## Bad choices for mathematics? (Eugene Gath)

**It has been widely agreed for some time that there is a serious problem regarding the teaching of mathematics in Irish schools. In response to a review of maths teaching, a new initiative called Project Maths was launched in 2008 and is being gradually rolled out across post-primary schools. It is often described as the main solution to the mathematics learning issues. Here Dr Eugene Gath, Lecturer in the School of Mathematics and Statistics in the University of Limerick, offers a different view.**

It is widely accepted that there is a crisis in Irish school-level maths, from early primary school up, including unqualified teachers, students leaving school innumerate, under-challenged students, low numbers taking Leaving Cert higher level 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 (new, new) maths for our schools.

What it attempts to do is essentially eliminate all choice from Leaving Cert Maths and to ask exam questions that are ‘unseen’, thereby stopping the cherry-picking of easy questions and reducing the rote learning that is currently rife. That said, it 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 40+ 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 required at third level.

What disappears 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 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 would reduce the amount of time spent doing detailed calculations even further. The engineering professional bodies have been supportive of Project Maths, but this was prior to the publication of the full syllabus. Do they realise the extent to which this syllabus runs counter to their goals? I wonder would they rather our Leaving Cert 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 (e.g. 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 etc.).

Project Maths is not the answer to most of the problems mentioned, it is seriously misguided and it could be very damaging.

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**Tags:** mathematics, Project Maths

February 14, 2011 at 2:36 pm

I am at an early stage in the process of trying to inform myself and form an opinion on Project Maths. I found one passage in the following document particularly striking, and am quoting it immediately below:

Report of the Project Maths Implementation Support Group – June 2010

Inside Classrooms – The teaching and learning of mathematics in social context

(2003) concluded that many students have negative attitudes to Mathematics because

of how it is taught, and that there was also a perception that mathematics teachers

present Higher Level Mathematics as being difficult and only accessible to the select

few. The study, which primarily focused on gender issues in mathematics, videotaped

teacher instruction in 20 lessons in 10 case study post-primary schools. The report

indicated:

“The Mathematics teachers we observed were respected and experienced

teachers in their schools, people who were deeply committed to their work.

While most of them believed that varying teaching methodologies and having

practice at the subject improved learning, learning of itself was most often

equated with the memorisation of formulae and procedures. It was not equated

with thinking creatively, being able to provide reasons for solutions, or

understanding how mathematics is used in the real world…

While a very small amount of time was devoted to outlining lesson aims and

homework in class, most time was spent on exposition by the teacher, followed

by a programme of drill and practice. Overall teacher initiated interaction

comprised 96% of all public interactions in the classes, and within this context

a procedural rather than a conceptual and/or problem solving approach to the

subject prevailed. Little time or attention was devoted to the problem solving

nature of mathematics, to the practical application of mathematics in the

physical world, to alternative methods of solving mathematical problems,

other than those prescribed by the text or the teacher. Teachers were far more

likely to use lower order than higher order questioning, and to use drill and

repetition rather than discussion type questions, to teach mathematical

concepts.”

====

An anecdote of my own. A couple of weeks ago I was giving second year computer science students a review of basic trigonometry (including a review of the derivation, from first principles, of the formulae for the sine and cosine of a sum of two angles, and all the ‘standard’ formulae that follow from the basic ones). I moved onto the formulae for differentiating sin kx and cos kx.

I asked the class if they had ever SEEN any sort of proof, justification or explanation of the result that the derivative of the sine function is the cosine function.

Just one student put up his hand!

(And before you jump in, I know that students regularly claim not to have seen results that you know they have covered in earlier courses.)

I then went through the standard derivation, relating the area of a sector to areas of right-angled triangles contained within and containing the sector in question.

But if students haven’t been exposed to ANY sort of explanation, no matter how vague and handwaving, that gives them some sort of feel for the rational basis underlying the basic formulae and rules of calculus, then the supposed ‘rigour of calculus’ just becomes a meaningless jumble of procedures for manipulating formulae to answer exam questions.

When I have browsed Leaving Certificate mathematics textbooks, the general style has seemed to be along the following lines: here is a recipe and formulae for solving standardized problem A so learn this off; here is a recipe and formulae for solving standardized problem B so learn this off; here is a recipe…

and so on for page after page. If this is how students have been taught to approach mathematics, no wonder they find it difficult!

Well, just first impressions at this stage.

February 14, 2011 at 2:39 pm

Just to be clear -these are my personal views and do not necessarily reflect the views of the Department of Mathematics and Statistics at University of Limerick or any other body to which I belong.

February 14, 2011 at 3:40 pm

I couldn’t agree more. Having taught physics to ‘general science’ first-years, there is no doubt in my mind that students are entering college without a sound grounding in mathematics. I watched in disbelief as large numbers in the class couldn’t solve simple problems of volume, struggled to convert units, and couldn’t manipulate simple algebraic expressions. The level of maths was, at best, Junior Cert. It was as if the final two years of secondary school had never happened.

Anecdotes aside, Project Maths is simply the newest step in a long history of ‘curriculum reforms’ that have only succeeded in stripping maths content from our schools. Even in Leaving Cert. physics it is now possible to obtain almost 50% of the marks by regurgitating definitions and memorising experimental setups.

I believe the solution is to make mathematics ‘more relevent’. To do that, we need to reintroduce maths content into subjects where appropriate. Off the top of my head I can think of mathematical applications/content that could be applied to biology, chemistry, physics, art, geography, music, business studies, classical studies, history, and probably a lot more besides.

What we have done is removed maths content from courses thereby placating students and allowing them to ‘avoid’ maths. We then set third-level minimum entry requirements so low that students can even enter mathsy courses like science without higher-level maths. Unsurprisingly, the students opt overwhelmingly for ordinary-level maths.

It is a perfectly vicious circle, in which maths is seen as unimportant and unneccessary for a college place because it is unnecessary for a college place and the subjects students choose.

February 14, 2011 at 6:14 pm

I fully agree with the posted article. Calculus is the core element required in first year Physical Science. Many students now have no experience of logarithms when coming to College. If they can’t manipulate simple algebraic expressions, differentiate and integrate then they will be in bad trouble very quickly if they study Physical Chemistry or Physics!

February 14, 2011 at 8:04 pm

First of all a tip of the hat to one of my former lecturers, who had me and a bunch of other Chemistry students in the mid-90s. I hope we weren’t too much of a burden 🙂

Removing Calculus would be a big mistake. It was my biggest weakness at LC level and led to my early departure from an Aero Engineering course I had scraped into, whereon I went back and redid LC maths, got to grips with calculus and had a somewhat easier time when I encountered Dr. Gath.

Essentially this plan will download responsibility for teaching basic maths to Universities (some might argue they are doing it already) and continue the process of downgrading undergraduate degrees in order to defer into the student’s 20s the reality of what they need to know rather than pushing students to accept these when their brains are more pliable.

@jfryar: “We then set third-level minimum entry requirements so low that students can even enter mathsy courses like science without higher-level maths.” Our host tried to us about a prior attempt to change this in another post:

https://universitydiary.wordpress.com/2010/10/29/understanding-why-students-drop-out/#comment-10356

February 14, 2011 at 8:05 pm

bah… that should have read “Our host told us…”

February 14, 2011 at 11:55 pm

Well there is one problem with trying to have an informed discussion on Project Maths right now.

All attempts to retrieve the Project Maths syllabus from the NCCA website current lead to the following:

HTTP Error 404 – File or directory not found.

My memory of some previous email correspondence is that Strand 5 is to include some calculus. This strand is apparently the last to be developed.

It seems unfortunate that the NCCA seems so coy as to what it indends in this area.

February 15, 2011 at 1:18 pm

I have been emailed the old syllabus and the new Project Maths syllabus for Leaving Certificate mathematics, with a request to place them somewhere on the Web.

Accordingly I am temporarily placing these documents amongst my own web pages for a short period.

Old syllabus: http://www.maths.tcd.ie/~dwilkins/tmp/lc_maths_sy.pdf

New syllabus: http://www.maths.tcd.ie/~dwilkins/tmp/LC_str1-5_sep10_ex12.pdf

I should be deleting these documents from my web pages at some point in the future.

I believe that the Project Maths syllabus ought to have been available from the following broken link:

http://www.ncca.ie/en/Curriculum_and_Assessment/Post-Primary_Education/Review_of_Mathematics/Project_Maths/Project_Maths.html

February 15, 2011 at 1:37 pm

The draft Project Maths syllabus for calculus is described, in learning outcome format, on page 36 of the Draft Syllabus for initial 24 schools.

Copying the relevant learning outcomes below.

At Ordinary Level students should be able to:

− find first and second derivatives of linear, quadratic and cubic functions by rule

– apply differentiation to

• rates of change

• maxima and minima

In addition Higher Level students should be able to:

− differentiate linear and quadratic functions by first principles

– differentiate the following functions

• polynomial

• exponential

• trigonometric

• rational powers

• inverse functions

• logarithms

− find the derivatives of sums, differences, products, quotients and compositions of functions of the above form

− apply the differentiation of above functions to solve problems

− recognise integration as the reverse process of differentiation

− use integration to find the average value of a

function over an interval

− integrate polynomials and exponential functions

− determine areas of plane regions bounded by polynomial and exponential curves

February 18, 2011 at 5:44 pm

David, for completeness I might list what’s gone (some of it I would not be upset about especially towards the end of the list):

Limits of functions, rules for sums, products and quotients.

Differentiation from first principles of all but linear and quadratic functions.

Derivatives of inverse tan and inverse sine.

Application of derivative to finding tangents of curves.

First derivatives of implicit and parametric functions.

Curve sketching using points of inflection, turning points and asymptotes.

Newton-Raphson.

ALL integrals other than integrals of polynomials and exp(kx).

Integration by substitution.

Integration by parts.

Ratio Test.

Taylor/Maclaurin Series.

ALL material on matrices, including uses for solving equations and in coordinate geometry.

ALL material on vectors.

ALL material on difference equations.

Parametric equations of line and circle.

Transformation geometry.

Telescoping series. Sum of n^2. Sum n*x^n

Derivations of some of the double angle formulae in trigonometry.

ALL group theory.

Ellipses (coordinate geometry).

February 15, 2011 at 4:34 pm

It has been drawn to my attention that the new Project Maths Leaving Certificate syllabus is now on the web at the following URL:

(The broken link from the NCCA Project Maths webpage at

http://ncca.ie/en/Curriculum_and_Assessment/Post-Primary_Education/Senior_Cycle/

needs to be updated.)

July 24, 2011 at 4:25 pm

Project maths in time will be shown to be the greatest mistake ever made in teaching maths. for the following reasons

(i)The syllabus is wrong too much Geopmetry and statistics and too little Calculus.

(ii)The “new” teaching methods are wrong problem based learning has been shown not to work ,Medical schools in Scotland have abandoned it!Findland which embraced problem based learing in 2003 stopped using the method in 2007.

(3)The NCCA the people who are in charge of implementing project maths are out of touch with reality that recent sample papers produced by the NCCA were rejected by SEC (state examinations commission) in fact the SEC insisted that they be removed from the NCCA website.

August 21, 2011 at 6:06 pm

I have reservations about this course also. The style of teaching might be funky but how will students cope with applied maths? Or machanics in physics? scary , that is whatt it is. classical problem solving is facilitated by algebra in the main. I would introduce algebra a lot earlier than ist year…as well as indices. easy little theorems wouldnt damage anyone in 6th class either.

October 22, 2011 at 11:22 am

As a teacher of leaving certificate higher level maths, I too have serious reservations about the new course. I worry about the content that has been removed and will be required in future studies or science and engineering etc. I take on board some of the comments made about standards and understanding of students but must state that those of us preparing students for the leaving certificate exam are under pressure to produce results and are therefore caught between doing the thing right and doing the right thing. Each year i receive a very simplistic statistical analysis of my results from management, who are only interested in how many A’s, B’s etc my students have received and not whether they are best prepared for their choices of further study. limited time and resources are having major impact on the amount of time I can spend on other than rote learning of routines, combined with a student body that are so under pressure for points that they baulk at spending any time on anything that is not directly exam related. The new syllabus is a dumbing down of maths standards and also seeks to eliminate students with maths skills and some language difficulties, from achieving higher levels. Inservice training is not addressing teachers worries, and indeed subjects us to large periods of time playing with bits of plastic rather than seeking to explain to us what really is required at higher level LC. I really enjoy teaching Mathematics and have been cery proud of the achievements of my students over the past 20 odd years but levels of frustration are at an all time high and I can see it pushing many very effective teachers out of the profession.

October 22, 2011 at 2:45 pm

People who promote Project maths state that it will eliminate rote learning.The opposite is the case .Strand 1 and strand 2 require lots of rote learning .

Definitions in statistics.

Axioms,Theorems,constuctions in geometry. All the new material is rote learning based.