A EARLY LGEB RA WI TH GRAPH ICS SOFTWARE
Se rg ei Ab ra m ov ic h
Sta te Un i v e r si t y of
Ne w Yor k a t P o t s da m , USA
a b ram o v s @po t s da m .e d u
The tea c h e r m u s t o ri ent h is w o rk n ot o n y e s te r d a y s d e ve l opm e nt i n the c h il d b u t o n to m o rr o w T s. L.S . Vy got sk y
de v e lo p in g a l g eb ra c u r r ic ulu m f or y oun g c h ild re n
A l ea r n in g en s t r u ct u r ed b y f ro m pa r tic ul s yn ta c tic a l l y a c tio n.
vir on m en t s hou l d b e th e du a l ity o f m ovi ng ar to g ene ra l and g u ided m a th e m at i ca l
Kap ut (19 9 5 ): Tne it he r the f or m a l is m s n o r t h e a c ti on s o n t h e m c an be vi a b l y l e a rn e d wi th o u t so me s em a n t i c st a rti n g p oi n t w he r e th e f or m a l is m s ar e in iti a l l y tak en to r e pr es e n t s o m et hin g in t h e s t u d en t es e xp e r i e nc e e .
Questions to be answered (see Report from PME27)
How c a n o n e m a k e m e a n i n g t hr o ug h the ac t of sy mb ol iz in g ?
T
How c a n fo r m al n o t a t ion a n d a cti ons o n it b e b or n f r o m la n g u ag e a n d g u id e d b y se m a n t ic s u se d ou t s id e o f m at h e m a tic s?
Pr o b le m 1 ( r e co mmende
d f or gra
de 2 ).
A pe t s to re s ol d o n ly bi r ds an d ca ts. One d ay t he s tor es o w n er a s k ed h is c le r k to co un t h ow m a n y a n i m a ls t he r e we r e in
t he s to re. The c le rk c ou n t ed 18 le g s. Ho w m a n y cat s a nd bee n? b ir ds mi ght t h ere hav e
Pr o b le m 2 ( r e comm
e nd e d fo r mi d dle sc h o o l ).
Ju lie
s o l d 125
f roz en ju i ce ba rs
a n d 150
ic e c re am c on es o n Sat u r d ay . She m ad e a tot a l o f $500 . Ju lie s ol d e ac h ic e cr e a m
co ne fo r $2 .25 . W r i te an e q u at io n yo u can use to f in d t h e co st of e ac h f r oz en j u ic e
b ar . So l v e t he e qua t io n yo u w r ot e to f in d t he co s t o f on e f ro ze n jui c e ba r.
Pr o b le m 3 ( r e c o mm en d e d
f o r h i g h s c h oo l ).
M ary
pu r c ha sed
12
pe n s
a nd
14
n ot eb ook s fo r $20 . Car l o s b oug h t 7 pens an d 4 n ot e boo ks fo r $7 .50 . Fin d t he p r ic e of o ne pen an d t he p r i c e of o ne no te boo k alg eb ra ica ll y .
Re cu r ring
m ath em a t ical s t r uc t ur e acro s s grad es
P ro b l em 2 .1 (Ad a p t e d f o r gr a d e 2) .
Ju l i e,
a pet
s t o r e es o w n e r , s o l d o n Sat u r d ay ;
t w o t y p es o f an im als
t w o of t h e m w e r e c a t s a n d t h r e e o f t h e m w e r e so m e o t h e r t y p e o f a n i m a l . Sh e c o u n t e d 2 3 l e g s. Ho w m an y l e g s di d t h e o t h e r h av e? a n im a l
P ro b l em 3 .1 (Ad a p t e d f o r gr a d e 2) .
M ar y , a p e t s t or e c le r k , s o d l fi v e t r im p s a n d tw o g r i m ps w hi c h h av e t he t o t al of 18 l e gs . Ca r los , t r im p s
an ot h er
c l e r k , s o l d fo ur
a n d t hr e e g r i m ps w hi c h h av e t h e t ot a l o f 20 l e gs . Ho w m an y l e g s
d o e s a t r im p h av e an d ho w m a ny l e gs d o e s a g r i m p h a v e ?
Con te x t u a l an d nume ric a l c oh er enc y o f mu lti p l i cat i v e s tr uc t u r e s
?
4
?
2
18
2nd grade appropriate
3
?
2
4
23
common structure
5
?
2
?
18
4
?
3
?
20
So l v ing i n d e te rm in at e l in e ar e qu a t io n in t wo u nk n o w ns
QuickTimeOE and a Cinepak decompressor are needed to see this picture.
Rud im e nts o f ad v a nc ed m a th e ma ti c a l t hin ki n g
Fi nd i ng o t h e r l a t t ic e p o in ts
QuickTime and a Cinepak decompressor are needed to see this picture.
Ho w d o we c ou n t by f o u rs ? e
e By a
s ki ng
a t ha t
qu e s ti on , he h as ,
t he in
ch il d f act ,
ind ic at e s
f or mul at ed a p la n to s ol v e t he ta s k be f o r e hi m , b u t i s un a b l e to p e r f or m a l l t he nec e s s a ry op e r a tio ns e . L.S. Vy got s ky .
Th ar p & Ga l li m or e:
t e a ch in g is a s s ist ed
p e r f or m a nc e a t
t h o se p oi n t s o f t h e z on e o f p r oxi m a l de v e lo pm e n t r e q u ir e d . w he r e as s is ta n c e is
So l v ing a sy s t e m o f e q ua ti ons as a p a t it i o n ing b ra in t e a se r
QuickTimeS and a Cinepak decompressor are needed to see this picture.
Mov in g fr o m ic on ic t o sy mb ol ic r e pr e se n t a t io n
QuickTime and a Cinepak decompressor are needed to see this picture.
De v e l o p ing...