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From lessons to lectures: NCEAmathematics results and first-yearmathematics performanceAlex James a , Clemency Montelle a & Phillipa Williams aa Department of Mathematics and Statistics , University ofCanterbury , Christchurch, New ZealandPublished online: 15 Oct 2008.

To cite this article: Alex James , Clemency Montelle & Phillipa Williams (2008) From lessonsto lectures: NCEA mathematics results and first-year mathematics performance, InternationalJournal of Mathematical Education in Science and Technology, 39:8, 1037-1050, DOI:10.1080/00207390802136552

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International Journal of Mathematical Education inScience and Technology, Vol. 39, No. 8, 15 December 2008, 1037–1050

From lessons to lectures: NCEA mathematics results and first-year

mathematics performance

Alex James*, Clemency Montelle and Phillipa Williams

Department of Mathematics and Statistics, University of Canterbury, Christchurch,New Zealand

(Received 27 November 2007)

Given the recent radical overhaul of secondary school qualifications inNew Zealand, similar in style to those in the UK, there has been a distinctchange in the tertiary entrant profile. In order to gain insight into this newsituation that university institutions are faced with, we investigate some of theways in which these recent changes have impacted upon tertiary levelmathematics in New Zealand. To this end, we analyse the relationship betweenthe final secondary school qualifications in Mathematics with calculus ofincoming students and their results in the core first-year mathematics papersat Canterbury since 2005, when students entered the University of Canterburywith these new reformed school qualifications for the first time. These findingsare used to investigate the suitability of this new qualification as a preparationfor tertiary mathematics and to revise and update entrance recommendations forstudents wishing to succeed in their first-year mathematics study.

Keywords: NCEA; tertiary study; New Zealand; calculus

1. Introduction

‘There is an overwhelming consensus among University teachers of Mathematics, Physics, andEngineering that the mathematical achievement of students entering University with given gradesis substantially worse than those entering with the same grades five or ten years ago . . .The cleardecline in analytical powers, algebraic manipulation and technical fluency of many under-graduates to a broad range of courses, including specialist mathematics and statistics courses,must be addressed’ (School Curriculum and Assessment Authority, 1996a p. 7.10 and 9.1) [1].

A common observation made by both professionals and educational researchers alike is aperceived decline in the abilities and preparedness of students entering tertiary institutions.Indeed, many fields testify to decline in general competencies, and in fact, mathematics canidentify specific areas of weaknesses, and the steady decrease in mathematical standardsand subject coverage has been frequently substantiated [2–5]. Furthermore, the increasingdisparity between tertiary-level curriculum content and expectations in mathematics andthe actual proficiency and skills of the incoming students is not just limited to specificcountries, but is seemingly a worldwide phenomenon.

*Corresponding author. Email: a.james@math.canterbury.ac.nz

ISSN 0020–739X print/ISSN 1464–5211 online

� 2008 Taylor & Francis

DOI: 10.1080/00207390802136552

http://www.informaworld.com

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There have been various institutional reforms to address this. In the UK, for example,there have been recent radical changes in the ways in which the exit qualifications ofsecondary schools are organized, assessed and managed. The introduction of G.C.S.E. in1992 and the so-called ‘Curriculum 2000’ and the implementation of the AdvancedSubsidiary (AS) award, all contributed to the massive overhaul of the existinginfrastructure, which in effect ‘modularised’ a more protracted single unit of study intosmaller discrete modular components. These changes have met with mixed reports ofsuccess [6,7] with institutional responses being quite polarized, as ‘. . . some talk about theopportunities for reinforcement of learning and for the increased ability to tailor the choiceof curriculum to the needs of the pupil, while others point to an over-reliance on shortterm learning goals’ [8]. To what extent these reforms are raising the standards ofmathematical proficiency in university entrants is still yet to be fully determined [8,9].

New Zealand followed suit, overhauling its formal secondary school qualifications ina similar fashion several years later. This change came none too late, as thequalifications it replaced had been relatively untouched for a considerable amount oftime: University Entrance (Year 13) and Sixth Form Certificate (Year 12) wereimplemented in the early 1970s, and the ‘School Certificate’ (Year 11) qualification in1945. Given its recent inception, it is timely to get an initial indication of how effectivethese changes have been in equipping students as they transition to tertiary study andwhether or not such changes may help mitigate the tide of general global decline in themathematical sciences. This study will also provide a New Zealand perspective of trendsconcerning the student profile in mathematics as they transition from secondary schoolto higher education, particularly those that are being presently done with the UK [1,4],the US [10–13] and Australia [14].

1.1. NCEA Mathematics with Calculus: general background

‘Tertiary providers set their own selection requirements for admission to restrictedcourses. . . .Tertiary providers are aware of the usefulness of merit and excellence grades andthe scholarship results in guiding school leavers into appropriate tertiary programmes’ [15].

NCEA or National Certificate for Educational Achievement, New Zealand’s mainnational qualification for secondary school students, was first introduced into schools in2002. Based on a three-tiered model, the final stage, i.e. Level 3, was implemented in2004, so the students who entered tertiary level education in 2005 did so with the newqualifications for the first time. Instead of arriving with a simple percentage grade fromthe previous Bursary qualification, the intricate structure of NCEA means that tertiaryproviders have an array of potentially useful information about the abilities ofprospective students. With respect to mathematics with calculus, school leavers nowarrive with up to 24 Level 3 credits organized into five separate achievement standards(Differentiation, Integration, Trigonometric Equations, Algebra and Complex numbersand Conic Sections), each with four possible results: Excellence, Merit, Achieved andNot Achieved.

As NCEA necessitated significant changes in the way in which the secondary schoolmathematics syllabus is organized, taught and assessed, new trends in the universityentrant population are quickly becoming apparent. To assimilate and incorporatethis changing student dynamic, tertiary level mathematics providers are interestedin the various issues raised by these new circumstances and how to respond to them.

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This preliminary study uses the recent data as guidance on issues such as determining the

type of advice to give to prospective undergraduate students based on their previous

achievements. These findings may also offer indirectly fresh strategies for attracting and

retaining this new student body, realigning course content to reflect evolving strengths,

identifying students’ ability to cope with university modes of assessment, and so on.At the University of Canterbury (UC), the overall structure for the mathematics degree

is modulated from the top (graduate) end down. Mathematics is a primarily cumulative

discipline, in that success in higher levels requires a strong degree of competency in the

elementary material. Thus, one of the main concerns for tertiary level providers is to

minimize the amount of disconnect in the transition from school to the university

environment while still maintaining the standards required for a university graduate. The

majority of UC’s first-year courses have open entry, but students are assisted with course

selection through departmental advice based on the student’s level of prior achievement,such as NCEA. The existing departmental entrance criteria are given in Appendix,

recommendations which were based largely on anticipated results and reasonable

expectations. With data now available, the practical offshoot of this study is to examine

and possibly revise these in light of recent trends.

2. Background and methodology

This study is the first of its kind in New Zealand since the new qualifications were

implemented in mathematics, though similar studies have been carried out in the UK [8].

Given the existence of negative publicity that has surrounded the changes to the

secondary school qualifications, studies such as these are vital so that tertiary-level

providers can independently review the impact these changes have had by examining the

relationships in a considered and researched-based approach. Indeed, despite the factthat there are only 2 years of results available for research, given the mixed reception,

this study is timely and is intended to initiate further research into mathematics and

related fields.The appropriate data were gathered in accordance with standard practice. Before any

analysis was conducted, each student was given a random unique identifier (separate from

their NCEA identity number and their UC student number) to ensure full anonymity.

Then their NCEA Level 3 mathematics with calculus results and their relevant

mathematics grades were retrieved from the database held by the Department ofMathematics and Statistics at UC. Students who had enrolled in first-year courses but

subsequently withdrawn were not included in the analysis. The results only include

students who completed either MATH105 in 2005 or MATH108S1 in 2006; other first-

year mathematics papers were not studied. Those students who received a V or a Y in their

NCEA grade have been included in the not-achieved category. All NCEA national

statistics are available on the NZQA website [16].For most students at UC intending to major in engineering, mathematics or physics,

from 2006 onwards the ‘core’ first-year mathematics papers are MATH108S1 followed byMATH109S2. MATH108S1 is therefore the entry point for the majority of students.

A second option is MATH108W, a whole year paper that covers the same material as

MATH108S1 but moves at half the pace. In rare cases, exceptional students are

encouraged to enter directly into the appropriate stage-two courses. In 2005, MATH108S1

and MATH109S2 were taught as a single full-year course, known as MATH105. The

majority of students take the appropriate first-year course as part of an engineering

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intermediate year, and one of the requirements for continuation into the engineering

programme of their choice is a Bþ average.Whilst there are no compulsory NCEA prerequisites for entering MATH108S1, the

current recommended entry criterion is as follows:

. At least 15 credits in NCEA Level 3 mathematics with calculus include both of the

achievement standards:

90635 Differentiation90636 Integration

With 2 years of first-year results for NCEA students now available, it is possible to get an

initial indication of the accuracy of this advice.

3. Results

3.1. NCEA results of students entering UC to study mathematics

In general, students entering first-year mathematics at UC had NCEA Level 3

mathematics with calculus results representative of the national average (available from

the NZQA website). From the national trends, the excellence level is accomplished by only

roughly 5% of students and merit by roughly 25% of students.Notably, consistently fewer students attained merit or excellence in the Integration

standard (90636) and Conic Sections standard (90639) over both years. In contrast, the

Trigonometric Equations standard (90637), which is internally assessed, showed a

markedly higher number of excellent achievements. Differentiation is the standard which

had a significantly lower not-achieved rate than all other externally assessed standards.

3.2. Total NCEA credits and first-year mathematics results

Table 1 compares the MATH105 (2005) or MATH108S1 (2006) grade (A, B, C, F)1 with

the total number of NCEA Level 3 mathematics with calculus credits. An A grade

indicates an overall mark of greater than 75%, a B grade is between 60 and 74% and a C

grade between 50% and 59%. The overall pass rate (i.e. all marks above 50%), shown at

the bottom of the table, for MATH105 in 2005 was 76% (472 students of 625 passed) and

for MATH108S1 in 2006 was 71% (502 students of 704 passed) and the number of

students achieving A grades was 32% (204 students out of 625) in 2005 and 31% (220

students out of 704) in 2006. In 2005, 329 of the 625 MATH105 students and in 2006, 386

of the 704 MATH108S1 students had done NCEA Level 3 mathematics with

calculus. The pass rate for the NCEA students was 78% in 2005 and 79% in 2006,

which is comparable with students from other backgrounds, e.g. Cambridge A-level,

International Baccalaureate and other international qualifications. The number of NCEA

students achieving A grades were 30% in 2005 and 33% in 2006, again comparable to the

overall rates.It is very clear that students entering first-year mathematics with515 credits are highly

unprepared for tertiary study, with failure rates of 58 and 67% in 2005 and 2006,

respectively. Very few of these students achieved high scores in first-year mathematics;

over the 2 years only 16 out of the total of 78 received a grade B or higher. Alarmingly, the

number of students entering tertiary level mathematics at the UC with this number of

credits doubled over this 2-year period. Students who had barely achieved the

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recommended criteria, i.e. those with 15–17 credits, also had a very high failure rate ofalmost 50% across both years. Again, the number of high achievers in this category waslow, only 2 students out of 52 in this category received an A grade and 13 received a Bgrade. As many students in first-year mathematics are part of the Bachelor of Engineeringintermediate-year programme, gaining a high grade is of utmost importance. Students whowish to continue in the engineering discipline of their choice are required to maintain aBþ average in this intermediate year.

Incoming students with between 18 and 21 credits had a substantially higher pass rate,73% in 2005 and 81% in 2006; both these figures are close to the overall pass rates. It isencouraging to note however that around 60% of NCEA Mathematics with Calculusstudents achieves all 24 credits and students with such qualifications had about an 80%likelihood of getting an A or B grade. These students had pass rates (88% in 2005 and 91%in 2006) significantly above the overall pass rates in both years. They also had many morehigh achievers than the average, with 40 and 49% in 2005 and 2006, respectively, receivingan A grade. Figure 1 gives a pictorial representation of the data contained in Table 1.As the results for each year are notably similar, they have been combined henceforth intoone larger set of results for improved data analysis.

Table 1. Total Level 3 Calculus credits with MATH105/108S1 grade.

Total Level 3Calculus credits

Grade inMATH105/MATH108S1

2005Frequency (%)

2006Frequency (%)

Less than 15 F 15 (57.7) 35 (67.0)C 3 (11.5) 8 (15.4)B 5 (19.2) 6 (11.5)A 3 (11.5) 2 (5.8)

Total 26 52

15–17 F 14 (45.2) 10 (47.6)C 9 (29.0) 4 (19.1)B 6 (19.4) 7 (33.3)A 2 (6.5) 0 (0.0)

Total 31 21

18–21 F 22 (27.2) 17 (18.9)C 14 (17.3) 22 (24.4)B 29 (35.8) 34 (37.8)A 16 (19.7) 17 (18.9)

Total 81 90

24 F 23 (12.0) 20 (8.9)C 29 (15.2) 23 (10.3)B 62 (32.5) 71 (31.7)A 77 (40.3) 110 (49.1)

Total 191 224

Total NCEA students 329 387All students F 153 (24.5) 202 (28.7)

C 92 (14.7) 108 (15.3)B 176 (28.1) 214 (30.8)A 204 (32.6) 220 (31.3)

Total students 625 704

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3.3. Individual credits and first-year mathematics results

Given the range of skills separately assessed, it is pertinent to reflect upon whichstandards, if any, can be identified as indicative of future success. The perceived wisdom,reflected in the UC entrance recommendations, is that the two most important standardsfor predicting success in first-year mathematics are Differentiation (90635) and Integration(90636). A casual examination of the data, combined with the prominent role these topicsplay in first-year calculus, seem to support this recommendation. Students who did not

Total calculus credits vs. MATH108S1 grade (2006)

0%

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F C B A F C B A F C B A F C B A

15–17 18–21 24

Total level 3 calc. credits

Total calculus credits vs. MATH105 grade (2005)

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<15

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B F B C A C A

Figure 1. Bar charts of data from Table 1.

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achieve either of these standards had a very low pass rate, only 25% in 2005 (and 19% in

2006). For one standard only, the pass rate improved to 57% (58%), and if a student

achieved both standards, the pass rate was over 70% in both years.However, upon deeper analysis, it was established that those students who failed one or

both of these standards necessarily had a lower number of NCEA credits and in fact,

similar statistics could be quoted for each of the other standards as well. This realization

led naturally to the following questions:

. Does ‘not taking’ or ‘not achieving’ a particular standard lead to poor

performance in first year? (and if so, which one(s) is it?)

OR

. Does ‘not taking’ or ‘not achieving’ a particular standard simply lead to a lower

number of credits, which leads to poor performance at first year?

That is to say, are some standards more useful than others for tertiary level mathematics

study, or rather is the best sort of preparation for a complete coverage and achievement of

all credits? In order to untangle these confounding factors, we calculated the average first-

year mark and the average number of credits of all students who achieved each standard

and those who did not achieve (N–A) or not complete (N–C) each standard. These results

are shown in Table 2.On its own, the data in Table 2 does not immediately resolve the above questions.

However, the data becomes significant when it is combined with the mean first-year result

dependent on number of credits achieved. If a certain standard was particularly indicative

of first-year success (or failure), one would expect students who achieved (or did not

achieve) that standard to lie significantly above (or below) the average first-year mark for

that number of credits.Figure 2 graphs the number of credits achieved against the first-year mark. The

points show the original data of total NCEA Level 3 mathematics with calculus credits

gained and subsequent first-year mark. It is immediately apparent that there is a wide

spread of first-year marks for any particular number of credits achieved and this spread

increases at the higher NCEA achievement levels. Conversely, it is also apparent that

very few students get a very high first-year grade (85%¼Aþ) with less than 18 NCEA

credits. A line of best fit has been added to the points to show the average first-year

Table 2. Mean number of NCEA credits and mean first-year results dependent on individualcredits.

Standard Credits (/24) First year (%)

Differentiation Achieved 21.66 63.92N–A or N–C 10.21 44.62

Integration Achieved 22.40 65.91N–A or N–C 12.68 44.97

Trigonometry Achieved 21.05 62.35N–A or N–C 14.48 66.44

Complex numbers Achieved 22.38 65.92N–A or N–C 14.09 47.67

Conic sections Achieved 22.12 65.80N–A or N–C 16.37 51.12

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mark for a given number of credits. The regression statistics (F¼ 190, p5 0.001) showthe excellent correlation between high achievement at NCEA and high achievement infirst-year. The crosses show the data from Table 2 where achievement in individualcredits is taken into account. These points all lie within a very close distance of the linearregression line, clearly indicating that there is no individual standard that is indicative ofsuccess in first-year.

An anomalous outcome from the analysis of individual credits is that data seems tosuggest that not sitting or not achieving the trigonometric standard is a good indicator ofsuccess in first-year mathematics! It would be absurd to suggest that this was curriculumrelated, but rather it must be an effect of the way in which the trigonometry standard isadministered and processed. The trigonometry standard is a standard that is internallyassessed. In a number of schools, students are in fact, able to resit this standard to get amore desirable outcome. Furthermore, under present policy, schools are not required toreport ‘not-achieved’ internal standards, with the effect that the pass-rate for thisstandard is abnormally high (here reflected by the fact that not one single student gaineda not-achieved!) especially when compared to externally assessed standards. Recently,these discrepancies have been recognized at a national level and in this year (2008)NZQA have announced a nationwide study of all subjects to compare the differencesbetween internally and externally assessed standards. One aim of this study is to identify‘problem subjects’, which fall outside national guidelines for differences in assessmentmethod [16]. However, further research may be needed to identify and clarify therelevant issues and determine if this is a subject-specific issue rather than one ofimplementation.

3.4. Total credits achieved at merit or excellence level and first-year mathematics result

As well as the number of standards was achieved, we were also interested in whether thequality of those achievements was indicative of future performance. Figure 3 shows themarks of students with different numbers of credits gained at merit or above and theirsubsequent first-year mark. The line of best fit is also shown and again there is a very

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Figure 2. The (lack of) effect of achieving or not-achieving individual credits on first-year mark.

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strong correlation between the number of credits achieved at a high standard and the first-

year result (F¼ 295, p5 0.001). Based on these results there is evidence of a clear

relationship between the quality of a student’s achievement in Level 3 NCEA Mathematics

with Calculus and success in first-year university mathematics papers. Once again it is

worth commenting that the spread of the marks remains large, and whilst the overall trend

is very distinct there are still a handful of students with over 20 credits at merit level or

above who have failed in first-year mathematics. Conversely, of the 499 students who

achieved one or more credits at merit or above, only 58 (12%) failed first-year

mathematics. This shows that achieving any credit at a high standard is an excellent

indicator of tertiary success in mathematics.

3.5. Mathematics with Calculus or Statistics?

Between 2005 and 2006, there was a significant decrease in the number of students taking

NCEA Level 3 Mathematics with Calculus. The largest drop was in the number taking

Conic Sections, which fell by 19% from 2005 to 2006, and similar drops were seen in the

numbers taking Differentiation and Integration (11 and 14%, respectively). Conversely,

the number of students taking NCEA Level 3 Mathematics with Statistics is rising. Over

the period 2005–2006, there was an increase of up to 11% in the number of students taking

each of the seven achievements that makes up the NCEA Level 3 Mathematics with

Statistics standard. Whilst the majority of students entering the UC first-year mathematics

courses have taken calculus, there are a significant minority who have taken statistics.

Are students with NCEA statistics at a disadvantage over their calculus peers or will the

general mathematical knowledge and techniques gained in NCEA statistics provide an

adequate preparation for tertiary study in mathematics?Figure 4 shows the results of students who took only NCEA Level 3 Mathematics with

Calculus or only NCEA Level 3 Mathematics with Statistics. At first glance, it

appears that the statistics credits are not a good preparation for first-year study.

However, when one looks more closely at the students who achieved all 24 statistics credits

and compares their results to students who achieved all 24 calculus credits, there is very

little difference (Table 3).

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Figure 3. Total credits at merit or above with first-year mark.

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It appears that provided a student achieves all 24 credits, there is little differencewhether these are in calculus or statistics. However, the caveat to this information is thatstudents who achieve less than the full 24 credits in statistics appear to do significantlyworse than students with the same number of calculus credits as can be seen in Figure 4.This result is based on a very small sample size and may in fact be distorted due to thedepartmental advice given to students for entry. Further years’ worth of results should beprocessed in order to explore this relationship more rigorously.

3.6. Options for weaker students

Students who fail the criteria for taking MATH108S1 are recommended to take the paperMATH108W. This option covers the same material but progresses at half the speed of108S1 (i.e. two lectures a week in MATH108W for a whole academic year, as opposed tofour lectures a week for half the year). Conventional wisdom amongst academics is thatthe whole year option allows less confident students more time to consolidate the materialand develop greater understanding than they could do on the faster course. Figure 5 showsthe number of credits achieved against the mark gained in MATH108W with theappropriate trendline. The previous trendline for the cohort taking the faster paperMATH108S1 is also shown. It is immediately obvious that there is very little difference inthe two sets of results. In particular, a student with any given number of credits is likely toachieve the same mark in the slower or the faster paper. This challenges the popular notionthat simply covering the same material at a slower pace will improve a student’scomprehension and learning.

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(a) (b)

Figure 4. First-year marks against number of NCEA credits for students taking only calculus (a) andstudents taking only statistics (b).

Table 3. First-year results of students with either 24 Statistics creditsor 24 Calculus credits.

First-yeargrade

24 CreditsStatistics only (%)

24 CreditsCalculus only (%)

F 16 12C 14 12B 28 31A 42 44

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4. Discussion and future recommendations

NCEA mathematics results are a strong indicator of success in tertiary mathematics study.Like studies in other educational systems [17], the evidence here shows that performance inthe first year of tertiary mathematics has a strong correlation with mathematicsperformance in the final year of secondary education.2

Many factors must be taken into account when discussing achievement at one’s firstyear of tertiary level study, as the transition is not just an academic one, but may involve awhole host of other factors such as adaptation to university life, new social groups andsituations, geographical relocations, new responsibilities and pressures, and so on[9,17,18]. Thus, the occasional student who has performed brilliantly at school will,contrary to expectations, function poorly at university. But, more importantly, when itcomes to formulating eligibility requirements, there are also a significant number ofstudents with poor or average NCEA results who shine at university. Hence, we aresatisfied with the current policy of making ‘recommendations’ to incoming students, ratherthan restricting enrollment.

The number of students taking NCEA Mathematics with Calculus in 2005 and 2006appeared to be a drop on previous figures, however current projections suggest that this isonly temporary. This drop corresponds to trends in other countries, for example the UK,and contributes to the smaller pool of students available for mathematically-based tertiarystudy programmes. Further consequences of this decline in numbers are discussed in [3].However, the achievement rates in NCEA mathematics are similar to those in othersubject areas, which are contrary to UK A-level mathematics, and which have beenassessed to be more difficult than many other subjects [1]. This difficulty has been observedto be a significant factor in the decline in numbers in the UK [3]. A further criticism ofA-level maths is that the high number of students achieving an A grade (the modal resultat almost 30%) does not allow highly able students to be challenged [19]. The proportionof students achieving an excellent grade in NCEA mathematics is commensurate withother subject areas at around 12%, so it appears that NCEA does not suffer from thisproblem.

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Figure 5. Results of students taking the slower paper MATH108W. The trendline from the 108S1data is shown as a dotted line for comparison.

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The modularization of assessment that new qualifications entail, and with it animplied modularization of teaching and delivery has not had the negative impact inNew Zealand, is surprising to many of its critics. This study of the initial effects findsno evidence of a fall in standards due to the changeover to the new modular system.Indeed, the tertiary courses studied did not change significantly to compensate for thenew secondary qualification, and the results of students taking the new NCEAqualification were comparable with those entering with other qualifications. This iscontrary to other education systems, which have seen a fall in competency levels inrecent years when a switch has been made to a modular system [4]. This conclusionraises some fascinating issues in pedagogy and educational theory. Have changes inassessment in fact affected the ways in which New Zealand teachers deliver theirmaterial, or has it only required some minor changes and adjustments in theircurriculum? Is this a phenomenon just applicable to mathematics? After all, mathematicsis a cumulative discipline that may be entirely suitable for modularization, unlike someother disciplines. Furthermore, is this initial study too early for full effects to berecognizable? And for those favourably disposed towards modularization, is the ‘statusquo’ result disappointing to the supporters and instigators of the new qualifications?Would they have preferred to see rather a marked increase in the abilities andcapabilities of the prospective tertiary level student, rather than a maintaining ofstandards? Further studies, particularly that compare mathematics to other subjectareas, must be done to explore this phenomenon further.

The above data suggests that the current recommended criteria for entry intoMATH108S1 could be strengthened. A proposed updated criteria for entry intoMATH108S1 would be 18 or more credits in Level 3 Calculus or at least one standardat merit or excellence level regardless of the number of credits achieved. For the 2005/06 students, only 16% of those who met this criteria failed, but 67% of those who didnot meet this criteria failed. Similarly, 36% of those who passed the criteria received anA grade whilst only 2% of those that failed the criteria received an A grade.

Students who do not meet the criteria are strongly encouraged to considerenrollment in MATH108W, which covers the material over the course of a wholeacademic year, rather than just a semester, although the wisdom of this advice shouldbe analysed further. Students should be reminded that ultimately academic success istheir own responsibility, and the rate at which the course proceeds may not compensatefor lack of application and dedication. This has clearly been shown by the comparativepass rates between the single semester and whole year courses. Enrollment intoMATH108S1 is accepted provided the students are aware that they must be preparedto put in extra effort to succeed. The criteria are useful to give to students and teachersas they raise the awareness of the strong predictors of success in the first-yearmathematics papers required for engineering, mathematics and physics. Hopefully, thedissemination of studies such as these will make students more conscious of theiracademic prospects and encourage diligence and self-motivation in order to succeed inthe path of study they self-select into.

Thus, despite some of the negative exposures surrounding the implementationof NCEA into secondary schools, this initial study concludes that in the field ofmathematics, NCEA is a solid preparation for tertiary study. In fact, a full quota ofNCEA Level 3 Mathematics with Calculus credits equips its achievers with a significantadvantage – students with all 24 credits have an overall pass rate higher than theaverage and get more A’s than the average. Thus, secondary level providers should be

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encouraged and supported as much as possible to continue this breadth in theirteaching.

Notes

1. Although the university does give out letter grades which are further modulated with‘þ’ and ‘�’, for simplicity’s sake these have been grouped together in the categories indicatedabove.

2. Although there are some discrepancies between internal and external methods of assessmentthat are being studied at a national level.

References

[1] A. Kitchen, The changing profile of entrants to mathematics at A level and to mathematical

subjects in higher education, Br. Educ. Res. J. 25(1) (1999), pp. 57–74.[2] Engineering Council (UK), Measuring the mathematics problem, Report, 2000.[3] N. Gordon, Crisis-what crisis? MSOR Connect. 5(3) (2005), pp. 1–10.[4] D. Lawson, Changes in student entry competencies 1991–2001, Teach. Math. Appl. 22(4) (2003),

pp. 171–175.[5] C. Stripp, The crisis in further mathematics and how MEI and Gatsby are working to address it,

Teach. Math. Appl. 20(2) (2001), pp. 51–55.

[6] Campaign for Real Educations, Observations on London (EDEXCEL) ‘A’ level mathematics

from 1960 to 2004, Report, 2004.[7] T. Graham, AS mathematics: the results of a survey of schools and colleges, Teach. Math. Appl.

21(1) (2002), pp. 11–28.

[8] K. Hirst and S. Meacock, Modular A-levels and undergraduates, Teach. Math. Appl. 18(3)

(1999), pp. 122–127.[9] K.L. Todd, An historical study of the correlation between GCE Advanced level grades and the

subsequent academic performance of well qualified students in a university engineering department,

Math. Today 37, (2001), pp. 152–156.[10] CUMP Panel, Report of the CUMP panel on calculus articulation: problems in transition from

high school calculus to college calculus, Am. Math. Mon. 94(8) (1987), pp. 776–785.[11] J. Ferrini-Mundy and M. Gaudard, secondary school calculus; preparation or pitfall in the study

of college calculus? J. Res. Math. Educ. 23(1) (1992), pp. 56–71.[12] G. Lewis, V. Lazarovici, and J. Smith, Meeting the demands of calculus and college life: the

mathematical experiences of graduates of some reform-based high school programs, Paper

prepared for the Navigating Mathematical Transitions Symposium, Annual Meeting of the

American Educational Research Association, 2001.[13] W. Robinson, The effects of two semesters of secondary school calculus on students’

first and second quarter grades at the university of Utah, J. Res. Math. Educ. 1(1) (1970),

pp. 57–60.[14] S.I. Barry and J. Chapman, Predicting university performance, ANZIAM J. 49 (EMAC2007)

(2007), pp. C36–C50.[15] New Zealand Qualifications Authority, University Entrance – what’s happening? QA News 38,

Wellington, 2001.[16] New Zealand Qualifications Authority (NZQA). Available at. www.nzqa.govt.nz[17] M. Evans and A. Farley, Institutional characteristics and the relationship between students’ first-

year university and final-year secondary school academic performance, Working paper 18/98,

Monash University, Australia, 1998.[18] H.J. Forgasz, The typical Australian mathematics student: challenging myths and stereotypes,

High. Educ. 36(1) (1998), pp. 87–108.

International Journal of Mathematical Education in Science and Technology 1049

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[19] S-M.R. Ting and T.L. Robinson, First-year academic success: a prediction combining cognitive

and psychosocial variables for Caucasian and African American students, J. Coll. Student Dev.

39(6) (1998), pp. 599–610.

Appendix

MATH100 recommended entry standards

These entrance criteria for MATH100 are recommendations only. Any student who meets standarduniversity entrance requirements may enrol in any of the level 100 courses (excluding MATH109,which has MATH108 as a prerequisite).

The first- and second-semester occurrences of MATH108 are intended primarily for studentsplanning to take MATH109. The whole-year occurrence covers the same material at a slower paceand hence has lower-entry recommendations. We recommend that students intending to take anyoccurrence of MATH108 or MATH101 do the pre-entry self assessment quiz.

Each year the department offers direct entry to level 200 mathematics/statistics to a fewoutstanding students.

MATH108 (single semester)

Suitable preparation may include:

. 15 points in maths with calculus NCEA Level 3, including both of the achievementstandards Calculus 3.1 and Calculus 3.2 (Differentiation and Integration, respectively). Theequivalent Unit Standards are also acceptable.

. 18 points in NCEA maths with statistics

. Scholarship in NCEA maths either with calculus or statistics

. 50%þBursary maths with calculus

. 60%þBursary maths with statistics

. Pass in MATH101

. For the second-semester occurrence of MATH108, concurrent enrollment in MATH101

. A or B in FOUN003 or A in FOUN015

. Pass in A Level or AS Level Mathematics; or C in AICE Mathematics: Statistics (halfcredit)

MATH108 (whole year)

Suitable preparation may include:

. 12 points in maths with calculus NCEA Level 3, including at least one of the achievementstandards Calculus 3.1 or Calculus 3.2 (Differentiation and Integration, respectively). Theequivalent Unit Standards are also acceptable.

. 14 points in NCEA maths with statistics

. 40%þBursary maths with calculus

. 50%þBursary maths with statistics

. Pass in MATH101

. Pass in FOUN003, FOUN015, PREP009 or PREP029

. Pass in AICE Mathematics: Statistics (half credit)

There are no entrance criteria for any other 100 level MATH or STAT courses.

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