This morning I saw this post from Ed Southall (@solvemymaths):
And thought, that looks an interesting puzzle. I’ll have a little go. I think you should too, before reading any further…
Ok, so this is how I approached it. First I drew a sketch:
I assigned the arbitrary variables r and x to the radii of the larger and smaller circles respectively and used the fact that tangents are perpendicular to right angles, and the symmetry of isosceles triangles, to construct two right angled triangles.
I wrote an expression for the required area in r and x. Used Pythagoras’s Theorem to find an expression for x in terms or r, subbed it in and got the lovely answer of 25pi.
An interesting little puzzle, did you solve it the same way? I’d love to hear alternative solutions.
This post was originally published here on Edustaff on 26th June 2015.
A while ago I received an email asking what I thought were the main challenges facing maths teachers, and it got me thinking.
The challenges fall into two main categories. There’s the problems all teachers face and then there’s the problems that are more specific to maths teachers.
For the wider profession, the main issues revolve around the negative views of teaching and teachers that seem to be all too prevalent in today’s world. I’m not sure where they stem from, but I don’t think they’re helped by the scorn the media throws onto us, especially during the strike action the other year.
These challenges seem exacerbated by the government rhetoric that has seen education secretaries refer to teachers as “enemies of hope”, “dealers in despair” and “enemies of promise”. Breaking these negative stereotypes is also not helped by the military analogies used. Recently the Prime Minister announced he was to “wage war” on coasting schools. I agree that no school should be coasting, but using terms like ‘wage war’ gives the positive change we are aspiring to a very negative spin.
I don’t know how we, as teachers, can alter this. We already work our socks off to ensure that the learners in our care get the best possible education… I guess we just keep up the hard work and hope the negativity drops.
I believe the main maths-specific issues also centre on perceptions. However, the challenges here are down to the perceptions of maths espoused by many people, rather than the perceptions of the teachers themselves.
I’ve heard a Head of English say she never understood algebra and it didn’t do her any harm; I’ve heard teaching assistants say “I’ve never understood maths”; I’ve heard a Deputy Head question year 11s as to why they’d consider choosing a “boring” A-level.
Astonishingly, I’ve even heard a teacher tell a class “I don’t know how to change the score out of 40 into a percentage, ask a maths teacher.”
Students hear this and get instantly turned off. They often hear these views at home from parents who either “couldn’t do” or “didn’t like” maths. I had a year 7 learner the other year who had an in-built learned helplessness on algebra. She used to say “my dad’s the cleverest person I know and he can’t do algebra, how do you expect me to.”
After a lesson on quadratics, I asked her, “How did you find that?”
“It was well easy,” she said, “it’s only algebra I can’t do.” It took all I had to explain it was algebra without laughing.
I think this example suggests that the parents in question were failed by their own maths teachers. I think the best way to combat this is to really make sure this generation love maths and do understand it, avoiding a similar situation in the future.
As for the teachers, we all need to support each other. If teachers of other subjects are struggling with basic maths, we can help with that, and if they can start to realise the damage they do when rubbishing our subject, surely they would stop! I’d never think of slagging off another subject to learners, and I wish everyone felt the same.
Today was Northern Rocks 2015, slightly over 1 year since Northern Rocks 2014. Last years event was very enjoyable and included lots of opportunities to be reflective and question things so I was hoping this year’s would be the same, and I was not disappointed.
The conference opened with a few words from organiser Emma Ann Hardy (@emmaannhardy) who set the scene and the tone for the day. She then passed over to co-organiser Debra Kidd (@debrakidd), who chaired a panel discussion involving Kevin Courtney (@cyclingkev), Laura McInerney (@miss_mcinerney), Melissa Benn (@melissa_benn), Jonathan Simons (@pxeducation), Michael Cladingbowl (@mcladingbowl) and Mick Waters. The discussion was great and covered topics including “how can we ensure creative subjects aren’t marginalised by Ebac?” And “To what extent is Government rhetoric responsible for the problems we have with teacher recruitment?”
There were many salient points made within this discussion. Kevin stated that he believes it is time schools take back the ownership of the curriculum and start doing what they think is best for the children rather than the league tables. Jonathan proclaimed that there’s actually nothing to stop creative subjects being taught as there are 3 free “bins” on the Ebac. Mick Waters suggested that English and Maths shouldn’t be taught distinctly but within other subjects, which made me think a bit and I think will be the subject of a future post. I was left at the end thinking “but Maths IS Creative!”
Government Rhetoric and Teacher Recruitment
Laura made the point that if prospective teachers were going to be put off by things Gove said then they’d not last long with a tough year 9 class. Jonathan claimed the government had never knocked teachers (I guess “the blob”, “Enemies of hope,” “dealers in despair,” and “enemies of promise” were meant as compliments?). Kevin put forward the view that perhaps government rhetoric couldn’t be blamed for the recruitment crisis, but it could be blamed for retention rates which are low on the main due to the massively overwhelming workload faced by teachers which is driven by government rhetoric.
The whole discussion was lively, invigorating and all panel members made me think.
Martin Illingworth (@MartinIllingwor) – Think before you teach
Martin is a former teacher who now works in teacher education. His presentation style was brilliant and left me thinking that he must have been an inspirational teacher. He started the session by reading an excerpt from his new book which describes a satirical academy made up of all the strange policies he has encountered in his time. It was amusing and made me think I might enjoy the book.
He then moved onto his presentation, he posed a series of questions about learning which got me thinking. He seemed to strongly hold the view that we shouldn’t be teaching anything other than how to learn. This seems a bizarre idea to me. His argument is that we live in a digital age and we all have a wealth of knowledge at our fingertips so don’t need to remember anything. This goes against the research into long term and working memory that suggests holding knowledge in long term memory allows more working memory to process what we are doing.
He suggested that we shouldn’t teach prescribed knowledge because we don’t know what people will need to know in 15 years. This seems a redundant argument, Michael Cladingbowl has spoken in the panel discussion on creative subjects about the common curriculum that has run for years. That there were many things taught to our parents and grandparents that we still teach today.
I think the biggest argument against this idea comes when I think of the innovators of tomorrow. How will we cure cancer if we have no knowledge of cellular biology? How will we make the links into the unknown if we don’t know the known?
Alastair Arnott (@Alastair_Arnott) and Mick Waters – Positive Psychology
The main theme of this session was one I could certainly get behind, that learning from your mistakes is key. I strongly believe that by failing and learning from your failures you can make the best improvements. Alastair spoke well on this from a psychological perspective and I’m interested to read his views in more depth. I try to create a culture in my classroom where learners feel they can get it wrong without ridicule because this is a key part of learning.
However, the main thing I will remember from the session was perhaps a throw away line from Alastair that irked me no end. “Knowledge is becoming obselete.” NO IT IS NOT. As mentioned above, we need knowledge to live the world forward. We need knowledge to move us forward, he’ll we need knowledge to live, to eat, to have a conversation. It is not, and never will be, obselete.
Jo Pearson (@jopearson3) – Teaching schools, supporting teacher development from the inside out.
This was an excellent session, Jo started by explaining in depth about teaching schools and their nature. Then went on to discuss some of the work her own teaching school alliance (TSA) has been up to. It’s an interesting topic and she said some worrying things about the system as a whole.
Her TSA is one that encourages collaboration and increasing outcomes for all no. She spike about aristotellian friendships and the mutually beneficial nature that can be garnered by these partnership. But she also warned of the sharks, the TSAs out for themselves who could consume rivals in a sort of municipal darwinism.
She also spike of some of the challenges she faced with funding. It seems that the funding dries up a little after three years and puts TSAs into a negative incentive systen where they can make considerably more funds from one day conferences with little effect than they can by deploying SLEs to have real long term effects. This is something that I feel needs addressing.
Phil Wood (@geogphil) – Initial reflections on a slow research approach
I think I enjoyed this session most. Which is good, because it was the hardest choice of workshop fir me with 5 speakers on that I would have really liked to have seen. Phil spoke about slow education and slow research which looks at the process as being as important as the outcome.
He highlighted a view that all teachers should have been involved in at least some small scale research to enable them to be more able to critically evaluate the wealth of edu research that is thrown at them. I can see how this would be effective.
He also highlighted the importance of discussion. Giving examples of innovation stalling in schools with no shared staffroom as there us nowhere for spontaneous pedagogical discussion to take place. This was a large focus in a recent masters assignment I wrote and had been a topic of discussion for me with a former colleague earlier in the day.
Phil also spoke about how some teachers find themselves in a position when engaging in research led to them holding views that differed from senior leaders, and that the best schools were where leaders were open to being part of a wider debate.
Phil’s session, as all the sessions, gave me plenty of food for thought. The day was great and I’m already eagerly anticipating next year’s event.
This post is now part of the June 2015 #blogsync on Northern Rocks. The other posts can be found here.
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.
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 ilfalls 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.
The obvious advantage of verbal feedback I’d 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.
Hattie JA (2012) Visible Learning for Teachers Routledge: Abingdon
Black P and Wiliam D (1997) Inside the Black Box Kings College: London
This post was first published 16th June 2015 on Labour Teachers here
I have just read this post from Duncan Hall (@doktordunc) on Labour Teachers (@labourteachers) outlining what he’d like the Labour Leadership candidates to say on education and I thought I would draft my own wish list on the topic.
Qualified Teacher Status
I’ve written about this before, and it’s still something that I feel is extremely important. Teachers should either be qualified or working towards a qualification. Teacher education gives a good grounding in pedagogy, ensures a minimum level of subject knowledge and ensures fair payment for the teacher.
Bring back our schools
For too long we’ve seen the dismantling of local authorities and the removal of schooling from state control. Some of the “freedoms” offered are worrying, no need for the national curriculum etc. I am also worried by big academy chains and their motives. I know from experience many small chains are run well, but as the bigger chains expand I worry about what will happen. I worry about a race to the bottom, how cheaply can we run this school, and ultimately a switch into a for profit nature of schooling. I’d go further and suggest all schools, including private schools, be brought into local authorities to ensure minimum standards are met and raised and all children received a top quality education.
For too long we have tried, and failed, to create an educational system that caters for all. Vocational education is something we have repeatedly got wrong. I’d love to see the leadership candidates discussing how we can get it right, make it a real alternative with the same rigour and esteem as it’s academic counterpart. I was intrigued by Tristram Hunt’s comments on this pre election and would have liked to see how they panned out.
I do worry about the lack of knowledge some school leavers show about politics and democracy. It’s something that has been looked at before, but has not improved in recent years. I’d like to see compulsory political education for all, giving a broad overview of the way our democracy runs and the history behind how and why it developed as it did.
I’d like to see the teacher shortage acknowledged and addressed. Consecutive government’s have thrown money at teacher recruitment but have not really addressed the issues. The first think I’d like to see is an end to “teacher bashing” from the political elite. Some of the criticisms Gove levied at the profession and the insults will have weakened public perception of teachers which is already lower than it should be. I’d like to hear candidates put forward ideas to improve the status as I think that’s the way we can recruit passionate people into the profession and hold onto the ones we’ve got.
This post was originally published on Labour Teachers here, on 12th June 2015.
I recently wrote about my views on Academies and Free Schools (you can find the post here). I know many people hold strong views for or against them but given the current government are majorly for them I feel trying to protest against them currently is a little bit pointless. The speed of the Academies project has been immense and it us rapidly growing and continuing.
Today I read this announcement regarding the Education and Adoption bill which:
“(is due) to close loopholes and speed up the turnaround of failing schools
“has been backed by leading heads as ‘very positive step forward for families’”
“is part of the government’s plan to give every single child the best start in life”
I worry about this bill and the ideas behind it. The assumption is that all academies are improvements on local authority schools. Now this is true for some schools but not all schools and I feel the assumption itself is dangerous. I’ve also seen some pretty strange judgments handed down to Schools in recent years where the rumours have abounded that pressure was being applied to force academisation.
It’s clear to me, reading this bill, that the Conservative government are desperate for all schools to become academies. I’m not sure their reasons for this, I assume it’s due to their ideological belief that the state should be minimised and everything handed over to a competitive free market. Although the cynical side of me wonders if they are trying to move education out of local authority control because Labour still control a vast amount of local authorities. Either way the electorate handed them a majority last month and they are now in a position to drive through these reforms.
I still hold reservations about academies and free schools, but one thing really bothers me about the current climate. Pretty soon academies will be in the vast majority, meaning very little money will be put into the local authorities and those schools doing well under local authority control will be put in a position which holds them back. They will be receiving less finding and less support.
I wish the government would be more honest about their reasons. It’s an ideological move, not one done to improve failing schools, and I wish they’d just come out and say it. Rather than bring in piecemeal bits of policy to take bits at a time, (1st all outstanding schools can be academies, then “all failing schools” now it seems to be all RI/”coasting schools etc) why not just tell us the real plan? All schools will be forced to become academies. Just get on with it, tell us the truth, tell us the plan and give us the time scale. At least then everyone would know where they stand.
Today I was going through an M1 question with a year 13 student and was surprised to see the method he had used. The question involved finding an angle in a right angled triangle given the opposite and adjacent sides. The learner had used Pythagoras’s Theorem to find the hypotenuse then used the sine ratio to find the angle.
Puzzled I questioned further, thinking he may have instinctively found the hypotenuse without fully reading the question then having all 3 sides so going with the first. This turned out not to be the case:
“I know sine equals opposite over adjacent innit sir, I have trouble remembering the other ones so I just always use sine.”
This was extra interesting as earlier I had come across a markscheme which suggested the way to resolve a force at an angle of 30 degrees was to use Fsin30 for the vertical and Fsin60 for the horizontal! Further checking showed this learner did that too.
I wasn’t too sure what to make of it. It’s mathematically correct, so there’s no issue there. The learner has a grasp of the other ratios but is more confident with sine so I can see why he would default to that position, although I hope the extra time it takes isn’t an issue tomorrow. I can’t fathom, however, why the markscheme would show it this way in the first instance. (Not the only time a markscheme has confused me recently!)
What do you think? Have you got any quirky methods like this? Have any if your students? Do you have an idea why a markscheme would default to this position? I’d love to hear your response.