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Wednesday 28 May 2014

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

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.

Tuesday 27 May 2014

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!


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:



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 

Sunday 25 May 2014

Developing Grit and a Growth Mindset



"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).



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.


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:



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.




10 Heads in a Row

What is the probability of getting 10 heads in a row from flipping a coin?


All is explained here:


Two excellent videos for introducing probability. Show the first video and ask the class how James did this. This often highlights misconceptions. Then show the full version. Now get pupils experimenting and calculating the cumulative frequency when they flip a coin 10 times (or if you have time, 100), which leads to further discussions. I have found this works with pupils of all abilities.

Monty Hall Problem

Monty Hall Problem


Suppose you're on a game show and you're given the choice of three doors. Behind one is a car, behind the others, goats. You pick a door, say No. 1, and the host, who know what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it your advantage to switch your choice? (wikipedia)

Below is a collection of the best youtube videos on the Monty Hall Problem.

Numberphile - Lisa Goldberg, an adjunct professor in the Department of Statisitics at University of California, Berkley


Monty Hall Problem by singingbanana (Dr. James Grime, Cambridge University)


The Fundamentals of Space-Time

The Fundamentals of Space-Time by Andrew Pontzen and Tom Whyntie. (TED Ed lessons)

Space is where things happen. Time is when things happen. And sometimes, in order to really look at the universe, you need to take those two concepts and mash them together. In this first lesson of a three-part series on space-time, hilarious hosts Andrew Pontzen and Tom Whyntie go through the basics of space and time individually, and use a flip book to illustrate how we can begin to look at them together.

Light always travels at a speed of 299,792,458 meters per second. But if you're in motion too, you're going to perceive it as traveling even faster -- which isn't possible! In this second installment of a three-part series on space-time, CERN scientists Andrew Pontzen and Tom Whyntie use a space-time diagram to analyze the sometimes confounding motion of light.


In the first two lessons of this series on space-time, we've dealt with objects moving at constant speeds, with straight world lines, in space-time. But what happens when you throw gravity into the mix? In this third and final lesson, CERN scientists Andrew Pontzen and Tom Whyntie explore what gravity means for space-time -- or rather, what space-time means for gravity.



Saturday 17 May 2014

Mathematics and the Movies

I am fascinated about real-world applications for mathematics. This week, I introduced the equation of a straight-line to some of my classes and the usual question emerged, "Why are we learning about this?". So I showed them this excellent video - Pixar: The math behind the movies by Tony DeRose TED-Ed talk.



A discussion followed with lots of questions about special affects. This video by Weta Digital's Academy Award Winners explains their tissue program:



These two videos by ILM Visual FX show the construction of virtual sets and characters (although to admit, they are more visual and give less spoken explanation).



Strategy and Mathematics of 2048

I've become hooked to 2048! I have managed to get the 8192 tile and my highest score so far is 109340.


My preference is to play at the top. My strategy is to fix the top line, with the biggest number in a corner. Then, I try to create a chain from that number across the top, down to the next row etc. To achieve this I group the numbers in the corner (with the highest number). 

The video below is a brilliant discussion on the strategy and mathematics of 2048 by the wonderful  +James Grime and +Steve Mould (@jamesgrime and @MouldS) - this blog is inspired by this.


My strategy develops when I create a chain of large tiles along the top. Rather than grouping the numbers in the corner of the highest number it switches to the opposite top corner.

Other strategies:
  • If the top row becomes unfixed, fix it immediately by placing tiles into it.
  • If the worst happens, and the largest tile is now not in the corner. The quickest way to get it back is to work with the larger chain rather than smaller e.g. top row 4 256 64 8 try to complete from the right, rather than the left.
+James Grime discusses an interesting subgame, lowest score (most efficient) to 2048 (under 20000 is the challenge).

Calculating the lowest score possible is complicated because approximately 10% of tiles generated are 4s.
(k-1)2^k-2/5(2^(k-1))
where 2^k is the highest tile

Getting under 20000 requires a bit of luck in generating 4s. 

Interesting variations:

Tuesday 13 May 2014

NTEN ResearchED York 2014 #NTENRED


“Every teacher needs to improve, not because they are not good enough, but because they can be even better.” Dylan Wiliam

ResearchED is a growing movement in education with its roots within the UK education Twitter community. It was established last year by Ben Goldacre (Author of Bad Science @bengoldacre) and Tom Bennett (Teacher/TES @tombennett71) and is for educators, researchers and anyone interested in evidence-based education. From this, groups such as Teacher Development Trust, NTEN and the Headteachers Roundtable have established themselves.

On Saturday, 3rd May I made the trip to Huntington School, York for my first ResearchED conference (cost of ticket £13), joining nearly 300 eager people. The biggest decision which needs to be made when attending a ResearchED conference is who to see. Yes, you get a choice! All speakers had given up their time for free and are regular bloggers so you have an understanding of them and their views before the event.

The keynote speaker was John Tomsett (@johntomsett). John is Headteacher of Huntington School and a member of the Headteacher’s Roundtable. John talked about removing the culture of fear (observations and judgements) and moving towards a model of coaching and using questions such as ‘How best can I observe you to improve your practice?’. He discussed using HQ video and sound to observe yourself. Underpinning everything is creating a learning community throughout the whole school and the pursuit of excellence. 

If you haven’t read it, you should:

Mary Myatt (an OFSTED inspector to admire @marymyatt) and David Weston (Chief Exec, NTEN & Teacher Development Trust @informed_edu) spoke about introducing research into schools at a micro level. Mary divulged the characteristics of research schools: energy, openness, enthusiasm, enjoyment and collaboration (more in her blog). David introduced the idea of 'Lesson Study' as a model for research in schools (see more on NTEN website)

‘Too many teachers waste time looking for the magic bullet’, John Hattie

ResearchED doesn’t believe it is the magic bullet but believes through research, study, collaboration and a bit of hard work we can create a superb educational structure within the UK. 

Mark McCourt (Chairman, Teacher Development Trust @EmathsUK) discussed building education from the ground up and asked us to consider ‘What is knowledge?’, ‘What is the purpose of education?’ and ‘What is the best model for education?’.

Martin Robinson (author of Trivium 21C @surrealanarchy) smoothly lectured on achieving balance in the curriculum between grammar, dialectic and rhetoric. If you haven’t read his book, it is fantastic - explaining the history of education and what he believes to be its ideal structure. (Recent article by Martin in the TES and blog "Engaging with a Humane Education")

Tom Bennett ranted entertainingly about psuedo-science in education and his particular bug-bears of Braingym, NLP (which I hadn’t heard of) and VAK. Have you rubbed your brain buttons recently? There is an obvious need for schools to work closely with researchers to find the best and most reliable evidence out there. (Read his article in the TES here)

Finally, Alex Quigley (Deputy Head at Huntington School @huntingenglish) spoke eloquently about his own research and his seven steps for a great explanation - gesture, connecting prior knowledge, visual cues, metaphor, making it concrete, storytelling and repetition. (Alex's blog on #NTENRED)

ResearchED was truly inspirational. I left with the feeling I needed to read more but with the satisfaction I was on a learning journey with everyone else. No one at the conference believed they knew everything (which is different from the usual imported ‘experts’ that spout their opinions at school CPD and cost a fortune to do so).  

My personal reading list includes, Graham Nuthall’s ‘The Hidden Lives of Learners’, Tim Harford’s ‘Adapt’ and Ben Golacre’s “Bad Science’.

If you missed it, here are some videos

Dafydd Collins (@mrdgcollins)

                                              Sunny day in York