Kepler, Galileo, Newton, Einstein: not a bad roll-call

The following is an edited extract from notes which I give to students before going through the derivation for the rather intimidating equation below.

You have just arrived at an equation which bookmarks a seminal moment in the history of science.


Around this time (16th century) an astronomer called Johannes Kepler discovered empirically (i.e. by analyzing data on the motion of planets) that the square of the periodic time of these planets (time for one complete orbit around the sun) is proportional to the cube of their distance from the sun.
Kepler actually stole the necessary data from a colleague, Tycho Brahe, but that’s nothing new in the world of Science. We will conveniently ignore that for now.

Later on Newton came along and was able to demonstrate this relationship mathematically, by combining a well known equation for circular motion on Earth with his own universal law of gravitation. We are about to follow in his footsteps and see exactly what he did and how he did it. Do not under-estimate the importance of this exercise (yes you have to know it for exam purposes, but that’s not why I consider it important).

This event had two very important consequences.
1. It showed that Newton’s Law of Gravitation must be valid in its own right, which was very important in securing Newton’s reputation as a giant of science, both at the time and for posterity.
2. Even more importantly, it demonstrated that ‘the heavens’ followed the same rules of science as those which operated here on Earth.
This meant that they were a legitimate area of study, and so Astronomy (which in turn led to Cosmology) was given an added respectability. Just to give a sense of what people believed at the time, Kepler had to spend much of his time during this period defending his mother of charges of being a witch.

I can think of no modern discovery which compares with this. Even if we discovered life on Mars it really wouldn’t be that big a deal. For up to this point the heavens were considered off-limits – the realm of God or the gods or whatever you’re into yourself. But now they could be shown to be just another series of objects which followed set rules, much like cogs in a complicated clock. So God was being pushed into the wings. You could see why neither Martin Luther or the Vatican Church would have been keen fans.

Kepler was following on the work of Nicolas Copernicus (known to science students down the ages as ‘copper knickers’), who in turned showed that the Earth revolved around the Sun, not the other way around.
Galileo’s run in with the Church was because he supported Copernicus’ view, so Galileo never actually made that discovery but was happy to use it to make fun of the church authority figures of the time. I think we all know how that worked out for him.

This was really the dawn of science, and progress was hindered by medieval views of the astronomers themselves. It took Kepler many years to realise that the orbit of the planets was elliptical in nature, not circular. He had assumed initially that the motion had to be circular because a circle was a perfect shape (harping back to the teachings  of Pythagoras and Aristotle, among others) and therefore would have been more pleasing to God who obviously had created the planets in the first place.

Similarly Newton, despite being heralded as one of our greatest ever scientists, spend up to 90% of his time trying to date the creation of Earth by tracing who gave birth to who in the bible.

But then Newton had another problem. He realised that Kepler was correct in stating that the planets traced out elliptical orbits, but even Newton’s equations didn’t fully match the path of the heavenly bodies; according to Newton’s equations the planets should slowly but exonerably drift from their current pathways. He couldn’t figure out why this didn’t happen – after all, his equations seemed to be perfect in every other way. And Newton believed that he was getting his ideas directly from God. Which doesn’t leave much room for admitting you made a mistake.

We now know that while Newton’s equations are very accurate, we actually need Einstein’s Theory of General Relativity to explain why they don’t precisely describe the motion of the planets.
It’s interesting to note that Newton’s explanation was that God must step in every so often to gently nudge the planets back into their preferred orbits. Now as you now know, invoking a deity to explain discrepancies in scientific observations is the antithesis of Science. So perhaps Newton wasn’t actually so mighty after all. This is partly why he is sometimes referred to as the last sorcerer rather than the first scientist.

So now we’re up to Einstein. His general theory of relativity suggested that the universe was expanding, but just like all of his predecessors he was a man of his time, and this coloured how he saw the world. It was believed at the time that the universe had always been the way it is now (this is referred to as the ‘Steady State’ theory). Einstein figured that there must be some mistake in his paper so he introduced what he called a ‘cosmological constant’ which basically amounted to a fudge factor which altered the implications of his calculations and prevented the universe from expanding.

Which was all very well until Hubble (he of the ‘Hubble’ telescope) showed that the universe was actually expanding after all.

Einstein referred to this as his greatest ever blunder.

So there you have it. This has been my attempt to put some context on the derivation that we are about to carry out. It is our chance to repeat one of the greatest moments in the history of science.

So you have two options; you can consider this exercise to be a pain in the ass or you consider it an incredible privilege to be in a position where you can follow in the footsteps of giants.

I think we know which option I go with.

And don’t be afraid to tell your parents this tonight; they may well throw their eyes up to heaven but if they do that’s a slight on them – not on you.

Listen to Keano: write only what the examiner wants to read

Back in 2013 Manchester United played Real Madrid in a Champions League match.
In the 56th minute Nani went into a tackle with his foot up high; the referee not only gave a free kick against United – he also sent Nani off.
United lost the game and as a result went out of the competition.

Afterwards the ITV commentary team were discussing whether or not the referee made the correct call. They replayed a clip of the action at normal speed, then in slow motion.
Roy Keane was on the panel but wasn’t saying much at this stage. Finally the host asked him what he thought. It’s very simple he said; the debate here shouldn’t be about whether or not the referee made the correct call – the discussion should be why Nani was daft enough to raise his leg that high in the first place. By doing so he was creating a situation where the referee was forced into making a call one way or the other. At that stage the damage was done. Nani should have known better.

Whenever I correct a test I usually get a couple of answers where it’s unclear whether or not they merit full marks.
My tendency now is not to award the marks. This highlights to the student the danger of putting the examiner in a situation akin to the referee in the story above.

And Keane’s point is just as applicable here – If at all possible avoid creating a situation where the examiner has to make a call as to whether or not to give you full marks. It may seem obvious, but if it’s the leaving cert then remember that you won’t be in the examiner’s house when he’s correcting your paper, so you won’t have the opportunity to explain to him (or her) what you meant by your answer. And even though your answer may make perfect sense to you, it may still not get full marks on the day.

The moral of the story? Give onto the examiner that which is his. If there is a standard answer to a commonly-asked question then just learn it. And make sure that this answer – and only this answer – is what you write down on the day of the exam. If you’re reading this as a parent then check that your child knows their definitions – and if they stray off course by putting things in their own words then don’t be afraid to give them a red card.
On a serious note, if their definition doesn’t make sense to you (if it doesn’t read as an english sentence should) then it probably won’t make sense to an examiner either.

As I mentioned in my last post this is all just a game.
And this is just one of the rules.
So if you want to play you have to learn the rules.
Make Keano proud.


This was the closest I could get to a video of the discussion. Unfortunately it kicks in just after Keane’s comment about Nani unnecessarily putting the referee in a situation where he had to make a judgement call.
But then again, Keane is always worth watching.

For what it’s worth, the clip also illustrates one of our inbuilt biases known in psychology as fundamental attribution error. It’s one of the most profound ideas in all of science because it tells us that our own view of reality is filtered in such a way as to make us seem to be better than we actually are. But maybe that’s for another day.

Aims and Objectives? I have but one: see science as a source of wonder

This is to serve notice that I am changing the Aims and Objectives of my Leaving Cert Physics subject plan.
The existing plan was cobbled together at short notice by copying and pasting from other schools courtesy of some nifty google searching.
But it’s pretty bland and therefore not really fit for purpose.

So what are my objectives?
Actually, there are very few:
I simply want students to appreciate science as a source of wonder.
Science, to paraphrase Feynman, does not diminish our sense of wonder – it can only enhance it.

Feynman: wonder in ScienceI want students to see science as a cultural activity – it is an integral part of what it means to be human.

The awed wonder that science can give us is one of the highest experiences of which the human psyche is capable… to rank with the finest that music and poetry can deliver.
Richard Dawkins

Science represents the best and worst of what humanity is capable of. We celebrate literature, poetry, art, dance, music as aspects of culture. We need to see Science in the same light.
And we need to stop portraying it as all good. Because it’s not. We’re on a one-way ride to global catastrophe as a result of global warming. It may well lead to the extinction of the human species in the not too distant future. And I’m pretty sure this wouldn’t have happened without Science and it’s hand-in-hand link with uncontrolled capitalilsm. But you’re not likely to see that in any school textbook.

Science tells us as much about where we have come from as it does about the world we inhabit. This must not be downplayed. In this context psychology is probably the most important of all the sciences and it is deeply unfortunate that psychology plays no part in traditional school science.

I want students to appreciate that Science not merely an accumulation of facts. The picture we portray of it in school is therefore not only incorrect but totally at odds with reality.
We should all apologise to our students for this.

Science is built of facts the way a house is built of bricks: but an accumulation of facts is no more science than a pile of bricks is a house

Do I want my students to go on and become scientists?
Not in the slightest. If they do then good luck to them, and I will help them if I can, but it’s not a priority. Does anybody seriously think that being a scientist is somehow any more noble than being a writer or a poet, an accountant or a tax official? How about a lawyer? Or for that matter a teacher?
So why should I push them in a specific direction?

Do I want to re-dress the gender balance?
Not for its own sake, no. I would like as many students as possible to appreciate the wonder of science, but I can understand why lots of girls are reluctant to take on Physics and/or Applied Maths as they are currently presented and I can’t say I blame them. Sticking up posters of token female scientists isn’t going to have much of an effect either, so please stop sending them to me.
If I’m being very honest what matters most to me is that we have enough students to justify two physics classes and one Applied Maths every year.
We get on average 40 – 45 students taking on Physics and anywhere from 15 – 24 taking Applied Maths.
So I’m happy on that score.

Do I think my students are going to become better citizens, or more informed in relation to science controversies than students who don’t do Science?
Not a hope.

Am I interested in how the students do in the Leaving Cert exam?
Yes, but really only in the sense that it’s all a game. And it’s not even my game; it’s their game.
But if I want them to play my game then it’s only fair that I play their game.

So I take both the syllabus and the past-papers apart and base the main section of my notes just on these.
And then I go off on all sorts of tangents based loosely (sometimes very loosely) on the topic at hand. But then when I’ve finished I go back and cross-check what I’ve done against the syllabus and questions from past papers and pick up the pieces that way. And I teach it just about as well as I possibly can.
I do appreciate that there are students in my class who are looking for an A1 and I know that I need to facilitate them as part of my bread-and-butter duties. And I’m happy to do so.
But I don’t stress over it. Once the students walk out of my class for the last time in May I wish them well but then take the stance that my job is done. So I don’t look at their results. In fact I believe strongly that this is actually a dangerous thing for any teacher to do. I accept that I’m in a minority here but I don’t need to see the students’ leaving cert results to find out whether or not I’m doing a good job. There are any amount of ways to find that out throughout the year, and adapt accordingly.

So that’s it.
Those are my aims and objectives or whatever the buzz phrase is these days. I see no reason to change this just for inspection purposes. If that makes me a ‘bad’ teacher in some folks’ eyes well, I guess I can live with that too.

For more recent blogposts on wonder in science see this link


Irish company Havok makes waves with its Physics engine

Havok is an Irish company.
This gives us a pretty good idea of what they are about:

Take a look at their leadership team to see the interdisciplinary nature of their combined skillset.
Havok make the news recently and the following was posted on an american physics-teachers’ forum.
I am reposting it here with permission from the poster – John Denker:

Here is a news report that mentions physics in a real-world context:

Every video game has a module called the “physics engine”.
A good physics engine is worth a bunch of money.
The price that Microsoft paid has not been announced, but I reckon it is in the neighborhood of a billion dollars. That’s based on the fact that Intel paid 110 million to acquire Havok back in 2007.

This is useful as part of the answer when students ask what physics is good for in the real world. The number of physics jobs in the computer graphics industry is not super-huge, but it’s more than the number of physics jobs at (say) CERN.

It’s also worth mentioning that most of the high-paying jobs are interdisciplinary. Expertise in physics alone is not nearly so valuable as expertise in physics /and something else/. Even more valuable is the skill we call life-long learning, i.e. being able to come up to speed in a new area quickly.

What do you do when Science equipment breaks down?

How do you tell Biology from Chemistry from Physics?
It it wiggles it’s Biology, if it smells it’s Chemistry and if it doesn’t work it’s Physics.

I don’t know of any teacher-training course which spends time training teachers on the finer points of being a technician. Yet when equipment does break down you’re expected to somehow just ‘know’ what to look for – and how to fix it.

Even better, if you’re the Physics teacher then you automatically become the ‘go-to’ guy (or gal) for colleagues (and not just Science colleagues) when they have something which needs fixing.
If you’re replacing somebody who just retired then the chances are good that this person is not going to come back in to help you become familiar with what does and doesn’t work. It’s quite possible that you’ve never even met this person, so you’re likely to spend the next few years finding pieces of apparatus in shelves without having any clue as to what their function is.
As a result the shelves in our labs are full of expensive equipment that just sits there gathering dust.

So what do you do when something breaks down?
One option which many are not aware of is to ring up your supplier of school science equipment and ask them if they can fix it. Many of them do have repair departments and should be able to give you a quote which you can then compare to the price of a replacement.
Another option is to ask a senior class if anybody there wants to have a look at it. Usually you will find somebody there who has more free time than you do (but obviously don’t allow them play with anything that could have health and safety implications).

Reduce teacher stress: Don’t look at your students’ results

Stress 39/365

Stress by Mike Hoff. CC BY-NC 2.0

Teacher stress is not something that many outside the profession think about. When it does strike we often feel that we can’t show it – certainly not to our students (which can be tough given that we need to ‘perform’ in front of them all day). But  we are also reluctant to acknowledge it to our colleagues – we are afraid of how it may be perceived. But stress is part of every single teacher’s life. My mantra for new teachers has always been to be very aware of stress creeping into your teaching and do all that you can to control and minimise it.

Eliminating stress is not an option. It is always there, lurking in the background, waiting for an opportunity to grow and fester. Some stress we may be able to do little about, but that which we can control we should.

And exam results fall into the latter category. Most teachers feel nervous coming up to exam-results day. Probably not as nervous as the students, but at least for the students it only happens once (or at most twice) whereas for teachers it is an annual event.

If it turns out that the results are not great then the specter of this hovers over the teacher all year.

Now comes the crucial question; does knowing the results increase the probability that you will become a better teacher?
How will you become that better teacher?
Now assuming (and it’s a big assumption) that you have identified what needs to be done to improve, why not just do this anyway?
By the way, I hope you don’t think that ‘working harder’ is a legitimate action. I was once in a school where we were told that we all need to work harder to improve results, without any follow-up or advice beyond that. Obviously instructions this vague only serve to increase stress and not efficiency. Maybe if our job was to dig a hole in the ground then ‘work harder’ would be self-explanatory, but teaching is a tad more complex. We need to work smarter, not harder. And usually it’s not at all obvious how to go about this.

The fact that you’re reading this blog suggests that you are interested (at least in principle) in being a better teacher.
There are many ways in which you can go about this:

  • One of the simplest and most effective is to encourage feedback from students throughout your lesson.
  • Another is to use assessment to enhance on-going learning of a concept (‘Assessment for Learning’) rather than what we usually do which is to use assessment as a dubious means of establishing whether or not a student has ‘learned’ something – whatever that means.
  • Read up on the psychology of how students learn (a relatively newand incorporate this into your teaching.

To see what else you could do, check out which was created as part of my own learning curve on this journey.

So repeat after me: “You don’t need to know your students’ results to make the decision to become a better teacher.”

Of course one other reason to look at students’ results is good old-fashioned curiosity.
You just need to offset the benefit of this against the possibility/ probability that knowing the results will introduce stress which could otherwise have been avoided and which will now most likely remain with you (albeit at a low level) throughout the year.

If you think I exaggerate then look at this extract from a recent post from Tom Sherrington – one of the most respected teachers in the UK today.

I woke up last Tuesday night at 2am with the worst headache of all time; piercing intense pain.  I had to run downstairs for the pain killers.  This was stress, pure and simple; subconscious anxiety in anticipation of GCSE results download day.   I’ve only been there a year but Results Matter – and in this age of hyper-accountability, they assume meaning far beyond the limits of their validity and reliability as measures of our students’ experience.

Now here in Ireland we don’t have the pressure of accountability that hovers over Tom and his colleagues. We should take advantage of this and not go looking for stress when it can be avoided.

Don’t get me wrong – I do need to know that my teaching is going well. I need to know that my students understand what I am teaching them. I need to know that they are revising (throughout the year, not just in the final term ‘when we have the course covered’). I need to know that they like being in my class. I need to know that they are well prepared for the final exam. But I can establish all the above without ever knowing their final mark.

And of course knowing their final mark won’t provide the answers to all the questions above anyway.

I will continue to try and improve as a teacher. I do not need to know my students’ results to do this.
And I will not judge myself on the basis of their results in state exams.


A response to the recent NCCA discussion document on Applied Maths

The NCCA have produced a discussion document entitled Draft Background Paper and Brief for the Review of Applied Mathematics. Link

ncca discussion documentThe syllabus hasn’t been reviewed since it came into existence 40 years ago so a review seems reasonable.

Two key messages feature in the report

1. The exam is so predictable in nature that it’s just regurgitation of material learnt off in class.

2. The subject needs to be removed from the curriculum and replaced with one that is more relevant.

I want to respond to the first point.
From the document:
[Students’] experience of learning in Applied Mathematics is often reduced to exam preparation with the examination regarded as predictable.
Supporting evidence:
“Almost every question which appeared in recent years was similar to at least one other question on an older paper – the natural conclusion being that if you cover all the old papers along with the recent ones you really should see very little new material in the leaving cert exam.”
The Physics Teacher website

Let’s leave aside the dismal referencing system (the quote wasn’t taken from my website – it’s from an old blog post)

As I mention in the post, I had been teaching Applied Maths for a few years using a textbook when I came across some old exam papers left by my predecessor in the school. I spotted that there was a resemblance between these and more recent questions.

It certainly did change how I teach the subject in that when looking for questions I now go to past exam questions rather than a textbook.

I wanted to pass this message on to other teachers of the subject, many of whom teach Applied Maths in complete isolation.
I have now incorporated all 40 years worth of questions from past papers from both higher and ordinary level into my class notes such that there is now an excellent scaffolding of the work from a gentle introduction using ordinary level questions right up to the most difficult of the higher level questions.

I ‘sell’ this notion that I have ‘cracked’ the system to the students. It helps create the sense that we are a team working against the system. It’s a fun idea and quite simpIy I will try try anything if it helps to make the subject easier to teach and easier to learn.

I penned that particular post deliberately to increase the uptake of students choosing the subject and to let other teachers know of the resource itself (complemented by a bank of solutions to all questions) which are freely available on my website also.

But now let’s add some context:
We’re talking about 40 years worth of questions, at higher and ordinary level, most or which are sub-divided into a part (a) and a part (b) which are completely independent of each other.
If we get through half the questions on any given topic in class we’re doing well.

Secondly, while there is a pattern to many of the questions, there is often a twist at the end of the question and there is an infinite number of ‘twists’ that can be asked, so learning off one won’t necessarily help you with the next one.

So it’s a ruse. And anyone who thinks otherwise has never taught the subject. Using the quote above out of context will create a very misleading impression of how I view Applied Maths. Doing questions from past paper can help significantly in your study but you simply cannot get an A just by doing past papers, no matter how many you do.
Anyone who thinks otherwise does not teach Applied Maths.

Now let’s compare this to Leaving Cert Physics.
Physics (and as far as I can tell, Chemistry) are subjects where you can get an A grade by learning everything off by heart. Biology even more so. There is almost no problem-solving in these subjects and there is quite a lot of choice on the paper so if you don’t like a question that requires you to think, you can always skip it and pick a different question instead.

This is not the case in Applied Maths. Thinking/ problem-solving is the name of the game and you simply cannot get an A grade by learning past questions off by heart.

I go to great pains to warn students of this from the first week in September (and even before then when I am giving a few taster classes in Transition Year). In particular I am conscious of the student who is interested in Science, is a great worker and wants to do very well in all leaving cert subjects. I tell them that hard work will most likely get them the A grade in Physics, but not in Applied Maths. To get an A in Applied Maths you need to be good at solving problems (see extract from TY booklet below).

Let’s also consider that you only need to cover past-paper questions in physics for the past twelve years (the course was introduced in 2002) to be practically guaranteed a replica question next June. In Applied Maths you have to go back forty years and the most you’re likely to see is a question similar in context, but which may well have a completely different ending, one which will cost you your A-grade if you can’t solve it.

I was aware at the time that the post caused a little stir, but was happy to leave it as it was because it helped stoke a debate which we very much need in this country about the purpose of education.

If the author of this report had taken the time and effort to see what I really thought he would have gone to website itself where he would have read the following advice which I give to interested TY students (the link to this document is on the homepage).

Who shouldn’t study Applied Maths?
This subject doesn’t suit students who just like learning things off by heart.
In fact the questions are designed to catch out those very students and whether that is fair or not is a moot point – you are being warned about it now so if you don’t like it you know what to do. You cannot come out of an Applied Maths exam and say to your teacher ‘we never did that question before –you never covered it with us in class’. It is my job to ‘train’ you to tackle problems which you haven’t come across before.

So Applied Maths suits people who like solving puzzles (we like to make it sound more impressive so we call it ‘problem solving’). This means being able to think for yourself, and because almost all of your secondary-school education encourages you to ‘learn the right answer off by heart’ it can make a lot of students uncomfortable. The ability to problem-solve is however a very important skill and is highly-valued by many employers. It is one of the reasons why you often see politicians and business people on the news saying that the country needs more scientists and engineers.

The NCCA report is scathing in its dismissal of teachers’ claims that we are teaching problem-solving skills.
From the document:
Leaving Certificate Applied Mathematics is also promoted as a subject that enhances students’ problem solving skills. However, with its emphasis on content as opposed to the development of skills and mathematical reasoning students’ [sic] are not problem solving per se but rather, learning to solve particular problem types in mathematical physics.

Where to start?
My mantra when telling students about this subject is as follows:

“Applied Maths is the one subject which teaches you what to do when you don’t know what to do.”

In other words, rather than put down your pen because you don’t know how to proceed, we teach you to look at all available information and choose the next step as if your life depended on it. You may not be 100% confident that it is the correct step, but chances are it will lead to an opening which contains a signpost which will lead you to your destination.
Yes we will instil in you an appreciation of knowing the procedure which you will need to follow for each topic, but this will only get you so far and is not to be confused with a solution which you learn off by heart like an essay you buy from The Institute. You will need to think for yourself.
Now I regard this as teaching problem-solving skills. This report thinks otherwise.

Of course there’s a problem in transferring skills learnt in one context into another area. But that’s not unique to Applied Maths. That applies everywhere in education and is a problem we have yet to find a solution to (the problem is compounded by our current mode(s) of assessment). But we get closer to real problem solving in Applied Maths than we do in any other leaving cert science subject. So if you want to develop higher order thinking it would seem strange that you choose Applied Maths as the one to criticise.

Again I refer you to just about every other subject on the curriculum in both the Junior Cert and Leaving Cert curriculums. Find me another subject that demands a similar level of analysing rather than parroting information (the new Project Maths course may indeed be an exception, but it’s still early days with that one).

I can’t for the life of me figure out who would have been responsible for compiling this report.
The assumptions made about the subject are so ridiculous that I can’t believe it was written by a teacher who ever taught the subject. But why would the NCCA ask somebody who never taught the subject to produce a report on it?
No matter what way I look at it, it doesn’t make sense.

But then I have often found the workings of the NCCA to be a mystery.

Coming back to the point made above that
“with its emphasis on content as opposed to the development of skills and mathematical reasoning students’ [sic] are not problem solving per se but rather, learning to solve particular problem types in mathematical physics.”
The author is obviously referring to the students’ lack of ability to transfer problem-solving skills. For some reason he chooses to bolster his argument with the following:
Chief examiner reports on state examinations in mathematics over a number of years have consistently pointed to the over-reliance by candidates on rote-learned procedures and the lack of deeper understanding of basic mathematics concepts. There is evidence that students are not able to apply and transfer mathematical knowledge and skills, except in the most practised way and in familiar contexts.

That’s all very well, but this is from a Chief Examiner’s report on Maths, not Applied Maths. There have been four Chief Examiner reports in Applied Maths over the years (2000, 2004, 2007 and 2012) and nowhere in any of these does it refer to a lack of problem-solving skills or an inability to ‘apply and transfer mathematical knowledge and skills’.
To imply otherwise is mischievous at the very least.

For what it’s worth, these are some of the comments/ recommendations made in those reports over the years:

The overall standard of answering was quite good, particularly in questions [. . . ] and it was similar to previous years’ standards.
“Candidates appeared to understand each question and there was little or no confusion as to what was required. “

“The regular practice of examples is an essential part of preparation for this examination.”

“Practising problems regularly is an essential part of preparation for this examination.”

“Practising problems regularly is an essential part of preparation for this examination.”

“Overall, candidates’ answering was satisfactory. In general, candidates showed a good level of ability to extract from the text of the given problems the mathematical equations necessary to lead to successful solutions.”

I regard that last comment as a vindication of our belief that we teach problem-solving. The fact that students leave the subject not being completely proficient at transferring mathematical knowledge and skills from one subject area to another may have less to do with it being redundant as a subject and more to do with the fact that this mode of learning is at odds with just about every other subject in the school. Having a go at Applied Maths in this respect is a cheap shot.
Think that’s a bit harsh? Well so’s this (taken from the Introduction):

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.

Eric Mazur is a professor of Physics and Applied Physics in Harvard University.
He knows the difference between ‘school’ problem-solving and ‘real’ problem solving better than almost anyone. But to go from the former to the latter he didn’t so much change the subject material as how he taught it. The problem in our schools today is also not what we teach – it’s how we teach it. Replacing Applied Maths with a different subject would be a step backwards, not forwards.

“Even now, if I give my students a problem on an exam that they have not seen before, there will be complaints: ‘We’ve never done a problem of this kind.’ I tell them, ‘If you had done a problem of this kind, then by definition, this would not be a problem.’ We have to train people to tackle situations they have not encountered before. Most instructors avoid this like the plague, because the students dislike it. Even at Harvard, we tend to keep students in their comfort zone. The first step in developing those skills is stepping into unknown territory.”

Recent exam papers
In the past two years the leaving cert exam simply went too far and the test ended up being too difficult for the average student. Applied Maths is a numbers game in more ways than one and we need to balance noble aspirations against the possibility of alienating potential students (who have little if any experience of thinking for themselves and may well be fearful of the prospect) from coming in the door in the first place.
And exam papers need to bear this in mind.

Already the subject is seen as ‘elitist’ by many and I suppose I have the option of using that to just attract the top students.
I choose not to.
In sixth year I have 20 students, in fifth year I have 26.

WIN_20141117_091356From the word go I make them familiar with my mantra: “Applied Maths teaches you what to do when you don’t know what to do”.

To emphasise this idea in fourth year I give students the following challenge: using only 30 paper straws and 1 metre of tape, build a free-standing tower as tall as possible such that it can hold up a marble for at least 5 seconds.
And that’s it. No other rules. There are limited resources, limited time and not necessarily any one best way. They have a ball. And then I tell them that Applied Maths is the mathematical equivalent of this.

IMG_6498This needs to be a fully thought-out discussion, listening to all interested parties, not a hatchet-job.

It may well be the case that we need other subjects on the curriculum; it’s hard to argue against the inclusion of proposed options such as computer programming, business mathematics and game theory. I’m just not sure that Applied Maths is the subject which needs to make way for this.

But whatever decision is made, let it be for the right reasons, and let that decision be made on the basis of evidence, not ignorance.

While I am a member of the Irish Applied Maths Teachers’ Association (IAMTA), all views are my own and are not necessarily representative of anyone else or any organisation.
Tomorrow the IAMTA hold their annual conference and the discussion document is on the agenda on the day.
iamta website

Hope to see some of you there.