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Math’s seems to be quite popular in the media at the moment, with three great articles printed in the last week or so! I thought I would share them here and provide a short commentary.

The Singular Mind of Terry Tao

The New York Times has published this great (long) article on the life of Field’s Medalist Terence Tao by Gareth Cook (GarethIdeas) The New York Times often seems to publish these kind of long articles, and luckily they have a policy of letting you read 10 articles a month from their website for free! 

I think this article nicely describes the life of a research mathematician, and I particularly like this quote

“The steady state of mathematical research is to be completely stuck. It is a process that Charles Fefferman of Princeton, himself a onetime math prodigy turned Fields medalist, likens to ‘‘playing chess with the devil.’’…”

The article is a very readable account of what it is to be a modern research mathematician, and in it Steven Strogatz sums up maths as a “… conversation with each other over the millennia”, emphasising the connections with mathematical discoveries across the generations.

The New York Times also has a puzzle from Terence Tao here.

John Horton Conway: The Worlds Most Charismatic Mathematician

This article by Siobhan Roberts appeared in the Guardian on Thursday 23rd July is an interesting portrait of the mathematician that Sir Michael Attiyah describes as the “most magical mathematician in the world”. John Conway is famous for his work in the 1970s on symmetry groups. I didn’t realise his commitment to maths education and how much time he spends at maths camps in the university holidays – not many academics would be willing to give up this precious research time. Conway is of course famou for inventing The Game of Life – I will write about this in the future I think, once I have coded it in Python.

This article is fascinating and well worth a read – I will definitely be buying the full book hen it comes out in September.

20 Mathematicians Who Changed the World

Walter Hickey has written this article for Business Insider. It is considerably shorter than the previous two and provides brief (a few lines) information on twenty famous mathematicians. I don’t completely agree with his choices – for instance what about George Boole?!? Who do you think is missing from this list? I’d be interested to know…

The Arithmetic Mean – Geometric Mean (AM-GM) inequality is possibly one of my favourite inequalities (another is the Cauchy-Schwarz inequality, see a later #summerblogchallenge post for more details on this).

Earlier this week I was honoured to have received an envelope of goodies from Ed (@solvemymaths). Alongside the very cool mathematical Mr Men stickers, there was a note from Ed with a lovely geometric illustration of the AM-GM for two lengths \(a,b\).

  In words, the AM-GM inequality states that the arithmetic mean of a set of (non-negative) numbers is always greater than or equal to the geometric mean of the set of numbers. More formally, the AM-GM inequality can be stated as follows: For a set of non-negative real numbers \(a_1,a_2,a_3,…,a_n\) the following inequality holds

 \(\frac{a_{1}+a_{2}+a_{3}+ \cdots  + a_{n}}{n} \geq \sqrt[n]{a_{1}a_{2}a_{3} \cdots a_{n}} \)

Example: For the set \(1,6,9,15\) the arithmetic mean is \(\frac{31}{4} = 7.75\) and the geometric mean is \(\sqrt[4]{810} \approx 5.3348 [\latex].

Proof: There are many different proofs of the AM-GM inequality. The excellent Art of Problem Solving website lists a few here . I particularly like the one using the rearrangement inequality as it is so concise.

The AM-GM inequality is often useful for questions on STEP and Maths Challenge papers, here is a typical question and a solution.

Question: Given that [latex]a,b,c\) are non-negative integers, show that

\((a+b)(b+c)(a+c) \geq 8abc\)

Solution: Given that \(a,b\) then applying the AM-GM inequality we have that

\(\begin{align} \frac{a+b}{2} &\geq \sqrt{ab} \\ a+b &\geq 2\sqrt{ab} \end{align} \)

Performing similar calculations with the other pairings we can obtain

\( \begin{align} b + c &\geq 2\sqrt{bc} \\ a+c &\geq \sqrt{ac} \end{align} \)

So, since the numbers involved are all positive we can multiply the inequalities to obtain

\( \begin{align} (a+b)(b+c)(a+c) &\geq 2\sqrt{ab}2\sqrt{bc}2\sqrt{ac} \\ &= 8\sqrt{abbcac} \\ &= 8\sqrt{a^2b^2c^2} \\ &=8abc \end{align} \)

as desired.

Can you come up with a nicer solution? Can you come up with similar questions?

The AM-GM inequality comes up in higher level maths too, a friend, Edward Hall of the University of Leicester has reminded me that it is used in a modified form when proving the continuous stability of the solutions to parabolic PDEs. He also mentioned that he has recently seen it used in a paper concerning Discontinuous Galerkin (more on these in the future) methods for quasilinear PDEs.

So here is the first of my 47 (if I can count….) blog posts as part of #summerblogchallenge – not quite as many as @MissNorledge!

I thought I would briefly write about rational functions which often come up in A Level and STEP papers. A rational function is any function \(y = f(x) = \frac{P(x)}{Q(x)} \) with \(P(x), Q(x) \) polynomials. Quite often you are asked to sketch such a function. To do this, the normal steps are to find the roots, i.e. where the function crosses the \(x-\)axis which is easily done by solving \(P(x) = 0 \). Then we can think about asymototes: For the horizontal asymptote consider the ratio of the leading coefficients of \(P(x)\) and \(Q(x)\), for the vertical asymptotes try to factorise \(Q(x)\) and find what values make \(Q(x)\) zero. It is the normal to mark on any local maxima and minima. Of course the usual way is to use calculus and differentiate, setting to zero and solving. However, sometimes we just need to know the approximate location of the maxima and minima  and the associated y values. This is due to the fact that we can often sketch the graph using this knowledge and examining what happens near the asymptotes.

I was shown this by another teacher who recently retired and had never considered it before, always using calculus. But with practise, this could be done mentally, pretty quickly I think.

The picture below shows the method for an example. Essentially you imagine a fixed horizontal line across the graph of your rational function, so \(y\) takes on a fixed value. We can then form a quadratic whose coefficients depend on \(y\). A maximum (or minimum) of our rational function will occur when the discriminant of our resulting quadratic is zero. Using this fact we can find the local maximum and minimum values of the rational function.

   

This post will be updated daily to reflect the latest posts that have been published. Anyone who is taking part in the #summerblogchallenge will have their posts listed here – as long as you use the hashtag #summerblogchallenge they will be picked up by me and @MissNorledge and indexed so everyone’s posts can be easily found.

My Posts (@DrBennison)

1. What is #sumerblogchallenge
2. Day 1 – Rational Functions
3. Day 2 – Restaurant Menus
4. Day 3 – AM-GM Inequality
5. Day 4 – Maths in the Media
6. Day 5 – The First #mathsjournalclub Article and Discussion Date
7. Day 7 – The Magic of the Micro: Part 1
8. Day 8 – Math(s) Teachers at Play 88
9. Day 9 – A Bit of Nostalgia
10. Day 10 – The Importance of Having a Break
11. Day 11 – My Twitter History
12. Bonus Post – My WordPress Anniversary
13. Day 12 – One of my Favourite Resources
14. Day 13 – An FMSP Extension Resource
15.  Day 14 – Barnsley’s Fern
16. Day 15 – Summer Further Mathematics Taster Questions
17. Day 16 – A Little Rant about Flash
18. Day 17 – A Few Thoughts on Negative Numbers
19. Day 18 – An Interesting Question from OCR
20. Day 19 – Calculators
21. Day 20 – Further Maths Taster Session for Year 10
22. Day 21 – FMSP Favourite Problems Number 3
23. Day 22 – A Picture of Tea
24. Day 23 – The Damn Quotient Rule
25. Day 24 – Very Short Introductions: Book Review 1
26. Day 25 – Whitstable Maths
27. Day 26 – One Week Till #mathsjournalclub
28. Day 27 – Parental Anxiety with Mathematics
29. Day 28 – For the Love of Books
30. Day 29 – A MathsJam Puzzle
31. Day 30 – Engaging with Academia
32. Day 31 – An Awful MyMaths Homework Question
33. Day 32 – A Bit More on Barnsley’s Fern
34. Day 33 – #mathsjournalcub TONIGHT!!
35. Day 34 – The First #mathsjournalclub Discussion
36. Day 35 – Anscombe’s Quartet
37. Day 36 – Timetable2calendar
38. Day 37 – Making an M1 Paper Harder
39. Day 38 – ZSL London and its Architecture
40. Day 39 – Twitter Chats
41. Day 40 – #mathsjournalclub Second Poll
42. Day 41 – Mathematicians’ Quote Posters
43. Day 42 – Suko
44. Day 43 – No Plan Starters
45. Day 44 – Further Maths Specification Mapping
46. Day 45 – Proof School
47. Day 46 – A VideoScribe Video

Christine Norledge (@MissNorledge)

1. 2015 #summerblogchallenge
2. Day 1 – 51 Days of Summer, The Blogpost Challenge and a Bit of Self Assessment
3. Day 2 – Lessons Learned Writing a Scheme of Work
4. Day 3 – Logarithmic Thinking, Evolution and Dinosaurs
5. Day 4 – Planning for GCSE 2015
6. Day 5 – How Does Bob Marley Like His Maths?
7. Day 6 – Quantity not Quality
8. Day 7 – Looking Back,Looking Forward
9. Day 8 – Exponential Growth and the Chessboard Problem
10. Day 9 – Pick of Twitter 26/07/15
11. Day 10 – the Importance of Trust
12. Day 11 – And Your Starter for Ten
13. Day 12 -Something Old, Something New
14. Day 13 – How to Bend Space and Time – The Joys of Twuffer
15.  Day 14 – The New Sudoku
16. Day 15 – It’s Only a Model…
17. Day 16 – It’s Only a Model – Part 2
18. Day 17 – My #summer10
19. Day 18 – My Twitter History
20. Day 19 – It’s a Marathon not a Sprint
21. Day 20 – Top 5 for Starters
22. Day 21 – Chinese School and the Photo Challenge
23. Day 22 – Photo Challenge (1) Infinite Mirrors
24. Day 23 – Photo Challenge (3) – Arches
25. Day 24 – Photo Challenge (3) Set with Playing Cards
26. Day 25 – Photo Challenge (4) It’s all Greek to Me
27. Day 26 – Photo Challenge (5) – Tiling Patterns
28. Day 27 – Photo Challenge (6) Triangular Bridge
29. Day 28 – Photo Challenge (7) More Arches in Ancona
30. Day 29 – I’m Back
31. Day 30 – Pick of Twitter 16/08/15
32. Day 31 – If This, What Next?
33. Day 32 – I’m Going Again…. (And Facebook)
34. Day 33 – Top 5 for Rich Tasks and Problem Solving
35. Day 34 – Top 5 for Worksheets
36. Day 35 – My Journey into Teaching
37. Day 36 – To 5 Stationary Must Haves
38. Day 37 – Miss, Do You Enjoy Teaching?
39. Day 38 – Pick of Twitter 24/08/15
40. Day 39 – An IFTTT and Pocket Update
41. Day 40 – Three New Blogs
42. Day 41 – Six Good Comics and One Good Thing
43. Day 42 – An Intriguing Use of Primes
44. Day 43 – Google Trends
45. Day 44 – Pick of Twitter 30/08/15
46. Day 45 – My #summer 10 Revisited
47. Day 46 – My Classroom, Part 1
48. Day 47 – Display Goodies
49. Day 48 – Command Word Posters
50. Day 49 – Back to School Essentials
51. Day 50 – Useful Things for Website Developers
52. Day 51 – (Almost) Failing at the Final Hurdle

Mark Wilson (@mwimaths)

1. Day 1 – Psyching Myself Up
2. Day 2 – Circle Theorem Toolkit
3. Day 3 – In Praise of Teacher Led Inset
4. Day 4 – Google Keep as a Task Management Tool
5. Day 5 – Mix It Up: A Group Work Strategy
6. Day 6 – Transitioning to BYOD
7. Day 7 – Time Management Video
8. Day 8 – The Digital Native
9. Day 9 – Digital Card Sorts
10. Day 10 – Assignment Workflow
11. Day 11 – Revision Carousel
12. Day 12 – I Didn’t Want to do Maths
13. Day 13 – Google Search Takeaways
14. Day 14 – Fermi Questions
15. Day 15 – Stand Off
16. Day 16 – Set Theory Extension
17. Day 17 – Book Marking
18. Day 18 – Fermi Questions: A Worked Example
19. Day 19 – Taking Sabbaths
20. Day 20: Statistical Stories
21. Day 21 – BYOD: Some Tools
22. Day 22 – Subtracting From Questions
23. Day 23 – Who’s Studying Further Maths?
24. Day 24 – Teaching Google Sheets
25. Day 25 – Incognito Mode
26. Day 26 – Sunday Sabbatical: Podcasts
27. Day 27 – The 5 Minute Lecture
28. Day 28 – Homework, Feedback and Forms
29. Day 29 – docAppender and Keeping in the Loop
30. Day 30 – Don’t Watch the News Today
31. Day 31 – Multistep Chains
32. Day 32 – Set Zero: An Appeal
33. Day 33 – Sunday Sabbaticals: Mathematical Pilgrimages
34. Day 34 – Doctopus and Homework Feedback
35. Day 35 – Classroom and Calendars
36. Day 36 – Loop and the Introduction to Mechanics
37. Day 37 – Analysing the Answers
38. Day 38 – In Defense of Rote
39. Day 39 – Keeping Track Using VLOOKUP
40. Day 40 – Sunday Sabbaticals: xkcd
41. Day 41 – Farewell to the #summerblogchallenge

Archbishop Sancroft High School Maths Department (@ASHS_Maths)

1. Day 1 – Vertical Tutoring
2. Day 2 – Year 6s Starting in July
3. Day 3 – Down Time

Jennifer Stice (@mathchick5)

1. Day 1 – Activity Based Approach and the Success in “Doing Math“
2. Day 4 – Getting to Know You: Back to School Activities Using Technology

Kim Thomas Lee (@kimThomasLee)

1. Day 1 – Why I Love Teaching Fractions
2. Day 2 – I’m Still Thinking About Fractions
3. Day 3 – A Fraction of the Fractions Lessons, that I Still Love Teaching
4. Day 4 – Don’t Call me Average… That’s Mean, But at Least I Get to Eat the Smarties
5. Day 5 – Numeracy, do the Statistics Add Up? I Think They Are Scary
6. Day 6 – Numeracy Across the Curriculum, How do I Manage to Use Sport in my Lessons?
7. Day 7 – Why do we Need to be Numerate?
8. Day 8 -Numeracy Within MFL
9. Day 9 – Did I Use Numeracy Today? Of Course I Did, I Use It Everyday
10. Day 10 – Never Underestimate the Value of Tea
11. Day 11 – Design Technology:Numeracy in the Real World
12. Day 15 – A Quick Reference to Golf
13. Day 16 – Numeracy in English Lessons
14. Day 17 – Encouraging Literacy in Maths Lessons
15. Day 18 – Why Does a Maths Teacher Read the Core Texts?
16. Day 19 – Learning Styles, Starting With a Critique of My Learning Style
17. Day 20 – Seating Plans and Collaborative Learning Structures
18. Day 21 – Flipped Learning for Revision
19. Day 22 – Art and the Aesthetics of Mathematics
20. Day 23 – Starting Year 7, My Wishlist
21. Day 24 – Do Music and Maths Add Up
22. Day 25 – Students Say the Funniest Things
23. Day 26 – What About Those Who Don’t?
24. Day 27 – Will Your Job Use Maths? An Interesting Clip
25. Day 28 – Talk for Teaching
26. Day 29 – The Best Maths Quotes
27. Day 30 – The Pure Mathematical Beauty of Game Theory
28. Day 31 – The Maths and the Morals of BOGOF, BOGO and BOGOHP
29. Day 32 – The Economics, or is it Numeracy of Pocket Money
30. Day 33 – Numeracy Challenge Days
31. Day 34 – What About Those Times When Technology Fails
32. Day 35 – Rewards
33. Day 36 – If the World was a Village of a 100, One of my Favourite Resources
34. Day 37 – The New School Year, the Cost of Returning to School

@funASDteacher

1. Day 1 – I’m Here Because No-one Else Will Have Me
2. Day 2 – Match Your Books
3. Day 3 – Make Us Wait
4. Day 4 – Push Your Own Boundaries
5. Day 5 – Ode to the Envelope
6. Day 6 – Learn Like a Pirate
7. Day 7 – Where to Sit
8. Day 8 – Let’s Get Writing
9. Day 9 – M is for Autism
10. Day 10 – Notice the Small Things
11. Day 11 – Special Interests
12. Day 12 – Lose the Glare
13. Day 13 – Change the Objective
14. Day 14 – The Devil is in the Detail
15. Day 15 – Inclusion or Exclusion
16. Day 16 – The Obsession with Time
17. Day 17 – Braving The Technology
18. Day 18 – Rome Wasn’t Built in a Day
19. Day 19 – Support the Support Staff
20. Day 20 – Show Me How
21. Day 21 – If in Doubt, Make a List
22. Day 22 – Teaching Like a Pirate? Really?
23. Day 23 – Photocopier Traumas
24. Day 24 – Factor in the Anxiety
25. Day 25 – Routines: Friend or Foe?
26. Day 26 – Ask Questions
27. Day 27 – Making Mistakes
28. Day 28 – Time to Reboot
29. Day 29 – 5 Point Scale
30. Day 30 – The First Day
31. Day 31 – Group Work

It would be amazing to be able to add you to the above list……

Last week I made the (what I thought) throw-away comment that I was going to try and write a blog post every day of the holiday. I said this mainly because I want to try and get more regular in my posting, and this seems the perfect incentive! 

Christine (@MissNorledge) seemed keen to take part and has worked out that she has 51 posts to write!! This is a definite incentive for me to manage the whole summer as she has a much bigger summer of important events planned than I do!

A few others are joining in too, please feel free to join in!

Tomorrow afternoon I will create a post where I will continually link to all the posts everyone writes, so please remember to tweet your posts with the hashtag #summerblogchallenge so that me and Christine can link them 😉

I break up tomorrow so I will be officially starting my challenge on Friday…….

I am hosting this months Maths Teachers at Play blog carnival and o would like your submissions please! 

The Maths Teachers at Play blog carnival is organised by Denise Gaskins (on Twitter as @letsplaymath) and every month gets hosted by someone different. This month I have the honour of hosting, and I’m aiming to post the carnival on 30th July.

 I’d love it if you would submit any articles for consideration; you can either do this directly to me or using the form located here. 

I really want to get a broad range of posts from all over the world and across all age ranges so I am looking forward to anything I receive……

I had seen pictures like this before as it is the classic way of representing how the different types of number relate, but I wasn’t totally happy with any of the ones online. 

So I decided to make my own.   

 I used this in a Year 10 Sixth Form taster session. I will post more about this later in the week. 

I created this image in Geogebra because of how easy it is to draw ellipses and use LaTeX symbols. The file is available here. 

Because of my numerical analysis background I am very passionate about teaching coding alongside mathematics. In particular, I don’t think the A Level syllabus really does justice to the numerical methods component of the course. As most real life problems have to be solved approximately I think that numerical methods are incredibly important, but that teaching them without coding them up is silly!

I have always found that once I have coded something my understanding of a method has vastly improved. Because of this, if there is time I think it is good to expose A Level mathematicians to a bit of programming.

Python is the ideal language for this as

1. It is freely available across Windows, Mac OS X and Linux.
2. The documentation is fantastic,
3. The syntax is relatively straightforward.
4. In it’s simplest form the Python interpreter can function as an advanced interactive calculator.
5. It contains all the functionality of a modern professional program language (i.e. it is not just an academic curiosity)

I believe that number 4 in the above list is one of the big advantages of Python over a traditional compiled language (such as my favourite language Fortran or C++) as this can make the whole concept of learning a programming language for the first time a little bit less terrifying. For students, being able to see almost immediately what the commands they have just typed is very powerful – it must have been awful learning to program with punch cards!

For my first programming workshop with my Year 12 Further Mathematicians I chose to use the IDLE interpreter that comes with Python due to it’s simplicity.

  
With the Python Shell (the rightmost window in the above picture) you can type commands one-by-one and explore Python interactively. Launching the editor window you are able to write programs and then run them with the interpreter. One restriction of Idle with my school’s setup is that I cannot import my own modules – because of this for the next Workshop I will be using PyCharm. 

I gave my students a 6 sided worksheet, with some notes and examples to work through. See below for an example of the type of exercises and the full worksheet is here. 

 
All my students seemed to really enjoy doing a bit of coding, and I was really impressed with how well they got on. I’ve seen 2nd year undergraduates struggle more when they are introduced to Matlab than they were 🙂 

If you fancy having a go working through the sheet, the codes for the exercises are all contained in a tarball which you can download here.

I will write again about the future workshops.

At the weekend I suggested the idea of a Twitter Maths Journal Club. MY intention is for this to run along similar lines to the fantastic Twitter Maths Book club (@MathsBookClub) (blog is here).

I have set up a twitter account specifically for this, so please follow @mathjournalclub to stay up to date.

The plan is as follows: every couple of months a journal article will be selected by a poll and we will then have a twitter discussion for an hour one evening, about a month or 3 weeks after the article has been selected. This will be on a day where there isn’t already some kind of mathschat or #mathsTLP taking place.

As a lot of academic articles are pay-wall protected our choice will be a little limited – so either articles that have open access for a particular journal issue, free to access articles or articles where there are high quality pre-prints available on the author’s website.

My intention is to allow people to suggest articles for the poll on the following month, but to get things started here are the articles on the first poll (together with their abstracts)

• How Ordinary Elimination Became Gaussian Elimination; Joseph F Grcar – Newton, in notes that he would rather not have seen published, described a process for solving simultaneous equations that later authors applied specifically to linear equations. This method — that Euler did not recommend, that Legendre called “ordinary,” and that Gauss called “common” — is now named after Gauss: “Gaussian” elimination. Gauss’s name became associated with elimination through the adoption, by professional computers, of a specialized notation that Gauss devised for his own least squares calculations. The notation allowed elimination to be viewed as a sequence of arithmetic operations that were repeatedly optimized for hand computing and eventually were described by matrices.
• Contrasts in Mathematical Challenges in A-Level Mathematics and Further Mathematics, and Undergraduate Examinations; Ellie Darlington (Teaching Mathematics and its Applications) – This article describes part of a study which investigated the role of questions in students’ approaches to learning mathematics at the secondary/tertiary interface, focussing on the enculturation of students at the University of Oxford. Use of the Mathematical Assessment Task Hierarchy taxonomy revealed A-level Mathematics and Further Mathematics questions in England and Wales to focus on requiring students to demon- strate a routine use of procedures, whereas those in first-year undergraduate mathematics primarily required students to be able to draw implications, conclusions and to justify their answers and make conjectures.While these findings confirm the need for reforms of examinations at this level, questions must also be raised over the nature of undergraduate mathematics assessment, since it is sometimes possible for students to be awarded a first- class examination mark solely through stating known facts or reproducing something verbatim from lecture notes.
• A Glimpse into Secondary Students’ Understanding of Functions; Jonathan Brendefur, Gwyneth Hughes & Robert Eley (International Journal for Mathematics Teaching and Learning) – In this article we examine how secondary school students think about functional relationships. More specifically, we examined seven students’ intuitive knowledge in regards to representing two real-world situations with functions. We found students do not tend to represent functional relationships with coordinate graphs even though they are able to do so. Instead, these students tend to represent the physical characteristics of the situation. In addition, we discovered that middle- school students had sophisticated ideas of dependency and covariance. All the students were able to use their models of the situation to generalize and make predictions. These findings suggest that secondary students have the ability to describe covariant and dependent relations and that their models of functions tend to be more intuitive than mathematical – even for the students in algebra II and calculus. Our work suggests a possible framework that begins describing a way of analyzing students’ understanding of functions.
• Bridging the Divide – Seeing Mathematics in the World Through Dynamic Geometry; Hatice Aydin & John Monaghan (Teaching Mathematics and it’s Application) – InTMA, Oldknow (2009,TEAMAT, 28,180-195) called forways to unlock students’ skills so that they increase learning about the world of mathematics and the objects in the world around them. This article examines one way in which we may unlock the student skills.We are currently exploring the potential for students to ‘see’ mathematics in the real world through ‘marking’ mathematical features of digital images using a dynamic geometry system (GeoGebra). In this article we present, as a partial response to Oldknow, preliminary results.
• Using Geometric Images of Number to Teach Mental addition and Subtraction, Peter Lacey (Mathematics Teaching) – no abstract available.

It would be great if you would like to get involved, if you do please complete the Poll.

I really hope you want to get involved, I think it could be a great thing to do. Please suggest articles to include for future polls.

Update: Poll closes on 24th July

A week or so ago Graham Colman (@colmanweb) asked, on Twitter, for a definition and people’s views on Mastery maths. 

Before I started teaching I found it slightly odd that people were talking about the “mastery curriculum” as a new thing…. Surely, no curriculum would have the intention of not building mastery of mathematics in students. My personal experience of school mathematics was entirely on top sets and I definitely did achieve mastery and fluency with the concepts – it was just a bit boring repeating all the topics almost every year until Year 11. I was quite shocked, to be honest, when I started teaching and saw how weak some students were when they came into secondary school and how little progress some students made throughout Key Stage 3. 

Because of this I was quite keen to get involved in developing a mastery based curriculum for Year 7 and 8. My school has been working in conjunction with others in the East Midlands West MathsHub (@EM_mathshub). The layout of our KS3 is similar to that from ARK’s Mathematics Mastery scheme but not totally the same:

 
I am hopeful that this approach will lead to a deeper conceptual understanding in all students and lead to improved results as time can be focussed on extending students knowledge over time and not just repeating things. 

For a definition of mastery in mathematics as asked for by Graham Colman I think I would say something like:

 “A student has developed a mastery in mathematics when they can apply seemingly disparate  techniques and concepts in novel ways  to solve an unseen problem”. 

In a sense this is what a research mathematician is expected to be able to do. 

Can we expect this of school students?

I think we can, within the confines of the school curriculum anyway – geometrical knowledge could be applied to tackle an algebraic problem and vice versa for example. 

Spending more time and delving into topics in more detail allows students to be more critical, giving them a chance to learn how to evaluate strategies and choose the most appropriate given a particular problem. 

For this to be successful though I fundamentally believe that students need to be yet with the basic properties of number – being able to decompose calculations into stages that facilitate calculation, deep knowledge of number bonds and times table facts. Some people argue that these skills aren’t so important now that everyone has a relative good calculator on their phone – I disagree! As well as being quicker than finding a calculator and then keying in a calculation these skills also allow you to mentally check the whether a result is sensible. 

Last month I read Ian Davies (director of curriculum at Mathematics Mastery) post “Mastery – What it is and what it isn’t!” with interest. I liked how he explained why he felt that he had mastered addition, but not integration (in fact can anyone truly master integration?) He also quotes Helen Drury’s definition of mastery of a mathematical concept – the importance of being able to move between different representations stands out.