I saw that on a Numberphile video...

Last week, I was listening to a couple of year 7s discussing maths in class when I heard one of them say, "I saw that on a Numberphile video..." This was one of the highlights of my term because the video in question was not one I had promoted to the class. This term, I have introduced Google Classroom into my teaching and one of the excellent uses is to subtly promote Numberphile and other fantastic mathematical resources.

I'm going to go out on a limb here. I think Numberphile is the most inspirational resource I have seen for Mathematics. By using youtube, Numberphile has opened the door to complex Maths that to this point remained hidden deep and difficult to find for the vast majority of people. It is ground-breaking and has the potential to be as influential  as Martin Gardner's work in bringing maths to the masses and deepening understanding of Mathematics beyond basic numeracy. It goes way beyond other attempts to open up mathematics, such as Johnny Ball in the UK, who promoted mathematics in an entertaining manner, but failed to look deeply and demystify the subject. Numberphile embraces the complexity of mathematics and challenges the grey cells, and with the creation of Numberphile 2 (with its focus on the 'hard' maths) this looks to continue strongly.

A big thank you to Brady Haran (producer, editor and thoughtful interviewer of the Numberphile videos) and the inspirational contributors, including the fantastic and amusing Dr James Grime and Matt Parker (two of the main contributors to the channel). Dr James Grime has his own youtube channel singingbanana, which is full of mathematical gems. Matt Parker has been touring the country with the best show ever, Festival of the Spoken Nerd, his talk on spreadsheets zooms into the heart of computing and mathematics.

Other excellent youtube channels for maths include:
Vi Hart
Smarter Every Day (although there is some Science in this one as well!)

Khan Academy

I used Khan Academy for the first time with one of my teaching groups today. I have previously explored Khan Academy, from a student's point of view, trying out different tasks etc (I would strongly advise you do this before hand) and felt the time was now right to become a 'coach' and investigate its usefulness within the classroom. 

The group I chose was a GCSE retake group. I had managed to permanently book an IT room for once a week with this group and this made an opportune moment to experiment. I was pleasantly surprised how well the lesson went. It was very easy to set up a Khan Academy Classroom. I had already established a Google Classroom, so I highlighted every student and clicked to email them, then copied their email addresses and pasted this into Khan Academy. It took a couple of minutes to create a classroom and then for students to join by clicking the link sent to them via email.. 

I didn't set up any particular tasks or missions but invited the class to focus on the foundation area of Mathematics. With little instruction, students were quickly able to complete tasks. They became motivated by the points and awards. Soon this became competitive with some students. All students engaged and were on task. Students began asking for help when needed - I haven't yet started emphasising the online video support, but will start doing that from next lesson. Some students who had been less motivated suddenly started working very well. The class lost track of time and worked into their break time and at the end of the day, a student emailed to see if he was number one (as he had been working on tasks during his lunchtime). I will see if this level of enthusiasm persists. Students seem to enjoy not only the competitive side but the independence Khan Academy gave them.

The data on students and classes is very useful. Progress of the class is easily monitored time and this can be varied. Putting a leader-board on the interactive whiteboard inspired some students to be competitive - students worked hard to be at the top and to avoid the bottom place. Individual data was also useful to see where students were struggling and also what they were able to do. Generally, I think students found it motivating that they were able to complete easier tasks, but this will need to be monitored to ensure they can complete more complex tasks needed for their examinations.

My next step is to create a playlist of topics that will be suitable for my students and enable me to focus their attention on particular units. I have quickly set up Khan Academy Classrooms for my other groups, but are not planning to use it within the classroom, but to encourage them to practise their maths skills in their own time. I will be curious to see how it will work in this context.

Khan Academy provides clear and comprehensive support for teachers to help them on this journey - Khan Academy Support Materials

Book Review - The Chimp Paradox by Prof. Steve Peters

The Chimp Paradox provides the reader with a simple but effective model of how the brain works. Prof Steve Peters is a consultant psychiatrist who works in elite sport - notably with the very successful Sky Pro Cycling Team and Liverpool FC. He has received personal accolades from a long list of famous sports people such as Victoria Pembleton and Sir Chris Hoy (Olympic Gold Medallists), David Brailsford (Principal at Sky Pro Cycling Team), Steve Gerrard (England footballer) and Ronnie O'Sulllivan (Snooker player).

Peters is careful not to wrap himself as a guru but emphasises the science and research behind his model. He is not afraid to admit his model is a simplification of how the brain works. The book offers advice and exercises to improve the effectiveness of how your brain work, but the real power behind the book is explaining how yours and other people's brains work.

The model Peters uses is dominated by the Chimp and how to manage it. The brain is split into three personalities: the Chimp, the Human and the Computer. Put simply, the chimp represents our emotions and therefore it is important to understand not only how to manage our own chimps, but others.

From an education point of view, Peter's model could promote marginal gains within schools and students. It helps teachers empower their students to think constructively and productively, guiding them to become happy, confident and successful people (whilst avoiding being hijacked by their chimps). It provides a useful guide on how to interact with students, especially when their chimps are evident, which I have already found personally useful when dealing with potential conflict or emotional students.

I would highly recommend this book for anyone who wants to improve and have a better understanding of not only themselves, but those around them and the world we live in.

Google Classroom - Month One.

This September, I have been using Google Classroom. The story so far...

Google Classroom is very easy to use. It was relatively straightforward to invite students or get themselves to join the classroom and as they become used to the technology the process will become natural and simple.

I currently am a strong advocate of mymaths. It provides a comprehensive package for all school aged students. The user interface becomes more complicated post-16 and I use it less with A-level groups. However, for the majority of students it provides an efficient and straight forward method of completing homework. Firstly, you cannot lose it. Secondly, if you are stuck (including parents) it provides useful online lessons. Next, it marks work and provides instant feedback. Then crucially, it allows students to have further attempts (modifying the homework in the process). As a teacher, it provides me with spreadsheets which allow me to see how individual students/classes are performing on particular topics and links this with the old level descriptors and GCSE grades (soon to be old). This data can be easily used for report writing and for discussing with students and parents.

So why the need for Google Classroom. Currently, I am mostly using Google Classroom as a noticeboard. I use it to keep students up to date with current homework tasks and I use it to extend their current understanding and deepen their knowledge of mathematics by posting videos such as this from Dr James Grime:

I am currently experimenting with creating a questionnaire and then posting this on Google Classroom. I created a questionnaire to find out more information about the food students ate. Forms does not directly link to Classroom (yet). Although the Form, told me who had completed it, this was not then updated to the assignment on Google Classroom. I'm waiting for all the results to be in and looking forward to analysing the results. Update findings in next review of Google Classroom.

ResearchED National Conference London 6/9/14

The National ResearchED Conference this year was held at Raine's Foundation School,

The day began with an historical journey into education from when Raine's was established to the present day, by the present Headteacher, John Bradshaw. The difficulties faced over the years remain remarkably similar. Tom Bennett followed up, by trying to ignite a polite rebellion against top-down CPD and encouraging teachers to have a healthy relationship with research - cautious and curious.

One of the best things about ResearchED conferences is you get a choice of speakers (although this can lead to a bit of a headache deciding who to go to see). So first up, Dr Jonathan Sharples from The Education Endowment EEF. Jonathan talked about current research projects and advocated research champions within every school. He provided guidance on how to perform your own research:

1. Decide what you want to achieve
2. Identify possible solutions
3. Give it the best chance of success
4. Did it work?
5. Securing and spreading change.
And back to number one...

Next, the popular Dylan Wiliam (name spelt correctly!). Dylan's lecture was the provocatively titled, "Why teaching will never be a research-based profession and why that's a good thing". His first argument was that the cupboard was bare in relation to ed-research. He highlighted the difficulty of establishing causality and only lucky experiments get published (i.e. those who show a correlation, especially if it is the positive/negative result the author wished to see).

Following Dylan, I stayed in the beautiful old school hall to watch Prof. Robert Coe. Robert started with a cautionary note about offering teachers feedback after lesson observations. He emphasised, "What you think is good teaching may not be". Just because a certain way of teaching works for one person does not mean this can be transferred to another teacher, teaching a different class. Feedback to teachers must be:

1. based on best evidence
2. reflect diversity of teachers
3. include protocols for demonstrating when they are met

To improve teaching a better understanding of evidence is needed that is based on research.

Phillipa Cordingly (CUREE) has researched exceptional schools in challenging circumstances and her compared them to other schools to try to discover why they are exceptional. Her comprehensive findings can be downloaded here.

Amanda Spielman (OFQUAL) provided useful insight and data into the examination process. She discusssed the pros and cons of the current examination system. People hugely underestimate variability in year-to-year exams. This can be as much as 12% in mathematics (which is generally straight forward to mark) and not surprising even higher other subjects (e.g. English 15%). This means OFSTED, Governors, Headteachers etc. need to be very cautious when analsying results as a sudden rise/fall may be down to natural variation and nothing to do with latest interventions.  When we look on an individual student basis, about 23% of students won't be allocated the correct grade (usually a grade either side).

More information about other talks and blogs about the conference can be found here:

Google Classroom

I'm excited about using Google Classroom. The design is simplistic and easy to use, providing quick access to particular classes. I like how it automatically creates assignment/class folders within google drive. The announcement section looks promising as a way of developing teacher/student dialogue (particularly useful for quieter students). Can't wait to try this out in September...

Promotional video by Google

Google Classroom Training Level 1

Google Classroom Teacher and Student View by +Roger Nixon

Google Classroom Community on Google+

Book Review - Bad Science by Ben Goldacre

This is a fascinating book into the world of research and statistics. Goldacre looks at topics including Homeopathy, MMR Vaccines and Drug Companies and investigates the flawed medical trials and the problems with media interpretation. For teachers, there is also a section on the folly of Brain Gym.

As a Maths teacher, the book is extremely useful becuase it provides great insight in to modern techniques and how to correctly carry out Medical trials and Research, by often highlighting misleading methods and statistics. It looks in details at topics such as blind testing, regression to the mean, placebo effects, etc. This book would be particularly useful to run with the new Core maths qualification at post 16.

I would highly recommend this book to not just maths and science teachers, but to everyone within education. It highlights the need for a solid research base in education and the need for greater knowledge in this area. Fads, like Brain Gym, not only damage the education of our students but are a huge waste of money and resources.

See more at badscience.net and section for teachers.

For more on Educational Research see - EEF (UK) or WWC (US)

Book Review - Why Don't Students Like School by Daniel T. Willingham

Willingham has written a thoughtful and insightful book about cognitive science. He describes current understanding of how the mind works and the possible impact of this research on life in schools and classrooms. 

"The mind is at last yielding its secret to persistent scientific investigation. We have learned more about how the mind works in the last twenty-five years than we did in the previous twenty-five hundred."

The main findings from this book indicate the importance of factual knowledge to children's development, "thinking well requires known facts".  Willingham discusses the importance of working memory and uses the imagery of allocated slots. It is important not to overburden the working memory. To achieve this, teachers need to organise deliberate practice for their students. This must be maintained over time and can be folded into more advanced skills. This moves knowledge from working memory to long term memory and avoids overload by becoming automatic (e.g.like driving a car).

I agree with the important message though that drives through this book, that we should make students think deeply, but in order to achieve this, shallow learning and practice is sometimes needed. Unfortunately in recent times in the UK this has been ignored in pursuit of fluff, show lessons that are thought to impress OFSTED, but do not aid the overall development of students and often distract students from thinking about what they are supposed to. Practise automatices knowledge from working memory to long term memory and allows the brain to work more efficiently and fluidly. The idea of conceptual understanding is not rubbished but he promotes a balanced approach between procedural and conceptual development. Connections between topics and previous learning are also important because they reduce the stress on working memory and develop proficiency. 

The ideas of hooks are also promoted. Interestingly, it is suggested to try these in the middle of the lesson rather than the usual start or end. Hooks are very useful in allowing the brain to make connections and help to develop a deeper understanding of topics.

He states we should, "strive for deep understanding in your students, not the creation of new knowledge" (that will come later). The vast majority of top mathematicians, scientists etc. have got there because they have developed core knowledge and then have worked hard to excel in their particular field. I have also considered it strange how some of the highest academics criticise the path to how they have excelled. It is like a walker crossing a log on a fast moving stream and then kicking it away so others cannot cross in the same way. This is an important message and one that slightly differs from other books I have read - particularly in the field of maths education, where the message is to try to get students to think like elite mathematicians. Often this has happened through the promotion of calculators or more recently WolframAlpha as a substitute for basic knowledge. Although these products are useful, ignoring the basics does not benefit students in the long term. Again it is important to emphasise, higher order thinking is important but must not be done at the expense of basic skills that underpin subjects.

Willingham's chapter about intelligence is very similar to the work completed by Dweck. (See previous post about Grit and a Growth Mindset). Intelligence although partly dependent on genetics and environment is not fixed. "Intelligence can be changed through sustained hard work". This message is extremely important to us as educators. For if it was fixed what would be the point of education? This has particular resonance in mathematics in the Western World where often there is a stigma around the subject and people strongly believe they simply cannot do it because of some inherited fault. Willingham highlights the folly of this idea and with hard work everyone can improve.

Willingham's last chapter focuses on teacher development. He believes teachers need to deliberately practice to improve. He provides a comprehensive approach to this using video, teacher feedback and other ideas. I liked the simplicity of an idea earlier in the book where two skills of teaching were highlighted: organisation and relationships. I look forward to trying out some of these ideas in the new academic year.

At the heart of teaching is to try to make students think. This book provides powerful insight in how to achieve this.

Book Review - 1089 + All That by David Acheson

David Acheson has written an excellent little book about a journey into mathematics. This book is an ideal read for any teacher or student. It is an excellent holiday read to maintain and top up mathematical knowledge. 1089 is written in short passages, making it easy to read but doesn't shy away from some relatively complicated mathematics and contains some gems of mathematical insight. Amazingly, this small book deals with a whole range of topics from pi to chaos theory.

Online Markbook

Online Markbook

I have been developing my markbook online. See a blank version here. I prefer an online markbook to a paper copy because it allows me to edit and move things around easily. I have a separate markbook for each class.

The markbook has been divided into 11 sections: Seating Plan, Information, SEND, Marking, Homework, Mymaths/Sam Learning, Assessment, Intervention, Medium Term Plan, Postcards and Parents Evening. I don't record a register on the markbook because the data is collected elsewhere. Data is also collected separately centrally on a department tracking sheet. Therefore, the purpose of this markbook is to allow me to stay on top of classes by providing me with information needed to teach and most importantly to make notes so I can monitor individuals and classes as a whole.

Most of the sections are self explanatory. This year, I have added an Intervention section. In this section, I record all interventions for each of my classes. Examples of these include behaviour incidents, contact with parents or other members of staff, any concerns or worries I have about individuals, positive news/rewards. The outcomes is crucial to this intervention page. Here I record what has happened since the intervention eg has an individual students behaviour/attitude/performance improved? I have found this to be extremely useful, because it allows me to look back and see how individuals have responded and hopefully improved. Every half-term, I also write a quick class report. This allows me to reflect on the progress of the class and individuals. 

For the marking and homework, I use a very simple colour scheme. Green for good, Red for bad. Over time this provides a quick visual of class and individual performance.

My online Markbook is created on Google Sheets. Sharing this with other members of my department or SLT is easy. Google Drive has substantially improved in recent years. It is very easy and intuitive to create and save files. The filing system is now clear and structured, allowing you to find files quickly. The big advantage of Google Drive is I can access it anywhere. Files can be viewed and edited on my phone and tablet, aswell as any computer. Plus, I can't lose it! 

Next year, I want to develop my recording of assessment and progress. I'm looking to move towards analogue data per topic and using various sources to record data (e.g. quizzes, diagnostic questions, end of term tests, homework).

My markbook is designed to be simple to use and easily adapted to individual needs. It is not the markbook to end all markbooks and is a work in progress. Feel free to use or adapt if you want. (Press File, Make a copy...)

Reading List for New Maths Teacher

I was asked recently what I thought would be a suitable reading list for a new UK Maths teacher. Below is my list. Have I missed anything out?

General Teaching:

• "Teach Like a Champion", Doug Lemov 
• "Mindset", Carol Dweck 
• "Why Don’t Students Like School", Daniel T Willingham 
• "An Ethic of Excellence", Ron Berger 
• "The Hidden Lives of Learners", Graham Nuthall 
• "Visible Learning and the Science of How We Learn", John Hattie 
• "100 Ideas for Secondary Teachers", Ross Morrison-McGill 
• "Teacher’s Tookit", Paul Ginnis 
• "Trivium 21C", Martin Robinson 
• "Adapt", Tim Harford 
• "The Secret of Literacy", David Didau 
• "The Behaviour Guru", Tom Bennett 
• "How to Teach", Phil Beadle 
• "The Perfect (Ofsted) Lesson", Ian Gilbert 
• "Teacher Proof", Tom Bennett 
• "Getting the Buggers to Behave", Sue Cowley
• "The Lazy Teacher's Handbook", Jim Smith
• "How Children Learn", John Holt

Shaun Allison has created an excellent CPD video library


Mathematics:

• "The Elephant in the Classroom", Jo Boaler 
• "Alex’s Adventures in Numberland", Alex Bellos 
• "Taming the Infinite", Ian Stewart 
• "Murderous Maths" series by Kjartan Poskitt
• "The Perfect Maths Lesson", Ian Loynd
• "Adapting and Extending Secondary Mathematics Activites", Stephanie Prestage & Pat Perks 
• Collection of Professional Development Materials for Maths Teachers by Mark McCourt at Emaths
• Cockcroft Report 1982 
• "Getting the Buggers to Add Up", Mike Ollerton
• "Key Ideas in Teaching Maths - Research-based Guidelines for ages 9-19", Anne Watson, Keith Jones, Dave Pratt
• "Nix the Tricks", Tina Cardone and MTBoS
• "How Children Learn Mathematics", Pamela Liebeck
• Collection of Publications by Professor Malcolm Swan
• "Children Discover Arithmetic", Catherine Stern
• "Understanding Mathematics for Young Children", Derek Haylock
• "1089 and All That", David Acheson
• Debates in Mathematics Education, edited by Dawn Leslie and Heather Mendick, for the more reflective practitioner

Collaborative/Cooperative Learning

"Collaborative or cooperative learning can be defined as learning tasks or activities where students work together in a group small enough for everyone to participate on a collective task that has been clearly assigned. This can be either a joint task where group members do different aspects of the task but contribute to a common overall outcome, or a shared task where group members work together throughout the activity. Some collaborative learning approaches also get mixed ability teams or groups to work in competition with each other, in order to drive more effective collaboration."
(Education Endowment Foundation)

My Practice

This year, I have begun to develop my practice specifically towards cooperative learning. In my classroom, students work together in small 'pods' (3 or 4 students) where the goal is to help and support each other to develop their learning and understanding.

(source: National Wildlife Federation)

Within each pod, I try to mix gender, ability and other factors. When needed, students have changed pods during the year. For a small minority students, it has been more of a challenge to settle them to work collaboratively within a pod and to find the most effective pod for them to work in. I have found older students are more likely to create a stable and effective pod which stays in tact for the whole year. I would suggest this is probably because their friendships with other students is more secure. Younger students are generally happy to work within pods, but these have to be altered more frequently throughout the year to achieve balance and harmony within the class.

I have not imposed a competitive element into my system yet. I am considering it carefully for next year and I'm thinking of trialing it with my KS3 classes (11-14). I am aware of the importance of getting this right so will need to do more research over the next couple of months.

I actively encourage students to work closely within their pods. Specific tasks are created to encourage discussion within the pod. When an individual seeks help from me, I tend to talk and involve the pod as a group, thus encouraging them to work and think as a team. This often ensures quieter members of the pod also get support. On this note, both introvert and extrovert students seem to work well within this system and neither type dominates. I encourage higher ability members to support their pod with explicit praise, both verbal and written.

Evidence

"Evidence about the benefits of collaborative learning has been found consistently for over 40 years and a number of research studies have been completed. In addition to direct evidence from research into collaborative learning approaches, there is also indirect evidence where collaboration has been shown to the effectiveness of other approaches such as mastery learning or digital technology. It appears to work well for all ages if activities are suitably structured for learners’ capabilities and positive evidence has been found across the curriculum."
(Education Endowment Foundation)

Reflections

Collaborative/Cooperative learning has helped to develop a positive atmosphere within my classroom. I have found the vast majority of students work well within the pods and this has helped to develop independent learning skills as they tend to ask each other first and only ask me if confusion or uncertainty remains. To help students, I move between pods, rather than individuals, which is more time-effective. This strategy seems to work very naturally with Mathematics.

For more information:

• Concept to Classroom
• Education Scotland
• Behaviour Advisor
• Carlton University
• Collaborative Learning Project

Alan Turing

Today, 23rd June, is Alan Turing's birthday.

"Alan Turing was a British Mathematician, logician, cryptanalyst, philosopher, computer scientist, mathematical biologist, and marathon and ultra-distance runner. He was highly influential in the development of computer science, providing the formalisation of the concepts of "algorithm" and "computation" with the Turing Machine, which can be considered a model of general purpose computer. Turing is widely considered the Father of Theoretical Computer Science and Artificial Intelligence." (Wikipedia)

Turing is famous for his work during World War II at Bletchley Park decoding the German Enigma machine. Bletchley Park has recently been revamped and reopened by the Duchess of Cambridge (see BBC article)

Below are some of my photos of Bletchley Park from a couple of years ago.

German Enigma Machine

Turing's Office at Bletchley Park

Colossus Computer

Alan Turing - Celebrating the life of a genius by Cambridge University (Dr James Grime)

SciShow - Hank introduces us to that great mathematical mind, Alan Turing.

Dr James Grime explains the Enigma Machine

The Mathematics of Alan Turing - Professor Angus MacIntyre (Gresham College)

For more information about Alan Turing's life see:
BBC - History - Alan Turing page
Alan Turing: The Enigma
Alan Mathison Turing - MacTutor
Royal Pardon for Codebreaking Turing

Visit Bletchley Park

La Salle's National Mathematics Conference

La Salle's National Mathematics Conference for Secondary and Primary Teachers, Kettering Saturday 14/6/14 - Sponsored by AQA 
#mathsconf2014

My Notes and Reflections

Mark McCourt - Chief Executive, La Salle Education

Mark introduced La Salle's new mathematics product, Complete Mathematics. This provides complete support for mathematics teachers by providing them with help with planning and assessment. I look forward to investigating this in more detail in the next couple of weeks.

Keynote speaker, Dr Vanessa Pittard, Assistant Director, Curriculum and Standards, DFE

Dr Pittard began her talk with an analysis of the PISA results for Mathematics in the UK. (Key findings for the UK). English students are good at data and number, but need to focus on shape and space, and problem solving. Not sure why then, the weighting of Geometry in the New GCSE is reduced? 

Dr Pittard moved on to discussing the new Maths Curriculum and how it is benchmarked against high-performing jurisdictions like Singapore and Massachusetts. The new A-level Mathematics will be introduced in 2016, along with the Core Mathematics qualification for post-16.

The New GCSE - Andrew Taylor AQA

Andrew Taylor provided a comprehensive introduction to the new Maths GCSE and gave an overview not just from AQA but all the exam boards.

The GCSE will be writtten papers only. Exams will be linear and summer exams for all. November entry is only available for post-16.

The papers will be 4 1/2 hours long and split 1/3 to 1/2 between calculator and non-calculator papers. Foundation Tier and Higher Tier remain (no return to the intermediate tier). Grades are from 1-9 (9 being the highest). Foundation Tier 1-5 and Higher Tier 4-9 (there is a safety net grade 3 on the higher-tier).

The Mathematics GCSE will carry a double weighting in the new accountability measures. (See Factsheet on Progress 8). Andrew described the new GCSE as a "Maths GCSE on steroids".

AQA have split the time allocation into 3 papers each 90 minutes long. The first paper in a non-calculator and the other two are calculator exams.

The big news about content is there is a large shift towards Ratio, Proportion and Change (largely at the expense of Geometry and Measure), which a number of people in the audience felt aggrieved about. Both the Higher Tier and the Foundation Tier will be skewed more towards the top grades than currently.

More information see Craig Barton's blog

Assessing Without Levels - David Thomas and Alan Gothard (both from Westminster Academy)

Westminster Academy's mathematics curriculum has beed designed around three principles:

1. Break the curriculum down
2. Define it with questions
3. Use Analogue Data

The curriculum is split into discrete topics e.g. adding and subtracting decimals. Per year, there are 15-20 topics and each one is assessed individually. For clarity, the topics are defined with questions from three categories: Use & Apply, Reason, Interpret and Communicate, and Solve Problems.

Data is used to create an Assessment for Learning rather than an Assessment of Learning model. Rather than an approach of can they do this? Yes or No, analogue data is used (a percentage score) so it is easier to define the students level of mastery. Data is collected and weighted from small quizzes (40%), homework (20%) and end of term tests (40%). This leads to clear reporting to parents of what their child's strengths and weaknesses are topic by topic. 

The main advantages of this system is that it can highlight when an individual student's performance drops or when a member of staff might need support with teaching a particular topic.

What I liked about this system is the simplicity and clarity it provides to students, parents and teachers. It is far more practical to report on how a students is progressing in particular topics though out the year, rather than focusing on meaningless sub-levels.

For more information, read "Assessing Without Levels" blog by David Thomas including presentation given at Conference.

Blindingly Obvious - Bruno Reddy, King Solomon Academy

Bruno's work is highly influenced by the Cognitive Scientist, Daniel T. Willingham (author of "Why Don't Students Like School?"). Why do pupils get stuck? Their working memory runs out of space. On average, people can have 7 working memory slots. However, some students may only have 4 and three of these might be taken up with listening/speaking, writing and remembering.. It is therefore important to reduce the amount of pressure on the brain.

King Solomon Academy focuses on mathematics. Students receive between 6 and 7 1/2 hours per week of lessons on mathematics. Classes are taught as mixed ability form groups and it is the teacher which moves classroom rather than students, to reduce transition time.

The curriculum focuses on longer studying fewer things. In year 7 in particular, a lot of time is focused on the foundations of maths, number and place value. Some of the more complex areas of the KS3 curriculum and left until KS4. Similar, but confusing, concepts are separated e.g. area and perimeter or median, mean and mode. 

Blindingly Obvious refers to how they approach lesson planning. The pitfalls of modern day maths textbooks and worksheets are highlighted. 

1. Minimally Different Examples - questions where just one variable changes. 
2. Trickle Feeding - a little bit of the same every day.
3. Building Automaticity - practising the basics until perfect.
4. Splitting the Steps - and practice each in turn
5. Take the first step last

Bruno introduced his product, Times Tables Rock Stars

A challenge was also laid down to beat the KSA's World Record for the largest number of people rolling numbers

For more information see Mathematics Mastery.

La Salle Education Conference materials - presentations and handouts.

My personal reflections - this was a fantastic day. Managed to get up to speed in latest developments in the Mathematics Curriculum. Listened to inspirational Maths teachers, Bruno Reddy, David Thomas and Alan Gothard. Finally, got to meet some amazing people and put names to twitter handles - El_Timbre, MakeMathsMatter, Just_Maths, Ms_Kmp and missradders. My only regret is I couldn't see the other speakers. Wow. What a fantastic day. 

Thank you Mark, La Salle and AQA.

PS Cakes were amazing!!!!

Maths isn't *just* beautiful... #mathsconf2014 pic.twitter.com/bVdj7YiEtf
— Em (@El_Timbre) June 14, 2014

Logical Fallacies

I regularly see arguments on Twitter and straw men get mentioned. Initially, I thought it was something to do with the Wizard of Oz and lacking a brain. However, after a little research found it to be a little more sophisticated.

A fallacy is an argument which uses poor reasoning. An argument can be fallacious whether or not its conclusion is true. An error that forms from a poor logical form is sometimes called a formal fallacy or simply an invalid argument. An informal fallacy is an error in reasoning that does not originate  in improper logical form. Arguments containing informal fallacies may be formally valid, but still be fallacious. (wikipedia)

A straw man, also known in the UK as an Aunt Sally, is a common type of argument and is an informal fallacy based on the misrepresentation of an opponent's argument. To be successful, a straw man requires that the audience be ignorant or uninformed of the original argument. (see more here on wikipedia)

The image below outlines the main logical fallacies:

(source: Sheffield Company - can also download bigger version on this site)

For more on this, read "An Illustrated Book of Bad Arguments" by Ali Almossawi (thank you to Tim Taylor for providing me the link).

Zeno of Elea

Zeno of Elea was a pre-Socratic Greek philosopher of Southern Italy and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best know for his paradoxes, which Bertrand Russell has described as "immeasurably subtle and profound". (Source and for more information - wikipedia) 

Further information can be found on the MacTutor History of Mathematics.

Collection of Youtube videos on Zeno's Paradoxes

Numberphile

Math Bites with Danica McKeller

Math Bites on the Nerdist Youtube Channel - a quirky and light-hearted set of youtube videos on Mathematics, starring Danica McKeller

McKella studied at UCLA and earned a Bachelor of Science with honours (summa cum laude) in 1998. As an undergraduate, she coauthored a scientific paper with Professro Lincoln Chayes and fellow student Brandy Winn entitled "Percolation and Gibbs states multiplicity for ferromagnetic Ashkin-Teller models on ". Their results are termed the 'Chayes-McKeller-Winn theorem'. Referring to the mathematical abilities of his student coauthors, Chayes was quoted in The New York Times as saying, "I thought that the two were really, really first-rate." McKellar's Erdos Number is four and her Erdos-Bacon number is 6. (source: wikipedia)

1. The Pi Episode - Join Danica McKeller as she explains the joys and mysteries of the wonderful world of Pi!

 

2. Math Head: DO MATH IN YOUR HEAD!

3. Binary Numbers. Danica McKeller demystifies the 1's and 0's of binary numbers

4. Percents: explains percents and the value they have in everyday life

5. World Math - Mathematical concepts may be universal, but the mathematics around the world varies.

6. Dance of the Sugar Pi Fairies - it's pi like you've never seen before! Look out for Simon Pegg...

McKellar has written a number of books on mathematics, mainly designed to pass on her love of mathematics to girls. Titles include: "Girls Get Curves: Geometry takes Shape", "Math Doesn't Suck", "Kiss my Math" and "Hot X: Algebra Exposed"

Book Review - The Simpsons and Their Mathematical Secrets by Simon Singh

This is a superb book for anyone who loves Mathematics and is a fan of The Simpsons. I found it fascinating discovering the mathematical expertise of the writers of The Simpson and how they sneakily embed mathematical concepts into episodes. Simon Singh writes from a fans point of view. He obviously enjoys The Simpsons and his love of mathematics is clearly evident. This book is a joy to read.

Talks at Google, Simon Singh, "The Simpsons and Their Mathematical Secrets".

Pi and Four Fingers - Numberphile

Follow Simon Singh on Twitter @SLSingh

Why is x the unknown?

Terry Moore's Ted Talk about why 'x' is used to represent the unknown.

Terry Moore directs the Radius Foundation in New York, which, as its website says, "seeks new ways of exploring and understanding dissimilar conceptual systems or paradigms - scientific, religious, philosophical, and aesthetic - with the aim to find a world view of more complete insight and innovation."

Book Review - Adapt by Tim Harford

A review of Adapt by Tim Harford and how it can be "adapted" for the world of education. (Inspired to read and reflect about this book after it was recommended by John Tomsett at NTEN ResearchED York).

Tim Harford composes a clear and structured argument on the importance of adapting, not just from an ecological point of view but as individuals, companies and other organisations. Mistakes are commonplace and inevitable. Harford uses a number of case studies from the development of the Spitfire, changing tactics in Iraq, the Credit Crunch to modern companies such as Google to explain and demonstrate the importance of adapting and the problems that emerge when this doesn't take place.

Harford draws deeply on the three Peter Palchinsky Principles:

1. Seek out new ideas and try new things
2. When trying out new things, do it on a scale that is survivable
3. Seek out feedback and learn from your mistakes

What is intriguing about this book is how Harford unravels the mystery of adaption by highlighting the traps and promoting the benefits of adapting.

Barriers to learning from mistakes include: bundling losses up with gains, reinterpreting failures as successes, denial and loss chasing. To counteract this, Harford perscribes working by the Palchinsky Principles. We need to listen, whether this is as a company to whistleblowers or as individuals to what is described as the "validation squad" of trusted individuals who will give us their honest opinions. Change needs to be completed on a scale that promotes reflection and honesty.

Why adapt? 

"The process of correcting the mistakes can be more liberating than the mistakes themselves are crushing, even though at the time we so often feel the reverse is true". Mistakes in life are inevitable. However, "a single experiment that succeeds can transform our lives for the better in a way that a failed experiment will not transform them for the worse - as long as we don't engage in denial or chase our losses".

What are the implications for teaching?

In teaching, there is a constant need to adapt and change. In the classroom, what works one moment can fail disastrously the next. Teenagers can sometimes be a little volatile. However, how do we know we are doing the right thing and not just bumbling along? One of the most important principles mentioned is listening - whether this is to a trusted colleague or a sensible student. 

Education is bombarded with new ideas and the hunt for the pot of gold at the end of the rainbow. The difficulty is sifting and finding the sensible ideas - the ones which benefit students and make the lives of teachers easier. Harford promotes the importance of trying out new ideas. However, this needs to be done in a controlled way and not gun-ho, devil-may-care and by the seat of the pants. Adaption needs to be on a scale where success can be measured. Allowing ourselves times to reflect and most importantly, to listen to the opinions of those we trust. By investing on a modest scale, the traps of chasing losses and denial are more likely to be avoided. 

The Palchinsky Principles should be above the door of every staffroom.

For Students:

Students need to be encouraged to try new things and to accept mistakes as a part of life. This links to a previous blog by me, Developing Grit and a Growth Mindset.

I really enjoyed this book. It has made me reflect on the implications adaption has on my life and the wider world around me.

Book Review - "Taming the Infinite" by Prof Ian Stewart

This is a fascinating book on the history of Mathematics. Stewart goes way back to over 4000 years ago and the early days of recording the number system using tallies etc to an in depth look at the most important topics of the 20th and 21st century, such as Chaos Theory. Along the way, Stewart provides interesting insight into the lives of the brilliant mathematicians who have contributed significantly to the subject. He looks at how mathematics has develop and its most important advances. He considers the effects of mathematics on other areas of study and how mathematics affects our daily lives.

This is a must read for any student of maths (including teachers!). I love this book and look forward to reading it again.

Book Review - "The Hidden Lives of Learners" by Graham Nuthall

I bought this book after hearing a number of very positive comments about it at NTEN ResearchED York.

Graham Nuthall's book "The Hidden Lives of Learners" is a study of learning from a student's point of view. It focuses on the highly influential world of peers, and the student's own private world and experiences. Nuthall has used a comprehensive array of research tools and collected data over forty years, bring this all together in this one book. What becomes clear is that just because a teaching, does not mean students are learning.

"Learning requires motivation, but motivation does not necessarily lead to learning."

Nuthall's book provides a simple check list to effective teaching. The book discusses memory in detail. He slams learning styles (e.g. VAK) and states to build memory, students need to make connections with new knowledge and known concepts. For knowledge to be retained students need "several different interactions with relevant content for that content to be processed int their working memory and integrated into their long-term memory in such a way that it becomes part of their knowledge and beliefs."

Assessment is a tricky topic. He sees the best strategy for assessment is for it to be "conducted individual by individual, and embedded in a programme that fully considers individual preferences." This has obvious difficulties from a time management point of view, but one that we as educators need to consider and think carefully about.

The one area I was not completely convinced by was his idea of "becoming involved in peer culture" and in fairness, Nuthall too was reserved about this idea. I understand the advantages of knowing your students, but I also consider it to be healthy for their to be a little professional distance in the teacher/student relationship (especially in the age category I teach 11-18).

For me personally, I like to strive for the optimum seating arrangements in my class to develop co-operative learning. I think about peer groups but the driving force is finding which individual students work well with who.  I try to create pods of 3 or 4 students (preferably of varying ability and other factors). Most importantly, I look at my assessment results and if I believe a pod is not working effectively, I look to change it.

I prefer the "alternative culture" within the classroom by creating a "learning community". A complex idea but, in my opinion, this should be at the heart of any school and run through it like a river.

I was particularly interested in the strong evidence shown for the effects of students managing their own learning. This ideas seem to run alongside ideas of a Growth Mindset, grit and determination, and developing independent learning. I try hard in my teaching of mathematics to open this door, by using Mymaths,  Hegarty Maths, or BBC Bitesize. I also like to stretch their knowledge and understanding by looking at advanced material e.g. Numberphile or Vi Hart.

For more information, visit The Graham Nuthall Classroom Research Trust website.

Polya - How to Solve It

George Polya's book provides clear guidelines on how to solve problems. I came across Polya's book last year whilst completing a MOOC, "How to Learn Math", by Prof. Jo Boaler, Stanford University. Polya's work is useful for students by providing them with a framework and helping them to scaffold problems themselves.

Basic Outline of Polya:
1. Understanding the Problem
2. Devising a Plan
3. Carrying out the Plan
4. Checking and Looking Back

Summary here University of Utah

I've used the diagram below to help students when working on investigational problems (diagram is stuck in the back of each students book). It gives a structure and hopefully it helps develop resilience. When a student gets stuck, I ask them to look at the diagram. If they continue to be stuck, I prompt them, using the diagram, by asking questions such as, "Have you done this?" or "What about..?" etc. and point out various parts.

source: math-mom.com

This has worked well with Bowland Assessment Tasks, Nuffield's Applying Mathematical Processes, cre8ate Maths Resources and NRICH activities.

Collection of OU's 60 Seconds in Thought

No. 1 - Achilles and the Tortoise


No. 2 - The Grandfather Paradox


No. 3 - The Chinese Room

No. 4 - Hilbert's Infinite Hotel


No. 5 - The Twin Paradox


No. 6 - Schrödinger's Cat

The Infinite Hotel Paradox,, Other Infinity Paradoxes and Infinity...

OU's Hilbert's Infinite Hotel (60 seconds adventures in thought) - David Mitchell narrating

First video is a TED-Ed video by Jeff Dekofsky

Infintiy Paradoxes by Numberphile featuring Mark Jago

In the next video, Vi Hart explains the different kinds of infinity.

Desmos - the best Internet Graphical Calculator

Desmos is a simple to use but complex internet-based graphical calculator. It can be accessed on the majority of devices including i-Pads (i-Pad App here). Most importantly, it's FREE!

Introducing the Desmos Calculator - DESMOSINC

The screenshot below shows the basic functions:

A Desmos account is quick to set up, enabling you to save graphs. By connecting Desmos to your Google account, graphs can be quickly saved to Google Drive. Graphs can also be uploaded and shared to the wider Desmos community. Or simply, graphs can be used on your blog:

Where to start? DESMOS Quick Start Guide or try it's dedicated Youtube Channel with a number of How-to Videos to help you get started.

Desmos has a growing on-line community and is developing lots of resources for Mathematics teachers, via Desmos blog with resources by Dan Meyer and Fawn Nguyen amongst others. Further resources are available on youtube:

DESMOSLIVE: An Exploration of DESMOS by Mathalicious 

Desmos has a fun and creative side.

Stopwatch created by Ania Kuriata

See more at Desmos staff picks. These graphs can be saved to your area, allowing you to see how it was made. Just showing these pictures to younger students has allowed me to introduce and discuss more complex topics (e.g. equations of circles and ranges/domains) without frightening them off. It also allows for them to see applications with the real world, especially animation (see my previous blog, Mathematics and the Movies)

From a teacher's point of view, Desmos is very quick to pick up. It has a Projector Mode which makes class presentations clear for students. It is dynamic and simple to animate, leading to clear explanations. Graphs are easily saved and retrievable. Best of all, due to it being FREE, it has a growing community of teachers who share freely their resources. I LOVE DESMOS!

A special thank you to Eli Luberoff (Founder and CEO of Desmos)

Colleen Young has written extensively about Desmos on her excellent blog Mathematics, Learning and Web 2.0 

Developing Grit and a Growth Mindset

The Key to Success? Grit by Angela Lee Duckworth

TED Video

"Grit is passion and perseverance for very long term goals. Grit is having stamina. Grit is sticking with your future... and working really hard to make it a reality." Duckworth

source: patheos.com

But how do I build Grit in students? Duckworth suggests developing a Growth Mindset (Dweck) and make them understand it is okay to fail. Dweck talks about the brain 'growing' or making connections when misconceptions are corrected (with a Growth Mindset).

source: brainpickings.org

Teaching strategies I have used this year:

Model this myself - when I make a mistake and a student notices and corrects me, I award them a house point and thank them for pointing it out to me. This can often lead to a quick discussion about checking answers etc.

Use incorrect answers - when a pupil makes a mistake I use positive curiosity. I get them to explain their mistake and then talk about how useful or interesting their mistake was for the rest of the class and for me. Mistakes are fine as long as we learn from them.

Pause -  I've tried very hard to make sure I give pupils time to think about their answers. With some their may be an awkward long silence (eventually they do provide an answer). If they require prompting, I try to give them the minimum possible prompt. I never let them get away with not answering and sometimes ask them to simply make an educated guess. I have found this to be a very effective strategy to keep all engaged and raise expectations of all pupils.

Poor test result - discuss with class/pupil the correct response to a bad result immediately after being given it. E.g. rather than getting frustrated and wanting to tear it to pieces, write corrections and use it to help development and chat with me at the end of the lesson if you still don't understand.

source: pictures.picpedia.com

Life isn't easy - if it was it would be very dull. Imagine you got everything your own way. Every shot on goal you scored. You could instantly play any instrument perfectly. You knew everything. Where's the challenge? Where's the fun in that? My subject, mathematics, is hard, but that's the challenge, that's the fun. I explain to my class that I enjoy doing puzzles (e.g. Sudoku) in my spare time. I also love really hard maths questions and explain how, last week, with a particular A-level question I was stuck on it for a couple of hours. The longest I've spent mulling over a particular problem is several weeks or even months. Warning - some minds are blown at this stage!

I have no evidence to suggest any of the above strategies actually work. However, if we want pupils to have grit, determination and a growth mindset - I believe the most important thing we can do is model this behaviour ourselves and make this explicit to the children we teach.

Read more about this on this excellent blog by Joe Kirby, "Motivation and Mindset Anchoring". "This much I know about... developing a Dweck-inspired mindset culture" by John Tomsett. Interesting reflective blog by Stephen Tierney, "Growth Mindset: The Latest Silver Bullet". Finally, a balanced argument from David Didau, "Grit and Growth: who's to blame for low achievement?"

Using Mathematics to Make Pupils Less Gullible

I'm always amazed at how some students get taken in by videos like this:

The Haunted Doll - Richard Wiseman - Quirkology

Simply start the discussion with - why does this work? Can you explain it? Then give several minutes to come up with a carefully thought out solution. I have found this works very well as a plenary. This helps to develop reasoned argument and problem solving skills.

How Predicatable are you? (Richard Wiseman - Quirkology)

Response: How Predicatable Are You? by James Grime (IMA)