Noyes Mathematics Counts for What

8/12/2019 Noyes Mathematics Counts for What

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MATHEMATICS COUNTS FOR WHAT?

RETHINKING THE MATHEMATICS CURRICULUM IN ENGLAND

Andrew Noyes

University of Nottingham, UK

Abstract

This aer draws on !uroean and U" critica# (mathematics) educationtraditions to argue that the mathematics curricu#um in !ng#and is in urgentneed of reconcetua#isation if a more engaging and socia##y $ust mathematicseducation is to %e offered to young eo#e in the future. Throughout the

history of state schoo#ing in !ng#and there have %een cometing agendas forschoo# mathematics and its sacred osition as the gatekeeer to manyeducation, em#oyment and #ife oortunities is now firm#y esta%#ished. A%rief history and criti&ue of the curricu#um #eads to consideration of thecurrent and a#ternative curricu#um drivers. ' argue that more radica#traditions inc#uding critica#mathematica# #iteracy, mathematics for socia#$ustice, genera# (citienshi) education or allgemeinbildung shou#d %e centra#to a thorough rethink of the mathematics curricu#um in !ng#and.

Introduction: formatting mathematicsor many years, socio#ogists of mathematics education, a minority of scho#arsin this fie#d, have argued that mathematics acts as a gatekeeer. The oft&uoted *o#mink (+-), for eam#e, e#ains that mathematics /more thanany other su%$ect, has %een cast in the ro#e as an 0o%$ective1 $udge, in orderto decide who in society 0can1 and who 0cannot1. 't therefore serves as thegatekeeer to articiation to the decision making rocesses in society2. 3egoes on to say that /to deny some the access to articiation in mathematicsis then a#so to determine, a priori, who wi## move ahead and who wi## stay%ehind2 (.4+). This otentia# is one asect of what is descri%ed %y

"kovsmose (+5) as the /formatting ower of mathematics2, which has aninvisi%#e ro#e in the structuration of society (. +). "uch notions of societa#structuring are a centra# concern for many socio#ogists and 6ourdieu, whowrote etensive#y on the reroductive otentia# of educationa# systems(6ourdieu, +57 6ourdieu 8 9asseron, +::7 6ourdieu 8 "aint;artin, +:-),recognised the uni&ue ower of schoo# mathematics, articu#ar#y through theeamination system=

ften with a sycho#ogica# %ruta#ity that nothing can attenuate, theschoo# institution #ays down its fina# $udgements and its verdicts, fromwhich there is no aea#, ranking a## students in a uni&ue hierarchy ofa## forms of ece##ence, nowadays dominated %y a sing#e disci#ine,mathematics. (6ourdieu, +5, . ?5)

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A#though his contet was not !ng#and, 6ourdieu descri%ed such %oundaries asthe @enera# ertificate of "econdary !ducation (@"!) BC %order#ine as a/magica# thresho#d2 where%y two students, searated %y the narrowest ofmargins, have their future educationa# and #ife oortunities differentiated inan instant. "uch educationa# magic divides the /rofane2 ;grade C and %e#ow;

from the /sacred2 ;grades and a%ove (to use Curkheim2s terms). This is oneasect of the ower of mathematics as current#y constructed in thecurricu#um. The recent decision that mathematics and !ng#ish must %eattained at grade or a%ove if a student is to reorted with those attaining adi#oma (4 or more AD; grades) on#y serves to maintain the status ofmathematics as the gatekeeer. Eith an ar%itrari#y maintained #owerroortion of the cohort a%#e to achieve the re&uired grade in mathematics, aosition made ossi%#e %y the eretuated myth that mathematics isre#ative#y difficu#t, many more students wi## find mathematics to %e thestum%#ing %#ock for their future education and em#oyment #ans. To

i##ustrate, consider data from the ?FF+ @"! cohort.

Achieved AD; (G)

6oys @ir#s tota#

A## su%$ects H : :-

!ng#ish -H H? 4-

mathematics -H - -5

!ng#ish andmaths

I5 -H -?

!ng#ish notmaths

5 +H +?

maths not!ng#ish

5 I H

Fig 1: GCSE A*-C results from 2001 b gender

These figures do not te## us a%out students attaining 4 or more AD; inc#udingmaths and !ng#ish %ut they do indicate what might haen when they %ecomea necessary art of a di#oma system. !&ua# num%er of %oys wou#d noto%tain the di#oma due to not achieving the grade in maths or in !ng#ish.or the gir#s the icture is &uite different with mathematics %eing much more#ike#y to %e the stum%#ing %#ock. Ee might eect to see simi#ar ine&ua#ities if#ooking at c#ass or ethnic grous.

As a resu#t of dee#y em%edded cu#tura# %e#iefs a%out schoo# mathematics,generations of students have had a #ess than ositive eerience of thesu%$ect, #eading to ongoing discussion in the UK education ress, and %etween

a very sma## num%er of scho#ars, a%out whether or not mathematics shou#dremain a comu#sory comonent of the curricu#um. 3owever, if we fo##ow

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*o#mink2s argument then a##owing some students to not take a @"! inmathematics wou#d guarantee their consignment to the ranks of themathematica##y under&ua#ified. 6ut a%out ha#f of students wi## ro%a%#y notget a grade . "o, the argument fo##ows that for those who are very un#ike#yto get the @"! grade the curricu#um is sim#y not aroriate. This %egs

the &uestion of whether the curricu#um is aroriate for anyone, or is itsim#y a means of achieving the gradeJ

ne of these dee seated %e#iefs in the UK (as in many #aces in the wor#d) isthat mathematics need necessari#y %e taught in a%i#ity grous. These grousare %ased uon unre#ia%#e notions of a%i#ity (@i##%orn 8 oude##, ?FF+) whichdisguise initia# and ongoing ine&uita%#e access to the curricu#um, #arge#y onthe grounds of students cu#tura# and #inguistic resources (6ernstein, +::76ourdieu, +5). Leven%ergen (?FF+) has e#ored how these theoretica#ideas work in the contet of mathematics a%i#ity grouing to construct #earnerdisositions to mathematics #earning . A#though such streaming racticeshave now fi#tered down into !ng#ish rimary c#assrooms (even to HB: yearo#ds), the high oint of such grouing ractices is their structuring intoeamination sy##a%i. "ince their introduction in the #ate +5Fs, mathematics@"! has %een eamined at three #eve#s= higher, intermediate andfoundation. !ach of these #eve#s a##ows students to achieve a grade within aarticu#ar range. The foundation tier on#y a##ows for the maimum ossi%#egrade of C, so fa##ing short of the magica# thresho#d. After many years ofo#itica# de%ate and #o%%ying this has now changed with the introduction of atwo tier @"! that from ?FFH a##ows a## students to fo##ow a mathematicscourse which cou#d resu#t in a grade . At the same time schoo#s in !ng#andare increasing#y working with modu#arised courses that make this argumentsemi;redundant as students can work their way u through #eve#s throughtheir course. Ehi#st these are interesting deve#oments they are essentia##ya%out organisationa# and assessment structures and carefu##y steer c#ear of afundamenta# discussion a%out schoo# mathematics curricu#um and edagogy.oreover they sti## do not dea# with the &uestion a%out curricu#umaroriateness for the many students who wi## not achieve the grade.

'n order to e#ore this issue a%out aroriateness further we must considerwhose interests are served %y the current curricu#um and edagogyJ 6oa#er2s

(+:) study of two mathematics #earning cu#tures offers a cha##enge to thechamions of the increasing#y forma#ised and atomised curricu#um form thatseems reva#ent in !ng#ish maths c#assrooms %ut her ana#ysis is focused moreon c#assroom cu#tures than curricu#um structure and urose. 3aving saidthat, the more oen, task oriented forms of #earning that 6oa#er descri%esrovide the kind of environment that wou#d %e conducive to a more socia##y$ust, democratic form of mathematics #earning. "uch teaching is not s#ave towhat Cavis and "umara (?FFF) descri%e as an outdated !uc#idean form ofcurricu#um that atomises #earning to managea%#e arts. This is current#y seenin !ng#ish mathematics c#assrooms in an uncritica# o%session with stating#earning o%$ectives7 as if %y doing so there is some assurance of what wi## %e#earnt. Mather, /the art is not sim#y a fragment of the who#e, it is a fracta#out of which the who#e unfo#ds and in which the who#e is enfo#ded2 (Cavis and

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"umara, ?FFF, . 5?5). This metahoric shift in concetua#ising andunderstanding curricu#um has im#ications for disruting the %road#yreroductive rocesses of education and ' wi## return to this #ater.

9ossi%#e answers to this &uestion a%out curricu#ar and edagogic urose are

redicated on recognition that the formatting ower of mathematics runsdeeer than summative assessment rocesses to inc#ude the grouingractices and edagogies of mathematics c#assrooms mentioned a%ove. As ateacher and teacher educator ' have seen #enty of evidence of #earnerdisaffection and often %een cha##enged %y the veed &uestion= 0what2s theoint of doing thisJ1 The answers are not straightforward and must sure#ychange in accordance with the changing nature of society and work. 'n theincreasing#y techno#ogica# wor#d that aste##s (?FFF) refers to as 0theinformationa# society1 it is the roduction, management and distri%ution ofinformation that is understood to %e the core driver of the economy and therime source of ower. athematics is a critica# comonent of that wor#d.Unfortunate#y, whi#st society is changing, the mathematics curricu#um hasremained #arge#y unchanged. A## !ng#ish schoo# students must studymathematics for e#even years and most finish with #itt#e that is of va#ue tothem= around a ha#f wi## not have achieved a grade . 3eymann2s (?FFI)thoughtfu# ana#ysis of mathematics education in @ermany is re#evant here,motivated as it is %y the recognition that 0a#most everything that goes %eyondthe standard su%$ect matter of the first seven years of schoo#ing can %eforgotten without the ersons invo#ved suffering from any notica%#edisadvantages1 (. 5-). Cesite the differences %etween !ng#and and@ermany, the same cha##enge a%out secondary mathematics education can %emade here. 3eymann argues for a reorientation of schoo# mathematicstowards a c#ear#y articu#ated notion of genera# education and ' wi## return tothis idea %e#ow. 3is thesis right#y rovoked considera%#e discussion. Thedifficu#ty that many mathematics teachers have with such a notion is itsdissonance with common onto#ogies and eistemo#ogica# views a%out thesu%$ect. Teachers need to recognise that mathematics is not a%so#ute(!rnest, ++7 akoff 8 Nune, ?FFF7 erman, +F) or va#ue free %ut acu#tura# construct. The U" critica# mathematics educator, @utstein (?FFH),e#ains that mathematics shou#d %e used to /read and write the wor#d2. 3egoes further than 3eymann2s notions of /critica# thinking2 and /understanding

the wor#d2 to suggest that socia# $ustice shou#d %e a concern of themathematics c#assroom. The cha##enge from @utstein and 3eymann is to %emore radica# in thinking a%out mathematics edagogy and curricu#um.3owever, in order to do this effective#y it might %e he#fu# to understandsomething a%out how the current curricu#um has evo#ved to this oint. Anyana#ysis #ike this wi## of course %e necessari#y %rief %ut wi## he# to frame theensuing discussion.

A brief history of school mathematics

The +FF years from +:4F to +54F saw a dramatic eansion in the need forractica# a#ications and know#edge of science and mathematics (Mogers,

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+5). Universities that had for so #ong %een the custodians of mathematicseducation for the e#ite few were gradua##y accomanied %y other organisationsand institutions offering the more oen u%#ic education, often focused on thescience and maths re&uired in the new industria# society. These a#icationswere different from the historica# mathematica# tradition of !uc#id %ut the

more uti#itarian and eerimenta# a#ications of mathematics demanded %yindustry. The traditiona# mathematics was sti## the reserve of theuniversities and those incorrect#y named u%#ic schoo#s that su#ied them."o it was that the emerging demand for mathematics #earning he#ed toengrain a hierarchy that wou#d remain endemic to mathematics education=maths for the workers or mathematics for the e#ite. The de%ate has changed%ut the under#ying distinction is the same= the /go#d standard2 of A #eve# foruniversity study, or the /functiona# mathematics2 for em#oya%i#ity.

6y the start of the ?Fth century rimary education was avai#a%#e for a##chi#dren and focused on the I Ms= reading, writing and arithmetic. 3owever, in+F? the then onservative government set u oca# !ducationa# Authoritiesand the modern curricu#um %egan to take shae. 'n the +-Fs a tierededucation system was mandated with students streamed into grammar,secondary modern and technica# schoo#s through the ++O eamination.Cesite some resistance to this tiered system it was not unti# the +HFs thatcomrehensive education %ecame a rea#ity for most chi#dren in !ng#and.Throughout this deve#oment the mathematics curricu#um was at the disosa#of the schoo#s. Not that this was without its critics. The #andmark ockcroft(+5?) reort cites many eam#es through the #atter +thand ?Fthcentury ofcritica# reorts on the state of mathematics instruction and chi#dren2sknow#edge. 3owever it was not unti# the end of the +5Fs that we first had aNationa# urricu#um in !ng#and. 'nitia##y met with strong criticism (foream#e, Cow#ing 8 Noss, +F) it has remained as the comu#sorycurricu#um, a#%eit with regu#ar modification, and now a generation of teachersare entering the rofession whose entire mathematics education was framed%y such a N. 6ut concerns a%out the mathematics curricu#um and studentattainment have not su%sided and with the increasing ressure arising frominternationa# comarisons, emerging #a%our markets and shifts in internationa#trade atterns, the economic drivers of a uti#itarian curricu#um havestrengthened. o##owing c#ose#y %ehind the N the Nationa# Numeracy

"trategy and ramework for Teaching athematics (Cf!!, ?FF+) addededagogic direction to the curricu#um and a#though on#y descri%ed officia##y as/guidance2, the money sent on the rimary and then secondary schoo#incarnations of the "trategy #eave no;one in any dou%t as to its status. oream#e, re;service teachers have to %uy their own N %ut get a coy of theramework for Teaching athematics for freeP

Why do we teach mathematics?

!ven from that short account it is c#ear that mathematics is art of the

o#iticied know#edge of schoo#ing. As such there is a need to make sense ofwhose interests are served %y the curricu#um as it stands and in what ways.

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"ome of the many reasons that have %een given for teaching mathematicsare #isted %y Cavis (+I)=

we teach it for its own sake, %ecause it is %eautifu#7 %ecause it revea#sthe divine7 %ecause it he#s us think #ogica##y7 %ecause it is the #anguage

of science and it he#s us to understand and revea# the wor#d7 %ecauseit he#s our students to get a $o%, either direct#y, in those areas of socia#or hysica# science that re&uire mathematics, or direct#y, insofar asmathematics, through testing, acts as a socia# fi#ter, admitting to certainrofessiona# ossi%i#ities those who can master the materia#. Ee teachit a#so to reroduce ourse#ves %y roducing future researchmathematicians and mathematics teachers. (. +F)

6e#ow ' wi## e#ore some of the guiding rinci#es that various grous use forositioning mathematics education. ertain of these are current#y dominantand some, as ' wi## e#ain, are in urgent need of greater attention. !ach of

them wou#d %enefit from more sace %ut here they are sim#y intended tooen u curricu#um and edagogic thinking sace.

1. Mathematics education for the academy

The academy and those rofessiona# mathematicians in science and industryhave a#ways %een one of the dominant forces in shaing the schoo#curricu#um. uch of the concern a%out the state of schoo# mathematicssurrounds the aarent oor &ua#ity of mathematica# understanding ofscience, techno#ogy, engineering and mathematics ("T!) undergraduates7the oor su#y of these graduates into science and industry7 the downturn in

A #eve# utake of mathematics courses and the a##eged inade&uacy of currentcomu#sory schoo# mathematics courses to reare students for A #eve# study.That these are serious issues is not in &uestion as it is the wide#y he#d %e#iefthat future economic roserity does re#y uon the roduction of sufficientnum%ers of ski##ed "T! graduates (see Eo#f (?FF?) for a counterargument).As the UK chance##or @ordon 6rown has said, 0science is the %edrock of oureconomy1. Ehi#st this might %e the case a more critica# ersective is neededhere. As 6eck (+?) e#ains, society cannot %e understood in a sim#emodernist sense as so#e#y a%out the distri%ution of /goods2. The economy(industry, %usiness, commerce, etc.) is not %enevo#ent and in theindividua#ised /risk society2 in which we now #ive there is a concern forminimising /%ads2 or risks. Not on#y does mathematics offer us the too#s toassess such risks %ut it a#so is used to generate the /ref#eive modernity2 ofwhich 6eck writes, in which the economy (and science) is not va#ue;free ornecessari#y /good2.

2. Mathematics education for employment

!m#oya%i#ity is a key o#icy driver for imrovements in schoo# mathematicsstandards. 3owever, the nature of em#oyment is a#so changing and theg#o%a# shift of economic markets is making for a different view of what itmeans to have a mathematica##y we## educated workforce in the ?+ stcentury.

'n the #atter art of the ?Fth century internationa# comarisons ofmathematica# cometence (T'"", 9'"A) contri%uted to a /%ack;to;%asics2

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neo;conservative trend in many arts of the wor#d. 6rown2s (+5) ana#ysisof 0the tyranny of the internationa# horse race1 high#ights the considera%#emethodo#ogica# f#aws in these kinds of internationa# comarisons as we## asthe fact that they have ro%a%#y had a more wide;ranging imact uonmathematics education than other su%$ects. 'f one retains a cu#ture;neutra#

view of mathematics then such comarisons might %e considered moreo%$ective#y %ut schoo# mathematics is dia#ectica##y re#ated to the cu#tura# andsocia# contet in which it is taught. 6y comaring student mathematics#earning on an internationa# sca#e the way is oened u for the owerfu##ayers to determine what is deemed to %e the imortant mathematica#know#edge for the ?+stcentury. This is a## high#y o#iticised and a#though someinteresting comarisons of the nationa# differences have %een made (e.g."tig#er 8 3ie%ert, +) there is sti## the need to criti&ue these discourses ofg#o%a#iation.

A#though the g#o%a# know#edge economy is imortant as we anticiate moreinternationa# migration of workers and work, #oca# know#edge is e&ua##y, if notmore, imortant. As %usiness and industry %ecome more secia#ised so toodo the mathematica# ractices integrated into those work saces. oreover,u%i&uitous techno#ogica# suort is changing the tyes of mathematicsractices yet further. "o the o#itica# argument for a mathematica##ycometent workforce is grounded in a uti#itarianism that suorts acurricu#um that is inaroriate for the increasing#y diverse know#edgesre&uired in modern society. 'n contrast, the schoo# mathematics curricu#umhas changed #itt#e in structure, content and de#ivery. Admitted#y there have%een attemts to reinvigorate curricu#um and edagogy in the #ast twentyyears %ut this has argua%#y had #imited imact (consider for eam#e 6rown eta#. (?FFI) ana#ysis of the Nationa# Numeracy "trategy). 'n some cases theimact has %een &uite the reverse. or eam#e, the "mith Meort (?FF-)descri%es the /disastrous2 imact of the urricu#um ?FFF reforms uonarticiation in ost; comu#sory mathematics study.

Ehi#st the academy2s demands and those of the em#oyers have so far %eena%out the we## &ua#ified graduate, there has %een a ara##e# concern a%out the%asic ski##s of the UK workforce. 6ynner and 9arson2s (+:) high#ighted there#ationshi %etween oor #eve#s of numeracy and unem#oya%i#ity and that

this corre#ation was stronger than %etween oor #eve#s of adu#t #iteracy andem#oya%i#ity. The a%our government has sent huge sums of moneyseeking to rectify this ski## shortfa## %ased on the %e#ief that these eo#e can%ecome more economica##y roductive if their mathematica# ski##s areimroved. That these initiatives have %een strong#y criticised is erhasunsurrising. To imagine that after ++ years of comu#sory schoo#ing thosewho #eft schoo# with #itt#e or no mathematics &ua#ification cou#d %e /uski##ed2so easi#y is naive. Ehat has haened through this time is the increasedmomentum of the ski##s agenda and this is at the heart of the Tom#insonrecommendations on the future of the +-;+ curricu#um (Cf!", ?FF-). 'n thisreort the notion of numeracy has morhed into something ca##ed /functiona#mathematics2. 'f numeracy was a s#iery term then this notion offunctiona#ity, rooted in the uti#itarian metahor of mathematics %eing a too#set

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for work and #ife, is erhas even more so. Ehat function does this mathserformJ The dou%#e meaning reinforces how mathematics is not on#y usefu#to the one who has it, %ut that schoo# mathematics does things to eo#e.

"o are schoo# mathematics c#assrooms sim#y roduction #inesJ Are schoo#s

$ust training em#oyees of the futureJ A#though this might %e the unsokenyet agreed urose of the curricu#um as it stands, ' suggest that schoo#mathematics teaching shou#d %e more de#i%erate in its aim to reare citiensfor active articiation in democratic society. "o whereas schoo# mathematicsis c#ear#y imortant to future em#oyees and the academy it shou#d have othere&ua##y imortant riorities. The %roadening of curricu#ar and edagogicurose is key to reinvigorating interest and engagement in the #earning ofmathematics.

. Mathematics education for general education

3eyman2s e#oration of the &uestion 0why teach mathematicsJ1 makes astrong case that it %e art of a genera# education. 3e e#ains that=

A great num%er of chi#dren, ado#escents, and adu#ts encounterenormous difficu#ties with mathematics. or these eo#e, thedifficu#ties are intrinsic in the distinctive characteristics of the su%$ectmatter. 'n many cases, the mathematics which they are o%#iged to#earn in schoo#s on#y attains the status of know#edge re&uired foreaminations ; #earned suerficia##y and, corresonding#y, &uick#yforgotten again. (3eymann, ?FFI, . +)

any teachers and students of mathematics in !ng#and wou#d share his view.

3e goes on to assert that 0conventiona# mathematics instruction in schoo#sdoes $ustice neither to foreseea%#e societa# demands nor to the individua#needs and &ua#ification interests of a ma$ority of ado#escents1 (. ?).3eymann is critica# of the way in which mathematics educators can %ecome sofocused on the detai# of their own disci#ine that they fai# to take account ofthe #arger educationa# and socia# contet in which mathematics teaching issituated.

Ehi#e 3eymann retains the notion of /rearation for #ater #ife2, understoodmore %road#y than $ust em#oya%i#ity, he a#so suggests that the /romotion ofcu#tura# cometence2shou#d %e a core theme of mathematics education %ased

uon a mode# of genera# education. Third#y, mathematics is used to /deve#oan understanding of the wor#d2 that goes %eyond the fa%ricated contets seenin so many c#assrooms and tets. ourth#y, and more focused on c#assroomedagogy, he suggests that mathematics shou#d romote /understanding,cognitive ski##s and critica# thinking2 a#though ' think that use of the wordcritica# is not the same as that discussed %e#ow. ina##y his #ist of five corethemes moves on to consider the c#assroom environment and the imact thatthis has uon the #earner. 3e envisages a c#assroom in which the /wi##ingnessto assume resonsi%i#ity, communication and cooeration, enhances thestudents2 se#f;esteem2. 3e oints out that none of these e#ements are new

%ut that together they might offer something of a new rofi#e for mathematicseducation. Asects of this rofi#e have %een referred to a#ready and others

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wi## %e icked u %e#ow. The notion of cu#tura# cometence is a theme of theNationa# urricu#um %ut rare#y gets taken u %y mathematics teachers.

!. Mathematics education for citi"enship

Ehereas we are not used to ta#king a%out genera# education in !ng#and, other

than in contrasting rimary schoo# genera#ism and secondary #eve#secia#isation, we do have a #oose#y re#ated notion of citienshi. 'n rea#itythe idea of citienshi education is a contested one. 'n the UK in recent yearsthere has %een much concern a%out o#itica# engagement and fo##owing therick Meort (+5) recommendations were made regarding the inc#usion ofcitienshi education in schoo#s. or the two years schoo#s have had amandatory o%#igation to /de#iver2 citienshi using the new#y introducedNationa# urricu#um for itienshi. "choo#s adoted one of a num%er ofim#ementation mode#s, either em%edding the work across the curricu#um ormore common#y as a stand a#one taught curricu#um. 3aving a#ready

high#ighted the tendency of maths teachers to deny the va#ue;#aden nature ofmathematics, it is erhas unsurrising that most mathematics teachers didnot see education for citienshi (as constituted in this curricu#um) as theirdomain.

This form of citienshi education is not what ' am descri%ing here. 9ovey(?FFI) has offered a criti&ue of this o#icy from a mathematics educator2sersective. "he argues that education for citienshi is c#ose#y re#ated toeducation for socia# $ustice and that as such c#assroom mathematics needs to%e more ref#eive. 'n this way we can %egin to see how mathematics is used%y, and on, various mem%ers of society and in c#assrooms. "he makes the

imortant oint that 0to harness mathematics #earning for socia# $usticeinvo#ves rethinking and reframing mathematics c#assrooms so that %oth there#ationshi %etween articiants and the re#ationshi of articiants tomathematics (as we## as the mathematics itse#f) is changed1 (. 4H). "oa#though ' am arguing for a rethinking of curricu#um she reminds us that thiscannot haen aart from a reconcetua#isation of what such a c#assroommight #ook #ike and what kinds of edagogies are rivi#eged therein. Ac#assroom where mathematics and citienshi education run in ara##e# isdeve#oing a more socia##y $ust ethos through its ractices as we## as in thecontent and de#ivery of the curricu#um. The same concerns were art of

3eymann2s core rinci#es.

#. Mathematics $ducation for %ocial &ustice

The concet and ractice of critica# mathematics education ("kovsmose,+-) has much in common with education for citienshi. There is ane#icit aim to make one of the foci of c#assroom mathematics activity thecriti&ue of societa# ower re#ationshis. 'n these c#assrooms mathematics isused to make sense of the socia# and scientific dimensions of the wor#d inways that uncover in the va#ue;#aden nature of mathematics. !#sewhere,!rnest (?FF-, . I+H) inc#udes 0emowerment of the #earner as a high#y

numerate critica# citien in society (emowerment of socia# $ustice concerns)1as one of his si aims for the mathematics curricu#um.


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@utstein2s notion of /reading and writing the wor#d with mathematics2 ise#icit#y re#ated to socia# $ustice education in the mathematics c#assroom. 3edescri%es from his own c#assroom ractice two interre#ated curricu#ar goa#s=teaching for socia# $ustice and teaching mathematics. Ehi#st there are certaintensions %etween these goa#s, @utstein sees considera%#e va#ue in ho#ding

them together. At the same time he recognises that in his work withdisadvantaged students he is not on#y to ena%#e them to /read and write2 withmathematics in order to criti&ue what might %e termed /g#oca#2 socia#conditions of in$ustice, %ut a#so they must accrue mathematica# ower. "uchmathematica# ower is what is needed to secure traditiona# success in high;stakes testing. 3owever, schoo# systems tend to infer such ower on thosewho arrive a#ready having it (for eam#e, 6ourdieu 8 9asseron, +::), whichis not the case for re#ative#y socio;economica##y disadvantaged students.Understanding societa# ower re#ationshis means that the ro#e ofmathematics and of schoo#s more genera##y must %e made e#icit and the

/ru#es of the game2 honed, articu#ar#y for those students who have notac&uired this sense of the game from their fami#y contets.

't has %een suggested that the #ack of mathematics graduates may have anegative imact on future economic roserity. Though we cannot deny theimortance of mathematics in the fa%ric of modern society we must %e c#eara%out the fact that the socia# advancement is not $ust a%out economic growth.The economy and mathematica# know#edge uti#ised within it does not a#ways#ead to %etter #ife circumstances for the mem%ers of that society or othersocieties. Through various a#ications of mathematics (science, techno#ogyand engineering) we have imroved transort, the design of #ife saving drugsand emai# rivacy. n the other hand there is a g#o%a# arms trade, digita#fraud, increasing income divides, etc. @utstein offers come##ing ethnograhicaccounts of the transformative imact of socia# $ustice mathematics tasksuon students.

'. Mathematics $ducation for the Information Age

This %rief overview of some of the main uroses for mathematics educationhas moved from the very we## documented and current, taken;for;grantedem#oya%i#ity and ski##s rationa#e to the more ro%#ematic (for a neo;conservative administration) one of critica# thinking and teaching for socia#

$ustice. n the %asis of the future wor#d scenarios of the !C, theCeartment for !ducation and "ki##s has %een considering what schoo#s might%e #ike in the year ?F?F. Ee are in a eriod of sustained and considera%#echange in the education system in which !very hi#d atters (Cf!", ?FFI)#egis#ation and the move to more schoo# autonomy suggest that furtherchange is $ust around the corner.

Ehat does a## of this mean for mathematics education in the know#edge orinformation societyJ Ehat kinds of mathematica# know#edge ski##s andunderstanding wi## %e desira%#e in ten or twenty yearsJ To what degree and inwhat ways wi## increasing#y owerfu# techno#ogies imact uon schoo#s,

#earning and in articu#ar mathematicsJ These &uestions need to %e a art ofthe de%ate a%out curricu#um and edagogy and though this is haening to an

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etent amongst the research community it is evident from time in schoo#sthat the rea#ities for students of mathematics are often very different.

These various inf#uences out#ined a%ove are not new. !rnest2s (+?) ana#ysisof the origins of the athematics N showed a simi#ar set of inf#uences. 3e

e#ained how the /o#d humanists2 (those in the academy) and /industria#trainers2 (the em#oyers) dominated the new curricu#um, margina#ising thechi#d;centred /rogressive educators2. ne grou ; the /u%#ic educators2 ;were not given a say whatsoever. These u%#ic educators

Qreresent a radica# reforming tradition, concerned with democracy andsocia# e&uity...to emower the working c#asses to articiate in thedemocratic institutions of society, and to share more fu##y in theroserity of modern industria# societyQQreresent radica# reformers who see mathematics as a means toemower students= mathematics is to give them the confidence to osero%#ems, initiate investigations and autonomous ro$ects7 to critica##yeamine and &uestion the use of mathematics and statistics in ourincreasing#y mathematied society, com%ating the mathematica#mystification reva#ent in the treatment of socia# and o#itica# issues.(.IH)

These are the critica# educators7 teachers who are interested in a more radica#citienshi education and education for socia# $ustice in the mathematicsc#assroom.An aside: numeracy and functional mathematics

3aving considered these curricu#um drivers ' want to consider %rief#y the

terms numeracy and functiona# mathematics. A#though ' have used themfair#y #oose#y so far their introduction and evo#ution in curricu#um and o#icydiscourse is imortant for understanding the uroses for schoo#mathematics. ike many terms in everyday use they can have a wide rangeof meanings and numeracy is no ecetion. The rowther Meort of +4gave an introductory definition of numeracy as

An understanding of the scientific aroach to the study of henomenaR o%servation, hyothesis, eeriment, verificationQthe need in themodern wor#d to think &uantitative#y, to rea#ise how far our ro%#emsare ro%#ems of degree even when they aear as ro%#ems of kind. (.

?:F cited in Noss, ?FF?, . II)

rom a review of su%missions to the ockcroft committee the conc#usion wasmade that 0the words SnumeracyBnumerate have changed their meaningconsidera%#y in the #ast twenty years1 to denote #itt#e more than an a%i#ity to0erform %asic arithmetica# oerations1 (ockcroft, +5?, . ++). Thecommittee eressed the view that %eing numerate shou#d mean theossession of two attri%utes=

The first of these is an /at;homeness2 with num%ers and an a%i#ity tomake use of mathematica# ski##s which ena%#es an individua# to coe

with the ractica# mathematica# demands of everyday #ife. The secondis an a%i#ity to have some areciation and understanding of informationwhich is resented in mathematica# terms, for instance in grahs, charts

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or ta%#es or %y reference to ercentage increase or decrease. (o cit.,++)

The first attri%ute is grounded in the notion of uti#ity where%y maths is usefu#in everyday #ife. This dimension of ockcroft2s numeracy is now at the heart

of the functiona# mathematics that was introduced in the Tom#inson reortand reinforced in the su%se&uent white aer . The second attri%ute of thenumerate seems articu#ar#y re#evant in an era where information %om%ardsus dai#y and we are ca##ed uon to make some critica# sense of this.Unfortunate#y, the osition of data hand#ing in the curricu#um, articu#ar#y inthe +-;+ hase, seems to %e at risk. The "mith reort (?FF-) /akingathematics ount1 recommended a radica# rethink of the #ocation ofstatistics and data hand#ing which wou#d 0%e %etter removed from themathematics timeta%#e and integrated with the teaching and #earning of otherdisci#ines (for eam#e, %io#ogy or geograhy). The time restored to themathematics timeta%#e shou#d %e used for ac&uiring greater mastery of coremathematica# concets and oeration1 (.:). This wou#d %e very concerningfor a num%er of reasons. irst#y the agenda here is that of higher education,in articu#ar "T! su%$ects, rather than on what might %e %eneficia# in agenera# education for future citiens. "econd#y, the kinds of data;awarenessnecessary might not %e %est served in these c#assrooms a#though generating%etter cross;curricu#ar mathematics work is high#y desira%#e.

(e)eloping a more socially *ust mathematics curriculum

'n this aer ' advocate a shift from a redominant#y academy and em#oyer;driven curricu#um which direct#y %enefits a minority, to something that wi##

engage a## students in deve#oing not $ust mathematica# ower %ut whatrankenstein (?FF4) terms /critica#mathematica# #iteracy2. This form of#iteracy has four goa#s (.+)=

understanding the mathematics

understanding the mathematics of o#itica# know#edge

understanding the o#itics of mathematica# know#edge

understanding the o#itics of know#edge

9ut in a different way @utstein (?FFH) descri%es what it means to read andwrite the wor#d with mathematics as=

To use mathematics to understand re#ations of ower, resourceine&uities, and disarate oortunities %etween different socia# grousand to understand e#icit discrimination %ased on race, c#ass, gender,#anguage and other differences. urther, it means to dissect anddeconstruct media and other forms of reresentation. 't means to usemathematics to eamine these various henomena %oth in one2simmediate #ife and in the %roader socia# wor#d and to identifyre#ationshis and make connections %etween them. (. ?4)

Noddings (?FF-) tries to address the same ro%#ems a%out the need for amore $ust curricu#um %y suggesting three routes of mathematica# study #inkedto a) the humanities %) the socia# sciences and c) the natura# sciences. or a##

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of these there wou#d need to %e /cu#tura##y rich2 andBor /connected to currentsocia#Bo#itica# ro%#ems2. Ehi#st ' am not suggesting that this is a current#yimagina%#e way forward in the !ng#ish contet such an aroach does offerinteresting ossi%i#ities. or Noddings (as with 3eymann, 9ovey and @utstein)the who#e of this curricu#um design shou#d %e redicated on democratic

rinci#es of c#assroom articiation (Noddings, +I), which wou#d resent aconsidera%#e cha##enge for mathematics education in genera#.

'f it is not ossi%#e to devise more socia##y $ust Nationa# urricu#um given theo#itica# nature of schoo#ing, cou#ed with the common dissociation of o#iticsand mathematics, then erhas it is he#fu# to think of the twin edagogies ofaccess and dissent (orre##, ?FF4, cited in @utstein ?FFH, . ?FF). 'fedagogies of access strive to oen u future educationa# and em#oymentathways then edagogies of dissent seek to disrut the hidden structuringrocess of schoo# and society %y e&uiing students with critica# know#edgeand strategies for socia# agency. The &uestion remains then how these might%e incororated into mathematics teaching. 't is some of these concerns foro#itica# engagement that were the motivations for the inc#usion of itienshieducation in the curricu#um a#though one susects that critica# thinking andsocia# activism might not %e the referred outcomes in a schoo# system thatgenera##y suresses such activity (from %oth teachers and students)

'n considering the nature of citienshi and the #ace of mathematicseducation in rearing a future citienry the notion of allgemeinbildung offersan a#ternative to the U" critica# edagogies. !#mose and Moth (?FF4) deve#othis idea, which rough#y trans#ates to genera# citienry or genera# #iteracy, as

invo#ving 2cometence for se#f;determination, constructive articiation insociety, and so#idarity towards ersons #imited in the cometence of se#f;determination and articiation2 (. ?+). They see three ways for educationto deve#o (fundamenta#ist, retraditiona#ising or democratic) and argue thatmodern risk society re&uires greater invo#vement in co##ective decisionmaking. As such allgemeinbildung !inc#udes not on#y so#idarity in sirit, i.e.knowing that others are oor, knowing that there are greenhouse gases, %uta#so so#idarity in action, i.e. knowing for2 (. I+). There is considera%#ecommon ground %etween the ideas of those from a critica# edagogy traditionand the socio#ogists of risk society. Again the &uestion remains as to how this

is %ui#t into future mathematics curricu#um design. @utstein conc#udes thatthere is considera%#e work to %e done in such a curricu#um design (and that isin the very different U" contet). A#though there are #enty of eistingmateria#s and tried edagogic aroaches it is a very different matter toeand this to a com#ete curricu#ar aroach and this wou#d re&uiresignificant wi## and effort.

+inal comments

onc#uding his ana#ysis of the UK mathematics Nationa# urricu#um, @i##(?FF-) writes that

the current curricu#um for mathematics fai#s to meet the c#aims madefor it in mathematica# terms and a#so fai#s to contri%ute to the overa##ethos of the Nationa# urricu#um contained in the Aims and "alues.

+I


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Nothing #ess than a com#ete overhau# is necessary if it is to serve ourui#s and the society they, and we, #ive in. (. ++4)

Ehat ' am advocating here is such an overhau#. Ehi#st ma$or changes tomathematics education are %eing discussed amongst owerfu# grous

(academics, o#iticians and civi# servants) they do not rea##y get anywherenear what is %eing considered here. "eaking of U" reforms of mathematicseducation Ki#atrick and "tanic (+4) remarked that /true reform,unfortunate#y, may re&uire doing something not %etter %ut different2 (.+4)."uch a different way of thinking is what is needed in !ng#and if themathematics education of future generations is to %e more worthwhi#e.3owever, much of the considera%#e amount of time and money current#y%eing invested in mathematics teaching in !ng#and is driven %y standardsagendas and not those that foster the deve#oment of critica# ref#eivitythrough the curricu#um. The #andscae of mathematics education is not easyto change (Noyes, ?FF-) and so it wou#d re&uire a concerted effort to effectmeaningfu# change in the direction suggested here. 'n fact, such a changemight not %e ossi%#e, for it wou#d re&uire those with the ower to makedecisions regarding the form of the curricu#um which might not %e in theirinterests. As @utstein oints out, /deve#oing socio;o#itica# consciousness isanathema to those with ower who wou#d have those without maintain theirignorance, si#ence and assivity2 (@utstein, ?FFH, . H-). 9erhas it is morerea#istic to think a#ong the same #ines as 3eymann who considered that

the ath to instruction oriented more strong#y toward genera# educationcannot %e enforced from eterna# sources...%ut can on#y consist of sma##

stes invo#ving many articiants for whom these stes make goodsense. (3eymann, ?FFI, . 5-)

't seems that whi#st the tradition of critica# education is visi%#e e#sewherethere is otentia# for deve#oing curricu#um and edagogy that wou#d suorta %roader mathematics education that inc#udes critica#mathematica# #iteracyand education for socia# $ustice as we## as traditiona# mathematics instruction.Again @utstein (?FFH) is he#fu# here when he descri%es three tyes ofknow#edge that need to form art of students mathematics #earning=community, critica# and c#assica#.

6etween the continenta# and U" critica# traditions there are considera%#etheoretica# resources for deve#oing a radica##y different mathematicseducation in !ng#and, one that wou#d %e more engaging, re#ate to issues ofrea# concern to students, deve#o active articiation as citiens and deve#omathematica# ower. Ehether or not this cou#d ever %e organised on anationa# sca#e or for a #arge num%er of students is a moot oint. 3owever,considering the oor image of mathematics, difficu#ties in recruitment to ost;comu#sory mathematics courses, the genera# #ack of o#itica# engagement inthe ou#ace and amongst young eo#e, and so on, it seems time#y toconsider how a different mathematics curricu#um might make a contri%utionto addressing some of these issues.

,eferences

+-


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6eck, U. (+?). #is$ So%iet: to&ards a ne& modernit. ondon= "age9u%#ications.

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+4


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Documents

Noyes Mathematics Counts for What