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.





15 February 2018

Old Textbook Exercises

This is the last in my series of three posts sharing the work of a team of teachers who have been busy typing up exercises from a 1950s algebra textbook.

In today's post I've provided links for two topics:
A. Directed Numbers 
B. Factorising and Expanding. 

For each exercise I've included an extract so you can preview the type of questions covered. The idea is not necessarily to use these exercises in their entirety. They are provided in Word format so it's easy to copy and paste extracts and examples to use in class.

1. Use of Directed Numbers - with thanks to Dee Vijayan (@DeeVijayan)
2. The Products of Directed Numbers - with thanks to Alice Dwyer (@littlemissdwyer)
3. Quotients of Directed Numbers - with thanks to Dee Vijayan (@DeeVijayan)
4. Easy Brackets I - with thanks to Michael Devine (@FitzchivalryF)
5. Miscellaneous Easy Brackets - with thanks to Dan Rodriguez-Clark (@InteractMaths)
6. Factors by Grouping - with thanks to Paul Rodrigo (@PaulRodrigo2718)
7. Miscellaneous Factors - with thanks to Dan Rodriguez-Clark (@InteractMaths)

I will add these exercises to my resource libraries. Huge thanks to the teachers who typed these up, I really appreciate it.

Here are my two previous posts from this series:

I hope these are helpful. 






11 February 2018

5 Maths Gems #83

Welcome to my 83rd gems post. This is where I share some of the latest news, ideas and resources for maths teachers. Many of us are on half term this week, which is a great relief as usual. Time to catch up on some sleep, spend some time with my children, and get on top of my growing to do list. If you haven't had much time for Twitter lately, this post features some things that you might have missed.

1. Find Me... Problems
Mr Hill (@MathsWithMrHill) has shared a collection of lovely Find Me.. Problems. There are 30 tasks so far, covering a wide range of topics. These are easy to use and easy to adapt.
2. Algebra Starter Pack
Ed Southall (@solvemymaths) shared a great set of 'introduction to algebra' lesson activities on TES.
3. A Level Questions and Assessment
I first mentioned Markit in Gems 79. I've now explored it in more detail and have realised that it's a really useful website for teachers of the new A level. Register for free and you can access a large selection of well written questions for both the new and old A level specifications.
Each question comes with very detailed worked solutions. I've been using the questions as examples in class but they can also be assigned online to students. For free, you can set three online worksheets per class - this is about three hours of work for each student. By upgrading, you unlock all the other content.

In related news, Edexcel's new mock papers were due to be published in late January but have been delayed. Given that my school has Year 12 mocks straight after half-term, this meant I ended up making my own assessment. It took hours to make and I drew questions from a number of sources - I am happy to email it to anyone else in the same position! Just let me know if you need it.

Note also that CRASHMaths recently published a large set of new A level practice papers. My students are using these to revise for their upcoming mocks.

Finally, whilst I'm on the subject of A level, you might be interested in the outcome of this Twitter poll regarding teaching time for Further Maths.
4. Primary 'Topics in Depth'
I have previously blogged about the importance of subject specific CPD. I think that maths teachers need CPD that focuses on misconceptions, approaches, methods and activities for individual topics. I started designing CPD like this last year and although I'm still working on it, my progress has been very slow because my workload is so heavy this year and these things take a long time to produce.  Thankfully I have a wonderful collaborator called Nikki Martin (@nikki_nzmartin), who I met at an ATM/MA event last summer. She has been busy producing Topics in Depth packs for Year 6 and Year 2 topics, and piloting them in the primary schools she supports. She has very generously shared her work on my Topics in Depth page.
5. Shape Transformations 
Every month I get an update of new resources from MathsPad and it's always full of great stuff. Some of it is freely available, including a few of these interactive tools and worksheets for transformations. The tracing paper overlays work really well here, enabling the teacher to demonstrate while pupils complete the matching questions.

Update 
In my last post I shared a set of old textbook exercises that have been typed up for use in lessons. There's some great questions in these exercises so do check them out.
Last week my blog had its three millionth visit. Thank you to everyone who visits regularly - I'm so glad people find it useful. I really appreciate all the support. 

Last week I also got a new job! In September I'll be joining the Harris Federation, supporting maths teachers in a number of Harris schools. I'll also be getting involved in their Maths Hub work. I'm really sad to leave behind my lovely school that has been so good to me, and my colleagues and students who I'll really miss, but excited about this next step in my career and all the opportunities it will bring. Huge thanks to Colin, Bruno, Craig, Ed and Tom for all their advice on this big decision.

It's only four weeks until the Kettering conference now - I hope to see lots of you at the Friday night drinks. If I don't catch you at the drinks, do come and say hello at the conference - I'll be on the MA stand when I'm not in workshops. The line up of workshops is fantastic so I'm really looking forward to it.

There are another couple of new events coming up that are now open for bookings:
  • Stuart Price is hosting his third annual 'Maths in the Sticks' event for A level teachers at his beautiful school in the Surrey Hills. 
  • Equals, the MA's magazine for SEND in mathematics education, is hosting an event in London in June. Details here.
I've added both of these events to my conferences page.

Finally, as part of my role as the Chair of the MA's Publicity and Media Committee, I've made a selection of excellent MA journal articles available for free. If you enjoy these articles, I hope this encourages you to join the MA.

I'll leave you with this factorisation puzzle, shared by @YohakuPuzzle. Fill in cells to get the products in each row/column.







7 February 2018

Algebra Exercises

A few weeks ago I published a post called 'Equations Exercises' in which I shared some exercises that have been typed up from a 1950s algebra textbook. In my opinion these are well written exercises with a good level of stretch and challenge. The idea is not necessarily to use these exercises in their entirety. They are provided in Word format so it's easy to copy and paste extracts and examples to use in class.

In today's post I've provided links to another set of exercises from the same textbook. They cover simplifying algebra, including index laws and algebraic fractions. There's also an exercise on substitution and one on writing algebraically. For each exercise listed below I have included an extract so you can preview the type of questions covered.

1. Notation, Addition and Subtraction I and II - with thanks to Katie Pollard (@takepi21)

2. Notation, Multiplication and Indices I - with thanks to Phil Bruce (@pbrucemaths)

3. Notation, Multiplication and Indices II - with thanks to Daniel Udall (@DJUdall)

4. Simple Powers - with thanks to Larissa Alsford (@lolsford)

5. Notation - Division - with thanks to Caroline Beale (@cbealemaths)

6. Simple Multiplication of Fractions - with thanks to Alice Dwyer (@littlemissdwyer)

7. Simplification of Fractions by Factors II - with thanks to Sandra Douglas (@mathsbox1)

8. Easy Fractions, Substitution - with thanks to Steve Bishop (@stevebishop100)

9.  Translation I and II - with thanks to Ben Gordon (@mathsmrgordon)

I'll publish one more set of exercises next week - some on directed numbers, and some on factorising and expanding.

Huge thanks again to the teachers who typed these exercises up. I really appreciate the time and effort that has gone into this. Excellent teamwork!






24 January 2018

5 Maths Gems #82

Welcome to my 82nd 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.

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.


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:

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.