## 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.

## Patient problem solving

I recently read a piece by D Pearcy called “Reflections on Patient Problem Solving”, from Mathematics Teaching 247. It was an interesting article that looks at how teachers need to allow time for students to try their own ideas out while problem solving, rather than just coax them along in a “this is how I would do it” kind of way.

Pearcy’s definition of problem solving is looking at something you have never encountered before that is difficult and frustrating at times, takes a reasonable amount of time, can be solved more than one way and can be altered or extended upon easily. He then goes on to ask whether this is actually happening in classes or if teachers are just walking students through problems, rather than allowing them to problem solve.

He quotes Lockhart (2009) – “A good problem is one you don’t know how to solve” and states that it follows that if you give hints then it defeats the point of setting problems. He goes on to say that maths advocates talk of the importance of maths as a tool to problem solving – but that this isn’t actually happening if students are not being allowed to get frustrated and struggle through to a solution.

He explains how he finds it difficult not to give hints when students are struggling, both because it is in most teachers nature to help, and because of the external pressure to get through the syllabus quickly. This is something I too have encountered and something I have become increasingly aware of as I try to allow time for struggle. Other factors at play are maintaining interest, and increasing confidence. If we let students struggle too much they may lose interest and confidence in their ability – thus it is important to strike a balance between allowing the struggle but not letting it go too far. This is certainly something I keep in mind during lessons, and I feel it is something that we all should be aware of when planning and teaching.

This is an interesting article that looks at a specific problem and allowing students time to struggle and persist. This importance of this is paramount, in my view, and this is also the view expressed by the author of the article. I find it very hard to not offer hints and guidance when students are struggling. One way I manage to combat this at times is by setting problems I haven’t solved yet, thus leaving me a task to complete at the same time. This can work well, particularly at A Level and Further Maths level as then I can take part in the discussion with the students almost as a peer. This is a technique I have used often with my post 16 classes this year.

I have been reading a lot about problem solving recently, and a recurring theme is that teachers can often stifle the problem solving they are hoping to encourage by not allowing it to take place. This is something we need to be aware of, we need to have the patience to allow students the time to try out their ideas and to come up with solutions or fall into misconceptions that can then be addressed.J

.

*Have you read this article? If so, what are your thoughts on it? Have you read anything else on problem solving recently? I’d love you to send be the links if you have and also send me your thoughts. Also, what does problem solving look like in your classroom? Do you find it a struggle not to help? I’d love to hear in the comments or via social media.*

**Further Reading on this topic from Cavmaths:**

Dialogic teaching and problem solving

**References:**

Pearcy.D. (2015). Reflections on patient problem solving. *Mathematics Teaching. ***247** pp 39-40

Lockhart, P. (2009). *A Mathematician’s Lament*. Retrieved from: https://www.maa.org/external_archive/devlin/LockhartsLament.pdf

## Dialogic teaching and problem solving

I recently read an article entitled “Flowery math: a case for heterodiscoursia in mathematics problems solving in recognition of students’ authorial agency” by K. von Duyke and E Matusov. It was an interesting article that looked at some student teacher interaction in a lesson where students were asked to solve a mathematical problem dividing one dollar between three people. They had found an interesting exchange between the teacher (“John”) and a student who had approached the problem topologically and has a correct solution using physical coins but hadn’t calculated the amount each person had. This has irked and perplexed the teacher – seemingly because she hadn’t come up to the solution he had in mind. This is an interesting revelation and one that we, as maths teachers sometimes fall into. There can often be many ways to solve a problem in mathematics and all are equally valid. My view is that we need to be looking at solutions presented to us by our students with an open mind before telling them they are wrong. In this case the student had come up with her own approach and had the correct solution – an outcome that feel should be celebrated.

The authors use this as a starting point for a discussion on various pedagogies, suggesting that to really allow this sort of maths to thrive in the classroom teachers need to take a dialogic approach – to discuss with the students where their thinking has come from and help them refine their models. They also suggest that the reason John was keen to dismiss this valid reasoning in this case was due to his favouring of a more rigid pedagogical structure. I tend to agree with the researchers. We are there to help students make their own meaning, their own links, in mathematics. Obviously we need to pass on the relevant subject content, but in an open ended task like this it is important to ensure all solutions are explored and refined.

This leads me back into a discussion I had recently regarding the purposes of assessment in mathematics which came about from this blog that I wrote on a question with multiple solutions. John R Walkup (@jwalkup) said that we should be assessing all methods to ensure that students can do it. I think that to an extent he has a point. We do need to test that our students can complete the content, and we should be doing this with low order questions where they are directed to practice and recreate skills. However, maths is about making links, making your own links, and solving problems that are unfamiliar – trying the methods you know to see if you can find a solution to a problem, you have never seen.

It is the latter that is increasingly being tested in our terminal external exams in the UK as we move to the new specification GCSE and A Level tests, so we need to be preparing our students to be successful in this type of question. I think that the dialogic approach mentioned here is an extremely powerful tool in this quest. It allows us to help students explore their thinking and create their own links. I heard a colleague recently explain to a student that maths was about “finding shortcuts, and finding tricks” this worried me a little at first but then he continued “we all have hundreds of tricks and shortcuts that we have developed over years of doing maths. If Mr Cavadino and I were to teach you our tricks they wouldn’t make sense to you and it would overwhelm you.” I can understand this point – if a student notices that d = s x t can be rearranged simply in a triangle because they understand how to rearrange that equation then they will save themselves time. If they learn the technique without understanding what is taking place they open themselves up to the possibility of more errors.

In the article the authors use the term heterodiscoursia, which means legitimate simultaneous diverse discourse. The suggestion is that as part of the dialogic teaching teachers should be allowing discussions and methods to abound and thrive in the classroom. They suggest that this mix of discourses allows students to bounce ideas, allows the teacher to correct any misconceptions and helps build meaning making and engagement. Their suggestions are certainly in line with my observations from my own lessons that have allowed these types of discussion to develop and I think that it would be beneficial to explore how this can be allowed to grow with my other classes.

The authors have some practical suggestions for us maths teachers. They suggest that we need to be familiar with the fact that there are often different solutions and be able to develop them. We need to allow the students to frame the question into a context that works for them then use that context to find a solution which is salient, and we need to be able to question our preconceived notion of the solution. This sounds like sound thinking, I feel that these are things we should all do while we are trying to build the problem solving capacity of our students.

The authors go on to discuss dialogic pedagogy and how this can in effect allow teachers to discuss the problem almost as a peer – I find that this is a great tool when working with A Level students. If i can find problems that I haven’t done before then I can share my thinking with them and we can work through them together. This has been very successful when developing problem solving strategies.

The authors go on to discuss dialogic pedagogy and how this can in effect allow teachers to discuss the problem almost as a peer – I find that this is a great tool when working with A Level students. If i can find problems that I haven’t done before then I can share my thinking with them and we can work through them together. This has been very successful when developing problem solving strategies.

**Reference:**

Von Duyke, K. and Matasov, E. 2015. Flowery math: a case for heterodiscoursia in mathematics problems solving in recognition of students’ authorial agency.* Pedagogies: An International Journal.* **11**:1. pp 1-21. [accessed 23/5/2016] http://dx.doi.org/10.1080/1554480X.2015.1090904

## 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.

## 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.

## Passivity in the maths classroom

Today I managed to find a few minutes to browse the latest issue of Maths Teaching, the ATM journal. One article that caught my eye was the “from the archive” section, where Danny Brown (@dannytybrown) introduced an article that was first published in 1957. The article was written by Ruben Schramm and is entitled “The student’s passive attitude towards mathematics and his activities.”

The article discusses mathematics teaching, particularly the nature of students who often, for whatever reason try to find an algorithmic method to follow to solve a problem, looking to recognise the problem and answer it in a similar way to how they have answered questions before. This is a problem that was obviously prevalent in the 1950s, as evidenced in the paper, but it is still prevalent now, and I feel the nature of our exam system must at least hold a portion of the blame. The questions on exams tend to be very similar and students will learn methods to answer them whether the teachers like it or not. This is one issue I hope will be dampened a little with the upcoming changes to the exams.

Schaum suggests that this passivity in maths, this tendency to look for algorithms, is in part down to how students see mathematics. He suggests that when they see teachers solve problems on the board by delivering a slick, scripted solution they can get a feeling that it is via “witchcraft” and see the whole process of uncoordinated steps, rather than a series of interconnected mathematical ideas. The latter would encourage the students to drive the mathematics from their internal ideas, and this would lead to them being more able to apply their knowledge in new contexts. If we can develop this at all levels then I feel we really would be educating mathematicians – ie giving students the skills to be able to apply their knowledge in new contexts, rather than teaching them to follow a recipe to answer a question.

Schaum goes on to discuss authority, the infallible authority that students see in their teachers and in the mathematical theorems and formulae. It is suggested that students see these theorems as infallible, and as such they reach out for them in their memories and try to apply them to problems. This can mean that the problem they are applying them too is only vaguely similar to the problem the theorem or method is actually there to solve. Schaum calls these “analogy mistakes”, and suggests that it is down to how comfortable with the content students feel that mean they revert to them. I feel that this is true in part, but that also the pressure of exams can lead students to confuse things in their head if they have opted to learn algorithms rather than looking to develop a deeper understanding.

I’ve had a couple of examples of these “analogy mistakes” in lessons and exams recently. A year 12 student came to an afterschool elective as she was trying to solve some coordinate geometry problems involving tangents. She had gotten herself really confused because in her notes she had written tangent gradient is perpendicular (when discussing circles) but she didn’t think it should be perpendicular because a tangent at a point should have the same gradient as the curve. I spend a little time discussing where her misconception had come from (her notes should have said “perpendicular to the radius”) and discussed how she could remember this more easily if she has thought about the graphs and sketched them.

Another example was in a recent exam one of my students had answered part of a question on alternative from incorrectly, she had done the alternative form bit well and the answer was 25 Sin(x + a), but it then asked her for the maximum she had written -25. When I questioned her about this after it seems she had fallen victim to an “analogy mistake”, she had remembered that “maximum is positive” when discussing second derivatives and in the pressure of the exam this memory had taken over, rather than the rational thought process that should have flagged up that the maximum or the function would be 25, which is definitely bigger than -25.

In his preface Danny Brown suggested that one way to counteract this would be by questioning and discussion, if we remove the authority from the discussion and don’t validate the answers by issuing statements saying they are correct or incorrect, but rather open them as conjecture to the class who then can discuss this, then we can allow students to develop their own mathematical ideas. Lampert (2001) also discussed this idea and suggests that as teachers we need to be striking the right balance between allowing students to discuss and conjecture and ensuring they understand what is important and aren’t making mistakes. This is something I strive for in my own classroom, and something I am currently working on trying to improve.

*This post was cross posted to Betterqs here.*

## Impossible scorelines

Yesterday I was watching Exeter Chiefs vs Saracens in the premiership final. During the first half I was thinking about scorelines and how they are constructed and I thought that you could come up with some interesting activities around scorelines.

My first thought was “what scorelines are impossible?” – in Rugby Union there are a variety of ways to score, you can kick a penalty goal or a drop goal for 3 points each, you can score a try for 5 points and if you score a try you get a chance at kicking a conversion for an extra 2 points. From this we can see obviously that 1, 2 and 4 are impossible but I wondered briefly if any others were. I don’t think there are as you can make a difference of 1 between an unconverted try and 2 penalties, however that’s not really a strong proof. I may think about how to prove, or disprove, it later.

I then thought about the 4 4s challenge, and the variety of related challenges based around the year etc. I thought this might be interesting to attempt with rugby scores. It would be nice to investigate how many ways there are to make each score too, and to see if there were any patterns to it.

My thoughts turned to rugby league, the scores in that are 1 for a drop goal, 2 for a kicked goal and 4 for a try, thus all scores are possible, but it still might work for a 4 4s type challenge or an investigation into how many ways each score can be made.

I considered other sports too, football would of course be pointless, basketball would provide a simpler version which could be good for embedding the 2 and 3 times tables and that was as far as I managed.

Have you considered any of these activities or similar? Do you know of any other sports with interesting scoring systems that could be investigated? I’d love to hear in the comments or on social media if you have.## Share this via:

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