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That Scottish Higher Maths Paper 1…

Posted in Education, mathematics with tags , , , on May 15, 2026 by telescoper

Talking of Examinations, I saw an article on the BBC website about a recent Higher Maths paper in Scotland which has generated complaints and a petition because it was allegedly unfair. The introduction to the petition states:

This is not a complaint that the paper was too hard. Students expect to be challenged. The problem is that the 2026 Higher Maths Paper 1 used language and phrasing that was confusing, ambiguous, and inconsistent with every past paper students had revised from. Questions were not simply difficult — they were worded in ways that made it genuinely unclear what was being asked.

Past SQA Higher Maths papers have followed a recognisable style: clear command words, standard notation, and questions that test understanding rather than the ability to decode unusual phrasing. The 2026 Paper 1 departed from this in ways that penalised well-prepared students simply because the wording did not match the conventions they had been taught to expect.

Numerous other news outlets have covered the story too. It is frustrating that most of the pieces focus on what people said about the paper but don’t actually include a link to the paper itself, making it impossible to make your own mind up.

So you can make your own mind up here is a scan of the actual paper (obtained from here):

2026_Higher_Maths_Paper_1_ScanDownload

Bear in mind that the Scottish examination system is not the same as in England & Wales – the “Highers” are not as advanced as A-levels and are more similar to the Irish Leaving Certificate.

My opinion, for what it’s worth having neither taught nor studied in the Scottish system, is that there is nothing out of the ordinary with this paper. There is a lot to do in just 75 minutes – for 12 questions that’s just over 6 minutes a question. I don’t like examinations that are speed tests.

That said, the questions look well structured and the “command words” are without exception on the list here. Some questions are easy and others harder: I think Question 12 is the most difficult, but I think that’s intentional – to stretch the stronger students. The only thing I would quibble with is the wording of 11(a) (ii):

The second sentence is redundant. How can one possibly “explain why” without giving “a reason”? The reason is basically that the quadratic remaining after you have taken out the factor (x+2) does not factorize.

I looked at the 2025 Paper 1 and it seems a similar level, though the questions are phrased in a terser fashion. Here it is for reference:

NH_Mathematics_Paper1-Non-calculator_2025Download

There may well be context that I’m missing, however, so I’d welcome comments on the diffculty and/or fairness through the box below.

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John Venn’s Bowling Machine

Posted in Cricket, mathematics with tags , , on April 14, 2026 by telescoper

Most readers of this blog will be aware of the existence of Venn Diagrams, but not a lot of people know that their creator, John Venn, invented a mechanical bowling machine for use by cricketers in batting practice. Recently some folks at Cambridge University built a replica of this machine, presumably with the aid of some sort of diagram, and here’s a fascinating video about it.

Modern bowling machines fire the ball at you like a cannon, but this machine has the virtue of the batter being able to see the arm coming over which must help a little. This creation is also able to impart quite a bit of spin, in either direction but it looks like it should be quite easy to pick. The examples shown in the video also demonstratte that batting is only partly about hand-eye coordination – a key element is how to move your feet!

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Ahead of a Four-Day Week…

Posted in Artificial Intelligence, Biographical, Education, mathematics, Maynooth with tags , , , , , , on March 29, 2026 by telescoper

It’s Sunday 29th March – Palm Sunday in fact – and Friday 3rd April is Good Friday, which is followed by a break of a week for Easter, so I’ve been looking at what I have to do in the four days between Monday and Thursday.

On Tuesday afternoon part of my 3rd Year Computational Physics class will be doing a supervised test in the computer lab. I (foolishly) promised to ensure they would get their grades before Easter, so I’m going to have to mark them straight away. This is a larger group than usual because some students who would normally be in the lab on Thursdays swtiched to Tuesday so they could go on a trip to Armagh. Anyway, this is the third lab test and at least I have graded the first two tests for all groups in time for the arrival of the new ones.

There will be one more of these lab tests after the Easter break but after that the students will be working full-time for 3 weeks or so on mini-projects. That is the part they usually enjoy most and I’m very happy to see that some have already started work.

Then, on Wednesday I have the second class test for my 4th year Particle Physics module. This is the second such test, and it will be held during a tutorial session. This is a pen-and-paper test rather than a coding test to be done in the lab. For such tests I allow students to bring whatever they like on paper but phones, laptops and tablets are banned. This is the easiest way I could think of to avoid students using AI to solve the problems. In previous years I gave take-home assignments for this module, and I still hand out exercise sheets to be gone over in tutorials, but these are for formative purposes only. The summative assessments are the class tests. There will be three of those, which means they will have to endure one more after Easter. In a normal week I would have a Particle lecture on Friday, but that won’t happen because it’s Good Friday and my lectures apparently aren’t good enough to happen on that day.

As well as the Computational Physics lab test and the Particle Physics class test, next I have two lectures, both at 9am – one on Tuesday and one on Thursday – and another lab session on Thursday which is not a test, but a practical session about solving ODEs.

Then it will be the Easter Break. After that, according to my calculations, there will be four more teaching weeks before the examination period. The last day of teaching is May 8th. Between that and the examinations there is a gap of a week during which I will have to mark all the completed Computational Physics project reports, as well as giving some revision classes if there is demand for them.

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A Quarter Term

Posted in Biographical, Education, mathematics, The Universe and Stuff with tags , , on February 20, 2026 by telescoper
I searched for a free stock image using “quarks” as a search term and found this photo by Anh Nguyen on Pexels.com.

On most Friday afternoons there is a seminar in the Physics Department at Maynooth University, and I got it into my mind that there was one this afternoon and set aside an hour to attend it. It turns out that there isn’t a seminar so I have time to write a quick blog post before I head to Dublin for a concert.

We’ve (almost) reached the end of Week 3 of the Semester which means we’re about a quarter of the way through the term. Yesterday we did the first Lab Test of four in Computational Physics, which went off without any major problems. The first class test in Particle Physics will be next week and I hope that goes equally well.

Today’s the day students officially received their provisional first semester results. When I arrived at my particle physics lecture this morning the students were discussing their exam marks. Most seemed relatively happy, which is good because this is the final year for most of these students so their grades matter more now than in previous years.

This morning’s lecture was quite amusing. I was discussing electrostatic interactions between quarks in nucleons and in the course of that I asked the class to calculate (2/3)×(-1/3) +(-1/3)×(-1/3)+(-1/3)×(2/3). It took a surprisingly long time to arrive at the right answer! To make matters worse, when I announced the correct answer I got the sign wrong*.

It’s been a long week.

Anyway, next week I’ll be starting on the Dirac Equation, and on the basis of today’s events I wonder about the wisdom of having a lecturer who can’t do minus signs teach relativistic quantum mechanics to students who struggle to do simple arithmetic with fractions!

*ANS=-1/3

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The Next Semester

Posted in Artificial Intelligence, Education, mathematics, Maynooth with tags , , , , , , , on January 26, 2026 by telescoper

There’s just a week to go before the next Semester at Maynooth University so I’ve been looking at my calendar for the weeks ahead. Actually, I won’t start teaching again until Tuesday 3rd February, because Monday 2nd February is a national holiday. As it turns out, however, I don’t have any lectures, labs or tutorials on Mondays anyway so I won’t be missing a session either on February 2nd or on May 4th, another holiday. I will have to miss one on Friday 3rd April (Good Friday), though.

The Timetable has given me two 9 o’clock lectures a week for the forthcoming Semester, one on Tuesdays and the other on Thursdays. I don’t think the students like 9am lectures very much, but I don’t mind them at all. I find it quite agreeable to have accomplished something concrete by 10am, which I don’t always do. This schedule might mean that I defer publishing papers at the Open Journal of Astrophysics on those days. I usually do this before breakfast, but I might not have time if I have to be on campus and ready to teach for 9am.

As usual, Semester 2 is a stop-start affair. We have six weeks until the Study Break, which includes the St Patrick’s Day holiday, then we’re back for two weeks (minus Good Friday) before another week off for Easter. We return on Monday April 13th to complete the Semester; the last lectures are on Friday 8th May and exams start a week later. This arrangement creates no problems for lecture-based teaching, but it takes some planning to organize labs and project deadlines around the breaks. I’ll have to think about that for my Computational Physics module.

A more serious issue for Computational Physics is how to deal with the use of Generative AI. I’ve written about this before, in general terms, but now it’s time to write down some specific rules for a specific module. A default position favoured by some in the Department is that students should not use GenAI at all. I think that would be silly. Graduates will definitely be using CoPilot or equivalent if they write code in the world outside university so we should teach them how to use it properly and effectively.

In particular, such methods usually produce a plausible answer, but how can a student be sure it is correct? It seems to me that we should place an emphasis on what steps a student has taken to check an answer, which of course they should do whether they used GenAI or did it themselves. If it’s a piece of code to do a numerical integration of a differential equation, for example, the student should test it using known analytic solutions to check it gets them right. If it’s the answer to a mathematical problem, one can check whether it does indeed solve the original equation (with the appropriate boundary conditions).

If anyone out there reading this blog has any advice to share, or even a link to their own Department’s policy on the use of GenAI in computational physics for me to copy adapt for use in Maynooth, I’d be very grateful!

(My backup plan is to ask ChatGPT to generate an appropriate policy…)

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Marking Over

Posted in Artificial Intelligence, Biographical, mathematics, Maynooth with tags , , , , on January 19, 2026 by telescoper

Well, that wasn’t too painful. I’ve completed my marking duties. The fact that it has been pouring with rain most of the day made it easy to concentrate on this task. I was going to have a break for lunch, but I decided to keep on going until I was finished, though I did have to take a break for a telecon this afternoon. I also had to dash out to the shops, primarily to replenish my stock of food for the garden birds but also to get some groceries for myself. Having skipped lunch I bought myself something nice for supper.

Looking at the departmental database I see that I appear to be the first member of staff to have finished and uploaded all their Semester 1 examination marking. Normally I’m just happy if I’m not the last!

It feels good to have finished this task. It’s definitely a weight off my mind. I wouldn’t want to leave any loose ends when Trumpageddon comes.

I can’t say anything about the results of course but the change I made to continuous assessment, from take-home assignments to class tests, does not seem to have had a negative effect on either group of students I have been teaching. The opposite may indeed have been the case, as the class tests perhaps provide better preparation for the final assessment than the previous method. I think some other lecturers might make a similar switch in future. Anyway, I definitely plan to do something similar for my Semester 2 module on Particle Physics.

Now I have a couple of weeks before teaching resumes so I can get on with other things. For the rest of this week my priority is to finish revising a paper that I hoped to do before Christmas. I’ll see how that goes before deciding what to do next.

I’ll also have to prepare teaching for Semester 2. That shouldn’t be too difficult, as I’ve taught both modules before, but I do have to give some thought as to precisely how I’m going to word the instructions on the use of AI for my Computational Physics module. That can wait a little while, though, as it mainly affects the mini-project to be done towards the end of the Semester. In the meantime I’ll be thinking about other things…

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Marking Progress

Posted in Biographical, mathematics, Maynooth with tags , , , on January 16, 2026 by telescoper

The last day of a week dominated by examination marking found me briefly back on campus to return the batch of scripts I have finished corrrecting and collect the next set (which, happily, is much smaller):

There are 14 scripts in the pile for my second paper, for my 4th year Mathematical Physics module on Differential Equations and Complex Analysis, around one-third of those for my Engineering Mathematics module Differentiaol Equations and Transform Methods. Based on the total number of examinations I have to mark I am therefore now 50% complete, but based on the number of scripts I’m about 75% through. I should be able to finish the latest batch in a day, but there’s no desperate rush so I’ll do them on Monday. I’m not going to start them now as I am off a concert – my first of 2026 – this evening and I prefer not to work at weekends unless I absolutely have to.

I finished the first set of marking yesterday, and spent most of this morning uploading and checking the scores and the conflation of exam marks with coursework scores. Satisfied that all is OK, I returned the scripts to the office for storage until our Examination Board meeting in about 10 days. I wasn’t on campus long, but there was a fire alarm in the Science Building while I was there. As usual, it turned out to be a false alarm.

Anyway, I should be finished with examination matters by Monday evening, which gives me four days next week to get on with other things. I’m looking forward to the change.

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Everything is a Simple Harmonic Oscillator

Posted in mathematics, The Universe and Stuff with tags , , , on December 13, 2025 by telescoper

Anyone who has studied theoretical physics for any time will be familiar with the simple harmonic oscillator, which I will call the SHO for short. This is a system that can be solved exactly and its solutions can be applied in a wide range of situations where it holds approximately, e.g. when looking at small oscillations around equilibrium. I’ve often remarked in lectures that we spend much of our lives solving the SHO problem in various guises, often pretending that the difficult system we have in front of us can, if looked at in the right way, and with sufficient optimism, be approximated by the much simpler SHO. Cue the old joke that if all you have is a hammer, everything looks like nail…

That rambling prelude occurred to me when I found this little problem in some old notes. It is a cute mathematical result that shows that the Friedman equations that underpin our standard cosmological model can in fact be written in the same form as those describing a Simple Harmonic Oscillator. In what follows we take the cosmological constant term to be zero.

The resulting equation is the SHO equation if k>0. I’m not sure whether this result is very useful for anything, but it is cute. It also goes to to show that, if looked at in the right way, the whole Universe is a Simple Harmonic Oscillator!

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Nine Teaching Days to Christmas

Posted in Biographical, mathematics, Maynooth, Television with tags , , , on December 8, 2025 by telescoper

This week is the penultimate week of teaching term at Maynooth and, as usual at this stage of the Semester, we’re getting busier and busier. The examinations for January have been sent off for printing and are (presumably) ready to go, and I’m up to date with all my coursework gradin so I am, miraculously, on schedule as far as teaching is concerned. I should finish covering the respective syllabuses by Tuesday 16th, with the remaining teaching sessions devoted to revision. I don’t have any lectures on Fridays this term, so my teaching ends, a day before the end of term, on Thursday 18th December. To celebrate the end of term I’ll be presenting the students in the last session of my Engineering Mathematics module the gift of a final Class Test. I’m not sure when I’ll get to correct it. Oh, and our Department Christmas Dinner is on Wednesday 17th.

I’ll soon have to decide when to pause the publication of new papers at the Open Journal of Astrophysics and prepared for next year’s Volume 9. A certain person is insisting that I take a complete break for at least a week, so I think we’ll probably stop on Christmas Eve and begin again in the New Year.

In the meantime, term goes on. I have three lectures to get ready for tomorrow. Incidentally, my mid-Semester feedback suggested that I start each lecture with an introduction to say what I’m going to be covering. Here are some examples of what I’ve been doing in response:

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A month to go

Posted in Artificial Intelligence, Biographical, Education, mathematics, Maynooth with tags , , , , , on November 25, 2025 by telescoper

I’ve been a bit preoccupied these recent weeks so it was with a shock that I realised that we’re into Week 9, which means just four weeks (including this one) until the end of term and just a month before Christmas. Teaching finishes here in Maynooth on Friday 19th December, but I don’t have any lectures on Fridays so in my case it will finish the day before (with a tutorial). I don’t know how many students will be there, but the module concerned is my 4th year Mathematical Physics module and the students are very hard-working, so I think most will attend. After such a busy term I’m sure that they will need a break as much as I will.

I had to rejig the schedule for both modules I am teaching this semester to accommodate the introduction of in-class tests to replace take-home assignments (for reasons I outlined here). I’ve also been handing out voluntary exercises for practice, not counting towards the module mark but for formative reasons. Both modules are mathematical in nature, and I think the best way to learn mathematics is by doing it…

Despite the changes with respect to last year, I am still roughly on track. In my Engineering Mathematics module I’ve just finished Laplace transforms, and will start Fourier methods tomorrow. With the mathematical physicists, I am in the middle of complex analysis, having done complex differentiation and conformal mappings and starting complex integration next week.

I still have a couple more class tests to get through. On the positive side, the students are turning up for them and have expressed approval for the fact that they don’t have compulsory homework to do off-campus. This form of assessment is undoubtedly harder work for the students, it’s also better preparation for the examination that take-home assignments.

We’ve just received the draft examination timetable for January, and I’m pleased that both of the examinations for which I am responsible will take place quite early in the examination period (on 12th and 15th January, respectively) so I should be able to get them corrected in time to have a break for some research before teaching resumes at the start of February.

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