26 July 2021

Angle bisection, incircles and reasoning with ratios

This is the sixth article written by Anne Watson in the 'Dose of Don' series. She posted it on her blog here and I have replicated it word for word. For the background on this series, please see the post Lines and Angles on Square Grids. My thanks go to Anne for giving me permission to share her writing here.

Dose of Don 5: Substitution

This is the fifth of a very irregular series of writings in which I (and, I hope, others) delve deeply into the collection of tasks on Don Steward’s blog and pull out threads about key ideas in mathematics that run through several of his tasks. Where possible I give you a direct link to the tasks; where I have extracted part of a task I direct you to the ‘parent’ from which it came.

Don was very generous with his tasks and I hope that you will return this generosity in the way he requested before he died, namely to donate to justgiving.com/fundraising/jessesteward.

This ‘Dose of Don’ is different in flavour to my previous posts. Instead of following what has been stimulated for me in Don's work I am following a thread of his own inquiry.  For a workshop on angles he presented some tasks that depended on defining an angle by its tangent ratio. I talked about this in my first ‘Dose of Don’ blog as he had a theory that if you approached angle by limiting it to those that could be expressed on a square grid, then many angle and trigonometric facts, and other geometrical insights, could be deduced in special cases and that possible generalisation to all angles could be explored.  The angles that can be expressed on a square grid and those whose trig ratios can be constructed on the lattice points, e.g. in triangles for which some of the side lengths can be expressed as rational multiples of each other. To simplify this he limited exploration at first to those angles whose tangent ratio was ‘on the grid’, so angles are an inverse of ratios, not yet expressed in degrees or radians.

OK so far?

Then he began to explore angle bisectors.  I had forgotten this direction but found it again while tidying my desk and finding my scribblings. Most of what I found is at : https://donsteward.blogspot.com/search/label/angle%20bisector

He starts by suggesting you use a compass/straight edge approach to bisecting an angle and observe whether and where your bisector passes through lattice points.  There is then the following summary slide about what you might have found (a typo for tan B/tan 2B is easy to spot).

Now the reason I had put this to one side for over a year is because my knowledge of double angle formulae is robust even 60 years after I first met them and this seemed to be getting in the way of imagining how learners might answer the question about a relationship between K and 2K. Could I honestly reconstruct a relationship I knew without using it as a ‘goal’? In other words, could I treat the K question as a goal-free problem? I could pick out features of the diagram and confirm them by using existing knowledge. I don’t recall, however, ever using the tangent double angle formula to find the tangent of a half-angle – the information I need to bisect angles on the grid. After some manipulations I found out how to do this and realised that this is how Don must have developed the particular examples he offered and why (you may have noticed this) they seemed to need Pythagorean triples to ‘work’. But I have not answered for myself the question of how anyone who was not familiar with double-angle formulae might approach the bisection question.

Another way to approach the bisection question is to use knowledge of incircles to do some reverse reasoning: if angle bisection gives me the incentre, then the incentre will give me clues about angle bisectors. Triangles on grids, particularly right-angled triangles with their legs on the grid, offer several reasoning routes and – hey presto! – the tangent ratio for the half-angle plops out before your very eyes!

This line of reasoning depends on some lines of thought that might be more familiar than inverse tan and Don’s slide number 30 can take you there.  It is a free-standing exploration that depends on knowing about areas of triangles and Pythagoras.

A diagrammatic approach I particularly like follows:

I found it is possible to reason the relationship between the half angle and the full angle without thinking about incentres but instead by reasoning with ratios. To show that the two half angles are equal (and therefore must be halves of the full angle) I only have to show that their tangents are equal, i.e. that their defining ratios are equal. The unit lengths in which the ratios are constructed don’t matter. I love this. For me, it is about seeking similar triangles using lattice points.

During our time with Don, John Mason and I raised an issue that has been hanging around for years: reasoning with ratios can be very powerful but we rarely see diagrams from which ratio ‘jumps out’ obviously as an important underlying relationship. We cannot ‘see’ ratio; we have to reason it out. I suppose the same can be said for multiplicative relations more generally; we cannot ‘see’ them in the same way as we can ‘see’ addition or difference. We think Don worked on this in his grid tasks but never had a conversation with him about it. However, I have found four diagrams that suggest he had found something. I cannot find them on his website but maybe have not been looking in the right place. I have reconstructed them here, because my copy is covered with my scribbled workings-out which could be a distraction (e.g. ‘can’t see ratios’, ‘try 3:2 along hyp’ etc.)

Finally, if you have got this far, a pedagogic question: how do the variation and invariance in these four diagrams help or hinder understanding and generalising the underlying relationships?

27 June 2021

5 Maths Gems #145

Welcome to my 145th gems post. This is where I share some of the latest news, ideas and resources for maths teachers. 

1. Purposeful Maths
A new website purposefulmaths.com has been created by @pbrucemaths and @mrgreen_maths. This website contains questions which promote mathematical thinking. Everything is free to access, including over fifty 'I do, we do, you do' questions and around 40 accompanying worksheets. A blog post explains how to use the resources.

2. NCETM Checkpoints
Thank you to Richard Perring (@learningmaths) for making me aware of the new Checkpoints resources from the NCETM.

Checkpoints are diagnostic activities that will help teachers assess the understanding students have brought with them from primary school, and suggest ways to address any gaps that become evident.

There’ll be enough for three Checkpoints a week across the whole school year. Professional development seminars will accompany the release of each batch of Checkpoints - the NCETM website explains how to sign up to these seminars. 

3. Angles Resources
Since I totally changed the way I teach angles in parallel lines following the research I did for my Topics in Depth project, I am delighted when I see quality resources for this topic! 

@MrDraperMaths wrote a blog post about reasoning with angles in parallel lines, identifying which line segments are parallel, and questions where you need to add extra parallels in. 

@b_karadia shared some of the slides she uses when teaching angles in parallel lines. These are very clear. I have a similar approach, emphasising the relationship with the transversal. Watch my CPD session on this for an explanation. 

And finally, @ShivMcKenna55 shared her latest curriculum booklet which focuses on angles. This can be accessed via Chris McGrane's website startingpointsmaths.com.

4. Revision Booklet
Thanks to @beckyreedmaths for sharing her Year 7 revision booklet.

5. Statistical Charts Resource
Thank you to Jamie Copus of Llanwewn High School for emailing me a great statistics resource. It's an exercise in reading information from a variety of charts rather than focusing on each one individually. Students are asked to extrapolate from pie charts, vertical line charts, composite and comparative bar charts. The activity is football based with fictional statistics. There are higher order questions to supplement the task and it can be completed by students unsupported or with guidance as to which chart to use for each question. You can download the resource from TES.

It's been a while since I last blogged. The second half of the summer term is definitely my busiest time of the year because I do the school timetable, and I run assessment week and I'm responsible for end of year reports.

I've been marking all weekend and (for once!) I am really pleased with how my Key Stage 3 students have done in their end of year assessments. I've changed my approach to Key Stage 3 teaching quite significantly over the last few years - I will speak about this in my upcoming mathsconf talk.

My school changed the way we do end of year assessments this year, using single tier assessments in Year 7, 8 and 9 for the first time. By having every student in each year group do the same assessment, we now have consistent, high quality data to help us determine next year's groupings. I was always really unhappy using two tiers of assessment at Key Stage 3 as I felt it led to a lack of mobility (i.e. students in the 'foundation' classes getting stuck there). We were worried that the single tier papers would be difficult to write, but it has worked really well. The range of attainment in a comprehensive school is vast, so assessments need to be accessible yet contain the right amount of challenge.


Did you catch my last two blog posts?

If you're a Key Stage 3 maths teacher, please don't forget to complete my survey, which will close at the end of term. Some of the results so far are rather surprising, and I will be sharing them in my #mathsconf26 workshop on 10th July. You can get a ticket here.

Over May half-term I actually managed to meet up with some of my favourite Twitter maths teachers in person! It was lovely. I know we're all a long way from being 'back to normal' but this felt like a step in the right direction. A few of us have tickets to go to ResearchED in September - I really hope it goes ahead! I've missed in-person conferences so much.

I've also booked tickets to see mathematical poet Harry Baker in October. He was the after-dinner speaker at the MEI Conference a couple of years ago and I loved it, so I am very keen to see him again. He's also available to visit schools.

In other news, at the end of May Ofsted published their mathematics research review. There's a lot to read but it's definitely worth taking the time to do so, particularly if you lead a maths department.

Subscriptions to resourceaholic.com
Google is shutting down Feedburner, the service through which over 2000 people currently subscribe to my blog via email. So I have had to switch to Follow.it for email subscriptions. If you were already a subscriber, I will transfer your subscription over. Hopefully you won't notice any difference other than a slight change in email format. If you aren't a subscriber yet, click here to subscribe. This is a great way to keep up with my blog posts. You'll only get emailed when I publish a new post, so that's once or twice a month.


I'll leave you with this graphic showing the evolution of the word “hundred” in Indo-European languages, which is from this beautiful blog by @JakubMarian.

31 May 2021


This is the fifth article written by Anne Watson in the 'Dose of Don' series. She posted it on her blog here and I have replicated it word for word. For the background on this series, please see the post Lines and Angles on Square Grids. My thanks go to Anne for giving me permission to share her writing here.

Dose of Don 5: Substitution
This is the fifth of a very irregular series of writings in which I (and, I hope, others) delve deeply into the collection of tasks on Don Steward’s blog and pull out threads about key ideas in mathematics that run through several of his tasks. Where possible I give you a direct link to the tasks; where I have extracted part of a task I direct you to the ‘parent’ from which it came. 

Don was very generous with his tasks and I hope that you will return this generosity in the way he requested before he died, namely to donate to justgiving.com/fundraising/jessesteward.

John and I hosted a day’s online workshop on ‘Substitution’ recently. Don attended our day workshops regularly and always had something extra, interesting and challenging to offer so we co-opted him onto our team. Hence the workshops we have hosted since May 2020 have always reminded us of him.  So I was wondering how Don used substitution in his tasks – whether he made a Big Deal of it or not.  We posed the question at our workshop: ‘Is substitution a Thing?’  Our answer in the day was ‘Yes’ as substitution crops up all over mathematics; the purposeful substitution of one expression for another to simplify, gain insight, clarify, test, modify, make new things possible, etc. etc.

Shortly after this Richard Perring posted a tweet asking the same question but from a different perspective. His question was about exercises in subbing numbers into algebraic expressions in early algebra.  You know the kind of thing, the internet is full of them, e.g. ‘If p = 2; q = -6; r = 10, calculate -pq2r3’. The exercise is not about algebra, it is about calculating with negative numbers once you have understood the syntax of the symbol system. It changes algebra into numerical answers. By contrast this task: ‘If p = 2; q = -6; r = 10, find at least five different algebraic expressions whose value is 4 using as many of the letters and mathematical signs as you need’ focuses on turning arithmetical understandings into algebraic expressions and launches ‘what if …?’ questions.  

So I began to search Don’s collection of tasks to find places where substitution gives useful mathematical perspectives and handles.

I didn’t have to look very far: https://donsteward.blogspot.com/2020/04/two-types-of-sum.html

Think of substituting expressions for consecutive numbers into a,b, and c (which are generalisations but not very helpful ones) and you have the beginnings of a ‘proof’ for conjectures that arise from doing the calculations. A dance begins between generalisations, structure, examples and relationships that is typical of mathematical exploration and the associated questioning: ‘What can I write instead of ….?’  ‘Can I test that with an example?’ is often about substitutions that are helpful in revealing or expressing structure.

And a later slide of Don’s gives:

This gives a reason for becoming more fluent with such manipulations - ‘doing’ algebra with a purpose. There’s more but you’ll have to go to his website for that while I indicate some other things I found once I had ‘substitution’ in my sights.

I have to force myself to open files with titles like ‘decimal subtraction’ but here goes: https://donsteward.blogspot.com/search/label/decimal%20subtraction.

This does not disappoint; I have had some great fun showing only the first two lines to people and seeing what happens. I got myself hooked on wondering about the cyclic nature of what appears and rewriting numbers as sums of powers of ten, i.e substituting the separate place values into the ‘answers’.  This is  the ‘undoing’ of what is done in primary school to build up multi-digit numbers from the place value components and is the basis for many mental methods in Vedic mathematics and Trachtenberg methods (see Google for these) and an old method that was taught in schools in the 18th and 19th century called ‘casting out nines’. Some people immediately substituted ‘nine tenths of…’ for the left hand sides, which explains something about the answers you get but not (for me) the cycles. The immediacy of this response was impressive but they were all people ‘of a certain age’ for whom expressing rational numbers as fractions retains some of the manipulability that can get lost in decimal notation. So there are two kinds of substitution at work here, both about equivalent numerical structures, used to explore and then explain the generic behaviour seen from the specific examples.

This reminds me of a feature of substitution that exposes its purposefulness. Firstly, it is a two-way action in which some things are gained and some are lost: generality/specificity; approximation/accuracy; manipulability/visualisation etc.  This last duo comes from thinking about modelling phenomena, but also from substitutions that change bases, such as are done in order to integrate functions.

Don’s website holds many tasks in which substitution is more explicit than what I have offered so far, see https://donsteward.blogspot.com/search/label/substitution. Martin Wilson of Harrogate is credited with some of  the ideas.  There are several of these that can be explored by trial and adjustment, i.e. purposeful substitution to get a ‘feel’ for what is going on and also raw material for later reflection – why these numbers?  Both the sets below use structures which we hope will become familiar for learners but also have extra features to think about.  I could imagine learners being asked to ‘make up some of your own like these’ and hence writing algebra for themselves, having used substitution to test their inventions – some two-way number/algebra thinking.

You might be wondering about a place in the curriculum where the word ‘substitution’ is used explicitly – the solution of simultaneous equations. In our workshop, and also in some of Don’s tasks, a powerful use of substition turns out to be the use of equivalent algebraic expressions, or temporarily equal expressions to simplify the use of variables. A really simple version of this reasoning is: if a = b and a = c then b = c, and a, b and c can be substituted for each other. Here is a development of that, where expressions rather than individual letters can be manipulated to ‘reduce’ the number of variables in a situation (https://donsteward.blogspot.com/2016/03/find-expressions.html ).  The task is to express a, b and c in terms of n. Rather than using the language of moving terms to and fro over the equals sign like deranged chess pieces the language of logical arithmetical reasoning can be used. For example, in the first set: ‘if b – c = n, then I also know that c = b - n; does that help?’ The standard question: ‘if I know …. then what else do I know?’ kicks in big time when transforming algebraic expressions. New expressions for a and b can be substituted into the first equation.

This realisation, that so much mathematics depends on substituting one expression for another when building expressions, equations, mathematical models and so on, seems to get lost in formulaic approaches to simultaneous equations. The idea of substituting expressions into other expressions has recently made more sense to me than the traditional ‘elimination and substitution’ language of methods. For example Don’s ‘where do the lines meet?’ tasks (https://donsteward.blogspot.com/search/label/simultaneous%20equations) cry out ‘substitute for y in the second equation’ rather than the ‘rearrange and match coefficients and subtract the equations’ that appears in some textbooks.

I am not saying that this kind of substitution can be used to solve all such problems, but the awareness of the power of substituting one expression for a variable or another expression pervades mathematics, so a common pedagogic question could be: ‘is there a substitution that can be made from the given information that gives insight/simplifies/gives some traction?’Being a bit fanciful – this could even be used in angle-chasing situations. I suspect that if John and I wrote ‘Questions and Prompts for Mathematical Thinking’ today we might include a question of this kind that could turn many procedural tasks into something more creative.

In my next Dose of Don I shall return to finding inspiration in his tasks rather than imposing my own perspective on them.

23 May 2021

Autonomy vs Consistency

I'd like to present a hypothesis. Secondary maths teachers generally experience a relatively high degree of autonomy over the structure, style and content of their lessons. 

At my own school, I think the maths department is the only department where lessons aren’t centrally planned. That's not to say that lessons are fully scripted in other subjects, just that there's a sequence of lessons, planned by one individual, that every teacher in the department follows. Individual teachers make their own adaptions to these lessons, but for most part there's a high level of consistency over what is taught. 

Take history for example: all students in each year group will experience the same history lessons each week, regardless of who their teacher is. There will be variations in the style of delivery, naturally, but the content taught and the activities used will generally be the same in every classroom. 

Contrast this to maths: we are all teaching very different groups who work at different paces (because our students are taught in sets) so one centrally planned lesson would not be suitable for all. Also, we are a group of teachers who have different styles and preferences when it comes to pedagogy. So every teacher in the maths department plans their own lessons. I'm not saying we all plan every lesson from scratch every time - there are plenty of banks of lessons we can draw on, both provided by the MAT and available online - but individual teachers are free to decide the sequencing within each topic as well as the explanations and activities within each lesson. If you popped into maths lessons at my school at any given time, you'd probably see the same topic being taught in every classroom but you definitely wouldn't see the lesson being taught. 

Is it the same in other schools? Well I can't speak for other subjects, but a recent Twitter poll suggests that the perception amongst most maths teachers is that they do have a high level of autonomy. 

Of course, this Twitter poll doesn't give us enough information for a proper analysis. Perhaps those with a high level of autonomy are highly experienced and highly competent. Perhaps those with less autonomy are trainees, or perhaps they are teachers who need additional support. Perhaps if this poll had included non-Twitter users, the results would have been different. 

Regardless of the profile of respondents, I think it's remarkable how many of the 1653 teachers who responded to this poll feel that they are highly autonomous.

There are mixed views on whether this is a good thing. Here are some examples of thoughts on this:

Read the full thread for all the responses.

An Autonomy Spectrum
There is clearly a spectrum of autonomy in the classroom, which looks something like this: 

At one end of the spectrum teachers have absolute autonomy over both what to teach (i.e. no curriculum to follow) and how to teach. Though a wonderful idea in theory, even in the most extreme circumstances it's unlikely that this happens in practice. We have a highly prescriptive national curriculum, and we have public exams at age 16. Only in the case where a teacher neither follows the curriculum nor enters students into public exams would they have full autonomy over what to teach.

Just along from that end point on the spectrum we have schools that follow the national curriculum. For practical reasons they typically have an internal scheme of work that dictates the order of teaching to some extent (usually relating to internal assessment i.e. to ensure that students are all in a position to take the same assessment at the end of each academic year). As long as teachers teach the designated content (for example, teach angles in the summer term of Year 7), it's up to them how they do it. I'm calling this pedagogical autonomy. This means that no one tells the teacher what to do in the classroom. There are no school or department policies regarding lesson structure or style. Teachers plan all their lessons independently and without collaboration. Lesson style, structure and activities are determined by teacher preference, for example some teachers will use mini whiteboards, whilst other won’t, and some will use manipulatives, whilst others won’t.

In the middle ground of the teacher autonomy spectrum, there are a wide range of approaches. These may include one or more of the following:
  • Department polices on certain 'non negotiables' (e.g. use of retrieval practice, frequency of feedback etc).
  • A bank of centrally planned lessons that may be adapted by individual teachers.
  • Department policies on methods, representations and vocabulary (for example 'all students should see the grid method for expanding double brackets').

There are many more approaches I could add to this list, but you get the idea. Basically we're talking about measures to support consistency of experience for students, but still allowing teachers to plan (or adapt) their own lessons.

Towards the other end of the spectrum we have a situation where all teachers deliver centrally planned lessons. There's been an increase in this in the last couple of years as some schools have adopted curriculums such as Ark or White Rose to the fullest extent. 

At the extreme end of the spectrum we have zero autonomy. This is where teachers have absolutely no control over what they teach or how they teach it. They have no opportunity to add any of their own style or ideas. The only way I can see this happening is through fully scripted lessons where teachers are literally told what to say. 

I want to be clear up front that I hugely value the autonomy I experience in my own lesson planning. I certainly don’t create everything from scratch - I’m happy to borrow from resources and lessons created by others - but I take a lot of pride in my lesson planning. To me, planning lessons is one of the most enjoyable aspects of the job. I value being allowed to have my own unique style. Pedagogical autonomy is my happy place, and having the opportunity to be creative is one of the things I love about this career compared to my previous career. That's not to say that I have total autonomy. For example, department policy at my school dictates that my teaching does have to feature aspects of retrieval practice. But I would have done that anyway, so it's hardly an imposition! There's also a school-wide feedback policy that says my students need to receive feedback in maths twice every half-term. This means my teaching features end of unit tests, but I would have done that anyway too! All things considered, I think I probably sit about a third of the way along the autonomy spectrum. 

Although I very much enjoy having a high degree of autonomy myself and would be resistant to that autonomy being reduced, I am well aware that there are some very strong cases to be made for consistency across a department. In the remainder of this post I’m going to outline some of the arguments around the varying degrees of autonomy. I don't think there's a 'correct answer' here. To some extent, a school's approach will depend on the level of experience and expertise in the maths department. But there are certainly some considerations that all heads of department should be thinking about.

Workload and Wellbeing
It's hugely inefficient for every teacher to plan their own individual lessons. The workload associated with excessive amounts of lesson planning may have a detrimental impact on staff wellbeing. 

In my own school (we currently only have Years 7, 8 and 9), the weekly planning workload for a history teacher might involve reviewing, tweaking and preparing resources for six centrally planned lessons which will each be delivered to multiple classes. The weekly planning workload for a maths teacher might involve fully planning and preparing twenty different maths lessons. That's quite a contrast. 

I notice it myself - as an Assistant Principal with a large number of whole school responsibilities I really struggle to keep on top of my lesson planning. Though I only plan twelve lessons a week, this is four times more lesson planning than the other Assistant Principals on my team. 

Recruitment and Retention
For the teachers who enjoy planning and delivering their own lessons, autonomy brings job satisfaction and motivation. A department offering little autonomy may struggle to attract and retain expert teachers. 

At the same time, a department providing high quality centrally planned lessons (for example an 'off the shelf' mastery package) may attract and retain quality teachers who value work-life balance over autonomy.

Skill Development
Planning lessons and reflecting on their effectiveness is an important aspect in the development of the expertise of a teacher. I think we should be wary of de-skilling teachers by taking away their opportunity to make pedagogical decisions for themselves. At the same time, you could argue that moving the focus from planning to delivery means that teachers have the opportunity to perfect more specific aspects of their pedagogy. It's easier to refine your delivery if you're not overwhelmed by other aspects of your role.

Student Experience
Let's imagine Year 8 are learning about sequences and we take a look at what's happening in the maths classrooms one day. This is what we might see:
  • One lesson where the students are engrossed in a sequences investigation involving multilink cubes
  • One lesson where the teacher is at the board leading example-problem pairs 
  • One lesson where the students are silently working through a pile of Ten Ticks worksheets
  • One lesson where students are playing a bingo game involving sequences 
  • One lesson where the students are holding up mini-whiteboards while their teacher questions them and addresses misconceptions
  • One lesson where students are enthralled by a lecture about the history of Fibonacci
  • One lesson where students are being taught to find the nth term of an artihemetic sequence using the ‘shifting times tables’ method
  • One lesson where students are being taught to find the nth term of an arithmetic sequence using the ‘zeroth term’ method
  • One lesson where students are being taught to use the formula a + (n - 1)d
  • One lesson where the teacher has decided that arithmetic sequences are too easy and is instead teaching geometric sequences with surds

Do you read this and think 'what a wonderful diversity of experience', or do you think 'what a terrible inconsistency for students'?

I do think that there is a case for some consistency in methods, language and representations across a maths department. There may even be a case for some consistency in pedagogical style. But at the same time, who gets to decide what style is best? And who gets to decide the methods everyone has to use? What research are you basing that decision on? Choice of methods and representations often comes down to preference. If these things are going to be dictated or advised by department policy, I think it wise that the process of writing that policy is collaborative. I have seen some good work being done over the last couple of years on developing 'method policies' in some maths departments, and the best examples of these have always been developed in discussion with the whole department.

Groupings and responsiveness
In a school that sets in maths, it's logical for teachers to adapt the content and style of their lessons to best suit the class in front of them. Arguably, in 'mixed attainment' settings, there may be more of a case for 'one size fits all' lessons.

That said, a feature of all 'good' maths teaching, whether in sets or mixed attainment, is that it's responsive to students. Centrally planned sequences of lessons don't necessarily allow teachers to respond to their ongoing assessment of students' understanding. This is why planning lessons far in advance never works well in maths teaching - we can only really plan a lesson after we have taught the previous one, when we are in a position to determine the next step. 

Mathematical progression
In most schools, students have a different maths teacher in Years 7, 8, 9 and 10. There are many advantages to this, from the perspective of both students and teachers, particularly if there is a range of expertise in the maths department.

Some people argue that a disadvantage to students of changing teacher every year is that different teachers may 'do maths differently'. This could be confusing to students and may hinder their progression. On the other hand, this variety might actually be beneficial to students. I'm not convinced that "Mrs Morgan sets her working out like this and Mr Brown does it a different way" is necessarily a terrible thing, as long as Mrs Morgan and Mr Brown know each other's ways and acknowledge the different approaches with students. 

A typical example is when students have been taught to draw a 'wall' down the middle to separate the two sides when solving an equation. It's not a big thing, but I'm always a bit surprised when I see one of my students doing this.  
My students won't ever see me do this on the board (I don't have a problem with it, I just think the equals sign suffices to separate the two sides... I also don't like the way it makes the equals signs look like 'not equals' signs...). I always make sure I acknowledge with students that some teachers or tutors may draw a vertical line, and I don't mind students drawing the line if that's what they're used to. 

There's no point in trying to undo previously taught approaches or habits, unless they are mathematically damaging. It's fine for people to do things differently to me! I don't think these minor details need micromanaging.

Teachers do need to communicate well with the other teachers in their department though, and ideally pop into each other's lessons when they can. If teachers know what their colleagues do then they can at least recognise and acknowledge the different approaches in their teaching.

Quality of teaching
To ensure that all students benefit from high quality teaching, you could argue that novice or developing teachers should experience less autonomy than more experienced teachers, particularly where subject knowledge is lacking. This might come in the form of lessons always being co-planned with a more experienced member of the department, or perhaps a mentor checking the suitability of all lessons plans. A more extreme measure, which might be particularly suitable in a department with a high proportion of teachers who have been identified as needing support, is to insist that all teachers teach lessons that have been centrally planned. 

The big question here is: who's writing the shared lessons? And who's assessing the quality of those lessons? After centuries of maths teaching we are, as a profession, still a long way from universal agreement on what constitutes effective maths teaching. So someone claiming 'my lesson is best and everyone else should teach it' makes me a bit uncomfortable.

A few years ago, a friend of mine joined a school with scripted lessons. He was a highly experienced and knowledgeable teacher and was told to deliver scripted lessons written by a far less experienced teacher. This approach seems flawed. 

If absolute consistency is our aim, then we want all teachers delivering the highest possible standard of lesson. But I'm not totally convinced we truly know what that looks like.

Further reading 
If you want to read more about maths teacher autonomy, here are some articles you may find interesting:

I have not done adequate research to be able to support my claim that secondary maths teachers generally experience a relatively high degree of autonomy. But I hope that this post has given you a lot to think about. What is the optimal balance of autonomy vs consistency? I don't know the answer, but I know it's not an easy one. 

A plea
For an unrelated piece of work, I’d really appreciate it if you’d complete my survey if you teach Key Stage 3 maths. Many thanks!

15 May 2021

5 Maths Gems #144

Welcome to my 144th gems post. This is where I share some of the latest news, ideas and resources for maths teachers. 

1. Tasks
I've spotted loads of great new tasks on Twitter in the last few weeks. 

@giftedHKO shared a nice area task - perfect timing for me as I'm teaching area to Year 7 next week. This task features amongst a collection of fantastic area ideas here.

@DanielPearcy's Year 10 class were struggling to remember negative indices and so he made this homework. It's so unusual to see negative indices in these contexts.

@ticktockmaths shared a lovely bounds lesson. I am a big fan of the format and structure of his slides as well as the content.

Another resource shared by Richard was this great starter Two or a half?.

Richard also shared a task where students are asked to find coordinate pairs to fit into Venn diagrams. What a great introduction to simultaneous equations! Richard got the idea for this task from @mpershan's excellent book Teaching Math with Examples

2. Self-Explanation Prompts
Speaking of tasks inspired by Michael's book, @karenshancock also shared an idea that came from Teaching Math with Examples: self-explanation prompts.

Karen has shared more examples of self-explanation prompts on her Twitter feed. To quote Michael's book, "The idea is to push students towards explaining ideas they might not even have realized they don’t understand. It’s also a useful chance to ask students to dig deeper and answer some “why” questions that can connect a procedure to concepts".

3. MathsPad
For many years MathsPad has been my first port of call for resources when planning lessons, and it's getting better and better. I have blogged before about their new range of topic booklets which are a bit like textbook chapters, but with far more depth and a greater range of activities than typical textbooks. They have published booklets for six topics so far: place value (which is free), calculations, negatives, fractions, indices and introduction to algebra. Here's an example of a task from the indices booklet:

4. A Level
In response to the disruption caused by the pandemic, the AMSP (@Advanced_Maths) has been working on six Support Packages for maths teachers and students. They bring together loads of resources, PD opportunities and support. AMSP material is always top quality and I'm sure these resources will be very helpful to students and teachers in Years 10 to 13. 

Thank you to @naikermaths for sharing six Year 13 statistics practice papers on their website.

And thank you to @Miss_Jethwa for sharing ten new A-level Further Maths papers on jethwamaths.com.

5. Methods
I love these creative methods that @staceymaths' students came up with to solve equations involving fractions. Clever stuff.

Mathematics in School
I am incredibly proud to have guest edited the latest edition of ⁦the MA's journal⁩ Mathematics in School with ⁦Ed Southall. It’s a tribute to Don Steward. If you're not a member of the MA then join now and you will be sent a copy. It's packed full of excellent articles.

Did you see the Gem Awards post I published in April? If not, do check it out for a huge collection of amazing resources, ideas and websites to visit.

In the Easter holidays I attended a GLT Book Club meeting to chat about a chapter from my book A Compendium of Mathematical Methods. The recording of the session is here. I was really impressed by the quality of professional development in these book club meetings. 

Did you see @Mr_Rowlandson's blog post Thinking About Probability Trees? He is one of my favourite maths bloggers and his posts are always worth a read.

Don't forget to buy a ticket for the next La Salle maths conference which takes place on 10th July. I will be speaking about how to teach for depth instead of rushing ahead.

I'll leave you with a Don Steward task. Though it may look like a typical angles exercise, it was exactly what I needed recently when I was looking for a set of questions that only involved basic angle facts but had a good level of challenge. It reminded me of what an absolute treasure trove Don's blog is.

18 April 2021

Gem Awards 2021

Next week it's resourceaholic.com's seventh birthday! It's become a tradition for me to mark the anniversary of my blog by publishing an annual 'Gem Awards' post. Here I look back at all the ideas I've shared in my gems posts and choose some of my favourites. 

1. Subject Knowledge Award
I love nothing more than finding opportunities to develop my subject knowledge. This doesn't mean being 'better at maths'. It means having a greater depth of knowledge about the content I'm teaching, from a pedagogical perspective.

The last time I handed out a Subject Knowledge Award was in 2017 - it went to Ed Southall for his book 'Yes, But Why?'. Although there have always been books about maths education, Ed's excellent book was the first one I knew of which focused on pedagogical subject knowledge. Since then a few more subject knowledge books have been published, including my own book about methods. Ed's book remains an essential read for maths teachers, and it's great to see that there's a forthcoming second edition.

This year the Subject Knowledge Award goes to Sudeep Gokarakonda of Boss Maths for the brilliant vocabulary resource I featured in Gems 121. The idea of learning more about the technical language we use in the classroom was relatively new to me when I first started blogging back in 2014. It's now something that I regularly incorporate into my lessons. Sudeep's vocabulary resource is ideal for extending my subject knowledge. Every single page is fascinating! It's perfect for use both in lessons and in maths department meetings.

Highly commended in this category is Nathan Day for his classroom displays featuring famous mathematicians of the world and mathematical quotations. The content of these lovely displays is valuable for both teachers and students seeking to expand their knowledge of the field.

Also highly commended in this category is James Tanton for his fabulous video 'The Story of the Vinculum'.

2. Bright Idea Award
This award goes to Miss H for her induction booklet idea that I shared in Gems 129. Last year our incoming Year 6 students were unable to join us for their usual induction days so she came up with the idea of an induction booklet. The version she shared included loads of activities for her new joiners to do over summer, plus an introduction to the maths teachers in her department. Inspired by this, I created my own version - I used loads of Miss H's ideas, so she saved me a lot of time, and I know countless other teachers did the same. This is why sharing ideas on Twitter is so valuable. 
Highly commended in this category are the teachers who collaborated to produce the fantastic set of Guided Reading Activities I featured in Gems 137. I love the idea of maths comprehension exercises - it's not something I'd considered before. The design and content of the resources in this collection are perfect. Using a template shared by @MrHand__, these exercises have been created by Katie Pollard, Andrew Baxter, Nicola WhistonVandana Sethuraman and Nix. Katie and Andrew's guided reading sheets can now be downloaded from TES.

3. CPD Award
This award goes to the team at La Salle Education for the incredible conferences they have run since the start of the pandemic. They quickly adapted when the world moved online, developing a highly impressive conference platform with an outstanding user interface. This has resulted in a consistently high quality experience for both delegates and speakers. 

In addition to their regular conferences, La Salle's CPD offering is extensive. It includes their Teacher CPD College which offers over 120 courses for a cost of £7 per month.

Highly commended in this category are:
  • Loughborough University Mathematics Education Network (LUMEN) for their provision of incredibly high quality professional development videos which are freely available to maths teachers.
  • Teacher Tom Manners who during lockdown shared numerous free maths CPD videos for secondary and primary teachers on his website. This includes his #ResourceFULL series in which he interviews guests about maths resources. 
  • Rhiannon Rainbow and Dave Tushingham for #GLTBookClub. The format of this CPD is highly engaging and effective. In the expertly run book club meetings, teachers have the opportunity to participate in incredibly rich discussions. They reflect on the ideas presented in the books they've read, and discuss the application of those ideas in the classroom. 
  • Craig Barton for the many things he does to support maths teachers including his fabulous podcast and his excellent online CPD courses. I already gave him a Gem Award in 2017 for his podcast so I can't give him another one, but it's still by far the best education podcast there is.

4. Pedagogy Award
This award is for the big thinkers. It's for the teachers whose insights and ideas encourage us all to reflect on our practice in the classroom. 

This year the award goes to Chris McGrane for his ongoing work on curriculum and task design. In addition to the publication of his brilliant book Mathematical Tasks, he regularly shares his thoughts on Twitter and through his website startingpointsmaths.com. On his website he publishes tasks that he has written himself, such as this FDP task which is designed is to help students reinforce connections between division and multiplicative representations in decimals, percentages and fractions. 

Chris also publishes numerous tasks contributed by other authors, such as this excellent powers task from Kyle Gilles:

Highly commended in this category are:

5. A Level Award
This award goes to Devina Jethwa. I featured her new A level website jethwamaths.com in Gems 133. This website features lots of useful new content for A level teachers - resources include worksheets, practice papers, topic tests and calculator tutorials.
Highly commended in this category is Seb Bicen. On his excellent YouTube channel he has videos and accompanying resources for Edexcel A level Maths and Further Maths. What makes the videos unique is that they are recorded live with his classes, so they have an authentic classroom dynamic. His explanations are very clear, so these videos aren't just useful for sharing directly with students, they could also be useful CPD for inexperienced A level teachers who are preparing to teach their own lessons.

Also highly commended in this category is Jack Brown. He has been running his excellent website TLMaths.com for eight years now. He has shared over 4200 videos and had over 16 million views, and continues to make new content to support students and teachers with their A level studies. Jack is also active on Twitter, supporting and advising A level teachers. 

6. Task Design Award
This award goes to Miss Konstantine for the incredible collection of tasks she has published over the last few years. She is constantly producing wonderfully creative tasks and sharing them for free on her blog mathshko.com. Her tasks are always a delight. She thoughtfully listens to feedback from teachers as well as reflecting on her own experiences of using her tasks in the classroom.

In the bar model task shown below, she has clearly reflected on the underlying skills that students need in order to work fluently with bar modelling techniques. I haven't seen many other tasks that specifically address these skills.

Here's another example of her tasks: this one aims to get students thinking about negative numbers. In the blog post that accompanies it, she explains her rationale for creating the task. I'm sure many of us have encountered students who find it difficult to see that 7 – 2x is the same as – 2x + 7.

I could share dozens of Miss Konstantine's tasks, but here's just one more example to give an idea of the variation in her style and approach. Here's a lovely pattern colouring activity for number properties.

Highly commended in this category is Ashton Coward who creates really cleverly designed tasks and shares them through his website. One example is his gap fill task for significant figure rounding.

7. Interactive Tool Award
I don't like to give an award to the same people three times, but this award is going to have to go to MathsPad yet again! Well done to James and Nicola for creating the brilliant Place Value interactive tool that I featured in Gems 143. Freely available to all, this tool gives teachers the opportunity to fully explore place value with students to really deepen their understanding. It's very cleverly designed. Do check it out, along with all their other outstanding interactive tools.

Highly commended in this category is the author of sineofthetimes.org Daniel Scher for his Zoomable number line activity. I loved this so much when I did it with my Year 7s this year, I immediately wrote a blog post about how brilliant it is.

Also highly commended in this category is Mathigon's Multiplication by Heart tool. These virtual flash cards use spaced repetition to teach multiplication facts. The tool is free to use, and is just as high quality and beautiful as all the other wonderful treasures on Mathigon.

8. TES Author Award
This award goes to Andy Lutwyche who continually publishes excellent resources on TES. It feels like he's been doing it forever! His generosity is incredible, with over 2000 resources uploaded to TES including popular collections such as 'Erica's Errors', 'Spiders', and codebreakers with terrible punchlines. An example of one of his many resources is his Non-Examples Reasoning Tasks in which students are asked to identify examples and non-examples and explain their thinking. I featured this resource in Gems 118.

Another example is this excellent resource for the quadratic formula which I featured in Gems 130.

Highly commended in this category is TES author cparkinson3 who I wrote about in Gems 110Gems 116 and Gems 117. His collection of free resources features bundles of high quality slides and activities for numerous topics. Pictured below is a vectors task from his shape transformations collection.

9. Teacher's Website Award
This award is designed to celebrate classroom teachers who dedicate their spare time to producing or collating resources for others. By doing so they save many teachers a significant amount of time in planning lessons. I try to showcase some of the less well-known websites in this category. 

This award went to Dr Frost back in the Gem Awards 2016 when his website was relatively unknown (compared to now!). 

This year the award goes to GoTeachMaths.co.uk which has over 4000 free resources for Key Stages 2, 3 and 4. I first featured this website in Gems 109. A number of teachers have since contacted me to recommend it, telling me that the vast collection of well-organised resources saves them lots of time. Shown below is a short extract from their measuring angles PowerPoint.
This the is the sort of website that seems to have an activity for practising every skill you can think of! For example, here's an extract from a worksheet on ratio and angles.

Highly commended in this category are the following:
  • Mrs Jagger for her website jaggersmaths.co.uk which features lessons, teaching resources, enrichment activities, tutor time tasks, revision materials and much more. She has generously shared her full five year scheme of work too.

  • Amanda Austin for her website draustinmaths.com which features a collection of KS3 and KS4 IGCSE and GCSE Maths resources, as well as resources for AQA Level 2 Further Mathematics.
  • mathsteacherhub.com which features an extensive bank of differentiated resources. There are starters, exercises and homeworks at multiple levels of difficulty for Key Stage 3 and 4.

Apologies if I've missed anyone's favourite website here! There are many more I could have included. 

There are a number of new websites that have been launched over the past year - keep an eye on my gems posts for updates on these sites as they develop.

10. Lifetime Achievement Award
The 2021 Lifetime Achievement Award goes to The Mathematical Association. This year they celebrate 150 years since they were first established in 1871 to challenge the way geometry was taught.

150 years is a really long time in education. For a subject association established in Victorian times to still be actively supporting maths teachers today is quite remarkable. I love their origin story, and I love what they do, which includes (among many other things):

  • running a brilliant Twitter account which keeps maths teachers updated on all the important stuff happening in maths education
  • running the Primary Maths Challenge and First Mathematics Challenge, getting children excited about maths from a young age
  • representing members in important consultations regarding developments in maths education
  • running brilliant conferences and CPD events, including those run through its many branches around the country
  • publishing numerous excellent teaching journals and books
  • owning (and making available to view) a library comprising nearly 11000 books and 700 runs of periodicals from many different countries, plus over 1100 older or rarer items going back to the sixteenth century. The MA's library is a unique primary source for the history of the mathematics curriculum in the UK. It's incredible!

The members of the MA's Council and committees are volunteers. Many have spent their entire working lives (and beyond!) working hard to ensure that the MA continues to support maths teachers. They are very worthy winners of one of my Lifetime Achievement Awards.

Congratulations to all the winners of the Gem Awards 2021!

And thank you to every single member of the maths teaching community who shares ideas, collaborates, produces resources, gives advice and supports other maths teachers. There are many people who I've not mentioned here who have helped to fill my gems posts with resources and ideas. I've been on Twitter for seven years now and continue to feel privileged to be part of such a supportive and generous community. Good work, Team Maths.

If I'm still blogging in 2024 (the ten year anniversary of my blog!) then I hope I will be able to host an in-person glitzy awards ceremony. We can all get dressed up and drink champagne and I can hand out proper awards to say thank you.

If you're new to my blog and you enjoyed this post then visit my Gems Archive you'll find an index of 143 gems posts - they are all full of great ideas and resources. You might also want to check out the Gem Awards 2019Gem Awards 2018, Gem Awards 2017, Gem Awards 2016 and Gem Awards 2015 to see who has won awards previously. 

Happy 7th birthday resourceaholic.com. Thank you to my readers for all the support!