ESH 250 Lecture Notes - Lecture 10: Number Sense, Decimal Mark, Radio National
9 Jun 2018
School
Department
Course
Professor
Reading
---Van$de$Walle$et$al.,$2010
•
Introduction
You$should$have$achieved$the$following$learning$outcomes:
---Recognise$that$the$meanings$of$each$operation$on$fractions$are$the$same$
as$the$meanings$for$the$operations$on$whole$numbers
---Identify$the$limitations$and$affordances$of$different$models$used$to$teach$
fraction$and$decimal$concepts
---Identify$that$decimal$fractions$are$another$way$of$writing$fractions$and$that$
maximum$flexibility$is$gained$by$understanding$how$the$two$symbol$systems$
are$related
---Identify$how$to$diagnose$decimal$misconceptions$and$what$you$can$do$to$
address$them
•
Equivalent3Fractions
•
---As$Van$de$Walle$et$al.$(2013)$explains,$the$general$approach$to$helping$
students$create$an$understanding$of$equivalent$fractions$is$to$have$them$use$
contexts$and$models$to$find$different$names$for$a$fraction.$Area$models$can$
assist$with$this.
Comparing3Fractions
•
---It$is$important$to$establish$a$feel$for$the$relative$size$of$the$fraction$-
remember$that$fractions$are$numbers$in$their$own$right$and$occupy$a$place$
on$the$number$line.$We$can$help$students$gain$'fraction$sense'$by$focusing$on$
what$the$fraction$actually$looks$like$and$using$referents$to$compare$and$
order$fractions.
---For$example,$if$we$consider$7/8$-we$should$be$able$to$recognise$that$it$is$
almost$1$as$there$is$only$small$part$to$make$8/8$which$is$1$or$that$we$have$7$
parts$out$of$8$which$is$very$close$to$one.$5/8$on$the$other$hand,$is$still$a$fair$
way$away$from$1$whole,$but$we$could$say$it's$a$bit$more$than$1/2$as$it$is$more$
than$4/8.$Again,$while$there$is$a$'rule$ for$comparing$two$fractions,$the$rule$
often$requires$no$thought$about$the$size$of$the$fractions.$According$to$Van$de$
Walle$et$al.,$(2013),$if$students$are$taught$the$rules$before$they$have$had$the$
opportunity$to$think$about$the$relative$size$of$various$fractions,$there$is$little$
chance$that$they$will$develop$any$familiarity$with$or$number$sense$about$
fraction$size.
Fractions…
Developing3an3Understanding3of3Decimals•
----Understanding$and$interpreting$decimals$is$very$much$part$of$being$
numerate.$Decimals$are$used$in$a$variety$of$contexts,$such$as$reading$metric$
measures,$calculating$distances,$reading$calculator$displays$and$interpreting$
sports$statistics.$They$are$also$critical$to$many$occupations,$such$as$nursing,$
pharmacy$and$engineering.$It$is$an$area,$however,$in$which$there$are$a$
number$of$misconceptions$and$many$students$and$adults$have$difficulty$with$
understanding$the$relative$size$of$decimal$numbers$and$computing$accurately$
with$decimal$numbers.$Indeed$a$study$of$nurses$in$the$USA$found$that$in$a$
five$year$period,$almost$2000$people$were$accidentally$killed$and$almost$10$
000$were$injured$through$nurse$error.$Over$400$people$were$killed$by$nurses$
wrongly$programming$drug$infusion$pumps,$which$regulate$the$flow$of$
medicine.$This$kind$of$calculation$error$is$so$prevalent,$according$to$the$
investigation,$that$nurses$call$it$death$by$decimals."$(Quote$
from“Background$Briefing” ABC$Radio$National,$8th July,$2001).
Decimal3Big3Ideas
---The$base-ten$place$value$system$extends$infinitely$in$two$directions$-to$
tiny$values$as$well$as$to$large$values.$Between$any$two$place$values,$the$10-
to-1$ration$remains$the$same.
---The$decimal$point$is$a$convention$that$has$been$developed$to$indicate$the$
units$position.
---Decimal$fractions$are$another$way$of$writing$fractions;$maximum$flexibility$
is$gained$by$understanding$how$the$two$symbol$systems$are$related.
----Percents$are$simply$hundredths$and$as$such$are$another$way$of$writing$
both$fractions$and$decimals.
(Van$de$Walle,$2013,$p.$338)
•
Decimal3Point
---The$decimal$point$marks$the$location$of$the$ones$place.$Students$also$need$
to$see$that$adding$zeros$to$the$left$of$a$whole$number$does$not$change$the$
value$of$the$number$and$adding$zeros$to$the$right$of$a$decimal$fraction$will$
not$change$the$number.$In$terms$of$using$the$correct$language,$Van$de$Walle$
et$al.,$(2013)$recommends$that$decimals$are$read$as,$e.g.,$five$and$two$
tenths,$rather$than$"5$point$2"$as$it$can$result$in$a$disconnect$to$the$fractional$
part$that$exists$in$every$decimal.
•
Decimals3and3Money
---Money$is$often$used$as$a$context$in$which$to$teach$decimals.$Be$wary$of$
doing$this$-can$you$think$why?$In$our$money$system,$we$have$only$two$
decimal$places$-this$can$lead$to$a$misconception$that$you$can$only$have$two$
decimal$places.$Siemon$et$al.,$(2015)$also$caution$against$using$money$as$a$
context$as$the$language$of$money$treats$dollars$and$cents$as$two$whole$
number$systems$side$by$side.$In$our$system$the$money$context$is$not$so$
helpful$since$the$removal$of$one$and$two$cent$coins$-furthermore,$they$have$
no$physical$relationship$to$the$'whole'$and$they$are$referred$to$using$whole-
number$language.
•
Decimal3Materials3and3Models•
---According$to$Siemon$et$al.,$(2015t)$the$materials$and$models$used$to$
support$the$teaching$and$learning$of$fractions$and$decimal$fractions$can$be$
problematic.$Place$value$charts,$e.g.,$help$explain$why$decimals$are$written$
the$way$they$are$but$do$not$teach$decimals.
---Linear$Arithmetic$Blocks$(LAB) are$suitable$materials$for$teaching$decimal$
concepts.$They$are$made$of$washers$and$PVC$conduit$pipe$or$you$can$
purchase$commercially$produced$versions$called$decipipes.$If$making$your$
own,$start$with$the$washers$(thousandths),$then$cut$the$lengths$of$conduit,$
ten$washer$thicknesses$long$(hundredths),$cut$tenths$(10$x$hundredth)$and$
make$whole$(20$tenths$long).$For$further$instructions$see here.
Decimal3Misconceptions•
Vicki$Steinle$and$Kaye$Stacey$from$the$University$of$Melbourne$have$
conducted$a$lot$of$research$around$common$errors$and$misconceptions$that$
students$exhibit$when$comparing$and$ordering$decimals.$They$highlight$the$
following$misconceptions:
---Longer$is$larger$(an$incorrect$application$of$whole$number$ideas$-e.g.,$.375$
is$larger$ than$.97$because$375$is$larger$ than$97.
---Shorter$is$larger$(related$to$thinking$of$numbers$to$the$right$being$very$
small$-so$0.4,$eg.,$would$be$larger$than$0.97$because$a$tenth$is$larger$than$a$
hundredth
---Less$than$0$(some$think$that$decimals$are$negative$numbers$because$they$
are$'down$the$other$end$of$the$number$line')
---Reciprocal$thinking$(connect,$e.g.,$0.4$to$1/4$and$0.6$to$1/6$so$would$think$
0.4$is$larger$than$0.6$because$1/4$is$more$than$1/6)
Decimals…
Week$10-Operating$with$Fractions,$Decimals$and$
Percentages
Wednesday,$ 2$August$2017
12:09$am
Reading
---Van$de$Walle$et$al.,$2010
•
Introduction
You$should$have$achieved$the$following$learning$outcomes:
---Recognise$that$the$meanings$of$each$operation$on$fractions$are$the$same$
as$the$meanings$for$the$operations$on$whole$numbers
---Identify$the$limitations$and$affordances$of$different$models$used$to$teach$
fraction$and$decimal$concepts
---Identify$that$decimal$fractions$are$another$way$of$writing$fractions$and$that$
maximum$flexibility$is$gained$by$understanding$how$the$two$symbol$systems$
are$related
---Identify$how$to$diagnose$decimal$misconceptions$and$what$you$can$do$to$
address$them
•
Equivalent3Fractions•
---As$Van$de$Walle$et$al.$(2013)$explains,$the$general$approach$to$helping$
students$create$an$understanding$of$equivalent$fractions$is$to$have$them$use$
contexts$and$models$to$find$different$names$for$a$fraction.$Area$models$can$
assist$with$this.
Comparing3Fractions•
---It$is$important$to$establish$a$feel$for$the$relative$size$of$the$fraction$-
remember$that$fractions$are$numbers$in$their$own$right$and$occupy$a$place$
on$the$number$line.$We$can$help$students$gain$'fraction$sense'$by$focusing$on$
what$the$fraction$actually$looks$like$and$using$referents$to$compare$and$
order$fractions.
---For$example,$if$we$consider$7/8$-we$should$be$able$to$recognise$that$it$is$
almost$1$as$there$is$only$small$part$to$make$8/8$which$is$1$or$that$we$have$7$
parts$out$of$8$which$is$very$close$to$one.$5/8$on$the$other$hand,$is$still$a$fair$
way$away$from$1$whole,$but$we$could$say$it's$a$bit$more$than$1/2$as$it$is$more$
than$4/8.$Again,$while$there$is$a$'rule$ for$comparing$two$fractions,$the$rule$
often$requires$no$thought$about$the$size$of$the$fractions.$According$to$Van$de$
Walle$et$al.,$(2013),$if$students$are$taught$the$rules$before$they$have$had$the$
opportunity$to$think$about$the$relative$size$of$various$fractions,$there$is$little$
chance$that$they$will$develop$any$familiarity$with$or$number$sense$about$
fraction$size.
Fractions…
Developing3an3Understanding3of3Decimals•
----Understanding$and$interpreting$decimals$is$very$much$part$of$being$
numerate.$Decimals$are$used$in$a$variety$of$contexts,$such$as$reading$metric$
measures,$calculating$distances,$reading$calculator$displays$and$interpreting$
sports$statistics.$They$are$also$critical$to$many$occupations,$such$as$nursing,$
pharmacy$and$engineering.$It$is$an$area,$however,$in$which$there$are$a$
number$of$misconceptions$and$many$students$and$adults$have$difficulty$with$
understanding$the$relative$size$of$decimal$numbers$and$computing$accurately$
with$decimal$numbers.$Indeed$a$study$of$nurses$in$the$USA$found$that$in$a$
five$year$period,$almost$2000$people$were$accidentally$killed$and$almost$10$
000$were$injured$through$nurse$error.$Over$400$people$were$killed$by$nurses$
wrongly$programming$drug$infusion$pumps,$which$regulate$the$flow$of$
medicine.$This$kind$of$calculation$error$is$so$prevalent,$according$to$the$
investigation,$that$nurses$call$it$death$by$decimals."$(Quote$
from“Background$Briefing” ABC$Radio$National,$8th July,$2001).
Decimal3Big3Ideas
---The$base-ten$place$value$system$extends$infinitely$in$two$directions$-to$
tiny$values$as$well$as$to$large$values.$Between$any$two$place$values,$the$10-
to-1$ration$remains$the$same.
---The$decimal$point$is$a$convention$that$has$been$developed$to$indicate$the$
units$position.
---Decimal$fractions$are$another$way$of$writing$fractions;$maximum$flexibility$
is$gained$by$understanding$how$the$two$symbol$systems$are$related.
----Percents$are$simply$hundredths$and$as$such$are$another$way$of$writing$
both$fractions$and$decimals.
(Van$de$Walle,$2013,$p.$338)
•
Decimal3Point
---The$decimal$point$marks$the$location$of$the$ones$place.$Students$also$need$
to$see$that$adding$zeros$to$the$left$of$a$whole$number$does$not$change$the$
value$of$the$number$and$adding$zeros$to$the$right$of$a$decimal$fraction$will$
not$change$the$number.$In$terms$of$using$the$correct$language,$Van$de$Walle$
et$al.,$(2013)$recommends$that$decimals$are$read$as,$e.g.,$five$and$two$
tenths,$rather$than$"5$point$2"$as$it$can$result$in$a$disconnect$to$the$fractional$
part$that$exists$in$every$decimal.
•
Decimals3and3Money
---Money$is$often$used$as$a$context$in$which$to$teach$decimals.$Be$wary$of$
doing$this$-can$you$think$why?$In$our$money$system,$we$have$only$two$
decimal$places$-this$can$lead$to$a$misconception$that$you$can$only$have$two$
decimal$places.$Siemon$et$al.,$(2015)$also$caution$against$using$money$as$a$
context$as$the$language$of$money$treats$dollars$and$cents$as$two$whole$
number$systems$side$by$side.$In$our$system$the$money$context$is$not$so$
helpful$since$the$removal$of$one$and$two$cent$coins$-furthermore,$they$have$
no$physical$relationship$to$the$'whole'$and$they$are$referred$to$using$whole-
number$language.
•
Decimal3Materials3and3Models•
---According$to$Siemon$et$al.,$(2015t)$the$materials$and$models$used$to$
support$the$teaching$and$learning$of$fractions$and$decimal$fractions$can$be$
problematic.$Place$value$charts,$e.g.,$help$explain$why$decimals$are$written$
the$way$they$are$but$do$not$teach$decimals.
---Linear$Arithmetic$Blocks$(LAB) are$suitable$materials$for$teaching$decimal$
concepts.$They$are$made$of$washers$and$PVC$conduit$pipe$or$you$can$
purchase$commercially$produced$versions$called$decipipes.$If$making$your$
own,$start$with$the$washers$(thousandths),$then$cut$the$lengths$of$conduit,$
ten$washer$thicknesses$long$(hundredths),$cut$tenths$(10$x$hundredth)$and$
make$whole$(20$tenths$long).$For$further$instructions$see here.
Decimal3Misconceptions•
Vicki$Steinle$and$Kaye$Stacey$from$the$University$of$Melbourne$have$
conducted$a$lot$of$research$around$common$errors$and$misconceptions$that$
students$exhibit$when$comparing$and$ordering$decimals.$They$highlight$the$
following$misconceptions:
---Longer$is$larger$(an$incorrect$application$of$whole$number$ideas$-e.g.,$.375$
is$larger$ than$.97$because$375$is$larger$ than$97.
---Shorter$is$larger$(related$to$thinking$of$numbers$to$the$right$being$very$
small$-so$0.4,$eg.,$would$be$larger$than$0.97$because$a$tenth$is$larger$than$a$
hundredth
---Less$than$0$(some$think$that$decimals$are$negative$numbers$because$they$
are$'down$the$other$end$of$the$number$line')
---Reciprocal$thinking$(connect,$e.g.,$0.4$to$1/4$and$0.6$to$1/6$so$would$think$
0.4$is$larger$than$0.6$because$1/4$is$more$than$1/6)
Decimals…
Week$10-Operating$with$Fractions,$Decimals$and$
Percentages
Wednesday,$ 2$August$2017 12:09$am
Document Summary
You should have achieved the following learning outcomes: --recognise that the meanings of each operation on fractions are the same as the meanings for the operations on whole numbers. --identify the limitations and affordances of different models used to teach fraction and decimal concepts. --identify that decimal fractions are another way of writing fractions and that maximum flexibility is gained by understanding how the two symbol systems are related. --identify how to diagnose decimal misconceptions and what you can do to address them. --as van de walle et al. (2013) explains, the general approach to helping students create an understanding of equivalent fractions is to have them use contexts and models to find different names for a fraction. --it is important to establish a feel for the relative size of the fraction - remember that fractions are numbers in their own right and occupy a place on the number line.
