Dominick Donnelly’s submission to the NCCA on the Draft Background Paper for the Review of Applied Mathematics

Dear Sir / Madam,

 

I write to you with regard to the consultation on the Draft Background Paper and Brief for the Review of Applied Mathematics.  I have filled out the survey online on the NCCA website, but found the scope of its questions somewhat limited, and so I have also submitted the following detailed submission.

 

About Dominick Donnelly

I have been a secondary school teacher since 1990, having previously obtained a degree in Mechanical Engineering from TCD.  In the early years of my career I taught in British curriculum schools, both in the UK itself and in British schools in Latin America, teaching Mathematics, Applied Mathematics (as part of various ‘A’ Level Mathematics syllabi) and Computer Studies/ Science (Cambridge IGCSE and International Baccalaureate).  Since my return to Ireland in the late 1990’s I have taught in a variety of schools, mostly in the Cork City area, teaching a wide range of subjects including Mathematics, Applied Mathematics, IT, Physics, Technical Graphics, Science and Geography.  Throughout all my career I have taught Applied Mathematics.  I currently teach Applied Maths, Physics and Maths in two different schools in Cork City, as well as teaching Applied Maths as an extra subject outside of normal school hours to a number of students.  In 2010 my text book for Leaving Certificate Applied Mathematics was published, one of three currently available, and it is now in its 2nd edition.  For six years I was an examiner for Higher Level Leaving Certificate Applied Mathematics with the SEC.  I have also given a number of training courses for teachers of Applied Mathematics through the auspices of the PDST, IAMTA and IMTA.

 

 

Initial Response

My immediate response on reading the Draft Background Paper and Brief for the Review of Applied Mathematics (hereafter referred to the Background Paper) is that there is a definite attempt to combine what should be in three different subjects (Applied Mathematics, Further Mathematics and Computer Science) into one subject, and three into one just won’t go.  A fundamental decision must be made to decide what this syllabus review is about.  I do not believe that the proposed combination of what should be in three different syllabi is a good thing, and in trying to fix a number of problems which are nothing to do with Applied Mathematics something precious and beautiful will be destroyed.

 

Much of what is in the proposals within the Background Paper has almost nothing to do with applied mathematics per se, but is being lumbered in in response to various criticisms of the overall Leaving Certificate syllabus.  What is proposed in the core part of the syllabus is largely there because there has been much criticism, much of it from various departments within the third level colleges, over various aspects of mathematics which have been omitted in the Project Maths syllabus, and it is a blatant attempt to placate those voices.  If there are issues to be dealt with within the Project Maths syllabus, that is where they should be dealt with.  If this leads to the creation of a Further Mathematics syllabus, that is fine.  I fundamentally disagree with the rationale of lumbering this subject with them by means of a content heavy core, which has little or nothing to do with problem solving.  I have in the past taught part of an ‘A’ Level Further Mathematics syllabus, and a very esoteric beast it was with very limited appeal.  The number of second level students capable of operating mathematically at that level is tiny, so I would not be in general in favour of the development of a Further Mathematics syllabus.  That said, a number of the proposed components of the core are actually already deeply embedded within the current Applied Mathematics syllabus, and if these aspects need to be enhanced that can be done, while maintaining a problem solving approach.  I deal with this later in more detail.

 

With regard to the inclusion of Computer Science as an option within this syllabus, this is clearly another attempt to placate those voices who bemoan the lack of such a subject within the Irish secondary school system.  Should there be a proper Computer Studies / Computer Science course within the Irish secondary school curriculum – absolutely yes.  Should it be located as an option within a course primarily about mathematics and problem solving – absolutely not.  There has been a crying need for a Computer Studies / Computer Science syllabus at both junior cycle and senior cycle within the secondary curriculum for the past number of decades, but to propose to resolve that deficit here is merely tokenistic and shows a lack of understanding of both subjects.  Computer Science, which I have taught abroad in the past, is a very practical subject where a large element of the assessment which should be project based.  While certain elements of computer science could be incorporated into a mathematics based syllabus, such as algorithm design involving some of the fundamentals of programming such as iteration and recursion, this would not negate the crying need for a proper Computer Science syllabus across the Irish secondary school curriculum, and even into the primary school curriculum.

 

While the existing Applied Mathematics syllabus is forty years old, there is nothing fundamentally wrong with it, and while it absolutely should evolve to incorporate other areas of applied mathematics other than mechanics, it must retain its central emphasis on applied mathematics and problem solving, and not become too laden down with content which should really be covered in other syllabi.  There is an inherently contradictory theme running through the Background Paper, which is that the current syllabus is too content-heavy, which it isn’t, and yet the proposed changes lumber it with loads more content.  There is also a fundamental lack of recognition and understanding within the Background Paper as to what the current subject of Applied Mathematics is:  it teaches problem solving using various mathematical techniques through aspects of mechanics.  This can and should be developed to incorporate more areas of applied mathematics apart from mechanics, but it fundamentally must be built around a problem solving approach which does not encumber the student with too much content.

 

In all the official reports on mathematics and education, there is a huge and correct emphasis on problem solving.  This is essentially what the existing syllabus is.  This can be enhanced and developed by a revision of the syllabus, but it is vital that the problem solving element is kept to the fore in all decision making around the syllabus review.  Deficiencies in the current overall syllabus, both perceived and actual, must only be considered for inclusion within this syllabus review on the basis that at their core they fundamentally fall within the remit of problem solving using applied mathematics.  Anything else should have no place here, and any such deficiencies can and should be dealt with elsewhere.

 

Given the tone of much of the commentary within the Background Paper, I fear that within the NCCA and the authors of the Background Paper, there is a fundamental lack of understanding of what the current subject of Applied Mathematics actually is, and this does not fill me with confidence that this syllabus review process will result in a positive outcome for the overall quality of mathematical education in Irish secondary schools.  I sincerely hope that my fears are misplaced.

 

I fully recognize that there are a number of competing claims for attention with this syllabus review, but I think a fundamental question must be asked of each of these claims – does it belong here or should it be dealt with elsewhere?  There is a huge potential risk to get this very wrong, and thereby destroy something good and worthwhile.  The old adage “a camel is a horse designed by a committee” very much comes to mind on reading the Background Paper, and as proposed this syllabus has the potential to look far more like a lumpy camel than a sleek and elegant horse, unless great care is taken.

 

Analysis of the existing syllabus, and the proposed mechanics options

Within the Background Paper, there is very little analysis of how the subject is actually currently taught within the schools.  Any such analysis that is there is glib and condescending.  For example the second sentence on page 1 of the Background Paper states:  “With its emphasis on content and in the absence of any aim or rationale, it is difficult to ascertain what group of students’ needs the syllabus aims to meet.”  This suggestion that the syllabus is content-heavy is echoed repeatedly throughout the Background Paper.  For example at the top of page 3 it says:  “However, with its emphasis on content as opposed to the development of skills and mathematical reasoning students’ are not problem solving per se but rather, learning to solve particular problem types in mathematical physics.”  I can, and have in the past, summarised the current syllabus on one side of an A4 sheet of paper, for revision purposes.  This is very far from being content heavy, particularly in relation to many other subjects at Leaving Certificate.

 

I fundamentally disagree that the current syllabus is not really teaching problem solving.  Any teaching of problem solving requires a context, and in this case the context is aspects of mechanics.  The problem solving skills learned in studying Leaving Certificate Applied Mathematics can be, and are, transferred by students to other areas of problem solving in their further studies and careers.  I have in the past taught Applied Mathematics in four different countries, to a number of different syllabi, and the fundamentals of the current Irish syllabus are no different to the others I have taught.  This fundamental misunderstanding of the current subject is, in my opinion, a huge obstacle to progress in the development of the new syllabus if it is not addressed.

 

On the perception that the current syllabus is an easy way to get points, that is not borne out by experience.  While the A rate is higher in Applied Mathematics than for the majority of subjects, the cohort of students studying Applied Mathematics has very much tended to come from the top end of the spectrum, so to speak.  I know the SEC has done analysis on this over the years, and for example in 2013, of those who did both Higher Level Applied Mathematics and Higher Level Mathematics, on average the candidates’ marks in Applied Mathematics were 11% lower than their marks in Mathematics.  That is a hugely significant difference.  If people want to say Applied Mathematics is too easy, what does this say about Mathematics?  I know the SEC has done similar comparisons over the years with Physics also, and the analysis has never shown, as far as I know, that the students on average do better in Applied Mathematics than in Maths or Physics.  The SEC would be able to provide far more detail of their analysis, obviously.

 

Many of the issues raised in the Background Paper are not particularly issues with Applied Mathematics, but are more to do with Project Maths and maths education in general.  For example on page 5 the Background Paper states:  “Post-primary mathematics education in Ireland features a highly didactic pedagogy, with mathematics being taught in a procedural fashion that places relatively little emphasis on problem solving.”  It also states:  “Chief examiner reports on state examinations in mathematics over a number of years have consistently pointed to the over-reliance by candidates on rote-learning procedures and the lack of deeper understanding of basic mathematics concepts.”  The inclusion of this commentary within the Background Paper suggests that these issues also apply to Applied Mathematics, which I would argue they fundamentally do not.  Over my many years teaching the subject, I can think of numerous students whose strength was in rote learning, who took up Applied Mathematics, and who either abandoned their studies in the subject or who did not fare as well in the subject as they did in their other subjects.  Of all the subjects in Leaving Certificate, I would argue that Applied Mathematics lends itself least of all to a rote learning approach.  Those who do well in Leaving Certificate Applied Mathematics do so because they have developed a high level of problem solving skills.

 

It is probably worth considering what we mean by the phrase problem solving.  I would explain it as follows.  In solving a normal mathematical problem, there are probably at most three or four different steps or stages needed to get from the problem to the solution, and in general it is not that difficult from the start of the problem to see your way to the solution.  When we come to problem solving, that number of steps or stages increases substantially, and can be well over ten stages, and in many cases it is not clear at the start what an appropriate pathway to the solution might be.  While each of the stages may in themselves be quite straightforward, the complexity and skill in problem solving comes from being able to identify and sequence the relevant stages from question to solution, and also in having the ability to explore different potential pathways from question to solution.  In many cases, an actual pathway to the solution only becomes apparent after exploring a number of potential possibilities.  This has been very clearly apparent in my time marking actual scripts at the Leaving Certificate, where students, when they have enough time to do so, explore a number of possible ways to solve the problem, until they hit a correct one.

 

In the Background Paper there is no analysis of the current content of the Applied Mathematics syllabus, and no indication of what aspects might be retained in the proposed options in the new syllabus.  This is a huge deficiency and merits some detailed exploration.  The current syllabus document, which is included in Appendix 1 of the Background Paper, is so brief as to be pretty well meaningless beyond giving the vague headings under which problem solving can be explored.  The new syllabus must address this issue to provide clarity, if nothing else.  Countless times I have been asked by other teachers of Applied Mathematics, particularly those relatively new to the subject, “Do I have to cover …?” or “How much of … should I do?”.  The lack of clarity in the current syllabus is a huge obstacle.  For example, if you take a topic like Statics, you could probably spend half a year trying to cover Statics completely, and still miss something out.  It is an almost unending topic, and the new syllabus must be clear on each topic what is, and what isn’t, included.  There is still ample scope for there to be new and challenging problems posed within clearly defined specific syllabus content.

 

The current syllabus consists of ten general topics, as outlined on page 34 of the Background Paper.  Of these, 6 are studied by almost all students, namely questions 1 to 5 and 10, with the remaining four being much less studied.  I would advocate for the retention of questions 1 to 5 without major revision, except to be far more detailed in the syllabus as to what the scope of each topic is.  There is plenty of scope to increase the use of vectors and differential calculus with most of these topics, and I do not see the need for vectors and calculus to be treated within a core section, but rather maintain the applied nature of the course by dealing with these topics within a problem solving context.  With regard to question 10, I believe there is scope here to substantially increase the current scope of the syllabus.  Below I argue for the omission of the core and option structure of the syllabus, and I believe there is huge scope to incorporate more calculus and differential equations throughout the new syllabus, with an emphasis on a problem solving approach.

 

Of the remaining four questions, some serious revision is probably needed if they are to be retained.  If question 6 is to be retained, which it probably should be, I think the Circular Motion and Simple Harmonic Motion aspects should be separated, as it is really too big a topic for one question at present in relation to most of the others.  There is huge scope within this topic for really challenging problem solving, with plenty of scope to incorporate calculus and vectors.  If the Statics topic in question 7 is to be retained, it needs serious revision.  It is currently by far the longest and most open ended of the topics, and if it is retained, which it should be, it should be treated as at least a double topic in relation to most of the existing topics.  Again the use of vectors can be emphasised within this topic easily.  Rigid Body Rotation, or question 8, is probably the most esoteric of the current topics, and I would advocate its removal.  Finally Hydrostatics, question 9, is traditionally the least answered topic on the course, and while I am a huge fan of the topic, it could be safely left out.  It does not appear in many applied maths syllabi internationally.  From my knowledge of other Applied Mathematics syllabi, or the Applied Mathematics components of other syllabi, I don’t see any particular need to include other topics from mechanics outside of the current range.

 

 

Student responses to studying applied mathematics

I think it is worth noting the responses to some of my former students from over the years.  While for many it is a struggle at first, at the end of the day the majority grow to love the subject.  There are always students who take up the subject but do not see it through.  These generally perceive it as too difficult, and opt for something they perceive as easier.  This is particularly true for those who take it on as an “extra” subject, not in their main timetable.  It is a hugely admirable thing for a student to take on Applied Mathematics as an extra subject, and given that over half of those sitting the exam have done it as a non-timetabled subject in some way or other, this is hugely impressive.  The question must be asked why so many see it through to the end, given the perceptions (and reality) of the difficulty of the subject, and the time constraints they are put under by their other subjects.  They must be getting something useful from the subject, and while some are inevitably doing it for the “points”, I believe most are not.  They recognize the benefit of studying the subject, particularly if they are intending to study one of the technical disciplines at third level.  Many also relish the challenge and difficulty of the subject, and derive great personal satisfaction from succeeding.

 

As an experienced Applied Maths teacher, it is normally fairly easy to see those students who are going to succeed at the subject fairly early, and those who are not.  Most students of a reasonable level of mathematical ability can achieve at the subject to a reasonable level.  Those who achieve better than the average at the subject usually have a combination of the following characteristics:  a tenacity and stubbornness that does not allow them to give up on a problem, an ability to think in a non-linear way around a problem, and most of all an ability to sit back from a problem and analyse their options.  These are all inherent traits of good problem solvers, and this subject definitely helps develop and enhance these skills.  The ones who do not succeed well at Applied Mathematics are what I call the “straight-line thinkers”, and while their ability to problem solve can be improved with practice, it is very difficult for them to learn to see around corners.  Some of these students can do exceptionally well in other subjects, thus demonstrating that there is an inherent difference in the skills required to succeed at Applied Mathematics.

 

I have also done some work over the years with Engineering students at third level, both at university and IT level, who are really struggling with Applied Mathematics, and sometimes also Mathematics.  Invariably these students have not studied Applied Mathematics at school level, and most of the IT students have only done Ordinary Level Mathematics.  They find it a huge challenge at third level to get to grips with the problem solving nature of Applied Mathematics, and while they may have excelled at school at subjects such as DCG, Construction Studies or Engineering, the level of mathematics required at third level is a huge step up for them.  Some of these over the years have expressed that they wished that they had done Applied Mathematics at school, as it would have made their lives much easier in college, and I believe there is a substantial section of the cohort for whom this is the case.  Thus the numbers taking the subject has huge potential to grow, by bringing it into schools where it has never been, and by trying to counter the elitist attitude to the subject, that it is only for the “really smart.”

 

 

The overall structure and name of the new syllabus

On the subject of naming the subject, I am firmly of the opinion that the name Applied Mathematics should be retained.  The important essence of the subject, namely problem solving using mathematics, is best summed up by that title.  If a name such as Further Mathematics is used, that would de facto mean that more mathematical content has been added in, and the problem solving nature of the subject has been diminished, and that to my mind would be a very retrograde step, and contrary to just about any education policy document you care to cite.  It does what it says on the tin, and if it ain’t broke, don’t fix it!

 

With regard to the structure of the new syllabus, I am vehemently opposed to the proposed core and options structure.  What is in the proposed core is largely further mathematical content, and while some of it can incorporate some problem solving methodology, this approach is inherently content driven, rather than being driven by a problem solving approach, and this is, apart from everything else, contrary to the stated aims of this very syllabus review.  The first option proposed, where the core would make up 70% of the course is, in a word, terrible.  The second option proposed, where the core is 40%, is better, but still bad.  The third option is very vague, and I would have to have a much more detailed explanation of it in order to judge my opinion of it.  I will come back to how I would propose to structure the syllabus at the end of this section of my submission.

 

There are five proposed topics for the core section of the new syllabus.  I will deal with their merits / demerits individually, and as to how some of the them could be (or already are) incorporated into an applied mathematics syllabus.  The first one listed is matrices and their applications.  I really do not see the benefit of incorporating this into the course.  While matrices have a number of relevant applications, in any sort of real problem solving involving matrices, once the matrices get to any sort of realistic size, solving problems using them becomes very tedious and time consuming.  While a basic understanding of the mechanics of how to use matrices is definitely useful in a number of areas, in practice, if matrix calculations are performed manually they are turgid and not very illuminating.  They could fit very well in a Computer Science course, as they are the fundamental basis of a number of computerised calculations, and could be incorporated into a course on algorithm design, for example.  There is also a possibility of incorporating them into a course on graph theory.  It is not that difficult to understand the mechanics of how matrices work – it is just very tedious to do any sort of real manual calculations with them, and therefore any sort of real assessment of the students’ understanding of matrices becomes very tedious very quickly.

 

Vectors and their applications is already deeply embedded within any course on mechanics, and does not need separate treatment elsewhere.  Anybody who has studied, for example projectiles and collisions, and particularly relative velocity, has a deep understanding of vectors and vector algebra.  The only aspects that aren’t there already are the dot and cross products.  These could be incorporated if required, though I am not sure of the particular merits of that, and would need to see further justification of the merits of their inclusion.

 

Linear Programming would almost certainly best be dealt with via a course on Game Theory, or even Business Mathematics as a method of optimisation, and could incorporate matrices if required.  However again I am not sure of the overall benefits of incorporating this into a syllabus such as this, and I would need to see further justification as to the merits of its inclusion.  To do any sort of realistic problem solving using linear programming would pretty well inevitably require the use of computers, and could probably be best dealt with within a section on computational mathematics within a full course on Computer Science (see below).

 

The final two components of the Core are listed as Further Calculus and Differential Equations.  There is a substantial amount of calculus within the current Applied Mathematics syllabus, and we have a whole question on Differential Equations.  Any material that is required on calculus in general can be incorporated in an applied context easily.  There is a lot of calculus there already within a number of the existing topics on mechanics, and there is plenty of potential for lots more, if required, both within the existing mechanics topics and also in some of the other proposed options.  I do not see the merit of separating out the calculus components.  Of all the mathematical tools at our disposal, surely calculus most of all is inherently about problem solving, and should really be taught in that context.

 

Therefore overall I do not see the merits of having a core section to the course.  Much of it is already there within the existing course, particularly vectors and calculus, and these could easily be enhanced.  The other proposed components of the core, namely matrices and linear programming, are not so fundamentally critical, and if required, there are still plenty of ways they could be incorporated into the course via some of the proposed new option topics.  I really do see the introduction of the proposed core as a backwards step, as it is not motivated by enhancing the opportunities for teaching problem solving, but is really there to silence the critics of the current Project Maths syllabus, particularly from the third level sector, who see these as glaring omissions.  Please do not try to overcome any perceived shortcomings in the Project Maths syllabus by dumping them on Applied Mathematics instead.

 

With regard to how I think the syllabus should be structured, I would suggest two options.  The first option is that all topics be given an equal waiting, and have one question each.  By this I mean there would be a certain number of the existing topics from mechanics, as discussed earlier, each with one question, and also a certain number from the other proposed areas of applied mathematics.  This would be very similar to the current format, but with perhaps a reduction in the number of mechanics topics, and the addition of questions on some or all of the proposed new options.  The students would have to answer on a specified number of questions.  This has the merit of simplicity, and it also facilitates the gradual branching out of existing teachers into new areas.  They could take on, say, one new question per year, and in that way gradually increase their coverage of the course into new areas.  It must be remembered that a large section of the current Applied Mathematics teaching population are in effect largely self-taught, and that they have gone to great lengths to build up their expertise on the material in the current syllabus.  This expertise must not be discarded or rendered redundant lightly, but rather enhanced and built upon.

 

A further suggested possibility is to divide the overall into a number of sections, with, say, five topics in each section, on which the student would answer, say, three questions from each of two different sections.  I would propose that there would be two sections on mechanics, and one or two sections to cover the newer options.  This option is not as flexible as the previous one, but has the benefit of being more themed, as the topics within each section should have some level of relationship and coherency with each other.

 

I generally favour my first proposal for the structure, rather than the latter, as this facilitates a gradual shift in focus for teachers (and thereby students) away from just the traditional mechanics into the newer areas of applied mathematics.  For a teacher to take on, say Game Theory, as a topic to which he or she is completely new, and where that is worth up to a third of the total course, is a huge risk for the students of that teacher, and most wouldn’t dare take such a risk, and rightly so when the future studies and careers of their students may depend on it.  This militates dramatically against people taking on the new areas, no matter how fabulous they are.  Whatever structure is decided upon, it must facilitate the gradual shifting of emphasis away from just the traditional mechanics into the newer areas of applied mathematics.

 

 

The proposed new options

On the whole I am very excited by the possibilities offered by the proposed new non-mechanics options for the study of applied mathematics.  As I have already outlined, I would not see these as mere options per se, but really they are the choices as to the content of the course proper, alongside the existing mechanics topics.  I have already dealt in detail with the mechanics topics in a previous section, so I will not repeat it here.  Let it be clear that I fully endorse the expansion of the syllabus into some or all of the proposed areas of applied mathematics.  I am not however convinced that the proposal to have five or six equally weighted options, as the Background Paper proposes, is the best way to proceed.  I am not convinced that some of the new proposed topics should have the same weighting as mechanics, or even half that of mechanics.  I do not see that there is enough suitable material within them in general to justify such a large component for each.  There is also the reality that all of these are completely new to most teachers, and a better way would be to give each of them a weighting equivalent to maybe a quarter of that of mechanics, as this would facilitate people actually taking them on.  Otherwise they run the serious risk of being completely ignored.

 

If the overall structure is to be divided up into sections, I would propose that the new options be grouped together, rather than standing on their own.  For example I think that Game Theory and Business Mathematics would sit together quite well, if required, and likewise Computer Programming and Networks and Graph Theory would work well together.  If for example something akin to my second proposal for the structure of the paper was required, with four different sections on which the student must study at least two, then if there were four sections along the following lines:  (i) Mechanics A, (ii) Mechanics B, (iii) Computer Programming and Networks and Graph Theory, (iv) Game Theory and Business Mathematics.  Alternatively there could be just three sections, with two sections of mechanics and the new topics all together in a third section.

 

With regard to the individual proposals for the options, I will deal with Game Theory first.  Having never actually studied Game Theory, and only having a passing knowledge of what it is about, I am not qualified to comment in detail on it.  From what I do know however, I would welcome its inclusion in the new course, but it must be given a limited weighting, and if it is not introduced carefully it will likely be completely ignored.  It is not widely taught at second level around the world, so it is hard to find suitable curriculum material appropriate to a second level audience.  Much of the material currently available is definitely aimed at a third level audience, and much of it a highly specialised audience at that.  A couple of topics could be incorporated however, as long as it is reasonably light on content and theory, and is approached from a problem solving perspective.  The most likely area of application is in the field of economics, though game theory can be applied to a very wide range of applications.

 

The next option is Business Mathematics.  This is a fairly vague title, and could cover a multitude.  I am taking it to mean certain aspects of applied statistics and probability, with maybe some business calculus and optimisation.  There is scope to introduce certain topics in this area, but again the same precautions must be taken as for Game Theory, it must be relatively light on content and must focus on problem solving.  Again there is a serious risk that if it is not introduced carefully, it could be completely ignored.  I am not entirely convinced of the benefits to exposing students at second level to material of this kind, but again it is not an area in which I have much expertise, and so there might well be merit in its inclusion.

 

The option on Networks and Graph Theory for me is the area that excites me most, though how much can be done without extensive use of computers I am not sure.  There is obviously a huge linkage here with Computer Programming, with much of graph theory being based on algorithms.  This could be incorporated using algorithms written in pseudocode, which I think would be preferable to using any specific programming language.  There are many different possible fields of study to which graph theory could be applied, such as critical path analysis, optimising routes, project management, and so on.  It could also be applied to various problems in business and economics, with the possibility of incorporating linear programming and optimisation type problems.  Again there is huge potential to get overloaded with content, but I believe it should be possible to find a suitable course of study, which would give the students a feel for problem solving in this area without getting overloaded on content and theory.  I will deal with Computer Programming separately in the next section.

 

 

Computer Science should be its own subject

First of all let me state very clearly that I absolutely support the proper introduction of a computer studies / science syllabus throughout the curriculum, at primary level, and at post-primary level at both junior cycle and senior cycle.  There must be a substantial amount of computer programming as part of this.  In my view, it is criminal that this has not already happened, and it must be regarded as an absolute priority.  All the largest computer companies in the world have substantial operations here in Ireland, and it is an utter disgrace that computer science education has not been properly integrated into our school curricula.  With the number of children now involved in coderdojo projects around the country, these children are going to need a structured and suitable curriculum to follow, to guide and enhance their fabulous enthusiasm for coding.  If the inclusion here of Computer Programming is aimed to placate the calls for the proper introduction of Computer Science as a full subject at Leaving Certificate, then absolutely no, not nearly good enough.  If the inclusion of Computer Programming here is a pathway to the full introduction of a Computer Science syllabus, then fair enough.  So be it.  The introduction of a full Computer Science course at Leaving Certificate, as well as appropriate supporting courses at Junior Cycle and Primary Level is a matter of absolute priority.  I was teaching International Baccalaureate Computer Science twenty years ago in Latin America, in countries that we would regard as less developed than our own.  The fact that all these years later there is still nothing here is nothing but an utter disgrace and a complete failure of those in charge of our children’s education.

 

That said there are great possibilities to introduce useful topics in this area, as long as it is kept to the forefront that this subject is applied mathematics, and that problem solving using mathematics is the name of the game.  I think I may have been the very one to suggest its inclusion as an option in the new syllabus in a discussion on the subject at an IAMTA conference a number of years ago, when the talk of revising the Applied Mathematics syllabus first started.  What I proposed then, and what I still propose now, is to have a section on algorithm design using pseudocode, which should include all the basic tools of algorithm development such as recursion, iteration, and so on, and could go into the areas of computational mathematics.  There would be scope to link this in to such mathematical topics as matrices and linear programming, as well as huge possibilities for cross-fertilisation with Networks and Graph Theory, and also probably Game Theory and Business Mathematics.  The component of the International Baccalaureate Computer Science course on algorithms would be a very good template to use.  To this end I think calling the section Computer Programming is very dangerous, as anything called that would have to be an awful lot more substantial, with a large and vital practical element, and that does not sit well with the rest of this course.  It would probably be better called just something along the lines of Algorithm Design.

 

In the long run it would be absolutely brilliant if there was a full and comprehensive computer science curriculum in place across all the levels of education, and the component on algorithm design within applied mathematics could complement this very nicely, dealing with the more mathematical aspects of algorithms in a problem solving approach.  That for me would be the ideal eventual outcome on this.

 

 

Ordinary Level must be maintained

I think the proposal to have this as a Higher Level subject only is just plain wrong.  There are so many negatives to this.  First of all it would make the subject completely elitist, as it would be the only such subject in the Leaving Certificate, and it would almost certainly diminish the actual take-up of the subject, as it would be akin to putting up a sign saying “only the geeks and nerds need apply.”  This is so contrary to what our education system should be saying as to be risible.  I understand that the suggestion is coming from the motivation to turn this into a Further Mathematics course rather than an Applied Mathematics course, and I hope I have already made my views on that topic plain and clear.

 

There is also huge potential for growth of this subject among a cohort of students who traditionally have not gone near it.  These students, who are mostly male, are the sort of students who are heading for a variety of courses on engineering or IT at third level, probably at Institute of Technology level rather than University.  Many of these students excel at the practical subjects such as DCG, Construction Studies, Engineering or Technology, and studying an applied mathematics course, almost inevitably at Ordinary Level as most of these students do Ordinary Level Mathematics, would give them a vital grounding in mathematical problem solving which would serve them very well in their later studies.  Many such courses at third level have a huge fall-out rate of students, and the biggest reason for this is their lack of ability in applied mathematics.  Over the years I have spoken to many lecturers in Engineering courses at different colleges around the country who have said they would love to make Applied Mathematics a mandatory requirement for entry to their courses, but they cannot do this because so many schools around the country do not offer it.  I deal with how to grow the uptake of the subject in a later section.  I have brought a small number of these sort of students through Ordinary Level Applied Mathematics in recent years, and I hope that number will grow in years to come.  I have started to develop a new text book “An Introduction to Applied Mathematics” based on the existing Ordinary Level course, aimed specifically at these students, and also as a way of introducing the subject in Transition Year.  It aims to introduce the key problem solving skills without getting bogged down in more complicated mathematics which would be beyond a student of Mathematics at Ordinary Level.  This project is on hold for the moment, while this syllabus review is underway, as it would obviously be futile if the Ordinary Level got scrapped in a couple of year’s time.

 

Overall I think if this subject were to become Higher Level only that would be a very negative step, which could easily kill the subject off completely, or at least make it so esoteric as to be pretty well irrelevant.  The push to go down this road must be resisted.

 

 

How should the new syllabus be assessed?

The section on ‘Perspectives on Assessment’ on pages 21 and 22, with the accompanying appendix on pages 35 and 36 is possibly the most alarming of the whole Background Paper.  It is rich in educational theory of assessment, which largely applies to the assessment of mathematics, but does not really relate to applied mathematics, and this section seems to completely miss the point altogether that applied mathematics is about problem solving, and involves almost no rote learning whatsoever, bar learning off a couple of formulae.  The approaches to assessment suggested in appendix 3 have the potential to kill the subject absolutely stone dead.  It must be stated loud and clear that a majority of the current applied maths students study the subject as an extra subject.  They see their teachers only once or twice a week, and in many cases those teachers are not being paid for their time, or if they are it is minimal.  In many schools the subject has gone off-timetable in recent years due to general cutbacks.  Many teachers have to fight to keep the subject alive in their schools, and the introduction of the huge burden of project based assessment will quite simply stop a number of schools offering the subject.  I know that a number of schools are already looking at this syllabus review very closely, with the view that if it is not handled correctly, and the uptake in their schools fall off, then the subject could go off-timetable or die completely.

 

For the past number of years I have had a substantial involvement in the student assignment part of the DCG course.  The hours that the students have to put in to this, along with their teachers, to complete a satisfactory assignment are just colossal.  My own sons have done the subject in recent years, and the hours spent in school after hours working on their assignments has been enormous.  This has definitely become an obstacle to many students taking up the subject of DCG, as the amount of time that has to be put in to the student assignment definitely takes substantial time away from their other subjects.  DCG is a practical subject, and should have an assignment component.  Applied Mathematics is about using mathematics to solve problems.  While in theory it sounds nice to have a practical / project / assignment component, the reality is very far from the theory in this case.  When I first taught in the UK, there was what was called an investigation as part of the GCSE Mathematics.  This was very much in line with the Extended Modelling and Problem Solving Task outlined on page 35.  With the best will in the world, and with some fabulous teachers and students, this was nothing more than a chore, with very little in the way of beneficial learning outcomes for the vast majority of students.

 

I do not believe there is anything fundamentally wrong with the current method of assessment.  One argument is that the syllabus is 40 years old and tired.  If you turn that argument on its head however, you have a syllabus, including its method of assessment, that has stood the test of time, and in that time has continued to grow, and is really cherished by many.  At the time I sat Leaving Certificate Applied Mathematics in the 1980’s there were about four or five hundred doing the subject.  That is now approaching 1500, and is growing year on year, with absolutely minimal support from officialdom.  The question must be asked why it is growing?  The glib answer is that students are doing it for easy points, but in reality there are far easier subjects out there for getting easy points.  The reality is that the subject has an intrinsic value as it is currently constructed.  It is not an easy subject.  It is a subject that most students find challenging, and it is in overcoming those challenges that the students find reward, and it is this reward that brings most students into the subject.  They recognise the value of the struggle to learn those precious problem solving skills, skills which stay with them for the rest of their lives, and which are transferable to a wide range of contexts.  The growth has largely been driven by some of the most fabulous teachers you will encounter anywhere in any subject, teachers with real passion and expertise.

 

From my experience with the investigation element in the UK, and also from my experience with the student assignment in DCG, the students that succeed best in these circumstances are the meticulous and diligent – the ones who dot their i’s and cross their t’s.  These students already have ample opportunities to shine within the existing Leaving Certificate subjects.  The type of student who succeeds well in Applied Mathematics is often a very different beast.  They think on the hoof, are probably a bit sloppy and messy, but have minds that think their way around a problem, and arrive at solutions to problems.  These are the people that do some of their best work “on the backs of envelopes”.  When marking an Applied Mathematics script, the best bits are often in the margins, as the student works their way around the problem to find a solution.  In my opinion any sort of project / assignment / investigation / coursework would be merely a chore to be overcome, and I do not think it provides any real opportunities for developing the problem solving skills of the students.  It has the serious potential to put many of these students off taking up the subject.

 

The suggestion is in the Background Paper that the Core would be assessed in the traditional way, with the Options being assessed using some sort of coursework component.  Since the Core as presented is largely about content, there would be little or no opportunity to assess the problem solving skills of the students in the Core.  If the Options are to be largely or completely assessed by means of coursework, this is not really assessing problem solving, but as said earlier is really assessing how meticulous and diligent the student is, and which student can literally put the most hours into the task.  If that is the outcome, something wonderful and precious has been completely and utterly destroyed, and replaced with what?  I am not sure what real skills such an approach is rewarding.  The subject has stood the test of time as it is.  Please do not destroy it.  It works as it is.  It doesn’t need fixing.  It can evolve and grow without losing what is its essential essence – problem solving and the ability of students to think their way through a problem.  The key word there is think.  Coursework is essential in practical subjects.  This is not a practical subject.  It is a thinking subject.  The assessment method is basically fine as it is.

 

 

The role of technology

I do not see a huge role for the use of technology in the subject as it is.  Obviously that could change considerably if the components on Computer Programming and Graph Theory are implemented in a certain way which requires use of computers.  As it is, this is a subject that is very much suited to traditional chalk and talk, or more likely the students’ own pencil and paper, and there is nothing wrong with that.  There is a certain role for technology in the demonstration of some of the concepts behind the content, but generally in terms of problem solving, this is best done on a traditional board, as it is the method by which the solution unfurls is the most important aspect.

 

One method of the use of technology I have explored is to make a few short video lessons, where a particular problem is worked through on paper in front of a camera.  I believe these will have a role, but I have not got the quality of the presentation up to scratch yet.  I would intend to make them available on my website, http://www.appliedmathematics.ie, when I am happy with their quality.  The main purpose of these is to assist the quite considerable number of students who study Applied Mathematics with little or no input from a teaching professional.  Many are effectively taught in isolation by a relative or friend, in various parts of the country where there are no classes available in Applied Mathematics, and I have assisted a number of these students remotely over the years.  The video lesson idea is mainly aimed at them, as via video, you can demonstrate how a solution is arrived at.  In print, it can only be presented complete really.  These videos may have a role in the traditional classroom, though I do not think they would really replace the skills of an experienced teacher.

 

 

Managing Change

One aspect of this syllabus review which must be considered very carefully is how change is to managed and implemented.  As already highlighted, there is an enormous amount of good will already in existence in the teaching of Applied Mathematics.  If there is a very steep learning curve imposed on those teachers by the new syllabus, many will probably just up sticks and stop teaching the subject, and that would be disastrous.  There is already a very steep learning curve for new teachers teaching the subject for the first time, and I have worked with a number such teachers over the years, and the open ended problem solving nature of the subject definitely creates a much steeper learning curve for the majority of new teachers than is the case with other subjects.  The reality is that many teachers come to teach Applied Mathematics with only a limited background in the subject themselves, and they must not get overloaded with unnecessary content.  I was one of the lucky ones.  I studied the subject at school myself, with an excellent teacher, and I then went on to study Engineering in University, and thus had more exposure to the subject.  At heart I am an applied mathematician.  While I welcome wholeheartedly the expansion of the subject beyond mechanics into other areas of applied mathematics, this must be done in a way that these new topics can be taken on board one at a time, bit by bit.  If there is too much new material to take on board at once, and in most of the proposed new options that would be the case for almost all teachers, then there is a serious barrier there, which may prevent it happening at all.  This is the principle reason I favour a form of assessment very much along the lines of what is currently there.  This would facilitate the gradual taking on board of new material by teachers, which they can then start to teach to their students.

 

The idea of a rolling syllabus could be considered, where the changes are introduced incrementally.  When I was teaching Cambridge IGCSE Computer Studies, the syllabus changed in small ways every year, with new additions being inserted into the syllabus in italics each year, and any deletions being blacklined.  The new syllabus was delivered about two years before the relevant examination, and any changes were very clear.  I know this is not how syllabus change has traditionally occurred in Ireland, but I do believe it should be considered.

 

 

The Teaching Council and Qualification of Teachers

I would have a serious question mark about who might be considered qualified to teach this subject by the Teaching Council if the syllabus becomes too wide.  To illustrate this point starkly, by own degree is in Engineering.  While this degree actually incorporates an ordinary degree in Mathematics, the Teaching Council only recognizes me as qualified to teach Applied Mathematics.  They do not regard me as qualified to teach, say Mathematics or Physics, of which there were very substantial parts within my degree.  With the proposed broadening of the subject outside the traditional area of mechanics, the Teaching Council may decide that since my degree does not contain any Game Theory or Business Mathematics, that I am no longer qualified to teach the subject, and if I am not qualified to teach Applied Mathematics, then pretty well nobody is.  I believe there would have to be serious consultation on this matter with the Teaching Council, both in regard to existing teachers of the subject and for new entrants.  The incorporation of Computer Programming is a particular concern in this regard.  If the subject is to become a mandatory requirement for entry into certain courses at third level, which is mooted in the Background Paper, then it my belief that the Teaching Council will almost certainly much tighter guidelines as to who is qualified to teach the subject than is currently the case, and this could be a huge obstacle.  If the Teaching Council were to apply the same rigorous conditions to Applied Mathematics as they do to Mathematics and Physics already, then the number of teachers qualified to teach the subject would be absolutely tiny.

 

 

How to broaden the appeal of the subject

One of the key concerns in the Background Paper is that the number of students taking the subject currently is very small, only about 2.5% of the cohort.  It is my sincere belief that the number of students who should be taking the subject is considerably larger than that, but the reality is that there are huge tracts of the country where there are no schools teaching the subject, and no teachers teaching it privately.  A more useful figure to gauge the potential of the subject would be to find out the percentage of students in the Cork City area, where the subject is available in the vast majority of schools, and for those where its isn’t there are a number of private teachers (including myself) available.  Cork City probably has the highest penetration of the subject nationally.

 

I have recently started teaching a weekly private class in Co. Tipperary, as it came to my notice that I think there are only two schools in the whole county teaching the subject, and no private teachers.  This situation is replicated in a number of locations and counties around the country, and a suitable strategy to spread the subject into those areas should be devised.  The situation is improving nationally all the time, with new schools coming to the subject year on year.  For example a number of years ago there was one teacher teaching a private class in Dundalk.  Now it my understanding that there are four schools in Dundalk teaching the subject, along with that private teacher.  This has all happened in the space of a couple of years, and is very heartening.  I believe a serious and co-ordinated programme of identifying blackspots, finding suitable teachers and training those teachers is required.

 

Another obstacle to be overcome in the spread of the subject is the complete misinformation put out by many guidance counsellors on the subject.  A serious engagement on the strengths of Applied Mathematics, and the sort of students who should be studying it is required with guidance counsellors.

 

With regard to the introduction of a short course at Junior Certificate, I would fully support this proposal.  A course based around the current Ordinary Level syllabus, or material of a similar standard, would introduce the ideas of problem solving without needing to cover too much extra mathematical content, and would be very appealing to a considerable section of the cohort.

 

 

Conclusion

To conclude this lengthy submission, I ask that great care be taken in this syllabus review to not try to fix what is not broken.  It is clear from the Background Paper that there are pressures coming from outside of Applied Mathematics, and that there is an attempt to foist problems or omissions from other areas into this new syllabus.  All I ask is that the fundamental question be asked of all content in the new syllabus:  is this about problem solving using mathematics?  If the answer is anything other than a clear yes, then it should be rejected.  As it is Applied Mathematics is a fabulous subject, with a talented and passionate group of teachers teaching it.  Any changes should be designed to enhance this, not destroy it.

 

 

I apologize for the length of my submission, but I have a lot to say on the subject.  I am available to assist the process in any way you see fit.  I wish you well in your deliberations.

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