5 Maths Gems #82

Welcome to my 82^nd gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. GCSE Questions by Category
Jamie Frost (@DrFrostMaths) has started to create a set of "Full Coverage" GCSE revision worksheets. So far he's published question compilations for four topics: bounds, proof, functions and direct and inverse proportion. These compilations contain one example of each different category of past GCSE question for each topic. I expect I'll use these in revision lessons this summer.

2. Cover Me
The lovely people at MathsPad have made some new interactive 'Cover Me' puzzles for eighteen different topics. Four are free. I've enjoyed having a play with these.

3. GCSE Resources

David Morse (@maths4everyone) has shared a large number of free GCSE and iGCSE resources on TES - you can find them through maths4everyone.com. His collection includes a really helpful set of topic booklets and topic review sheets. I like the questions on these review sheets and have added a link to my 9 - 1 Revision Resources post.

Some of these would work well for Year 12 too.

4. Number Election
@MathsEdIdeas shared a lovely set of resources for a 'Favourite Number Election'. The idea is that students campaign for their favourite number and an election is held, perhaps to mark Pi Day or the NSPCC's Number Day. 

5. Visuals

I've recently spotted a few animations on Twitter that are worth sharing. @brilliantorg shared a visual representation of a difference of two squares.

You’ll never forget the difference of squares once you’ve seen this visual representation. pic.twitter.com/fwVLGJAvKq

— Brilliant.org (@brilliantorg) 19 January 2018

Tim Brzezinski‏ (@dynamic_math) shared an interactive animation showing how to find the surface area of a cylinder, which is something students often seem to forget.

This tweet about representing an equation, and the excellent Twitter conversation that followed, is worth a look.

Here are some ways to represent 2x+3 = 17. Any others you can think of? #msmathchat #mtbos pic.twitter.com/XQF2uW2sdq

— Kent Haines (@KentHaines) 3 January 2018

Also check out new website MathIsVisual.com which was created to assist in building a better conceptual understanding of mathematics through the use of visuals.

Update
I recently wrote another two posts about what I've found in old textbooks: Equations Exercises and Lost Vocabulary. I'll be sharing more old textbook exercises soon. In the meantime, if you enjoy browsing through old textbooks (my new hobby!) then here are a few links:

• Adams's New Arithmetic
• Ray's New Practical Arithmetic
• Elementary Algebra for Schools
• Calculus Made Easy (thanks to @FitzchivalryF for this link - it's an entertaining read!)

The big news in the world of maths education last week was the publication of Craig Barton's book 'How I Wish I'd Taught Maths'. I first read a review copy a couple of months ago and knew it would be a huge hit. It's great to see all the incredible feedback that Craig is getting. People love it. I was unbelievably chuffed when I saw my name on the back cover. My daughter keeps telling everyone that "mummy is on a book!". You can read my review here.

I'd like to say a belated thank you to Jamie Frost for inviting maths teachers on Twitter to festive drinks at his house earlier this month. I had a lovely evening.

I'm looking forward to the Kettering conference on 10th March. I'm not speaking at this conference but will be helping to run the MA book stand so do come along and say hello. I'll also be at the pre-conference drinks at the Premier Inn on the Friday night.

Also, don't forget that the early bird rate for BCME tickets ends next week - book now! It's the biggest maths conference of the year and takes place in the Easter holidays. I wrote a post about it here.

I'll leave you with this joke shared by @PG_Zan.

How I Wish I'd Taught Maths

Craig Barton's new book 'How I Wish I'd Taught Maths' is a game changer... in a game that very much needs changing. It's absolutely superb. I genuinely think it might have a huge impact on the way maths is taught. Or at least, I hope it does.

Craig's teaching has transformed significantly in recent years. Through a charming and witty narrative, Craig explains how he used to teach - back when he was a highly successful 'Ofsted outstanding' advanced skills teacher - and why he teaches totally differently now that he has properly engaged with education research. He jokes about how ineffective his previous approaches were, and this may make uncomfortable (but worthwhile) reading for maths teachers who still teach in this way.

Teachers are trained pretty quickly in this country, particularly those on school based schemes. New teachers are often thrown into classrooms with no knowledge of cognitive science whatsoever. As Craig writes, "a teacher not considering how their students think and learn is kind of like a doctor not being overly concerned about the workings of the body, or a baker taking only a casual interest in the best conditions for bread to rise". Thankfully Craig's book gives both new and experienced teachers the opportunity to remedy this.

Craig's anecdote about 'The Swiss Roll Incident' (in which his students remembered swiss rolls instead of maths after a messy jam-filled lesson) perfectly illustrates the notion that "students remember what they think about". Throughout the book Craig's anecdotes and reflections beautifully exemplify some of the common mistakes that teachers make. Each anecdote comes with a description of Craig's key takeaways from the relevant research and an explanation of what he now does differently. This provides a clear way forward for maths teachers looking to improve the effectiveness of their practice.

In his chapter on deliberate practice, Craig writes about identifying sub-processes and isolating skills. He shares examples of short activities which allow him to pick up on specific misconceptions and address them before they get wrapped up in more complex processes. The example below is part of a series of short activities for teaching students how to add fractions.

I'm not sure that this kind of activity is currently common practice, but like many approaches featured in Craig's book, I think it may become recognised as a highly effective approach over the coming years. It's great that Craig has shared so many specific examples of activities, resources and explanations. These are incredibly helpful to teachers. 

In summary, I love this book! Not only has Craig put all the relevant education research in one place, which is perfect for overworked and exhausted teachers like me, he has also interpreted broad education research specifically in the context of maths teaching. Most of the research isn't new, but Craig adds so much value by describing how it relates his own practice that even teachers who keep up to date with the latest research will benefit from reading Craig's interpretations. His advice is easy to understand and instantly transferable to the classroom.

Craig's book is a real pleasure to read and had me giggling throughout ("keep this quiet, but I flipping hate 3D trigonometry!"). It has the potential to have a huge impact on the way maths is taught. It's delightfully controversial at times and I'm certain that it will spark lots of interesting discussions in maths departments all over the country. Once you've read it, do let me know what you think.

Equations Exercises

I was looking at some of the old maths textbooks at my school and noticed than even as recently as the 1980s, textbook exercises contained a lot more practice questions than modern textbooks. Below is an example comparing the same topic in a 1980s textbook ('Negative Numbers and Graphs' by Heylings) and a current GCSE textbook ('Edexcel GCSE Maths Higher' published by Oxford University Press). The exercises cover the same skill but the first exercise is double the length of the second. I guess modern textbooks have to fit an entire GCSE course into a single book, restricting the amount of practice questions they can include.

From 'Negative Numbers and Graphs', first published 1984

From 'Edexcel GCSE Maths Higher', first published 2015

Last month I wrote a post about a 1950s algebra textbook called 'A Classbook of Algebra'. The questions in this textbook are generally more challenging than questions in most modern textbooks. In response to this post a number of very generous volunteers stepped forward offering to type up some of the exercises so that teachers can use them in lessons. I am very grateful for the time and effort that has gone into this. In this post I have provided links to all the exercises typed up so far which relate to the skill of solving equations. I have an additional eighteen exercises on other topics which I will collate over the next few weeks.

These exercises are all rather long and the idea is not necessarily to use them in their entirety. Because they have all been typed up in in Word, teachers will easily be able to edit the exercises or cut and paste particular questions to use as examples in class.

Most exercises include answers. I will edit this post if anything is updated. For each exercise listed below I have included a small extract so you can preview the type of questions covered.

1. Very Easy Equations (four exercises) - with thanks to Caroline Beale (@cbealemaths)

2. Easy Equations I - with thanks to Claire Willis (@MissWillisMaths)

3. Easy Equations II - with thanks to Caroline Beale (@cbealemaths)

4. Equations Involving Fractions - with thanks to Michael Allan (@mrallanmaths)

5. Equations Involving Brackets - with thanks to Jane Appleton (@JaneAppleton24)

6. Equations with Brackets and Harder Equations with Brackets - with thanks to Justin Thompson 

7. Equations Involving Directed Numbers - with thanks to Caroline Beale (@cbealemaths)

8. Easy Literal Equations - with thanks to Sandra Douglas (@mathsbox1)

9. Miscellaneous Simultaneous Equations - with thanks to Fee Wilson (@fionajw)

10. Miscellaneous Equations - with thanks to Dan Rodriguez-Clark (@InteractMaths). 

I hope this is useful. Thank you again to everyone who has worked on this. And of course full credit to Sidney Trustram, the original author of these exercises which, 70 years on, are still making us think. Look out for my next posts in which I'll share exercises on simplifying algebra, expanding, factorising, writing algebraically and working with directed numbers.

Lost Vocabulary

While reading the Victorian maths textbook 'Elementary Algebra for Schools' I spotted quite a few words and phrases which are rarely used in modern secondary schools. I'm not saying that this vocabulary has disappeared from the field of mathematics, but I doubt you will hear these terms in GCSE lessons.

Let's take a quick look at some interesting words and how they were used in the 1800s.

Unity
Meaning (in this context): the number one.

• "When the coefficient is unity it is usually omitted. Thus we do not write 1a, but simply a".
• "If the product of two quantities be equal to unity, each is said to be the reciprocal of each other".
• "Subtract 3x² - 5x + 1 from unity, and add 5x² - 6x to the result".
• "We proceed now to the resolution into factors of trinomial expressions when the coefficient of the highest power is not unity".

Resolve into Factors
Meaning: factorise (UK) or factor (US) - see Colin Beveridge's post 'Factorise or Factor'.

• "Resolve into factors x² - ax + 5x - 5a".
• "The beginner should be careful not to begin cancelling until he has expressed both numerator and denominator in the most convenient form, by resolution into factors where necessary". 
• "Resolve 4a²(x³+18ab²) - (32a⁵+9b²a³)  into four factors". 

Diminished
Meaning: made smaller or less.

• "The sum of -3x, -5x, -7x, -x is -16x. For a sum of money diminished successively by £3, £5, £7 and £1 is diminished altogether by £16".
• "Divide 105 into two parts, one of which diminished by 20 shall be equal to the other diminished by 15".

Whence
Meaning: as a consequence of which.

• "Whence the result follows".
• "Whence x = 1 is the only solution"

Involution
Meaning: multiplying an expression by itself.

Evolution
Meaning: the operation of finding the root of an expression.

Antecedent and Consequent

Meaning: the first and second term of a ratio respectively.

• "The ratio a:b is equal to the ratio ma:mb; that is, the value of a ratio remains unaltered if the antecedent and the consequent are multiplied or divided by the same quantity".
• "A ratio is said to be a ratio of greater inequality, of less inequality, or of equality, according as the antecedent is greater than, less than, or equal to the consequent. 

[There are loads of interesting terms in the ratio chapter - including commensurable and subduplicate.]

Transposition
Meaning: The act of transferring something to a different place.

[This hasn't disappeared, but the use of the phrase 'by transposition' or 'transposing' in worked examples is something we don't see in schools anymore].

There are many more words and phrases in this book which seem to have disappeared from the daily mathematical vocabulary of secondary schools - far too many to list here. Whence I will save the rest for another post. I hope you find all this as interesting as I do. No doubt someone will now contact me to say that they use all these words in their classroom on a regular basis...!

5 Maths Gems #81

Happy New Year! Welcome to my 81^st gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Number Properties
In Gems 79 I shared this lovely puzzle from Chris Smith (@aap03102):

Since then I've seen two great resources based on a similar idea. The first is the interactive 'Consecutive Number Types' puzzles from Jonathan Hall (@StudyMaths):

And the second is this free worksheet 'Consecutive Chains' from MathsPad.

I think these would work really well at Key Stage 2, 3 or 4 for exploring number properties.

2. Online Textbooks 
After I blogged about a 1950s textbook last month, I gathered together a group of volunteers to type up old algebra exercises into Word and work out the answers. Thank you so much to everyone who has contributed. The project is ongoing and I will blog about it soon. In the meantime, you can see our progress here. Also, thank you to @EmporiumMaths for sharing some 1950s algebra questions from past London O level papers here.

Huge thanks to @NetNym for sharing a beautiful Victorian maths textbook 'Elementary Algebra for Schools' that has been fully digitised. This really is lovely and well worth looking through (I'm a bit addicted to it!). The zoom function works well so the exercises can be used in class if desired - answers are included, and the whole book can be downloaded as a PDF.

I think that we have a lot to learn from the explanations and examples in this textbook. I've started writing a post about this which I hope to publish next week.

Meanwhile, a teacher in Australia (@adaprojectnet) has created a website adaproject.net which is an open online textbook for all to use, covering topics from Key Stage 1 to 5. It's worth having a look at this project which is growing all the time. 

3. Primary Resources

It's great to see John Corbett (@Corbettmaths) publishing lots of new primary 5-a-day content alongside his very popular GCSE 5-a-day collection. Each day there are five KS2 SATs style questions at four different difficult levels.

Also for primaries, Dr Frost (@DrFrostMaths) has now added Primary Maths Challenge questions to his website with the help of @Mathematical_A. You can browse by topic or by paper.

Primary Maths Challenge questions

4. A Level Resources
StudyWell (@_StudyWell) has published a couple of practice papers for the new A level - these are free for a limited time.

Tom Bennison (@DrBennison) shared a new Christmas Calculated Colouring for A level. If you didn't use this at Christmas, the set of questions may be helpful for Year 12 revision later in the year. Tom also published a longer Christmas Calculated Colouring in 2015.

5. Place Value
Thanks to Jonathan Hall (@StudyMaths) for sharing a new interactive place value chart. This is really helpful for demonstrating the effect of multiplying and dividing by ten. You can easily duplicate rows which saves you writing the same digits on the board multiple times.

Update

In case you missed my recent posts, they were:

• New GCSE: Ratio
• Jo's Blog Posts in 2017
• New this Christmas
• Algebraic Fluency - 50s Style
• Mock Season

You can read the latest eNews from the Mathematical Association here.

I'm excited that I've now booked a place at BCME - the biggest maths teacher conference of 2018. BCME conferences only happen once every four years - if you're not sure what BCME is then read my post about it. The price goes up after 31st January so book quickly!

I also hope to see lots of you at #mathsconf14 in Kettering on Saturday 10th March, and for drinks the night before. I had a wonderful time at all the maths events I attended in 2017 - if you've not been to a maths conference before, why not have a go in 2018? All events are listed here.

I'll leave you with this lovely graph, shared by @simongerman600, showing what people really mean when they use vague terminology describing the likelihood of an event. There are other cool graphs here. I think this one would make an excellent discussion point when teaching probability.

Have a fantastic term maths teachers! And all the best for 2018.