engaging prospective teachers in peer assessment as both.rtf

8/9/2019 Engaging prospective teachers in peer assessment as both.rtf

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Engaging prospective teachers in peer assessment as both

assessors and assessees: The case of geometrical proofs

Ilana Lavy

The max Stren Yezreel Valley College

Atara Shriki

Oranim- Academic College of d!cation

Abstract

One aspect of professional development of mathematics teachers relates to the development ofassessment skills. The aim of this study is to examine the effects of engaging prospective

mathematics teachers in peer assessment, both as assessors and assessees, on the

development of their assessment skills in general and assessment of geometrical proofs in

particular. The research was conducted within a Method course in which peer assessment

activities were employed. Sixteen prospective mathematics teachers participated in the

research and had to act both as assessors and assessees.Analysis of the research data reveals

that during the various phases of the study the prospective teachers developed skills

concerning the selection of categories and weights for the assessment of their peers work. !n

the criteria set they selected for the peer assessment, they referred to meanings and roles of

mathematical proof. !n their reflections, the prospective mathematics teachers also referred

to the effects of the peer assessment on their mathematical knowledge, asserting that by being

exposed to different solution strategies and new problems they were able to widen their

mathematical knowledge.

"ey#ord$% &eer a$$e$$ment' &rofe$$ional develo&ment' &ro$&ective teacher$' geometrical

&roof' (#hat-if-not)( $trategy

1. Introduction

*any i$$!e$ relating to vario!$ a$&ect$ of mathematical &roof$ have +een inve$tigated d!ring

the la$t t#o decade$, Among the$e i$$!e$ are the role of &roof in mathematic$ cla$$room$

.er$h' 1//0' $t!dent$2 diffic!ltie$ in &roviding &roof$ *oore' 1//3' $t!dent$2 &erce&tion$

of &roof .arel 4 So#der' 1//5 and the con$tr!ction of &roof$ 6e+er' 78819 6e+er 4

Alcock' 7883, .o#ever' the a$$e$$ment of mathematical &roof$ ha$ received little attention'

e$&ecially from the $t!dent$: &er$&ective$ Alcock 4 6e+er' 788;9 *amona-


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eer a$$e$$ment &rovide$ feed+ack that can red!ce error$ and have &o$itive

effect$ on learning #hen it i$ received tho!ghtf!lly and &o$itively, The active engagement of

learner$ in the a$$e$$ment &roce$$ i$ regarded a$ im&ortant' and &eer a$$e$$ment ha$ +een

fo!nd to +e an effective tool in $elf-im&rovement =o!d' 7888, ngagement in &eer

a$$e$$ment may al$o rai$e the $t!dent$2 motivation for learning .anrahan 4 I$aac$' 7881,

=eing a#are of the im&ortance of develo&ing the >*T$2 a$$e$$ment $kill$ and of the +enefit$of the &eer a$$e$$ment a&&roach' #e +elieve that engaging >*T$ in &eer a$$e$$ment of

geometrical &roof$ can $!&&ort their &rofe$$ional develo&ment, *oreover' engagement in

&eer a$$e$$ment' +oth a$ a$$e$$or and a$$e$$ee' may ex&o$e $t!dent$ to +oth $ide$ of the coin

and hence may re$!lt in a dee&er com&rehen$ion of the im&ortance of +oth the mathematical

con$tr!ct$ geometrical &roof$ and the &roce$$ of their a$$e$$ment,

In thi$ &a&er #e de$cri+e a $t!dy de$igned to examine the effect$ of ex&eriencing &eer

a$$e$$ment of mathematical &roof$' +oth a$ a$$e$$or$ and a$$e$$ee$' on the develo&ment of

a$$e$$ment $kill$ of >*T$ and their mathematical kno#ledge, In im&lementing the &eer

a$$e$$ment a&&roach on >*T$ #e addre$$ed the follo#ing ?!e$tion$%

i .o# do $t!dent$ choo$e' a$$ign #eight$ and @!$tify categorie$ for the a$$e$$ment &roce$$)

ii .o# do they react to feed+ack received from their cla$$mate$ a+o!t their o#n #ork)

iii .o# do they refer to the role$ and meaning$ of &roof in their a$$e$$ment &roce$$)

iv In #hat #ay$ doe$ $t!dent$: ex&o$!re to their cla$$mate$: #ork affect their o#n

mathematical #ork)


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2. Theoretical Background

A$$e$$ment method$ can +e !tilized in a $!mmative or a formative #ay, 6hen an a$$e$$ment

method i$ !$ed in a $!mmative #ay' the learning o!tcome$ and &rod!ct$ are +eing a$$e$$ed at

the end of the learning &roce$$, ormative a$$e$$ment' on the other hand' i$ intended to hel&

$t!dent$ &lan their o#n learning' identify their o#n $trength$ and #eakne$$e$' target area$ forremedial action and develo& meta-cognitive and other &er$onal and &rofe$$ional tran$fera+le

$kill$ =o!d' 7888, A$$e$$ment #hich $&ecifically indicate$ $trength$ and #eakne$$e$ and

&rovide$ fre?!ent con$tr!ctive individ!alized feed+ack lead$ to $ignificant learning gain$' a$

com&ared to traditional $!mmative a$$e$$ment =lack 4 6iliam' 1//5,

In #hat follo#$' #e &re$ent a +rief literat!re $!rvey regarding role$ and meaning$ of

mathematical &roof$' &eer a$$e$$ment' &eer a$$e$$ment in higher ed!cation' and a$$e$$ment of

mathematical &roof$,

2.1 Roles and meanings of mathematical proof

Be$earcher$ have +roadly di$c!$$ed the meaning and role$ of &roof in the mathematic$

&ractice >olya' 1/;39 =ell' 1/D9 de Villier$' 1//8' 1//19 .anna' 7888, ir$t of all the role

of a &roof i$ to verifyEconfirm a given $tatement, The &roof al$o &rovide$ in$ight into #hy the

$tatement i$ tr!e ex&lanation, >roof may al$o &rovide the organization of vario!$ re$!lt$ into

a ded!ctive $y$tem of axiom$' conce&t$ and theorem$ $y$tematization,

finding a &roof' ne# o!tcome$ may emerge, The &roof al$o ha$ an im&ortant role in

comm!nication of the mathematical kno#ledge +y the tran$mi$$ion of it' and the &roce$$ of

looking for a &roof i$ often an intellect!al challenge,

According to the FCT* 7888 $tandard$' rea$oning i$ a &ractice of mind and $t!dent$ need

o&&ort!nitie$ to engage #ith rea$oning and &roof in many context$' incl!ding co!r$e$ in

#hich technology &lay$ a $ignificant role Thom&$on' 788/, .arel and So#der 1//5

defined a $t!dent:$ &roof $cheme to +e the &roce$$e$ $t!dent !$e$ to +ecome certain of the

tr!th of a mathematical $tatement' and to convince other$ of thi$ certain tr!th, *any ed!cator$and re$earcher$ have inve$tigated $t!dent$: learning and a+ilitie$ #ith &roof conce&t$ #hen

!$ing a dynamic geometry dra#ing tool e,g,' Chazan' 1//09 .ada$' .er$hko#itz 4 Sch#arz'

78889 La+orde' 7888, St!dent$ #ho are acc!$tomed to technology in all a$&ect$ of their live$

may con$ider &roof$ done via technology a$ &roviding evidence to +oth a$certain' and

&er$!ade other$ of' the tr!th of a claim,

6e +elieve that in order to +e a+le to &ro&erly a$$e$$ a mathematical &roof' >*T$ $ho!ld +e

a+le to con$tr!ct a $!ita+le criteria $et, S!ita+le criteria $et i$ one that take$ into acco!nt the

role$ and meaning$ of mathematical &roof, x&o$ing >*T$ to an a$$e$$ment of mathematical


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&roof' in #hich they act +oth a$ a$$e$$or$ and a$$e$$ee$' may hel& them develo& a$$e$$ment

$kill$ needed for &ro&er a$$e$$ment of mathematical &roof,

2.2 Peer Assessment

In ed!cational context' a$$e$$ment i$ defined a$ the &roce$$ in #hich $t!dent$2 &rod!ct$'

#hich are re$!lt$ of their kno#ledge and $kill$' are mea$!red To&&ing' 7880,

Peer assessment' i$ a &roce$$ #here+y $t!dent$ a$$e$$ a$$ignment$ of their cla$$mate$ +a$ed

on a given criteria li$t *cari$ 4 >ari$' 7881,St!dent$ of all age$ react differently to feed+ack from

&eer$ and from ad!lt$ Cole' 1//1,

A$ to >*T$' &roviding effective feed+ack from &eer a$$e$$ment i$ a cognitively com&lex ta$k

re?!iring !nder$tanding of the goal$ of the a$$e$$ed ta$k and the criteria for $!cce$$' and the

a+ility to make @!dgment$ a+o!t the relation$hi& of the &rod!ct or &erformance to the$e,

ormative &eer a$$e$$ment i$ likely to involve intelligent ?!e$tioning' co!&led #ith large $elf-

di$clo$!re and there+y a$$e$$ment of !nder$tanding, >eer a$$e$$ment co!ld ena+le the

identification and analy$i$ of error and mi$conce&tion, Thi$ co!ld lead to the identification of

kno#ledge ga&$' andEor engineering their clo$!re thro!gh ex&laining' $im&lification'

clarification' $!mmarizing and cognitive re$tr!ct!ring To&&ing' 1//5, eed+ack #hich i$

corrective andEor confirmatory andEor $!gge$tive co!ld +e more immediate' timely' and

individ!alized, Thi$ might increa$e reflection and generalization to ne# $it!ation$' &romoting

$elf-a$$e$$ment and greater meta-cognitive $elf-a#arene$$, Cognitive and meta-cognitive

+enefit$ might accr!e +efore' d!ring or after the &eer a$$e$$ment, >eer a$$e$$ment demand$$ocial and comm!nication $kill$' negotiation and di&lomacy Biley' 1//;' and can develo&

team#ork $kill$, Learning ho# to give and acce&t critici$m' @!$tify one2$ o#n &o$ition and

re@ect $!gge$tion$ are all !$ef!l tran$fera+le $ocial and a$$ertion $kill$ *arco!lide$ 4

Simkin' 1//1,

>eer a$$e$$ment involve$ $t!dent$ directly in learning' and might &romote a $en$e of

o#ner$hi&' &er$onal re$&on$i+ility and motivation .anrahan 4 I$aac$' 7881, Giving &o$itive

feed+ack fir$t might red!ce anxiety among the a$$e$$ee$ and im&rove acce&tance of negative


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feed+ack, >eer a$$e$$ment might al$o increa$e variety and intere$t' $elf-confidence' and

em&athy #ith other$ for +oth a$$e$$or$ and a$$e$$ee$,

In the ca$e of >*T$' &eer a$$e$$ment i$ the eval!ation of >*T$ +y other >*T$ from the $ame

gro!&, The goal of the &eer a$$e$$ment &roce$$e$ i$ to verify #hether the #ork $ati$fie$ the

acce&ted $tandard$' a$ #ell a$ to &rovide con$tr!ctive feed+ack #hich incl!de$ $!gge$tion$for im&rovement$ .erndon' 788D, The !$e of &eer a$$e$$ment i$ +a$ed on the a$$!m&tion

that &eer$ can recognize each other2$ error$ ?!ickly and ea$ily' and that a larger and more

diver$e gro!& of &eo&le might find more #eakne$$e$ and error$ in a #ork, In their literat!re

revie#' alchikov and Goldfinch 7888 &ointed to the increa$e in $t!dent$2 involvement in

a$$e$$ment acro$$ the $&ectr!m of di$ci&line area$, Thi$ a&&roach' ho#ever' i$ rarely

im&lemented in higher ed!cation Heven+ergen' 7881,

Be$earcher$ al$o re&orted on di$advantage$ d!ring the im&lementation of &eer a$$e$$ment

alchikov 4 Goldfinch' 78889 *c


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a$$e$$ment &romote$ the exchange of idea$ and $erve$ a$ a +a$i$ for g!iding academic

di$co!r$e =erkencotter' 1//; and the interchange of idea$ and methodologie$ re$!lting in a

more refined &rod!ct Bee$e-

g!arantee that the feed+ack &rovided #ill +e acc!rate and val!a+le, A $imilar &ro+lem

concern$ &eer grading alchikov 4 Goldfinch' 7888, It a&&ear$ that the $!+@ectivity

a$$ociated #ith $coring $t!dent$2 #ork i$ infl!enced +y $t!dent$2 different $tandard$ of #hat a

n!merical $core $ho!ld con$i$t of Con#ay' "em+er' Sivan 4 6!' 1//0, ven #hen

$t!dent$ are &rovided #ith clear criteria for eval!ating and marking' there can $till +e a

&ro+lem of $!+@ectivity or exam&le' Heven+ergen 7881 examined &eer a$$e$$ment +y

mathematic$ &ro$&ective teacher$ of &o$ter$ created +y their &eer$ and identified an

o+$erva+le trend among high and lo# achiever$ and the $core$ they gave to their &eer$2 #ork%lo# achiever$ a$$igned higher than average mark$ and often &rovided very generic comment$'

#herea$ high achiever$ tended to a$$ign lo#er than average mark$ and &rovided more

in$ightf!l and critical comment$,

2.4 Assessment of mathematical proofs

Among the vario!$ $kill$ mathematic$ teacher$ have to develo& d!ring their teaching training

are a$$e$$ment $kill$ in general and a$$e$$ment of mathematical &roof$ in &artic!lar,

Be$earcher$ fo!nd that teacher$ and $t!dent$ enco!ntered diffic!ltie$ in &roce$$e$ of &roof$2

eval!ation for correctne$$ Alcock 4 6e+er' 788;9 Selden and Selden' 7880, *ore

$&ecifically' "n!th 7887 fo!nd that in-$ervice high $chool teacher$ acce&ted geometrical

&roof$ according to their format regardle$$ their content, *eaning that' t#o col!mn format

&roof$ of claim$ and @!$tification$ #ere acce&ted a$ correct regardle$$ their content,

Altho!gh a$$e$$ment $kill$ are f!ndamental to effective teaching' teacher$ rarely have

o&&ort!nitie$ to engage in a$$e$$ment de$ign 6e++' 788/, According to Ak$! 7885'

teacher$ have negative attit!de$ to#ard$ the idea of !$ing alternative a$$e$$ment techni?!e$

$!ch a$ formative' &eer or $elf-a$$e$$ment +eca!$e of their traditional tho!ght$ and fixation$,

Ak$! i+id, arg!e$ that one of the rea$on$ for the a+ove &henomenon originate$ in

in$!fficient ex&o$!re of the teacher$ d!ring their training &eriod to the$e techni?!e$, *any

re$earcher$ $!&&ort the idea of incor&orating engagement in alternative a$$e$$ment

techni?!e$' $!ch a$ formative' &eer and $elf-a$$e$$ment activitie$' in &rofe$$ional


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develo&ment &rogram$ of teacher$ Lovemore 4 *T$: a+ility to develo&

a$$e$$ment $kill$ in the context of geometrical &roof, 6e con@ect!re that the >*T$:

engagement in &eer a$$e$$ment of geometrical &roof$ a$ a$$e$$or$ and receiving feed+ack on

their o#n &roof$ &rovided +y cla$$mate$ a$ a$$e$$ee$' can hel& them +oth develo& a$$e$$ment

$kill$ and #iden their mathematical kno#ledge,

3. The tud!

In thi$ $ection #e &re$ent information a+o!t the $t!dy &artici&ant$' a de$cri&tion of the

*ethod co!r$e in #hich the $t!dy took &lace' and an o!tline of the vario!$ &ha$e$ of the

$t!dy, In addition' #e &re$ent the methodology !$ed for the data collection and analy$i$,

3.1 The stud! participants

Sixteen >*T$ from an academic college of ed!cation &artici&ated in the $t!dy, The >*T$gro!& #a$ heterogeneo!$% nine #ere reg!lar >*T$' $t!dying in their third year o!t of fo!r

to#ard a =,A, degree in mathematic$ ed!cation and com&!ter $cience$ or &hy$ic$ for middle

and high $chool, o!r >*T$ #ere in-$ervice mathematic$ teacher$ #ho cho$e to com&lete a

teaching certificate in mathematic$, Of the$e fo!r' t#o had 7 year$ of teaching ex&erience and

the other t#o had over 18 year$ of teaching ex&erience, The remaining three >*T$ #ere

ad!lt$ #ho #ere con$idering a $econd career a$ mathematic$ teacher$,


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3.2 The "ethod course

The method co!r$e in #hich the >*T$ ex&erienced &eer a$$e$$ment #a$ a t#o-$eme$ter

co!r$e and i$ the $econd mathematic$ method co!r$e the >*T$ #ere re?!ired to take, In the

fir$t co!r$e the >*T$ $t!died and di$c!$$ed teaching method$ and a&&roache$ for the

&racticing of vario!$ high $chool mathematic$ to&ic$, In addition' they +ecame ac?!ainted#ith +a$ic &rinci&le$ concerning the a$$e$$ment &roce$$ of an exam' $!ch a$ ho# to $et &ro&er

categorie$ for an exam a$$e$$ment' ho# to +!ild an indicator for the eval!ation of the exam2$

an$#er$' and ho# to eval!ate ta$k$ given to $chool $t!dent$, It i$ #orth noting that thi$

&roce$$ #a$ &erformed on a real exam of a $t!dent ho#ever' the $t!dent did not receive the$e

a$$e$$ment$, In the $econd co!r$e' in #hich the re$earch took &lace' the >*T$ #ere mainly

engaged in mathematical ex&loration$ thro!gh the &roce$$ of &ro+lem &o$ing and &eer

a$$e$$ment activitie$, Since the &roce$$ of &ro+lem &o$ing yielded familiar and !nfamiliar

&ro+lem$' they had to a$$e$$ +oth $ol!tion of familiar a &ro+lem they had already $olved and

!nfamiliar &ro+lem$ &ro+lem$ &o$ed +y their cla$$mate$ that #ere not familiar to them,

3.3 A ten#phase process

The $t!dy &artici&ant$ #ere engaged in a ten-&ha$e &roce$$ #hich la$ted one $eme$ter, The

&ha$e$ #ere a$ follo#$%

1, The >*T$ #ere a$ked to $olve an identical given &ro+lem in #hich they had to con$tr!ct

geometrical &roof$ ig!re 1 and to $!+mit their $ol!tion$ +y e-mail,

7, The >*T$2 $ol!tion$ #ere $canned and $ent thro!gh the co!r$e for!m mail for eval!ation,

ach >*T received t#o anonymo!$ $ol!tion$ for eval!ation and $coring, The >*T$ #ere

a$ked to form!late a li$t of eval!ation criteria' to a$$ign each criterion a n!merical #eight' and

to &rovide @!$tification for each criterion and #eight, In order to receive a !nified form' the

$t!dent$ had to com&lete a form that #a$ &re&ared +y !$ A&&endix 1, In addition' the >*T$

had to $&ecifically $core the #ork according to each criterion and ex&lain the !nderlying

rea$on$ for each $core, The$e eval!ation$ #ere e-mailed to !$,0, ach >*T #a$ e-mailed the t#o anonymo!$ eval!ation$ and $coring of hi$ or her #ork and

#a$ a$ked to reflect on them, In order to receive com&rehen$ive and effective feed+ack' the

$t!dent$ #ere a$ked to addre$$ eight ?!e$tion$ A&&endix 7 concerning vario!$ a$&ect$ of the

eval!ation received from their &eer$ and it$ contri+!tion to their f!t!re #ork,

3, J$ing the (6hat If Fot)( 6IF $trategy =ro#n and 6alter' 1//0 and +a$ed on the

initial &ro+lem' the >*T$ had to $tart an in?!iry &roce$$, The 6IF $trategy i$ +a$ed on the

idea that modifying the attri+!te$ of a given &ro+lem can yield ne# and intrig!ing &ro+lem$'

#hich event!ally may re$!lt in $ome intere$ting inve$tigation$, In thi$ &ha$e' the >*T$ #ere


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a$ked to &rod!ce a li$t of the initial &ro+lem2$ attri+!te$' $!gge$t alternative$ for each of them'

and choo$e one alternative or a com+ination thereof for f!rther mathematical inve$tigation,

Again' the >*T$ e-mailed !$ their ta$k$,

;-D, >ha$e$ 7 and 0 #ere re&eated' +a$ed on the o!t&!t of #ork &rod!ced in >ha$e 3,

, The cho$en ne# &ro+lem #a$ $olved' incl!ding a f!ll #ritten de$cri&tion of the con@ect!re$'$ol!tion$' indeci$ion$' and $o on,

5-/, In order to ena+le the >*T$ to ex&erience a &eer a$$e$$ment &roce$$ for !nfamiliar

&ro+lem$' >ha$e$ 7 and 0 #ere re&eated' +a$ed on the o!t&!t of #ork &rod!ced in >ha$e ,

18, ach >*T #rote a $!mmative eval!ation and final reflection on the entire &roce$$,

3.4 The initial problem

*id&oint$ of $!cce$$ive $ide$ of the &arallelogram A=C< form the &olygon G.,

a,6hat i$ the o+tained fig!re of G.)

+,6hat i$ the ratio +et#een the area of G. and the

area of A=Cand 5-/, The fir$t and the third ta$k$ #ere $imilar in nat!re, In the$e ta$k$ each $t!dent had to

con$tr!ct a criteria $et' $et #eight to each criterion and a$$e$$ t#o geometrical &roof$ done +y

cla$$mate$ according to each criterion and ex&lain the !nderlying rea$on$ for each $core,The

$econd ta$k #a$ different - in thi$ ta$k the >*T$ #ere a$ked to &rod!ce a li$t of the attri+!te$


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of the initial &ro+lem' $!gge$t alternative$ for each of them' and choo$e one or a com+ination

of a fe# alternative$ for f!rther mathematical inve$tigation,

ollo#ing analytic ind!ction Taylor 4 =ogdan' 1//5 and content analy$i$ Fe!endorf'

7887' the data collected from the fir$t and the third ta$k$ #ere analyzed according to the

follo#ing focal &oint$%

i

val!ation criteria% the nat!re of the criteria and it$ relation to the role$ and meaning

of geometrical &roof,

iiF!merical $core$ and feed+ack% an examination of the $core$ a$$igned to each

criterion +y t#o different cla$$mate a$$e$$or$ and the &rovided feed+ack,

iii Beflection$ on the #hole &roce$$ from +oth &er$&ective$% a$$e$$or$ and a$$e$$e$,

iv ffect$ of the &eer a$$e$$ment &roce$$ on the >*T$: mathematical kno#ledge,

4. Results and %iscussion

A$ &revio!$ly mentioned' d!ring the #hole &roce$$' the >*T$ #ere engaged in three &eer

a$$e$$ment ta$k$, Since the $econd &eer a$$e$$ment &ha$e$ ;-D i$ different in nat!re from

the other$' the data analy$i$ foc!$e$ only on the fir$t and the third &eer a$$e$$ment$ ta$k$, In

#hat follo#$' #e &re$ent re$!lt$ and di$c!$$ion of the data received from &ha$e$ 7-0 fir$t

ta$kand 5-/ third ta$k,

4.1 The peer assessment process

The &roce$$ of the fir$t and the third &eer a$$e$$ment$ com&ri$ed the follo#ing% a

con$tr!cting eval!ation criteria for the &eer a$$e$$ment of the geometrical &roof$ and $etting a

n!merical #eighting for each criterion9+determining the &ro&er n!merical $core$ according

to cho$en criteria9 c incl!$ion of @!$tification$ for the $coring &roce$$9 and d reflecting on

the a$$e$$ment &roce$$ +oth a$ a$$e$$or$ and a$$e$$ee$,

ig!re 7 ill!$trate$ a$$e$$ment relation$ #ithin a certain $!+gro!& of >*T$ a certain >*T i$

marked +y KSk, The notation Ai'@ indicate$ that Si a$$e$$e$ the #ork of S@, It $ho!ld +e

mentioned that the $it!ation de$cri+ed in ig!re 7 i$ only ill!$trative and can +e different,

Famely' S1can +e a$$e$$ed +y S0and S7,

In order to examine the develo&ment of a$$e$$ment $kill$' #e looked at the ga& +et#een the

t#o a$$e$$ment$ given +y t#o different >*T$ to the $ame ta$k, A ga& +et#een a$$e$$ment$

refer$ to% i the difference +et#een the con$tr!cted criteria for the a$$e$$ment of the $ame

ta$k for exam&le' the criteria incl!ded in A1'0 and A7'0 9 ii the ga& +et#een the relative

#eight$ a$$igned to t#o identical criteria9 iii the ga& +et#een the $core$ a$$igned to t#o


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identical criteria9 and iv the difference$ +et#een the &rovided feed+ack, *oreover' the

feed+ack #ere com&ared in order to check #hether allegation$ rai$ed +y the a$$e$$ed >*T

a+o!t checking hi$ #ork' #ere $imilar or not to the com&laint$ of the >*T$ that #ere

a$$e$$ed +y him, 6e cho$e to !$e the$e indicator$ $ince red!ction of the$e of ga&$ might

im&ly on the develo&ment of >*T$: a$$e$$ment ca&a+ilitie$ Lavy 4Yadin' 7818,

A S

SA

S0 AD'7

A7'0

7 S

ig!re 7% Schematic de$cri&tion of a$$e$$ment relation$

6e al$o com&ared +et#een the a$$e$$ment done +y each >*T for exam&le' A1'0' and the

a$$e$$ment$ $Ehe received in thi$ ca$e' A3'1and A;'1' a$ #ill +e ela+orated in $ection 3,7,

*oreover' in order to examine the a&&ro&riatene$$ of the con$tr!cted criteria' #e com&ared

the >*T$: criteria $et #ith the one con$tr!cted +y !$, ach one of !$ con$tr!cted her o#n

criteria $et to each of the ta$k$ and after di$c!$$ing the difference$ +et#een o!r criteria li$t$'

#e arrived at a criteria $et that #a$ acce&ta+le +y the t#o of !$, inally' the $core$ given +y

each >*T #ere examined in reference to their academic achievement$ in &revio!$

mathematic$ co!r$e$,

4.1.1

electing e&aluation criteria

A total of 08 a$$e$$ment$ t#o +y each of 13 >*T$ and 1 +y each of the other t#o >*T$

#ere o+tained in the fir$t and the third a$$e$$ment ta$k$, Analy$i$ of the$e a$$e$$ment$

revealed that the >*T$ !$ed 17 different eval!ation criteria in the fir$t ta$k and 5 criteria in

the third one Ta+le 1,

Ta+le 1 $!mmarize$ the li$t of the eval!ation criteria &rovided +y the >*T$ in the fir$t and

3'1

A1'0

;'1


12/34

the third ta$k organized according to vario!$ a$&ect$ of &roof li$ted in col!mn 1, The

n!m+er of >*T$ that $elected the criterion fre?!ency and the average #eight A6


13/34

17

a$$igned to each criterion' and the $tandard deviation$ Sroviding the $horte$t 1D,7; A6M18challenge &o$$i+le &roof

Sy$tematization 10, 6ell-defined &ro+lem / ;D,7; A6M1,S*T$ in the fir$t and the third ta$k$

Aspects of Eval$ation criterion %irst tas& Third tas&

proof %re'$enc Average %re'$enc Average

Str!ct!re 1, C!$tomary &roof 11D5,; A6M08 11 A6M17,51

organization and clarity S


14/34

In the fir$t ta$k each >*T con$tr!cted 0-; criteria and in the third ta$k each >*T con$tr!cted 0-D criteria


15/34

10

>*T$ i$ o+$erved in the third ta$k #ith com&ari$on to the fir$t one concerning the

con$tr!ction of the follo#ing criteria $haded ro#$ in Ta+le 1% 2ea$y and !nder$tanda+le

$ol!tion2 criterion ; and 2form!lation of correct concl!$ion$2criterion 18, 6e attri+!te the$e

difference$ to the different nat!re of the ta$k$, 6hile in the fir$t one the >*T$ had to a$$e$$ a

&roof of a &ro+lem familiar to them' in the third ta$k' they had to a$$e$$ a &roof to a &ro+lemthat #a$ not familiar to them' a$ #a$ reflected in $imilar #ord$ +y five of the >*T$% while

reading the proof to the problem suggested by my classmate ! had difficulties in following his

line of thought, hence, ! included this criterion "#$ in my criteria list2, A$ regard$ to criterion

18 one >*T #rote%2 the problem ! had to assess was %uite surprising and it was important to

me to see whether my classmate was able to draw the problems conclusions in a clear and

correct manner2,

A$ evident from Ta+le 1' 2correct $ol!tion2 criterion 7 i$ the mo$t common criterion in +oth

ta$k$, Thi$ might im&ly on the >*T$: a#arene$$ of the $ignificant role of verification in the

context of mathematical &roof,

The fir$t criterion refer$ to the $tr!ct!re of &roof, The relatively high n!m+er of >*T$ #ho

con$tr!cted thi$ criterion' +oth in the fir$t and the third ta$k$ i$ in line #ith *artin and .arel

1/5/ and "n!th 7887 #ho fo!nd that many &re-$ervice teacher$ acce&t the validity of a

geometry &roof mainly according to it$ $tr!ct!re, Famely' they #o!ld @!dge it valid if the

&roof #a$ in the $tandard t#o-col!mn format and invalid if it #a$ in &aragra&h form'

regardle$$ of it$ mathematical content,

2Correct ?!otation of geometrical theorem$2 criterion 3' 2a$y and !nder$tanda+le $ol!tion2

criterion ;' 2Adding a clear $ketch of the &ro+lem2 criterion D' 2Clear !$e of mathematical

notation$2 criterion and 2Additional ex&lanation$ to claim$2 criterion 5 refer to the

comm!nication role of &roof, 6hile criteria 3' ; and are &hra$ed in a general manner'

criteria D and 5 are more $&ecific, Criterion ; doe$ not $&ecify #ay$ the >*T ha$ to !$e in

order to make the &roof more !nder$tanda+le' ho#ever criteria D and 5 refer to $&ecific

attri+!te$ that might rai$e the !nder$tanda+ility of the &rovided &roof,

2Bea$ona+le organization and clarity2 criterion /' 2orm!lation of correct concl!$ion$2

criterion 18 and 2*T$ recognized the im&ortance of incl!ding

criteria #hich examine the $y$tematic $tr!ct!re of a mathematical &roof in the fir$t ta$k #hile

eleven of them did in the third ta$k, Thi$ increa$e may &oint to the >*T$ recognition of the

$ignificance of the $y$tematization role of &roof d!ring the &eer a$$e$$ment &roce$$, Thi$

increa$e can +e al$o attri+!ted to the difference +et#een the fir$t and the third ta$k, They


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13

fo!nd it ea$ier to follo# and a$$e$$ a &roof' to a &ro+lem #hich i$ not familiar to them' that

#a$ #ritten in a $y$tematic manner than other#i$e,

One >*T &rovided 2>roviding the $horte$t &o$$i+le &roof2criterion 17 #hich refer$ to

intellect!al challenge, It $ho!ld +e noted that the >*T$ had tro!+le generating criteria that

#o!ld reflect the con$ideration of non-mea$!ra+le a$&ect$' $!ch a$ creativity and originality'and the clear ma@ority of >*T$ did not $!gge$t $!ch criteria at all,

The decrea$e in the n!m+er of the con$tr!cted criteria from the fir$t to the third ta$k might +e

ex&lained +y the different nat!re of the ta$k$' &revio!$ly de$cri+ed,

The criteria $et for the third ta$k can +e divided into t#o $!+gro!&$% the fir$t refer$ to

characteri$tic$ of the cho$en &ro+lem and the $econd to the characteri$tic$ of the &rovided

&roof,

A$ to the fir$t $!+gro!&% 26ell-defined &ro+lem29 2Intere$ting &ro+lem2 and 2$!gge$tion$ for

ela+orationEgeneralization2 criteria 10-1; refer to the &ro+lem cho$en +y the >*T$, To more

than half of the >*T$ it #a$ im&ortant that the cho$en &ro+lem to +e #ell defined $ince

thro!gh thi$ criterion the >*T$ checked the mathematical validity of the &ro+lem, It #a$ al$o

im&ortant to them that the &rovided &ro+lem #ill incl!de 2$!gge$tion$ for

ela+orationEgeneralization2, .o#ever to only one >*T #a$ it im&ortant #hether the &ro+lem

i$ intere$ting,

A$ to the $econd $!+gro!&' mo$t of the >*T$ cho$e 2Correct $ol!tion2 criterion 7 #hich

refer$ to the verification role of &roof, Be&etition of thi$ criterion in +oth ta$k$ 1 and 0' &oint$

on it$ &erceived high im&ortance, Concerning the comm!nication role of &roof in the third

ta$k the >*T$ con$tr!cted 2Adding a clear $ketch of the &ro+lem2 criterion D and 2Clear !$e

of mathematical notation$2 criterion , To the $y$tematization role of &roof' many >*T$

&rovided 2orm!lation of correct concl!$ion$2 criterion18, A$ $tated +y de Viller$ 1//8 and

.anna 7888' comm!nication of the mathematical kno#ledge i$ among the main role$ of

mathematical &roof, To facilitate the comm!nication of mathematical arg!ment$ in general

and geometrical &roof$ in &artic!lar' it i$ im&ortant to add $ketche$ and !$e correctly

mathematical notation$,

4.1.2 %etermining the proper scores according to the chosen criteria

The &eer $coring of the fir$t ta$k i$ &re$ented in ig!re 0, ach >*T received t#o $core$

from t#o different cla$$mate$, It can +e clearly $een that in mo$t ca$e$ there i$ a con$idera+le

ga& +et#een the $core$ each >*T received average ga&% 1/,; &er cent' S*T$ had no &revio!$ ex&erience in

&eer a$$e$$ment and fo!nd it diffic!lt to &rovide $core$ according to a $et of eval!ation


17/34

1;

criteria they had to con$tr!ct, The high variance S*T$

received from their cla$$mate$, The$e large ga&$ might al$o re$!lt from the $ignificant

difference$ of n!merical #eight$ the >*T$ a$$igned to $imilar eval!ation criteria, It $ho!ld +e

noted that $ince there #ere t#o >*T$ #ho$e ta$k' for technical rea$on$' #a$ $cored +y onlyone >*T' they #ere not incl!ded in the data &re$ented in ig!re 0,

ig!re 0% >eer $coring of the fir$t a$$e$$ment ta$k &ha$e$ 7-0

The follo#ing are $ome reflection$ regarding the $coring &roce$$ according to the $elected

criteria and n!merical #eight%

B!th% !t was a difficult decision & should ! give points for the correct portions or

deduct points from the total for the mistakes'

No$ef%! could see from the work that the solver knew the solution. The final answer

was correct. The method, however, was unclear. So ! wondered whether ! should

give points for the solution or deduct points for the method(!t was difficult to give

a low score, because ! felt that s)he knew(

=oth B!th and No$ef &$e!donym$ are !$ed thro!gho!t the &a&er referred to the i$$!e of the

$coring techni?!e, 6hen a certain n!merical #eight i$ a$$igned to a criterion' !$!ally it $tand$

for the f!lfilling of thi$ criterion in the eval!ated $ol!tion, =!t #hen the criterion i$ not

com&letely f!lfilled in the $ol!tion' the ?!e$tion of $etting the &ro&ortional &art of the

n!merical #eight to thi$ criterion i$ rai$ed,


18/34

1D

correct &art$ or in rever$e red!ce &oint$ from the total n!merical #eight a$$igned to thi$

criterion) According to No$ef' the $coring &roce$$ +ecome$ even more diffic!lt #hen the

a$$e$$or feel$: that the a$$e$$ee ha$ a certain !nder$tanding +!t fail$ to convey hi$ idea$,

ig!re 3 &re$ent$ the &eer $coring of the third ta$k,

ig!re 3% >eer $coring of the third a$$e$$ment ta$k &ha$e$ 5-/

or the third ta$k' the average ga& +et#een $core$ #a$ 11 &er cent and the $tandard deviation

S*T$ to the $ame

cla$$mate #a$ $ignificantly red!ced, Thi$ re$!lt may im&ly that the >*T$ enhanced their

a$$e$$ment $kill$,

xamination of the $core$ given +oth in the fir$t and the third ta$k and the academic

achievement$ in mathematical co!r$e$ of the a$$e$$or$ $ho#$ that lo# achiever$ tended to

give higher $core$ than high achiever$, Thi$ finding i$ in line #ith Heven+ergen 7881' #ho

fo!nd a $imilar &henomenon among mathematic$ >*T$ #ho #ere engaged in &eer

a$$e$$ment,

4.2 The P"T' re(ections on the feedback

eed+ack i$ among the mo$t critical infl!ence on $t!dent learning .attie 4 Tim&erley'

7811, .ence' a$ &art of the a$$e$$ment &roce$$' the >*T$ #ere a$ked to reflect on the

feed+ack they &rovided to the $coring a$ a$$e$$or$ and on the feed+ack they received on their


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1

o#n #ork a$ a$$e$$ee$, Content analy$i$ Fe!endorf' 7887 of the >*T$: reflection$ revealed

that thi$ feed+ack can +e divided according to their nat!re and their mathematical content,

Thi$ analy$i$ revealed different ty&e$ of feed+ack%

1, eed+ack com&ri$ing of the follo#ing $ym+ol$% 2V2 for a &ro&er &roof9 2V2 for a &artially

correct &roof' and 2P2 for a fa!lty &roof,7, eed+ack com&ri$ing of ver+al ex&lanation$ #hich can +e characterized +y their nat!re and

content,

4.2.1 )eedback comprising of s!mbols

In the fir$t ta$k t#o >*T$ &rovided feed+ack merely +y !$ing $ym+ol$, or exam&le%

Yael% One of the #ork$ I had to a$$e$$ #a$ very good and I @!$t marked the 2v2 $ign at

the +ottom of it,

6hen teacher$ a$$e$$ their $t!dent$2 #ork$ and the #ork$ are correct' many of them areacc!$tomed to react +y adding the 2v2 $ym+ol, .ence' $ince $t!dent$ are !$ed to receive $!ch

comment$ they react $imilarly, Lortie 1/; termed thi$ &henomenon a$ 2a&&rentice$hi& of

o+$ervation2 #hich de$cri+e a $it!ation #here+y &re-$ervice arrive for their training co!r$e$

having $&ent many ho!r$ a$ $choolchildren o+$erving and eval!ating their teacher$ in action,

According to Lortie i+id thi$ a&&rentice$hi& i$ mainly re$&on$i+le for +ehavior$ de$cri+ed

a+ove and many of the &reconce&tion$ that &re-$ervice teacher$ hold a+o!t teaching, It $ho!ld

+e mentioned that #hen the >*T$ enco!ntered a &artially correct or a fa!lty &roof' all of

them added a ver+al ex&lanation to the a$$e$$ment they &rovided,

4.2.2 feedback comprising of &erbal e*planation

In thi$ $ection #e ela+orate on feed+ack com&ri$ing of ver+al ex&lanation$, Thi$ kind of

feed+ack can +e characterized +y it$ nat!re and content,

4.2.2.1 +ature of feedbackat$reof the$e feed+ack refer$ to i$$!e$ $!ch a$ #hether the ex&lanation i #a$ &hra$ed in a

&o$itiveEnegative manner9 ii #a$ $hort andEor general or detailed andEor $&ecific9 iii

incl!ded reinforcing comment$ in ca$e$ #here the &roof i$ correct9 iv Incl!de alternative

$!gge$tion$ in ca$e$ #here the &rovided $ol!tion #a$ not correct or incl!ded g!ideline$ that

might hel& the a$$e$$ee im&rove herEhi$ $ol!tion, In #hat follo#$ #e &re$ent and di$c!$$

re&re$entative ?!otation$ from the >*T$2 reflection$ according to the a+ove categorie$ from

+oth &er$&ective$% a$ a$$e$$or$ and a$ a$$e$$ee$, In develo&ing feed+ack ty&ology' T!n$tall 4

Gi&&$ 1//D identified different ty&e$ of feed+ack #hich &art of them are $imilar to the one$


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15

identified +y !$, The feed+ack ty&e$ that are $imilar to the one$ #e identified are% ver+al and

non-ver+al9 di$tinctly &o$itive or negative9 feed+ack #hich #a$ +a$ed on the !$e of ex&licit

criteria,

Phrased in a positive/negative manner

rom an a$$e$$ee2$ &er$&ective%

*T$

to learn ho# to refer to mi$take$ a$ an o&&ort!nity for learning rather than a$ a fail!re Bace'

1//5,

Include alternative suggestions in cases where the provided solution was not correct or

include guidelines that might help the assessee improve her/his solution

rom an a$$e$$ee2$ &er$&ective%


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1/

$&ecify g!ideline$ that might hel& the a$$e$$ee to correct hi$ #ork' claiming $he acted

differently a$ an a$$e$$or, A certain fr!$tration #a$ ex&re$$ed in ca$e$ #here the >*T$ felt

they acted differently from their a$$e$$or$,

Include reinforcing/harsh comments

rom an a$$e$$ee2$ &er$&ective%Yael% I #a$ very &ro!d to receive the comment% 2xcellent #ork' I en@oyed reading

yo!r #ork2,

B!th% I felt h!miliated #hen I received the follo#ing% 2Behear$e the relevant $!+@ect

matter and com&lete yo!r $ol!tion' ela+orate on the vario!$ $ol!tion $tage$ and add

@!$tification$2, It co!ld +e &hra$ed more gently,

rom an a$$e$$or2$ &er$&ective%

Yael% One of the #ork$ I had to a$$e$$ #a$ very good and I @!$t marked the 2v2 $ign at

the +ottom of it, After I received my #ork I realized that I $ho!ld have acted

differently,

B!th% I read again and again the @!$tification$ I gave to avoid a $it!ation in #hich thea$$e$$ee co!ld +e offended,

In her reflection' +oth a$ a$$e$$ee and a$$e$$or' Yael refer$ to another a$&ect of the nat!re of

the @!$tification #hich refer$ to the ?!e$tion #hether or not to incl!de reinforcingEhar$h

comment$ in the @!$tification $aying that $he #a$ &ro!d to receive &rai$e on her excellent

$ol!tion and recalling $he did not act $imilarly a$ an a$$e$$or, rom the other $ide of the

$&ectr!m' B!th com&lained a+o!t the har$h nat!re of the @!$tification $he received on her

#ork $ince the$e comment$ conveyed an ex&licit critici$m concerning her ma$tery of the

$!+@ect matter,

All the a+ove reflection$ refer to vario!$ a$&ect$ concerning the nat!re of the &rovided

@!$tification$, 6hile $ome of the >*T$ re&orted acting differently from the #ay they #ere

treated a$ a$$e$$ee$' other$ realize they acted $imilarly a$ a$$e$$or$ and +ecame a#are of their

fa!lt$, A$ a$$e$$ee$' #hen referring to the nat!re of the received @!$tification$' the >*T$

referred mainly to affective i$$!e$, or exam&le' they rai$ed a$&ect$ $!ch a$ general and $hort

@!$tification that can leave the reader #ith the feeling that hi$ #ork #a$ not treated $erio!$ly,

The lack of reinforcing comment$ or the a+$ence of alternative $!gge$tion$ to reach the


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78

correct an$#er left them #ith feeling$ of fr!$tration, Acting +oth a$ a$$e$$or$ and a$$e$$ee$

confronted the >*T$ #ith their +ehavior a$ a$$e$$or$ and #ith their ex&ectation$ from their

cla$$mate$ #hile acting a$ a$$e$$or$,

4.2.2.2 ,ontent of feedback

*ontent of the feed+ack refer to a$&ect$ of role$ and meaning$ of geometrical &roof a$

follo#$% i correct $ol!tion9 ii !nder$tanda+le $ol!tion9 iii correct ?!otation of geometrical

theorem$9 iv adding a clear $ketch of the &ro+lem9 v clear !$e of mathematical notation$9

vi additional ex&lanation$ to claim$9 vii form!lation of correct concl!$ion$9 viii

demon$tration of connectedne$$ +et#een mathematical area$, Bevie# of the feed+ack a$

regard$ to their content revealed they #ere e$&ecially from the a$$e$$or &er$&ective,

Correct solution

lla% ir$t of all I have to verify that the &roof I have to a$$e$$ i$ mathematically

correct and clear,

lla' a$ many of the other >*T$ in +oth ta$k$' i$ a#are to the fact that a+ove all the #ork $he

ha$ to a$$e$$ m!$t +e correct, It i$ al$o in line #ith the re$earch literat!re a$$erting that fir$t of

all the role of a &roof i$ to verifyEconfirm a given $tatement >olya' 1/;39 =ell' 1/D9 de

Villier$' 1//8' 1//19 .anna' 7888, Thi$ criterion i$ a$$ociated #ith the category of

verification,

Understandable solution

*T$ incl!ded thi$ criterion in their criteria $et #hile none did in the

third ta$k, The &o$$i+le ex&lanation to thi$ &henomenon i$ that $ince in the fir$t ta$k all the

>*T$ $olved the $ame &ro+lem' #hen they had to a$$e$$ a $ol!tion #hich #a$ different from

the one they did' they enco!nter diffic!ltie$ in !nder$tanding it, In the third ta$k each $t!dent

foc!$ed in a different &ro+lem' $o they tried to !nder$tand the &rovided $ol!tion #itho!t

having in mind the &rior $ol!tion for it, To facilitate the !nder$tanding of one:$ &rovided

&roof' $Ehe ha$ to #rite it in an !nder$tanda+le #ay, .ence thi$ criterion can +e cla$$ified

!nder the comm!nication category,

Correct quotation of geometrical theorems

B!th% In ca$e the ?!otation of geometrical theorem #a$ incorrect' I directed thea$$e$$ee to the &ro&er &lace in the $!+@ect matter,


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71

ive $t!dent$ in the fir$t ta$k incl!ded thi$ criterion in their criteria $et #hile none of them in

the third ta$k, Thi$ can +e ex&lained +y the fact that the >*T$ admitted they cho$e the criteria

$et afterthey read the cla$$mate$: $ol!tion$ and the a+$ence of thi$ criterion in the &roof$ of

the third ta$k $!gge$t$ they fo!nd it !nnece$$ary, Thi$ criterion can +e attri+!ted to the

category of comm!nication $ince in order to convince other$ that the &rovided &roof i$correct' one $ho!ld e$ta+li$h hi$ claim$ +y adding correct ?!otation of the relevant

geometrical theorem$,

Adding a clear sketch of the problem

*T$ #ill incl!de $!ch a criterion in their criteria $et, A &o$$i+le

ex&lanation of thi$ &henomenon i$ the >*T$2 lack of ex&erience in a$$e$$ment of geometrical

&roof$' or that +eca!$e they &erceived the $ketch to +e an o+vio!$ &art of the &roof' they

a$$!med it #o!ld +e &re$ent in each geometrical &roof, The !nder$tanding of one2$ &roof +y

other$ can +e facilitated #hen the &rovided &roof incl!de$ a clear $ketch of the &ro+lem,

.ence thi$ criterion can +e a$cri+ed to the category of comm!nication,

Clear use of mathematical notations

Gila% It i$ very diffic!lt to trace a &roof co!r$e that i$ #ritten only +y

mathematical notation$, One $ho!ld add ex&lanation$ to make it more reada+le,

Only a fe# >*T$ fo!nd thi$ criterion im&ortant to incl!de in their criteria $et in +oth ta$k$,

Since a geometrical &roof can +e #ritten +oth in a formal #ay +y !$ing mathematical

notation$ or in an informal #ay +y !$ing #ord$' and +oth #ay$ are !$!ally acce&ta+le +y

in$tr!ctor$' mo$t of the >*T$ did not recognize the need to incl!de $!ch a criterion, Thi$

criterion can +e recognized a$ +eing &art of the category of comm!nication $ince the main

&!r&o$e of the #ay the &roof i$ &hra$ed i$ to comm!nicate #ith the mathematical comm!nity

#hich in thi$ ca$e incl!de$ cla$$mate$ and the cla$$ in$tr!ctor,

Additional explanations to claims

Bonit% I en@oyed reading one of the &roof$ $ince each $te& of it #a$ clearly@!$tified +oth +y relevant mathematical theorem$ and +y additional ex&lanation$,


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77

Only t#o of the >*T$ incl!ded thi$ criterion in the criteria $et for the fir$t ta$k and none for

the third ta$k, Since the >*T$ read the &roof$ they had to a$$e$$ and did not find them in

mo$t ca$e$ diffic!lt to !nder$tand' they did not find thi$ criterion im&ortant to +e incl!ded in

their criteria $et, Additional ex&lanation$ to claim$ enhance the !nder$tanding of the &rovided

&roof +y other$ and hence are &art of the comm!nication category,ormulation of correct conclusions

lla% The fir$t t#o thing$ that #ere im&ortant to me in the &roof I had to a$$e$$

#ere that the &roof #ill +e correct and the concl!$ion$ at every $tage #ill +e al$o

correct and @!$tified,

6hile only one of the >*T$ fo!nd thi$ criterion im&ortant to +e incl!ded in hi$ criteria $et in

the fir$t ta$k' 11 >*T$ did in the third one, A$ #a$ $tated +efore' the t#o ta$k$ differ in their

nat!re, In the fir$t ta$k they had to a$$e$$ a &ro+lem they had already $olved and it$ correct

concl!$ion #a$ o+vio!$ for them, .o#ever' in the third ta$k each >*T cho$e a different

&ro+lem to $olve' hence the a$$e$$or$ #ere not familiar #ith the &ro+lem$ they had to a$$e$$9

they fo!nd thi$ criterion to +e cr!cial $ince thro!gh it$ !$e they co!ld a$$e$$ the validity and

the mathematical correctne$$ of the !nfamiliar &ro+lem, Thi$ criterion can +e a$cri+ed to the

category of $y$tematization $ince one of the re?!ired attri+!te$ of a $tr!ct!red &roof i$ the

&roviding of correct concl!$ion$,

!emonstration of connectedness between mathematical areas

*T$ referred to vario!$

a$&ect$ of geometrical &roof $!ch a$ verification' comm!nication and $y$tematization, A$ #a$


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70

$tated +y the >*T$' many of the criteria #ere added to the criteria $et after they read the

&roof$ to +e a$$e$$ed,

4.3 The e-ect of the peer assessment on the P"Ts' mathematical ork

the fir$t ta$k' made them realize there are many different #ay$ to $olve the &ro+lem, Second'

o+$erving a $ol!tion #hich i$ more elegant and effective from the one they had &rovided'

made them rethink the $ol!tion they had &rovided and look for #ay$ they co!ld im&rove it, In

the third ta$k' the a$$e$$ment &roce$$ ena+led the $t!dent$ to +e ex&o$ed to variety of

&ro+lem$ and $ol!tion$ #hich ena+led them to ex&and their mathematical kno#ledge,

J$!ally the a$$e$$ment &roce$$ i$ done +y the teacher #ho$e mathematical kno#ledge i$ #ay

+eyond that of her $t!dent$ and a$ a re$!lt the a$$e$$ment &roce$$ i$ often done offhand

.an$en' 1//19 *c*illan 4 or$yth' 1//1, In o!r $t!dy the $it!ation i$ different - the >*T$

act at the $ame time +oth a$ a$$e$$or$ taking the teacher:$ role and a$ $t!dent$ +eing

a$$e$$ed +y cla$$mate$, 6hen taking the teacher:$ role in thi$ $it!ation' they #ere ex&o$ed to

different $ol!tion $trategie$ #hich ena+led them to vie# ne# mathematical idea$ they did not

think of, Thi$ $it!ation #a$ de$cri+ed +y Bee$e-

reflected +y 10 of the >*T$ in $imilar #ord$%

Bonit% 6hen I read one of the &roof$ I had to a$$e$$ I #a$ im&re$$ed +y it$

$im&licity #hich made it very ea$y to follo#' and d!ring the #hole reading I

tho!ght #hy I co!ldn:t think the $ame #ay,

Thi$ ex&o$!re #a$ inten$ified in the third ta$k in #hich each >*T had to a$$e$$ t#o ne#

&ro+lem$ that #ere totally different from the one $Ehe cho$e to foc!$ on' a &roce$$ that

ena+led herEhim to ex&and herEhi$ mathematical kno#ledge, A$ #a$ reflected +y $even of the

>*T$ in $imilar #ord$%

B!th% 6hen I read the &ro+lem$ of my cla$$mate$ in the third ta$k it made me

think a+o!t the variety of the h!man thinking and the richne$$ of the mathematic$,

I #ondered ho# $t!dent$ in my cla$$ #ith $imilar kno#ledge to mine can come

#ith $o intere$ting idea$ #hich are $o different from the one$ I cho$e,

Six of the >*T$ referred to the follo#ing i$$!e%


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*T$: do!+t$ regarding

their a+ility to a$$e$$ the mathematical correctne$$ of $ol!tion$, Three of them referred to thi$

i$$!e in $imilar #ord$%

Gila% In the third ta$k I had to a$$e$$ ne# &ro+lem$ that #ere not familiar to me

and I #a$ not $!re #hether the &rovided $ol!tion$ #ere mathematically correct $o

I t!rned to the co!r$e in$tr!ctor for hel&,

In her reflection Gila referred to the &ro+lematic $it!ation in #hich the a$$e$$or doe$ not

&o$$e$$ the re?!i$ite mathematical kno#ledge to @!dge the correctne$$ of the $ol!tion, In

ca$e$ like thi$ the >*T$ cho$e to +e hel&ed +y the co!r$e in$tr!ctor, Thi$ &ro+lem can occ!r

#hen $chool $t!dent$ are engaged in &eer a$$e$$ment activitie$' ho#ever $ince in o!r ca$e the

&eer a$$e$$ment #a$ done +y >*T$ that #ere on the verge of +ecoming mathematic$

teacher$' #e +elieved that the occ!rrence of $!ch ca$e$ #ill +e infre?!ent,

4.4 Assessment of the /rst task &s. the third task 0 P"Ts' e*periences

After the third ta$k #e a$ked the >*T$ to reflect on the #hole &roce$$ and to refer to the

a$$e$$ment$ they had to &rovide and receive, Their reflection$ refer to three main i$$!e$% the

con$tr!ction of the criteria $et9 their ex&erience a$ a$$e$$or$ and their ex&erience a$ a$$e$$ee$,

4.4.1,onstruction of the criteria set

*ore than t#o third$ of the >*T$ re&orted on enco!ntering diffic!ltie$ in con$tr!cting

criteria $et in the fir$t ta$k a$cri+ing their diffic!ltie$ to their lack of &rior ex&erience, A$ #a$

reflected in $imilar #ord$%

B!th% I had diffic!ltie$ deciding #hether the categorie$ $ho!ld relate to the

method$' techni?!e$' line of tho!ght' or &erha&$ the final $ol!tionR, .o# to

#eight them) 6hich i$ more im&ortant) 6hich de$erve$ a higher #eight)

rom the a+ove ?!ote it a&&ear$ that the >*T$ #ere !ncertain a$ to the nat!re of the criteria

$et - #hat $ho!ld they a$$e$$) The >*T$ al$o had he$itation$ concerning the #eight they

$ho!ld a$$ign to each category' a$$!ming that the greater the criterion2$ #eight i$' the greater


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7;

i$ it$ im&ortance to the &roce$$ of $olving the geometrical &ro+lem, The >*T$ arg!ed that the

#eight it$elf (bears a certain message(9 namely' (if a teacher wishes to educate his students

to work in a certain way, s)he should weight specific categories in accordance(,

Feverthele$$' in +oth ta$k$ the fir$t and the third the >*T$ refer to role$ and meaning$ of

&roof #hile con$tr!cting their criteria $et, In the fir$t ta$k the >*T$ con$tr!cted 17 criteriareferring to role$ and meaning$ of mathematical &roof #hile in the third ta$k there #ere only

nine criteria, A$ #a$ &revio!$ly mentioned' in +oth ta$k$' the >*T$ re&orted that the

con$tr!ction of the criteria $et #a$ done after they had read the &roof$ they had to a$$e$$,

.ence thi$ co!ld ex&lain #hy &art of the criteria of the fir$t ta$k did not a&&ear in the li$t of

the third ta$k e,g, criterion 0, O+$ervation of all the &rovided &roof$ in the third ta$k

revealed that they did not incl!de n!merical val!e$9 hence' the >*T$ did not find the criterion

referring to calc!lation error$ to +e nece$$ary, A$ to criteria 5' / and 11 #e #ondered #hy the

>*T$ did not incl!de them a$ #ell in their criteria $et for the a$$e$$ment of the third ta$k, A

&o$$i+le ex&lanation i$ that the$e criteria #ere !$ed only +y >*T$ #ho fo!nd reference in the

&roof$ they had to a$$e$$ in the fir$t ta$k +!t not in the third one,

After the third ta$k' ho#ever' there #ere no reference$ in the >*T$2 reflection$ regarding the

enco!ntering of diffic!ltie$ #hile con$tr!cting the criteria $et, The fact that the >*T$ did not

$&ecify any diffic!ltie$ regarding the con$tr!ction of the criteria $et for the third ta$k may

im&ly on the develo&ing of the con$tr!ction of criteria $et $kill$,

A$ #e mentioned earlier' #e al$o con$tr!cted o!r criteria $et for the a$$e$$ment of the fir$t

and the third ta$k$ and com&ared the >*T$ criteria $et of each ta$k #ith the one con$tr!cted

+y !$ re$&ectively, The$e com&ari$on$ revealed that +oth li$t$ #ere very $imilar, .o#ever'

$ince not all the >*T$ con$tr!cted the $ame criteria $et $ee the fre?!ency col!mn in Ta+le 1'

#e +elieve that there i$ more to +e done in the &roce$$ of the >*T$2 develo&ing their a+ilitie$

to con$tr!ct criteria $et according to the role$ and meaning$ of &roof for the a$$e$$ment of

geometrical &roof$,

4.4.2The P"Ts' e*perience as assessors

The >*T$ referred in their reflection$ to the feed+ack they &rovided a$ a$$e$$or$, The >*T$

#ho re&orted that they changed their feed+ack from the fir$t to the third ta$k #ere tho$e #ho

had received feed+ack on the fir$t ta$k that #a$ $ignificantly different from the feed+ack they

had &rovided a$ a$$e$$or$, A$ #a$ reflected in $imilar #ord$ +y five >*T$%

lla% After I read the feed+ack on my #ork' #hich #a$ detailed and incl!ded

hint$ for im&rovement' I realized that the feed+ack I &rovided #a$ not effective,


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.ence in the third ta$k I inve$ted many effort$ to &rovide a detailed and a +etter

feed+ack,

In ca$e$ #here the $it!ation #a$ o&&o$ite' namely' feed+ack &rovided a$ a$$e$$or #a$

$ignificantly +etter than feed+ack received a$ a$$e$$ee' the >*T$ reaction #a$ ex&re$$ion$ of

fr!$tration, They ex&ected feed+ack $imilar in nat!re to #hat they had &rovided and #eredi$a&&ointed, x&eriencing di$a&&ointment made them recognize the im&ortance of the

feed+ack a$ #a$ reflected +y fo!r >*T$ in $imilar #ord$%

B!th% In the a$$e$$ment$ I gave I did my +e$t to @!$tify each $core caref!lly and

altho!gh my #ork did not receive a $imilar attention' I $till +elieve that a$$e$$ment of

$t!dent$ $ho!ld +e done $erio!$ly $ince they can learn a lot from the feed+ack they

receive,

rom the >*T$2 reflection$ after the third ta$k a$ a$$e$$ee$' #e can learn that com&laint$

a+o!t the nat!re of the received feed+ack +ecame rare, In thi$ ca$e' #e can infer that the

>*T$ +ecome a#are to the con$e?!ence$ of giving a &oor or negative feed+ack in the

a$$e$$ment &roce$$,

4.4.3The P"Ts' e*perience as assessees

Beceiving detailed and !$ef!l feed+ack or har$h or minimal feed+ack led to t#o kind$ of

reaction$ from the a$$e$$e$, A$ to the fir$t kind' >*T$ #ho received $!ch a feed+ack &rai$ed

it $&ecifying it$ merit$, Three >*T$ $aid in $imilar #ord$ that (such a detailed feedback

helped me to understand why the score ! received was lowand to accept it. Moreover, it gave

me the feeling that my work was treated seriously(, On the other hand' har$h or minimal

feed+ack #a$ &erceived +y the >*T$ a$ negligence, The >*T$2 reaction$ to the har$h

feed+ack they received a$ a$$e$$ee$ #ere ex&re$$ion$ of fr!$tration and di$a&&ointment, They

inter&reted har$h feed+ack to +e a re$!lt of their a$$e$$or$2 lack of tolerance to#ard$ their

#eakne$$e$ and diffic!ltie$ a$ #a$ reflected +y one >*T( the intolerant expressions in one

of the feedback ! received left me with the feeling that the student who had to assess my work

had no empathy with my difficulties+.

Looking at the reflection$ after the third ta$k reveal$ that there #ere no indication$ regarding

the receiving of har$h feed+ack, 6e +elieve that the >*T$ ex&o$!re to +oth $ide$ of the coin

Qacting at the $ame time a$ a$$e$$or$ and a$$e$$e$ confronted them #ith $trength$ and

#eakne$$e$ of the a$$e$$ment &roce$$ and a$ a re$!lt ena+led them to develo& their $en$itivity

to vario!$ a$&ect$ of the a$$e$$ment &roce$$,

Feverthele$$' revie# of the >*T$2 a$$e$$ment of the fir$t ta$k reveal$ that mo$t of the

feed+ack #a$ &hra$ed in a &olite manner, In the >*T$ reflection$ on the &eer a$$e$$ment

&roce$$ mo$t of them referred to the &ro+lematic $it!ation in #hich they have to a$$e$$ and +e


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a$$e$$ed +y their cla$$mate$, .aving thi$ $it!ation in mind' made them choo$e their feed+ack

caref!lly in a #ay that' on the one hand' #ill not +e offen$ively &hra$ed +!t' on the other

hand' #ill ena+le the correction of the a$$e$$ed $ol!tion, Thi$ i$ in line #ith alchikov 1//;

#ho fo!nd that $t!dent$ have diffic!ltie$ in a$$e$$ing their friend$ e$&ecially in $mall $ocially

cohe$ive gro!&$,xamination of the >*T$2 a$$e$$ment$ of the third ta$k $ho#$ an im&rovement in the

ex&lanation$ &rovided, Thi$ im&rovement can +e o+$erved +y the decrea$e of the ga& +et#een

the criteria $et &rovided +y the >*T$ and the one con$tr!cted +y !$, There #ere no com&lain$

a+o!t receiving har$h feed+ack and mo$t of the >*T$ re&orted the feed+ack they received

incl!ded reference$ to the correct &art$ of the &roof' and only then $!gge$tion of alternative$

to correct the fa!lty &art$ of it,

$. ,oncluding Remarks

rom the a+ove re$!lt$ and di$c!$$ion it can +e concl!ded that%

ir$t' com&ari$on +et#een the fir$t and the third a$$e$$ment ta$k$ revealed that the >*T$

develo&ed their a+ilitie$ to $elect a criteria $et that addre$$e$ the role$ and meaning of

mathematical &roof and a$$igned a n!merical #eight to each criterion according to it$

im&ortance, A decrea$e in the n!m+er of the criteria and in the $tandard deviation of the

a$$igned n!merical #eight$ #a$ o+$erved +et#een the fir$t and the third a$$e$$ment ta$k$, In

addition' a decrea$ing tendency #a$ o+$erved in the ga& +et#een the $core$ each #ork

received from t#o different >*T$, Al$o' +et#een the fir$t and the third ta$k$' a decrea$ing

tendency #a$ o+$erved in the ga& +et#een the criteria $et con$tr!cted +y the >*T$ and the

one con$tr!cted +y !$,

Second' the >*T$2 reflection$ on the a$$e$$ment &roce$$ $ho# that their engagement in &eer

a$$e$$ment hel&ed them develo& a $en$e of the role$ and meaning$ of geometrical &roof$ and

to +e $en$itive to the feed+ack they &rovide a$ a$$e$$or$, Thi$ develo&ment may indicate an

im&rovement in their a$$e$$ment $kill$ that are e$$ential to their &rofe$$ional develo&ment a$

f!t!re mathematic$ teacher$ Lovemore 4 *T$ refer to the feed+ack they &rovided a$ a$$e$$or$ and

received a$ a$$e$$ee$, rom a$$e$$or:$ &er$&ective$' the >*T$ referred to the categorie$

according to #hich they a$$e$$ed their cla$$mate$ #ork, *o$t of the$e categorie$ refer to

role$ and meaning$ of geometrical &roof, .o#ever' from a$$e$$ee$: &er$&ective$' the >*T$

referred mainly to the nat!re of the feed+ack, Famely' the #riting $tyle of the feed+ack' theincl!$ion of har$hEenco!raging comment$ and $o on, rom the >*T$2 reflection$ on the #hole


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&roce$$ #e may $ay that +y ex&o$ing the >*T$ to &eer a$$e$$ment in #hich they had to

&rovide and receive feed+ack $im!ltaneo!$ly' hel&ed them develo& their $en$itivity to vario!$

a$&ect$ of a$$e$$ment,

o!rth' the richne$$ of the ex&re$$ed reflection$ indicate$ the &o#erf!l im&act the

engagement in &eer a$$e$$ment' +oth a$ a$$e$$or$ and a$$e$$ee$' had on the >*T$,ifth' the effect of the &eer a$$e$$ment on the >*T$: mathematical #ork came to fr!ition in

them realizing that there are many different #ay$ to $olve the &ro+lem and in rethinking of the

$ol!tion they had &rovided' looking for #ay$ they co!ld im&rove it, They came to the fir$t

concl!$ion after o+$erving different $ol!tion$ to the &ro+lem in the fir$t ta$k #hich #a$

different from the one they had &rovided, They came to the $econd concl!$ion after o+$erving

$ol!tion$ #hich #ere more elegant and effective than the one they had &rovided, In the third

ta$k' the a$$e$$ment &roce$$ ex&o$ed $t!dent$ to a variety of &ro+lem$ and $ol!tion$ #hich

ena+led them to ex&and their mathematical kno#ledge, =y revie#ing the #ork$ of their &eer$

they gained in$ight$ into their o#n &erformance =erkencotter' 1//;,

inally' in order to $!+$tantiate the finding$ of thi$ $t!dy' thi$ re$earch $ho!ld +e re&eated

#ith a larger gro!& of >*T$,

ReferencesAk$!' .,., 7885, A $t!dy on the determination of $econdary $chool mathematic$ teacher$2

vie#$ on alternative a$$e$$ment,umanity - Social Sciences ournal' /7' 5/-/D,

Alcock' L, 4 6e+er' ",788;, >roof validation in real analy$i$% inferring and eval!ating

#arrant$,ournal of Mathematical 0ehavior' 12 7' 17;-103,

=angert-


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=ro#n ' S,I, 4 6alter' *,I, 1//0, >ro+lem &o$ing in mathematic$ ed!cation, In Ste&hen I,

=ro#n 4 *arion I, 6alter d$,'9roblem 9osing: 3eflection and Applications &&, 1D-

7..ill$dale' FN' La#rence rl+a!m A$$ociate$,

Cole'


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"n!th' , 7887, Secondary $chool mathematic$ teacher$2 conce&tion$ of &roof, ournal for

3eseach in Mathematics 4ducation' // ;' 0/-38;,

Lavy' I, 4 Yadin' A, 7818, Team-+a$ed &eer revie# a$ form of ormative A$$e$$ment - the

ca$e of Sy$tem$ Analy$i$ and ari$' A,., 7881, Cla$$room a&&lication$ of re$earch on $elf-reg!lated

learning,4ducational 9sychologist' /5 7' 5/-181,

>olya' G, 1/;3,Mathematics and plausible reasoning, >rinceton Jniver$ity >re$$,


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Bace' >, 1//5, >ractical &ointer$ in &eer a$$e$$ment, In S, =ro#n d,'9eer Assessment in

9ractice &&, 110-177.=irmingham' S

Biley' S,*, 1//;, >eer re$&on$e$ in an SL #riting cla$$% St!dent interaction and

$!+$e?!ent draft revi$ion,;issertation Abstracts !nternational' #5' 0801,

Selden' A, 4 Selden' N, 7880, Validation$ of &roof$ #ritten a$ text$% Can !ndergrad!ate tell

#hether an arg!ment &rove$ a theorem)ournal for 3esearch in Mathematics 4ducation'

/5 1' 3-0D,

Sl!i@$man$' rin$' , 788D, A conce&t!al frame#ork for integrating &eer a$$e$$ment in

teacher ed!cation, Studies in 4ducational 4valuation/1' D-77,

Stylianide$' A, N, 4 Stylianide$' G, N, 788/, >roof con$tr!ction and eval!ation'4ducationalStudies in Mathematics, 71' 70-7;0,

T!n$tall' >, 4 Gi&&$' C, 1//D, Teacher feed+ack to yo!ng childten in formative a$$e$$ment%

A ty&ology, 0ritish 4ducational 3esearch ournal, 113' 05/-383,

Taylor' S,N, 4 =ogdan' B, 1//5,!ntroduction to @ualitative 3esearch Methods.Fe# York%

Nohn 6iley 4 Son$,

To&&ing' ", 1//D,4ffective peer tutoring in further and higher education' Sa&er /;

=irmingham' S!+li$her$'


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6e+er' ", 4 Alcock' L, 7883, Semantic and $yntactic &roof &rod!ction$, 4ducational

studies in Mathematics' #5' 78/-703,

Heven+ergen' B, 7881, >eer a$$e$$ment of $t!dent con$tr!cted &o$ter$% a$$e$$ment

alternative$ in &re$ervice mathematic$ ed!cation, ournal of Mathematics Teacher

4ducation, 2'/;Q110,

Appendi+ 1:

Wor& assessment form , phase 1

A$$e$$or name%

engaging prospective teachers in peer assessment as both.rtf

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