3 April 2021

5 Maths Gems #143

Welcome to my 143rd gems post. This is where I share some of the latest news, ideas and resources for maths teachers. 

1. Place Value Tool
When making the CPD course Marvellous Maths 2 I struggled to find a virtual manipulatives tool that really effectively conveyed the relative sizes of tenths, hundredths, thousandths and so on. Luckily James and Nicola from MathsPad watched the course and this prompted them to go off and make one!

What they have produced is incredible. Freely available to all, this tool gives you the opportunity to fully explore place value with students to really deepen their understanding. 


Do have a play with it. It's awesome.

James and Nicola have also started publishing their Curriculum Booklets which are packed full of brilliant activities. Read their latest update for more information about their excellent booklets. 


2.  Powers
I really like the ideas explained in this Twitter thread from Sam Blatherwick (@blatherwick_sam). When teaching fractional powers, he gives his students a power chart like this:


Students can use these charts to answer questions like this:


Read Sam's thread for the full description of how he uses this chart to develop understanding. 

I really like this. It reminded me of my favourite indices resources - Mental Mathemagician from yummymath.com, and 3 Power Line and 5 Power Line from Don Steward.

3. New App
I don't often feature resources that aren't free but this one caught my eye. The Arc Maths app (@ArcMathsApphas been developed by a maths teacher and looks rather good. 


It's aimed at students aged 11 - 16 and gives users a highly personalised experience. Schools with access to iPads can subscribe and make use of this app in the maths classroom, in tutor time or in intervention. 

I don't know many state schools with class sets of iPads, but it's worth knowing that parents can subscribe to this app for their child at a cost of £3.49 a month. This may be something that schools can advise parents to invest in if they ask for ideas of how to help boost their child's maths grade.

4. CPD
The collection of free CPD videos from the Loughborough University Mathematics Education Network is brilliant. Maths teachers looking for CPD for either themselves or their department would benefit from exploring the videos on offer. The latest addition is Improving Language Use in Maths by Dani Quinn.



5. Simple Linear Graphs
As I mentioned in Gems 141, Dan Draper (@MrDraperMaths) has published loads of great blog posts lately. His post 'x=a, y=b: When?' looks at curriculum and concept development through a series of well-designed tasks.





Update

I enjoyed the first day of the MA conference. Delegates who attended my session can download my slides here. And you can listen to the post-conference podcast with me and Craig Barton here.

Did you see my recent blog post? I wrote about the The Power of Modelling and Exemplars.

If you enjoy my blog posts then you can subscribe here. You will only be emailed when I publish a new post, which is normally once every three or four weeks.

I'm very glad to be on my Easter break after a crazy Spring term. As well as attending the three-day MA Conference, my holidays will mainly consist of hanging out with my lovely daughters, and completing endless Only Connect style puzzle grids from the website puzzgrid.com (which I am a bit addicted to). I'm trying to keep work to a minimum this Easter because I desperately need a rest, but I do have a few things coming up over the next two weeks:

  • I'll be reading Michael Pershan's book 'Teaching Maths with Examples' which looks excellent.
  • I'll be recording a podcast with Ben Orlin, chatting about bygone maths symbols
  • I'll be attending #GLTBookClub on 13th April. We'll be talking about Chapter 18 (angles in polygons) from my book A Compendium of Mathematical Methods.

I'll leave you with the news that Tarquin are now taking orders for empty protractors. I spoke about this idea in my Angles in Depth CPD. A lot of the mistakes made in measuring angles come from students relying on reading scales on protractors (which they often misread) rather than using reasoning. Protractors without numbers help students think logically about the measure of turn. 




 

20 March 2021

The Power of Modelling and Exemplars

Last week I observed an art lesson which featured expert use of modelling. The teacher wanted students to use a particular technique to create a piece of artwork. Before the lesson she'd used a visualiser to record herself performing the task. The video was shot from above, recording only her hands. She played the recording to the class while she narrated what she was doing and pointed out the difficulties she had encountered along the way, advising students of how they could overcome those difficulties when it was their turn to have go. She then left recording running on a loop on the screen while the students performed the task, meaning they could look up and refer to it throughout. What an excellent technique! 

I occasionally do something similar in maths - when teaching constructions I leave gifs playing on a loop on the board so students can refer back to them whilst practising:



I think this is really powerful in topics like constructions. 

It is very easy to leave animated written examples running on a PowerPoint while students practise (like in the example below). But here it's probably more helpful to instead leave all the steps and solutions static on the board, rather than an animated version. It's the same idea though. We model how to do the process and we show students what the final outcome looks like. And then we leave that with them to refer to, rather than hide it away and expect them to remember it.

In generic whole school CPD, I'm often surprised to hear people talking about the importance of modelling examples as if it's a new or unusual idea. But perhaps it is, in other subjects. For maths teachers it's just so ingrained in everything we do. We are always modelling. Students are constantly getting to see us do 'live' maths. From our modelling, students can see what the final outcome should look like. 

Students don't get the same benefits from PowerPoints which are clicked through to show animated step-by-step solutions as they do from live modelling. They need to see 'pen and paper' modelling, done during the lesson by their teacher, whether on a whiteboard or a blank PowerPoint slide, or under a visualiser. Because otherwise, how can they possibly know what their work should look like?

A few years ago there was a trend for using 'WAGOLL' techniques in many subjects ('what a good one looks like'). This is a logical thing to do - if you want students to produce something of a certain standard, how can they achieve that if they are not shown what that standard looks like? It's like when we follow a recipe to bake a cake or cook a meal - we start by looking at a picture of the final outcome, so we know what we are aiming for. 

There are a number of commonly used techniques for showing students 'what a good one looks like' in many subjects, including the live modelling I've described. Another technique is using a visualiser or photo to share examples of excellent work by other students. For example if a student's book is laid out immaculately and you want other books to look the same, simply show the class what that good book looks like. Just moaning at a student that their workings are a mess won't help them improve. Find a good example and show them.

In maths, the exemplar response materials provided on Edexcel's Emporium are perhaps one of our most useful tools for showing 'what a good one looks like' in terms of the maths itself. 

Take this Edexcel exam question for example. What would a good answer look like?

Edexcel has provided us with an excellent solution given by a student in their GCSE exam. Note that reasons are given throughout, that workings are clear and presented vertically down the page in a logical order, and they have even used a 'therefore' symbol (not essential, but nice to see!). This student knows what they're doing.


When teaching circle theorems, it is so useful to show examples of answers gaining full marks. This helps students know what they need to do to get marks in questions like this, particularly in terms of the wording of their reasoning. 

Edexcel also provides examples of student answers to the same question that did not gain full marks. As a class you can discuss where students went wrong and where their misconceptions lie. Looking at real student answers and seeing how marks can be gained and lost is powerful stuff. 

Analysing exemplar responses is also incredibly useful for maths teachers. Edexcel very helpfully provides accompanying comments which explain the misconceptions and tell us where students lost and gained marks.

I could show you endless examples here, but I will just share one more. This is an angles questions from an Edexcel Foundation paper. The first solution gains full marks. The second got one out of three.

Can you guess what the second student did to come up with the angles of 120 and 150? 

The comments from Edexcel tell us that this student appears to have measured the angles with the protractor. We are urged to remind our students that diagrams are not drawn accurately and they should not be measuring anything (unless specifically asked to do so!). This is not something I would have anticipated students doing in this question.

I'm sure you'll agree that exemplar answers are an incredibly useful teaching tool, not only for showing students 'what a good one looks like', but also for our own CPD purposes. You can find Edexcel's brilliant exemplar resources on the Emporium:


We are very fortunate in maths to have access to such useful resources to support our teaching.




13 March 2021

5 Maths Gems #142

Welcome to my 142nd gems post. This is where I share some of the latest news, ideas and resources for maths teachers. 

1. Times Table Packs
Thank you to Sarah Farrell (@SarahFarrellKS2) for sharing a set of times table packs. Each one has a 25 different activities aimed at targeting conceptual knowledge and quick recall in each multiplication table. These will be very useful to primary teachers, and I can imagine that they'll also be a helpful resource in secondary interventions.

2. Website
Thank you to Matt Woodfine (@PiXLMattTheApp) for sharing his website
mathswhiteboard.com. This website features examples pairs, mini whiteboard activities, retrieval practice, a worksheet generator, starter activities, class interactive keypads and more. It's all free and easy to use, and Matt has done a lot of work on it recently.

3. Tasks
There have been loads of great tasks shared on Twitter recently. I have probably missed many of them, but here's a selection:

 





4. Virtual Escape Rooms
Thank you to Grant Whitaker for sharing three online maths escape rooms that he has made: one for Key Stage 1, one for Key Stage 2 and one for Key Stage 3.
Some of my readers have asked about maths escape rooms before. Grant asked me to share a link to an online course where teachers can learn how to make an online escape room. 

5. Foundation Booklets
Thank you to Mr Kingsley (@KingsleyMaths) for sharing a set of Foundation GCSE booklets. Each task contains ten 1/2 mark questions, which can be used as lesson starters. They build up in difficulty in both calculator and non-calculator topics.

I have a page of GCSE revision resources here.

Update
I am having a tough time at the moment to be honest. I was so excited to get back in the classroom and I spent a long time preparing to teach some awesome topics, but ridiculous lateral flow test policies have sent many of my students home already. This has upset me. On top of that, being the member of SLT in charge of cover in a year with high staff absence has finally broken me. But Easter is round the corner (a much-needed break after my Covid-filled Christmas) and I'm sure things will improve in the summer term.

Before my teaching of Pythagoras to Year 8 was interrupted, I'd delivered a lovely lesson revising squares and square roots (with and without a calculator) - this is a really important pre-requisite skill in this topic, so I feel that it was time well spent.

In case you missed them, my most recent blog posts were:

If you haven't already booked, don't forget to get a ticket for the MA's April conference which is coming up soon. If you're not sure, just check out the workshops on offer! They are fantastic, and it's amazing that you can access them on demand for only £10.

Thank you to La Salle for running #mathsconf25 today, and to all the speakers. This was the seventh year in a row that I have attended a maths conference on Pi Day weekend!  Thank you to everyone who came to my session on GCSE Topics: What and How?. I focused mainly on simultaneous equations. The video will be released by La Salle soon. 

Thank you also to people who have bought a copy of my book A Compendium of Mathematical Methods over the last fourteen months. It took a lot of time and effort to write a book, and it's hard to be a female author in a male-dominated field, so I really appreciate the support. I probably don't tell people about my book as often as I should, because promoting your own book seems to bring disapproval from many. I need to stop shying away from it though - the tiny amount of extra income is helping me slowly save for a small loft conversion so my daughter can have her own bedroom, so it is important to me. If you want to 'try before you buy' then there's a sample chapter here, and a free Seneca course here which covers two chapters.

I'll leave you with this fun iceberg drawing tool from @JoshData. I love this!




 

8 March 2021

Geometrical Reasoning Within Constraints

This is the fourth 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 4: Geometrical reasoning within constraints
This is the fourth of an irregular series of writings in which I (and, I hope, others) delve deeply into the collection of tasks on Don Steward’s blog donsteward.blogspot.com 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




Nichola Clarke’s DPhil research investigated the mathematical reasoning of students in lower attaining sets in upper secondary school. She found some students who were perfectly capable of ‘if … then … because…’ reasoning in geometrical situations but took ages to then calculate angles; they appeared to be struggling with reasoning when they were actually struggling with arithmetic. You might think that things would be different now that arithmetical fluency appears to have a higher profile in primary mathematics, but because the emphasis is sometimes on column methods instead of on recognising number bonds and relationships the same obstacles are likely to apply. Ori Golan had the insight, when first teaching geometrical proofs, that removing arithmetic by teaching algebraic and logical representation of angle relationships would fasttrack his students to reasoning rather than a dependence on calculating, and it did.

I didn’t have this in mind when looking for hidden threads in Don’s geometry resources but found it in his sequence of work on the regular dodecagon (confusingly posted under the heading ‘area’).  He had told me of the existence of this sequence, and made it available to participants of our workshop on angles (see pmtheta.com ‘PMTheta@home’ series) but my mind had been elsewhere and I did not pick up its potential significance. A superficial look shows pretty patterns and I thought of it as ‘how you apply angle-reasoning’ rather than as a sustained introduction to reasoning with angles.

His sequence is partly inspired by David Wells’ book: ‘Curious and Interesting Geometry’ but I cannot find that book to check what is in it (after Covid I will return to downsizing my library to passing visitors – but Wells seems to have already gone).

In several of his resources Don realised that working with a limited ‘palette’ of numbers would focus learners’ minds on relationships, properties and theorems rather than on calculation. In this case the palette involves 360, 180, 90, 45, 60, 30 and various multiples, sums and differences of these together with the idea that one revolution is 360° and the interior angle sum of a triangle is 180°. Even this list has some redundancies but you have to start somewhere. I found it useful to employ the ‘angle subtended at the centre of a circle by an arc is twice the size of the angle at the circumference subtended by the same, or a congruent, arc’ but this fact could also be deduced within the sequence.

Preparation for the sequence could consist of a significant time finding out ‘what numbers can be made from adding, subtracting, halving, doubling etc.?’ as a toolkit and sharing it as a public resource in a classroom. As with many number facts, it is useful to be so familiar with these that they can be recognised, e.g. ‘how might 120° be made?’ rather than seeking probable components with a calculator or p&p calculations.

OK, so armed with some familiar angle relationships what can be said about this?:


The more relationships you know about the more ways there are of answering this, but both can be deduced from the angles round a point, the angle sum for triangles, and symmetry. Other reasoning chains are possible of course, for example a turtle argument for an exterior angle would also be a good start.  Anyone who has played successfully with the coding for ‘Frozen’ would have ways to think their way into this.

I would follow up with this inquiry:

What are the minimum assumptions necessary to sort this diagram out? I am imagining a ‘facts board’ collecting the facts that are necessary and those discovered as they emerge from students’ work. I am afraid I cannot recall the name of a teacher who had a ‘conjecture’ list and a ‘proved’ list on show in the classroom for work such as this. Is it OK to use results that are only conjectures? Would a class-owned digital collection give the same availability for students to browse when they become stuck?

You might think me unadventurous in my choice of a second slide, given the goodies that Don has offered, but I am thinking that this second slide gives an opportunity to do some preparatory work on how chains of reasoning might be talked about and represented. These communication tools need to be established before moving on.

Here is a flavour of more complex slides:


There are 47 slides in all for this sequence and I have worked through them, but am stuck on a couple. The central feature of my working was always the methods of reasoning; usually several pathways are available. Some of the reasoning uses general facts and geometry-theorem-development through reasoning about structure; some of the reasoning uses the specific properties of specific angles.

I presented the benefits of using a limited collection of numbers also in Dose of Don 1, in which I showed tasks where he had used grids to constrain the variation of angles, often represented by ratios.

But what I set out to do for this Dose of Don was look at area, and it was only because the above ideas had been posted under ‘area’ that I found them again. I had intended to collect some of Don’s ‘cutty-uppy’ tasks (otherwise known as proofs by dissection or proofs without words) to identify some of the thinking behind them. Conservation of area is a concept that most children arrive at when very young, and this sense is enhanced by playing with sets of 2-D shapes. Indeed, understanding of area as a concept is difficult since it is not such a lived experience as linear measure is (by moving things) or volume (by pouring things or trying to hide under chairs). In my experience, this is why many students grab onto a formula such as A = l x w for area questions, often incorrectly, instead of using conceptual understanding. Cutting paper shapes and moving them around builds on an intuitive idea of conservation and also provides content for ‘if … then … because …’ reasoning. I am afraid that watching shapes move on a screen does not give a sense of personal spatial manipulation. Deciding where to cut and move involves imagination plus reasoning; using dynamic geometry software limits these decisions to those that are conventionally and geometrically useful, such as using mid-points, perpendiculars, and so on; origami offers a palette of useful outcomes that can be achieved by folding. Watching someone else, or a prepared animation, pre-decides the moves.

I find that with some of Don’s ideas I have to distinguish between what can be used for initial learning, what for the development of ideas through sequences of examples that draw attention to characteristics or complexify those ideas, and what for application and extension to other ideas. I am offering some that can be used for initial learning about area of certain shapes alongside using manipulation as an exploratory tool.

Consider this question, which appears to be about the area of triangles but only needs the concept of conservation of area. Don offers eight different proofs in 'quartering a parallelogram' and I am particularly interested in those that build on a cutty-uppy approach:


Note how the final proof that I have selected labels angles to transform a cutty-uppy approach into the use of congruence because certain kinds of movement also conserve angles. To know that congruence is not only about angles and sides but also area (therefore) seems to be a relevant fact.  Can you find this explicitly in a school textbook?

Can congruence be deduced from the equality of one side, of one corresponding angle, and of area? Ditto two pairs of equal angles plus area? And so on.

Don provides a sequence about the area of a rhombus from an initial idea to one of the possible algebraic representations in 'grid kites and rhombuses'. The sequence uses cutty-uppy with constraints by limiting the exploration to a square-dotted grid.


I shall leave the algebraic development to readers.

 






19 February 2021

150th Birthday!

Every time I read anything about the history of mathematics education in England, I am struck by the pivotal role played by The Mathematical Association. Originally called the 'Association for the Improvement of Geometrical Teaching', the MA was founded 150 years ago in 1871. Education looked very different back then.


The MA was the first ever teachers' subject association to be formed in England. It paved the way for many that followed - both in maths and other subjects. Subject associations exist to support teachers and to give them a voice. 

The MA has played an absolutely crucial role in both reforming the maths curriculum and in supporting classroom teachers for 150 years. 150 years is a really long time in education. 

I think we all know that subject associations need all the help they can get at the moment. The rise of the internet has meant that over the last decade teachers have found new ways of supporting each other. It would break my heart if the MA had to shut down. Imagine that - the oldest subject association in England, the original reformer of maths education, closing because it couldn't convince maths teachers to part with £2.50 a month to support it.


You can join here by the way. It's super cheap, and even cheaper for NQTs and trainees. And it's important. But anyway, I am not here to talk to you about joining the MA - I have done that before in my post 'Strength in Numbers'. I am here today to tell you about some of the exciting stuff that will be happening this year for the MA's Sesquicentennial. 


Conference!
I absolutely adore maths conferences. I was really sad that last year's MA Conference had to be cancelled (it was meant to be at a spa hotel! And I was going to do a keynote!). 

I can't put into words how much I miss the real-life human interaction that comes with attending in-person conferences. But I do see the multiple advantages of virtual conferences. From a speaker perspective, the large audiences are great! Now I've got the hang of presenting online, I really like it. And from a delegate perspective, virtual conferences are more affordable and more widely accessible. So I am a big fan of these online events. I hope that once Covid is over, a mix of both in-person and virtual maths conferences will be available.

The MA's Easter conference usually takes place over three days, but three days in a row is a long time to spend online, so the upcoming virtual conference has been carefully designed to ensure that delegates don't suffer from screen fatigue.

The first day takes place from 12.20pm to 5.50pm on Thursday 1st April (for most teachers, this will be the first day of the Easter holidays). The programme for this day is amazing. The keynote will be from Dr Nira Chamberlain and the plenary will be presented by Charlie Gilderdale and Liz Woodham from NRICH. Throughout the day there are numerous 50 minute workshops across a variety of strands, including primary, secondary and post-16. I'm looking forward to presenting on task design in the secondary strand. 

The second and third days of the conference take place on Friday 9th and Saturday 10th April. Both days start at 12.40, so no need to get out of bed early in the holidays! These days are jam-packed with brilliant workshops, plus unmissable plenary sessions from current MA President Hannah Fry, and future MA President Colin Foster.

The full programme for all three days can be accessed on the conference website. There's so much to choose from!

There will be evening events at the end of the first and second day of the conference. And the workshop recordings will be available to access after the event, which means that for a mere ten pounds you get access to the full live conference experience, plus a bank of over 60 conference sessions to access if and when you want to. It's actually crazy that you get all this for just ten pounds - it's worth booking even if you just intend to attend a couple of sessions! And for those of you wishing to attend the whole conference, you are welcome to pay more for your ticket if you wish (bear in mind that the MA is a charity and the extra support is much appreciated). 

What else is happening this year?
There are even more exciting things planned for the MA's 150th anniversary year. 

One thing I am particularly looking forward to is an upcoming edition of the excellent journal Mathematics in School which I am guest editing with Ed Southall. We have decided to make this edition a tribute to the late Don Steward. I am delighted by the high quality articles submitted for inclusion. If you wish to receive a copy and you are not currently a member of the MA, then join now and make sure you choose to receive Mathematics in School along with the standard subscription. 

Another lovely thing that the MA has done to mark its 150th birthday is share a hand-picked collection of papers from across the history of The Mathematical Gazette. You can download these for free here. I absolutely love the article 'A century of textbooks'. 

If you're not familiar with the Gazette - it's quite something. It's a leading journal in its field, with global readership, and has been published since 1894. There's a sample booklet here.



Follow @Mathematical_A to see what's happening in this important year in the MA's history. And don't forget to get a conference ticket! See you there.







7 February 2021

5 Maths Gems #141

Welcome to my 141st gems post. This is where I share some of the latest news, ideas and resources for maths teachers. 

1. Tasks
Task designer @ashtonC94 has made a new website mrcowardmaths.wixsite.com to collate his excellent resources. He is gradually populating it with tasks, so do check it out. 

Ashton has shared an activity on plotting coordinates that draws attention to scales in order to tackle common misconceptions. An extract is shown below - click here for the full task.

I also really like Ashton's excellent task on significant figures
Both of these tasks have been added to my resource library.

Maths Gems regular @giftedHKO has also continued to share a number of excellent resources recently, including this task on gradient of a line segment, and many more on her website.

2. Booklets
@ChrisMcGrane84 has started sharing a series of curriculum booklets that he has been working on in conjunction with Siobhan McKenna (@ShivMcKenna55). Read Chris's post for more information. If you were at #mathsconfmini then you can watch Chris discuss how these booklets form an integral part of the design and delivery of his maths curriculum in his talk Designing an Effective Curriculum in Early Secondary

These booklets are packed full of carefully selected tasks from a range of excellent task designers. The booklets are used instead of a scheme of work - they define curriculum order and pedagogical approach.

3. MS Forms Quizzes
Thank you to @timdolan for creating a Padlet for maths teachers to share maths quizzes made in Microsoft Forms. This should save teachers a lot of time, as they don't have to create everything from scratch anymore. If you have been making MS Forms quizzes then please share your quizzes here (as a template) so that others can borrow them. I've added mine!


It's interesting to see a variety of quiz types. My quizzes feature short-answer questions, or multiple choice questions (which I borrow from diagnosticquestions.com) which are automatically marked. But there's a lot more we can do with these quizzes. I received an email from maths teacher Alison Page who turned my Algebra True or False activity into a Forms quiz. Looking at this, it occurred to me that it would be easy to turn an 'always, sometimes, never' activity into a Forms quiz, with spaces to write explanations.  Another option for turning existing worksheets and activities into online versions is teachermade.com

If you are a MS Forms user then you might be interested in this post from @andrewkbailey13 about giving feedback to students. My team have been giving lots of feedback but we have been concerned that our students haven't been reading it - this post answered our questions about how it works from a student perspective!

4. A Level Example-Problem Pairs
Thanks to @acutelearning for sharing a large set of A Level Maths and Further Maths example-problem pair slides with answers. These are freely available to download from TES.

5. Mr Draper's Blog Posts
Mr Draper (@MrDraperMaths) has been busy writing a series of excellent blog posts lately. They all feature cleverly thought-out tasks and teaching approaches. For example, this post on Understanding Angle Labels addresses a number of difficulties that students commonly experience when learning to solve angle problems.
Another example is this excellent post on reasoning with frequency trees:

And a great post about perimeter:

Read Dan's blog for lots more like this.

Update
It's been a busy time for everyone. I don't know about you, but my online lessons are still taking ages to plan, so I'm spending entire weekends working. I'm not doing anything fancy in my lessons, but explanations and tasks need really careful thought. I also have a lot going on in my other roles at school - I'm responsible for reports, parents evenings, Year 9 options and so on - and all of that has been really full-on in the last few weeks! Roll on half-term.

In case you missed it, here are my most recent blog posts:

I recently recorded a podcast with Craig Barton where I discussed what I am doing for online lessons. Do have a listen!

I also presented at two conferences: at #mathsconfmini I did an evening presentation on teaching quadratics, and at the White Rose Secondary Maths Brunch I presented on task design. 

I hope to do a longer version of my task design presentation at the MA Conference this Easter, if my proposal is accepted. Also, following a lot of thinking recently about how to introduce probability in an accessible way to my Year 9s, I have submitted a proposal to speak about probability at #mathsconf25 in March. 

Finally, I recently produced a Topics in Depth CPD video for Loughborough University Mathematics Education Network which you can access here. It covers curriculum, history, misconceptions and resources.

I'll leave you with a pie chart that I spotted recently, which is probably one of the worst pie charts I have ever seen.



Stay safe, maths teachers!







23 January 2021

Squares

This is the third 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 3: Squares
This is the third of an 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.



In Dose of Don #2 I quoted a remark Don had made in his blog that ‘legitimately going from one statement to another (is kind of what maths is about)’ [his brackets]. It was said in the context of working with straight lines and linear expressions, so I decided to continue with squares as both geometric and algebraic objects. How does Don ‘legitimately go’ from one statement to another?

Squares as geometric objects have the properties of equal sides, right angled vertices, several kinds of symmetry and equal diagonals that bisect each other at right angles. Any of these properties can be expressed algebraically. Observation and measurement of components of squares can support deductive reasoning about other properties. For example, any straight line segment going through the point at which the diagonals intersect cuts the square in half by area. Area is a very useful feature of squares because it is the square of the side length (hence the similarity in the names) and that provides a bridge into meaningful algebra. But before I go over that bridge I will stay for a while with the equality of sides. If learners have internalised the reasoning power that depends on these equalities – the legitimate journeys from ‘these sides are equal’ to some less obvious statements – they are on the road towards mathematical reasoning. I propose that reasoning with squares based on their property of equal sides is good preparation for later geometric reasoning and a suitable arena for engaging in mathematics that is not primarily about calculation.

There are classic types of problem that can be solved using that property, see https://donsteward.blogspot.com/search/label/squares%20inside%20rectangles. The numbering of the squares indicates the length of the side.



These two diagrams appear in his ‘number’ task sequence. Both of these depend on reasoning about equal sides. The first diagram can be completed with verbal reasoning: ‘this must be … because …’. The second diagram is less straightforward: ‘if I knew this length, then I would also know that length’ and therefore needs algebra – the labelling of the unknown. It doesn’t even have to be a letter for younger learners, it could even be a coloured sticky dot to stand for an unknown value. It might be astonishing that many ‘square in rectangle’ problems can be resolved from very few initial measurements and the labelling of only one unknown.

Another version of the second diagram appears in the ‘equations’ task sequence on the same blog. This starts with the 7 and 28 being given. One feature of these puzzles is that a choice has to be made about which square to label as the unknown. In this diagram it is not the smallest that he has chosen to label as d. There is also some potential confusion about whether a length or a square has been labelled so you have to work through it to see what he was thinking. I have always thought it is important for learners to see someone else’s ‘work in progress’ so they understand that – yes - maths can be done messily and then it has to be tidied up to communicate it to others.

One interesting thing about this task sequence is that he called it ‘equations’ but there are no actual equations written out, only expressions.  Several ‘opposite side’ equations have been in his head in order to generate the expressions, and then the expressions can be gathered together and organised into equations from which the unknown will be found. Here, height can be expressed as  5d + (d + 7) and also as (42- d) + 28.  Simpler examples are available.

The algebraic manipulations that arise in these puzzles are purposeful and meaningful. In my view these problems do not need prior experience of gathering like terms or dealing with brackets – they can themselves generate a need for these tools so could be a starting point rather than an end point of a ‘gathering terms’ teaching sequence.

Don offers a few simpler examples, and making them up takes a great deal of patience to ensure that solutions are always integers (there is no point in complicating matters with fractions when introducing deductive reasoning and algebraic formulation).

Let’s suppose learners have done several of these and become adept at ‘reasoning from the equality of sides’. Where can we go from here?  For me an obvious direction is towards the use of area models – those two-dimensional representations of algebraic relationships that relate the word ‘square’ as a shape to the word ‘square’ as the second power. This model depends on the internalisation of equality of sides of a square, of the equality of opposite sides of a rectangle, and of expressing area of rectangles as the product of side lengths, which in squares gives x2.  I am not being patronising to list these, rather I am drawing attention to the need for learners to have internalised them not as verbally memorised facts but as components of the network of concepts that go with squares and rectangles. If they have to think hard to recall these they may not be in a suitable state to use such diagrammatic representations – they are the necessary tools for thinking.

So when and how does Don use area models? There is an example of his creative insight at https://donsteward.blogspot.com/2020/03/two-2-digit-multiplications.html which caught my eye. In this task sequence Don explores what happens when 35, 45, 55 … are squared. He asks the learner to work these out without a calculator and then hopes they will notice the appearance of 25 as the rightmost digits. The diagram he offers to explain this is:

There is a sense in which the leftmost digits of the answer are out of the way of the ‘25’ because of … well why?

He then generalises this particular family of numbers to ‘10n + 5 squared’ showing that what is being represented by the diagram can also be represented, and calculated, by multiplying each term of the second bracket by each term of the first bracket and hence getting  100 as the coefficient for both n and n2. Why might we be interested in these rather special cases? One possible answer is the availability of suitable grid paper to facilitate the transformation between head and pencil-&-paper. Another possible reason is that Don the offers a further sequence about 312 and 31 x 29 which are represented as extensions of  the use of area diagrams, and from which can be learnt something about the formation of the middle term of the polynomial format of a quadratic (other formats are available) and difference between two squares, and more, under the general heading of ‘a add b squared’: https://donsteward.blogspot.com/search/label/a%20add%20b%20squared.

As always, he leaves the progression up to the teacher, but I can’t help thinking that the presence of these tasks on his website means that he is favouring a meaningful, purposeful ‘slow burn’ approach to using and manipulating algebra by using spatial awareness.  That would fit with his commitment to the ideas of Dina van Hiele who classified the components of mathematical understanding as: visualisation, analysis, abstraction, deduction and rigour. These are often regarded as hierarchical and indeed ‘seeing’ is a first response to a new bit of mathematics while imagination and imagery are key experiences of doing maths, not only features of good pedagogy. While the order might be hierarchical in terms of abstraction it is not hierarchical in terms of learner-age. In my next ‘Dose of Don’ I will explore some more of his commitment to spatial reasoning. 



9 January 2021

Tools for Online Lessons

I've been reluctant to write a blog post about teaching online lessons because I am far from being an expert on this matter. In the first lockdown I mainly set work on Hegarty Maths. I don't believe in reinventing the wheel. With good quality videos and tasks at my disposal, I was very happy to use them. This time round my school has changed its approach in order to improve the student experience and (hopefully) maximise engagement. So now I do some live lessons (we run these as Microsoft Teams events, which are recorded for students who need to watch at different times). We will shortly be introducing asynchronous practice lessons in maths too, for which I will mainly be using Hegarty Maths.

So compared to many other teachers I'm relatively new to live lessons. Thankfully the expertise of maths teachers on Twitter is already immense. Tweets about live lessons are constant and amazing, with teachers all over the world giving each other practical advice and inspiration. I thought it might be helpful to share some of that advice here for my readers who aren't on Twitter. I'm not featuring the well-known websites here - we are very lucky that there are absolutely loads for maths (Hegarty Maths, Dr Frost, Corbett Maths, MyMaths, Mathigon, MathsWatch, MathsPad, White Rose, Times Table Rockstars, Desmos and so on). Here I'm featuring tools and websites that might be less well-known. This should give you a starting point to investigate further if you are looking for a particular solution for your online maths lessons. 

1. If you use Google Classroom...
I don't work in a Google school but if I did then I would make use of @Philmaths314's free Self Marking Google Sheets. As the name suggests, these are sheets that mark themselves. ⁦‪The site is very easy to use: choose a sheet, create your own copy and then assign to your pupils. When they input answers, they get an instant score. I featured this resource in Gems 131.

The sheets can be downloaded as spreadsheets too, so can also be used outside of Google Classroom.

The other thing that Google users might want to explore is flippity.net.



2. If you want online exercises without logins...
In the first lockdown there were times when I wanted to set my students an independent practice task at the end of a topic, but they had already done all the tasks for that topic on Hegarty Maths. One option was to set them an exercise from a website like Corbett Maths, asking them work on paper and mark their own work. Another option was the use the free interactive tasks on the CIMT website. These tasks look a bit dated but still work beautifully. I love it that examples are provided before the exercises, so support is built-in for students. No login is required so access is quick and easy. The exercises are self-marking so students get instant feedback. The other thing I like is that there are no adverts. I can't stand it when there's advertising on student-facing websites. Here are the links:



There's also interactive material for Years 3 to 5 here.


3. If you want to use mini-whiteboards...
Teachers who are heavy users of mini-whiteboards in their normal lessons might be interested in the various online mini-whiteboard tools that are now available, allowing you to see your students' work in real-time. There's been a lot of tweets about whiteboard.fi this week. Read the threads for more information:


 

The online whiteboard tool on drfrostmaths.com is also very popular.

Apparently Teams at my school has a built-in function call NearPod which has whiteboard capabilities but I haven't explored it yet. 


4. If you want to quiz students through Teams...
Each of my live lessons ends with a short online quiz to check what students have learnt. Because I work in a Teams school I use Microsoft Forms to conduct these quizzes, stealing most of my questions from diagnosticquestions.com. I schedule the quizzes in advance to pop up as an assignment near the end of the lesson. These quizzes are massively informative, guiding my planning of the next lesson and allowing me to give whole-class feedback and address misconceptions. It's like a self-marking exit ticket, which is awesome. 

In case you happen to be teaching rounding, here are four of my rounding quizzes from this week that you are welcome to borrow. I will add lots more quizzes to this document over the coming weeks. If we all start sharing our Forms quizzes, we will eventually have a big bank of them that we can all borrow and adapt, which will save a lot of time.

From Twitter I can see that many other teachers are using Forms quizzes very effectively in their Teams lessons. Here are some tweets that might be of interest.






5. If you are using PowerPoint...
My online lessons are the usual mix of explanation, modelling examples, questioning and practice tasks. They are pretty similar to my normal lessons. Apart from my quizzes, everything else I do in my lessons is in PowerPoint - and apart from a pen mouse so I can write on my slides, I don't use any special technology or online tools. The only thing I want to mention is the timers that I use in my PowerPoints. To minimise faff during my live lessons, I make use of the lovely transparent timer gifs shared by @DrStoneMaths. They're just so easy to use: I put a task up on the PowerPoint, the timer starts automatically, and the students start working. Easy! I featured these timers in Gems 134.


Just bear in mind, when you see teachers on Twitter using loads of fancy interactive tools, a simple PowerPoint containing examples and a series of tasks works really well when presented by an enthusiastic and confident teacher. Sometimes we see other people doing different stuff and assume we're not doing enough. Let me reassure you - if your students are getting clear explanations and regular opportunities to practise then you're doing a great job.

I hope that this post is helpful. I know there are dozens more tools and ideas that I haven't featured - I'm trying to avoid overwhelming people.

Feel free to add your own advice in the comments. 

My husband works in a hospital so when I'm not in school looking after keyworker children, I'm home alone trying to homeschool my six year old and nine year old whilst simultaneously teaching lessons, responding to hundreds of messages and attending dozens of online meetings. Stressful is an understatement! Many teachers are struggling at the moment, so it's more important than ever that we support our colleagues during this difficult time.

Well done on surviving a killer first week. It will get easier!