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 Volume 28, Issue 1, 2009
SuidAfrikaanse Tydskrif vir Natuurwetenskap en Tegnologie  Volume 28, Issue 1, December 2009
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Volume 28, Issue 1, December 2009

Die ontstaan en ontwikkeling van melaatsheid in die antieke tyd s
Authors: Francois Retief and Louise CilliersSource: SuidAfrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 28, pp 1 –12 (2009)More LessIn die geskiedenis was daar waarskynlik geen ander siekte wat die mensdom met soveel vrees en weersin vervul het as melaatsheid (lepra) nie. Die Hebreeuse Bybelverwysing na tsara'at (Lev. 1314) het gedui op 'n oënskynlik niedodelike skilferige velsiekte, wat as 'n ritueelonrein toestand deur die priesters met verbanning uit die gemeenskap hanteer is. Alhoewel feitlik sekerlik nie melaatsheid nie, is tsara'at in die volksmond mettertyd met hierdie siekte geassosieer. In die 5de / 4de eeu v.C. is 'n soortgelyke onskuldige veltoestand deur Hippokrates lepra genoem, en in die Griekse vertaling van tsara'at na lepra in die Septuagint, 3de eeu v.C., is hierdie naam moontlik uit die Hippokratiese Corpus ontleen. Ware melaatsheid, 'n kroniese siekte deur Mycobacterium leprae veroorsaak, begin as 'n velsiekte maar mag mettertyd prominente verminking van die liggaam veroorsaak. Te oordeel aan geskrifte van die tyd, mag dit moontlik reeds in die 2de millennium v.C. in Mesopotamië en Egipte, en 'n millennium later in China en Indië teenwoordig gewees het. Dit is moontlik in die 4de eeu v.C. deur die leërs van Alexander die Grote uit Indië en Mesopotamië na die Middellandse seegebied gebring. Die vroegste osteoargeologiese bewys van melaatsheid is in Egiptiese skedels wat na die 2de eeu v.C. terugdateer, gevind. Siektebeskrywings, herkenbaar as melaatsheid, is in die 3de eeu v.C. deur Strato in Alexandrië opgeteken. Hierdie toestand, wat as elefantiase / elefas bekend geword het, is daarna deur verskeie geneeshere beskryf, en het onder andere na Griekeland en Italië en daarna deur die Romeinse Ryk versprei. In Europa het dit eers in die Middeleeue 'n epidemiese omvang aangeneem. Alhoewel dit aanvanklik met standaardterapie van die tyd behandel is, is die rituele isoleringsmaatreëls wat vir tsara'at gegeld het mettertyd na elefantiase oorgeplaas. Stigmatisering van die siekte as 'n Goddelike straf het op laasgenoemde van krag geword, en die naam is van elefantiase na lepra (melaatsheid) verander.
The origin and development of leprosy in antiquity
Through the ages leprosy has filled mankind with awe and horror. It still remains one of the unconquered infectious diseases, although the World Health Organisation reports a decrease in its prevalence (18 million to two million new cases annually over the past 20 years). For many, leprosy's origins are to be traced back to the Hebrew Bible and the condition of zara'ath mentioned in Leviticus 1314. This was a lightcoloured scaly skin lesion which rendered the patient ritually unclean. Such a person was banned from society by a priest, and could only return on being pronounced clean. Zara'ath was almost certainly a benign skin lesion and not leprosy. When the Hebrew Bible was translated into Greek (the Septuagint) in the 3rd century BC, zara'ath was translated as lepros / lepra, possibly after an apparently comparable disease described in theHippocratic Corpus (5th  4th centuries BC). The Hippocratic disease was clearly a benign, scaly skin eruption, and not leprosy as we know it. The fact that leprosy, as a very chronic progressive disease with a characteristic clinical picture, was not described by Hippocrates, almost certainly means that it did not occur in the Greek community of the time.
True leprosy is an infection caused by mycobacterium leprae, and manifests initially as lightcoloured skin macules. With prominent bodily immunity against the organism the skin lesions enlarge slowly, later become scaly with a numb surface, and are complicated by nervous infiltration and atrophic degeneration of the extremities (tuberculoid leprosy). With low immunity, progressive nodular infiltration of skin and underlying structures result in extensive deformities (e.g. the typical "leonine facies"), subcutaneous abscesses, destruction of nerves and other tissues, blindness,deafness and testicular atrophy (lepromatous leprosy). Medical writings of ancient civilisations show that a leprosylike disease was recognised in Mesopotamia by the 2nd millennium BC, and possibly in India and China in the 1st millennium BC. It has been suggested that leprosy was brought to the Mediterranean region by Alexander the Great's armies, 4th century BC.

Krieketprestasiemaatstawwe s
Author Hoffie LemmerSource: SuidAfrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 28, pp 13 –34 (2009)More LessDie doel van hierdie oorsigartikel is om 'n samevatting te gee van 'n verskeidenheid maatstawwe wat oor 'n periode van agt jaar ontwikkel is om die prestasie van kolwers en boulers te meet. Omdat sekere formules wat gebruik word groothede bevat wat uit spelerdata van die betrokke periode bereken is, word sodanige groothede bygewerk sodat huidige prestasie beter daarin weerspieël word. As 'n speler op plaaslike vlak goed presteer, is dit belangrik om sy prestasie met dié van spelers in die nasionale span te vergelyk. Gewigte is uitgewerk wat dit moontlik maak om kolwers sowel as boulers se prestasies te vergelyk ongeag elkeen se mate van deelname aan internasionale wedstryde. Enkele verbeteringe, gebaseer op die gebruik van die maatstawwe, word ook aangebring. Die bygewerkte maatstawwe word gebruik om ranglyste van kolwers en boulers in Eersteklas en Lys Awedstryde op te stel. 'n Ander baie belangrike probleem is om die prestasies van spelers na afloop van 'n toets of eendagreeks te vergelyk. Spesiale metodes is hiervoor ontwikkel. Die nuutste vorm van die spel, naamlik Twintig20 krieket, word ook kortliks bespreek.
Cricket performance measures
The purpose of this paper is to present a summary of batting and bowling performance measures that have been developed over a period of eight years. The measure of batting performance (BP) consists of three components. Firstly, an exponentially weighted average (EWA) is calculated in which recent scores have higher weights than scores further back in time in order to get a measure that adequately reflects the batsman's present form. EWA is then adjusted up or downwards depending on the batsman's consistency and strike rate. This measure was subsequently generalized to BPW which includes weights for runs scored against specific countries in home or away matches separately. In the case of bowling, the first challenge was to develop a single comprehensive measure of bowling performance, the combined bowling rate (CBR) for limited overs matches and the dynamic bowling rate (DBR) for unlimited overs matches. Hence, by also taking bowling consistency into account, the current bowling performance measure (CBP) was developed. This measure was also extended to include weights for wickets taken and runs conceded against each international team in home or away matches. These measures contain parameters that were based on players' scores up to the specific date of calculation. Before using the latest available data for the present presentation, it is logical to update these parameters. This has been done in the present study. If a player performs well at local level, it is important to be able to compare his performance with those of players already in the national team. Weights have been determined for runs scored by batsmen and also for runs conceded and wickets taken by bowlers in local matches, whereby the performances of players can be compared irrespective of the proportion of international matches played by each player. After having used the measures on various occasions the need arose to revise some of them slightly. The updated measures are then used to give rankings of South African batsmen and bowlers in First Class and List A matches alike. Methods have also been developed to compare the batting and bowling performances of players after a short series of matches. It was shown that the traditional average could have unrealistic values in the case of a batsman who had a large percentage of not out scores in a short series. Alternative measures have been developed and shown to give much more realistic values. In the case of bowling it is important not only to count the number of wickets a bowler has taken, but to distinguish between the wickets of top and middle order batsmen on the one hand and lower order batsmen on the other. A system has been established whereby weights are attached to the wickets according to the batting position of the batsman. Instead of just counting the number of wickets taken, the sum of the weights of the wickets taken by the bowler is used in the calculation of CBR or DBR. Finally, the latest form of the game, Twenty20 cricket, is also discussed.

Skedulering van gadevermydende gemengdedubbels rondomtalietennistoernooie : navorsing en oorsigartikels
Authors: A.P. Burger and J.H. Van VuurenSource: SuidAfrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 28, pp 35 –63 (2009)More LessBy die opstel van 'n gadevermydende gemengdedubbels rondomtalietennistoernooi van orde n word daar gesoek na 'n spelskedule waarvolgens n getroude pare op só 'n manier in gemengdedubbels tennispotte kragte meet dat geen speler saam met sy / haar eggenoot in 'n span afgepaar word nie, geen speler teen sy / haar eggenoot te staan kom nie, elke speler presies een keer teen elke ander speler van dieselfde geslag te staan kom, elke speler presies een keer saam met elke speler van die teenoorgestelde geslag (behalwe sy / haar gade) in 'n span afgepaar word, en elke speler presies een keer teen elke speler van die teenoorgestelde geslag (behalwe sy / haar gade) te staan kom. Hierdie potte moet boonop in die kleinste moontlike aantal rondtes ingedeel word sodat geen speler in meer as een pot per rondte meeding nie, en sodat die aantal potte per rondte dieselfde en 'n maksimum is. Spelskedules vir sulke toernooie word vir n ≤ 20 opgestel en ter wille van naslaandoeleindes op 'n gebruikersvriendelike manier gedokumenteer.
Scheduling spouseavoiding mixed doubles roundrobin tennis tournaments
The problem of scheduling a spouseavoiding mixed doubles roundrobin tennis tournament (SMDRTT) of order n involves finding a playing schedule for n married couples in such a way that no player teams up with his / her spouse, no player opposes his / her spouse, each player opposes every other player of the same sex exactly once, each player teams up with every player of the opposite sex (except his / her spouse) exactly once, and each player opposes every player of the opposite sex (except his / her spouse) exactly once. These mixed doubles tennis matches have to be partitioned into the smallest number of rounds so that no player plays more than once per round and so that each round comprises the maximum number of matches. If n is even, then each player may be scheduled to compete in every round and hence an SMDRTT of even order n comprises n1 rounds, each containing n / 2 matches. However, if n is odd, then one man and one woman must necessarily receive a bye during each round and hence an SMDRTT of odd order n comprises n rounds, each containing (n  1) / 2 matches.
The notion of an SMDRTT may be attributed to the director of the Briarcliff Racquet Club in New York, who sought such a schedule for his club in 1972. His motivation was that spouses know each other too well and hence may have an unfair advantage with respect to anticipating elements in each other's play. Although it is known that results from the mathematical subdiscipline of design theory may be used to construct SMDRTTs of virtually any order, neither these techniques nor the application thereof is easily accessible to administrators of tennis clubs, who are typically not mathematicians. The aim in this paper is therefore twofold: (I) to investigate which techniques from design theory are applicable in the construction of playing schedules for SMDRTTs, and
 (II) to apply these techniques in the construction of playing schedules for SMDRTTs of order n ≤ 20, and to document the resulting schedules in a way that is easily accessible to nonmathematicians.

Optimale toewysing en skedulering van werkers en take op 'n monteerbaan s
Authors: Lieschen Venter and Stephan VisagieSource: SuidAfrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 28, pp 64 –84 (2009)More LessDie optimale aanwending van kruisopgeleide werkslui in 'n produksielyn word ondersoek. Kruisopgeleide werkslui het die vermoë om haas enige taak in die produksieproses te verrig. 'n Wiskundige model om die optimale toewysing van werkers aan take te bepaal, word geformuleer. Die invloed van vaardige of onvaardige tydelike werkslui op die produksielyn word ook ondersoek. Die finale model neem verskeie doelwitte in ag. In die eerste fase word 'n nutsfunksie van produksie gemaksimeer terwyl werkers aan take toegewys word. Daarmee saam word die verbetering van die werkers se vaardigheidsvlakke gemaksimeer terwyl die effek van die leer en vergeet van vaardighede in ag geneem word. Die arbeidskoste van produksie word ook geminimeer terwyl die tipe werkers en die manier waarop hulle aangestel en afgelê moet word, bepaal word. Laastens word die totale produksietyd geminimeer terwyl die optimale volgorde van die take in die produksieproses bepaal word. Die praktiese bruikbaarheid van die model word ook met behulp van 'n gevallestudie gedemonstreer.
Optimal assignment and scheduling of workers and tasks on an assembly line
In this paper the assignment of crosstrained and temporary workers to tasks on an assembly line is investigated. Crosstrained workers are skilled to perform more than one task on the assembly line in the production process. Temporary workers are viewed as either trained or untrained and may be hired or laid off as required. The solution procedure may be divided into three parts. During the first part a model is formulated to determine an optimal assignment of the workers to the production tasks.
During the second part the model is extended to determine the effect of the assignment of both trained and untrained temporary workers to the tasks on the assembly line. During the final part of the model an optimal sequence of tasks in the assembly line is determined that minimises the resulting execution times of these tasks. During the first part the objective is to maximise the total production utility. This is achieved by implementing a twophase model. The first phase maximises the utility of production by minimising labour shortage in the assembly line. During the second phase the improvement of the workers' levels of skill is maximised while the effect of the learning and forgetting of skills is taken into consideration. A learnforgetcurve model (LFCM) is implemented to model the effect of this human characteristic on the master model. This approach ensures that the advantageous crosstrained nature of the workers is maintained and optimized, without a large deviation from the solution determined by the first phase.

Versoek om bydraes : Spesiale uitgawe van SA Tydskrif vir Natuurwetenskap en Tegnologie (28(4), Desember 2009 : Die uitdaging van ontoereikende Wiskundeprestasie : Fokus op probleemoplossing : Aankondiging
Source: SuidAfrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 28, pp 85 –86 (2009)More LessDaar bestaan wêreldwyd kommer oor ontoereikende prestasie in wiskunde en SuidAfrika is geen uitsondering op hierdie reël nie. Inteendeel  SuidAfrikaanse kenners spreek toenemend hul kommer uit oor veral Graad12leerders se ontoereikende prestasie in Wiskunde en die impak hiervan op sowel tersiêre studie as die nasionale ekonomie (Simkins, Rule, & Bernstein, 2007). Die negatiewe effek van ontoereikende Wiskundeprestasie op die ekonomie is immers haas onberekenbaar. Tensy daar genoeg beroepslui soos ingenieurs opgelei word, kan die ekonomie eenvoudig nie vinnig genoeg groei nie.

Die uitdaging van ontoereikende wiskundeprestasie : fokus op probleemoplossing : voorwoord
Author J.G. MareeSource: SuidAfrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 28, pp 261 –264 (2009)More LessDaar heers wye kommer oor ontoereikende prestasie in wiskunde. Hoewel hierdie probleem ook in ander dele van die wêreld voorkom, spreek SuidAfrikaanse kenners al vir jare hul besorgdheid uit oor veral Graad 12leerders se ontoereikende prestasie in wiskunde en die impak hiervan op tersiêre studie sowel as die nasionale ekonomie. Tewens, oor die afgelope jaar wil dit voorkom of hierdie toestand vererger het  universiteitsdosente meld dat eerstejaars se punte swakker is as in die verlede. Kommer word ook uitgespreek dat leerders vanjaar minder as in vorige jare blyke gee van insig in basiese wiskundevaardighede. Verder is hulle nie in staat om basiese probleemoplossingsvaardighede toe te pas en aan te hou probeer om moeilike wiskundeprobleme op te los en nie voortydig moed op te gee en na hulp te soek nie. Tensy daar genoeg beroepslui soos ingenieurs, rekenmeesters en wiskundeonderwysers opgelei word, kan die SuidAfrikaanse ekonomie eenvoudig nie vinnig genoeg groei nie. Dit verg nie veel insig om die volgende afleiding te maak nie : Om in wiskunde te presteer beteken nie net oorlewing vir individuele leerlinge nie, maar ook vir enige ontwikkelende land. Wiskunde is immers by verre die belangrikste vak vir toelating tot gesogte studierigtings.

Die uitdaging van ontoereikende wiskundeprestasie : fokus op 'n metabenadering
Author J.G. MareeSource: SuidAfrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 28, pp 265 –287 (2009)More LessSoos in talle ander lande die geval is, bestaan daar in SuidAfrika diepe besorgdheid oor veral Graad12leerders se ontoereikende prestasie in wiskunde en die impak hiervan op sowel tersiêre studie as die nasionale ekonomie. Om in wiskunde te presteer, beteken immers nie net oorlewing vir individuele leerlinge nie, maar inderdaad vir enige ontwikkelende land. In hierdie artikel val die soeklig op oplossings vir die probleem van ontoereikende prestasie in wiskunde. Eerder as om nogeens slegs op "probleemareas" of geïsoleerde fasette van die uitdaging te hamer, word in hierdie artikel gefokus op 'n metabenadering; die uitdagings en faktore wat op die makro, meso, asook mikrovlak manifesteer, word onder die loep geneem. Hoewel besef word dat daar gewis nie 'n sogenaamde silver bullet bestaan nie, 'n soort towerstaf wat maar geswaai kan word en waarmee probleme eensklaps uit die weg geruim kan word nie, kan baie gedoen word om die situasie ten minste betekenisvol te verbeter. Die betoog sentreer daarom in die eerste plek om die rol wat die staat behoort te speel, insluitend die verskaffing van infrastruktuur, asook die opheffing van armoedige gemeenskappe om sodoende die nodige omstandighede te skep waarbinne leerders kan presteer. Op die mesovlak verdien die opleiding van onderwysers indringend aandag en op die mikrovlak word gewys op die belangrikheid daarvan om ouers by die leergebeure in wiskunde te betrek  met ander woorde om ouers as't ware toe te rus om hul kinders te kan begelei; daar word voorts gewys op die belangrikheid daarvan om in wiskundeklaskamers 'n klimaat te skep wat optimale leer moontlik maak. Die oorkoepelende ideaal is dat leerders nie slegs beter sal presteer in wiskunde nie, en dus toelating sal verkry tot meer gesogte studierigtings nie, maar dat hulle skool ná Graad 12 beter toegerus sal verlaat om tipiese uitdagings wat met universiteitstudie gepaardgaan, te kan aanpak.
The challenge of inadequate achievement in mathematics : Focus on a metaapproach
As is the case elsewhere in the world, all stakeholders in South Africa are deeply concerned about the level and scope of under achievement in mathematics, not only at Grade 12 level, but, indeed, at University, University of Technology and Further Education and Training levels. These concerns assume a deeper dimension in light of the fact that inadequate achievement in mathematics inevitably will have a ripple effect on the academic situation in any country: inadequate achievement in mathematics precludes learners from applying for admission to soughtafter fields of study, which, in turn, prevents numerous learners from realising their true potential and, eventually, from being happy and successful in careers that they might otherwise have been able to execute successfully. It goes without saying that inadequate achievement in mathematics will impact negatively on the overall economic situation in any country (even more so in a developing country such as South Africa). Truth being, achievement in mathematics amounts to equipping oneself with survival skills. In this article, the spotlight shifts from a narrow and outdated focus on problems that are associated with inadequate achievement in mathematics to possible solutions for this disconcerting situation and the implied challenge it raises. The focus is thus on three levels that collectively underpin and impact on achievement in mathematics, viz. the macro level, the meso level and the micro level.
The macro level refers mainly to the input by the national government (and, by default, the National Department of Education). In the first instance, it is the responsibility of the state to provide adequate schooling facilities for all learners, irrespective of where they fi nd themselves. Furthermore, it is the duty of the state to ensure that every learner has access to basic facilities, including food, water, sanitation and housing. The state (via the National Department of Education) is also obliged to ensure that the basic philosophy that underpins mathematics education in the country is scrutinised continuously and that changes be made to existing teaching philosophy should these be recommended by the majority of stakeholders. Case in point: the implementation of OutcomesBased Education (OBE) in South African mathematics classrooms has now already been under the spotlight for a number of years and there seems to be general consensus that it is essential to facilitate a number of basic changes to this philosophy and (especially) to the way in which it is implemented in South African classrooms. Teachers, for instance, constantly complain about matters such as an administrative overload, unacceptably high stress levels (brought about by factors broadly associated with OBErelated issues) and the fact that the laudable philosophy underpinning OBE is not consistently realised in practice. At the meso level, the spotlight falls on factors related to teacher training. For example, it seems highly advisable to optimise teacher training in mathematics, to facilitate training in emotional intelligence, to conduct a national audit on the number of mathematics teachers currently teaching mathematics (in terms of how many teachers are currently in the system, where these teachers find themselves, their level of training, etc.) to determine training needs in mathematics and to facilitate a more equitable distribution of teachers across the country. For example, it is proposed that all graduating teachers be compelled to do community service in an effort to facilitate a better understanding of the challenges that teachers in various parts of the country face, thereby breaking down barriers between people. At micro level, the emphasis is on measures that might be taken to provide guidance to parents on how to assist their children in mathematics on the one hand and on practical ways in which to help learners in mathematics perform better in mathematics and leave school better equipped to deal with typical challenges at tertiary level on the other.
It is hoped that this article will contribute to an improvement in the disconcerting situation to be found in mathematics classrooms across South Africa. I sincerely hope to have sensitised readers to the need not only to talk about the situation in mathematics in South Africa, but instead to start acting and in so doing to impact on the situation in practical ways.

Is studente wat in 'n uitkomsgerigte onderrigbenadering opgelei is, gereed vir universiteitswiskunde? : navorsings en oorsigartikel
Authors: Johann Engelbrecht, Ansie Harding and Patrick PhiriSource: SuidAfrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 28, pp 288 –302 (2009)More LessUitkomsgerigte onderrig (UGO) het 'n groei en hersieningsfase beleef en is vir die afsienbare toekoms hier om te bly. Die 2009inname van studente op universiteit is die eerste matrikulante wat slegs binne die nuwe bedeling onderrig ontvang het. Hierdie matrikulante se graad 12uitslae is deur onnatuurlik hoë punte in wiskunde gekenmerk. Die vraag wat gevra word, is hoe die 2009inname van studente in wiskunde op universiteit vaar wat prestasie, algemene vaardighede, wiskundige vaardighede en inhoudverwante vaardighede betref. Dit blyk dat alhoewel hierdie studente beter toegerus is wat persoonlike eienskappe soos selfvertroue betref, hulle in baie gevalle swakker daaraan toe is wat wiskundige en inhoudverwante vaardighede betref. Voorstelle word gemaak om die oorgang van sekondêre na tersiêre wiskunde gladder te maak.
Are students who have been educated in an outcomesbased approach prepared for university mathematics?
Following the political changes of 1994 in South Africa, the decision was taken to replace the traditional skillsbased education system at primary and secondary school level (Grades 1  12) with an outcomesbased education system (OBE). The OBE approach, referred to as Curriculum 2005, was introduced into schools in 1998. The implementation of the OBE system did not occur without problems, giving rise to revised initiatives and a fair amount of criticism. The 2009 intake of students at universities is the first group of students that had been subjected to the OBE approach for their entire school career. This is also the first group of students for whom some form of mathematics was compulsory up to Grade 12 level in the form of mathematics or mathematical literacy. These students were characterised by the fact that their mathematics marks for Grade 12 were exceptionally high and that many more students qualified for university entrance.
This article reports on the impact of this new education system on the mathematics preparedness of students entering university. The study involves an empirical analysis of the students in the firstyear mathematics course for engineering students at the University of Pretoria as well as an analysis of a questionnaire completed by experienced lecturers at this university.
The question addressed in this article is how the 2009 intake of students cope with mathematics at university level with regard to Performance
 General attributes
 Mathematical attributes
 Contentrelated attributes
The study also indicates that these students seem to be better equipped with regard to personal attributes such as selfconfidence and the will to work. However, in many instances, their general mathematical attributes such as algebraic manipulation skills and their general mastery of mathematical writing are worse than those of students in the past. There are also areas where their content knowledge is either lacking or unexpectedly shallow. It therefore appears that these students have improved personal attributes but not necessarily the knowledge and mathematical skills to back them up.
Some recommendations are made with regard to handling the situation. It is clear that the new school system necessitates changes at school level with a view to university level in order to ensure a transition that is surmountable.

Leer deur probleemoplossing in wiskundeonderwys
Author Piet HumanSource: SuidAfrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 28, pp 303 –318 (2009)More LessDrie vorme van wiskundeonderwys word onderskei: tradisionele onderrig deur mededeling, demonstrasie en inoefening van prosedures; wiskundig strenge deduktiewe onderrig (die sogenaamde "moderne wiskunde beweging"); en probleemgesentreerde wiskundeonderwys (onderrig in die konteks van, vir en deur probleemoplossing).
Na die mislukking van die "moderne wiskunde"beweging van die sestigerjare, en die toenemende frustrasie met die swak uitkomste van tradisionele onderrig, het probleemgesentreerde wiskundeonderwys in samehang met sosiaalinteraktiewe klaskamerpraktyke (die sogenaamde vernuwingsagenda ("reform agenda")) na vore getree as 'n broodnodige en (volgens huidige insigte) enigste alternatief vir proseduregerigte onderrig deur mededeling, demonstrasie en inoefening. Sedert die laat sewentigerjare is talle suksesvolle eksperimentele onderrigprogramme, wat op probleemoplossing as dominante klaskamerpraktyk steun, van stapel gestuur. Sedert die eeuwending word die vernuwingsagenda in die nasionale kurrikulumdokumente vir wiskunde van baie lande gespesifiseer.
Probleemgesentreerde, sosiaalinteraktiewe wiskundeonderwys stel aansienlik hoër eise aan leerkragte se wiskundige kennis en hul sosiale, diskursiewe en pedagogiese vaardighede as onderrig deur mededeling en demonstrasie. Suksesvolle implementering van die vernuwingsagenda, ook in SuidAfrika, is dus sterk afhanklik van doeltreffende indiensopleiding en professionele ondersteuning van leerkragte.Learning via problem solving in mathematics education
Three forms of mathematics education at school level are distinguished: direct expository teaching with an emphasis on procedures, with the expectation that learners will at some later stage make logical and functional sense of what they have learnt and practised (the prevalent form),
 mathematically rigorous teaching in terms of fundamental mathematical concepts, as in the socalled "modern mathematics" programmes of the sixties,
 teaching and learning in the context of engaging with meaningful problems and focused both on learning to become good problem solvers (teaching for problem solving) and utilising problems as vehicles for the development of mathematical knowledge and proficiency by learners (problemcentred learning), in conjunction with substantial teacherled social interaction and mathematical discourse in classrooms.
Parallel with the problemsolving movement, over the last twenty years, mathematics educators around the world started increasingly to appreciate the role of social interaction and mathematical discourse in classrooms, and to take into consideration the influence of the social, sociomathematical and mathematical norms established in classrooms. This shift away from an emphasis on individualised instruction towards classroom practices characterised by rich and focused social interaction orchestrated by the teacher, became the second element, next to problemsolving, of what is now known as the "reform agenda".
Learning and teaching by means of problemsolving in a sociallyinteractive classroom, with a strong demand for conceptual understanding, is radically different from traditional expository teaching. However, contrary to commonlyheld misunderstandings, it requires substantial teacher involvement. It also requires teachers to assume a much higher level of responsibility for the extent and quality of learning than that which teachers tended to assume traditionally.
Over the last 10 years, teaching for and via problem solving has become entrenched in the national mathematics curriculum statements of many countries, and programs have been launched to induce and support teachers to implement it. Actual implementation of the "reform agenda" in classrooms is, however, still limited. The limited implementation is ascribed to a number of factors, including the failure of assessment practices to accommodate problem solving and higher levels of understanding that may be facilitated by teaching via problem solving, lack of clarity about what teaching for and via problem solving may actually mean in practice, and limited mathematical expertise of teachers. Some leading mathematics educators (for example, Schoenfeld, Stigler and Hiebert) believe that the reform agenda specifi es classroom practices that are fundamentally foreign to culturally embedded pedagogical traditions, and hence that adoption of the reform agenda will of necessity be slow and will require more substantial professional development and support programs than those currently provided to teachers in most countries. Notwithstanding the challenges posed by implementation, the movement towards infusing mathematics education with a pronounced emphasis on problem solving both as an outcome and as a vehicle for learning seems to be unabated. Substantial work on the development of more effective means for professional development and support of teachers is currently being done.

Die moontlikhede van 'n modelleringsperspektief vir skoolwiskunde
Author Dirk WesselsSource: SuidAfrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 28, pp 319 –339 (2009)More LessDie resultate van die Trends in International Mathematics and Science Study (TIMSS) se ondersoeke na die stand van probleemoplossing en die toepassing van probleemoplossingstrategieë in skoolwiskunde in ongeveer 50 lande 'n paar jaar gelede het die wêreld geskud. SuidAfrika het elke keer onder die laastes van die deelnemende lande geëindig. Uitkomsgebaseerde onderwys, wat in die Westerse wêreld al hoe meer algemeen word, laat al die deelnemers in daardie stelsels beter voel oor wiskunde, maar verbeter nie noodwendig wiskundeprestasie nie. Modellering in die wiskundeklaskamer blyk 'n belangrike middel te wees om die leer van wiskunde meer effektief te maak. Die bemeestering van probleemoplossing en modelleringstrategieë en vaardighede sal beslis die oriëntasie, bevoegdhede en prestasies van leerders op elke skoolvlak verbeter. Dit blyk uit die navorsing dat leerders daarvan hou om wiskunde se toepassingskant te sien en te beleef. Egte probleme behoort gekies te word en daarom is 'n modelleringsperspektief in die onderrig en leer van wiskunde noodsaaklik. Modellering wat ook integrerend aangebied word in die wiskundekurrikulum het ook die vermoë om leerders op alle skoolvlakke ten volle betrokke te kry, om hulle beter toegang tot matematisering te gee, die wiskundige taal meer vloeiend te laat gebruik, probleemoplossings te doen, argumente te kritiseer, bewyse te vind en wiskundige konsepte te herken en te abstraheer uit 'n realistiese situasie. Modellering behoort 'n volle kans gegun te word in wiskundeonderwysersopleiding en in indiensopleiding sodat die leerders die volle voordeel daarvan kan beleef. Dit sal die wiskundeprestasies van ons land se jeug op 'n stewiger grondslag plaas en hulle so hopelik meer mededingend in die toekoms maak.
The possibilities of a modelling perspective for school mathematics
The findings of the international TIMSS investigations of a few years ago into the position and application of problem solving strategies in school mathematics in about 50 countries caused serious concern globally. During each survey South Africa was found to be among the poorest performers of the participating countries. The main problem was that the majority of school learners in South Africa do not have the ability to solve mathematical problems; in fact, it would appear that they lack the total spectrum of mathematical problem solving competencies. The present school system does not develop their mathematical abilities or competencies. While Outcomesbased education, which became very popular in the Western world, has the ability to improve participants' affective values of mathematics, it proved to be inadequate in improving the quality of their mathematical performances.
Mathematics teachers are unsuccessful in teaching in a manner that will make a difference with respect to the way learners do, learn or perform in mathematics. The pedagogical and mathematics content knowledge of the teachers are lacking in conceptual depth, clarity and connectedness (integration). The language proficiency of the learners is poor, which means that they do not understand what they should do with a problem and how to interpret, present and verify their findings. Learners still do not know how to handle mathematics and how to utilise mathematics in order to solve problems. They seriously lack the ability to approach problems in a meaningful and constructive way. Reallife and openended problems are being perceived as huge obstacles to most learners. Teachers are not trained and educated to assist their learners in bridging this gap.
The teaching methodology that will make a difference in the classroom falls in the broad category of problem solving. The daytoday teaching method should be the problemcentred teaching and learning approach. This rather complex teaching methodology requires indepth thinking about the role of the teacher, the role of the learner, the nature of the classroom culture, the nature of the negotiation of meaning between the teacher and individuals or groups, the nature of selected problems and material, as well as the kind of integrative assessment used in the mathematics classroom. Modelling is closely related to the problemcentred teaching approach, but it also smoothly relates to bigger and longer mathematical tasks. This article gives a theoretical exposition of the scope and depth of mathematical modelling. It is possible to introduce modelling at every school phase in our educational sytem.
Modelling in school mathematics seems to make the learning of mathematics more effective. The mastering of problem solving and modelling strategies has definitely changed the orientation, the competencies and performances of learners at each school level. It would appear from research that learners like the application side of mathematics and that they want to see it in action. Genuine real life problems should be selected, which is why a modelling perspective is so important for the teaching and mastering of mathematics. Modelling should be integrated into the present curriculum because learners will then get full access to involvement in the classroom, to mathematisation, to doing problems, to criticising arguments, to finding proofs, to recognising concepts and to obtaining the ability to abstract these from the realistic situation. Modelling should be given a full opportunity in mathematics teacher education so that our learners can get the full benefit of it. This will put the mathematical performances of learners in our country on a more solid base, which will make our learners more competitive at all levels in the future.

Meesterwiskundeonderwysers as mentors in ondervoorsiene en benadeelde skole
Author Annemarie HattinghSource: SuidAfrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 28, pp 340 –354 (2009)More LessHierdie artikel het ten doel om oorsigtelik 'n insae te bied in 'n mentorskapinisiatief wat geïmplementeer is om onderwysers in hul eie wiskundeklaskamers en komplekse sosiale skoolkontekste te ondersteun en professioneel te ontwikkel. In die geval van die Onderwysermentorskapsprogram (OMP) is wiskundeonderwysers gemonitor wat jare se ervaring het, maar wat self óf nie oor voldoende kwalifikasies beskik nie, óf, wat nie deur hulle eie toedoen nie, nooit blootgestel was aan uitnemende wiskunde onderrigpraktyke nie. Die impak van die vierjaarlange program word bondig statisties weergegee, terwyl die kwalitatiewe temas wat onderwysers self versoek het, soos vakkonseptuele en assesseringsondersteuning verhelder word. Laastens word 'n teoretiese raamwerk vir realistiese professionele ontwikkeling aan die hand van die konsep "Sone vir Proksimale Ontwikkeling" verken.
Master mathematics teachers as mentors for underperforming and disadvantaged schools
The work of teachers has a signifi cant role both with regard to learners' achievement and their opportunities in life. In the context of a developing country, particularly with respect to township and rural schools, it is a common occurrence that many practising teachers are un or under qualified for the teaching of mathematics. What kind of professional development would be effective for upgrading the content knowledge and professional competence of such teachers? It is the aim of this article to give an overview of a professional development approach, called mentorship by master mathematics teachers, which had a positive effect both on teacher learning and their learners' performance. Where mentorship usually focuses on the induction of beginning or student teachers, the Teacher Mentorship Programme (TMP) in this case, had as its target experienced teachers who had been teaching mathematics for years in township schools, but without the appropriate qualifications. The mentee teachers had never previously, due to no fault of their own, been exposed to excellent teaching practices as learners, student or practising teachers. The TMP was an initiative of a university's Faculty of Engineering, that collaborated with private engineering companies and the Department of Education with the aim of preparing more and better equipped Grade 12 leavers who enrol for studies in engineering and technology related fields. The intended outcomes of the TMP were: To improve teachers' content knowledge, pedagogical competence and attitude towards mathematics and science.
 To broaden teachers' knowledge of careers related to the mathematics, science and technology fields.
 To improve learners' attitudes towards mathematics, science and related careers.
 To increase the number of learners who enrol for mathematics on the higher grade which is the equivalent of "core" mathematics in the new National Curriculum Statement.
The impact of the four year programme (20032006) is briefly provided through statistical lenses, while the qualitative themes that emerged as the biggest learning and developmental needs as reported by teachers themselves are narrated. Teachers mostly needed support with subject conceptualisation, pedagogic content knowledge (how to teach challenging concepts) and portfolio assessment. A detailed account of results can be found in Fricke (2008). In conclusion, the article offers a possible theoretical framework for designing and negotiating an individualised professional development plan. The suggested theoretical frame departs from the Zone of Feasible Development (ZFD), analogous to Vygotsky's ZFD which describes the "distance" between actual performance on the one hand and the idealised performance an individual can achieve under the guidance of an expert on the other.

Wiskundige modellering : vanaf skool na universiteit s
Author Ansie HardingSource: SuidAfrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 28, pp 355 –365 (2009)More LessDie uitkomsgerigte onderwysstelsel (UGO) word deur kontroversie gekenmerk en die 2008 matriekuitslae in wiskunde wat aan ongewoon baie studente toelating tot universiteit verleen het, het nie daartoe bygedra om kritici te stil nie. Die eerste studente wat hulle volle skoolopleiding in die UGOstelsel beleef het, het universiteite in 2009 bereik en hulle gereedheid vir universiteitswiskunde en prestasie op universiteitsvlak is belangrik as sukseskriterium van die UGOstelsel.
In 'n vorige studie is studenteprestasie in wiskundetoelatingstoetse vir 20052007 ondersoek en bevind dat studente wat gedeeltelike blootstelling aan UGO gehad het, slegter vaar in die kategorie van modellering en verhoudingsprobleme as wat hulle voorgangers gedoen het. Na aanleiding hiervan word hierdie studie onderneem om te ondersoek hoe die 2009inname van studente in 'n modelleringskursus op universiteitsvlak vaar. Daar word verslag gedoen oor studenteprestasie in die kursus, probleme wat ondervind is, die effek van regstellende intervensie op prestasie en of die UGOstelsel se studente voldoende vir modellering op universiteitsvlak voorberei is.Mathematical modelling: From school to university
The outcomes based education (OBE) system is characterised by controversy and the 2008 matric results that rendered admission to an unusually large number of students did nothing to silence critics. The first students who completed their full cycle of school education in the OBE system entered universities in 2009 and their preparedness for university mathematics as well as their performance at university level are important as indicaters for estimating the success or otherwise of the OBE system.
In a previous study student performance in mathematics admission tests for 20052007 was investigated and it was found that students who had had partial exposure to OBE performed worse than had been the case with their predecessors in the categories of modelling and ratio problems. As a result, this study was conducted to investigate how the 2009 intake of students performed in a modelling course at university level. A report is presented which deals with student performance in the course, problems experienced, the effect of remedial intervention on performance and whether students of the OBE system are adequately prepared for mathematical modelling at university level.
This study focuses on performance in a first year course in mathematical modelling at the University of Pretoria. The course is problem based and is technology intensive, requiring use of the software package Matlab. For investigative purposes the papers of semester tests 1 and 2 of 2005 were used unchanged for tests in 2009. Students of 2009 did not have access to the 2005 papers and the same lecturer taught students of both groups. The lecturer also noted personal experiences in respect of students and was able to draw reasonable comparisons between the 2009 students and previous groups because of her years of involvement with the course. The entrance requirement of 60% for matric mathematics in 2005 was increased to 70% in 2009.
Results indicate that the pass percentage decreased in both semester tests from 2005 to 2009. The percentage of students with less than 40% also increased in both semester tests from 2005 to 2009. A surprising observation was that the percentage of students who excelled increased remarkably from 2005 to 2009 in both semester tests. Extra assistance was requested by students after the first semester test in 2009 and resulted in a week of intervention during which revision and extra problems were attended to.
The results of the two semester tests of 2009 did not live up to expectations of the matric results of the group . Despite the exceptional matric results, performance in the modelling course was decidely poorer. Results indicate that most students were not adequately prepared for the course  as could be predicted from the previous study.
The 2009 group was characterised by two aspects  excellent matric results on the one hand and full school education within the OBE system on the other. It is difficult to distinguish between the influence of the two aspects. It would be convenient to blame the substantial size of the "under 40" groups in both semester tests of 2009 on the better than average matric marks. Yet the increased admission requirements should have compensated for the high matric marks. If the matric results are considered to be reliable then the possibility cannot be excluded that the OBE system had had an influence on the underpreparedness of underperforming students in the course. By contrast, in the instance of top achievers if these students had been influenced by the OBE system it had a positive rather than a negative effect on their performance.
It would appear that OBE may have had a polarising effect between good and poor students. Good students performed exceptionally well and poor students did worse than expected. It would, however, be unfair to summarily reject the OBE system. Personal impressions of the lecturer include that the OBEgeneration asks for assistance when they need it, they are willing to try and use their resources to their own advantage.

Projekte om wiskundeprestasie aan 'n tegniese universiteit te verbeter
Author C.J. LouwSource: SuidAfrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 28, pp 366 –377 (2009)More LessDosente aan tersiêre instansies spreek al 'n geruime tyd hul kommer oor die kwaliteit van wiskundestudente uit. Die blaam word dikwels voor die deur van die sekondêre skole gelê, maar deurvoersyfers is van kardinale belang vir staatsubsidie; derhalwe moet tersiêre instansies voortdurend aksies en intervensies onderneem om studente se wiskundeprestasie te verbeter. In hierdie artikel word verslag gedoen oor die uitslae van twee sodanige studies, met verwysing na moontlike verdere projekte. Die eerste studie behels 'n aksienavorsingsprojek met 'n eksperimentele ontwerp wat aan die voormalige Technikon Noord Gauteng uitgevoer is. Die studie het die implementering van 'n tutoriaalstelsel behels en het bepaalde tendense uitgewys. Die tweede studie het 'n ondersoek na assesseringspraktyke in eerstejaarswiskunde aan die Tshwane Universiteit van Tegnologie (TUT) behels. Respondente is aan vernuwende assessering blootgestel, met die oogmerk om hul kennis en vaardigheid in kontemporêre assesseringspraktyke te verbeter en gestalte aan uitkomsgerigte onderrig en leeromgewings te gee. Hierdie studie het nuwe probleme aan die lig gebring en hernieude pogings moet vervolgens aangewend word om wiskundeprestasie te verseker.
Projects to improve mathematics performance at a university of technology
Lecturers at tertiary institutions have been expressing concern about the quality of students in mathematics for a long time now. Blame is usually placed at the door of secondary schools, but through put figures are still determining state subsidies and as such necessitate institutions to constantly revisit their efforts in terms of improving performance in mathematics. In this article, the results of two studies to improve the mathematics performance at a technical university are reported. The first study took place at the former Technikon Northern Gauteng and was an action research project with an experimental design. The study entailed the implementation of tutorial sessions and it revealed certain traits of tertiary mathematics education. Data were collected through structured observation, questionnaires and focus group interviews. A standardized questionnaire, Study Orientation Questionnaire in Maths (SOM), was introduced and "study habits" was the field in the test that featured most significantly in both the experimental and the control groups. The results revealed, inter alia, that timely assessment is needed to detect and correct misconceptions as soon as possible. It also pointed out that attendance of lectures (and tutorials) impacted strongly on performance. Lastly, it was found that students preferred communicating mathematics through the medium of English and not by using their mother tongue. The second project was conducted at the Tshwane University of Technology (TUT) and consisted of an investigation into assessment practices in fi rst year mathematics. The study was executed with action research as strategy and data were collected through interviews, questionnaires and focus group interviews. Qualitative and quantitative data were collected and respondents were exposed to contemporary assessment strategies as suggested by OBE. The aim of the study was to enhance the respondents' insight into contemporary assessment strategies and to empower them to execute their assessment in a more accountable way. The merger of the institution was still underway at the time, which left many respondents with questions and uncertainties about their future. Some of the findings were, inter alia, that some respondents were convinced that new assessment strategies would lower the standard of teaching; some respondents mentioned the poor quality of students and were convinced that the Tshwane University of Technology would benefit from reviewing their admission criteria, and respondents cited large class groups, lack of marking assistance and ignorance about Outcomes Based Education (OBE) as reasons for failing to undertake assessment renewal. In the mean time class groups got even bigger and a new solution needs to be found as a matter of urgency.

Studieoriëntasie en basiese woordeskat in wiskunde in die laerskool
Author Marthie Van der WaltSource: SuidAfrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 28, pp 378 –392 (2009)More LessRelatief min navorsing is in SuidAfrika gedoen met betrekking tot Graad 4 tot 7leerders se ontoereikende prestasie in wiskunde. Gedurende die afgelope 15 jaar fokus navorsing op onder meer die invloed van sosiale, kognitiewe en metakognitiewe, konatiewe en affektiewe fasette op prestasie in wiskunde. Drie gestandaardiseerde vraelyste, naamlik die Studieoriëntasievraelys in Wiskunde (Primêr), die Wiskundewoordeskat (Primêr) en die Basiese Wiskunde (Primêr), is ingeskakel. Die doel van hierdie studie was om ondersoek in te stel na die mate waartoe prestasie in studieoriëntasie (SOW(P)), én kennis van basiese woordeskat in wiskunde (WW(P)) van Graad 4 tot 7leerders hul prestasie in wiskunde (BW(P)) voorspel. Graad 4 tot 7leerders in Noordwes (n = 1 103) het aan die studie deelgeneem. Resultate van die data, wat deur die berekening van interkorrelasies en stapsgewyse regressieanalise bekom is, dui daarop dat leerders se prestasie in wiskunde in die praktyk voorspel/verklaar kan word uit hul prestasie betreffende kennis van basiese woordeskat, wiskundeangs, studiehouding en studiegewoontes in wiskunde. Indien onderwysers leerders se ontoereikende studieoriëntasie en kennis van woordeskat in wiskunde in die vroeë skooljare kan identifiseer, is dit moontlik dat eersgenoemde voorkomend sou kon onderrig en/of remedieer ten einde laerskoolleerders se prestasie in wiskunde te verbeter.
Study orientation and knowledge of basic vocabulary in Mathematics in the primary school
Whatever the reason, underachievement in mathematics in South Africa is endemic and tantamount to a national disaster. Despite the transformation of education in South Africa, failure rates in mathematics at school and university remain unacceptably high, and the number of learners who leave Grade 12 with a pass mark in both mathematics and physical science is unacceptably low. Relatively little has been written about inadequate performance of Grade 4 to 7 learners in mathematics in South Africa, and even less about possible solutions to the problem. South African primary school learners' lack of basic mathematics and vocabulary skills in particular is a source of major concern. In the first national systemic evaluation of learners' skills in English, mathematics and science in 2001 Grade 3 learners achieved an average of 30% in mathematics. In the followup studies, Grade 6 learners achieved a national average of 27% in mathematices, in 2004, while nationally eighty percent of Grade 3 and 6 learners achieved less than 50 percent for mathematics and Languages in 2008. The finding that so many primary school learners today are not numerate or literate has a direct influence both on the teaching and the learning of mathematics. Everything possible needs to be done to change this situation. During the past 15 years, the research focus in mathematics has shifted to an examination of the influence of social, cognitive and metacognitive, conative and affective factors on achievement in mathematics. In this regard, it is of particular importance that an ongoing investigation into "other" aspects that impact on achievement in mathematics is launched, rather than to restrict the investigation to mere assessment of objectives that are aimed at continually evaluating cognitive progress in mathematics. There is sufficient empirical evidence that an adequate orientation to the study of mathematics correlates positively with high achievement in mathematics on secondary and tertiary levels. The aim of this research was to investigate the extent to which the performance in study orientation (Study Orientation questionnaire in Mathematics (Primary)) and knowledge of basic vocabulary/terminology in mathematics (Mathematics Vocabulary (Primary)) (vocabulary as one aspect of language in Mathematics) of Grade 4 to 7 learners predict performance in mathematics (Basic Mathematics (Primary)). Three standardised questionnaires were administered, namely the Study Orientation questionnaire in Mathematics (Primary), or SOM(P), Mathematics Vocabulary (Primary) or (MV(P), and Basic Mathematics (Primary) or BM(P). The participants consisted of learners in Grade 4 to 7 (n = 1 103) in NorthWest Province with respectively Afrikaans, English and Tswana as their home language. Results from the data, by calculating intercorrelations and stepwise regression, confirmed that learners' performance in mathematics (BM(P)) can be predicted through their performance in the knowledge of basic vocabulary in mathematics (MV(P)), their "maths" anxiety, study attitude towards and study habits in mathematics (SOM(P)). The results can be implemented to improve learners' performance in mathematics when teachers identify inadequate knowledge of basic vocabulary in mathematics as well as study orientation (for example, "maths" anxiety, study attitude towards and study habits in mathematics) in the early years of schooling. Learners' scores can be checked to identify those requiring aid, support, remediation and / or counselling. An analysis of individual answers (particularly those where learner's replies differ significantly in respect of the answers usually given by good achievers in mathematics) could be extremely useful. Enculturing learners to the vocabulary of mathematical language is an aspect of instruction that needs specific attention. The three questionnaires, which are administered in this research, provide mathematics teachers with standardised tools with which to make a simple systematic analysis of a number of important background particulars, feelings, attitudes, habits and customs with regard to the learner's academic orientation in mathematics, as well as to their knowledge of basic vocabulary in mathematics that could be remedied when inadequate.

SuidAfrika se Wiskundekrisis : innoverende oplossing nou nodig : standpunt
Author Alet RademeyerSource: SuidAfrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 28, pp 393 –398 (2009)More LessOorgenoeg is al gesê oor SuidAfrikaanse leerlinge se swak prestasie in wiskunde. Nie net vaar hulle internasionaal power nie, maar ook op die vasteland steek lande wat ekonomies swakker as SuidAfrika daaraan toe is, ons die loef af. In 2007 het die nasionale onderwysdepartement besluit om aan die Trends in International Mathematics and Science Study (Timss) te onttrek nadat die land in 2003 die swakste van 50 lande gevaar het. Die internasionale gemiddeld vir wiskunde was 467 uit 800 teenoor die 264 uit 800 wat SuidAfrika behaal het.

Verspreiding van die inheemse mossel Pisidium costulosum Connolly 1931 (Bivalvia : Sphaeriidae) met betrekking tot temperatuur, reënval en water tipe in SuidAfrika
Authors: Kenne De Kock and Corrie WolmaransSource: SuidAfrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 28, pp 399 –413 (2009)More LessDie genus Pisidium sluit van die kleinste mosselspesies in, kan bykans alle varswaterhabitats benut, is bodembewoners en fi ltervoeders en beskik oor 'n relatief lang lewensduur  eienskappe wat bydra tot hul geskiktheid vir die monitering van omgewingstoestande. Hierdie artikel fokus op die geografiese verspreiding en habitats van P. costulosum, 'n spesie wat tot SuidAfrika beperk is en die omvangrykste geografiese verspreiding het van die ses Pisidiumspesies wat in die databasis van die Nasionale Varswaterslakversameling (NVWSV) opgeneem is. Meer as 60% van die 428 monsters op rekord, is in spruite en riviere versamel en in watertoestande wat as standhoudend, staande, helder en vars beskryf is. Resultate dui daarop dat waterliggaamtipes, temperatuur en hoogte bo seevlak die belangrikste rol gespeel het in die daarstelling van die geografiese verspreiding van hierdie spesie. Landswye versameling van varswater Mollusca deur staatsinstansies is in die vroeë 1980's getermineer en die meerderheid vindplekke is sedertdien nie weer besoek nie. Dit bring mee dat hul gedokumenteerde verspreiding gevolglik verouder is en dat min oor hul bewaringstatus in SuidAfrika bekend is. Omdat P. costulosum koeler klimaattoestande en standhoudende habitats verkies, kan verwag word dat klimaatverandering wat met aardverwarming gepaardgaan 'n negatiewe effek op die voortbestaan van die spesies mag hê. Pogings behoort aangewend te word om die geografiese verspreiding van minstens medies en veeartsenykundig belangrike spesies op te dateer en hul bewaringstatus te bepaal. Terselfdertyd sou die vordering van eksotiese indringerspesies in hul besetting van nuwe gebiede gemoniteer kon word.
Distribution of the indigenous clam Pisidium costulosum Connolly 1931 (Bivalvia: Sphaeriidae) in relation to temperature, rainfall and watertype in South Africa
Little is known of the functional role of freshwater bivalves and particularly the ecology of burrowing species has been inadequtely studied. This is an unfortunate situation in view of the fact that freshwater biotopes are losing biodiversity at an alarming rate. The genus Pisidium includes some of the tiniest bivalve species known to man, it has a worldwide distribution and its origin is considered to be Mesozoic. Due to the fact that Pisidium can exploit virtually almost any freshwater habitat, that it is a bottom dweller and filter feeder and is also relatively long lived, it is important to study species of this genus because they could be used for monitoring environmental conditions. This article focuses on the geographical distribution and habitats of P. costulosum, a species considered to be endemic to South Africa and which is the most widespread of the six Pisidium species on record for this country in the database of the National Freshwater Snail Collection (NFSC).
Details pertaining to the habitats of the 428 samples of P. costulosum as recorded at the time of collection were extracted from the database of the NFSC. The 187 loci (1/16 th square degrees) in which the 428 collection sites were located, were distributed in preselected intervals of mean annual air temperature and rainfall, as well as intervals of mean altitude, to illustrate the frequency of occurrence within specific intervals. A temperature index was calculated for all mollusc species in the database from their frequencies of occurrence within the selected intervals and the results used to rank them in order of their association with low to high climatic temperatures. To evaluate the significance of the difference between frequencies of occurrence in, on, or at the various options for each parameter investigated, chisquare values were calculated. In addition an effect size value was calculated to determine the contribution of each parameter towards establishing the geographical distribution of this species based on the data in the database. A multivariate analysis in the form of a decision tree was also constructed from the data which enabled the selection and ranking of those variables that maximally discriminated between the frequency of occurrence of P. costulosum in, on, or at the various options for each parameter as compared to all other bivalve species in the database.
Our records show that this species is well represented in Gauteng and the central and southern areas of Mpumalanga. It is relatively continuously spread through the north eastern parts of the Free State and all along its eastern and southern borders, sparsely distributed in the Northern Cape and poorly represented in Limpopo.
Although it was reported from 13 of the 14 waterbodies represented in the database, more than 60% of the samples were recovered from streams and rivers and from water conditions described as perennial, standing, clear and fresh and more than 45% came from habitats with a predominantly muddy substratum.
From the effect size values calculated for the various parameters it can be deduced that type of waterbody, temperature and altitude played the most significant role in establishing the geographical distribution of this species in South Africa.
On account of the fact that countrywide surveys by government and local health authorities were discontinued in the early 1980's the current range of the geographical distribution of freshwater mollusc species in South Africa is rather uncertain. The data in the database were compiled over a period of approximately five decades and due to the fact that the majority of sampling sites have not been revisited since, little is known of the conservation status of the freshwater molluscs of South Africa. Its close association with cooler climatic conditions as indicated by its temperature index and its dependency on perennial habitats, suggest that P. costulosum could be vulnerable to climatic changes due to global warming and that this could have a negative impact on the range of its geographical distribution. It is suggested that efforts should be made to update the geographical distribution of at least those mollusc species that are of medical and veterinary importance and at the same time to monitor the progress of exotic mollusc species in their invasion of waterbodies in this country. A comparison of the results of such surveys with the long range of data in the NFSC database could be valuable to assess the current conservation status of at least some of the indigenous freshwater molluscs of this country.

Dieetsamestelling van tuinduiwe in die Bloemfonteinse stadsgebied, sentrale Vrystaat
Authors: Pieter Le Roux, Ordino Kok and Hennie ButlerSource: SuidAfrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 28, pp 414 –422 (2009)More LessOndersoek is ingestel na die kropinhoud van 333 volwasse tuinduiwe (Columba livia) en 49 neskuikens wat oor 'n tydperk van 'n jaar in die Bloemfonteinse stadsgebied, sentrale Vrystaat, versamel is. Daarvolgens word die dieet deur natuurlike plantegroei uit die onmiddellike omgewing, waarvan boomvruggies van die inheemse witstinkhout (Celtis africana) die belangrikste is, oorheers. Vergelykenderwys is die bydrae van invertebraat prooiitems, hoofsaaklik larwes en papies van die Diptera en Lepidoptera, onbeduidend. Die teenwoordigheid van 'n beduidende hoeveelheid kropmelk hou waarskynlik verband met die uitgebreide broeiseisoen van die duiwe, terwyl die gereelde voorkoms van anorganiese materiaal op die belangrikheid van klipgruis by die fisiese verwerking van harde plantdele op 'n latere stadium in die spiermaag dui. Die kropinhoude van neskuikens impliseer dat verbouingsgewasse soos koring, mielies en sonneblom veral gedurende die broeiseisoen deur volwasse tuinduiwe benut word. Hoewel die voëls onder eksperimentele toestande 'n duidelike voorkeur aan mielies en die geel skakering van die ligspektrum verleen, blyk die benuttingsfrekwensie van spesifieke gewasse eerder op hul relatiewe beskikbaarheid gebaseer te wees.
Dietary composition of feral pigeons in the Bloemfontein urban area, central Free State
Crop contents of 333 adult feral pigeons (Columba livia) and 49 nestlings collected over a period of a year in the Bloemfontein urban area, central Free State, were investigated. According to the abovementioned data the diet is dominated by natural vegetation from the immediate vicinity, with small fruit from the endemic stinkwood (Celtis africana) being the most important. Comparatively the contribution of invertebrate prey items, mainly larvae and pupae of the Diptera and Lepidoptera, is negligible. The presence of a significant amount of pigeon's milk can probably be related to the extended breeding season of the birds, while the regular occurrence of inorganic material points to the importance of grit in the physical processing of hard plant parts at a later stage in the gizzard. The crop contents of nestlings imply that cultivated crops such as maize, sunflower and wheat are utilised by adults particularly during the breeding season. Although the pigeons showed a clear preference for maize and the yellow shade of the light spectrum under experimental conditions, the utilisation frequency of specific crops is based more on their relative availability.

Siegfried Passarge (18661958) en die begin van 'n SuiderAfrikaanse geomorfologiese diskoers : baanbrekers
Author Barnie BarnardSource: SuidAfrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 28, pp 423 –432 (2009)More LessSiegfried Passarge se loopbaan as aardwetenskaplike het oor drie tydruimtelike segmente verloop : sy opvoeding in aartskonserwatiewe Pruise tussen 1875 en 1892, sy veldwerk in SuiderAfrika tussen Junie 1896 en Januarie 1899 en die lang skof van 1908 tot 1935 as professor in Geografie in Hamburg. Hierdie artikel is veral gerig op die resultate van sy SuiderAfrikaanse verblyf.
Passarge is in 1896 deur 'n Britse firma gekontrakteer om vir edeldelfstowwe in die noordweste van Betsjoeanalandprotektoraat te prospekteer, 'n soektog wat teen die einde van 1898 vrugteloos geblyk te gewees het. Hoewel hy sy pligte nougeset nagekom en spoedig leier van die eksplorasiespan was, het hy homself eintlik as 'n wetenskaplike beskou en inligting oor 'n wye veld van die aardwetenskappe versamel. Terug in Duitsland het drie monografieë uit sy pen verskyn: Die Kalahari (1904), 'n massiewe fisiografie van 822 bladsye, Die Buschmänner der Kalahari (1907), 'n saaklike etnografie, en Südafrika (1908), die eerste streeksgeografie van die subkontinent wat deur 'n professionele geograaf geskryf is.
As verkenning het Die Kalahari die temas aan die orde gestel wat vandag nog die geomorfologiese diskoers oorheers: klimaatsverandering, fossieldreineerstelsels, eilandberglandskappe, hardekorse, die Platorand en die stratigrafie van die Kalahari Opeenvolging. Südafrika het 'n nog wyer bestek gehad en sy geomorfologie was grof in resolusie, maar Passarge kon van sy vertolkings herbevestig. Nogtans is hierdie vertolkings vinnig deur ander skrywers opgevolg, aanvanklik net so oorgeneem maar later gewysig en verwerp.
In Hamburg het Passarge se belangstelling gaandeweg van geomorfologie na "landskapskunde" oorgeswaai en sy uitsprake het al hoe meer kontroversieel geword. SuidAfrikaners onthou egter sy werk van die eerste dekade van die eeu.Siegfried Passarge (18661958) and the opening of a Southern African geomorphological discourse
Siegfried Passarge's career as an earth scientist passed through three segments of spacetime : his education in archconservative Prussia between 1875 and 1892; the fieldwork in Southern Africa between June 1896 and January 1899 which made him the leading German arid geomorphologist of the time; and, finally, his stay in Hamburg where he was a controversial professor of geography from 1908 to beyond retirement in 1935. This paper is primarily concerned with the results of his brief sojourn in Southern Africa.
In 1896 Passarge was contracted by a British firm to search for minerals in the northwest of Bechuanaland Protectorate. Diligently performing his duties Passarge regarded himself a scientist rather than a prospector, gathered extensive data on the Kalahari environment and people and, back in Germany, published his research results in three benchmark monographs : Die Kalahari (1904), a massive physiography of 822 pages, Die Buschmänner der Kalahari (1907), a concise ethnography, and Südafrika (1908), the first regional study of the subcontinent by a professional geographer.
As a reconnaissance Die Kalahari opened up the themes which still dominate the Southern African geomorphological discourse: climate change, fossil drainage systems, inselberg landscapes, duricrusts and the stratigraphy of the Kalahari Sequence. Südafrika was ingeniously structured but its geomorphology was coarse in resolution and its interpretations facile. Nevertheless Passarge's work left a long trail in the South African earth sciences. W.M. Davis based his arid cycle on Passarge's explanation of the plateau plains, A.L. du Toit included his fivefold division of the Kalahari sediments in The Geology of South Africa, while John Wellington's Southern Africa was an intentional successor to Südafrika.
At Hamburg University during the 1920s and 1930s Passarge exchanged the earth sciences for "Landschaftskunde" and as a result became an embarrassment to the German community of geographers because of his virulent support of the National Socialist regime.