The proof behind one of the most famous theorems in mathematics

Vishaali, Year 10, looks behind the proof of one of the most famous mathematical theorems – that of Pythagoras’ theorem.

 

What is the difference between a theorem and a theory?

A theorem is a mathematical statement that has been proven on the basis of previously established statements. For example, Pythagoras’ theorem uses previously established statements such as all the sides of a square are equal, or that all angles in a square are 90°. The proof of a theorem is often interpreted as justification of the statement that the theorem makes.

On the other hand, a theory is more of an abstract, generalised way of thinking and is not based on absolute facts. Examples of theories include the theory of relativity, theory of evolution and the quantum theory. Take the theory of evolution; this is about the process by which organisms change over time as a result of heritable behavioural or physical traits. This is based on undeniable true facts, but more from experience and from an abstract way of thinking.

It is also important not to confuse mathematical theorems with scientific laws as they are scientific statements based on repeated experiments or observations.

The proof behind Pythagoras’ theorem

You have probably all heard of Pythagoras’ theorem, one of the simplest theorems there is in mathematics, that is relatively easy to remember. Given that it’s so easy to remember and to learn, wouldn’t it be an added bonus to know exactly how this theorem came to be?

The theorem, a²+b²=c², relates the sides of any right-angled triangle enabling you to find the lengths of any side, given you have the lengths of the other two.

This whole theorem is based on a triangle like this:

These four right-angled triangles are exactly the same just rotated slightly differently to create this shape:

Two shapes have been made by putting these triangles in this order. A big square on the outside, and another slightly smaller square in the middle. As all these triangles are the exact same you can label them A, B and C.

You can tell from the labels the triangles have been given, that the bigger square would have the sides (a+b), and the smaller triangle in the middle will have sides of c. Therefore we know the area of the smaller square is c² :

Using the exact same four triangles, we can rotate and translate them to create a slightly different shape:

Now two more squares have been added to this shape. We can call them  a² and b².

Thinking back to the shape we made before, we can also see that the length of this shape is also (a+b). As we know we used the same four right-angled triangles for the shape before and now, we can infer that the two squares  a² and b² are exactly the same as the square from the first shape, c². Hence we get Pythagoras’ theorem, a²+b²=c²:


References:

https://www.livescience.com/474-controversy-evolution-works.html
https://www.askdifference.com/theorem-vs-theory/

Forging new relationships; the value of school partnerships – 28/09/18

Mr Richard Bristow, Director of Music here at WHS, looks at school partnerships and how external groups can enhance the academic and co-curricular programme, discussing a new partnership between WHS and the Jigsaw Players.

Partnerships have become increasingly important to schools since the turn of the Millennium, with a significant number of schools in both the State and Independent sectors working together in various ways. Broadly speaking there are two different types of school partnership: formal and informal.

Formal partnerships

A formal partnership will often involve a strategic merger between two or more schools, who might operate under the same trust with a central CEO or Executive Headteacher. The GDST, of which Wimbledon High is proudly a part of, is in this type of partnership with 25 schools (including 2 Academies) across the UK working closely together to provide the very best education for girls. In the State sector, this might involve the merger of an Academy Trust with several different schools working under the same central leadership team; a local example would be the Harris Federation, where 47 different academies – Primary and Secondary – operate within the same charitable trust.

The GDST Network in numbers

Informal partnerships

The informal partnership, however, involves smaller links between schools that retain their autonomy and own decision-making at a strategic level. This could be between two similar schools – for example the OWLS project between Oxford High and Wimbledon High (OWLS standing for Oxford and Wimbledon Leading Scholarship). This is where two schools work closely together to craft a vision to enhance an aspect of their shared goals, sharing resources, good practice and enabling the pupils and staff to develop their skills.[1]

These informal partnerships also exist between Independent and State Schools, as detailed below.

Teach Together

In late 2014, the Department of Education granted a significant amount of money to various different partnerships across the UK, focusing specifically on the primary curriculum. Various different projects occurred throughout the country, from developing coding skills to organising MFL challenge days. Wimbledon High was involved in this project, forming a Teach Together partnership with St Boniface RC Primary School to engage pupils with the science curriculum through storytelling and narrative. This partnership continues to this day with weekly links between the schools with our Enrichment programme.

This partnership has significant benefit to both schools, and this is essential for the partnership to work effectively. Both schools need to put in and get something out of the partnership to avoid it from lacking equality. In this case, WHS girls from Year 11-13 consolidate their scientific knowledge and understanding by teaching scientific concepts in a new way to Key Stage 2 pupils. This not only helps the pupils they are teaching, but develops the older pupils’ ability to communicate with others, encouraging them to look outwards, to support others and be ready to shape the society in which they live. Thus partnership work also meets one of Wimbledon High’s key aims. By ensuring both sides of the arrangement are getting something they require out of the partnership, it is far more likely to succeed. If it was a one-sided agreement, where only one side was gaining from the arrangement, the chances of success would rapidly diminish.

When asked the question ‘Have you seen notable progress?’ the feedback is overwhelmingly positive from both sides, including

  • From WHS Staff: “Yes, in interest & excitement in science. Pupils have produced projects which reflect the time they have spent to continue on these themes & also class room displays linked to our visits.”
  • From WHS Pupils: “I get to see the delight of the pupils in learning new things… developing my confidence and resilience” and “[I have more] confidence in my abilities as I am able to fully teach new concepts to children in maths. [I have] an insight into how far I have come with maths as I reflect “
  • From St Boniface Pupils: “The lesson I enjoyed the most was when we went to Wimbledon High School and learned about Light. I have enjoyed going outside to try new experiments”

SHINE

WHS also hosts the nationally-recognised SHINE programme. This is an education charity seeking to turn potential into success, and at WHS this is presented as ‘Serious Fun on Saturdays’, with 24 Year 4 and 5 primary pupils coming to WHS to learn a range of topics based around the idea of ‘Reaching for the Stars’. Some of the activities include making frisbees in DT, learning to bake, understanding more about astrology in Geography and learning how to perform as an ensemble in Music. Each pupil is given a WHS mentor from Year 12, allowing these pupils to develop their mentoring skills.

External agencies

These links between schools – where skills and resources are shared to develop both sides of the partnership – are of vital importance. However, schools are also increasingly offering new partnerships using external agencies and providers which are open to the whole local community.

A new partnership from September 2018 is the partnership between Wimbledon High School Music Department and the Jigsaw Players. The Jigsaw Players are a Not-for-Profit concert series based in Wimbledon, performing world-class chamber music and jazz. They run educational projects for local children, sponsor young up-and-coming jazz and classical ensembles, and heavily subsidise all their concert ticket prices, to help ensure music is accessible to all in Merton.[2]

WHS and the Jigsaw Players

This accessibility is increasing further with this new partnership with WHS. The Jigsaw Players will host four different events throughout the academic year 2018-19 focusing on composition skills and female composers via workshops and concerts. These are completely free to attend and are open to all.

The workshops will allow pupils from year 9-13 from WHS and local schools to understand more about how to write for chamber forces – specifically string quartet – enabling a higher quality of composition work required for GCSE and A Level Music courses. With numbers of pupils studying the subject across the country in sharp decline[3], schools are either struggling to offer Music as an academic subject or have small numbers doing so outside of the timetable. As the numbers are small, funding can be hard to secure as the impact lacks large-scale focus. Against this backdrop, these partnerships are of even more importance as they offer a chance for all schools – state and independent – to engage with curriculum enrichment at zero cost.

Composition is frequently the area of compulsory study at GCSE and A Level which is the most complex to teach and learn and is the area where examiner marks are frequently debated owing to the more ‘subjective’ nature of composition. This will not change as long as composition is a compulsory part of GCSE and A Level Music, but what we can do as a school is to create a time and space for teachers, pupils and professional musicians to come together to discuss the challenges and work together on finding potential solutions. This collaboration gives confidence and allows for networking – something vital for a subject like Music which are often staffed by only one teacher for the entire school.

Free tickets to the workshops can be booked below:

Workshop 1 https://www.trybooking.com/uk/book/event?eid=4118& 3rd October 4:15-6:15pm M11

Workshop 2 https://www.trybooking.com/uk/book/event?eid=4121& 14th February 4:15-6:15pm M11

The concerts are also open to all, focusing on the chamber music of female composers. This clearly links the chance to hear professional musicians with the overall ethos of girls’-first education, championing music which often struggles to find a voice in the canon of Western Classical Music. This type of cultural enrichment is universal and has significant benefits to overall academic progress[4].

Free tickets to the concert can be booked below:

Concert 1 https://www.trybooking.com/uk/book/event?eid=4120& 3rd December 7pm Senior Hall

Concert 2 https://www.trybooking.co.uk/4122 7th May 7pm Senior Hall

Summary

The most effective partnerships are ones characterised by a shared vision and passion between the schools and agencies agreeing to work together. Without this shared goal, partnerships become forced and subsequently lack effectiveness, reducing impact. Honesty, openness and clear communication are central to ensuring success for all stakeholders.

The new partnership with the Jigsaw Players is an exciting opportunity to work with local professional musicians and other GCSE and A Level pupils and staff, allowing new networking opportunities on a staff and pupil level and encouraging all-important discussions about Music as an academic subject. Whether you would like to attend as an active participant in the workshops or simply as a member of the audience listening to the music by composers past and present, you are warmly invited to become part of our shared passion for all things musical.

[1] See OWLS Quarterly here http://www.wimbledonhigh.gdst.net/userfiles/wimbledonhighmvc/Documents/Sixth%20Form/OWLS/OWLS%20Quarterly-First%20Edition%2C%20February%202018.pdf

[2] http://www.jigsawplayers.co.uk/about-us/

[3] https://www.economist.com/britain/2018/03/01/the-quiet-decline-of-music-in-british-schools

[4] https://www.sciencedaily.com/releases/2018/03/180326140244.htm

 

The value of spontaneous speech

Claire Baty, Head of French at Wimbledon High School, considers the importance of spontaneous speech in the Modern Foreign Language classroom.

Like many a Modern Foreign Language teacher at this time of year, I have spent the last few weeks    conducting oral exams. I like this opportunity to work with my students individually. I see oral exams as the chance to shine, to show off all the hard-work done during the year. It’s the only exam that the student is in control of, that they can steer down their path of choice. So why then, do the students have such opposing views? They hate oral exams. Clearly nerves are natural, let’s face it, no one relishes being put on the spot and being recorded at the same time. But the problem goes deeper than simply being nervous about an exam. There is often a big discrepancy between the quality of written work compared to the quality of spoken work. A student who can write at length using a range of subordinate clauses and move comfortably between time frames, reverts to simplistic sentences or one-word utterances when asked to speak spontaneously. Many a time has a parent expressed frustration at their daughter’s inability or unwillingness to speak in the target language when on holiday, despite their excellent grades in MFL at school.

Why is any of this important, you might ask. If the students are getting good results, then does it matter? I would argue that yes, it matters a lot. As a French teacher it is my job to enable my students to communicate, and true communication is not about writing an essay, learning it off by heart and reciting it under exam conditions. True communication is the desire to share experiences and ideas with others. In the words of Nelson Mandela “If you talk to a man in a language he understands that goes to his head. If you talk to a man in his language, that goes to his heart”.  Only by speaking to someone in their own language can we truly begin to understand them, their identity and culture.

 

“If you talk to a man in a language he understands that goes to his head. If you talk to a man in his language, that goes to his heart”.

 

Moreover, speaking another language enables you to experience different attitudes to life, relationships, food and environment. Adam Jacot de Boinod makes this point succinctly in the books “The meaning of Tingo” and “Toujours Tingo”. He states that words change the way we see the world. Take for example that in Hawaiian kapau’u means ‘to drive fish into the waiting net by striking the water with a leafy branch’. In Inuit, pukajaw means ‘firm snow that is easy to cut and provides warm shelter’. Jacot de Boinod realised that sometimes a dictionary can tell you more about a culture than a guidebook. (The meaning of Tingo, Adam Jacot de Boinod, p.ix, Penguin Books 2005). My students know this; they want to travel the world and be international citizens. They can see the importance of linguistic knowledge and cultural awareness in the workplace, and it is our job to guide them towards a more spontaneous approach to communication.

Whilst teenage anxiety plays a part in making students reluctant to speak in the target language, it is by no means the only cause. Confident, able students resist our advice to move away from pre-learnt speeches. There is a notable disjuncture between the language needed for a speaking exam in Key Stage 4 and the phrases that would be helpful as a tourist in the country. Whilst the new GCSE specifications go some way towards addressing this, the lack of requirement for transactional language means our students can converse at length about their holiday plans or environmental problems in their local area, but struggle if someone stops them in the street and asks for the time, they panic if the waiter in the café doesn’t stick to his side of the role play when ordering a chocolat chaud! Infrequent lessons in an artificial classroom environment further compound this difficulty, not at all helped by the fact that when the students do muster up the courage to speak in French, the French have a tendency to reply in English.

Guy Claxton, an academic, a cognitive scientist and leading educationalist specialising in well-founded ways of enhancing intelligence, spoke to us recently about The Learning Power Approach and the importance of teaching learners to teach themselves. He made a number of points that resonated with me in terms of their pertinence in this discussion. Firstly, that it is our job as language teachers to equip the girls with the traits and skills to cope outside of the classroom, to be independent enough to flourish. Secondly, if we want students to be able to do something, then we have to coach it, build it up gradually by providing them with structured opportunities to speak spontaneously:

  • Key phrases on the wall and learning mats to help build complexity into spontaneity
  • The use of listening exercises to model the language we want the students to use rather than just to assess their understanding of a topic
  • Effective vocabulary Focussing on verbs to enable responses in complete sentences.
  • Scaffolding is necessary in Y7 and 8 to build confidence.
  • Target language use needs to be built into classroom routines and bravery must be rewarded.
  • Support should be gradually withdrawn in Y9 so that by Y11 spontaneity is more likely.

These practices are essential to good language teaching, they are the bones of our lesson plans. But whilst they might encourage pupils to talk, do they encourage spontaneous speech? Gianfranco Conti, PhD and co-author of ‘The Language Teacher toolkit’, winner of the 2015 TES best resource contributor award and founder of www.language-gym.com would argue that true spontaneous speech is without prompts. “[speaking tips] refer but to the very embryonal stage of spontaneous talk, what […] I refer to as the ‘imitative’ stage. However, in order to bring our learners from the ability to ‘parrot’ phrases on the wall or on writing mats to what applied linguists call ‘autonomous speaking competence’ it takes way more”. We need to expose our students to native speakers, whether that be through trips, language assistants, other bilingual students or simply authentic audio and visual material such as radio, music and films. We need to encourage peer to peer interaction and exercises that focus on the communication of a message and less on the accuracy. Cloze exercises, speed dating conversations and dictagloss should be prioritised.

Here at WHS, Modern Foreign Languages are hugely popular. Students enjoy the variety of an MFL lesson and the satisfaction gained from learning something brand new. French, German and Spanish continue to be popular choices at GCSE and A Level. Speaking exams will always be nerve-wracking but I would like my students to feel proud when they can confidently develop their answers. I would like them to experience a sense of joy at being completely understood when speaking French in France. I want to enable them to speak with the same enthusiasm and conviction with which they express ideas in their mother-tongue. Above all I would like my pupils to want to communicate at every given opportunity. Ce n’est pas la mer à boire, non?

Should standardised exams be exchanged for another form of assessment?

wimbledon logo

Jasmine (Year 11) explores the merits and weaknesses of exams as the formal assessment of intelligence, discussing whether an alternative should be introduced that suits all students.

Exams – the bane of existence for some but an excellent opportunity to excel for others. Thought to have been founded in China, with the use of the standardised “imperial exam” in 605 AD, they are the education system’s way of assessing the mental ability and knowledge of students whilst also creating a practical method of comparison to others in the country. They are therefore an important factor and indicator for employers. But does this strict, tight method really work for assessing intelligence or is it just a memory game that is only achievable for a select few?

I asked 80 students in a survey if they think that exams should be exchanged for another form of assessment and the results concluded that 78% agree that they should. However, when asked about their reasoning, it was mostly due to stereotypical dislike for the stressful period. Some who agreed with the statement also mentioned the unrealistic exam conditions that would not occur in daily life. An example was set forth that during a language oral exam a great amount of pressure is put on the students causing them to become nervous and not perform to their best ability. However, in a real-life conversational situation they would not have to recite pre-prepared answers and the pressure would be taken off so the conversation would flow more naturally. This shows that although someone may have real fluency and talent for the language, their expertise will not be notified and rewarded accordingly

Among many students, examinations are accused of being memory tests that only suit a certain learning style; and the slow abolishment of coursework at GCSE level is contributing to this. This could be shown by the fact that many people in the country have learning difficulties such as dyslexia. These students may be particularly bright and diligent workers however, their brains do not function in the way exams rely on them to. Nonetheless, if they are put in front of a practical task that they have learned to do through experience, they are deemed to be far more knowledgeable and perceptive. Studies show that by learning something consistently for a long period of time it stays in our memory but though it is important to ingrain essential facts into our brains, especially at GCSE level, GCSEs are mostly comprised of learning facts over a period of around 2-3 years and then a final exam at the end; which does not particularly show consistent learning and is more just an overflow of information.

Stress levels caused by the lead-up, doing, and waiting period for results that subsequently follows are also a major factor in the argument that traditional standardised tests should be augmented. According to the NSPCC, from 2015-2016 there was a 21% increase in the likelihood of counselling sessions being for 15-18 year olds affected by exam stress many of whom would be doing GCSEs and A Levels. Some say that the stress these tests cause is necessary for success and mimics the stresses of the real world; but how essential are some of these exams like non-calculator Maths papers when nowadays most people of have calculators on their phones? Exams are also said to create healthy competition that prepares people for the struggles and competitive nature of the modern working world and also motivates students, but can’t this be done with another form of assessment that is more suited to the individual student?

However, the use of different approaches to examination may, in fact, lead to the risk of the test being corrupted. This would mean that grading would be mainly subjective and there would be more scope for unfair advantage for some rather than others. The restrictive nature of our exams today with a set time, set paper and set rules does ensure that fairness is a priority but is the actual exam really the most equal way to test so many different students?

Standardised exams are not the best way of determining the knowledge and intelligence of students around the world. This is due to the stress and pressure they cause, the fact that they are only appropriate for certain learning styles and their ill comparison to real life events in the working world. Changing the form of these assessments may, however, cause grades to be unreliable. My suggestion would be smaller and more practical examinations throughout the course that all contribute to the final grade as this puts less pressure on the students and helps those who rely on different learning strategies to excel and demonstrate their full potential.

The creative-academic problem: why we should value the creative curriculum.

Richard Bristow, Director of Music at WHS, looks at recent developments in the creative curriculum.

News report:

‘Creative industries worth almost £10 million an hour to economy’

Dept. of Digital, Culture, Media & Sport, January 2016

 

‘Creative industries grow twice as fast as UK economy in 2015-16, making up 5.3% of the economy’

Economic Estimates for 2016 Report

 

‘The government is aiming for 90% of Year 10 pupils to be studying the EBacc…by 2025’

Telegraph April 2018

 

 ‘Arts education should be the entitlement of every child’

Nick Gibb, Schools Minister, April 2018

 

 

‘Music could “face extinction” in secondary schools in England, researchers have warned’

BBC, March 2017

 

“A combination of cuts to school budgets and the consequential loss of specialist teachers has created a skills loss”

Prof Colin Lawson, Director of RCM, March 2018

 

‘How to improve the school results: not extra maths but music, and loads of it’

Guardian, October 2017

 

‘Axe looms for county music service: 7000 school instrumental lessons impacted’

Sussex Express, April 2018

 

The creative-academic problem:

The news reports above pose a dilemma. On one hand, Creative Industries in the UK have had a celebrated few years, adding significant value to the UK economy; on the other, cuts to creative (and specifically Music) education in secondary schools in England paint a bleak picture of an emerging skills gap, threatening this very success.

A recent BBC survey with data collected from over 1200 schools – some 40% of all secondary schools in England – revealed a damming 90% of schools have made cuts to staffing, resourcing or facilities to at least one creative arts subject over the last year. Music, Drama, Art and Design and Technology all find themselves squeezed because of a growing need to teach ‘academic’ subjects – a key feature of the new English Baccalaureate (or EBacc for short), which has become a compulsory part of state education in England. This division of academic and creative is a central problem in education. After all, we want creative solutions to scientific problems, and an academic approach to art allows for increased understanding. Yes, the Theory of Relativity is complex, but so is Schenkerian musical analysis. Why do we have to choose? Why can we not value both?

90% of schools have made cuts to the creative arts over the last year

BBC Survey

The EBacc

Introduced from 2010, the EBacc seeks to counter the fall in numbers of pupils studying foreign languages and sciences (see here) by measuring pupil progress in English, maths, the sciences, a language and either history or geography.

EBacc Subjects:
  • English Literature
  • English Language
  • Maths
  • Combined or Triple Science
  • History or Geography
  • A Language – ancient or modern

The Government originally set a target of 90% of Year 10 pupils studying the EBacc to be achieved by 2025. This however has recently been reduced to 75% in the latest Conservative manifesto – not to allow for a broadening of the EBacc subjects, but because there are not enough Modern Foreign Language teachers to allow the original target to be met. The cuts to language teaching are a little more established than the more recent cuts to creative subjects, showing the ‘boom and bust’ approach to education in the UK in the 21st Century (see here for more information on MFL provision).

 

However, despite the significant press coverage of schools closing their music departments (see here) and some schools even charging pupils to study Music at GCSE (see here) the data seems to show a mismatch. A New Schools Network report (which can be viewed here) analysing data for all state school GCSE entries between 2011/12 and 2015/16 actually shows a rise in the number of pupils sitting at least one creative subject at GCSE over the period and confirms that pupils who achieve the very best EBacc grades are likely to have also achieved well in a creative arts subject. However, it also shows other issues:

  • A reduction in the funding in the creative arts in secondary schools, suggesting state schools have ‘misunderstood’ the requirements of the EBacc to prioritise named subjects at the expense of non-named subjects
  • The impact of this funding reduction has not yet impacted achievement, but it may well do so in the future
  • That the Government can be more enthusiastic about the value of the creative arts
  • That the biggest decline in the take up of creative subjects was in the Independent Sector, who are not required to follow the EBacc, recording a 12.9% fall in take up of at least one creative arts subject at GCSE from 2011/12 to 2015/16
  • That the independent sector has seen a 30% fall in total GCSE and IGCSE entries over the 2011/12 to 2015/16 period

These points add to the confusion. If the picture is as positive as this report suggests, why are we seeing reports suggesting the ‘extinction’ of subjects like Music in state schools? If the picture is one where the evidence shows the EBacc has not declined the provision of arts education, why are Music departments and Music hubs closing? How can pupils access an arts curriculum if the department is not physically there?

Perhaps the biggest problem with the report is that it does not give data for individual subjects. It might show a rise for the pupils studying the creative arts, but it does not show a rise in all creative subjects. So whilst the numbers studying Art or Design and Technology (part of the STEAM initiative) might have risen sharply, this might be at the expense of Music or Drama, who might have seen a strong decline in education provision. This data is needed to truly understand the impact of the EBacc on individual creative subjects.

Partnerships

The NSN Arts Report also calls upon Art Providers to be more active in helping to engage pupils in the creative arts (see Kendall et al, The Longer-Term Impact of Creative Partnerships on the Attainment of Young People). This might be via art organisations setting up free schools, or more likely to encourage art organisations to engage with a cultural education programme with school pupils.

One such example is the Philharmonia Orchestra who have recently completed their Universe of Sound and 360 Experience Project. This exciting project uses virtual reality to film the Philharmonia Orchestra performing Holst’s The Planets allowing people to experience and learn more about the symphony orchestra. They have also commissioned new music by Joby Talbot to give a contemporary interpretation to writing music to represent time and space. I have been lucky enough to do some work on the education resources for this programme over the Easter break, and it is hoped that this experience can offer pupils, parents and teachers a way into linking Music to other STEAM subjects, rising cultural engagement and musical understanding. If you would like to learn more about the project, please visit here for more information.

Final thoughts

If we are to view subjects by their perceived academic worth, then it can be useful to view how the subject has been taught through history. Whilst many would view Music as now being a creative (and not academic subject), it is important to remember that Music as an academic and theoretical subject was one of the Ancient Greek seven liberal arts and a part of the quadrivium (arithmetic, geometry, music and astronomy) which was taught after the trivium (grammar, logic and rhetoric). Rather than being marginalised as a non-academic subject, we should relish the fact that Music can and has informed scientific understanding throughout history. Practical study of Music is obviously a useful skill, but it is the academic and theoretical knowledge that comes from advanced study of the subject that can really inspire the very best musicians. Perhaps we should redefine STEAM to STEAMM – Science, Technology, Engineering, Art, Maths and Music.

Or maybe we should lose the hierarchy altogether; perhaps, instead of putting subjects against each other in some fruitless competition, we should value passion, enjoyment and the love of learning, seeing the subjects as having equal worth. As Ian McEwan states:

“Science, the humanities and the arts are all forms of investigation,

driven by curiosity and delight in discovery.

The child who flourishes in one should flourish in the others.

The best, the liveliest education, would nourish all three.”

Richard Bristow, 21st April 2018

Follow @Music_WHS on Twitter

Why being bad at Maths just doesn’t add up

By Helena Rees, Head of Maths.

Many still see people who are good at maths as slightly weird, geeky, uncool. Why is this? Why should we study maths?

A couple of years ago Professor Brian Cox hosted ‘A Night with the Stars’ on the BBC. From the lecture theatre of the Royal Institution, he undertook to explain among other things how diamonds are made up of nothingness and how things can be in an infinite number of places at once. He took the audience, made up of famous faces, celebrities and scientists, through some of the most challenging concepts in physics, using maths and science experiments as he went along. It was a truly fascinating programme and if nothing else demonstrated the power of numbers and the speed with which they can make a grown man cry. Jonathan Ross (43 mins approx) was invited to assist Brian Cox in a maths calculation using standard form. The look of sheer panic on Ross’s face, followed by him saying, “This is the worst thing that’s happened to me as an adult” and “I’m sweating”, just about sums up many people’s attitude towards maths.

Mrs Duncan spoke to the whole school this week and used this example. Imagine going out for dinner with six friends and the bill comes. When the time comes to split the bill between seven, the bill is shuffled to the maths teacher or accountant with a slightly shame-faced look saying, “I am rubbish at maths” or “I couldn’t do maths at school”. Imagine, however, that same group of people sitting down to order and someone asking for the menu to be read out because they can’t read it. Few will admit that they can’t read as the stigma of this would be hugely embarrassing. Yet no such reservations exist for maths with individuals almost boasting about their lack of maths ability. Why is this?

Many still see people who are good at maths as slightly weird, geeky, uncool. A PhD in Maths or Physics at the end of a name tends to conjure up images of social awkwardness — people more to be pitied. On the whole surveys of attitudes over the past 50 years have shown that the cultural stereotype surrounding ‘scientist and mathematician’ has been largely consistent — and negative. However, things are changing, in November 2012, President Obama held a news conference to announce a new national science fair. “Scientists and engineers ought to stand side by side with athletes and entertainers as role models, and here at the White House, we’re going to lead by example,” he said. “We’re going to show young people how cool science can be.” The idea that scientists, mathematicians and engineers could attain iconic status is exciting.

The popularity of television shows such as ‘Think of a Number, ‘Countdown’ and more recently the use of numbers in ‘Numb3rs’, and ‘How Do They Do That?’ have boosted the public’s perception of Maths. CSI has done more for boosting number of students of forensic science than any careers fair. The Telegraph recently reported that students who had a Maths A Level earned on average £10,000 more than a student without. Perhaps statistics like these would encourage more students to take the subject seriously. A report by think-tank Reform estimates that the cost to the UK economy between 1990 and 2008 of not producing enough home-grown mathematicians was £9 billion, such is the value of maths expertise to business.

Marcus du Sautoy, second holder of the Charles Simonyi Chair in the Public Understanding of Science at the University of Oxford says he can’t understand the pride there is in being bad at Maths. “It’s bizarre why people are prepared to admit that because it’s an admission that you can’t think logically. Maths is more than just arithmetic. I would rather do business with someone who admits they’re good at Maths. You don’t get that in the Far East. In Korea or China they’re really proud of being good at Maths because they know the future of their economies depend on it, their finances depend on it. Mobile phones, the internet, Playstations and Google all depend on Maths,” he says. “If people realised that, then they wouldn’t poke fun at it so easily. In today’s information age, Mathematics is needed more than it ever was before – we need Maths. Problem solving skills are highly prized by employers today. There is an increasing need for Maths and the first step needed is a change in our attitudes and beliefs about Maths.”

It is true that many of us will not do another quadratic equation or use trigonometry in our daily lives. However, Mathematics is more than just the sum of subject knowledge. The training to become a scientist or an engineer comes with a long list of transferable skills that are of enormous value in the ‘outside world’. Communication skills, analytical skills, independence, problem-solving skills, learning ability — these are all valuable and at the top of Bloom’s taxonomy. But scientists, mathematicians and engineers tend to discount these assets because they are basic requirements of their profession. They tend to think of themselves as subject-matter experts rather than as adaptable problem solvers.

We have all heard of Pythagoras and his famous theorem. The theorem states that the sum of the squares on the two shorter sides of a right angle triangle sum to the square on the hypotenuse, more commonly shortened to a2 + b2 = c2. In 1637 Pierre de Fermat postulated that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. For example to a3 + b3 = c3 After his death, his Fermat’s son found a note in a book that claimed Fermat had a proof that was too large to fit in the margin. It was among the most notable theorems in the history of mathematics and prior to its proof, it was in the Guinness Book of World Records as the “most difficult mathematical problem”.
(https://plus.maths.org/content/fermats-last-theorem-and-andrew-wiles ) However, in 1994 Andrew Wiles, published a proof after 358 years of effort by Mathematicians. The proof was described as a ‘stunning advance’ in the citation for his Abel Prize award in 2016. You can watch an interview with Andrew Wiles by Hannah Fry where he was interviewed this week in the London Public Lecture Series organised by Oxford University.

In a recent article Wiles commented “What you have to handle when you start doing Mathematics as an older child or as an adult is accepting the state of being stuck. People don’t get used to that. They find it very stressful.” He used another word, too: “afraid”. Even people who are very good at Mathematics sometimes find this hard to get used to. They feel they’re failing. “But being stuck, isn’t failure. It’s part of the process. It’s not something to be frightened of. Then you have to stop. Let your mind relax a bit…. Your subconscious is making connections. And you start again—the next afternoon, the next day, the next week.”

Patience, perseverance, acceptance—this is what defines a Mathematician.

Hilary Mantel, novelist and writer of Wolf Hall writes “If you get stuck, get away from your desk. Take a walk, take a bath, go to sleep, make a pie, draw, listen to music, meditate, exercise; whatever you do, don’t just stick there scowling at the problem. But don’t make telephone calls or go to a party; if you do, other people’s words will pour in where your lost words should be. Open a gap for them, create a space. Be patient” Perhaps Mathematicians and novelists are so different after all?

When it comes to Mathematics people tend to believe that this is something you’re born with, and either you have it or you don’t and this is the common refrain at parents evenings. But that’s not really the experience of Mathematicians. We all find it difficult. It’s not that we’re any different from someone who struggles with Mathematics problems in junior school…. We’re just prepared to handle that struggle on a much larger scale. We’ve built up resistance to those setbacks. A common comment on parents evening is to delegate the Maths homework to dad as that is ‘his thing’. What message does this give our girls of today? That this is a subject that boys are good at.

Luckily for us here at Wimbledon High School we have a strong culture of doing well in Maths. We have excellent results at iGCSE and there are over 50 girls this year in year 12 alone studying some form of post 16 Mathematics qualification with a view to a STEM career. The new Steam room is an exciting initiative to be part of. A recent article in the National Centre for the Excellence in Teaching of Mathematics journal, asked how can we get more girls to study A Level Maths. The answer at WHS? Keep doing what we are doing well and continue to be excited and positive about the beauty and the magic of numbers.

 

The long and winding road: how factual recall tests can effectively support linear examination courses

Wimbledon High History

By Emily Anderson, Head of History.

Think back, if you can, to your own History studies at school, whether these were months, years or perhaps decades ago. For most, the content covered becomes, over time, increasingly hard to recall. My current grasp of the French Revolution, for example, which I studied at AS Level, is embarrassingly basic now, almost 15 years later, as it is something I have rarely had need to revisit. At Parents’ Evening, parents smile wryly at vague memories of the Corn Laws or the Spinning Jenny (not meaning to undermine their importance, but their ubiquity in the collective memory of British adults is truly extraordinary) and voice envy at the breadth of opportunities available in the current History curriculum.

Instead, it is the broad conceptual understanding of, say, the nature of power, as well as the skills that remain, and these which lie at the heart of the purpose of History education for our department here at WHS. Empowering our students to participate in the academic discourse of History is our core aim, to enable them to engage critically with the world around them in their future lives. It is, however, impossible to participate in this discourse without what has been termed ‘fingertip knowledge’ as well as more conceptual ‘residual knowledge’: to secure meaningful progress in History, both need to be developed (Counsell, 2000). As argued recently in Teaching History where dialogue around cognitive psychology is increasingly evident, ‘fluent access to a range of types of knowledge is what enables historians to participate in some of the more sophisticated forms of historical discourse’ (Fordham, 2017).

Recent changes to A Levels (AL) have brought how we secure this fingertip knowledge into focus. The nature of the new linear exams mean there is more demand for a greater volume of content to be retained over a longer period of time. The importance of detail is evident both from reviewing past papers and from our experience in examining at AL last summer.

To approach this, we reflected on our experience of nurturing fingertip as well as residual knowledge at GCSE, where the linear model is, of course, long established, as is our practice of setting factual recall tests at the end of each topic. Our evaluation of the latter is below:

Advantages Disadvantages

It is classic retrieval practice, which results in stronger storage and retrieval strength (Fordham, 2017).

It encourages an extra stage of revision early in the course before more high stakes testing kicks in for mocks and terminal exams, reducing the pressure on Year 11.

It helps lead to great results (above 75% A* in the past three years).

Our tests were much too challenging – becoming notorious amongst our students and sapping morale.

They were no longer fit for purpose – pupils would never need to recall such specific detail, especially after the reform of the CIE IGCSE Paper 4 in 2015 which removed such questions.

 

Therefore, we have changed the structure of our tests to open ended questions. At IGCSE these are in the style of 4 mark recall questions. At AL I am experimenting with questions taking the form ‘cite two pieces of evidence which could be used to support an argument that…’, or similar. To try to tackle the issue of relevant but vague answers, I have awarded bonus marks at AL for detail to encourage both a conscious choice in selecting evidence (as pointed out by Foster & Gadd (2013)) and in-depth revision. All are now out of a uniform mark – 20 – to encourage comparison across topics and at different stages of the two years.

Furthermore, we have used the new AL structure to rethink when we test, in order to support maximum recall over the two years. Here, we currently have two approaches: retaining end of topic testing at GCSE in order to keep the advantages identified above, but utilising spaced tests at AL (the benefits of which are argued by, amongst others, Laffin (2016) and Fordham (2017)) by revising and testing existing knowledge on a topic before the next stage of it is covered. This lends itself particularly well to the unit on the British Empire from c1857-1967: in the past few weeks, my Year 13 class have sat tests on the increasing independence of the Dominions and on India, both in the period from c1867-1918, before studying inter-war developments. Students then complete their own corrections, consolidating the learning and identifying areas for development. During the revision period at AL, they can also undertake the same test several times citing different evidence. My 2017 cohort had, at their own suggestion, a star chart to record how many times they had undertaken a test for each area of the course, broadening their evidence base each time.

Whilst I hope that this gives a snapshot of the department’s current and very fledgling thinking, I would be mortified if it was taken to show that we are overly focussed on factual recall testing in the department. We are not. Tests of course never can and never will be the ‘be all and end all’ in terms of assessing student progress, but approaching them critically can only be a good thing.

References and further reading

Counsell, C. (2000). Historical knowledge and skills: a distracting dichotomy . In James Arthur and Robert Phillips, Issues in history teaching (pp. 54-71). London: Routledge.

Fordham, M. (2017). Thinking makes it so: cognitive psychology and history teaching. Teaching History, 166, 37-43.

Foster, R., & Gadd, S. (2013). Let’s play Supermarket ‘Evidential’ Sweep: developing students’ awareness of the need to select evidence. Teaching History, 152, 24-29.

Laffin, D. (2016). Learning to like linear? Some ideas for successful introduction of the new A Levels. Historical Association Conference workshop.