## The Problems with Student Loans

*This post was originally published on Labour Teachers here. On 22nd December 2015.*

I believe that higher education should be free at the point of use, and for this reason I see many flaws with the current system which allows people to access it. I can see the argument that it should be those who benefit from it that fund it, but the current system is drastically flawed. A much better system, in my opinion, would be the much talked of graduate tax. The current student loan really is a graduate tax, it only kicks in when you earn a certain amount and is taken at a proportion of your earnings, but that’s not easily recognisable from the name. Perhaps it’s a rebranding exercise that’s needed. Either way there is a negative view of the loan among many of the population, normally stemming from the word loan being included.

I knew a student a few years back who wanted to go to uni but was against the idea of a student loan as he didn’t want to be indebted to anyone. He was worried about the idea of a loan, but once he investigated it an realised it’s true nature he did go and succeeded at higher education.

Another flaw is how it’s calculated. Mature students are fine, it’s calculated on their household earnings and worked out against what they need. For 18 year olds it’s calculated against parents earnings, which means that some young adults are unable to access university as their parents won’t support this decision. This seems wrong to me, I agree that it’s fine for the majority of cases cut not everyone has parents who are willing to support them. Some parents turf their children out at 18, some even earlier, some refuse to see the value in education, and some use this as leverage to keep control over their children in an unhealthy manner.

I’ve known peoe who didn’t go to uni for these very reasons, or who had their degree choice and uni choice dictated to them, dropping out because they didn’t want to study the subject they ended up in. Some have gone back as mature students, but not all, and this is a shame.

I feel that removal of the monetary value of debt and replacement with grants, that cover tuition and cost of living, would be a far better situation and combat both the issues outlined above. If this is then funded by a graduate tax worked out in a similar manner to loan repayments then the funding should look after itself. Another added benefit would be that it would eliminate the potential marketisation of universities which is something we are likely to see more of over the coming years.

## Newton Raphson

It’s Christmas! An important day because it marks the anniversary of the birth of a man who changed the world! That’s right, Isaac Newton! *(Ok, now I know at least one of you is about to comment with “but since the calendar changed from the Julian to the Gregorian his birthday should be moved to January 6th. I can see your point, but no one suggests moving Christmas so why would we move Newton’s birthday? Anyway, I tend to celebrate both.)*

Recently a couple of people have asked me about the Newton – Raphson method for finding roots of equations, and why it works, so in keeping with the festive spirit here is a brief overview of why. Incidentally, I’m working this up into a help sheet for my further maths class so any additional input would be great.

**Newton Raphson**

A nice little numerical method if finding the root of an equation. You start with an approximation (often referred to as Xo) and then you take away the ratio of f (Xo)/f’ (Xo) to get a better approximation. You keep going until you get an approximation which is correct to a suitable degree of accuracy.

**But why? What is this witchcraft and why does it work?!**

No, it’s not witchcraft, and it’s relatively simple and based on our old favourite “right angled triangle trigonometry” *see, I told you triangles were the saviours of everything….*

Let’s start with a sketch *(always a good start)*:

Here is a curve, as you can see I’ve drawn a tangent to it at C, a perpendicular from C to the x axis (which meets it at B) and labelled the point where the tangent intercepts the x axis as A. Already we have our right angled triangle!

Now we all know that f’ (Xo) or f'(B) will give us the gradient of the tangent to the curve at that point(C in this case). The gradient of that line is the same as the tangent ratio of the angle CAB, as you can see from the sketch (as the gradient is difference in y / difference in x). The opposite side in this case is f (Xo) so the adjacent side is f (Xo) / f’ (Xo) – ie is the opposite side over the tan ratio.

This shows us what’s going in here, we are taking away the adjacent side each time and getting closer to the actual root (when it does converge that is!).

So, a festive look at Newton Raphson. Merry Christmas.

## Microteaching

Microteaching: you know, splitting a lesson between a number of people each delivering 5-10 minutes. It was always an activity I enjoyed during my ITT course and it is something I’ve used well come revision times over the course of my career, although not for a while to be honest.

Then, on Monday when I was discussing with one of my Y13 classes the plan for this week and the work I would like them to complete over Christmas to pepare for January’s “Pre public examinations” one of them asked if they could do it. He didn’t use the term “microteaching”, he just asked if they could split up the unit we’ve just covered and each deliver a short revision session on it. The class are fantastic and I was pleased to give this the go ahead.

Today was the time for that lesson and when it arrived none of them had prepared anything, I suspected that this might happen and had brought so work as a contingency plan, but the class were keen to deliver the sessions anyway without planning. To “wing it” as it were. I was amazed by their willingness, as I know experienced teachers who would freeze on panic if they were asked to deliver even a short lesson like this without a set of pre prepared slides.

The sessions themselves were great, they led to some great discussions around the topic “differentiation”, and any errors made were picked up by others in the class. They also made some fantastix links to other topics and other areas of maths. There was also a decent amount of comedy in some of the sessions, including a little at my expense, but that just made it more interesting.

Definitely an activity I would run again, especially with this class.

*Cross-posted to “One good thing” here.*

## How are we questioning our students?

This month’s maths journal club is based in the article “Contrasts in mathematical challenges in A – level mathematics and further mathematics, and undergraduate mathematics examinations.” By Ellie Darlington

I found the article quite interesting overall. It looks at the differences in examination questions between A level mathematics and undergraduate mathematics. It starts off with the idea that A level mathematics is tested in a manner that involves routine questions and that as such this doesn’t prepare students for undergraduate mathematics, which it presumes is tested in a higher level. I think this is one of the issues with A level mathematics and I hope that when the new curriculum appears this will have been addressed. The problem is even worst at the transition point between GCSE and A level though, but again, I have hope that the new specification will address this.

Interestingly, my own experience of undergraduate mathematics was that there were a lot of courses that were tested in a routine manner, and that learning the lecture notes by rote and practicing the past papers for a course could allow people to score well despite not understanding what was going on and not being able to apply their knowledge in other contexts. There were some of my peers who had no conceptual understanding of some of the modules yet still scored high enough to achieve firsts.

That said, I still feel that the procedural nature of the GCSE and A level papers is a massive problem. In recent years we have seen a change in the A level papers towards questions that are not answerable in a routine manner, but it needs to go even further.

There are many problems with these procedural questions. My main issue is they allow students to score well without understanding the mathematics behind the questions. This in turn can allow teachers to skip teaching for a relational understanding and just teach an instrumental or procedural understanding, which lets down the learners, especially if they are hoping to go into mathematics or another mathematical based subject at higher education.

**So what can we do?**

Well, rather than waiting for the changes we can be implementing these questions in our classrooms, ensuring that we are teaching for relational, or conceptual, understanding rather than teaching purely procedures. Take the time to ask the questions that require application in new contexts. Take the time to teach the concepts, the why behind the what. Enrich the curriculum with tasks that involving thinking outside the box and questions framed in a way that the correct method isn’t always immediately obvious, perhaps try some of these puzzles?

**Other points in the article**

There was a lot early on that I thought I already knew, but it was nice, and useful, to see references and studies to back up some of the ideas.

The MATH taxonomy in this explicit form is new to me and I’m interested to look further into it and see how I can apply it myself.

I was a little purplexed to see that the article stated that questions can change their position on the MATH taxonomy with time, but then have no explanation of how these questions were classified in the research.

*All in all a very interesting read that I will re-read and digest in more detail later. I’d love to hear your thoughts on it also.*

*This post was cross posted to the BetterQs blog here *

## Finally….

I’ve long been at pains to try and hammer into my Y13s the importance of diagrams (see this and this). The amount of times I’ve told them I won’t help them until they’ve sketched the problem is ridiculous. One thing they struggle a bit with is finding the range of a function from an equation. They can easily do it from diagrams and have not yet realised that they can sketch them themselves!

Today, however, two of them were working together and they were trying to find the range of f(x) = 2lxl + 3. They were talking about it and struggling to solve it. I pointed out that neither of them had sketched it yet. They sketched it. They got it.

Both of them saw instantly that the range was in fact f (x) is greater than or equal to 3. Then one of them said “these are really easy if you sketch them, but really hard when you just look at the equation.” It’s finally starting to sink in….

*Cross-posted on One Good Thing here.*

## A puzzle with possibilities

Brilliant’s Facebook page is a fantastic source of brain teasers, they post a nice stream of questions that can provide a mental work out and that I feel can be utilised well to build problem solving amongst our students.

Today’s puzzle was this:

It’s a nice little question. But when I use it in class I will only use the graphic, as I feel the description gives away too much of the answer. Without the description students will need to deduce that the green area is a quarter of a circle radius 80 (so area 1600pi) with the blue semicircle radius 40 (so area 800pi) removed, leaving a green area of 800pi.

I find the fact that the area of the blue semi circle is equal to the green area is quite nice, and in feel that with a slight rephrasing the question could really make use of this relationship. Perhaps the other blue section could be removed or coloured differently and the question instead of finding the area could be find the ratio of blue area to green area.

Another option, one I may try with my further maths class on Friday, could be to remove the other blue section and remove the side length and ask them to prove that the areas are always equal, this would provide a great bit of practice at algebraic proof.

*Can you think of any further questions that could arise from this? I’d love to hear them!*

*This post was cross-posted to the blog Betterqs here.*

## Problem solving triangles

Brilliant – a lovely puzzle app and a source of many little puzzlers if you follow their Facebook page. The other day, I came across this one:

It looked like it might be interesting so I screen shot it and thought, “I’ll have a go at that later, when I’ve got a pen. It’s bound to be nice using a bit of trigonometry and angle reasoning.”

But as I thought about it I realised I didn’t need paper. The hypotenuse of the large triangle is easy enough to find (6rt2) using Pythagoras’s Theorem. You can deduce the size of the green square is then 2rt2 as the big triangle is isosceles meaning the angles are 90, 45 and 45, as the square is only right angles then the little blue triangles in the 45 degree corners must also be isosceles. Thus the two blue and the green segments of the hypotenuse are equal.

The area of the square is then way to find (8) by squaring 2rt2. A nice easy puzzle.

My first thought had been that it would take a bit of working out, but it didn’t, it was a very straightforward question once I got going. That got me thinking, problem solving is something that I would love my students to get better at and I’m hoping to launch a puzzle of the month in January. This sort of puzzle is ideal. It will require then to build their perseverance skills as well as their problem solving skills and will give them a mental workout. I’m going to use this as a starter this week to warm them up.

*This post was cross posted to the BetterQs blog here.*

## The small things

This week has been hellish. I’ve felt more ill than I can ever recall feeling but because we had an external review on Thursday I dragged myself in each day, compounding the tiredness and the run down feeling that goes hand in hand with feeling crap. Today was Friday, the end of the week and the chance to try sleep some of the grogginess off was in the offing so I felt positive about it. However a couple of incidents of poor in periods 2 and 4 managed to take the sheen off it a little.

But then period 5 arrived. Period 5 Friday is one of my favourites as I teach year 12 further maths, and today we were talking topology. We were looking at route inspection problems and discussing Euler in great depth. The discussion wandered to the traveling salesman problem and how it can be mapped do all NP problems, meaning that a solution to the traveling salesman problem able to sole any in polynomial time would be enough to prove that P = NP *ie that any problem that’s easy to verify can be solved quickly also* and earn the mathematician in question a million dollars.

This piqued their interest no end and we had a nice discussion around the millennium prize problemsand the other great unsolved maths problems. This also led onto a discussion of Fermat and his last Theorem. It was really great to see young people so enthused about mathematics and the different types of problems involved, and to see them trying to get their head round such complex ideas as the Riemann zeta function.

On top of this amazing fact they also completed some great work collaborating on a set of topological problems I’d set them. I ended the day felling extremely positive, and despite the week I’ve had I can’t wait to get back into it on Monday, although I do hope I feel better by then!

*This post was cross posted on the blog “One good thing”, here.*

## The long way round

Today one of my Y12s was looking through a C3 paper he found on my desk. *(For those unaware A level maths, studied in Y12 and Y13, ie from 16-18, is currently modular. There are 4 Core Pure modules known as C1, C2, C3 and C4. The first two are studied in Y12 and the second two in Y13.) *He came across this question:

While looking at it he said, “are you sure this is a C3 question?” I told him it was and he then said “But I can answer it.”

I looked at the question, all the main skills it tests are taught at C1 and C2, but the chain rule for differentiation isn’t taught til C3. I thought about it and realised that yes, with the application of the binomial expansion (a C2 skill), or indeed a long winded brackets expansion, it would give him a polynomial he could differentiate.

Then it occurred to me that it was in fact a brilliant question to set my Y12s as revision. It allows them to see links between the things they’ve learned, allows them to practice important skills from C1 and C2, namely the differentiation, the coordinate geometry involved finding te equation of a tangent and the binomial expansion, and to solve a problem using those skills.

It took them longer than it would have taken someone who knew about the chain rule, but it was time we’ll spent and I got some perfect answers from them. I didn’t tell any of them how to do it, they managed to talk each other through it, and I only had to pick up on one slight error when one of them had a slight hiccup with a power. I think I need to have a good look through some more higher level papers to see if I can find any other gens to test the earlier skills.

*This post has been cross-posted to Betterqs here.*

## A look back at 2015, and forward to 2016

New year’s eve, a natural time for reflection, my timehop has shown me my reflections from last year and the year before, and the Cbeebies pantomime is currently on its second consecutive showing on my telly so I thought I’d share some thoughts from this year and some hopes for the next.

My 2015At home2015 was an amazing year. I got married and continued to see my daughter grow. It’s amazing what she picks up and learns, and how fast she does it. She’s writing her own name, doing basic sums and even showing an interest in Star Wars.

The blogThis has been a continuing source of enjoyment, reflection and discussion. There has been more hits this year than the last, although not many more. I’ve also gotten involved with some

group blogs, Labour Teachers, One good thing and Better Questions. Do check them out.

Studies and CPDI’ve been to some good events this year, I enjoyed northern rocks, and have continued to enjoy studying for my masters. This year is the final year for that and I’m currently in the midst of writing a literature review for my dissertation. I’ll be sad when it’s over, but intend to continue to read and research. I may even seek out further study opportunities.

One of my favourite new things this year has been the launch of the twitter “Maths Journal Club“, where teachers read then discuss a journal article on a piece oaths educational research.

TeachingThis year saw me end my first year and start my second at my current school. I feel more established now and I have a fantastic timetable this year with some excellent classes (including this one and this one). The department was strengthened in the summer transfer window, although there has been some tough times towards the end of the year. I’ve been mentoring again this year, which is something I missed last year, and I’ve been able to teach some of the new GCSE and more of the core maths qualification.

I was pleased to see some of my Y13 students from last year go on to study mathematics and maths related degrees at uni and it was excellent to hear that a former student is thoroughly enjoying his maths degree.

MathsI’ve managed to read a few more maths books, although not as many as I had hoped. And the celebration of maths event in February was awesome, I not only got to meet Marcus Du Sautoy in the flesh but also for the see him talk about some of the Maths that excites him. I also got to meet some friends in the flesh who had previously only been twitter avatars.

I’ve been reading some interesting stuff on quaternions, something I enjoyed learning about in my final year at uni, and I hope to investigate them further – if you know any good texts on them then do let me know.

2015 in EducationGone are the days of Gove, where education policy was at the forefront of every political discussion and a constant source of front page news. 2015 has seen a further roll out of academies which left me questioning why the government weren’t just honest about the plans to academise all schools.

There’s been a lot of talk of Shanghai, and an exchange programme. I have slight concerns over the politics behind this but am excited by the prospect of learning from another culture. I’ve not seen much written about this that has been able to give any insight into the outcomes yet, but hopefully this will come. I feel that the amount of non contact time given to teachers in Shanghai to allow for planning, marking and collaboration is key to their success.

There was a general election, it’s was excellent to read the potential new directions in education policy being championed by all parties, although we did end up with the same party and the same ed sec. I took the opportunity to have a look at the last few years against the aims they set out in 2010.

We have found out more about the new specification for GCSE, A level and core maths, although I still feel that mot enough material has been released by the boards, especially for Core Maths, which will be assessed this summer and the new GCSE which will be assessed in 2017.

What about the future?I have enjoyed 2015, and I hope 2016 brings more of the same. I hope to be able to find more time to read and more time to play my guitar, as well as continuing to spend time with family. I hope to be able to see friends more often too.I want to continue to develop as a teacher and a leader at school, by being reflective, evaluative and engaging with the research in a critical manner. I hope to produce and excellent dissertation and to continue to study.I hoe we continue to gain clarity over the new specifications and that we gain it early enough to ensure our students are prepared. I’d like to see the massive inequalities in our society and our Education system wiped out and I’d really like to see some changes to the regulator.I hope you have enjoyed reading this, I’d love to hear your thoughts on your own 2015, your take in education in 2015 and your hopes for the future## Share this via:

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