Points of Inflection

As MEI has just had its new A-Level Maths specification accredited, I thought I would glance back over their sample assessment materials. On one of the papers for the full A-Level, I came across this question:

Now I stared at this question for a while. I hope I'm not the only one, but if I am I'm willing to take the hit.

Either because my A-Level Maths teachers didn't realise, or they didn't teach it because it was not something that would be tested, I was led to believe that points of inflection were just types of stationary points, where the curve has either positive or negative gradient on both sides of the stationary point. A classic example is y = x^3 with its stationary point at the origin.

Checking with my colleagues and researching the actual definition online, it turns out that points of inflection on a curve are where the curve changes from being concave to convex (or vice versa), and this does not have to be a stationary point.

So the two points I have circled on the graph is where these points of inflection appear.

To find them, you have to find the second derivative of the curve and find where this is equal to zero, giving you in this case two values of x (the x-coordinates of the points of inflection).

I have certainly never seen a question like this on an exam paper before, and this could easily have sneaked past me if I hadn't read through the questions on the specimen paper carefully. I then looked up points of inflection in the content statements and found:

So it is there, and can be examined. I'm happy with that - I've learnt some new maths and another use for the second derivative, other than for checking whether a stationary point is a local maximum or a local minimum. It always made me think the second derivative was a bit useless, because if it's zero at a stationary point you've learnt nothing. Why not just look at the gradient either side of the stationary point?

Then I kept looking. One more search of the word inflection through the content statements found this:

Now what had points of inflection to do with the normal distribution? I took a moment to think about the shape of the bell curve and clearly it has two points of inflection, one either side of the mean and the centre, but what relevance does this have to the distribution?

So here's the equation of the normal distribution curve:

Now we can't integrate this algebraically to find areas - that's why we use statistical tables and calculators. But to find the points of inflection, we need to differentiate. In fact, we need to differentiate twice to find the second derivative. This we can do, using a combination of the Chain Rule and the Product Rule.

It's not easy, however, and I made an initial mistake with my Chain Rule as you may be able to spot:

But it's a great result at the end and definitely worth the effort. It turns out that the points of inflection of a normal distribution curve appear a single standard deviation away either side of the mean.

This is definitely an extension problem I will set to my second years in the linear A-Level. The problem can be differentiated and made more approachable if you focus your students' attention on the standard normal distribution, with the mean = 0 and the standard deviation = 1.

If anyone has any other ways of extending this problem and/or further problems involving points of inflection that aren't stationary points, I would be very interested.

A-Level Maths Google Classroom: Two Weeks In

So it's been two weeks since I set up a Google Classroom for my two AS classes and my two A2 classes. I'd had experience using it before because I use it with my Core Maths group, but I had not ever used the Assignment function - that was what I was really trialling here.

Previously I'd been taking in a piece of homework from each student in my four classes each week. As these could be several pieces of paper long, this is a lot of paper to keep account of and sift through, and is especially problematic if they forget to write their name at the top (still an issue for Level 3 students I can assure you!). Classroom erases this problem, but that's not to say that it's been clear sailing.

The students complete their weekly homework on paper and the idea was that instead of handing this straight in to me, they would take a picture of each page and upload it to Classroom. This sounds like a simple enough idea given the number of pictures students take on their phones what with Facebook and Snapchat, but reality sometimes has its way of surprising you. 

One of the problems is that the Google Classroom app is not always reliable. However, there is more than one way to work around this problem. Many students were not aware that when you take a picture on your phone that you could then email it to yourself - a few students don't even use their college email account and hence cannot be contacted except through face-to-face discussion.

Then there was another solution: scanning their work. Some had scanners at home and didn't think about figuring out how to use it. All of the printers in college are also scanners and they didn't know how to use them, and seemed reluctant to try.

In the majority, students were able to get their work uploaded, but for those thinking it would be an easy thing for everyone to do, think again. I am convinced, however, that this is just initial teething problems and I shouldn't have assumed my students' abilities with technology (this is difficult to do when they're always on phones, tablets and desktop computers - I assumed they would be proficient).

As students were able to hand in their work throughout the week, rather than just in lessons, I could monitor who did what during the week, and I saw that many left their work until the last possible day to hand in their work. This was disappointing to see as it gave me the impression that this was all the maths they were doing each week outside of lessons, when really I expected it as the bare minimum.

It has been a revealing experience using Google Classroom, and I certainly wouldn't go back from using it. What it has taught me is that many of my students need to be taught key technology skills, as well as time management skills, from the beginning of the year. Building student independence cannot just come from teacher expectation, it should be physically built in to their routine from the start of the year.

A-Level Maths Google Classroom: Day One

I have had several hours of training on Google Classroom and, like many maths teachers I initially discarded it. The first problem was (and still is) with Google Docs, which has an equation editor worse than that from Word '95. Google Slides doesn't even have an equation editor, so I certainly won't be using it anytime soon. Google Sheets is probably the most useful of all three, but we use it mainly within the department for results analysis and it isn't something we incorporate into lessons (maybe next year with the new specification).

So without the collaboration side really being any use for us at the moment, Google Classroom looked like all it could be was a dumping ground, i.e. another Moodle. But there are restrictions even on that idea as there is currently no option to copy a class as a template, so each teacher would have to recreate the same environment for each of their classes, which seems like a complete waste of time.

So maybe it's a bit of a surprise that I have set up two Google Classrooms over the weekend, one for AS and one for A2, and have enrolled all of my students in them today.

Once I was told you could set up topics and Classroom organised them neatly down the side, this got me interested again. It doesn't sound very interesting, but if Classroom could be organised, I thought, it might be worth using. The idea of just having the Stream as just a stream of consciousness of things sounded horrible.

In each topic, I created an announcement that included all of my teaching videos on that topic.

They also have an announcement that includes each of the Powerpoints I made for that topic that we used in class.

I then downloaded each of the Integral worksheets and their solutions and posted them in the relevant topics as assignments without a date. This would allow students to complete them as and when they wanted to and keep track of what they've attempted.

I can now set the weekly homework as an assignment on Google Classroom, and I now won't have to take in a mountain of bits of paper each week to check through.

My students won't be using Google Docs to turn in their weekly homework of course. It would be pointless to get them to type up all their equations - they're sitting a pen-and-paper exam at the end of the year, after all. So students will take a picture of their work, either on their phone or tablet, or by scanning their work in, and then upload it to Classroom.

I have also added links to Desmos, Mathway, textbooks, calculators and revision guides. It may feel like something that can be done with Moodle, but Classroom looks and feels a whole lot better.

There are still deficiencies of course. There's still no gradebook, but then the real grades I care about during the year are three summative assessment points and the final grade at the end of the year. If it makes the juggling from my end easier and it improves my students' outcomes, I will keep at it. After all, it's only Day One.

Further Maths students: Mixing or Splitting?

We've just had a week of mocks, our 2nd big assessment point in the year, and my brain is still a little frazzled after finishing the great pile of marking. This year I'm teaching two AS classes and two A2 classes, both of which have Further Maths students, and all four of the classes have quite polarised results.

When I've had discussions with colleagues before about Further Maths students, it has tended to solely be whether they should be taught in parallel or series. If taught in parallel, as has been the way in both sixth form colleges I have worked at, then the Further Maths (FM) students mix with those just studying Maths (M).

The idea is that M students benefit from having the extra expertise in the room and, as FM students are generally more focused, they have the FM students as role models. The FM students benefit as they have the opportunity to help teach the M students, and this builds their strength on the core principles. I have always believed in this concept, but is it really working, and will the situation just get worse in the future?

In my classes I feel a significant gap has grown between the strongest and weakest students in the groups, and I'm not quite sure how to tackle it. I recognise that there's always that grasping-at-straws feeling after marking mock papers to try and understand precisely what the results mean, but I'm wondering if we're shooting ourselves in the foot as this polarising has got me concerned.

There's a lot of material to cover in AS Maths and not a lot of time to cover it in. It's got to be taught at a fair pace and more often than not you've got to deliver it to the top end (they all sit the same exam paper after all, and you can't just miss out reduction to linear form, for example, because the weakest students will never really get it). This would still be true if the FM students weren't in the class.

However, would the M students feel more open about asking questions and asking for help? Or would they be losing a significant resource? When you're teaching a group that can range from the top A* student to those that may not pass, the differentiation involved stretches the teacher to the limit. There's likely to be an even wider stretch next year, where students who would have studied and been very successful with Use of Maths will have to study the traditional Maths course. The bottom end are likely to be even weaker, unless the new GCSE manages to make a host of stronger students, which is probably unlikely.

Some institutions logistically have no way of teaching FM students except by mixing them in classes with M students, and the concept of having a two-tier system still doesn't feel quite right. But we're going to have exactly the same problem if we offer AS Maths alongside A-Level Maths - it is very likely these will have to be taught separately as well. So if it could be scheduled, we could potentially have three different first year classes running, and that just sounds like a logistical nightmare.

For those who teach FM in series, what benefits do you see, and do you feel there is any detriment to your M students because of it?

Assessment Structure Overview

In the interest of sharing, Twitter handle @MsSteel_Maths recently published a blog on her workplace's assessment, and I thought I'd do the same.

First of all, whether an assessment policy or procedure will be effective really depends on your place of work. I work in a 6th form college with over 600 students studying AS / A2 Maths on 3 x 1.5hr lessons a week, with an average class size of approximately 24. There will be cases where something that works in an institution where there are class sizes of 10-12, but this may not scale-up well to those of 24 and would be a struggle to adapt.

Students are expected to focus on maths outside of class for 4.5hrs a week (the same amount of time they get for lessons). Part of this is a weekly homework, while the rest of the time is undirected. Students can use this time to build revision cards and crib sheets, work through textbook exercises and practice papers, watch teaching videos, or attend one of our daily lunchtime workshops that are run by one of the maths team on a rota. How many students actually focus on maths for the full 4.5hrs a week is unknown and can't be checked, and I'm sure that the majority will not follow this religiously. However, I believe that to go some way to building independent learners, we need to have the trust to let students organise part of their own revision.

Our assessment policy currently looks like this:

• Students sit three summative assessments (graded mocks) during the year: one in October, one in January, and one in March. These dates are set by the college to coincide with subject reviews and parent evenings. These are marked by the teacher and the grades are inputted into the college's database.
• Students complete a weekly homework that is made up of rigorous practice and past paper questions, which is accessed, with the solutions, from the VLE. Students work through the problems and self-mark their work. Any problems the students have should be brought to the lunchtime workshop prior to hand-in (if they managed to be organised enough not to do it the night before). Teachers take these in for a quick once-over, before recording on the database whether it has been completed (2 - all present and correct, 1 - parts missing or not marked and corrected, 0 - MIA), and then handed back to the student.
• Students are tested once a week to assess how confident they were with the homework. The test is short, between 10-15 marks, and takes only about 15-20 minutes of the start of a lesson. These are taken in and then marked by the teacher. The scores are recorded on the database and the tests are then returned to the students.

The reason I started investigating other colleges' practices was due to the workload. Although the weekly test is a quick and easy thing to mark by itself, when scaled up to 5 classes of 24 students each, that's 120 tests a week to mark and then record alongside all of the pieces of homework. It can take up to an hour a week to record all of the homework completion and the scores for 120 tests on the college database, and that's without actually marking them! This is time that could be devoted to planning, which sadly takes a backseat in order to keep on top of this weekly routine.

Please share any suggestions you may have that would help. What are we missing?

The New A-Level Specification: Order of Teaching

The new spec moves A-Level Maths from a modular to a linear course: you now have a blank canvas. Do you want to teach Differentiation all in one go? You can if you like, but then what topics will the students need to have met already in order to do this?

I recently attended training where we had cut-outs of all the detailed content statements. In pairs, we tied a length of string across the room and tried to pin each statement up in what we believed to be the correct teaching order. We only spent half an hour on this activity and only got a small portion of the way through, and with the number of different interpretations around the room from different pairs, it was clear this was going to be more complicated than I had originally anticipated.

There’s also another problem: AS Maths. Many institutions won't offer it but there's bound to be students that enrol on the two year course that would be more suitable for just AS. This is particularly inevitable for institutions that have offered the outgoing Use of Maths and have an entry grade for A-Level Maths at a 6. For these it may well be suitable to have a similar order of teaching for AS and A-Level up until Christmas, say, allowing for transfers between the two, which limits that blank canvas somewhat.

Another problem is that we don't have long. It doesn't have to be perfect, but I'd like to have a relatively good idea about what we'll be teaching when for the next two years by September. On top of everything else we have to do between now and then, we could each of us write the order of teaching from scratch, but why should every institution take on this huge job individually? Some institutions will find this easier than others, having department teams of 10 to 20, while some have teams of just 1. Why can't we all just use one big shared Google Doc? Or would it be too many cooks?

Thankfully AQA has designed a route map for both AS and A-Level and it's an excellent place to start as a reference, but I'm a little reluctant to blindly follow it (perhaps only because I've been promised a blank canvas and I've had no input into its construction).

Two ideas:

• Firstly, I'm building a playlist of teaching videos to go through the new spec, but instead of making them in teaching order, I'm going through them via the detailed content statements. That way, teachers can easily find them and insert them into schemes of work.
• Secondly, I would use as a quick teaching frame your current scheme of work. Take out the topics that have gone, add in the weeks you gain from not having to revise for AS exams, then put in the new topics you need to cover. It's not a perfect solution, but remember you can edit as you go.

Introduction

Twitter Handle: @TLMaths

This blog is born out of a blog post by Twitter's @mathsjem Resourceaholic point 8, about wishing there were more maths teachers blogging about A-Level. The whole Level 3 scene is getting a major shake up and I hope that my experiences over the next few years will be worth sharing.

I've taught A-Level Maths for over 6 years now, and I'm currently the Subject Leader for A2 Maths. I am also the first to trial the new Core Maths Level 3 Certificate qualification at my college, so I will hopefully be able to discuss changes from all sides.

I have not written a blog before as I have focused on my YouTube channel that I started over three and half years ago. It's taken a lot of time but it's been very rewarding for my teaching and my students' progress. My next big project is to create a playlist for the new specification for A-Level Maths for first teaching Sep 2017.

I said to myself that if I was going to do this, I would lay out some ground rules:

• My views are my own. The last thing I want is for this to ever be a place to rant, and I'll never aim to be controversial. Boring eh?
• From time to time I might discuss actual maths, but more to do with how I approach teaching certain topics. I'm not a big fan of puzzles, and even though I have a Master's Degree in Mathematics, I dislike Group Theory.
• I'm going to TRY to keep posts under 500 words. You don't have time to read anything longer and I don't have time to write them.

Comments and suggestions are more than welcome!