Ways%of%Thinking%About%Mul4plica4o n%Many%solvers%think%they%should%divide/subtract%on%Problem%2%Interes4ng%empirical%result:%Success%on%Problem%2%is%usually%35D40%%less%than%on%Probl em%1$1%MTE%494,%Arizona%State%University%What%might%explain%this%result?%A%plausible%explana4on:%repeated'addi)on%is%oPen%the%only%meaning%for%mul4plica4on%that%students%learn%and%retain.%%First%meaning%aSached%to%“4x12”%is:%4x12%=%12+12+12+12.%Similarly,%“3x8.71”%means%8.71+8.71+8.71.%%For%students%“axb”%usually%means%“do%something”,%namely%add%b%a%4mes:%axb$=%b+b+…+b,%where%there%are%a%bs.%%2%MTE%494,%Arizona%State%University%ReDread%the%two%problems;%does%the%idea%of%mul4plica4on%as%repeated%addi4on%fit%both%problems?%Explain.%3%MTE%494,%Arizona%State%University%So'what’s'the'problem?'• Overemphasizing%mul4plica4on%as%repeated%addi4on%to%the%exclusion%of%other%interpreta4ons%leads%many%students%to%unwi`ngly%develop%the%following%misconcep4on:%Mul-plica-on$always$makes$bigger$??'When'does'mul)plica)on'NOT'make'big ger?'When%first%factor%in%the%product%axb%is%a%whole%number,%repeated%addi4on%meaning%makes%sense.%F urther,%mul4plica4on %do es%“make%big ger”%when%the%first%factor%is%a%whole%number%>1%%4%MTE%494,%Arizona%State%University%??'When'does'mul)plica)on'NOT'make'big ger?'BUT%what%about,%say,%“( ¾)x200”? %Does%it% makes%sense%to%think%about%adding%200%threeDfourths%4mes?%A'more'unifying'meaning:%Mul4plica4on%as%imagining%mul4plici4es%of%things/amounts%and%making%some%number%(including%a%frac4onal%part)%of%copies%of%things/amo unts.%%“axb”%means%imagine%a%bs$or%a%copies%of%amount%b.%The%amount%you%get%by%making%a%copies%of%b$is$a%4mes%as%much%as$b.%%5%MTE%494,%Arizona%State%University%Under%this%meaning%“(¾)%of%an%amount”%is%a%partDofDan%amount—more%specifically,%it%is%3%4mes%as%large%as%oneDfourth%of%the%amount.%%Thus,%(¾)x200%is%an%amount%that%is%3%4mes%as%large%as%oneDfourth%of%200%[the%result%of%making%3%copies%of%oneDfourth%of%200].%In%this%concep4on,%mul4plica4on%definitely%does%not%make%big ger.%%%%Students%who%thought%they%should%subtract%or%divide%in%Problem%2%are%thought%to%be%missing%this%“partDofDan%amount”%interpreta4on%of%mul4plica4on.%%Their%thinking:%”0.73%is%less%than%1%lb,%so%it%should%cost%less%than%$2.19.%So%I%must%do%some%calcula4on%that%gives%less%th an%$2.19.”%If%mul4plica4onDmakesDbigger%was%their%guide,%then%the%opera4on%of%mul4plica4on%is%not%an%op4on.%%6%MTE%494,%Arizona%State%University%Summary%• Some%concep4ons%of%mathema4cal%ideas%are%more%powerful%and%genera4ve%than%others%%• We%can%approach%mathema4cs%instruc4on%from%an%“engineering”%perspec4ve%to:%1.%Design$mathema4cal%concep4ons%(i.e.,%specify%meanings%and%ways%of%understanding%a%mathema4cal%idea)%to%target%as%learning%objec4ves;%2.%Create%and%engage%students%in%ac4vi4es%d
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