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
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:
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:
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.
Corporate Affairs Division
State Examinations Commission
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,
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,
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
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
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
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.
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.
The following have expressed concern about Project maths in there interim report October 2011
Professor Emeritus P. D. Barry,
Dr Cora Stack
DR. BRENDAN GUILFOYLE.
Prof Ted Hurley
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 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,
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 email@example.com
John Brennan will be answering your questions.