21 June 2016

5 Maths Gems #58

This is my 58th update from the world of Maths EduTwitter. Here I summarise some of the latest ideas and resources for teaching maths.

1. New GCSE Revision Resources
Thanks to @pixi_17 for sharing a fantastic set of revision resources for new GCSE students. This well designed set of 9 - 1 revision booklets for four different grade levels (eg 'Aiming for a Grade 5') has come in handy for my Year 10s this week, as they revise for their upcoming end of year exams.
2. Whack a Prime 
I love the Is This Prime? game but some of my students get disheartened by low scores. It's quite a stressful game to play! Slow it down with this alternative 'Whack a Prime' game from @prehmet.
3. Polygons Angles
This new tool from MathsPad is awesome. I've seen similar tools before but this is the best yet. Click 'play' below for a demonstration.

4. Key Stage 3 Scheme of Work
At this time of year many schools are thinking about their schemes of work for next year. I really like the Year 7 Scheme of Learning from the White Rose Maths Hub. Schools looking to change their approach at Key Stage 3 (less 'spiral', more 'mastery') will find this helpful. There's primary schemes here too.
5. Interesting Exam Questions
Did you particularly like any of the questions in this year's GCSE papers? Share them on Twitter! For example this question was in the Edexcel Linked Pair Methods 2 paper - it will come in handy when teaching/revising recurring decimal proofs in future years.
This circle theorems question from Edexcel Methods 1 was a bit harder than usual - there's lots to do here! Have a go at it.
For A level teachers, this MEI question is really nice - it was sent to me by a student who got stuck on it when revising.
Did you catch my latest posts? One was about my day at researchED Maths and Science and the other was about maths school trips.

On Monday I presented at LIME Oldham, focusing on mathematical methods. My slides are here.

Because I'm an A level teacher, exam period is still in full swing for me. The last exam is S2 next Monday - phew! I'm looking forward to Edexcel's conference in Warwick on 2nd July, I should be far less stressed by then!

I've got a date in mind for #christmaths16 - Thursday 22nd December - please keep it free! You can read about last year's event here. This year it will probably have a slightly different format - less CPD, more enrichment - but I will maintain the 'Christmas party for maths teachers' social element. Watch this space!
Finally, I'll leave with you with this nice activity from James Tanton that I found, alongside other interesting tasks, in his 9th curriculum essay.

18 June 2016

Maths School Trips

This post provides a list of maths school trip destinations. I've included five recommendations here and I hope that readers will add their own recommendations and experiences in the comments below. I'd like to build up a comprehensive list of trip destinations all over the UK and perhaps include a few overseas trips too.

Although I've led a handful of maths trips over the years, I've never taken a whole year group out of school. All of the trips I've planned involved relatively small numbers of students. It can be hard to justify the cost and time involved in a cohort-wide maths trip. That's not to say that enrichment isn't important, but it may be more practical to bring maths enrichment into school (see my post about in-school speakers and workshops for details).

1. Legoland
I recently visited Legoland Windsor for a demonstration of their Lego Robotics - Space Challenge workshop. I was absolutely blown away by how brilliant it was. This maths and programming workshop is aimed at Key Stage 2 and 3. It was really fun and I think most kids would come away from this workshop excited about coding. Legoland is expensive for families to visit so I thought it wouldn't be a viable option for school trips, but I stand corrected. School trips are only £12.25 per head for secondary school children at peak time, plus £2.50 for a workshop and £3 for lunch, so you're looking at under £20 a head plus travel (full price details here). The robotics workshop takes 45 minutes and students spend the rest of the day enjoying themselves at Legoland, which is one of the country's top attractions.
2. Racecourse Days
Racecourse Days take place at 59 racecourses all over the country. Examples worksheets and schedules are available here. I've heard great things about these trips. Activities include visiting the Weighing Room, which is where jockeys prepare themselves before each race. Students find out about the relationship between weight and performance in racing and how the handicapping system works. In the 'Photo Finish' activity, students discuss distances, the condition of the going and other factors that may affect a horse’s performance. They also learn about the technology of the photo finish and winning distances, plus judge some close finishes for themselves. Sounds fascinating to me - and it's all free! So the only costs involved are travel, lunch, insurance and school administrative fees.

3. Bletchley Park
Bletchley Park, near Milton Keynes, was Britain's main decryption establishment during World War II. It's a popular destination for school trips. It offers a variety of facilitated one hour workshops, suitable for Key Stages 3, 4 and 5, with specific focus on history, mathematics and computing. The cost is £10 per head. Information about secondary workshops is available here. The maximum capacity for students is 180 so if you come from a large school (mine has 240 in a year group) you'd have to split the trip over two days.

4. London Museums
There are loads of destinations for maths school trips in London. I have listed a few here:
The Bank of England Museum. 
This is my personal favourite (I did the graduate training scheme at the Bank of England back in 2002). The museum's one hour 'Pounds and Pence' talk is aimed at Key Stage 2 and 3 and encourages students to think about the value of money and prices and their spending and saving decisions. 'Keeping on an even keel' is aimed at Key Stage 4 and 5 - it explains what the Bank does to an even keel keep inflation low, maintain trust in its banknotes and keep the financial system stable. Admission and presentations are all free of charge. Note that these trips are only suitable for smallish groups (up to 50). 
Royal Museums Greenwich (including the Royal Observatory, National Maritime Museum and the Queen's House) are good destinations for maths school trips. We visited them during my PGCE course and I was particularly taken with the Queen's House - I loved its wonderful Great Hall (a perfect cube).
Workshops at the Royal Museums Greenwich include 'Maths and the Milky Way' for Key Stage 3 and 4, in which students explore the scale and variety of planets in our Solar System and in other planetary systems in our Milky Way Galaxy using a range of mathematical techniques.
The British Museum
The British Musuem's two hour teacher-led Maths Challenge looks like fun. Groups of students rotate through up to nine activities in different galleries, completing challenges which focus on developing students’ mathematical thinking. I particularly like the task in which students look at a colossal granite arm in the Egyptian sculpture gallery and attempt to determine the size of the statue that this arm came from. The maximum group size is 70. 

The Mathematics Gallery at the Science Museum
The new maths gallery is due to open in December 2016 and will undoubtedly be an excellent destination for maths school trips. Similar locations further afield include the Museum of Mathematics in New York and the Mathematikum in Germany.

5. Maths Lectures
The most inspiring thing I've done since becoming a maths teacher was attend a day of Mathematics in Action lectures. I took 20 Year 12s along and it was brilliant. Maths Inspiration events are similar. This national programme of interactive maths lecture shows for 14-17 year olds features an awesome line-up of speakers such as Matt Parker and Hannah Fry.
The Royal Institution runs Masterclasses all over the country. These events typically take place on Saturday mornings, with schools sending a small group of their keenest Key Stage 3 mathematicians.

The best school trip I've ever been on was actually organised by my previous school's Physics Department. I paid to go along. We chartered a plane, took off from Gatwick in the evening and flew over the Shetland Islands. I saw the Northern Lights (amazing!) and awe-inspiring constellations. Our plane circled for a while then flew home the same night. It was one of the most wonderful things I have ever experienced, demonstrating the power of a good school trip.
The view from my window...

Please add a comment below to let me know your school trip recommendations and experiences. Thanks!

12 June 2016

researchED Maths and Science 2016

People rave about Tom Bennett's researchED events so I was very pleased when he announced a conference devoted to maths and science. I am a staunch advocate of subject specific development for teachers - I find that generic training (the type delivered at school INSETs - 'how to differentiate', 'how to do AfL', 'ideas for revision') is rarely of direct relevance to maths teachers.

The programme for researchED was outstanding - interesting sessions, excellent speakers and a well structured day. Unfortunately I had to leave at 2pm to go to a wedding so I was only able to attend four sessions. I've provided summaries of those sessions here.

Thankfully I had the opportunity to briefly catch up with a number of Twitter friends throughout the day. This event marked the start of a busy conference season for me - over the next two months I'll be speaking at four events - first up is LIME Oldham on 20th June where I'll be speaking about quadratic methods.

Comparing exam questions
The first session I attended was run by representatives of Ofqual. They spoke about their work in comparing the difficulty of exam questions, and the sources of bias in judgements made.

Remember when the new GCSE sample assessment materials were first released by the exam boards? Many maths teachers said that the Edexcel papers were notably harder than the AQA papers. After Ofqual had accredited the papers, concerns raised by teachers prompted them to do a study of exam question difficulty. They asked 43 maths PhD students to do a comparative judgement of 2,150 questions. Their task was to judge 'Which question is the more mathematically difficult to answer fully?'. In each case the PhD students were given two questions side by side on a screen and had to indicate the more difficult of the two. Later these questions were attempted by 4,000 pupils (presumably these pupils had actually been taught all the topics they were being tested on - it would be pointless to give pupils a question on a quadratic inequality if they hadn't yet been taught that topic).

The questions shown below are those that pupils found harder than the PhD students expected. Here we have a multiple choice decimal/percentages question and an order of operations question. People generally, and often incorrectly, perceive multiple choice to be easy.
The circle theorem question shown below is one that pupils found 'easier' than the PhD students expected. Of course, this doesn't mean this is an 'easy' question - it could well mean that teachers have anticipated the challenges and taught this topic well. At the end of the session someone suggested that this indicates that teachers are doing something right, a comment which was laughed off but is actually rather important.
I was frustrated by the decision to use PhD students instead of experienced teachers in this study - surely teachers have a far better idea of what pupils will find hard. Ofqual realised this too, and revised the process accordingly. Prior to accreditation of the new science GCSE, they asked science teachers to undertake a comparative judgement. Interestingly it turned out that the science teachers' judgements weren't as 'accurate' as we might expect. Ofqual identified a number of sources of bias that might explain the discrepancy. For example questions with a large word count were often judged to be more difficult than pupils actually found them.

I do wonder whether it is helpful to judge a question by simply glancing at it - surely one needs to complete a maths question to properly assess how hard it is. Twenty seconds a question doesn't sound like enough.

A similar project will be undertaken over the coming months for the new A level sample assessment materials. This time Ofqual will ask maths teachers for their judgements prior to accreditation. Given the use of grade boundaries as a levelling mechanism, the exercise is of limited value in the grand scheme of things. It's interesting though.

Kris Boulton is an excellent speaker and I very much enjoyed his session on Engelmann's ideas. Engelmann's work provides detailed guidance on how to teach - scripts in some cases. I'm rather opposed to the scripting of lessons but his 'my turn, your turn' stuff is intriguing.

We rarely question whether our instructional methods are effective. I want to learn more about different techniques for explaining mathematical concepts. I think it's really important.

Kris talked us through how he has used minimally and maximally different visual examples to help students understand concepts - he showed us slides including triangles and surds and it occurred to me that teachers would benefit from sharing a large bank of these slides. I have dabbled in this kind of explanation recently but I'm not an expert in creating a logically faultless sequence of examples.
Times tables
I like Times Table Rockstars very much. I think that every primary school should use it. It works well at Key Stage 3 too. Bruno talked us through his ongoing research from his vast data set - 1.5 million questions answered in a day! There's lots to explore.
On the slide shown above we can see which questions are commonly answered incorrectly (shown in red). Interestingly children get 9 x 3 right more often than 3 x 9. When the bigger number comes first they are more likely to answer correctly.

Bruno has recently surveyed students who use Times Table Rockstars, using a selection of questions taken from the recent PISA study. The results suggest that as students get older, their intrinsic motivation drops (eg measured by responses to questions such as 'I do maths because I enjoy it') as does their self-concept (ie whether they think they are good at maths). Maths anxiety is relatively low.
The highlight of Bruno's session was the delightful group of students who were there to demonstrate how Times Table Rockstars works and talk about their experiences. These highly competitive students are amongst the country's fastest times-tablers. Their demonstration was jaw-droppingly awesome. I can not believe how fast they were able to answer the questions. It was amazing.

The students were very articulate, and rightly proud of their achievements. They all spoke about enjoying the competitive nature of the programme and how their desire to be faster than their friends motivated them to practise a lot. I suppose the big question is whether this fluency in times tables gives significant advantages later on. Will these students all go on to be amazing mathematicians?

I sat next to Nisha de Alwis who pointed out to me that all the students in attendance were boys, which raised the question of whether this indicates that boys are 'better' at times tables. Or perhaps this particular programme (ie the gamification and competitive nature) appeals to boys. Bruno shared some interesting data from the survey he conducted and will continue to explore the gender question.

I got to have lunch with these lovely people. Pictured left to right are Jemma (@jemmaths), Hilda (@frimymaths), Nikki (@mathszest), Julia (@tessmaths), Nisha (@nishadealwis), Laura (@mathsatschool), Andrew (@AQAMath) and Tom (@DrBennison).
Primary Maths
In my post 'Knowledge Gaps' I wrote about my desire to learn more about how maths is taught at primary school. There seems to be a disjointedness in the way things currently work between primary and secondary. Jack Marwood is a primary school teacher with a maths degree and I attended his session '10 things you should know about primary maths'. The slide below shows the potential confusion arising from the use of 'remainders' in division in primary school.
Jack recommended the work of Hung-Hsi Wu at Berkeley, particularly The Mathematics K-12 Teachers Need to Know.

Jack's slides from his reasearchED presentation are available here.

I look forward to reading other bloggers' write-ups of the sessions they attended. It was very much one of those 'I want to be in three places at once' sort of days. Thanks to Tom Bennett and Oxford University Press for an excellent event.

Finally, I must mention the venue! The University of Oxford Mathematical Institute is wonderful - look at the beautiful Penrose Tiles outside. The building is named after Andrew Wiles, the mathematician who proved Fermat's Last Theorem. I had only been there once before, representing the Bank of England at a careers fair, and I actually bumped into Stephen Hawking that day! Amazing. Maths is awesome, isn't it?

5 June 2016

5 Maths Gems #57

This is my 57th update from the world of Maths EduTwitter. If you're not a tweeter, or if you're wondering whether you've missed anything, here's a summary of some of the latest ideas and resources for teaching maths.

1. Artful Maths
Clarissa Grandi (@c0mplexnumber) has created the delightful website artfulmaths.com. It provides resources and inspiration so that children can experience the beauty of mathematics. There's a large section on how to deliver arty maths lessons on topics such as Celtic Knotwork, Mazes and Labyrinths, and Insect Symmetry. There's also display ideas for maths classrooms, guidance on running an origami club, recommendations of puzzle games and lots more. Do check it out.

2. GCSE Paper 2
This gem has a limited shelf life but is still worth sharing for those of you who see (or email) your Year 11s between now and their second GCSE paper on Thursday. Numerous helpful teachers have created 'best guess' papers for the upcoming exam, based on the topics that came up in Paper 1. Superstar Emma Bell (@EJmaths) has collated them here. The collection covers both Higher and Foundation for Edexcel, OCR and AQA. It's well worth sharing the relevant papers with your students so they can focus their exam preparation on the topics that are most likely to be tested.
Some of these papers are from new website onmaths.com - this website instantly marks students' answers so is particularly useful for independent study. Thanks to Nikki (@mathszest) for first making me aware of this website.

3. Odd + Odd 
I just love this cute 'proof' shared by @ProfSmudge.

4. Maths Quickies
The Twitter account @MathsMastery tweets a daily #MathsQuickie like the example shown below.

Thanks to Mark Greenaway (@suffolkmaths) for pulling together these questions (and their answers) in PDFs and PowerPoints - they can be downloaded here

5. Parallel Lines
Don Steward has shared yet another fantastic resource on his wonderful blog Median. His PowerPoint and tasks for angles in parallel lines are excellent.
If you're doing any revision with GCSE, A level or Certificate of Further Maths students over the coming weeks, do check out my growing collection of revision clocks.

If you're currently teaching Year 10 then you might be interested in my recent post: Useful GCSE Questions from Linked Pair Papers.

Tom Bennison's summer event for A level teachers now has a website. The East Midlands KS5 Mathematics Day is on 9th August 2016 - it looks like an excellent event and I hope to see you there.

Have you signed @MathsMrCox's petition to show your opposition to the proposed Year 7 SATs resits yet? It's important that we do whatever we can to make our views known, rather than stand back and grumble about bad decision making. Please sign!
I've had a lovely half term celebrating my youngest daughter's second birthday. We were very lucky to benefit from a free trip to Legoland, in return for having a quick look at their maths school trips. I was hugely impressed by the workshop we saw (they got my 4 year old programming robots!) and will write a blog post about it soon. Although a family day out at Legoland is rather pricey, the school trips are very reasonably priced.

I'll leave you with this lovely puzzle from @YohakuPuzzle. Fill in the cells with different factors to get the products in each row/column.

31 May 2016

Useful GCSE Questions from Linked Pair Papers

If you attended any training on the new GCSE last summer then you may have been advised to look through Linked Pair papers for questions on new GCSE topics. It's a nice idea, but what teacher has time to trawl through past exam papers?

A year later, I've now taught Linked Pair (for the first and last time) so I have a better idea of whether the Linked Pair papers are actually helpful. Before you start searching through papers, here are a few caveats:
  • There are loads of topics in Linked Pair (spreadsheets, flowcharts, AER, linear programming, moving averages etc) that are not in the new GCSE.
  • There are some new GCSE topics that are not in Linked Pair such as functions, iteration and quadratic inequalities.
  • There are a few topics in Linked Pair that are in the new 9 - 1 GCSE - so this is where the papers are helpful (ie Venn diagrams, quadratic sequences, gradient of a curve and area under a graph). 
  • There are also some lovely challenging questions in Linked Pair papers - on familiar topics such as bounds, angles and forming equations - which might be helpful in lessons. I've shared some examples below. 
The easiest way to access past exam questions - rather than trawling through papers - is to use a test building tool (AQA's Exampro, Edexcel's ExamWizard, OCR's ExamCreator or Eduqas's Question Bank). I think these tools are really useful for creating resources such as homeworks and end of year exams, so a worthwhile investment for a maths department.

The topic 'compound measures' has grown in the new GCSE. It can be hard to find resources for teaching density so you'll be pleased to hear that there are some good questions in Linked Pair exams. There's a nice example below. I've put a worksheet on TES containing this question and similar density questions.
Applications 2 Higher, Edexcel, June 2015, Question 16

Compound measures links really nicely with bounds - it's good to see an exam question that combines these two topics.
Applications 2 Higher, Edexcel, June 2013, Question 17

Forming expressions
Linked Pair GCSE papers contain a lot of questions in which students have to form expressions. This question takes some thinking - it really stumped some of my students:
Applications 1 Higher, Edexcel, June 2014, Question 20

This question tests understanding of angles in polygons:
Methods 2 Higher, Edexcel, Specimen Paper, Question 7

My class struggled with this angles question in their mock exam this year:
Applications 1 Higher, Edexcel, November 2015, Question 2

Did you like Hannah's sweets? Try this one!

Methods 1 Higher, Edexcel, November 2015, Question 24

Percentages and Ratios
The Applications module in Linked Pair contains quite a lot of wordy questions that EAL students may struggle with. Here are a couple of examples. These might be worth trying with new GCSE classes.
Applications 1 Higher, Edexcel, Specimen Paper, Question 15

Applications 2 Higher, Edexcel, Specimen Paper, Question 7

I have loads more examples of interesting Linked Pair questions, but in order to keep this post to a manageable length I'll leave it there!

What are your favourite GCSE questions? I'd love to see them.

26 May 2016

5 Maths Gems #56

This is my 56th update from the world of Maths EduTwitter. If you're not a tweeter, or if you're wondering whether you've missed anything, here's a summary of some of the latest ideas and resources for teaching maths.

1. Euro 2016 Maths
@dooranran makes lovely maths resources. I think teachers will really like the activities he has prepared for Euro 2016. Topics include vectors, Pythagoras, trigonometry, angles on parallel lines, time, pie charts and estimating the mean.

2. Pythagoreanmath.com
I discovered a fabulous website, pythagoreanmath.com, which has lots of cool content including an amazing section on Euclid's Elements. Do check it out. I also really like their 'Complete Explanation for Area of a Circle Formula'.
3. Find someone who can
@rondelle10_b shared an interesting idea which Phil Bruce (@pbrucemaths) adapted for maths. In 'Find Someone Who Can', students ask other members of the class to answer questions using the sheet pictured below. This develops students' mathematical communication skills as they are required to explain solutions to classmates. You can download the PDF from Phil's page of revision resources.
4. Colin Foster's Activities
I discovered Colin Foster's 'Instant Maths Ideas' books a while ago. I enjoy browsing through them every now and then - they're packed full of excellent tasks and questions.

For example, in Possible or Impossible? students are asked to decide whether each statement in the table below is possible or impossible. If it’s possible they are asked to give an example and if it's impossible they have to say why.
Another great activity, this time from Colin's chapter on rounding, is this task which looks at various scenarios in which students have to calculate 17 ÷ 5. In each case they have to think carefully about how to round their answer.
5. More from MathsPad
Another awesome free resource from MathsPad: Using a Calculator - Odd One Out. This activity addresses lots of common errors and misconceptions.
I've written quite a lot of blog posts since my last set of gems. Here they are in case you missed them:

I've also added some A level revision resources to TES - check out my TES Shop to download these for free.

There's loads going on in June and July - see my conferences page for details. First up for me is researchED Maths and Science in Oxford on 10th June - the programme is excellent so I'm really looking forward to it.

Bookings are now open for my session at LIME Oldham on 20th June where I'll be talking about quadratic methods. I also recommend that you book a place at Edexcel's Warwick Conference where I'll be presenting a session on 'The Wonderful World of Maths Resources'

Finally, I'll leave you with this set of maths exam meme posters - I put these up in my maths corridor a couple of weeks and they went down really well with students in every year group. They were created by Paul Collins (@mrprcollins) - you can download them here.