Irish Math. Soc. Bulletin 67 (2011), 27{55

Mathematics Education and Reform in Ireland: An Outsider’s Analysis of Project Maths.

The following are some extracts from Sarah Lubienski Paper.

**2.2.1. ****Textbooks and Project Maths.**

First, the issue of textbooks in

Ireland seems politically sensitive, with Project Maths leaders seem-

ingly afraid to say anything positive or negative about any particu-

lar book. They appear to be circumventing textbooks as opposed to

leveraging them, as illustrated by these quotes:

I deliberately have not seen any of the textbooks.

I haven’t seen any [texts] so I don’t know what’s

out there | and the best thing to do is not look.

**2.3. ****Project Maths Vision?** The more I delved into Project Maths,

the less sure I became about what, exactly, its instructional vision

is.

**2.1.5. Credibility of the exams**.

The final question I raise about the

exams system stems from teachers’ responses to my questions about

the probable impact of Project Maths on students’ Leaving Cert

scores. I received several responses suggesting that the SEC will

just make the results come out,” or in other words, that the exam

results will show whatever the DES and NCCA want them to show.

Hence, I began to wonder what checks and balances there are in

the Irish exam system, how much trust the Irish people have in the

exam scores, and whether bridging studies would be used to compare students’ results on the old and new maths exams.

Problem based learning :

In the U.S. reform movement, the push has been toward problem

solving as the primary means of learning mathematics [12, 13]. That

is, students are given a problem (or a carefully designed sequence of

questions), and through the process of solving and discussing, they

gain understanding of intended mathematical ideas. The Project

Maths teaching and learning plans I examined were consistent with

this approach. However, after interviews with key Project Maths

players, I became less sure about the role of problem solving and

discovery learning in Project Maths.

**Sarah Theule Lubienski is a professor of mathematics education in the**

**Department of Curriculum and Instruction at the University of Illinois,**

**Urbana-Champaign****, USA****. She studies mathematics instruction, reform**

**and equity. Her research has included analyses of large-scale data on U.S****.**

Projectmaths.com has been contacted by many students and teachers regarding answering the maths exam in the spaces provided in the answer booklet.

The main ponts made were (i) Not enough space provided.(ii)Not comfortable writing on squared paper(iii)Some questions were spread over 4-5 pages as a result they had to flick back and forth to refer to the original question/diagram.

A SEC circular(S77/2011 )_ was sent to all schools stating ” Candidates must write their answers on this booklet, and marks may be forfeited if they do not do so”.

We asked the question “Can you clarify what marks will be forfeited if a students does not answer the questions in the booklet but chooses to use the supplied extra lined paper to answer all questions of the project maths paper 2 ?”

The following is the reply received , while we agree with many of the points raised . It is hard to believe that the integrity of the exam is compromised by a student choosing not to answer on the booklet! We also note no mention of the sanction! **On balance we would advise students to use the answer booklets the one major advantage we feel is that students will be able to see parts of questions that they have not completed**.

Reply from SEC:

I refer to your recent telephone call and subsequent correspondence in relation to the 2012 Project Maths examination. The position is as follows. The new Mathematics/Project Maths examination papers are provided as a combined question-and-answer booklet. Candidates are instructed to write their answers into the spaces provided in the booklet. If they run out of space, or need to cancel an answer and repeat it, they are free to continue their work either in the additional space at the back of the booklet, or on supplementary four-page answer-books.

This format was introduced as a result of consultation between the State Examinations Commission, the National Council for Curriculum and Assessment and the Department of Education and Skills. It was piloted in the 24 initial schools as part of the trialling of the draft sample papers for the 2010 examination in the 24 initial schools, following which feedback was sought and received, and the format finalised. This combined question-and–answer booklet has been successfully used in the Leaving Certificate Mathematics examinations for the 24 initial schools since 2010. Consequently, the sample papers for the 2012 examinations as issued to all schools were also presented in this format. It is worth noting that the question-and–answer booklet format has been a common feature of Junior Certificate examinations at all levels across a range of subjects for several years. Accordingly, it is not unfamiliar to candidates.

This format is widely regarded as candidate-friendly, for a number of reasons:

- Candidates’ time can be saved by, for example, asking them to complete a partially filled table, rather than asking them to “copy and complete” it. As well as saving time, this avoids the potential for transcription errors by candidates.
- The completion format facilitates giving additional guidance to candidates regarding how to begin an answer, and drawing candidates attention to the key elements to be addressed in the answer. For example, where candidates are expected to give both an answer and a reason, the labelling of the response areas with the headings “Answer:”…”Reason:”.

- The spaces allocated for answering gives candidates a general indication as to the length and/or complexity of the answer expected, while allowing them to continue elsewhere if required.
- The squared-paper format of the response areas gives flexibility to candidates in readily combining text-based and graphical approaches to solving a problem
- Experience indicates that candidates are far less likely to accidentally miss a part of a question when answering in this format.
- There is a further advantage to adhering to the same format at all three levels, as it maintains a familiar examination format for candidates who change levels late in their programme.

The format also facilitates a greater flexibility in the type of questions that may be asked, and the efficiency with which particular skills can be tested. As indicated by some of the material that has appeared on sample papers and on the examination papers for the initial schools, examples include the following:

- The capacity of candidates to interpret and read graphs, showing their work on the graph, can be tested without first having to ask the candidates to construct the graph in question. If graphs are to be drawn, it facilitates providing the candidates with scaled and labelled axes when appropriate.
- Selected-response questions, such as multiple-choice and item-matching questions, can be presented in a way that is straightforward to respond to.

Candidates are required to follow the instructions given and to answer the examination in the format in which it is being presented. Deviating from the SEC’s instructions in this examination increases the risk level associated with the SEC’s work in subsequently collating, processing and evaluating their work properly.

For all of the reasons outlined above, the State Examinations Commission wishes to discourage, in the strongest possible terms, candidates from deviating from the given instructions. School authorities and teachers have an important role in ensuring that examination procedures are complied with, in order to maintain the integrity of the examination system. In the best interests of candidates, and in order to ensure fairness between all candidates, we anticipate that teachers and schools will co-operate with us in seeking to ensure that all candidates follow the instructions as given. For these reasons the SEC would be gravely concerned if it transpired that teachers were encouraging candidates to disregard instructions given on the examination paper.

If you have any further query in relation to the above please don’t hesitate to contact me.

Regards,

Corporate Affairs Division

State Examinations Commission

Cornamaddy

Athlone

Co Westmeath

From: UCD Maths Department Distribution List on behalf of Stack, Cora – Lecturer of Maths

Sent: Thu 5/24/2012 08:48

To: MATHDEP@LISTSERV.HEANET.IE

Subject: maths: From Oli Martio on project mathemaitcs etc in Finland. Too good not put put out on public record sorry…..

Dear Cora,

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 the type of project maths is used in teaching mathematics but luckily its scope is limited compared to some other countries. Project maths also means different things in different countries. In the Finnish school text books proofs and definitions play a minor role – in fact, about 10 years ago their role was even less. In the latest curriculum basic things have been emphasized more than in the previous curriculum. For instance, trigonometry was very much neglected but it is back now.

I think that problem solving without a decent background is a waste of time. If there is no background, then the problems tend to be so trivial that there is no connection to problems the students meet later. Actually a look at some school textbooks confirms this – the problems have nothing to do with real life although some people believe so. One should remember that problems have always been solved in mathematics and there should be a balance between theory and what is called “problem solving”. During the last 40 years the emphasis has been on the side of “problem solving”. There is a common belief that this is more useful but this has not been criticized enough.

The previous school curriculum in Finland contained a lot of applied statistics. This was a serious error. 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.

The Finnish school system ends at the matriculation (student) examination which is taken by all gymnasium students. Mathematics is not a compulsory subject in the examination but a majority takes it.

I have worked 14 years at the board of the matriculation examination and have been responsible for mathematics about a decade. This has given a good view to see the effects of the curricula changes.

Unfortunately the examinations are organized in Finnish and in Swedish only. Some English translations are available but it takes some time and effort to collect them.

I hope that these lines are for some help to you. I think that mathematics curriculum is a serious matter and in many countries math teaching has been destroyed by adopting a curriculum which does not make any sense.

With best regards,

Olli Martio

Professor, Secretary General

Finnish Academy of Science and Letters

Matrices, Vectors and an Opportunity for Project MathsOne of the key reasons for having students take mathematics at second levelMATLABthroughout the discipline), physics (from vectors for Newtonian physics toProject Mathsdoes not include an introduction to vectors or matrices as part of the syllabus. The

is that it is a discipline that is simultaneously practical and theoretical. Mathematics

and its areas of application have, for centuries, benefited from an ongoing

dialogue between the theoretical and the practical. Problems arising in science and

engineering require the development of new mathematical tools and techniques for

solution; these new developments in turn lead to new directions for mathematical

theory; new ideas from the theoretical domain then serve to expand the horizons

of scientific inquiry, and the cycle continues. Mathematicians know this, and so

do mathematically inclined scientists and engineers. Perhaps it’s a good idea for

Ireland’s students at second level to learn this too.

Linear algebra provides a particularly fruitful example of that productive dialogue

between theory and applications. Matrices, vectors and their associated rich

and powerful theory are ubiquitous throughout engineering (Laplacian matrices for

electrical networks; stable matrices in control theory, the matrix laboratory software

Hamiltonians in quantum networks), mathematical chemistry (Huckel theory and

adjacency matrices), quantitative ecology (Leslie–type matrix models for demography),

computer science (coordinate geometry for graphics, spectral methods for

analysis of networks), economics (Leontif models) and sociology (actor–event matrices),

to name just a few such areas of application.

In view of that, it’s curious indeed that the current incarnation of

current syllabus itself provides several natural points of entry into that material. For

instance the geometry and algebra components already have the students solving

systems of simultaneous linear equations, but neither deals with vectors, matrices,

1

or Gaussian elimination. This, despite the fact that students at second level are well

able grasp these ideas, at least in the case of systems of small order. Further, for

stronger students, linear algebra can move very quickly from basic, computation–

based considerations (row reduction, linear systems, matrix–vector multiplication)

to more abstract concepts (subspaces, linear independence). It’s an efficient way of

taking mathematically inclined students from the concrete to the abstract in short

order.

A little work with matrices can also be used to connect different components

of the syllabus. For example a short segment on matrix techniques in difference

equations (or even just the example of the Fibonacci numbers) could be used to

enhance the material on sequences. The coordinate geometry section can also benefit

from some work with matrices – one might introduce rotations or reflections in

three dimensions via matrix–vector multiplication, allowing for a stronger link to be

made between algebraic and geometric/trigonometric concepts. As a byproduct, the

approach above would reinforce the central message that, at its best, mathematics

is a rich network of interconnecting and mutually informative ideas, rather than a

discrete collection of hermetically sealed techniques with no flow between them.

Generations of frustrated mathematics students have posed a challenge to their

instructors:

linear algebra as part of the

opportunity to provide a concrete answer to that question, because matrix and vector

based methods have reshaped our world.

pages on the internet is a massive matrix computation on eight billion rows and

columns (and counting); consequently, notions from linear algebra have altered not

only the volume of information available to us, but also way that we access and

interpret that information. Increases in computing speed and power have facilitated

2

the discretisation and linearisation of problems across science and engineering, with

the result that fast, accurate, and stable algorithms for matrix computations are a

cornerstone for the numerical solutions to those problems. If

that Irish students at second level have some exposure to matrices and vectors,

then we can tell the students all of this. They’ll then see how the content of their

leaving cert syllabus is vital, relevant and influential on their lives.

Steve Kirkland

Hamilton Institute

NUI Maynooth

In defense of retaining calculus on the schools honours curriculum

(The project syllabus contains less than 40% of the old course syllabus)

Calculus is one of the pillars of mathematics. Apart from its ramifications in virtually all areas of mathematics, e.g. geometry, probability and number theory, it also has numerous applications in applied sciences, notably, physics and chemistry, geology, mechanical engineering, electrical and electronic engineering, as well as economic theory and finance.

It may be useful to mention a few examples of such applications.

The applications in physics are obviously too numerous to mention. The origin of calculus was of course in Newton’s formulation of mechanics, but in modern quantum mechanics it is even more crucial, it being based on Schroedinger’s wave equation, which is a (partial) differential equation.

(Schroedinger was professor of theoretical physics in Dublin from 1939 until 1955. He devised his equation well before he came to Dublin, however.)

In chemistry, calculus plays a role in rate equations for reactions, and of course in thermodynamics. In geology, models of fluid flow are given by differential equations, and crystal growth is usually also modelled as such. In mechanical engineering fluid flow also plays a role as well as mechanics of course. The bending of a beam is another application. Electrical and electronic engineering is filled with examples: the voltage across a inductive coil is proportional to the time-derivative of the current, the charge of a conductor is the integral of the current, the motion of electrons in a transistor is given by the diffusion equation, etc., etc. And, indeed, all of the behaviour of electrons is derived from Maxwell’s equations, as is the behaviour of electromagnetic waves. In economics ,everybody talks about the ‘rate of inflation’ which is a differential quotient, and financial theory depends on advanced concepts of probability which themselves depend strongly on calculus.

This small list of examples already illustrates the importance of calculus for applications. Note that these are real-life applications, which are used in practice, not articially constructed ones. But learning about calculus is not just useful, but also very satisfying for the mathematically inclined. Many mathematicians had their curiosity aroused in the first place in learning about calculus. To see, and understand, that the area under the curve of x^2 is exactly 1/3 that of the enclosing rectangle has a fascination all of its own as it is rather more surprising than the area of a triangle being half that of the rectangle. No cutting and pasting will demonstrate this satisfactorily.

All the more reason therefore to teach this subject at school. It attracts students because of its remarkable and elegant results and prepares them for many future careers. However, it may be argued that it is simply too difficult and as a result discourages many to take up mathematics. This, I believe is a fallacy. First of all, in the past this does not seem to have been the case, and moreover, in many other countries teachers succeed, for example, Korea, Japan, Singapore, Russia, Iran. Ireland needs to compete with other countries economically and cannot afford to lag behind in the technological development, for which mathematical knowledge is so important. Indeed, the fact that a subject may be difficult should not lead to the conclusion that it should not be taught to those that are able for it.

Besides, I believe that on the contrary, honours students, i.e. about 50% of the leaving certificate cohort, should be well able for calculus including methods of differentiation and integration. Note that this short piece is entitled ‘In defense of calculus’, not ‘mathematical analysis’. The precise mathematical definitions of the concepts of analysis are not easy: I am not arguing for epsilon-delta arguments. But the methods of calculus are quite accessible, even for average students. Once the method has been explained, it simply requires lots of practice, working through many similar examples. The advantage is that doing many examples builds confidence and with that comes enjoyment. It is quite similar to practicing for tennis or playing the flute. Note that this is different from learning by rote: the examples are plentiful and can be endlessly varied, and the correct method has to be selected in each case. In this way a number of skills is acquired. And these skills are useful in many applications as argued above. This approach which is more like that of a carpenter’s apprentice, is more effective, esp. for the less gifted, because they can be given firm guidelines along which to proceed, rather than the rather haphazard approach to ‘problem solving’ advocated by Project Maths. In addition, in many cases, understanding develops after practice. A large part of ‘understanding’ is in fact familiarity, and practice breeds familiarity. Practical problem solving is usually simply a variation on a theme, i.e. a known skill. The better the skill is embedded, the easier it is to find a variation or application.

**Prof Tony Dorlas (DIAS)**

he following is a list of 3rd level maths professionals who have express concern about project maths and the claims made by NCCA.

**From UCC**

The following have expressed concern about Project maths in there interim report October 2011

**Professor Emeritus P. D. Barry**,

**IT Tallaght **

**Dr Cora Stack **

**IT Tralee**

DR. BRENDAN GUILFOYLE.

**NUIG**

**Prof Ted Hurley **

Dr Madeeha Khaleed

By Dr Eugene Gath Department of Mathematics and Statistics, University of Limerick

Wednesday, March 21, 2012

This dumbed down syllabus is a distortion of the mathematics required to equip our students for third-level education, writes Dr Eugene Gath

IT is widely accepted that there is a crisis in school-level maths, from early primary school up, including unqualified teachers, students leaving school innumerate, under-challenged students, and low numbers taking higher-level Leaving Certificate maths, not to mention the low standard of maths among many of those students who actually do get an honour.

Many readers may be aware of Project Maths, either through their own children or professionally. It was set up to address this crisis and is essentially the new maths for our schools.

One possibly welcome feature is that it eliminates all choice from Leaving Cert Maths and attempts to ask exam questions that are “unseen”, thereby stopping the cherry-picking of easy predictable questions and reducing the regurgitation without understanding that is currently rife.

That said, Project Maths is, in my view, a retrograde move. The main reason is that the proposed syllabus constitutes a major dumbing down of the current syllabus, as well as a sea-change in emphasis.

There are five strands — one of which is classical geometry (which disappeared from the syllabus 40-plus years ago), and another is probability and statistics, the content of which has been at least doubled. The syllabus is a complete distortion of the mathematics that is required to equip our students for a third-level study of the subject.

What disappears under Project Maths is most material on calculus, a lot of differentiation, almost all integration, as well as all vectors, all matrices, discrete maths and much more.

This material is the bread and butter of engineers, scientists, economists, financiers, computer scientists and, not least, statisticians. Yes, it is difficult, but almost every country exposes their students to the intellectual training and rigour of calculus at second level; soon our students will not know the integral of cosine.

The universities assume familiarity with this material in first-year maths classes: the impact will be to force the dumbing down of first-year university courses, not just in maths but also physics, applied maths, mechanics etc, thereby, for example, pushing topics such as Laplace Transforms, vector analysis and PDEs much later into the curriculum.

Today some of our best students have difficulty sustaining an algebraic calculation over a few lines; the new syllabus reduces the amount of time spent doing detailed calculations even further.

Do the engineering professional bodies realise the extent to which this runs counter to their stated goals? I wonder would they prefer for our Leaving Certificate students be well-versed in theorems of Euclid and conditional probabilities or in simple integration, vectors and matrices?

Another matter of concern is that Project Maths is very resource-intensive. It is more hands-on and uses lots of “laboratory” equipment that will be needed in every school (for example, students will be throwing dice to learn about probability). It will also require the retraining of most maths teachers.

Even if it results in higher participation rates, at what cost in terms of content and standards? Surely there are better ways to spend any additional funding of mathematics. The Government would do well to incentivise maths teaching as a career, as in other countries. The attitudes of students would change with a proper rewards system (such as bonus points, compulsory questions and so on).

Project Maths is not the answer to most of the problems mentioned above. It is seriously misguided and it will be very damaging in the long run.

* Dr Eugene Gath, Department of Mathematics and Statistics, University of Limerick

The following is a letter from Brendan Guildea showing the extra time required to teach engineering students as a result of topics removed from LCert syllabus.

Hi y’all,

The meetings I attended with lecturers and engineers made me think seriously about the PM changes from the point of view of maths courses for engineers post leaving certificate.

I met with some colleagues and text book writers to discuss the proposed changes and what it means from an engineering viewpoint. We arrived at the following conclusions:

1 Matrices totally off the ProjectMaths course,4/5 hours class teaching time

2 Vectors totally gone, 8/9 hours class teaching time

3. Integration by substitution and parts gone, 6/8 hours class teaching time. This was the most difficult to get agreement on.

4. Calculus, elimination of implicit, parametric Newton-Raphson etc 5/6 hours class teaching time.

A total of 23/28 hours teaching time postponed until third level. These are not propaganda estimates, if anything they may be too low.

This takes no account of the situation for ordinary level where product, quotient and chain rule in calculus are totally banished. That and other horrors that are, thankfully, outside the scope of this brief note.

I made these calculations two different ways

(i) by counting classes required ( a compromise of various teacher opinions) to teach and revise each topic

(ii) making an evaluation of teaching time as a proportion of value to the candidate in the exam ie a question in the exam worth 10% is allocated 10% of the available teaching time. No opinion here, simply the hard facts.

PM will attempt to poo poo these estimates. However I would be prepared to go anywhere for a discussion to back up my position. Spin, spoof and waffle prevails at present, this cannot continue.

Brendan

These teaching times are all on the low side. The situation may need adjustment upwards by a factor of up to one third.

The following is a letter to the Irish times from Dr Cora Stack lecturer in maths at IT Tallaght

Sir, – If you look at the papers for project mathematics on the Department of Education website, and this is a very worthwhile exercise, you will notice a huge reduction in areas such as calculus and linear algebra, as well as corresponding significant increases in areas such as applied statistics and geometry in comparison to previous years.

I couldn’t see any question on group theory. I see very little on sequences and series, for example. I think the new syllabus should be called practical mathematics!

The demonstration of the intrinsic beauty of mathematics, which is often illustrated in techniques required to work out difficult integrals or inherent in what might be considered tricky mathematics (which always appealed to me), is being sacrificed in favour of this more practical approach.

I am always in favour of being able to apply mathematics in a practical way where feasible: and there are many students who may prefer this approach and who appreciate the value of the subject more by being exposed to these types of practical examples.

However, the inclusion of practical mathematics should not lead to the exclusion of a substantial amount of fundamental theory.

Students with high aptitudes for mathematics could find this course intellectually undemanding, inferior and unchallenging. This point was strongly made by recent students’ letters in this newspaper.

Excluding the introduction of fundamental concepts in calculus and algebra from the honours Leaving Cert syllabus may not be in the best interest of the better or more theoretically minded students.

Perhaps a choice of equally valued mathematics courses should be offered to reflect different types of learners in mathematics. – Yours, etc,

Dr CORA STACK,

Lecturer in Mathematics,

Institute of technology,

Tallaght,

Dublin.

Rte 2 FM countdown 606 leaving cert program will be answering your questions on maths and in particular project maths. Please email your questions to countdown@rte.ie

John Brennan will be answering your questions.