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,

How the ideas and language of algebra K-5

set the stage for Algebra 8-12

MSRI, May 15, 2008MSRI, May 15, 2008

E. Paul GoldenbergE. Paul Goldenberg

To save note-taking, To save note-taking, http://thinkmath.edc.orghttp://thinkmath.edc.org

Click Click download presentationsdownload presentations link link(next week)(next week)

Language vs. computational toolTo us, expressions like ((nn – – dd)()(nn + + dd)) 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 use algebraic notation the first two ways, as a computational tool, most most cancan use it the last two ways, as use it the last two ways, as language.language.

((nn – – dd)()(nn + + dd) = ) = nn2 2 – – dd22

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 rearrangeabilityw/o rearrangeability 3 + 5 = 8 3 + 5 = 8 can’t make sensecan’t make sense NourishmentNourishment to to extend/apply/refineextend/apply/refine built-ins built-ins

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

breaking arrays; describing whole & parts breaking 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 “learned” in math class.developmental, not “learned” in math class.

Developmental

Algebraic language, like any language, is

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

explicit definitions and lessons, but from explicit definitions and lessons, but from language used in contextlanguage used in context

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

Convention

Demand “does it work with kids?”

Algebraic language & algebraic thinking

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

A number trickA number trick ““Pattern indicators” Pattern indicators” Difference of squaresDifference of squares

Systems of equations in kindergarten?Systems of equations in kindergarten? Understanding two dimensional informationUnderstanding two dimensional information

Linguistics and mathematicsMichelle’s strategy for 24 Michelle’s strategy for 24 –– 8: 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 is 6, 4 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 linguistic idea (mostly)

Algebraic ideas

(breaking it up)(breaking it up)

Arithmetic knowledge

What is the “linguistic” idea?

28 28 –– 8 on her fingers… 8 on her fingers…

Fingers are Fingers are counterscounters,,good for good for grasping thegrasping the idea idea, , and and good (initially) for good (initially) for finding or verifying finding or verifying answers to problems answers to problems like 28 like 28 –– 4, 4, but…but…





Algebra as abbreviated speech (Algebra as a second Language)


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 number Subtract the number

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

How did it work?



How did it work?



How did it work?



How did it work?



How did it work?



How did it work?



How did it work?



How did it work?



Kids need to do it themselves…

Using notation: following steps

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 notation: undoing steps


510168 7 3 20

Dana

Cory

Sandy

Chris

Words

14

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

Pictures

Using notation: simplifying steps


510168 7 3 20

Dana

Cory

Sandy

Chris

Words Pictures

4

Abbreviated speech: simplifying pictures


510168 7 3 20

Dana

Cory

Sandy

Chris

Words Pictures

4 b2b2b +

6b +

3

Notation is powerful!

Computational practice, but Computational practice, but muchmuch more more Notation helps them Notation helps them understandunderstand the trick. the trick. Notation helps them Notation helps them inventinvent new tricks. new tricks. Notation helps them Notation helps them undoundo the trick. the trick. Algebra is a Algebra is a favorfavor, not just “another thing , not just “another thing

to learn.”to learn.”


A number trickA number trick ““Pattern indicators”Pattern indicators” Difference of squaresDifference of squares

Children are language learners…

They They areare pattern-finders, abstracters… pattern-finders, abstracters… ……naturalnatural sponges for language sponges for language in contextin context..

n 10n – 8 2

8

0

28

20

18 17

3 4

58 57

Go to index


A number trickA number trick ““Pattern indicators”Pattern indicators” Difference of squaresDifference of squares

Is there anything less sexy than Is there anything less sexy than memorizing multiplication facts? memorizing multiplication facts?

What helps people memorize? What helps people memorize? Something memorable!Something 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

Shhh… Students thinking!Shhh… Students thinking!

Take it a step further

What about What about twotwo steps out? steps out?

Shhh… Students thinking!Shhh… Students thinking!

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


What about What about threethree steps out? steps out?

What about What about fourfour??

What about What about fivefive??

1200

31 32 33

34

353029 403936 37 38

1225


What about What about twotwo steps out? steps out?

1221

31 32 33

34

353029 403936 37 38

1225

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

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

To do…To do… 5353

4747

I think…I 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 But nobody caresnobody cares if kids can if kids can multiply 47 multiply 47 53 mentally! 53 mentally!

What What do do we care about, then? 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 thinking 3 Keeping 2500 in mind while thinking 3 3 3 Subtracting 2500 – 9Subtracting 2500 – 9 Finding the patternFinding the pattern DescribingDescribing the pattern the pattern

Algebraic language

Algebraic/arithmeticthinkingScience

(7 – 3) (7 + 3) = 7 7 – 9

n – 3

n + 3

n

((nn – 3– 3) ) ( (nn + 3+ 3) = ) = nn nn –– 99((nn – 3– 3) ) ( (nn + 3+ 3))

Q?

Nicolina Malara, Italy: “algebraic babble”

(50 – 3) (50 + 3) = 50 50 – 9

Make a table; use pattern indicator.

2 4

4 16

5 25

Distance awayWhat to subtract

1 1

3 9

dd dd dd

((nn – – dd) ) ( (nn + + dd) = ) = nn nn

––((nn – – dd) ) ( (nn + + dd) = ) = nn nn –– dd dd

(7 – d) (7 + d) = 7 7 – d d

n – d

n + d

n

((nn – – dd) ) ( (nn + + dd))((nn – – dd))

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 it’s Gives practice. Helps me memorize, because it’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


5 4

Removing a column leaves

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


6 4

Replacing as a row leaves

with one left over.



6 4

Removing the leftover leavesshowing that it is one less than

5 5.






Systems of equations

Challenge: can Challenge: can you find some you find some

that don’t work?that don’t work?

in Kindergarten?!

5x + 3y = 23

2x + 3y = 11

Is there anything interesting about Is there anything interesting about addition 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 beginningsBack to the very beginnings

Picture a young child with Picture a young child with a small pile of buttons.a small pile of buttons.

Natural to sort.Natural to sort.

We help children refine We help children refine and extend what is already and extend what is already natural.natural.

6

4

7 3 10

Back to the very beginningsBack to the very beginnings

Children can also summarize.Children can also summarize.

““Data” from the buttons.Data” from the buttons.

blue gray

large

small

large

small

blue gray

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

AbstractionAbstraction

6

4

7 3 10

6

4

7 3 10

4 2

3 1

A Cross Number PuzzleA Cross Number Puzzle

5

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

21

8

13

912

7 6

3

Relating addition and subtraction

6

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





Two-dimensional informationThink of a number.

Double it.

Add 6.

51016

Dana CoryWords

4814

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, How many two-letter words can you make, starting with a red letterstarting with a red letter and and ending with a purple letterending with a purple letter??

a i s n t

Multiplication, coordinates, phonics?

a i s n t

asin

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 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 GoldenbergE. Paul Goldenberg http://thinkmath.edc.org/http://thinkmath.edc.org/

Quick recover

The idea of a word problem…

An attempt at An attempt at realityrealityA A situationsituation rather than a “naked” calculation rather than a “naked” 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 ELAState ELA tests test ELA

• • State Math tests test MathState Math tests test Math

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. In tests it is discriminatory!

and ELAand ELA

Attempts to be efficient (spare)

Stereotyped wordingStereotyped wordingkey wordskey words Stereotyped structureStereotyped structureautopilot 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 with irrelevant numbers, just to confuse kids.

Many numbers: +

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

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”

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 by creating 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 do? What I do? What cancan I figure out? I figure out?

Representing 22 17

22

17

Representing the algorithm

20

10

2

7


20

10

2

7

200

140

20

14


20

10

2

7

200

140

20

14

220

154

37434340


20

10

2

7

200

140

20

14

220

154

37434340

2217154220374

x1


20

10

2

7

200

140

20

14

220

154

37434340

172234340374

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

3725600

35925

x

140

Documents

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,