Polya – Let us teach guessing

This week I finally listened to a podcast I have been meaning to for a considerable time – Craig Barton’s podcast of his discussion with Anne Watson and John Mason.

As you can see from Craig’s notes the discussion is wide-ranging; so well worth a listen, this will leave you with plenty to think about.

I was particularly struck by Anne and John’s Big 3 (or Big 5!) (scroll down Craig’s notes). The only website they mentioned is Underground Maths; a personal favourite – Underground Maths pages here). John Holt’s How Children Learn was my first reading on my teacher training course decades ago – still a classic.

This morning, I have been distracted from writing watching Anne and John’s recommendation – Polya’s video “Let us teach guessing”. (Also on YouTube). To hear the man himself talking to his class is a joy. Not a traditional lecture – join in with his class as they guess!

Much food for thought – you can hear Polya’s summary of what is important in reasonable guessing at 54:16. Including the all-important reminder that we must test our guesses.

Polya Guessing from ihor charischak on Vimeo.

From the University of California, Berkeley see this summary of Polya’s problem-solving techniques – including a summary, in the Polya’s own words, on strategies for attacking problems in mathematics class from the book, How To Solve It, by George Polya, 2nd ed., Princeton University Press, 1957, ISBN 0-691-08097-6.

For a useful document on cutting space with planes, the problem discussed in Polya’s class, see this University of Toronto resource.

Seneca Learning

Seneca Learning

I included Seneca Learning in my latest sessions on Retrieval Practice (Retrieval Practice – it’s not all about quizzes); a free revision and homework platform, Seneca has applied research in cognitive neuroscience and psychology to provide an engaging environment for students. To answer the question What is Seneca? we can turn to this blog post from Stephen Wilks. The blog is helpful for learning more about Seneca, also note the Frequently Asked Questions which includes guides for teachers, students and parents. Further features are planned for teachers.

Exam Board specific GCSE content is available, my Year 10 students have tried this and reacted very favourably; this is certainly something I will be using with my students in the coming academic year; the step by step explanations and examples look very helpful indeed. My students certainly liked the AQA GCSE Maths sections we looked at – they also strayed into the many other subjects available, liking the content they saw.

Happily, Seneca has just announced that KS3 and A Level courses will also be free – such excellent news.
Free KS3 & A Level

Currently, several GCSE courses are available for UK Exam Boards.
GCSE Courses

Holiday Time

Looking back over some previous posts, I thought I would check and update some holiday-themed posts.

An Italian holiday included a visit to the wonderful National Cinema Museum in Turin – see the Mathematics in the Movies post (now with added songs!)

Milan Cathedral – photo by Sophie Young

Visiting Milan and wandering around the lovely Cathedral (one of the largest in Europe) I was struck by the geometry all around!

Some interesting sites on Mathematics and architecture:

From Tripbase, 9 Most Mathematically Interesting Buildings in the World and from Flavorwire, 10 Amazing Examples of Architecture Inspired by Mathematics. There is some overlap between these two lists, both mention quite rightly the cathedral in Barcelona Designed by Antoni Gaudi. 

Barcelona - Cathedral

Barcelona – Cathedral – photo by David Young

For some further reading Plus Magazine has an article on Perfect buildings: the maths of modern architecture by Marianne Freiberger.

WolframAlpha can be used for information on buildings. Note the different types of queries possible – simply type in the name of a building for information, for example, see Eiffel Tower or enter two buildings for a comparison. It is also possible to look up properties of buildingsdid you know that the Empire State Building has 102 floors?!


A great resource on TES for when we get back to school is Laura Wilson’s Plans and Elevations, a PowerPoint of images showing the plans and elevations of some famous buildings.

Holiday Pictures

All photographs by David Young, taken in the attic space of the Casa Milà Barcelona a building designed by Antoni Gaudí, constructed between 1906 and 1910.

cosh x Desmos image

Imaginary Exhibition


I do like to keep my eye open for mathematical pictures (Tetrahedral Numbers on Mathisfun), hence:

Neuwied Schloss cannons Photograph by David Young

Neuwied Schloss Cannons –  Photograph by David Young

PhET Simulations – Algebra

Algebra - Area model

Area Algebra Model – PhET Simulations

A new simulation in the excellent HTML5 collection – from PhET Interactive Simulations, we have Area Model Algebra, this is very simple to use. Rectangles of various sizes can be built showing the relationship between multiplication and area.

Also new in the collection, I like both Area Model Decimals and the Equality Explorer series.

Area Model - Decimals - PhET Simulations

Area Model Decimals – PhET Simulations

Equality Explorer - PhET SIMS

Equality Explorer – PhET Simulations


The ability to display or hide the variable and to take snapshots means working can be clearly shown.

Equality Explorer 2 - PhET SIMS

Equality Explorer – PhET Simulations

I have included a PhET Simulations page in the Lesson Planning series which notes several favourites I have used many times in the classroom.


Transition Time

It’s Transition time of year, with many students having completed their courses and about to embark on a new stage in their education. A reminder of some valuable resources to help their learning.

Mathematics for Students

As we come to the end of an academic year and look to a new one, it will be a time of change for many students. Perhaps you have completed GCSEs or equivalent qualifications (UK age 15-16) and are about to start on your A Levels or perhaps you have completed those and are about to start studying Mathematics at university.

To be in a position to begin your new courses well you should be thoroughly familiar with the essentials of the work you have studied to date. At whichever level you are studying your Algebra should be at a standard where you can manipulate expressions with ease.

Some resources to help you prepare and will be useful reference material for you during your course…

OCR Transition Algebra Fractions OCR – Bridging the Gap – A Student Guide

For students going on to A Level then the best thing you could do is use…

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