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.