The problems with Project Maths and the dire
consequences for Irish mathematics education and the knowledge society.
4 June, 2012
Background
Mathematics is an underpinning technology for modern society. It is, in fact, the underpinning technology. This is obvious to experts, but less so to the man in the street, who is commonly almost proud of the fact that he ‘was never any good at maths’. So the argument will bear indefinite repeating. Put simply, it is hard to think of any technology in which mathematics is not in some way inextricably involved. Weather prediction, climate change, flood prevention; electricity, water supply, sewage treatment; roads, buildings; airline scheduling, supermarket restocking; they all involve mathematics at some fundamental level, but not one which is necessarily visible to the consumer (or indeed the provider). Without mathematics, there would be no computers, no aeroplanes, no space programme. There would be no internet, no Google.
Almost all science is built around the concept of a mathematical model; a proper mathematical training at all educational levels is a sine qua non for a technological or “smart” economy.
What is Project Maths?
From the web site of the National Council for Curriculum and Assessment (NCCA, www.ncca.ie), we read that:
“The NCCA conducted a review of post-primary mathematics education. This review came at a time when there was concern about the uptake of Higher Level mathematics, particularly in the Leaving Certificate, and about the standards of mathematical achievement in state examinations and in international tests such as the Programme for International Student Assessment (PISA). Since mathematics underpins many other disciplines, including science and technology, a decline in mathematical knowledge and skills among school leavers can affect the potential of our society for future economic growth and development. The review also took place against the background of a revised curriculum in primary schools and revised Junior Certificate mathematics syllabuses. Mathematics forms a significant part of a student’s education, with all students taking this subject to Leaving Certificate. It is critical, therefore, that issues related to post-primary mathematics education can be widely discussed,
and that all voices can have the opportunity to contribute to shaping its future
development.”
In addition, on www.projectmaths.ie it is reported that:
“Project Maths aims to provide for an enhanced student learning experience and
greater levels of achievement for all. Much greater emphasis will be placed on student understanding of mathematical concepts, with increased use of contexts and applications that will enable students to relate mathematics to everyday experience.”
What is wrong with Project Maths?
So far so good. Project Maths appears to be a laudable attempt to update and modernise the Leaving Certificate syllabus: the NCCA clearly acknowledges the importance of mathematics to science and technology. The trouble is that the practice is different: Project Maths contains serious flaws in its syllabus and methodology. Several of these flaws were already pointed out in a preliminary report produced by mathematicians at the University of Cork, but their criticisms seem to have been largely ignored. Indeed, at the time of writing of this report, many of the details of Project Maths were still unclear, and as a result it was difficult to assess the precise content of the proposed curriculum. In the mean time, the project is being implemented without proper assessment. In fact, there is a significant community of senior mathematicians who think that the curriculum as well as the underlying philosophy is unsatisfactory. The main problems are:
The Finnish view
Typical propaganda (on www.educationmatters.ie) runs as follows:
“Good practice in mathematics teaching in other countries was also examined. The focus was on the countries that perform better in international studies (PISA, TIMSS). These countries are Finland, South Korea, Holland and Japan. NCCA has brought together best practice from around the world and tailored it for the modern Irish school. We have retained what is seen as valuable from the past and blended it with what is viewed as essential for the future.”
This sounds reasonable but, in a personal communication, Professor Olli Martio,
(Secretary General Finnish Academy of Science and Letters) informs us that:
“It seems that a new math curriculum for schools has created a major debate in
Ireland. The same phenomenon has happened in many countries. Usually the worst thing is that there is a strong group who is able to push their ideas forward
although the ideas do not stand reasonable criticism. Most likely this has
happened in Ireland although I do not know the background. In Finland a type of Project Maths is used in teaching mathematics but luckily its scope is limited compared to some other countries.”
Unfortunately the comments of Professor Martio seem all too appropriate in the
Irish context. The result, Project Maths, espouses a new philosophy of teaching mathematics at Leaving Certificate. In itself, this would result in significant upheaval as most secondary level teachers do not have the training to teach mathematics in this way. Moreover, as mentioned above, the syllabus was also radically transformed. In short, much of calculus (integral and differential) and all linear algebra (vectors and matrices) has been completely removed from the syllabus in order to introduce Euclidean geometry and to beef up probability and statistics. Perversely, this is against the wishes of most secondary school teachers who have years of teaching the omitted topics behind them but will need training to cope with the new topics. Moreover, despite claiming to have examined the Finnish model, Project Maths is ignoring the experience in Finland.
Professor Martio, when speaking of the Finnish experience, also notes that:
“The curriculum contained concepts the students had no feeling and no background to understand. As mathematicians know statistics is no easy subject and it is my definite opinion that, except discrete probability and some basic continuous distributions, it has no place at the school curriculum. The applied statistics has very much been dropped out in our present curriculum. The Finnish curriculum still contains some basic faults in teaching probability. Integration, for example, comes too late and without it continuous distributions cannot be understood.”
Indeed upon closer examination of the Finnish advanced syllabus it is clear that calculus and vectors are given very strong emphasis. In addition, there are a number of specialised courses on top of this syllabus.
In view of this as well as the comments made by Professor Martio on the inclusion of applied statistics into the mathematics curriculum, it is misleading to suggest that ‘the best of this syllabus was taken and adapted to suit the needs of the Irish education systems’.
As the NCCA also refers to the South Korean syllabus, it may be worth pointing out that their curriculum is in fact considerably more demanding than Project Maths. South Korea is known to have one of the best mathematical education systems in the world. A detailed comparison shows that there is in particular far more calculus in their secondary school syllabus than in Project Maths, and it is treated in remarkable depth.
Other countries like Canada, Singapore and the Netherlands have a much more differentiated curriculum. Parts of these do contains elements similar to Project Maths, but all have also more advanced options which concentrate on linear algebra and calculus, and these are compulsory for students who wish to study technical subjects at third level.
In a private communication one of the experts on mathematical education in Singapore, Professor Peng Yee Lee, remarks that ‘Whether you like it or not, calculus is the gateway to advanced mathematics‘. He also makes the following observation on the approach to teaching mathematics in Singapore: ‘Next to modern insights we also hold on to a rigorous approach of practicing. Problem solving is ok, but you have to be fluent in basic facts.’
What emerges, therefore, from an examination of the world’s top mathematical education systems, is that they are distinguished by a thorough treatment of the very topics, namely calculus ( differential and integral), vectors and linear algebra which in Project Maths have been removed to a large extent.
Is this sensible for Ireland?
Here in Ireland, we are preparing to make the same mistakes already made in Finland. Indeed, at a time when many of these mistakes have since been corrected in Finland, we are heading in the opposite direction and in a more radical way. (Note that, in the above quote, Professor Martio states that ‘integration’ comes too late: in Project Maths there is almost no integration at all!) It gets worse. The big users of mathematics in the universities are in engineering, the sciences and business. Calculus, linear algebra, vectors, are their bread and butter. Unfortunately Project Maths has dispensed with much of the mathematics that is regarded as being of key technological and scientific worth (calculus, linear algebra, vectors). Especially noteworthy is the serious reduction in calculus. Discovered by Newton and Leibniz, it is one of the pillars of mathematics, and an absolute requirement for any engineer, scientist or economics student at third level. Yes, these subjects are not easy, but almost every country exposes their students to the intellectual training and rigour of calculus at second level. Instead Irish Leaving Certificate students will study the geometry of Euclid and some applied statistics. To be sure, most of the geometry examined is of a very elementary nature, and should certainly be taught, but this is not sufficiently challenging for higher-level Leaving Certificate students to be internationally competitive. We note that the Finnish syllabus, in its various incarnations, has never excluded linear algebra, vectors or calculus.
Despite claiming to wish to “increase applications”, the syllabus pointedly omits much of the mathematics at the heart of applications in science and technology. There is no other secondary syllabus in an advanced country which de-emphasizes calculus to the extent that Project Maths does. The introduction of Project Maths, while being detrimental to the scientific education of our own third-level students, will also place them at a disadvantage when travelling abroad. A quick glance at a representative list of questions for entry at Cambridge (see www.thestudentroom.co.uk), for example, shows that Project
Maths will leave them wholly unprepared. There is little doubt that the watering down of the Leaving Certificate syllabus will have a knock-on effect at third level. A reduction in standards of the same magnitude is not occurring in Britain or among any of our European partners. The reputation of our universities will be severely damaged.
Is this really what we want for Ireland? On the face of it, this approach lacks logic
and is certainly not consistent with a knowledge-driven or “smart” economy. The ethos may derive from the principle of providing rigorous training for the mind without due consideration to the content. But if we follow such an approach to its logical conclusion,it would be simpler to stop teaching mathematics at all at second level and to train students in Logic and Philosophy. The alternative didactic view is that the particular content of the Leaving Certificate syllabus is important in its own right and the earlier that students are exposed to key mathematical concepts (such as calculus, linear algebra) the better.
The remedy
It is the thesis of this group that Project Maths is flawed in the extreme, both as regards the syllabus and as regards the underlying philosophy, which seems to assume that students will be able to analyse real-world problems without the necessary mathematical tools. This will have serious ramifications for students at second and third level as well as the economic competitiveness of the country. It is therefore necessary to devise a new (higher-level) syllabus with a more balanced and intellectually demanding content including calculus and linear algebra (vectors and matrices, e.g. the equation of a plane in 3 dimensions in vector form). In the interim the alternative syllabus could even be offered to schools as an alternative to Project Maths. In the longer term it would be desirable if the problem of the underqualification of teachers were addressed.
Summary
Mathematics is an underpinning technology at the heart of any smart economy. Project Maths comprises both a new approach to teaching and a new mathematics syllabus at secondary level. It is seriously deficient. Notably,
1. Project Mathematics is flawed in the extreme in its syllabus and methodology.
2. There has been little consultation with the wider mathematical community during the development of Project Maths.
3. Secondary teachers are not equipped to teach mathematics using the methodology envisioned by Project Maths.
4. The new syllabus, omitting much of the calculus, including sequences and series and all integration, and also all linear algebra (vectors and matrices) is completely insufficient to support third-level engineering and science education, and will put our third level graduates at a clear disadvantage internationally.
5. Project Maths will severely damage the reputation of our universities
6. The lack of preparation of Leaving Certificate students having taken Project Maths will result in a lowering of standards at third level, which will hurt economic competitiveness.
If we are to avoid serious damage being done to our mathematics education system it is imperative that new courses are designed as soon as possible to equip students with an appropriate mathematical foundation for third-level engineering, science and applied mathematics
We stress that the Project Maths syllabus is unique to Ireland, and below standard among the technologically advanced countries.
This Report was compiled by Dr Cora Stack ITT and other maths professionals.
Paper 1 (pilot schools)
Question 1 Algebra ; same as old course but worth 25 marks v 20 marks old course.
Question 2 Number/Complex numbers. A question based on the definitions of R,Q,Z, and complex numbers again a soft 25 marks.
Question 3 Complex numbers same as part b or part c off the old course.
Question 4 Induction /geometric series. Very very easy so far you could get 100 marks for very little knowledge! How do Q1,2,3 and 4 (100 marks) Compare to regular paper 1 Q1 and Q2 Much easier!
Question 5functions/calculus .Nice question very similiar to Q8c old course but worth 5 marks more!
Question 6: Calculus nice question if a bit tedious.
Question 7; Calculus;Interesting question very similiar to pre 1994 calculus questions .
Question 8;Problem : Nice question but it went on far too long and by the end most students had lost the will to live! Some of this question had been asked in an NCCA sample.
Question 9:Problem Again all over the place ,logs and indices being the main features.Very similiar to the earthquake question in the samples.
Comment all of the following have been removed from paper 1. In Calculus parametric/Implicit functions,differentiation of Trig functions from 1st principles , Integration of trig functions/integration by substitution, all sequences and series except Arithmetic and Geometric. About 50% of the for paper 1 content has been removed and not been replaced. Without a good knowledge of Calculus students will be at a major disadvantage competing with students educated in the north.
Paper 2 (all schools)
Question 1: Coordinate geometry of the line.Easy question just about Junior cert standard.
Question2:The circle .Nice question very similiar to old course.
Question3:The Circle:Generally regarded by teachers and students alike to be difficult (It actually can be done in about 5 lines) One pilot school teacher suggested that 10 marks should be given for the mid point of a line !
Question4; probability :Nice question most students would have been happy.
Question 5: The word tolerance may not have been understood by students many may not have seen it before!But otherwise not a bad question.
Question 6A : Geometry .Much nicer than expected but no theorem ! There is a lovely way to do the congruent triangles bit using transformations (whoops sorry they are no longer on the course!).
Question6B:Again a nice question much easier than last year but most students will not have attempted this question.
Question 7; Apart from the fact that it was tactless to ask a question on mortgage arrears when there are at least 150,000 people many of them have sons and daughters doing the LC . The two diagrams extracted from the Central Bank report are CROSS PLOTSnot SCATTER PLOTS ! When the NCCA were asked about this they replied that students would assume they were scatter plots !But scatter plots have a correlation coefficient cross plots do not ,and students could have stated that there was a strong correlation in part(c). apart from that the question was long winded and I am sure it will be marked easily. There was great concern expressed about direction of Causality the following definition was given by NCCA I presume ‘direction of causality’ in this instance means which one (interest rate; arrears rate) can be considered the ‘cause’ and which one the ‘effect’.
Question 8 : Trigonometry. Part a was very nice part a combination of Pythagoras and the sine and Cosine rules (will probably get at least 20 marks(b) was well constructed , but parts c and d were only for the brave and suited Applied maths students .
Comment : The main talking points were what was not asked or what has been removed.
Not asked(i)Independent events(ii)Set Notation in probability (iii)Empirical rule (iv)Mutually exclusive events (v) Trig equations,periodic functions(vi)Permutations/Combinations
Gone from the course ; All Coordinate geometry proofs,Linear transformations,Vectors,Integration by parts,Difference equations,Max and Min problems involving modelling ,Groups,Affine transformations.
At a recent meeting of the Dublin Branch of the Irish Maths Teachers Association the project maths papers had little or no support except from the three reps from the Pilot schools who thought the papers were fine ,fair and challenging in parts.They said there students were happy! The 30-40 students who posted comments to this site do not reflect this.
Project maths will be changed it is only a matter of time .Note no support for PM was given to Project Maths in the media from NCCA or the Minister of Ed following Brendan Guildea’s criticism on Drive time.
General comment too much English not enough maths.
PAPER 1; (PILOT SCHOOLS)
Question1;Money/tax/USC more or less same question asked on unior H.
Question2;Number very little work for 25marks.
Question3Complex numbers;Identical to old course.Question4 Algebr4a 25 marks for solving two equations.Same as old course.
Question5;Algebra/functions very similiar to old course see 1994/1995/LC Ord Q3P1.Question6 similiar to Q6orQ8 of the old course.
Question 7 (problem) This was a daft question to ask Ord level students 9 parts spread over 4 pages .Looking forward to seeing the marking scheme to see how much of the marks are front loaded!Lets see if they give 20 marks to part (a) and 5 to part (g).
Question8: Arithmetic series:Most Ord level students would not have managed this question parts f and g will end up being worth 5 marks each parts a and b will get the bulk of the marks.
Question 9; Differential calculus : Very similiar to old course Q7c students should have been OK.The examiners seem to have run out of ideas with part b students will probably be able to do b(i),b(ii) but will have difficulty with b(iii),b(iv)
Comment : Notice all of the following topics are gone ;(i) Solving cubic equations (ii)Product Rule,(iii)Quotient Rule(iv)Chain Rule(v)Geometric Series.
The material content of this paper in comparison to the old course is about 50% (ie students need to know 50% less maths
Question 9:
Paper 2 All schools.
Question 1 (probability) not a single number in this question.It was a very tedious question based on the probability whether three students would be boys or girls.
Question 2: Another probability question ,this one involved a pack of cards .The examiners assumed that students knew the composition of a pack of cards! Lets see how they mark this!
Question 3: Coordinate geometry of the line pretty basic question based on slope. But again students were asked to describe !
Question 4:Coordinate geometry of the circle ,nice question most students should have coped well. Syllabus says only tangents to a circle centre (0,0) can be asked.
Question 5A Geometry Probably too difficult for weak students ,the constructions had been well predicted.
Question 5B Very easy the cyclic quad was predicted ,just fill in the angles and the answer walks out!
Question 6:Statistics;The data was huge students would have struggled with the “suitable chart”.The parts a(ii),(iii),(iv),(v)b(i),(ii),(iii)were a very low standard and contained very little maths! For b(ii)what would students know about buying patterns they do not own or drive a car.Part c was fine. c(ii) Could have several collect answers.In part d(ii)They asked for central tendency totally predicted by most teachers .One big problem with this question that it spread itself over 4 pages as a result students constantly flicking back and forward.
Question7;Trigonometry. Most students will struggle to get any of this correct;
Question8; Area and volume not a bad question by far the easiest on the exam all students should have attempted this first
Hi John,
It was great to hear your comments on Project Maths on Countdown to 606 earlier. I completely agree with your analysis of the papers and your opinion of the Project Maths idea in general. In fact I would go further to say that this messing with the Maths syllabus is taking a very dangerous route.
A bit about me, I’m have just finished a degree in Actuarial Science at UCD with first class honours. I sat the Leaving Cert in 2008 with 600 points including A1s in Maths, Physics and Applied Maths. I’m not here to brag but having gone through a very mathematical route with my life, I feel what they’re doing to the system is very deterimental.
My sister is currently doing the Leaving but she’s doing Ordinary Level Maths. I sat down with her last night to do some papers and I was just horrified at what they had done to Paper 2. It’s like an English exam as you said yourself.
I thought the old course was very fair and the (a), (b), (c) format of the questions was very nice and approachable.
When I was studying for the leaving I did papers as far back as the early eighties or anything I could get my hands on.
Now obviously I’m somebody that enjoys Maths and I don’t represent everyone taking the paper (most hate it) but I believe Maths has to be made enjoyable for people to properly engage with it. This enjoyable factor and explaining maths in the real world needs to be thought in the classroom and not on the exam paper.
If Gauss, Laplace or Riemann were to see this Project Maths rubbish they’d turn in their graves.
And one final point… It’s certainly the case that hand calculations are being phased out by computers, but as you know yourself the computers didn’t invent themselves and won’t invent themselves in the future. We need bright minds and not just monkeys punching numbers into computers and staring blankly when something goes wrong not knowing what the computers are doing behind the scenes.
Thanks for reading this,
Kind regards,
Alan Benson
Students a full analysis of Paper 2 (H&Ord) will appear here on Monday after the exam .
Students and Parents are welcome to post their comments .
Please note comments with bad language or who mention peoople will not be piblished!
When posting a comment your email address is required.Also state whether you took higher or ord and if you were in a pilot school or regular school
The Comments of Project maths.com will be posted here .
Please feel free to add your comments.
Paper 1 project Maths H .
Q1 old course most students will be OK.
Q2Daft question taken straight of BAC .
Q3 very similiar to old course Q4 proof by Induction they did say something like this could come up, 4b very easy.
Q5Similiar to old course not too difficult. Q6 was akward but not difficult .The differential calculus course is very short.
Q7 Very long a,b,c, Ok the rest gave trouble by the way this was straight off the pre 1994 course.
Q8 Not a bad question but it went on forever again straight of the pre 1994 course.
Q9 Mindnumbing this ios not maths a desperate attempt to get all of the Syllabus into one question .
General consensus (from Pilot school pupils) Not too bad thank god no finincial maths .
Ordinary Level Project Maths Paper 1 .
Q1 Too complicated too much English.
Q2 (a,b)Could have been on foundation Junior cert . (c) Ok.
Q3 same as old course but part d was tricky .Q4 basically Higher Junior maths.
Q5 Very similiar to LC ord level 1994-1996 part c too complicated.
Q6 Nice question if you were ex H.Regular ord level no chance!
Question 7 ;How daft was this a question so complicated and so full of English a complete turn off for most Ord students .It was spread over 4 pages you had to keep turning back pages to see what was being asked..This question alone typifies the daftness of project maths! Having looked at this question more carefully it had no place on any leaving cert paper.
Q8 : Again Ord level students could not identify with this where are they going? No real maths here students found this question impossible!Again this question was spread over 3 pages.
Q9: Not a bad question very similiar to 7c on the old paper. Again spread over 3 pages?
General comment from Pilot school students Disaster!
What to do? parents of LC students should contact any of the following and let them know their feelings
Please use the link for UCC Report on Project Maths.
http://www.ucc.ie/en/euclid/projectmaths/ProjectMathsInterimReport_Nov2011.pdf
Sharp W531 Calculator and Statistics
The sharp w531 calculator can be used to find the mean and standard deviation as follows.
X | 1 | 2 | 3 | 4 |
f | 5 | 7 | 6 | 2 |
Method
(i) Press mode then pres 0 the screen reads STAT 0 (SD).
(ii) Input data as follows 1 (x,y) 5 Data (change) the screen reads DATA SET = 1 .
(iii)Then 2(x,y) 7 Data etc the screen will read DATA SET =2
When all the data has been entered the screen will show DATA SET = 4.
(iv)To find the mean press RCL 4 ( ) = 2.25
(v)To find the standard deviation press RCL 6 ( ) = 0.94207
To find the Correlation coefficient from a scatter plot using Sharp w 531 calculator.
X Score (independent variable) | Y score(Dependent Variable ) |
5 | 20 |
8 | 18 |
6 | 22 |
7 | 28 |
10 | 27 |
Method :
(i)Press mode then press 1 SCREEN READS STAT 1 (LINE).
(ii)Input data as follows 5(x,y) 20 data screen reads data set = 1 . (as above)
(iii)Repeat for the rest of the data .after the last imput screen should read DATA SET = 5.
(iv)To find the correlation coefficient press RCL (DIVISION SYMBOL) =0.417
(v)To find the equation of the regression line (line of best fit)
(vi)To find the slope of the regression line pres RCL ) (RIGHT BRACKET ) = .946
(vii)To find where the regression line cuts the y axis pres RCL ( (LEFT BRACKET) = 16.189
(viii)Therefore the equation of the line of best fit is y = 16.189+.946x
(use the line of best fit is often written as
Casio fx-83GTPlus calculator
Given the frequency table find the mean and standard deviation
X | 1 | 2 | 3 | 4 |
f | 5 | 7 | 6 | 2 |
(i) press Mode 2,1 you will see a table as below imput the data as shown .Make sure the frequency is on by pressing shift mode (set up) down curser 1.
(ii)To get the mean press AC
(iii)Then shift 1.(iv) then 4 (on 83ES use 5) then 2 for the mean = 2.25.
(v)Then shift 1.and 3 for the standard deviation = 0.94207.
x | y | |
1 | 1 = | 5= |
2 | 2 = | 7= |
3 | 3 = | 6= |
4 | 4 = | 2= |
Data from a scatter Plot
X Score (independent variable) | Y score(Dependent Variable ) |
5 | 20 |
8 | 18 |
6 | 22 |
7 | 28 |
10 | 27 |
x | y | freq | |
1 | 5 = | 20= | |
2 | 8 = | 18= | |
3 | 6 = | 22 = | |
4 | 7 = | 28 = | |
5 | 10 = | 27 = |
(i) Press Mode 2 2 you will see a table as above imput the data as shown .
(ii) When all the data has been entered Press AC
(iii) Press shift 1 then press (5 (var) then press 3 (r= 0.4174 ) this will give the correlation coefficient
(iv)To get the slope of the regression line press shift 1 then 5 then 1 (A)=16.19 this is the intercept of the regression line and the Y axis .Press shift 1,5 then 2 (B= 0.9459) for the slope of the regression line. The regression line is in the form
For Casio Fx-83ES
(iii) Press shift 1 then press (7 (reg)then press 3 (r= 0.4174 ) this will give the correlation coefficient (iv)To get the slope of the regression line press shift 1 then 7(reg) then 1 (A)=16.19 this is the intercept of the regression line and the Y axis .Press shift 1,7(reg) then 2 (B= 0.9459) for the slope of the regression line. The regression line is in the form y-A-bx