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Pattern ~~~~~~~~~~~~~~~~~~~~ LYNN ARTHUR STEEN "He just saw further than the rest of us." The subject of this remark, cyberneticist Norbert Wiener, is one of many exceptional scientists who broke the bonds of tradition to create entirely new domains for math- ematicians to explore. Seeing and revealing hidden patterns are what mathematicians do best. Each major discovery opens new areas rich with potential for further exploration. In the last century alone, the number of mathematical disciplines has grown at an exponential rate; examples include the ideas of Georg Cantor on transfinite sets, Sonja Kovalevsky on differential equations, Alan Turing on computability, Emmy Noether on abstract algebra, and, most recently, Benoit Mandel- brot on fractals. To the public these new domains of mathematics are terra incognita. Mathematics, in the common lay view, is a static discipline based on formulas taught in the school subjects of arithmetic, geometry, algebra, and calculus. But outside public view, mathematics continues to grow at a rapid rate, spreading into new fields and spawning new applications. The guide to this growth is not calculation and formulas but an open- ended search for pattern. Mathematics has traditionally been described as the science of num- ber and shape. The school emphasis on arithmetic and geometry is deeply rooted in this centuries-old perspective. But as the territory ex- plored by mathematicians has expanded into group theory and statis- tics, into optimization and control theory- the historic boundaries of mathematics have all but disappeared. So have the boundaries of its

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N~:~= ~ ~~y . no Io~ j~ the I~= of - ~~s ~ ~~:~, em IS :~ an e~ tO~ ~~r ba~ ~~g mm scions, ~ m~e ~.~ .. .~ ~ ~ ~ ~ . ~ . ~ ~ When v:~ed in this ~ X=~~ ~ s=- . -A ls n~ j~n about num~ and sh~ but ~ pat and ones of ~ ~^ ~~r =e but ~ of m~ media :nItch ;~emanc~ans ~~. Act~ paws is n~ him by who cad :~ ~ mth ~e, chews ni:~th~ :3aathe~n=;~ ~~s Dim Chaws am Po~care had <~end more ~ mung m~ Am.- mimes- m. "T see' Is ~~s had two Ad: meanings.* to~ perceive with ~e = =d to uMe~d why ~e ml:~. ~r centu:~.~ the m.~d ~S ~~d the ~ ln the hierai ~ ma:them~ic~ pr=~~ today the b~e :~s b~g =~d mamercies 6nd ~~ ~~s to =e pattems, both with the -~Ye and h the m~. C~ in the p~e of Hi forces w~.~= o~f m C=8ti~ ~~3tiOO. ~Ot ]~:St CO=~' ~t ~~O ~~ ~~:i~iOO$ 3~6 :~ew themes ~ ~ ~ 6~' It 3~ ~~" ~50 Wl~l Die 8~6 W~ U5iOg ~~ as ~ mutate too] :~ ~ Ieam ~ defeat ~~em~:~s than the~ fbr~the~. Scandal sow paid-=, mo~ ~ tra~ns ~~ are :. ~~ A ~ ~ ^~6 ~5 ~~ At,: to mathematical needs of ~eAria =~ Amp: in ~ prompt mcord of ma~em~ic~ e~n elm prid strong ~~es ~r cha~. Indeed' since new deVelopm=:ts amid on ~~l pnnci:~., It {~s pl=~, as m=~ ~~ arm. su~ gusts th~ one -A few ~ on r - ~~g ~;~h to dm - ~~d nda~nen~$ Ace em:~i~ on :ref6~s bash on chan~s ~n ~e Am practice of Mathews ~tics. ~ ~~ .. .. ........ .. ~ . ~ ~ . ~ ~ ~ ~ ~ tUOt:~C . for ~~g ba- sac cu~n=~a w~es tne mmom ot tte pa~the traditional sch~l :. if c~y ~~t and well Ieamed' prov~s sound prepa mt:~n bo~h for ~e wodd of wo~ and for a~c`~d By in mathe:~:~ imps ta~d 501ds The key :~e for mathe:~ti-~s education :~s not whether to t=~h ~~- dam=~s but Inch Of; to teach and how to teach t50=K Changes in ~e practice of mathematics do alter the balen~ of pnon ties among the many top~s that are important for numera - .~" C3~s ~n society, :n techno1~' in ^~samong othe~will hew great inn pa~ on what mI:t be possible in schoo:! ma:~-s in the :~:t Hi-. M! of th=e changes ~~} a~t the fundament~s of school mathematics. ~ develop Hi new ~~$ C~, OH~ must attempt to ~~e the :~.~:~ti=) needs of tomorrow ~ students. lt ls the

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PATTERN present and future practice of mathematics at work, in science, in researchthat should shape education in mathematics. To prepare ef- fective mathematics curricula for the future, we must look to patterns in the mathematics of today to project, as best we can, just what is and what is not truly fundamental. FUNDAMENTAL MATHEMATICS School tradition has it that arithmetic, measurement, algebra, and a smattering of geometry represent the fundamentals of mathematics. But there is much more to the root system of mathematics- deep ideas that nourish the growing branches of mathematics. One can think of specific mathematical structures: Numbers Algorithms Ratios or attributes: . Linear Periodic Symmetric Continuous or actions: Represent Control Prove Discover Apply or abstractions: Symbols Infinity Optimization Logic or attitudes: Wonder Meaning or behaviors: . Shapes Functions Data Random Maximum Approximate Smooth Model Experiment Classify Visualize Compute Equivalence Change Similarity Recursion Beauty Reality Motion Chaos Resonance Iteration Stability Convergence Bifurcation Oscillation

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4 off dicholomi^: Disc- as. ~~ndsuous Fights infinite at Tic as. ~istebti~ vs. ~e~i~ni~ic ~ , ~ ~xaclv~s Pimple ^~ Ha.\ cat Fee Dale ~~ mumble Be ~-l.exi~ of Rectums lb~ supper m~bemat~. ^^_^^ an ^~.^.^ a... . ~~ a_ ~ ~ ~ ~~ u~ ~ti-~ll~ that ~# Aging lhem Me r sit lop a Si~i,C=1 b~1i~ idea ham inky Loons of eye chilled ~ the 1b~ so Ed Age and~on into SC~li5C or Hem . ^ Be ma counter glib visuals a1I of Fed ~~ distal Yes ad ideas. liaditionaI school matbematics picks very fag ~ _ (e.g., Melba nicety> geometry, aIgebna) and an~I~eslb~m bodzon~]ly to form the cunic~um: [~1 arilbxnelic:1ben simple ~gebra~lben ~eomel~y~lben more Bra and h~ily=-as i~fil bare 1bc epitome of ~~atbemal~ic~ kno~ded~e--~ulus. ThL~yer OCR for page 1
PA TTERN s effect will be to develop among children diverse mathematical insight in many different roots of mathematics. FIVE SAMPLES This volume offers five examples of the developmental power of deep mathematical ideas: dimension, quantity, uncertainty, shape, and change. Each chapter explores a rich variety of patterns that can be introduced to children at various stages of school, especially at the youngest ages when unfettered curiosity remains high. Those who de- velop curricula will find in these essays many valuable new options for school mathematics. Those who help determine education policy will see in these essays examples of new standards for excellence. And ev- eryone who is a parent will find in these essays numerous examples of important and effective mathematics that could excite the imagination of their children. Each chapter is written by a distinguished scholar who explains in everyday language how fundamental ideas with deep roots in the math- ematical sciences could blossom in schools of the future. Although not constrained by particular details of present curricula, each essay is faithful to the development of mathematical ideas from childhood to adulthood. In expressing these very different strands of mathematical thought, the authors illustrate ideals of how mathematical ideas should be developed in children. In contrast to much present school mathematics, these strands are alive with action: pouring water to compare volumes, playing with pen- dulums to explore dynamics, counting candy colors to grasp variation, building kaleidoscopes to explore symmetry. Much mathematics can be learned informally by such activities long before children reach the point of understanding algebraic formulas. Early experiences with such patterns as volume, similarity, size, and randomness prepare students both for scientific investigations and for more formal and logically pre- cise mathematics. Then when a careful demonstration emerges in class some years later, a student who has benefited from substantial early in- formal mathematical experiences can say with honest pleasure "Now I see why that's true." CONNECTIONS The essays in this volume are written by five different authors on five distinct topics. Despite differences in topic, style, and approach, these essays have in common the lineage of mathematics: each is connected in myriad ways to the family of mathematical sciences. Thus it should

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A~ "~= =-^\r~Y mm,0 ~ ~ w~= ~ t~ amp. ~e,=~ ~~ .~ :~ conn~' both :n ~p s^~w and ~n ~ paunch ~~- ~ :~..am.~+ :;: ;s an idea Bawd =~v :n ~~ =~. ~~m e~ ~ ~~c -5 (~$ =~, ~me)' mth ~~ 9~ (~`e :~0~ ~~3els,) m~ ~~m wn~mn En ~ ~ ~3, =d mt:h ^~ ~~= (~# champ) ~! pme ~~ c~ to an~ a. ~ In- q uses "~w ~g is ~~.~" One =~ Him Rev ~~= t:h~ Ads q~= :s fit* it :s at o~ce ~~le yet s^~` ~~ ~ ~ffl~. `~s - o ~w up =~g the com~t of ~~t ~Y s likely to~ accept ~~:~Y ~ ~ ~0 =~ ~65 of nu~ ~ Mimi. ~~ ~w ~ I'll is ~ b~:~g of :: Sv~v :s another deep idea of I thM tums ~ or by, 7 I. ~~$C =~S 8~6 l~ 3~ :~5 0[ =~5 50~i=~3 it 7~:5 ~~ ~~ 0t t~6 - ~07 ~ 35 ~6 - ~0 (8 are 50 OOmC=~8 ~8t hard to count them a~. (~t with prows ~~, ~~g :~n us in ~ I: pea~ toothpick model -aim do it.) Other t:~= ~: is the ammo ~ the pa~s, as :n the =~th of named o~ts firm repet ~ Or ~~. In ~~l o~r cases l:t is symm;~ bmken, ~ in the bowing o~f ~ ~~:2i~31 beam. or ~e gmw~lh of i ~ ~ ~o ~ <~il:~3 asvmmetn~ ad~t an~! Unlike ~ - ~ god- lS SCIdO~ I =~;~h TO SChOO! 2~t 3,Wr ~ ~,r~l ~~t it ls equ~y ~~) as ~ m~! ~r explaln:ng ~~ms of such dI~ verse phenomena as ~e ~~c ~~= of na:~, ~~e s~e of c~, ~~h of o~lismS. the ~~tica:l con:" v 5~0x In- in many examples IO TOlS VOtU=0 8~G :15 0-~6 of the most mpi-~W ~~g ~~.s of ma~! and w:iend6c =~ search. The first step- in ~ta an~`s- ::s the visual d:~' of data to search ~r h:~n pattems. C)~s of v~s ~e`s pmv~ hi: d: - play of relations a~ f6~ion`~, they aw m~:v used t:hro~ghout science ~ ray the behavior of one ~r3~alb-lC (C 2. is ~ ~~:~n of another (~.~, If . . . ~ . . ~ ~ ~ #, ~ ~~n,5 to Ike m- trains ~ ~ ~ .. ~ a- ~ ~~ ~ ~/ ~~ _ ~ ~~ ~~ ~~ And map ma~ ~e us-cc ~ometr:c ~~es suc~ as :~ction to rep:~- sent I s=~s on ~ two~li~sional canvass or sheet of paper. ~w computer graphics auto~e these processes and iet us explore as well the p~s of shapes in hi~dimensio:~! space. Leami~ to- v~e ~~tical pa*~s e;~ll`~t OCR for page 1
~7~:~ :~::.A~ ~~ ~ ~~.~ ~~m ~~:er of m^~c~mo~:i~e~ - , ~~ ~rwe~s hut a: ~ ~~ day of ~ ~ ~ ~ .: ~ . . . . ~ ~ :~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ :~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ . ~ ~ , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ : ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ : ~ ~ ~ h ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Barb a p~m ~~c ~~ ~ ~ pm :~` - ~ ln~ ~~: ~ I: ~ :: ~::~:::~ : ~ ::: :::::::: it: I::::::: ::::: :: I: ::: :~: :::::: : ::: :::: :: ::::::::::::::: :::::::: ~::~:::::~:: :::::: :: ::::: :::::~:~:~:~:~::::~::~:~:~:::~:~:::~:~:~:::: ~~. ~~ E: so : =~r ~# = E I:" " -$ ~~ ~~n~ ~ ~ a" ~~ ~~.~ ~ ~ 0e~ echo ~ ~~ m~ ~~ con Lemmas m^~ ball ~~; ~stery~~ ~ Rho :~:~ A: ~~:~ _~bY~ ~ : ~~ it. ~ ~~ ~~ . :~ ~ ~ ~ ~ ~ ~ ~~ .. I. . ~~ . ~~ ~~ I. ~~ I. I.. .~ ~ ~ .. ~ . ~ ~ ~~ ~ I. .. ~~ ~ .. ~ ~~ ~~ ~ ~~ ~~ ~ ~ ~ ~ I. ~ ~ ~ ~: m~ Asia e~ ' s~ ~~ ~ ~.~ ~N it: ~~ ~~ bait:: ~~ :~ ~~ A m~ ~ ~ ~ ~~ ~ ~ ~ ~~ ~~ ~~ ~~ ~~ ~ ~ ~ . ~ ~ ~ . ~~ . ~~ ~ ~ ~ ~ ~~ ~~ ~ ~ ~ ~ . ~~ ~~ ~ ~~ ~~ ~ ~~ ~~ ~~ ~ ~~ ~ ~~ ~~ ~ ~~ ~ ~ . .~ ~ ~~ ~~ ~ ~ ~~ ~ ~ ~ a ~~ 1~ >~ bmIf ~mme~o~a~:Ina~ M=y ~~ ==e~ nor ~~ Ws~ ~~ - A s Of mapped m~ A.; has a~ Ad: ~~.~ ~ tn~~ ~~ am ~ a= ~ Ante nine of ~~n of: ~ ~ of I~ ~. ~s, em: A= ~~= ad ~p ~=~ ~s ~da~al Pet My m~s, beam bask ~~ ~ o~ ~M m~ m~= ~ mhem~ M~ ~~ ~ ~ sore ~~=s ~ Ad? p~: :n set why ~ mde barked of Spasm- and liti=^ HI Ply Newton ~ - his =~+ ~~t =1~us to the Ha w~ of kits p - sensors If ~ h=e ~~r :~: IS by ~~ ~ the ~~ of ~~- T - e - O d - ~~ m~s current ~: the twent~Y~lirst ~~ ~ :need similar bores sights t sm~ the t~e of Newton h~ m~cs chid as much ~s in recent ~. M~ - 7n I~ ~~ ~ the m~n of c~ puters' ~e nMu:re and pi of mathemadcs have been Amy Do- 57 fit 0~:~57 t00~57 appli~ons, and m~. ~ the wI~e of Gaines em that en~d the Newtonian revolut~, to- 6~:~$ CO=~: C5~t=~ItiOO~ V!~S 8~6 ~~=S ~~:~ of deeuh.~ held values. ~S it did t~= C0~$ 880 1~ t5C tt=SitlO~ trom =~= promS to~~an ~~' mathemat~ Once a~n is un - :~ing ~ ~~ rec=~tion of pmeedu~ ~~* ~~es of Emmy ~~ a~ :~n the r=~ ch i~re of :mathemati~ and :n pm~! =~lica:tions of mathem~i=l methods. :~+ =e ~~:n in the es=~s in this volu:~*

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Ns~ ~ : A 1s n~ ~~ SmCe ~W :~. fit phmomena :~:~:~:~m ~~ :: ~ I :is n~ ~~ a Her Stack aIm~ a~ me=s I: .~ O - ~~ ~~ ~ ~ .~ .~ . . ~ ~~n ~~ ~.e ~~ of ~~, .~ :~ cat Yis~ ~esem~= ~~;~ ;~s ~ amen = :~=en ~ ~^an~c approaches. ~ ~~ - ~ ~~e~.=~ - s B~ Ala;: ma~ d:~m errands o:E mat~emat~^ ~ =~:n . . ~ . ~ . ~ Hi. .. ~;ve on ~~=cures =d And 1~S" Returns m=~ts (~, ~~ ~7 Amy id: ^~n to ~ -me me in order to un-~M =~7 =~ 0~5 (~.~, ~~' , pmw) Deal -ids th~ one must develop -in o~r to ~ math~ em:~ ~:~, c=~s =d Al ~ tb.e nwns =d ve~s of the Togae of math~+ ~~:t bum=s do Oh the tankage of m~ ~s ~ desk p~ tems. Anti is an I; s=~:~= ~~ seeks to u~d e~ k~nd of p~empatwms t~ oc=r in nature, pa=ms metered bra the h~n m:~' =d wen panems ~~d by o~r patw=s To ~ =~i~77 Brim ~~ t~ 0~6 t~ ~ ~ ~ jam to t5~r o~ ]~ ~~= which Chow 08~ =C ~7 re~anty, a~ i: ~e es=~s -it fit vol:~e prmr~ide f~e -~d == swdI~ ~~t eX~ employ h~ thus can ~ done. Ot~r w~ =~ld jug as city h=~e deS~d ~~e or ten -A examples. The ~~s and a~= listed ow are replete m:th additional e=:~es of n~ -: ideas. Wh~ At: ::n the stu~ of Am :~s :~t so much which -4 ular strands -one explores, but ~e p=~ce :n these Ads -of gait:: eXa.~s of su~cien:t v~y aM dash to :~ pa.~. :~Y encour~ a~ng ~~:ts to explore patt-~ms that hwe proven thm: po~r and s~iEca3~, we o~r them broad shoulder= - m -which thev mI! see ~ 8~ ^~ ~~ ~ ~ ~ ~ A .~ are. =.~t t:~: ace. ~ p~~ ~ ~ /~ /~ W~4 ~~' MA~ Bitk~,"$0r B0~+ ~ 985- ~, ~3, id, :~6 5~< ~~ 4~ Oaf ^,$~c pmss~ 'i 988* . Sa msI ~ ~ lc~Xe) F. ~ D=~p Hacf al ~n smem ~ N~ YO~ ~ NYX ~ =~m te ~ 84~\ pt(~6 0t 81 i 565~^ {~e - ~~n ~~. ~ Y0~^ Oaf ~~}

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hi:- I: : :: : ~9 ~5~ ~~=ed~ Am ~~# #_1^, 6~is<~FbOlp~s~J~.~B ,~n~s~s~s~ ~s~sss~s~s~s~ I, : ~ ~~ ~B~966 ~~ ~ ~~ ~~ SO Sag ~~ IS :S~ ~ ~ ~ ,~. .~S~73~E~D~i~PbJ~#~d~H~b~s~ ~~S~_~^ ~8~^~S~ I ~9,i~^'~~ ~s~ ~^ I< bitt I: Imp ~~ APE, 1~989~. ~~ Emit' ~~ ~~ E~:~i~E~:~ ~~ ~:~:~ ~~SS~:~SS~S~;~:~S~ IS ~~S~S~ A ~ o ~ ~ ~ s ~ ~ ~ ~ ~ ~ ~ ~ ~ i ~ l ~ < ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ * ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ i _ , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ _ If ~~ A ~ Ii ~~ ~~ ~~<~ss~s~s~ ~~ ~~s Am ~~ ~1~1,~ ,~- -1~s~,~^~S 4- E~ ~~ ~~- &~ {~) 1~986. ~~ ~ ~~ ~~ ~~ ~~ ~~ as 1 13. ~~ ~~ ~~ ~ ~# #: ~~, 1988. 4. ~ ~ ~~, ~ ~613 ~ /S ~~( ~ ~~/ ~ ~~ ~~ ^ I 15. IS: ~~- ~ ~ ~~ ~~ ~ ~~ ~ ~~~ ~ ~ ~ ,~ 198. : ~ ~ :: ~ ~ ~~ ~ E 16. OR ~ ~ ~ J~ ~~~ I: Vita P 1 ASP. 17. On. Aim. ~ ~~ ~~ ~~ ~ I: Cat ADAM ~ 1971. ~.s.s ~ : ~ s: s ... . ... . : :: :::E ESSsSs :SS :~::: HESS :SSSSESSESSSSS S SSS:SESSSE~ So :Es~: ::SESS~ESSS SSS~:E:ESSSS::SSSSS~ ~~E~S~ I age :~S~: S ::S~ S :~:E :.:: SS~S'~E~ ~~ SE~S~E~ :: US: ~~:'~ss:~:~ :E ' ' 'a gum, _1 ~~ ~ ~ ~~ I. ~~ ~ ~ SAW. Inn ~~ Aft, 19. Runty, ~ E ^e ^~ ~~: ~ ~# ~ ~~# ~# ~3, 20. I, Our. 4~g He Ante ~ ~k ~ ~e=~ ~ ~^ LS ~~ ^~7 ~~r ~~ ^~ Divot, I: N~ ~^ I, 19~> 117-162. 21 tier, Philip Of ~~/ ~~ Ne~ ~ Ad: ^ ad, 1983. 22. ad, Boas. ^~_# ~~ ~~ ~ ~r ~~ ^~ Ha. Ha: Okay Unive~i~ PI 1985. 23. . ~~. E. I: SO ~~ 1985 24. And Ad. ~ ~~ ~~ ~~# ~ ^^ ^~ ~~ Ha. ha: _ 1985. . ma, Debug. #~ ~~ ^~ ^~ ~~ ~~. amid, Ah: Id, 1976 26. ~andelb~, Benoi1 B. ^~ ^~ ~~~ ~^ ~~ ad, ma: W.~. F~- ma, 1982. . Mat, David S. If ^~ ~~ ^~# ~~ ~~ few I, at: ~ H. ~~ 1983 28. godson. Pbilip and Moron, Pby~li$. ^~ ~ an. ~~w at, at: ~ienliSc ~~ ~^ 19S2. Dawn, Into a~ Ricb~ler. ales a. ~ ~~ ~~ :~ ~^ ~~7 #~ ~~ ark, BY: Sprin~^rle~ 1 986. 30. Prime, ~einz OCR for page 1
10 #~ ~ _ 31. ~ am. ^~1~ ~~ ~^ ~- . ~ ~ ~ ~ ~ 3~2. ~ ~~,~ Jo. ~~ ~~ ^~ I Nc~ ~ ad: ~bdd~ I 197~- 33. ~~ ~~ ~# ~~ ~~ ~ ^e ~~ ~~ ^~ ~ ~ ~~~ age, ~ ~~ 1982. 34. Rat ^~. ^~ ~- ^~ ~~ ~~ ~~e~ age, ~ ~ 1^ ~~, 1984. 35; ma, ~~ ad ~~ ~ ~~. ~ ~~ Arm an: Unfair ~ ~~_c-~ Pa 19~77. 36 ~~ ^~ ~ Raven, ~~- I ~~ ~^ ~ ~ ~~; ~~# ~ ^~ an, a: ~ Unfit Pa, ~981~. 37. as, -n Agog. ~~# ~ ~ #~ ~- ~~ a: -~ 1978 38. ~< ~~ ~~ me ~= ~ P_~- 240 (29 #~ 198~8}, 6 I 1- 616. 39. ~~ H. ~~= ^~ ~^ ~~ ha. #: ~ U~ an, 1983 ~ - - ~ - ^ ~ - . 41. ~$, ales b. ^~ '~ ~~~ Bang ~ ~~ ~~ & Cam, 1974~. Aim, #. ~~ ~~_~ ~ ~ Ad: 0~ polyp, 1987. 42. ~~ lag ~ ~ ~ ~7 ~ ~~ ~^ ago: ~11, 1989. 43. --- = Jag M.> al ~. (a.). ~1~^ ~ ~~ ~ 76~ aim, ^~ ace. 1989. 44. and ~~ a. ^~ ^~ ~~ ~l^zi~ ala. chit, an: Oasis Pa, 1983. Inning, ~~ j. ~~ ~# r4~ ~^ ~ age. ~on, ~ align ^~ ~ ~~= ~ Me~li=. 1975.