math summative assignment.docx

8/20/2019 Math Summative Assignment.docx

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Bowering 1

Summative Math Assignment

Adam Bowering

March 4, 2016

1. Teaching Mathematics to Students withLearning Disai!ities " Annotated

Bi!iogra#h$%%%%%.Page 2

Instructional Design in Mathematics for Students with Learning

Disabilities………...2-

Math inter!entions for students with learning disabilities" m#ths and

realities…....2-$

Mathematics and learning

disabilities…………………………………………….........................$In!estigating elementar# mathematics curricula" focus on

students with learning

disabilities………………………………………………............................$-%

Pre&aring teachers to teach mathematics to students with

learning disabilities…………………………………………………………………%-'

Mathematics instruction for elementar# students with learning

disabilities…………..'-(

2. A &ritica! 'eview o( )&TM Artic!es

*+ractions%%age -. 'eview o( Math

Mani#u!atives%%%%%%%%..Base-ten

Bloc)s…………………………………………………………………………………………

Pattern

Bloc)s…………………………………………………………………………………………..

Mone#……………………………………………………………………………………………

…………

*raction +iles…………………………………………………………………………………………….

+wo-Sided

,ounters…………………………………………………………………………………

4. Mathematics and

Techno!og$%%%%%%%%%.


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Smartboard lesson n Introduction to

*ractions………………………………………………

/ideo 0sing regrou&ing for addition and subtraction using base ten

bloc)s…….

Big boo) *all Into

,ounting………………………………………………………………………………

/. ther Areas o(

nterest%%%%%%%%%%%%%lementar# Mathematics course ournal

entries…………………………………………

Science obser!ation 3ournal gra&hs……………………………………………..

……………….

,ool 4ebsites and !ideo related to mathematics conce&ts………………………..

………………………………………………………………………………

1.+eaching Mathematics to ,hildren with

Learning Disabilities - nnotated Bibliogra&h#

Mathematics is regularl# seen as an area of struggle for man#

students. +his is es&eciall# true for students with learning disabilities.

Mathematics content can be o!erwhelming to these students since there are

man# strategies and s)ills the# must ac5uire in a relati!el# short amount of

time for their cogniti!e &rocesses. I belie!e it is u& to the educator to ta)e

note of these di6erences7 and ad3ust their instruction accordingl#.

+hroughout the following annotated bibliogra&h#7 I highlight some articles

that I ha!e found interesting that introduce a&&licable wa#s of teaching

mathematics content to children with learning disabilities.


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,arnine7 D. 188(. Instructional Design in Mathematics for Students with

Learning Disabilities. Journal of Learning Disabilities, 3092:7 1;-1$1.

+he author suggests that there are multi&le causes for low

achie!ement le!el for students with learning disabilities. cult# with memor#7 and strateg# ac5uisition. 4hen

conce&ts are introduced at a ra&id &ace that students are not

able to )ee& u& with7 these students often become o!erwhelmed

and disinterested in the material. Students with learning

disabilities bene?t more when the design is centered on big

ideas7 strategies7 e>cient use of time with mani&ulati!es7 clear

and e@&licit instruction7 and &ractice and re!iew. nother

strateg# for instruction is to use strand organiAation for lessons.

+he idea is to design lessons around strands to address di6erent

big ideas in %-1; minute &eriods. 4hen teachers a&&l# these

design &rinci&les7 it could ha!e a &ositi!e e6ect on achie!ement

for students with learning disabilities.

*leischner7 . .7 Manheimer7 M. . 188(. Math inter!entions for students

with learning disabilities" m#ths and realities. School Psychology

Review, 269:7 8(-$1.


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+wo t#&es of learning disabilities are discussed in this

&a&er. +he# include d#scalculia7 and !erbal learning disabilities7

which are referred to as Cgarden-!ariet# learning disabilities.

+he focus of this &a&er is to hel& educators understand e6ecti!e

instructional techni5ues that hel& students with disabilities.

Starting with children at the &recom&utation le!el7 methods for

whole number o&erations7 &lace !alue7 and fractional conce&ts

are discussed as well as algebra and &roblem sol!ing. n

e@am&le used for whole number com&utation is &ro!iding

students with wide-celled gra&h &a&er7 so that students who

ha!e di>cult# with &lace !alue can ha!e mar)ers that align

numerals.


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students with learning disabilities7 rather than a one-siAe ?ts all

class instruction.

Fear#7 D. 2;;$. Mathematics and learning disabilities. Journal of Learning

Disabilities, 3791:7 $-1%.

,hildren with Learning disabilities often ha!e trouble with

numbers7 counting7 and arithmetic. ,ounting &rinci&les and the

&rinci&les of &roblem sol!ing tended to be lac)ing in the children

used in this stud#. +he stud# also goes on to sa# that children

with learning disabilities continue to ha!e di>culties in retrie!ing

arithmetic facts from their long-term memor#7 which did not

im&ro!e o!er time. +he stud# found that children with LD had

interru&tions in attentional information &rocessing. +his resulted

in di>culties with information mani&ulation with math language.

Students with learning disabilities re5uire more time to &rocess

information7 which is 3ust as true for math content. Pro!iding

students with the time to &rocess new information allows these

students to ha!e a better understanding of conce&ts li)e

counting and basic arithmetic.

Gord7 ,.7 Eewton7 . . 2;1$. In!estigating elementar# mathematics

curricula" focus on students with learning disabilities. School Science

an !athe"atics, ##$9$:7 181-2;1.


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+he &ur&ose of this stud# was to e@amine curricula to see if

there was a wa# to &ro!ide more o&&ortunities for engagement

for students with LD. +he stud# focused on three mathematics

curricula that &ut a s&otlight on accessibilit# in regards to

challenges in wor)ing memor# for these students. Instruction

that was used in this stud# included the em&hasis of !isuals7

math con!ersations7 and de!elo&ing metacogniti!e s)ills. +he

?nding was that teachers must consider how their curriculum

outcomes &ro!ide e6ecti!e instruction for the storage and

organiAation of information for these students. +eachers must

also ta)e into account how the outcomes de!elo& students=

thin)ing &rocesses and their o!erall understanding of

mathematical conce&ts. +his s&eci?c stud# concluded that the

curriculum areas were e6ecti!e in teaching students with LD to

sol!e multi-ste& &roblems. Gowe!er7 su&&lemental materials

should be used in con3unction with curriculum to meet the needs

of students with LD.

Parmar7 H.7 ,awle#7 . 188(. Pre&aring teachers to teach mathematics to

students with learning disabilities. Journal of Learning Disabilities, 3092:7

1-18(.

Se!eral standards are &ut forth in this stud# which reJect

the wa# in which teachers need to &re&are to teach children with

learning disabilities. Standard one is to model good mathematics


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teaching. +eachers need to ha!e )nowledge of the curriculum7

and the !ariet# of instructional techni5ues a!ailable to them.

+eachers shouldn=t 3ust focus on one area to strengthen

their teaching7 the# must focus on a well-rounded a&&roach

when teaching mathematics. collaborati!e a&&roach between

the teacher and s&ecial education teacher is suggested when

teaching children with LD. n e@&ectation of this a&&roach would

be enhanced 5ualit# of education7 allowing students to reach

higher le!els of &roblem sol!ing7 reasoning7 and connections with

the mathematics content.

Standard two is for students to ha!e the a&&ro&riate

)nowledge of mathematics. +e@tboo)s ha!e gotten longer7 and

therefore more com&le@. Brea)ing the information down into

?ner details will decrease the time for co!erage of the content.

+eachers need a balanced a&&roach to &ro!ide students with an

understanding of the meanings7 &rocesses7 and &rinci&les of

mathematics.

+he stud# stressed that acti!ities need to be &lanned

around real-life &roblem sol!ing and big ideas. Eew teachers

must constantl# )ee& in contact with other teachers and sta67

creating a collaborati!e a&&roach when focusing on a&&ro&riate

instruction. +his stud# also cited the E,+M as one of the

organiAations to use as a reference for teachers.


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Thornton, C., Langrall, C., & Jones, G. 1997. Mathematics instruction for elementary students

with learning disailities. Journal of Learning Disabilities, 30!"#, 1$"%1'.

+his stud# highlights the need for teachers to constantl# be

5uestioning their long standing beliefs when it comes to

teaching7 learning7 and using curriculum. ,ommon &ractice in

toda#=s classroom reJects a narrow focus on com&utation. +he

&a&er s&ea)s about how the E,+M &ro&osed ?!e goals to rethin)

math teaching and learning. +he# include instilling !alue of math

on students7 increasing con?dence in their abilit# to do math7

become mathematical &roblem-sol!ers7 to reason

mathematicall#7 and to communicate using math. +eachers

should do this b# decreasing their &encil-and-&a&er com&utation

and rote memoriAation of the content. Students should be gi!en

multi&le wa#s to re&resent the same answer. +his could be

through illustration or mani&ulati!e use.

In the end7 the article suggests that teachers need to ha!e

a balanced a&&roach when teaching mathematics7 engaging

students in meaningful tas)s7 accommodating for the di!erse

wa#s in which students learn7 and to allow students to discuss

and 3ustif# their &roblem-sol!ing strategies. +eachers need to

&ro!ide students time to engage in and share their own solutions

to meaningful &roblems.


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2. A &ritica! 'eview o( )&TM Artic!es "

+ractions

Artic!e 1 "

D=mbrasio7 B. Kastberg7 S.. 92;12:. Building understanding of Decimal

*ractions. %eaching &hilren !athe"atics' 198:7 %%8-%'%.

ntroduction

BeatriA D=mbrosio is a &rofessor at Miami 0ni!ersit# in


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+his article discusses an in!estigation loo)ing at decimal

understanding amongst &re-ser!ice teachers and students. +he

in!estigators &resented &re-ser!ice teachers and students with

decimal grids and were told to illustrate and e@&lain their

thin)ing when ordering ;.';'7 ;.;'''7 ;.'7 ;7 ;.'' and ;.;';.

Se!eral of the &re-ser!ice teachers struggled with using the grid

&a&er to re&resent the decimal numbers. +he# had ?!e students

com&lete this 5uestion and the# used their answers to draw their

conclusion. *our out of the ?!e students struggled li)e the &re-

ser!ice teachers. B# anal#Aing the students wor) it re!ealed to

them three distincti!e sources of di>cult# of understanding

decimal numerals and their re&resentations of 5uantities. +he

?rst source of di>cult# is that students didn=t understand the

relationshi& between the !arious subdi!isions of a whole. +he

second source of di>cult# man# students did not see the

additi!e nature of the decimal &laces. +he last7 and most

im&ortant7 di>cult# is that rel#ing on &rocedures for ordering

decimals can mas) a student=s true understanding of the relati!e

siAe of decimal numerals. D=mbrosio and Kastberg o6er

solutions for these three di>culties the# unco!ered. *or the ?rst

di>cult# teachers should focus on onl# one grid at a time and

instruct students to use that &articular grid to show di6erent

decimal 5uantities. +he second solution is for teachers to &lace a

larger em&hasis on additi!e nature of the &lace-!alue s#stem.

+he last strateg# is to &ro!ide follow u& tas)s for students that

continue to build the students understanding of the relationshi&s

between and among subdi!isions. +he article concludes with the

recommendation that &rocedural routines that mas) students

misunderstanding of how to re&resent decimals can be a!oided if

teachers begin with idea of what ma)es decimals di>cult for


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Bowering 11

students to full# gras& and as the# teach to )ee& those

di>culties in mind.

resentation

ersona! 'es#onse

&onc!usion

Artic!e 2 *

Kent7 L.B.7 m&son7 S.B.7 Eielsen7 L. 92;1%:. +he richness of children=s

fraction strategies . %eaching chilren "athe"atics. 2292:7 %-8;.

ntroduction

Summar$

resentation

ersona! 'es#onse

&onc!usion

Artic!e "


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Mc,ormic)7 K.K. 92;1%:. Ma)ing *ractions Meaningful. %eaching chilren

"athe"atics. 229$:7 21-2.

ntroduction

Summar$

resentation

ersona! 'es#onse

&onc!usion

. 'eview o( Math Mani#u!atives


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Bowering 1

(ase)#0 bloc*s

Image source" htt&"NNwww.amaAon.comNLearning-Hesources-Plastic-Base-

StarterNd&NB;;;*/BB<

Base ten bloc)s are !er# !ersatile mani&ulati!es that can be used to

re&resent man# abstract conce&ts in mathematics. +he# can be used to

illustrate addition7 subtraction7 multi&lication7 and di!ision &roblems. I would

use base ten in an# grade7 since the# ha!e such a range of use. +he# can be

used to introduce &lace !alue to lower grades7 and decimal &oints in u&&er

elementar#. During m# student internshi&7 I introduced base ten to m# grade

2 class. I started with the basics. 0nits were ?rst7 followed b# rods which I

associated with being 1; units. +his was a tough conce&t for students to

gras& at ?rst7 since the# couldn=t 5uite understand the !alue of each unit. I

would use base ten bloc)s for an# of the abo!e mentioned to&ics.


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Bowering 1$

Pattern (loc*s

Image Source" htt&"NNwww.mathed&age.orgNanglesN

s the &icture illustrates abo!e7 a set of &attern bloc)s consist of a blue

rhombus7 green triangle7 tan rhombus7 orange s5uare7 #ellow he@agon7 and

the red tra&eAoid. +hese mani&ulati!es can be used to meet a number of

curriculum outcomes. Students can use &attern bloc)s to e@&lore basic

geometric sha&es or &ut them together to build more intricate or com&le@

sha&es. +hese bloc)s allow students to see that larger sha&es are com&osed

of smaller ones7 which relates to fractions. +hese bloc)s would also be a

great tool for students to create and identif# &atterns. I would use them to

indicate conce&ts7 li)e teaching about the &attern core to lower lementar#

students. +hese mani&ulati!es also &ro!ide students with a wa# of

identif#ing and disco!ering 2D sha&es in a hands on wa#. Pattern bloc)s are

an educational tool that students ha!e fun using7 which ma)es their learning

more engaging. +he# are !er# !ersatile and are found in most schools.


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!oney as !ani+ulatives

Image source" htt&s"NNwww.rainbowresource.comN&rodlist.&h&O

sub3ectMathematicsN1;categor#Mone#Q-QMaterialsQR2'QMani&ulati!esN2%'8

In a t#&ical set of mone# mani&ulati!es7 #ou will ?nd &ennies7 nic)els7

dimes7 5uarters7 loonies7 toonies and bills. Mone# is a tangible mani&ulati!e

since it is something students will use in their e!er#da# li!es. +he

mathematic s)ills related to mone# are necessar# for all ages.


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dollar. +he fact that students use this mani&ulati!e in a real life content

strengthens its abilit# to connect students to the content.

raction %iles

Image source" htt&s"NNwww.enasco.comN&roductN+B1%11+

s seen in the &icture abo!e7 a full set of fraction tiles consists of one

whole7 two hal!es7 three thirds7 four 5uarters7 ?!e ?fths7 si@ si@ths7 eight

eighths7 ten tenths and twel!e twelfths. *raction tiles are an e@cellent wa#

for students to e@&lore and understand how &arts ma)e u& a whole. I would

use the sim&ler fractions at ?rst when I am introducing fractions7 and


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graduall# introduce fractions with a greater denominator as students become

more comfortable with this t#&e of mani&ulati!e. +his mani&ulati!e also

furthers students understanding of addition7 subtraction7 multi&lication and

di!ision of fractions. +hese mani&ulati!es gi!e students a concrete

re&resentation of an abstract conce&ts. Since a lot of students belie!e that

each number in a fraction is di6erent7 these &ro!ide clarit# on that issue b#

getting students to see the set of numbers as a whole. Being able to see the

&ro&ortions of these fractions ma)es it easier for students to com&are7 order7

and e@amine e5ui!alent fractions. Some fraction tiles also ha!e decimal

notations on the o&&osite side7 which is e@cellent for seeing the relationshi&

between decimals and fractions which is bene?cial in the u&&er &rimar#

grades. I chose this mani&ulati!e to re!iew because I feel as though it

&ro!ides a great !isual for students when the# are learning fractions. It=s

im&ortant that students see conce&ts and are able to e@&lore them in a

hands-on wa#. Students struggle with understanding fraction siAes. Man#

children might belie!e that 1N( is bigger than 1N sim&l# because the ( is

larger than the . +he mani&ulati!e &ro!ides students a wa# to see these

fractions for what the# trul# are.

%wo)Sie &ounters


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Image source" htt&"NNwww.amaAon.comNLearning-d!antage-(21;-Magnetic-

,ountersNd&NB;;10F;B

+wo sided counters are another !ersatile mani&ulati!e that can be used

in so man# di6erent wa#s. *irst o67 the# hel& students with counting7

di6erentiating between sets of ob3ects7 addition and subtraction7 or

multi&lication and di!ision. I used counters in m# &eer teaching module to

di6erentiate between di6erent sets of fractions using ten frames. I as)ed for

students to use one colour to re&resent N1;7 and another colour to re&resent

2N1;. +his allowed me to bridge into the addition of fractions as well7 since

the two sets could be combined. I also use them to teach about the multi&le

wa#s a number good be re&resented using addition. *or e@am&le7 if students

were loo)ing for the number C'7 I would as) students to dro& their '

counters on their des)s to see how man# di6erent wa#s this number could

be re&resented.

I reall# en3o#ed using counters. +he# &ro!ide students with so man#

eas# wa#s to re&resent abstract conce&ts. I would use these as a regular &art

of m# math teaching in m# classroom.


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Bowering 18

4. Mathematics and Techno!og$S"artboar lesson - .n introuction to fractions

+he focus of the following lesson is how to introduce fractions at a

basic le!el. +his lesson is intended for a grade introduction. +he foundation

of fractional conce&ts is so im&ortant in math7 as these conce&ts e@tend into

other math areas. In this lesson7 I s&ea) about the a&&ro&riate !ocabular# to

use when tal)ing about fractions7 and the wa# in which students should be

associating fractions to real-life a&&lication.


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Bowering 2;

/sing !ani+ulatives for Regrou+ingBase-ten bloc)s are a fantastic wa# to illustrate addition and

subtraction &roblems using a techni5ue called regrou&ing. +his techni5ue

&ro!ides a sim&le and concrete wa# for students to ha!e a !isual

re&resentation of a &roblem. In the following !ideo7 I tal) about the

ad!antages of using this techni5ue7 and the ste&s to follow to com&lete this

techni5ue.


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Bowering 21

all into &ounting - (ig boo*

+he aim of this big boo) was to &ro!ide students with multi&le

re&resentations of counting using familiar ob3ects. s the authors7 we wanted

to include an element of interacti!it# with the boo) so that students could

use a hands-on a&&roach when counting. Students will also be e@ercising

their literac# s)ills7 re5uiring them to listen to hear the &roblem that the#


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Bowering 22

must sol!e on each &age. 4e also introduced a sim&le wa# to include

addition within some of the &roblems.

/. ther Areas o( nterest

Journal entries from my math methods course.

Discussion uestions

3amine $our own regions curricu!um document. 5hat are the

!earning e3#ectations (or numer (or chi!dren in indergarten

to grade 27 &om#are these e3#ectations with ideas #resented


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Bowering 2

in this cha#ter. 5hat ideas are stressed7 Did an$thing sur#rise

$ou7

+he learning e@&ectations are similar to those in this cha&ter. ,hildren are

e@&ected to s)i& count b# 2s7 %7 and 1;s7 count on from a number7 count b#

ones from a gi!en number and count forwards and bac)wards. ,hildren are

also e@&ected to count using coins. +his last e@&ectation sur&rised me since

mone# is an abstract conce&t. Students must ?rst learn to associate the

corres&onding worth of each coin to the coins &resented in a math &roblem.

+his could ser!e to be &roblematic at ?rst since children at this stage ha!e

onl# learned to count ob3ects based on how man# ob3ects there are. Eow7

ob3ects ha!e di6erent 5uantities associated with them7 &enn# being one7

nic)el being ?!e7 and a dime being ten. ,hildren ha!e been learning using

singular mani&ulati!es to this &oint7 so adding this conce&t of coins being

worth more than others ma# be confusing to children in the beginning.

Gowe!er7 this is an im&ortant s)ill since children will e!entuall# be using

mone# in the real world.

8ouve noticed that a student $ou are woring with is counting

o9ects with an accurate se:uence o( numers words, ut is

not attaching a numer to each o9ect. There(ore, the

students ;na! count is inconsistent and inaccurate. 5hat

wou!d $ou #!an to he!# this student deve!o# a etter gras# o(

one*to*one corres#ondence7


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Bowering 2$

+here a cou&le of strategies I might use to hel& this child with one-to-one

corres&ondence. +he ?rst strateg# would be to ha!e a set of Jash cards with

the corres&onding number. s the student counts the ob3ect7 the Jash card

with the correct number could be located underneath that ob3ect so the

student could associate the number with the ob3ect. nother strateg# would

be to use a calculator or counter. s the student counts each ob3ect7 ha!e

them record the number on the calculator or counter so the# are associating

that number with the ob3ect in the se5uence.

3#!ore a we*ased #rogram (or dri!!ing asic (acts. 5hat

(eatures does the #rogram have that are good7 )ot so good7


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Bowering 2%

answer would be. +he teacher would use the answer on the website to see if

it would be correct or not.

5hat is the im#ortance o( menta! mathematics and

com#utationa! estimation (or dai!$ !iving7 5hat are the

advantages o( eing a!e to estimate in #u!ic situations and

settings7

Mental mathematics aids in the e!er# math conce&t that we use in a real life

conte@t. 4e use math to count mone# when we bu# groceries from the store7

or we use mental math to count our hours we=!e wor)ed during the wee).

4e use mental math while &la#ing s&orts to add u& scores to determine who

defeats who. +he ad!antage of being able to estimate in &ublic situation

shows #our com&etenc# with being able to &erform sim&le math &roblems

without the use of a calculator or other de!ice that &eo&le tend to de&end

u&on. +his is es&eciall# true with the use of cell &hones. Most of these de!ice

come e5ui&&ed with a read# to use calculator7 ma)ing them eas# and

con!enient7 which ham&ers our abilit# to use our mental math s)ills.


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Bowering 2'

Science Observation Journal

+he following images de&ict gra&hs that I created in a &ro3ect that we

com&leted as &art of m# lementar# science course. +he ?rst &hotogra&h

shows a line gra&h7 which com&ares two di6erent sets of tem&eratures o!er

a s&an of 1% da#s. I gathered the data from a thermometer outside of Brian

Mulrone# Gall on the S+0 cam&us7 as well as from theweathernetwor).com.


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Bowering 2(

In this second &hotogra&h7 I chose to use a bar gra&h to show the di6erent

5uantities and t#&es of garbage I found in the habitat we were in!estigating.

+his )ind of gra&h gi!es a good re&resentation of the t#&e of data I collected.


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Bowering 2

*or this ?nal illustration7 we had to ma& the area we were stud#ing7 and

measure the total area using length width. I found this acti!it# eas#7 and

found that it would easil# be translated into a classroom acti!it#.

Cool websites and videos!

htt&"NNwww.a&lusmath.comN

+his website &ro!ides a ton of cool and interacti!e games to hel& students

with their math s)ills. +here are also wor)sheets7 digital Jashcards7 and

!ideosT

htt&"NNwww.coolmath-games.comN

http://www.aplusmath.com/http://www.coolmath-games.com/http://www.aplusmath.com/http://www.coolmath-games.com/


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Bowering 28

+his is another cool and fun interacti!e ?lled with information on !arious

to&ics in mathematics.

Uou+ube !ideo for decimals"

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