 173 4

58 57

See more onhttp://thinkmath.edc.org

Puzzles: What can I do?

I. I. I am even.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. II. All of my digits < 5All of my digits < 5III. h + t + u = 9

IV. I am less than 400.

V. Exactly two of my digits are the same.

432342234324144414

1 4 4

Algebra as abbreviated speech (Algebra as a second Language)

A number trick “Pattern indicators” Difference of squares

Is there anything less sexy thanIs there anything less sexy thanmemorizing multiplication facts?memorizing multiplication facts?

What What helpshelps people memorize? people memorize?SomethingSomething memorable!memorable!

4th grade4th grade

Math could be fascinating!Math could be fascinating!

Teaching without talking

Wow! Will it always work? Big numbers?Wow! Will it always work? Big numbers??

38 39 40 41 42

3536

6 7 8 9 105432 11 12 13

8081

18 19 20 21 22… …

??

1600

1516

ShhhShhh…… Students thinking! Students thinking!

Take it a step further

What about What about twotwo steps out? steps out?

ShhhShhh…… Students thinking! Students thinking!

Again?! Always?Again?! Always? Find some bigger examples.Find some bigger examples.

Teaching without talking

1216

6 7 8 9 105432 11 12 13

6064

?

58 59 60 61 6228 29 30 31 32… …

???

Take it even further

What about What about threethree steps out? steps out?What about What about fourfour??What about What about fivefive??

100

6 7 8 9 1054 151411 12 13

75

Take it even further

What about three steps out?What about four?What about What about fivefive??

1200

31 32 33 34 353029 403936 37 38

1225

Take it even further

What about What about twotwo steps out? steps out?

1221

31 32 33 34 353029 403936 37 38

1225

““OK, um, 53OK, um, 53”” ““Hmm, wellHmm, well……

……OK, IOK, I’’ll pick 47, and I can multiply thosell pick 47, and I can multiply thosenumbers faster than you can!numbers faster than you can!””

To doTo do…… 53 53×× 4747

I thinkI think……5050 ×× 5050 (well, 5 (well, 5 ×× 5 and 5 and ……))…… 25002500Minus 3 Minus 3 ×× 3 3 –– 9 9

24912491

“Mommy! Give me a 2-digit number!”2500

47 48 49 50 51 52 53

about 50

But nobody cares if kids canmultiply 47 × 53 mentally!

What do we care about, then?

50 50 ×× 50 (well, 5 50 (well, 5 ×× 5 and place value) 5 and place value) Keeping 2500 in mind while thinkingKeeping 2500 in mind while thinking 3 3 ×× 3 3 Subtracting 2500Subtracting 2500 –– 9 9 FindingFinding the patternthe pattern DescribingDescribing the pattern the pattern

Algebraic language

Algebraic/arithmeticthinkingScience

(7 – 3) × (7 + 3) = 7 × 7 – 9

n – 3 n + 3

n

(n – 3) × (n + 3) = n × n – 9(n – 3) × (n + 3)

Q?

Nicolina Malara, Italy: “algebraic babble”

(50 – 3) × (50 + 3) = 50 × 50 – 9

Make a table; use pattern indicator.

2 4

4 165 25

Distance away What to subtract1 1

3 9

dd dd ×× dd

(n – d) × (n + d) = n × n –(n – d) × (n + d) = n × n – d × d

(7 – d) × (7 + d) = 7 × 7 – d × d

n – d n + d

n

(n – d) × (n + d)(n – d)

We also care about thinking!

Kids feel smart!Kids feel smart!Why silent teaching?Why silent teaching?

Teachers feel smart!Teachers feel smart! Practice.Practice.

Gives practice. Helps me memorize, because itGives practice. Helps me memorize, because it’’s s memorablememorable!!

Something new.Something new.Foreshadows algebra. In fact, kids record it Foreshadows algebra. In fact, kids record it withwith algebraic language! algebraic language!

And something to wonder about:And something to wonder about: How does it work?How does it work?

It matters!It matters!

One way to look at it

5 × 5

One way to look at it

5 × 4

Removing acolumn leaves

Not “concrete vs. abstract”semantic (spatial) vs. syntacticKids don’t derive/prove with algebra.

One way to look at it

6 × 4

Replacing as arow leaves

with one leftover.

Not “concrete vs. abstract”semantic (spatial) vs. syntacticKids don’t derive/prove with algebra.

One way to look at it

6 × 4

Removing theleftover leaves

showing that itis one less than

5 × 5.

Not “concrete vs. abstract”semantic (spatial) vs. syntacticKids don’t derive/prove with algebra.

Algebraic language & algebraic thinking

Linguistics and mathematics Algebra as abbreviated speech (Algebra as a Second Language)

A number trick“Pattern indicators”Difference of squares

Systems of equations in kindergarten? Understanding two dimensional information

Systems of equations

Challenge: canChallenge: canyou find someyou find some

that donthat don’’t work?t work?

in Kindergarten?!

5x + 3y = 23

2x + 3y = 11

Is there anything interesting aboutIs there anything interesting aboutaddition and subtraction sentences?addition and subtraction sentences?

Start with 2nd gradeStart with 2nd grade

Math could be spark curiosity!Math could be spark curiosity!

4 + 2 = 6

3 + 1 = 4

10+ =7 3

Back to the very beginnings

Picture a young child withPicture a young child witha small pile of buttons.a small pile of buttons.

Natural to sort.Natural to sort.

We help children refineWe help children refineand extend what is alreadyand extend what is alreadynatural.natural.

6

4

7 3 10

Back to the very beginnings

Children can also summarize.Children can also summarize.

““DataData”” from the buttons. from the buttons.

blue gray

large

small

large

small

blue gray

If we substitute numbers for the original objectsIf we substitute numbers for the original objects……

Abstraction

6

4

7 3 10

6

4

7 3 10

4 2

3 1

A Puzzle: Problem Solving

5

DonDon’’t always start with the question!t always start with the question!

21

8

13

912

7 6

3

Relating addition and subtraction6

4

7 3 10

4 2

3 16

4

7 3 10

4 2

3 1

Ultimately, building the addition and subtraction algorithms

The algebra connection: adding

4 2

3 1

10

4

6

37

4 + 2 = 6

3 + 1 = 4

10+ =7 3

The algebra connection: subtracting

7 3

3 1

6

4

10

24

7 + 3 = 10

3 + 1 = 4

6+ =4 2

The eighth-grade look

5x 3y

2x 3y 11

23 5x + 3y = 23

2x + 3y = 11

12+ =3x 0x = 4

3x 0 12

Algebraic language & algebraic thinking

Linguistics and mathematics Algebra as abbreviated speech (Algebra as a Second Language)

A number trick“Pattern indicators”Difference of squares

Systems of equations in kindergarten? Understanding two dimensional information

Two-dimensional readingThink of a number.Double it.Add 6.

51016

Dana CoryWords48

14

Pictures

Naming intersections, first gradePut a red house at the intersection of A street and N avenue.

Where is the green house?

How do we go fromthe green house tothe school?

Combinatorics, beginning of 2nd

How many two-letter words can you make,starting with a red letter andending with a purple letter?

a i s n t

Multiplication, coordinates, phonics?

a i s n t

as in

at

Multiplication, coordinates, phonics?

w s ill

it ink

b p

st ick

ack

ing

br tr

Similar questions, similar imageFour skirts and three shirts: how many outfits?

Five flavors of ice cream and fourtoppings: how many sundaes?(one scoop, one topping)

With four different bottom blocksand three different top blocks,how many 2-block towers canyou make?

The idea of a word problem…

An attempt at An attempt at realityrealityA A situationsituation rather than a rather than a ““nakednaked”” calculation calculation

→TheThe goalgoal isis the the problemproblem,, notnot thethe wordswords•• Necessarily bizarre dialect: Necessarily bizarre dialect: low redundancy or low redundancy or veryvery wordy wordy

→The goal is the problem, not the words• Necessarily bizarre dialect: low redundancy or very wordy•• State ELA tests test ELA State ELA tests test ELA•• State Math tests test MathState Math tests test Math

The idea of a word problem…

An attempt at realityA situation rather than a “naked” calculation

“Clothing the naked” with words makes itlinguistically hard without improving themathematics.In tests it is discriminatory!

and ELAand ELA

Attempts to be efficient (spare)

Stereotyped wordingStereotyped wording→key wordskey words Stereotyped structureStereotyped structure→autopilot strategies

Key words

Ben and his sister were eating pretzels.Ben left 7 of his pretzels.His sister left 4 of hers.How many pretzels were left?

We rail against key word strategies.

So writers do cartwheels to subvert them.

But, frankly, it is smart to look for clues!This is how language works!

Autopilot strategiesWe make fun of thought-free “strategies.”

Writers create bizarre wordings withirrelevant numbers, just to confuse kids.

Many numbers: +Two numbers close together: – or ×Two numbers, one large, one small: ÷

But, if the goal is mathematics andto teach children to think andcommunicate clearly……deliberately perverting our wording to makeit unclear is not a good model!

So what can we do to help students learn toread and interpret story-based problemscorrectly?

“Headline Stories”

Ben and his sister were eating pretzels.Ben left 7 of his pretzels.His sister left 4 of hers.

Less is more!

What questions can we ask?

Children learn the anatomy of problems bycreating them. (Neonatal problem posing!)

“Headline Stories”

Do it yourself!Do it yourself!Use any word problem you like.Use any word problem you like.

What What cancan I I do? What do? What cancan I figure out? I figure out?

Thank you!

E. Paul GoldenbergE. Paul Goldenberghttp:http://thinkmath//thinkmath..edcedc.org/.org/

Representing 22 × 1722

17

Representing the algorithm20

10

2

7

Representing the algorithm20

10

2

7

200

140

20

14

Representing the algorithm20

10

2

7

200

140

20

14

220

154

37434340

Representing the algorithm20

10

2

7

200

140

20

14

220

154

37434340

2217

154220374

x1

Representing the algorithm20

10

2

7

200

140

20

14

220

154

37434340

172234

340374

x1

More generally, (d+2) (r+7) =d

r

2

7

dr

7d

2r

14

2r +

dr

7d +

14

2r +

14

dr + 7d

More generally, (d+2) (r+7) =d

r

2

7

dr

7d

2r

14

dr + 2r + 7d + 14

150

3725

600

35925

x

140