critical thinking

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.