## Consultation time again

Is it cynical of me to question the DoE’s repeated tactic of releasing consultations either just before the summer, when most teachers are in the midst of high stakes exam testing, or over the summer when a lot of teachers are either away or spending time catching up with their families who they haven’t seen through the heavy term time?

Anyway, this year they have released another one. It focusses around the new GCSEs, and more specifically the awarding of grades. The consultation states that for the first award there will be a heavier reliance on statistical methods to set the grade boundaries, allowing the same proportion of grade 4s as we currently have of grade Cs, likewise similar proportions of 1s to Gs and of 7s to As. The rest will be split arithmetically ie the boundaries in between will be equally spread. From Year 2 onwards it will revert back to examiner judgement, but use the statistical analysis as a guide as well as the national reference tests.

This immediately raises questions – how do we know that the first year to sit it should have a similar proportion of 4s as Cs? It seems that this has been decided without much thought about the prior attainment; the consultation certainly doesn’t mention it for the first year. It does going forward, but that doesn’t really explain how this prior attainment will be measured. I have been under the impression that the KS2 SATs are moving from level based assessments to assessments where the students’ scores will be reported as percentiles – surely then comparisons of prior assessment will always be the same? “This year, bizarrely, we saw exactly 10% score above the 90th percentile, what’s more bizarre is that is exactly the same proportion as last year!”

It seems strange to me to put such a heavy reliance on these prior attainment targets anyhow. We live (for now) in a society that has a fairly fluid immigration system, so the students who get to year 11 haven’t always been through year 6 in this country. There is also a question of the validity of the assumption that every year group will progress over the 5 years of secondary at the same rate.

The obvious elephant in the room is floor targets. By setting the boundaries so the same proportion of students get above a grade 4 as get above a C, but switching the threshold to a grade 5 you immediately drop the results of a whole host of schools down, what happens then remains to be seen, but I can imagine lot of departments will become under pressure and scrutiny for something that is statistically inevitable given the new grading formula.

This is all interesting, but it’s not much different to previous announcements and consultations, what is different is the formula for awarding grades 8 and 9. The formula looks to be a fair way of doing it, but it seems strange to me to use this formula just for the first year. Why then revert to examiner judgement about the grade standard? The government seem to be happy to use statistical analysis and similar grade proportions in parts of their grading system, but not in all of it, and that seems odd to me.

Have you responded yet? If not you can here (but hurry, the consultation closes June 17th). I’d love to hear other people’s views either in the comments or via social media.

## Is one solution more elegant?

Earlier this week I wrote this post on mathematical elegance and whether or not it should have marks awarded to it in A level examinations, then bizarrely the next day in my GCSE class I came across a question that could be answered many ways. In fact it was answered in a few ways by my own students.

Here’s the question – it’s from the November Edexcel Non-calculator higher paper:

I like this question, and am going to look at the two ways students attempted it and a third way I think I would have gone for. Before you read in I’d love it if you have a think about how you would go about it and let me know.

**Method 1**

Before I go into this method I should state that the students weren’t working through the paper, they were completing some booklets I’d made based on questions taken from towards the end of recent exam papers q’s I wanted them to get some practice working on the harder stuff but still be coming at the quite cold (ie not “here’s a booklet on sine and cosine rule, here’s one on vectors,” etc). As these books were mixed the students had calculators and this student hadn’t noticed it was marked up as a non calculator question.

He handed me his worked and asked to check he’d got it right. I looked, first he’d used the equation to find points A (3,0) and D (0,6) by subbing 0 in for y and x respectively. He then used right angled triangle trigonometry to work out the angle OAD, then worked out OAP from 90 – OAD and used trig again to work out OP to be 1.5, thus getting the correct answer of 7.5. I didn’t think about the question too much and I didn’t notice that it was marked as non-calculator either. I just followed his working, saw that it was all correct and all followed itself fine and told him he’d got the correct answer.

**Method 2**

Literally 2 minutes later another student handed me her working for the same question and asked if it was right, I looked and it was full of algebra. As I looked I had the trigonometry based solution in my head so starter to say “No” but then saw she had the right answer so said “Hang on, maybe”.

I read the question fully then looked at her working. She had recognised D as the y intercept of the equation so written (0,6) for that point then had found A by subbing y=0 in to get (3,0). Next she had used the fact that the product of two perpendicular gradients is -1 to work put the gradient of the line through P and A is 1/2.

She then used y = x/2 + c and point A (3,0) to calculate c to be -1/2, which she recognised as the Y intercept, hence finding 5he point P (0,-1.5) it then followed that the answer was 7.5.

A lovely neat solution I thought, and it got me thinking as to which way was more elegant, and if marks for style would be awarded differently. I also thought about which way I would do it.

**Method 3**

I’m fairly sure that if I was looking at this for the first time I would have initially thought “Trigonometry”, then realised that I can essential bypass the trigonometry bit using similar triangles. As the axes are perpendicular and PAD is a right angle we can deduce that ODA = OAP and OPA = OAD. This gives us two similar triangles.

Using the equation as in both methods above we get the lengths OD = 6 and OA = 3. The length OD in triangle OAD corresponds to the OA in OAP, and OD on OAD corresponds to OP, this means that OP must be half of OA (as OA is half of OD) and is as such 1.5. Thus the length PD is 7.5.

**Method 4**

This question had me intrigued, so i considered other avenues and came up with Pythagoras’s Theorem.

Obviously AD^2 = 6^2 + 3^2 = 45 (from the top triangle). Then AP^2 = 3^2 + x^2 (where x = OP). And PD = 6 + x so we get:

*(6 + x)^2 = 45 + 9 + x^2 *

x^2 + 12x + 36 = 54 + x^2

12x = 18

*x = 1.5*

Leading to a final answer of 7.5 again.

Another nice solution. I don’t know which I like best, to be honest. When I looked at the rest of the class’s work it appears that Pythagoras’s Theorem was the method that was most popular, followed by trigonometry then similar triangles. No other student had used the perpendicular gradients method.

I thought it might be interesting to check the mark scheme:

All three methods were there (obviously the trig method was missed due to it being a non calculator paper). I wondered if the ordering of the mark scheme suggested the preference of the exam board, and which solution they find more elegant. I love all the solutions, and although I think similar triangles is the way I’d go at it if OD not seen it, I think I prefer the perpendicular gradients method.

*Did you consider this? Which way would you do the question? Which way would your students? Do you tuink one is more elegant? Do you think that matters? I’d love to know, and you can tell me in the comments or via social media!*

Cross-posted to Betterqs here.

## Mathematical Style

Yesterday I read this post from Tom Bennison (@DrBennison). The post was written to start a conversation for a twitter chat that I unfortunately couldn’t make. It did, however, make me think.

He was questioning Wetherby mathematical elegance and style should be assessed at A level. Suggesting that solutions with more elegance should be awarded more marks.

Bizarrely, the example he used was almost exactly the same as a discussion if had with a year 13 class not long before I read his post. His example was finding the midpoint of a quadratic. He looked at two methods – completing the square and differentiation – and suggested that as CTS is more elegant that should be worth more.

I agree immensely that CTS is a preferable method with far more elegance, but I don’t think the marks should be different depending on the method you choose. I feel that we should be encouraging mathematical thought, trying to create young mathematicians who can apply themselves to a problem and find their own way through. I feel if we start assigning marks for elegance and style them we would be moving towards the “guess what’s in my head” style of assessment that I feel we need to be moving away from. The way to do well would be to spot from a question what the examiner wants, rather than to apply the mathematical tools at ones disposal and find a solution.

Back to that Y13 lesson I mentioned, we were looking back over some C3 functions work and one of the questions involved finding the range of a quadratic function – so obviously it was necessary to.find the minimum. A discussion ensued as to how to do this with students coming up with 3 valid methods. The two mentioned above, both of which I find quite elegant, although I do much prefer CTS. The third method was suggested by one student who said “it’s -b/a – you just do -b/a” I knew what he meant – he was saying that this was the x value where the minimum occurred and that you put that in to find y, but he didn’t really understand what it was or why. He’d come across the method online and has learned it as a trick. When I showed him it came from completing the square and looking at it as a graph transformation, I saw the light bulb come on.

It is an interesting discussion. Some methods are far more elegan, and some are just algorithmic tricks. I think that the lack of understanding with these tricks will lead to marks being lost. So perhaps this will self regulate.

*I’d love to hear your views in this, which way would you tackle finding the minimum of a quadratic? And do you think we should assign marks to elegance and style?*

## Examinations, Examinations, Examinations

*This post was first published on the 3rd May 2016 here, on Labour Teachers. *

Sometimes it feels like the government’s main three priorities are examinations, examinations and examinations, and this fact has certainly led to many people involved in education to express their disagreement and disappointment with the system.

Most recently, a large number of people with children of a primary school age have chosen to keep their children off school in protest against the new SATs test their children will sit. This has caused me to spend some time thinking about this, and try to put together some views.

**Exam factories**

One of the leading criticism of these tests is that it drives schools to shrink their curriculum and focus heavily on the content which will be examined – meaning subjects like art, music, history etc get widely ignored and children miss out on an important part of their education. I can certainly find agreement with this, however I think this is already an issue with the SATs as they stood, so it doesn’t seem to warrant the furore of the new tests, which can only compound an already prevalent problem.

**What are they for?**

This is a key question, and I think that a different answer to it would lead to a different outcome. The tests as a marker for informing future teachers of a students ability are very helpful. The tear that SATs were boycotted we saw real problems with the grades reported by primary schools as there were massive inconsistencies from school to school. However, this argument alone seems to be silly, as what we see often is that students primed and drilled from the test from September to May achieve well, but then do no more maths from May to September and often regress. If this was to be the sole reason then surely they could be abolished totally and secondary schools could complete diagnostic tests on entry?

The other answer to this question is to measure school performance, and this is a real can of worms. It is this exact fact that leads to the exam factory conditions and the gaming the system and as such causes a load of problems. The other side of it is, however, that there needs to be some way of ensuring that schools are doing what we expect them to do. I don’t know what the answer is, but I tend to think high stakes testing is not the answer.

**Is it just a problem with SATS?**

No, all the issues outlined above are transferable to GCSE and A level exams. Again, I don’t have an answer, but I think that there must be a better way to treat 16 and 18 year olds than to make them sit high pressure, high stakes, examinations at a time of increased hormones knowing that if they go wrong that could seriously affect their life chances.

I don’t have the answers, but I do feel that there are answers and our job in opposition is to find them and present them to the public, showing that if they vote differently in 2020 we can give them a better way.

## Some interesting questions on the new maths GCSE

I’ve written before about the SAMs (Sample Assessment Materials) for the new GCSE, and currently we are swaying towards Edexcel. We have recently given year ten the SAMs to see how they got on with them and a couple of questions that stood out for me. First was this one:

Students need to find change, fair enough, but them part b seems to be purely testing their understanding of the word “expensive”. This seems a really bizarre question in my opinion, and I’m not sure it fits well on a maths exam. It’s not even really a mathematical term.

Another that stood out was this:

I think this one is a great question that approaches the assessment of fractions knowledge in a new way, it requires a deeper analysis but I do think there is a limitation to it if it isn’t thought out. If this type of question is regularly asked about fractions, then it becomes a “when they ask this you say this” sort of question. This could be combated by asking this type of question about different topics. It’s certainly a question I enjoyed seeing, and is much better to assess deeper learning than he current GCSE, particularly the foundation tier.

The final question that caught my eye was this one:

It’s similar to the type of question on forming and solving equations we see how, but the interesting bit is the additional bit of reasoning students need to apply at the end, ie to work out if the amount of marbles that Dan and Becky have together is odd or even to work out if they can have the same amount.

*Have you noticed any interesting questions cropping up? Have your students attempted the SAMs? If so, how did they get on? I’d love to hear.*

*Cross-posted to Betterqs here.*

## Verbal or Written Feedback

I’m in the midst of completing an MA assignment on formative assessment. I wrote some preliminary thoughts on the topic a while ago but I’ve since had more time to read around the topic and digest ideas further.

Two of the strands I’ve been looking into have been verbal and written feedback and I’ve found some interesting things written about each that has made me think more deeply about my own practice.

**Written Feedback **

Most schools have a policy on written feedback, it normally involves looking at books at least every other week and issuing some feedback relating to the quality of the work and how it could be improved. This is based in sound principles, but I feel leaves a lot to be desired. I questioned the logic in a previous post and suggested that this was a little strange and that misconceptions would arise and be missed, learners would forget the work etc. Hattie (2012) agrees on this topic, suggesting that feedback should be instant to avoid learners learning something wrong.

Hattie’s findings in his meta analysis that feedback has a much larger effect size than any other intervention is often quoted as a headline and used to back up the necessity of a new marking policy. The problem is the headline is not fully representative of the findings. Hattie also says the biggest spread of effects in this meta analysis falls within feedback and that some isn’t paticularly good. He points out key principles of immediacy, relevance and structure of the feedback.

Hattie built on the work of Black and Wiliam (1997) which stated effective feedback was formative feedback which focused on the task, rather than the learner, and gave the learner help to get to where they are going. This is the premise behind most marking policies, but at up to two weeks after the exercise has been completed this can often be too late.

**Verbal Feedback **

The obvious advantage of verbal feedback is that it is immediate and instantaneous. There is no time to consolidate mistakes and learn misconceptions as they are picked up straight away. On top of this verbal feedback is done in the presence of the learner and as such enables a dialogue to take place. This means that the teacher can be totally sure of why the learner has gone wrong and try to fix the issue.

Another advantage is for learners who have completed the work correctly, a dialogic conversation between learner and teacher can allow the latter to check whether the work has been understood on s deep level or whether it has just been learned as a procedure.

Obviously there are issues, it’s hard to get around the whole class every lesson and I don’t have an answer for that. There’s also the matter of evidencing that this is taking place, this one is easier to fix, if you have your red pen with you you can highlight errors, model answers, scaffold or extend and mark what’s there while you talk.

I’ll write more about the assignment as I go along and after I’m done, the area of formative assessment is one that has really piqued my interest.

**References**

Hattie JA (2012) *Visible Learning for Teachers* Routledge: Abingdon

Black P and Wiliam D (1997) *Inside the Black Box *Kings College: London

## Dumbed down markschemes?

Today I was working through the June 2014 Edexcel M1 paper for my year 13s, noting down thought processes etc and I arrived at this question:

It’s quite a nice, general bouncing ball question that takes into account many of the rules of mechanics. When I finished the question I checked it against the markscheme so I could jot down where each mark came from and on part E I received a bit if a shock.

The weight of the ball is given as is the distance it’s dropped from and the distance it bounces to. In earlier parts of the question you have been asked to calculate the final speed of the bit before the first bounce and the initial speed after. Part E asks for the time between the ball being dropped and the second bounce. I split it into to, used s = ut + (1/2)at^2 (u=0, s=2,a=9.8) to find the time it took to drop, then the same equation for the time between bounces (this time s=0, u= the value already calculated, a=(-9.8)). A simple question, a simple solution and a nice answer. The markscheme, however, says this:

I don’t understand why they have felt the need to work out the time between the first bounce and the top then double it. It’s a valid mrthod, granted, but surely looking at the whole motion is sensible as acceleration is constant?

Worryingly, in the notes there’s no other solutions offered and I was left wondering if examiner’s might miss that thus method is not only correct,, but actually more sensible.

*Which way would you have done it? Do you think I’m correct thinking mines the mire logical sensible way?*