If you could design your own LC Maths curriculum, what would it look like?

Mr Pseudonym

As in the title, if you could design a curriculum to your own desires,what would it include?


This may be controversial, but I would remove calculus from the syllabus of the core, mandatory Mathematics subject.

The answer to the question, "Why, if one wants to study English Lit in uni, should they have to study mathematics in school?", is that skills useful to anyone are usually developed through studying maths. With that in mind, I think the core, compulsory subject of Mathematics should aim to teach those topics which can best develop cross-discipline skills of logic, critical thinking, formal approach, etc. If mathematics is the application of "tools" to the solving of logical problems, students are currently taught too much of the first quantity (i.e. different branches of maths), to the detriment of problem solving? I believe that, for one not intending to continue with a subject requiring Maths beyond school, no new skills are developed through the studying of calculus that couldn't have been developed through a less advanced topic. In fact, I believe that teaching calculus is actually detrimental for many averagely-performing students. It's not that my syllabus would be easy, just that, rather than spending time developing calculus, it would be used to apply and further explore the algebra and trigonometry that has already been taught. Those of you who are familiar with LC App maths will know that the maths topics are barely more advanced than JC (with the exceptions of Calc in Q6 and 10, Trig Formulas in Q7, and Vectors in Q2), yet, some tricky questions can be posed. I haven't thought long enough about what I think should be there instead. But, there is the possibility of including Computer Science (to enable any meaningful development, this would have to be introduced at JC), Groups, Discrete Maths topics like Game Theory, Logic, etc. I would base statistics on the non-calculus AP Statistics course. The study of Linear Motion could help to fuse graph theory and algebra. A crucial change would also be the method of learning. Project Maths has developed some of that, but, to name one shortcoming, it doesn’t have any ICT involvement (I use the outdated inititalism to make clear that I mean separate from Computer Science).

Obviously, not having calculus on the syllabus would greatly disadvantage talented students and those wishing to continue to "STEM" degrees in third-level. To facilitate those, as well as the core maths course, I would have an optional subject called Further Maths, similar to the case with A-level. This subject would not only introduce calculus, but develop it to a greater extent than it currently is (not hard!). Also included could be complex numbers, linear algebra, and application could be accommodated in a range of modules - mechanics, statistics, etc.

As an aside, I don’t think App Maths could continue to exist under my Maths/F Maths model, and while some content would be accommodated (Linear Motion to Maths; SHM and Diff Equations to F Maths; and Proj, Dynamics, Circ Motion, Collisions, and more advanced LM to Physics) the rest would be dropped.

In conclusion, my syllabus aims to enable those engaged and capable in mathematics to leave school having studied to a significantly higher level, while allowing those who are not engaged to study mathematics which will benefit and may even inspire them.

I would be greatly interested in anyone’s opinions or criticisms.


Edit: I should probably say that, though I was a successful Maths/AM second-level student, I am certainly not a mathematician. Therefore, some of the topics I suggest may be inappropriate, etc

equivariant

Mr Pseudonym wrote: »

As in the title, if you could design a curriculum to your own desires,what would it include?


This may be controversial, but I would remove calculus from the syllabus of the core, mandatory Mathematics subject.

I agree completely. I have spent many years teaching mathematics at various levels (both second and third level) and I have become more and more convinced that calculus should be hugely deemphasised at all levels up to and including first year university. I think that our current syllabus is a product of 19th century and early 20th century society when the ability to be able to differentiate and integrate by hand was much more important than it is now. We spend so much time trying to make calculus digestible by the majority that we largely ignore areas that have become much more important recently such as graph theory and discrete mathematics.

Blackpanther95

Adding in the further maths would disadvantage people not good at maths. I think it would be easier to get into Medicine if you were good at maths as if you got an A1 in further maths, the A1 in regular maths would be simple. It works well in England because you can take less subjects. Also the calculus is probably easier than a tricky algebra question. If someone wants to be a lawyer or some other non-STEM career its probably easier to learn of calculus than do a tricky algebra question. In fact some people I know who are studying law think that college maths is easy because in the LC the maths they did was mostly about memorizing and learning off rules.

Mr Pseudonym

Blackpanther95 wrote: »

Adding in the further maths would disadvantage people not good at maths. I think it would be easier to get into Medicine if you were good at maths as if you got an A1 in further maths, the A1 in regular maths would be simple. It works well in England because you can take less subjects. Also the calculus is probably easier than a tricky algebra question. If someone wants to be a lawyer or some other non-STEM career its probably easier to learn of calculus than do a tricky algebra question. In fact some people I know who are studying law think that college maths is easy because in the LC the maths they did was mostly about memorizing and learning off rules.

I think there are greater obstacles than the one you propose.

It could be argued that mathematically-inclined students are disadvantaged at present by having to take three linguistic subjects.

Although not quite a parallel with the double A-level award, taking App Maths renders most of the mathematical contents of Physics laughably simple.

I'm not sure what you mean by the calculus being easier than a tricky algebra question. Surely you're not suggesting that calculus is not a challenging topic?!

It's beyond question that LC maths is easier than college maths.

Iwasfrozen

Well it would have matrices and complex numbers on it that's for sure!

I'd also like to introduce more statistics onto the curriculum, probability densities and moment generating functions though the latter may be a tad advanced.

Mr Pseudonym

Between a quarter and a third of students sat the Higher paper last year. Only about a third of those got A1-B3 - anything less than that would suggest poor ability, IMO. The extent to which more advanced content can be added to the curriculum is limited by that.

eoins23456

Mr Pseudonym wrote: »

I think there are greater obstacles than the one you propose.

It could be argued that mathematically-inclined students are disadvantaged at present by having to take three linguistic subjects.

Although not quite a parallel with the double A-level award, taking App Maths renders most of the mathematical contents of Physics laughably simple.

I'm not sure what you mean by the calculus being easier than a tricky algebra question. Surely you're not suggesting that calculus is not a challenging topic?!

It's beyond question that LC maths is easier than college maths.

In first year college maths I would say they are on par. After that college maths is harder

Mr Pseudonym

eoins23456 wrote: »

In first year college maths I would say they are on par. After that college maths is harder

I'm surprised you say that, and presume you are/were a Maths student to be able to be so confident. I did a year of Science in Trinity, and the Maths in that, while markedly inferior to what was in Single Honours, was certainly more challenging (at least, the calculus was). If I remember correctly, SH Maths has over one hundred hours of calculus in first year; do you really think the content/difficulty/etc is on par with the current maths syllabus?

eoins23456

I think trinity might have one of the most difficult first years for maths heads. I've heard it goes fairly into detail. From my first year of maths science anyway I wouldn't rate any of the maths higher. I would say the calculus is on par as we'll as the statistics.

Also I wouldn't get rid of calculus. I would change the way it is taught though. It would be nice if you could build up your knowledge of calculus with maths programs such as mat lab or maple where you could use differential equations to solve simple world life problems and be able to graph every step of what you are doing so a student can view what they are doing as they are doing it. Learning off a load of differentiating rules as is currently done doesn't make much sense.

Mr Pseudonym

eoins23456 wrote: »

I think trinity might have one of the most difficult first years for maths heads. I've heard it goes fairly into detail. From my first year of maths science anyway I wouldn't rate any of the maths higher. I would say the calculus is on par as we'll as the statistics.

Also I wouldn't get rid of calculus. I would change the way it is taught though. It would be nice if you could build up your knowledge of calculus with maths programs such as mat lab or maple where you could use differential equations to solve simple world life problems and be able to graph every step of what you are doing so a student can view what they are doing as they are doing it. Learning off a load of differentiating rules as is currently done doesn't make much sense.

I respect your opinion, but disagree. For a variety of reasons (maturity, incentive to work, etc) university mathematics may not prove more challenging than LC maths. But, it is my opinion that it is certainly more technical, more involved, and for those, more difficult than LC maths.

Certainly, programmes like Math lab and Wolfram Alpha should definitely be incorporated into LC syllabus.

eoins23456

Mr Pseudonym wrote: »

I respect your opinion, but disagree. For a variety of reasons (maturity, incentive to work, etc) university mathematics may not prove more challenging than LC maths. But, it is my opinion that it is certainly more technical, more involved, and for those, more difficult than LC maths.

Certainly, programmes like Math lab and Wolfram Alpha should definitely be incorporated into LC syllabus.

I don't really see it as more technical. It gets more technical in second and third year when you are really delving into pure maths and statistics.

Mr Pseudonym

eoins23456 wrote: »

I don't really see it as more technical. It gets more technical in second and third year when you are really delving into pure maths and statistics.

Can I just clarify, what do you mean by not being more technical: the content is no more advanced or it is a natural progression from LC?!

In truth, I'm quite surprised that you're taking either position. Here are Trinity's Calculus papers from last year: 1 and 2; and here is last year's LC paper.

Obviously (!), one would expect the university maths papers to be substantially more involved. And I think that is borne out in the comparison I give.

Blackpanther95

College maths is objectively harder than LC maths, obviously. IMO calculus questions on the LC are much easier, its mainly just about being able to apply the rules that you have learned, whereas algebra isn't always that simple. Also statistics isn't a necessary element of mathematics, I do maths and do computers and applied maths instead of statistics, the core elements of modern mathematics are algebra and analysis. In older times it was geometry and algebra. There are many top mathematicians who didn't care for statistics. I don't see why people emphasise statistics so much in recent times.

Mr Pseudonym

Blackpanther95 wrote: »

Also the calculus is probably easier than a tricky algebra question. If someone wants to be a lawyer or some other non-STEM career its probably easier to learn off calculus than do a tricky algebra question. In fact some people I know who are studying law think that college maths is easy because in the LC the maths they did was mostly about memorizing and learning off rules.

Blackpanther95 wrote: »

College maths is objectively harder than LC maths, obviously. IMO calculus questions on the LC are much easier, its mainly just about being able to apply the rules that you have learned, whereas algebra isn't always that simple. Also statistics isn't a necessary element of mathematics, I do maths and do computers and applied maths instead of statistics, the core elements of modern mathematics are algebra and analysis. In older times it was geometry and algebra. There are many top mathematicians who didn't care for statistics. I don't see why people emphasise statistics so much in recent times.

I don't really understand what your position is. Do you not share your friends' beliefs?

When you say the calculus is harder than algebra, are you saying that the distinct Further Maths subject I proposed wouldn't be as hard?

Regarding the last: I imagine it's because analysing large sets of data is becoming increasingly useful. Statistics is applied to public policy, etc, so understanding basic statistical results is therefore important to the average citizen.

Blackpanther95

Mr Pseudonym wrote: »

I don't really understand what your position is. Do you not share your friends' beliefs?

When you say the calculus is harder than algebra, are you saying that the distinct Further Maths subject I proposed wouldn't be as hard?

Further maths doesn't need to be harder just extra. Much in the same way as solving differential equations need not be harder than linear algebra. However the case of having two subjects in the one area is unprecedented. I think all language and arts teachers would object whereas barely any mathematicians would object to the teaching of more mathematics. I was good at maths in school and would have probably done the proposed further maths rather than economics (which is what I did). However now that I do maths in college I actually appreciate the other subjects I did, especially economics so...

Mr Pseudonym

Blackpanther95 wrote: »

Further maths doesn't need to be harder just extra. Much in the same way as solving differential equations need not be harder than linear algebra. However the case of having two subjects in the one area is unprecedented. I think all language and arts teachers would object whereas barely any mathematicians would object to the teaching of more mathematics. I was good at maths in school and would have probably done the proposed further maths rather than economics (which is what I did). However now that I do maths in college I actually appreciate the other subjects I did, especially economics so...

It needn't be harder, but it would be! In case you're interested, these are examples of the three core A-level Further Maths papers (students also take either three or four modules in applied maths - in statistics and/or mechanics): FP1, FP2 and FP3.

What you say about it being unprecedented is categorically false:

According to this paper which analyses a number of foreign curriculums:

• "After finishing middle school South Korean students move to high school and can take a number of different mathematics courses. The highest level of these is called Mathematics 11"
  
• "A key feature of (Canadian) curricula is that they require students who wish to pursue studies in engineering and science to take particular courses designed to support their mathematical education need at third level"
  
• "In Singapore students who wish to pursue engineering and science must take a course called H2"
  
• "While there is a roughly equivalent syllabus to Project Maths in existence in the Netherlands, there is also a technological or engineering mathematics syllabus, including calculus, linear algebra, vectors etc., which is compulsory for students wishing to study the sciences and engineering"

As well as the above, the IB Diploma has an equivalent, US students can take Advanced Placement exams, and, as previously mentioned, there is the English/Welsh/NI A-level.

I'm not sure that citing having enjoyed or valued Economics is a good argument against my proposal.

Blackpanther95

Mr Pseudonym wrote: »

It needn't be harder, but it would be! In case you're interested, these are examples of the three core A-level Further Maths papers (students also take either three or four modules in applied maths - in statistics and/or mechanics): FP1, FP2 and FP3.

What you say about it being unprecedented is categorically false:

According to this paper which analyses a number of foreign curriculums:

• "After finishing middle school South Korean students move to high school and can take a number of different mathematics courses. The highest level of these is called Mathematics 11"
  
• "A key feature of (Canadian) curricula is that they require students who wish to pursue studies in engineering and science to take particular courses designed to support their mathematical education need at third level"
  
• "In Singapore students who wish to pursue engineering and science must take a course called H2"
  
• "While there is a roughly equivalent syllabus to Project Maths in existence in the Netherlands, there is also a technological or engineering mathematics syllabus, including calculus, linear algebra, vectors etc., which is compulsory for students wishing to study the sciences and engineering"

As well as the above, the IB Diploma has an equivalent, US students can take Advanced Placement exams, and, as previously mentioned, there is the English/Welsh/NI A-level.

I'm not sure that citing having enjoyed or valued Economics is a good argument against my proposal.

Yes I meant unprecedented in Ireland. I mean whose to say that others might not argue that there should be a further English and a advanced Gaeilge, or maybe splt French into oral and written etc. I mean what good is Shakespeare to a mathematician or engineer? My point regarding economics was that by introducing a further maths, you may entice potential mathematicians away from doing subjects which they may find beneficial or worthwhile in favor of an almost early specialization. Yes I'm familiar with the A-level standard for further mathematics. However I'm not sure if teachers in Ireland would even be able to teach that sort of stuff. My teacher used to struggle to teach the regular old stuff. I do like the idea of further mathematics being available to students but I think that the points system should be readjusted so further mathematics would only be beneficial (points-wise) to those wishing to pursue mathematics in university.

Mr Pseudonym

Blackpanther95 wrote: »

I mean who's to say that others might not argue that there should be a further English and a advanced Gaeilge, or maybe splt French into oral and written etc? I mean what good is Shakespeare to a mathematician or engineer? My point regarding economics was that by introducing a further maths, you may entice potential mathematicians away from doing subjects which they may find beneficial or worthwhile in favor of an almost early specialization. Yes I'm familiar with the A-level standard for further mathematics. However I'm not sure if teachers in Ireland would even be able to teach that sort of stuff. My teacher used to struggle to teach the regular old stuff. I do like the idea of further mathematics being available to students but I think that the points system should be readjusted so further mathematics would only be beneficial (points-wise) to those wishing to pursue mathematics in university.

The argument for Further Maths is that the gap between the highest and lowest performing students is greater than that of any other subject, and having a one-size-fits-all curriculum is detrimental to both extremes.

I pre-empted a parallel to the Shakespeare question when I said, "The answer to the question, "Why, if one wants to study English Lit in uni, should they have to study mathematics in school?", is that skills useful to anyone are usually developed through studying maths." I think it's a non sequitur anyway because those taking F Maths would still be required to take the core subjects.

There's an opportunity cost, as well, to taking Economics: stunted mathematical education or not having taken another subject (say, Art).

A practical criticism such as one regarding unsuitability of teachers is not in the spirit of this discussion!

I agree with your point about a flaw in the CAO, but that exists to some extent at present: an English friend was flabbergasted when I told her that one applying for, say, Physics needn't have the Physics grade contribute to their points-total - i.e. it could have been the seventh best subject. There would be several ways around what you mentioned - one of which would be that for one taking F Maths, the core Maths subject only counted as half a subject.

Blackpanther95

Mr Pseudonym wrote: »

The argument for Further Maths is that the gap between the highest and lowest performing students is greater than that of any other subject, and having a one-size-fits-all curriculum is detrimental to both extremes.

I pre-empted a parallel to the Shakespeare question when I said, "The answer to the question, "Why, if one wants to study English Lit in uni, should they have to study mathematics in school?", is that skills useful to anyone are usually developed through studying maths." I think it's a non sequitur anyway because those taking F Maths would still be required to take the core subjects.

There's an opportunity cost, as well, to taking Economics: stunted mathematical education or not having taken another subject (say, Art).

A practical criticism such as one regarding unsuitability of teachers is not in the spirit of this discussion!

I agree with your point about a flaw in the CAO, but that exists to some extent at present: an English friend was flabbergasted when I told her that one applying for, say, Physics needn't have the Physics grade contribute to their points-total - i.e. it could have been the seventh best subject. There would be several ways around what you mentioned - one of which would be that for one taking F Maths, the core Maths subject only counted as half a subject.

Ye the CAO points thing is the real issue. Further maths is a good idea but do you really need points to reward you? Most maths courses in third level are really easy to get into so I'm not so sure. If you ask me every third level course should have a characteristic way of calculating points and thus entry requirements.

Mr Pseudonym

Blackpanther95 wrote: »

Ye the CAO points thing is the real issue. Further maths is a good idea but do you really need points to reward you? Most maths courses in third level are really easy to get into so I'm not so sure. If you ask me every third level course should have a characteristic way of calculating points and thus entry requirements.

Certainly, the equivalence of subjects is preposterous: while it's probably wrong to compare the merits of an A1 in, say, Maths and English, History is patently more challenging than Geography, App Maths more challenging than Physics. A system that doesn't reward those who take more difficult subjects encourages what we now see - the predominance of mainly learning-based subjects like Biology, Geography and Business (three most popular non-core subjects). But, for the CAO system to be credible, I think it must treat all subjects equally. A system where grades in different subjects were attributed different points couldn't work. Having each course calculate points differently would complicate the system hugely. And it throws up lots of questions regarding whether appropriateness of a subject ought to trump difficulty: should a Physics course give more merit to LC Biology than LC English, even though the latter is certainly more challenging than the other (if only for there being two papers), and Biology, though a Science, isn't especially relevant to Physics; what would happen in the cases where a school doesn't offer a wide-range of subjects (and, in these cases, they are likely to offer the less challenging subjects); lack of uniformity - an A1 in App Maths is definitely more worthy than one in Physics, but is a C1?

I think this will be solved in two ways: the different admission routes that third-level colleges are discussing (and TCD is trialling) and the potential introduction of broad freshman years in university (i.e. where one applies to Arts or to Sciences, rather than to a course, and then decides their major later).

Blackpanther95

Mr Pseudonym wrote: »

I think this will be solved in two ways: the different admission routes that third-level colleges are discussing (and TCD is trialling) and the potential introduction of broad freshman years in university (i.e. where one applies to Arts or to Sciences, rather than to a course, and then decides their major later).

Ye the broad entry is a great idea. I think they should do the same for medicine and bio-medicine aswell... but thats another issue (please don't discuss).
To wrap up- I believe further maths is an excellent idea but simply superimposing it on the already inadequate system won't solve most of the problems pertaining to second level education. I think the CAO and universities need to go back to the drawing board and get rid of this almost video-games like points scoring counting and not counting etc of subjects, and bring in an apt way of discerning between 3rd level applicants, that takes into consideration alot more than the current system.

Mr Pseudonym

Blackpanther95 wrote: »

Ye the broad entry is a great idea. I think they should do the same for medicine and bio-medicine aswell... but thats another issue (please don't discuss).
To wrap up- I believe further maths is an excellent idea but simply superimposing it on the already inadequate system won't solve most of the problems pertaining to second level education. I think the CAO and universities need to go back to the drawing board and get rid of this almost video-games like points scoring counting and not counting etc of subjects, and bring in an apt way of discerning between 3rd level applicants, that takes into consideration alot more than the current system.

The question of the thread was a purely theoretical one. It was you who kept imposing reality on it! While I agree there would be several obstacles, I think you over-state the challenges of implementing it (were that the purpose of this thread).

In truth, almost everything needs reform: the curriculms, the exams, the third-level entry procedure, etc.