18 March 2018

18th Century Arithmetic

My fascination with old mathematics textbooks continues. I've previously blogged about an algebra book from the 1950s and a Victorian textbook from 1885. But now... well, hold the front page, because @BTNMathsJam sent me a link to a digitised arithmetic textbook from over 300 years ago. Bear in mind that this was way before The Industrial Revolution. In the early 1700s the population of England was only around 5 million and most people lived in poverty. Education in England was mainly for wealthy boys, and focused largely on Latin and Greek, morality, discipline and the Bible.
I've been reading the 22nd edition of Hodder's Arithmetick which dates from 1702 (there are also some earlier editions on Google Books). In this post I'm sharing some cool stuff from this fascinating book, just because I love it and you might find it interesting.

The Earliest Maths Textbooks
According to 'Early Schools and School-Books of New England' (Littlefield, 1904), the earliest arithmetic printed in English was Robert Record's 'The Ground of Arts: Teaching the perfect work and practice of Arithmetic, both in whole Numbers and Fractions, after a more easie and exact form then in former time hath been set forth'.  This book, with it's excellent snappy title, was written in 1540 in the form of "a dialogue betweene the master and the scholar; teaching the art and use of arithmetic with pen". This book remained popular for well over a hundred years, during which time a number of other arithmetic texts were published including works by Baker, Wingate and Oughtred. In 1661, the first edition of 'Hodder's Arithmetick' was published in London and was hugely successful, both in England and America. James Hodder was a master of a writing school in London.
In the 18th Century, arithmetic was taught in a similar way to writing. The teacher would provide a example from a book that students would work out on a separate sheet of paper and, once correct, it would be copied into a notebook known as a ciphering book. This sounds a bit like a modern approach involving students having a go on a mini-whiteboard before copying a neat worked example into their exercise book.

Much of Hodder's Arithmetick is devoted to 'vocational' arithmetic - working with money, measures and time (for example 'the addition of wine measures' which involves carrying hogsheads! Plus a whole chapter on 'The Rule of Barter').


The first thing that jumped out at me in Hodder's Arithmetick was the use of the word cypher for zero.
"Numeration is that part of arithmetick whereby one might rightly value, express, and write any number and sum propounded. To the attainment whereof, that all numbers are expressed by these characters following, whose simple value by themselves considered, you may here take notice of
one, two, three, four, five, six, seven, eight, nine, cypher.
 1       2       3       4       5       6       7       8       9       0
The Cypher serves to make up the number of places, but of itself signifies nothing"

It's fascinating to see the word cypher listed here. A century later, Victorian textbooks used the word zero. You may be aware that the word zero comes through the Arabic literal translation of the Sanskrit śūnya, meaning void or empty, into sifr. The word cipher or cypher was once commonly used for zero in the English language, but has come to refer to encoding.

Hodder goes on to explain place value.

'A prick with your pen between every three figures' - now known as a comma.

Multiplication and Division
The times tables are presented in Chapter 4:
Note the sensible lack of duplication (ie 7 x 6 is not listed because 6 x 7 has already been listed).

On learning multiplication tables, Hodder says "You must of necessity get it very perfectly by heart, before you can make any further progress in this art". I'm with him on that.

In describing how to do long multiplication, Hodder tells the reader to keep track of workings by crossing out digits in the multiplier when they've been dealt with.
"And having done with the first Figure of the Multiplier, cancel it with a Dash of the Pen, and proceed to the next..."
Cancel it with a dash of the pen! I love this.

The description of the method for division is a bit hard to follow. It's similar to the short division method we now use, though the digits are placed in different positions throughout the calculation (eg the answer ends up to to the right of the 'crooked line' instead of at the top).
Hodder later goes on to describe 'a more easy way of division, and with fewer figures'.
"I will not stand to shew you more of this common way of division, which is indeed very tedious and burdensome to the memory, and hath caused (to my knowledge) many to despair of attaining it, and to proceeding further in this art. But proceed by the method following, which will enable one to go on with far more ease and delight then commonly is seen". 
I tried to follow Hodder's 'easier method' but started to lose the will to live, so gave up.

The section on division ends with a glorious yet terrifying example which takes up an entire page.
The next chapter in the book is reduction, then we have fractions followed by 'The Rule of Three' (also known as 'The Golden Rule', which involves proportional reasoning). However, after reading the division chapter, my brain needs a rest! I will continue to share the delights of Hodder in a subsequent blog post. In the meantime, you can read the whole book here. Enjoy!

16 March 2018

5 Maths Gems #85

Welcome to my 85th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. SSDD and Venns
In case you haven't heard, Craig Barton (@mrbartonmaths) launched two new websites last week: ssddproblems.com and mathsvenns.com.

I first blogged about the 'Same Surface Different Deep' idea after Craig introduced it at the JustMaths conference last June. The idea is that we have a set of problems where the surfaces are similar (eg all isosceles triangles) but the deep structures (ie the topics) are different. It's brilliant to see this idea taking off in classrooms all over the world now that Craig has shared a large collection of SSDD problems.
Craig explains how and why to use these problems, and invites contributions, on ssddproblems.com. It's also worth reading Michael Pershan's (@mpershan) post "When is it helpful to make a bunch of different problems look the same?" and Karen Campe's (@KarenCampe) post "Looking Below the Surface".

Craig's second new website, mathsvenns.com, features a large collection of Venn-based rich tasks.

These tasks are excellent. Again, Craig is inviting contributions to this site so if you have a great Venn idea, email it to Craig for inclusion.

2. Pi Display
If you're looking for a display to brighten up your maths corridor, you might like this lovely 'First 1000 Digits of Pi' display from Jae Ess (@jaegetsreal). This display sparked some lovely conversations between students at Jae's school.
I've added this to my page of maths display resources.

3. Convince Me That
Daniel Kaufmann (@KauDan721) has created a set of 'Convince Me That' problems. Giving students the answer, rather than asking them to find the answer, allows them to focus on different aspects of the problem (the how, the why, the process).
Teachers are invited to contribute their own problems to this collection.

4. Minimally Different Questions
Jess (@FortyNineCubed) has created a new collection of minimally different problems. These are carefully structured so students can make connections between each question. Topics covered include solving equations with brackets, ratio, dividing negative numbers, multiplying negative numbers and solving equations.
5. Literacy
On Twitter I shared some word trees from membean.com that I use when talking to my students about vocabulary. For example when teaching percentages, I like to ask students where else they have heard the word cent. Their answers are always brilliant!
Other root trees with mathematical links include bi, uni, dia, equ, multi, tricircum and poly. I'm grateful to Maggie Harnew (@skillsworkshop) for sharing her lovely word maps for the numbers one and two. There's so much to explore here.
I had a great time at the Kettering maths conference last weekend - you can read my write up here and listen to my post-conference podcast with Craig Barton here.

In the podcast I promised that I'd set up a page to share knowledge organisers. My colleague Andy has been using knowledge organisers very effectively this year and he has given me permission to share some of his to get the page started. If you've made knowledge organisers that you're happy to share, please send me a link and I'll include them on this page.

On Monday I ran a 20 minute CPD session on GCSE revision for my colleagues (mainly focused on resources). My slides are here if you'd like to borrow them. Links are in the notes section on each slide.

In other news...
  • Dr Madas has published some IYGB papers for the AS pure content of the new A level (Paper L, Paper P, Paper Q and Paper R). Given how little exam practice material there is for our current Year 12s, these are really helpful!
  • Dr Frost is organising another maths teacher social event on Friday 13th April in Surrey. All welcome!
  • Chris McGrane wrote a great post on planning a lesson on integration which is worth a read if you teach A level.
  • If you're looking for an Easter-themed resource for the end of term, try Chris Smith's Easter relay
  • There are only a few residential tickets left for BCME!  It starts in two and a half weeks - I can't wait. 

I'll leave you this with excellent puzzle from Brilliant, shared by Mark Horley (@mhorley). 

11 March 2018


I had a wonderful day at #mathsconf14 in Kettering yesterday. Kettering is my favourite La Salle conference venue! Normally after a conference I write a detailed blog post about all the new things I learnt, but I've promised my daughters that I won't spend Mothering Sunday on my laptop, so today's post is very short.

I had a lovely time on Friday night, starting with celebratory drinks with my Twitter besties Craig, Ed and Tom (between us celebrating three new books and one new job).
I then had a lovely dinner with friends, including my former colleague Mariana who was attending her first ever conference. After dinner we joined the rest of the delegates for drinks - what a fantastic turnout! I was really grateful to Martin Noon who gave me another beautiful old textbook to add to my collection.

I had to arrive at the conference ridiculously early on Saturday to set up the MA bookstand, which I helped out on between workshops. We gave free goody bags to the first 100 visitors, and had a busy day selling lots of excellent books. I didn't get time to visit the rest of the exhibition but I was very pleased to meet Rob Eastaway on the Maths Inspiration stand, where I picked up a free truncatable prime pencil (when you sharpen it, the number remains prime!). Here's Lucy Rycroft-Smith modelling a giant version:
I loaded up on sweets to keep me going - thanks to Rob Smith for stocking the tuck shop with plenty of astro belts (my favourite!).

I enjoyed all four workshops I went to, and was gutted to miss the rest. It's always so hard to pick workshops. I won't go into detail about the sessions I attended here because I have already done so in my podcast with Craig Barton. Craig and I sat down immediately after the conference (well, after he'd finished giving autographs) and recorded a chat about what we learnt from each workshop. It's a relatively short podcast, so if you can spare 40 minutes then do have a listen.
If people like this 'Conference Takeaways' Podcast idea, we'll make it a regular thing, so do let us know your feedback.
I'm all fired up for BCME now, which is only three weeks away. Hopefully see you there, or at the next La Salle conference in Manchester.

Thank you to Mark McCourt, the La Salle team and everyone involved for another fantastic day.

1 March 2018

5 Maths Gems #84

Welcome to my 84th gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. GCSE Revision Checklists
Thanks to @mrchadburn for sharing his GCSE revision checklists for both Higher and Foundation tier in his post about GCSE preparation.
And for similar information in a totally different format, Miss Banks (@MissBanksMaths) has shared Higher and Foundation GCSE checklists, organised like mind maps.
For more resources for the fast approaching GCSE exams, see my 9-1 revision resources post (the most viewed post I've ever written!).

2. A Level Resources
Thanks to Sandra D (@mathsbox1) for creating a set of free AS Pure revision notes for the new A level.
Joe Berwick (@Joe_Berwick) has also been busy creating some excellent A level resources which are available on TES, including detailed notes and examples for the new Year 12 statistics content. These are written for students but might also be helpful for teachers who are teaching this content for the first time.

Finally, if you've got any students preparing for STEP this year then do check out the fantastic STEP Support Programme from the University of Cambridge.
3. Rich Tasks
Mr Knowles (@SK18Maths) has shared a number of activities on Twitter. This one is on sequences. Follow Mr Knowles for more tasks like this.
4. Variation Theory
You may have heard a lot about variation theory over the last couple of years. To use it correctly in the classroom and create appropriate resources, it's essential to have a strong understanding of the underlying principles. Naveen Rizvi (@naveenrizvi) has written an excellent blog post 'Resourcing: Applying Variation' that really clarifies how variation theory works - it's well worth a read.
5. Addition Pyramids
I observed an interview lesson in which the teacher used a classic starter activity. It's really simple and engaging, and works well at any age. It looks like it may have been based on this Nrich task.
I tweeted this activity and had lots of replies suggesting adaptations involving manipulatives, algebra and probability. A lot of people used this activity after I tweeted it and reported back that it worked really well. It made it to Mexico too.
I enjoyed reading Michael Jacobs'‏ (@msbjacobs) lovely blog post about similar puzzles that he's used with his students.

Thanks to Miss Banks (@MissBanksMaths) who has shared some free resources for this activity on TES.

In case you missed them, here are my latest blog posts:

If you enjoyed my post about Elementary Algebra for Schools, you might be interested in looking through the online textbook 'Elementary Arithmetic' to see how primary mathematics was taught a century ago.

If you're involved in the delivery of Core Maths or A level maths then I'm sure you'll be interested in this week's announcements regarding financial incentives for schools to increase uptake. I have mixed views on this!

The Kettering maths conference is next weekend! I'm looking forward to seeing everyone. I'm also really excited about BCME which is only one month away now.

I'll leave you with these puzzles from The Mathematical Gazette, one of the excellent publications of the Mathematical Association. The Student Problems will be made publicly available going forward. Prizes are available! The deadline for these two puzzles has passed but look out for the next set soon!

20 February 2018

5 Websites You Should Know... #5

In October 2016 I started writing a series of posts called '5 Websites You Should Know...'. I wrote four posts, covering Corbett Maths, Mr Carter Maths, MathsBot and MathsPad. These posts were based on a presentation I did at a TeachMeet. The fifth website in my TeachMeet presentation was resourceaholic.com, but it didn't make sense for me to write a blog post about my own website. So my '5 websites' series ended after four posts, leaving me with an unsatisfactory unfinished project.

Recently I discovered David Morse's resources and wrote about them in Gems 82. I feel like I should do more to spread the word about these resources, so I've decided they need a post in their own right. This is a good opportunity to finish off my '5 Websites You Should Know...' series!

David's resources are all freely available on TES and can be accessed through his growing website maths4everyone.com. There are over 250 resources, many of which I have linked from my resource libraries. David, an experienced maths teacher and Head of Computing, was the most downloaded new TES author of 2017. In this post I'll focus on David's worksheet grids.

I like these grids for five reasons:
  • The questions are well written.
  • Some sheets cover a single skill in depth, which is very useful when introducing a new skill for the first time (isolation of skills is all the rage at the moment, and makes a lot of sense to me... one thing at a time, please).
  • Other sheets are specifically designed for revision, so cover a whole topic in one place. Last year I wrote about how I run my Year 11 revision lessons after Easter - these grids will be perfect for the topic-specific element. 
  • The grid format is print friendly and student friendly. Plus there's no clutter, no mistakes, and the branding is unobtrusive.
  • Full solutions are provided.

Let's look at some examples...

1. Circle Theorems (First Steps) includes a whole sheet just on isosceles triangles in circles. I've never taught this explicitly before - it normally just comes up in amongst the other circle theorems. Next time, I'll slow down, teach it properly and use this resource.

This is what the solutions look like:

2. Calculating bearings has five worksheets of varying difficulty levels. GCSE students always seem to struggle with bearings - these questions really help develop fluency.
3. Area of a triangle using sine is an excellent set of questions on this topic, with a good level of challenge. I'd use these questions at both GCSE and A level.
4. Multiplying surds is one of a set of worksheets on surds. It specifically focuses on multiplying. As with the other resources, there's a sensible progression of questions here.
5. Expanding triple brackets provides straightforward fluency practice with a good level of challenge. There are similar sheets for expanding a single bracket, expanding and simplifying and expanding double brackets.

I've only provided five examples here but the collection of resources on TES is extensive and growing. Hopefully you get the idea - it's all standard fluency practice but well designed and user-friendly. These work well for classwork, revision, and cover. In my opinion these are really useful sets of questions.

David has shared way more than just worksheet grids so do check out his website to see the rest of his resources. Today I found his collection of challenging exam questions on vectors really helpful when planning a Year 11 lesson.

I'm very grateful to David for all his hard work in creating and sharing his resources. I hope you find them useful too.

17 February 2018

Elementary Algebra for Schools

I've really enjoyed looking through a fully digitised version of Elementary Algebra for Schools, a maths textbook which was first published 1885. I wish I had more time to do so. Unlike A Classbook of Algebra, a set of algebra exercises that I blogged about a couple of months ago, Elementary Algebra for Schools is a proper textbook featuring explanations, definitions and worked examples (plus over 3,500 questions for students to practise). I read the explanations with interest - it's fascinating to see some of the things that changed in the teaching of algebra over the course of the 20th century. For example algebraic division was previously taught very early on and is now not taught until A level.
In their introduction, the authors tell us that "The examples are very numerous, and have been compiled with great care". Pause for a minute and think about whether we can say the same about modern textbooks, in a world of ever changing curricula and pedagogical fads. Things aren't built to last anymore, including textbooks.

It's interesting to see the order in which topics are tackled. It is clear that the authors put a lot of thought into this. For example, in the preface they explain that the skill of factorising is purposely left until after "the student has acquired some freedom and readiness in the use of symbols". They go on to explain that leaving factorisation until students have developed algebraic fluency allows them to deal with it in greater depth, which is of course preferable to factors being "introduced and disposed of in one short early chapter".

The book also covers common misconceptions, which are rarely mentioned in modern textbooks. The authors say, "under the belief that prevalent mistakes are not sufficiently guarded against, we have given occasional notes to caution the reader against the blunders which experience shews to be almost universal amongst beginners". 

There is a lot I want to share from Elementary Algebra for Schools - in this post I'll feature a few things that may be of interest from the first four chapters. You can read the whole textbook yourself here, if you wish to do so.

Chapter I - Definitions. Substitutions.
We start with a set of very clear definitions and examples (extract below).
We're told that "the beginner must be careful to distinguish between coefficient and index" (eg know the difference between 2a and a2). It looks like a x a = 2a was just as a common a misconception 130 years ago as it is now. The first exercise in the book is specifically designed so that students can develop fluency in making the distinction between a coefficient and an index. This is done through substitution.
Modern textbooks have comparable explanations and exercises, though there's considerably less rigour.

To give a direct comparison, this is what the Victorian textbook says about writing the factors of a product in alphabetical order, compared to a modern day GCSE textbook.
Extract from 'Elementary Algebra for Beginners', 1885

Extract from 'Edexcel GCSE Maths Higher', OUP, 2015

I suppose we could argue that as long as teachers' verbal explanations are sufficiently clear and detailed, then it doesn't matter what the textbook says. These days, textbooks and worksheets are used primarily for practice rather explanations. So I suppose the brevity in textbooks explanations is acceptable, as long as the teaching itself is thorough and the exercises are well written.

In Elementary Algebra for Schools, it's interesting to see that guidance is given regarding the setting out of working. I expect we all wholeheartedly agree with the following points:
Chapter II - Negative Quantities. Addition of Like Terms.
The first chapter covered algebraic definitions and notation, and these fundamentals were practised through substitution exercises. The second chapter covers collecting like terms, focusing on how to deal with positive and negative quantities. Modern day teaching of algebra often takes these two concepts in reverse order, starting with simplification before later tackling substitution.
The first exercise on collecting like terms has a higher level of challenge and granularity than equivalent modern day exercises on the same topic. Here we have thirty questions where students are not required to identify whether the terms are 'like' or not - they are only required to decide whether to add or subtract each term.
In a GCSE textbook, we only have ten questions to practise this particular skill, before students move onto the next skill where like and unlike terms are mixed.

Chapter III - Simple Brackets. Addition.
In the third chapter of Elementary Algebra for Schools, we move onto simplifying expressions with a mixture of like and unlike terms. It is interesting that like terms are collected using a column addition method.
Note the explicit mention of descending powers - this convention regarding the ordering of terms is something I don't normally mention until I teach Binomial Expansions in Year 12.

Chapter IV - Subtraction.
There is an entire chapter on subtracting one expression from another. In my experience students these days don't spend any time on this skill at all, hence they often stumble in Year 12 when they are required to simplify 2x2 + 5x + 8 - (x2 + 6x - 7) for example, to find the area between two curves.
I wonder when, and why, questions of this form disappeared from most classrooms.

So that's the first four chapters: definitions, substitution, and collecting like terms. Forming algebraic expressions doesn't come until Chapter 9, but often comes earlier in modern day algebra teaching. The order in which skills are taught is really interesting, as is the isolation of particular skills. I don't know what the 'best' order is to introduce algebra, but it's certainly something I want to research further.

You may have noticed that I'm very geekily addicted to this old textbook thing now! If you're interested in this stuff too, look out for another post about Elementary Algebra for Schools in the near future.