My plan this year was to return in September part-time. I was looking forward to getting my teeth properly back into teaching now I have completed my family. Unfortunately the situation with Covid-19 means my children can't be cared for by grandparents, as had been the plan, and putting both children in nursery would have left me working for about half of minimum wage (I love my job, but not quite that much). So we made the decision in March that I'd take a year out and see where we were in September 2021; I'm officially a stay-at-home mum from next week.

It has surprised me just how much this has been in the back of my mind this week. There are definite advantages; for one, I'm not having my standard back-to-school stress dream (all my teeth falling out - apparently this is a common one?!). Despite my kids being fairly early risers, I'll still be able to catch more kip than when I returned to work after having my daughter, when getting us both ready for the day involved me getting up a full hour before her in the mornings. I won't have to spend my lunch breaks in the medical room expressing milk, which was a ridiculous chore for the first term I was back. And beyond the personal challenges, there's no denying that this academic year is going to be difficult for other reasons, and I have so much respect for everyone who has returned or is returning to the chalkface in these few weeks.

​That said, I am very much going to miss the break in my week from being "just mum". Prior to the Covid lockdown, I thought I was the sort of person who needed that break; I panicked a bit in the first couple of weeks when suddenly my daughter was with me 24/7, rather than spending days with grandparents or at nursery. This year has proved to me that I do fine without it, but I am planning to carry on with a little bit of education stuff on the side. I've already reworked the Number/Proportion and Algebra strands of my GCSE Resources by Topic collection, and Shape and Data will be done soon. I'm also going to get around to uploading a few more of my own resources, as well as blog a bit more about general maths-y stuff (warning: it's likely some might end up preschool focused as that's the level of maths I'm now working with on a day to day basis!).
​
​Finally, I'm not going to end my back to school post with resolutions, as these tend to get broken! I'm a planner by nature and I've struggled with the uncertainties of the last few months, but I am getting better at taking things one day at a time... as Nemo would say, just keep swimming!

Working backwards...​ 
Once students have got the hang of how these work, ask them to work the other way around and provide the product and sum to give the required answers. 

When dealing with negatives, I ask students to initially ignore the signs and just focus on finding two factors of the top number that they can "use to make" the bottom number. This way of phrasing ​the process helps them to get round the ever-present issues with negative numbers. We then discuss how the signs can be added to make the puzzle work.

Picking apart and reworking my lessons from last year, I noticed that I almost exclusively model worked examples by handwriting "live" rather than pre-typing. I don't recall making a conscious decision to do this, but following a discussion with my husband this weekend about print vs digital media, I thought I'd think more critically whether either style has any considerable advantage.

There are some interesting studies about reading information via print or digital media; this paper from the Research in Learning Technology journal looks specifically at educational materials and this article from Phys.org summarises some of the preferences people have for different media - generally, we prefer printed media for longer, more involved reads and digital media for shorter reads. I also found this study examining handwriting vs typing inputs for students when solving equations; the authors conclude that the only real difference in this scenario is the speed at which students completed the task, with typesetting algebra taking twice as long as handwriting solutions. However, I've not been able to find anything on the modelling of worked examples in Maths, so this blog post is written with the caveat that this is all personal theorising and is not research-based.

Time-gains while planning are significantly in favour of handwritten examples, particularly as questions become more complex towards the top end of GCSE and into A Level, and when complex algebra is involved. However, as handwriting recognition tools such as Microsoft Office's Ink Equation Tool are becoming increasingly effective, it might be interesting to see whether this remains true in five or ten years' time.

One advantage of working "live" is that the teacher can model thought processes more easily, verbalising each step in the working. This process is also instantly adaptable to the class - the teacher can speed through stages that students are confident with, or elaborate in more detail if a step requires further explanation. 

Again of particular importance for top-end GCSE and A Level, as problems and solutions become more complex, students increasingly need guidance with how to present their work in a coherent manner. Watching a teacher model this directly may be easier to translate to their own work than looking at a typed example.

When I complete worked examples with a class, I tend to ask for input from students at each step and see what they can work out logically or infer from prior knowledge. The rigidity of a pre-typed example makes it more difficult to include student responses, as many of them may go slightly off-script from the pre-planned work.

If I'm going to typeset solutions, I make damn certain the example I'm using illustrates a key point or highlights a common misconception. The need to work efficiently means I then select an example which serves many purposes and has decent mathematical weight. It also means that the numbers work properly - while I usually check all the examples I prepare in my lesson materials, occasionally the odd blooper slips in - the star culprit is quadratics that don't have real roots. While this can form the basis of an interesting discussion, it can also wrong-foot a class and damage confidence in the teacher.

While there are losses in the step-by-step process, the end product is often more coherent when digitised. There are times when I've stepped back from a worked example and actually looked at what's on the board - and because I've documented my thinking process or expanded on certain points, the solution can end up looking mathematically messy or sometimes confusing to students. I also have a tendency to use arrows and random bits of annotation which get very scruffy when handwritten, and my handwriting itself has deteriorated significantly over the last ten years as I get less and less daily practise.

Unless I am specifically modelling graph-plotting with Year 7, I no longer draw graphs by hand. It's tedious and time-consuming, and incredibly difficult to do when working right up against a whiteboard! Using a tool like Desmos is now pretty commonplace, but if I'm doing this I will often prepare graphs in advance rather than plot "live", as I can make sure that the graph is scaled appropriately. Diagrams or pictorial representations always look better when done digitally - for example, your standard "counters in a bag" when working with tree diagrams. 

This year I'm probably going to carry on handwriting most of my examples - I enjoy the process of working through problems collaboratively with a class, and I think typeset examples lose some of this. However, I am going to reconsider selecting some gold-standard examples for each topic to typeset; at GCSE, our students have small exercise books that are used to take important notes and write down illustrative examples, and these would seem the ideal candidates for a bit of deeper thought and clearer presentation.

In two weeks I'm returning to work after pretty much a year off on maternity leave. I had such plans - I was finally going to sort out my website, blog more, organise the stupid pile of resources on my memory stick, finish my #mathsconf blogs from 2017 (16, 15, ...) - the list goes on. These plans were made in the naivety of late pregnancy, when I'd decided that I had loads of time off and a sleeping baby would be dead easy to work around.

Unfortunately, they aren't, and I'd also seriously underestimated my need for a little bit of self-care time - the last thing I wanted to do once I'd got my daughter down for a nap was turn my computer on and start working - so none of my plans actually came to fruition. 

However, I did manage to finish the book I'd been working on for the Tarquin A Level course and resources (shameless plug but I'm so proud that I finally achieved what I thought was a complete pipe dream of becoming an author - although I have to confess in my head I was the next JK Rowling, not writing maths revision guides). I also wrote a guest post for TeachWire in 45 minute spurts during naps, so I'm not entirely convinced it's any good. Over the last couple of weeks I've made some attempt at deleting some old rubbish resources and trying to plan my first few lessons, but I'm aware that returning to work with a 9-month old (albeit part-time) will involve some serious expectation management on my part. Fortunately, over the last nine months I've become much more accustomed to "that'll do", and my plans for this academic year involve a significant amount of "work smarter, not harder".

When I sat down to write this post, I discovered a prompt saved in my drafts from two years ago titled "Stop, Start, Continue, Change", with nothing other than headers. Presumably that was a thing that was going round the blogosphere at the time, so I decided better late than never - so here's my Stop Start Continue Change for academic year 2018-19.

Homework generates a significant amount of my marking load, and I'm not sure it's always entirely effective. My homeworks are generally a set of problems similar to those worked on in class, giving pupils an opportunity for further practise and for me to identify individual misconceptions when marking their work. The trouble with this is that by the time the homework comes back in, is marked and returned with feedback, a week has passed (due to peculiarities in our timetable, we only see classes once a week) and the feedback becomes less effective as the immediacy is lost.

A few years ago I experimented with flipped learning with one GCSE group, to great effect. I'm keen to look at this again this year, perhaps using our MathsWatch subscription to create tasks to revise prior knowledge before beginning a new unit. I also want to look at more effective strategies for revision homework, short of "there's a test next week, revise" - as we're all aware, that's a week off for some pupils!

I admit it, I love a well-formatted resource. Unfortunately, this can occasionally border on obsession, with a tendency for reformatting other people's resources to fit my "style", and also reformatting all of mine every three or four years when I change whatever style I've picked, not to mention dealing with the whole Promethean/Smartbook/PowerPoint choice. This will definitely be stopping this year - I'm not sure how many weeks of my life I've wasted on this pointless task, but it adds next to nothing to pedagogy or pupil engagement, and I don't have the time to burn anymore. However, I will be making sure anything that I make new is in a consistent style with what I've already got and sticking with it!

This is a bit of a cheat, really - I'm not sure you can consider one blog post a proper return, but I am going to attempt to keep up with blogging and updating this very neglected site this year. I've got a few ideas bouncing around already, and I'm sure the challenges of working part-time and balancing childcare and my career will give me plenty to write about. A fortnightly blog or update sounds manageable, particularly if I keep it short and simple (unfortunately not my forte), so I'll be giving that a go!

At the weekend I read Mark McCourt's latest blog in his series on Mastery - of particular interest as I spent ages working on what I thought was an effective "Mastery curriculum" in my previous role. It's the only bit of CPD I've done this year, but it really is an excellently written piece (I almost want to say manifesto). The bit that struck a real chord with me was titled "Moving from Current Practice to Mastery Approach" and examines the pattern of a typical lesson compared with a suggested mastery approach. I'm not going to attempt to paraphrase too far, as it really does need to be read in full - but the gist is that more time is spend on impactful teacher direction, perhaps interspersed with pupils trying examples for themselves, and questioning is planned carefully at each stage.

There is also an interesting discussion about problem-solving - or as Mark titles it, "behaving mathematically" in the section "Phasing Teaching". My biggest takeaway from this was that pupils should not be asked to behave mathematically with content that they have just encountered, as they need time for this knowledge to mature in order to work flexibly with it; instead, pupils should work on problems linked to these topics but relying on earlier learning. As I've already decided that one of my performance management targets is to make use of open-ended or rich problems more effectively, I'll be considering some of these points when planning tasks.

The Prezi from the session and links to all of the applets and resources mentioned can be found on Tom's blog post.

Session 2 at #MathsConf7 was delivered by Christine Andrews, Andrew Taylor and Gary Wing, and looked at the developments and questions about the new Maths A Level for AQA. This blog is just my interpretation of their workshop, and shouldn't be taken as gospel, although I've tried to reproduce accurately what was said. It's also worth remembering that these materials are still in draft form, and have not been accredited by Ofqual yet.

The session looked at many more than the titular 10 issues, and provided a lot of food for thought - I'm looking forward to teaching the new A Level, but it will be hard to mentally move away from the modular system.

Teaching and examining using the "large data set"
Many of the Statistics questions will now be based on a "large data set"; AQA have chosen Family Food, which is published by the Office for National Statistics every December. Christine clarified that this selection would not change through the lifetime of the specification, but that the updated version would be used every year, and examination questions would be set using the most recent set of data; I took this to mean that exams in summer 2018 would use data published at the end of 2016. Christine also highlighted that teaching using the data set would not be compulsory, but that familiarity with the scenarios and language used would be advantageous in an exam situation (see example below: Paper 2, Q14 from the AS Level papers, available here). This approach would certainly make the teaching of statistics more coherent, and give lots of time for real in-depth analysis of one problem, as opposed to skimming over the surface of many different scenarios. Gary highlighted the opportunities to investigate these data further using technology such as Excel and MatLab.

Use of calculators; are pupils expected to purchase/use a graphical calculator?
Calculators used for the new specification materials are expected to have an iterative function and standard statistical calculations. The normal and binomial tables will no longer be provided in the formula booklet, so pupils will need to be familiar with use of calculators for these topics. Gary said that more reasonably-priced calculators are currently being developed, but we have no idea when these will hit the market.

Use of the equation solver on a calculator will be credited in the mark scheme if used well - such as using a calculator to solve a quadratic at the end of a mechanics problem (see examples below, Paper 3 Mechanics, Q9 from A Level Further, available here and Paper 3, Q4b alternative solutions from A Level). In such scenarios, the final solution would be worth one mark, and pupils are expected to use their calculators to do this; any marks for calculations like this have now gone.

Pupils should use the wording of the question as an indicator of whether a calculator solution is appropriate; "show that" would require pupils to demonstrate their method without using a calculator.

With regards to graphical calculators, Texas apparently have one out now and Casio are aiming to produce one by the end of November.

For Further Mathematics, calculators are also required to compute using 3x3 matrices. Gary highlighted that pupils studying FM may well want to invest in a graphical calculator, but Christine clarified that AQA would ensure that those who didn't have graphical calculators would not be disadvantaged in an exam.

Further Maths Paper 3 - choosing options
Andrew clarified that pupils should not attempt all three papers (Discrete, Mechanics, Statistics) in a timed exam, as this would disadvantage pupils. The decision about which papers to sit would be made at the point of entry, and is not a choice on exam day. Pupils sitting the exam will get the correct two papers out of three and use the examination time to answer these two papers.

Curriculum planning for AS Mathematics, A Level Mathematics and Further Mathematics
The AS and A Level are not linked (so results in AS will not count towards the final A Level qualification), but they are co-teachable.

When delivering A Level Mathematics and Further Mathematics, it's possible to deliver either side-by-side or in sequential order. AQA has permission to offer examinations for A Level Maths after the first year of teaching, so pupils could sit the full A Level after the first year, then sit Further Mathematics in the second year.

AS Further Maths will only require content from AS Maths, with the exception of radian measure, and can also be sat in June 2018.

Weighting of Pure and Applied in AS and A Level
Both A Level Mathematics and A Level Further Maths are weighted 2/3 Pure, 1/3 Applied - in the case of the single A Level, this is essentially 1/6 Mechanics and 1/6 Statistics.

AS Further Maths is weighted approximately 50/50. 50% of the content is prescribed and focuses on pure mathematics - Andrew explained that this had come from end-users such as universities and industry, who prefer confidence with pure mathematics over applications. There is a small amount of prescribed mechanics (simple and damped harmonic motion for application of differential equations). AS Further Mathematics offers more opportunity for pupils to study Mechanics and Statistics in more depth without completing the full A Level FM.

Topics appearing on certain papers
Papers will become more like GCSE, where any topic can be examined on any paper. This has been done to avoid fragmenting the A Level and providing pupils with a more holistic experience. Emphasis should be on developing a body of mathematical knowledge rather than learning particular topics for particular papers.

Textbooks
Textbooks are being produced by Cambridge, Oxford and Hodder, for (hopeful) publishing next spring. These depend on accreditation of specification materials. AQA plan to only endorse those that produce a full range of texts for Maths and Further Maths.

Practice papers and resources
AQA plan to produce comprehensive teaching guidance and a route map for teaching the new specifications. There will also be two additional sets of specimen papers, one after accreditation and one secure set early 2018, and a set of topic tests and an eLibrary of resources.

Peter Mattock's session at #mathsconf5 (Concrete Approaches to Abstract Mathematics) was great and gave me a lot of takeaways, so signing up for this one as my first session seemed like a no-brainer. I wasn't disappointed; this time, Peter looked in depth at fifteen topics that can lead to misleading assumptions by pupils, and challenged us to come up with examples that "broke the mould".

We started with a little game; Peter asked us to come up with examples we could use to teach particular topics, such as a solvable linear equation or a diagram for teaching parallel line properties. I could sort-of see where this was going, but I decided to embrace the task as it was designed and go for my first idea for each example. We were then awarded points for the predictability of our answers - I scored a whopping 17/20 on my first go with some completely textbook examples. Peter then asked us to improve these examples to come up with a problem set that would score zero. I've neatened up (but not censored!) and scanned my notes below, with the original example on the left and improved examples on the right.

Happily, I'd predicted a few of these as I was working through them. The issues around linear equations in particular caused considerable consternation with my Year 10s a few months ago - they seemed perfectly happy to solve 2x + 5 = 15, but were really confused by 15 = 2x + 5. Many of them decided that they preferred to re-write the equation if it was "backwards" before solving, which worked fine when dealing with all positive terms, but re-writing something like 15 = 5 - 2x caused some of them even more issues, with 2x - 5 = 15 being the most common error. 

Personally, I think I sometimes avoid these more complicated problems, particularly when trying to boost confidence with pupils who struggle with mathematics, but this is something I need to work around for myself - pupils need to see more "unfamiliar" examples, particularly now we're entering the realms of a new, less predictable GCSE. 

Peter highlighted the importance of being aware of limitations on our own thinking; while I had predicted some of these assumptions, I hadn't thought of tilting a simple parallel line diagram or presenting a right angle in a non-standard orientation from the off - this is something I would address later on, often in the context of problem-solving or multi-step questions. However, maybe there is room for this when introducing a new concept, possibly presenting more examples of the right angle L-shape but in non-standard orientations, and specifically pointing out to pupils that these are all right angles.

As I promised my first #mathsconf7 blog before the en d of Monday, I thought I'd better crack on, which means turning pages of tiny scribbled notes on an old Osiris pad into something a little more coherent.

Much like @taylorda01, I've spent a lot of time in Leeds over the last decade (three years of university, followed by two years travelling up and down nearly every weekend, followed by a further five years living there and working in Yorkshire) and have never been to the Royal Armouries, but I concur with him that it worked well as a venue, and the unrelated event going on in the Leeds Dock area certainly added a bit of a party atmosphere every time I walked from New Dock Hall to the main Armouries building.

I thoroughly enjoyed the keynote from Mike Askew, which was a mini-workshop in itself. Mike set up the address by highlighting key recommendations from research into mathematics teaching, evaluating each before seguing into a discussion of problem-solving models. I particularly enjoyed the way he completed his list before presenting the level of evidence for each, reminding me that I have a duty to think critically about what I'm told and not accept things because "an expert" has recommended them.

The recommendations each offered significant food for thought, and although there was nothing new here, it's nice to be reminded of important principles at the start of a day like this.

1. Space learning over time
I completely agree with this principle, and many teachers and planning documents subscribe to this notion. One point on this was a comment from Mike that seemed to imply that "current mastery ideas" and spaced learning are incompatible - I would disagree, and wholeheartedly believe that a workable scheme contains elements of both. I detest use of "mastery" as a buzzword; as many commentators have pointed out, "mastery" is the way that good mathematics teaching was done until introduction of the National Strategies, and isn't new or revolutionary. For my money, a good scheme of work should allocate large blocks when introducing a big new concept, but that doesn't mean that this concept will not be revisited in other contexts later on - i.e.  when working on area of 2D shapes, include problems requiring pupils to multiply fractions and decimals as well as positive integers.

2. Interleaved examples and problems
This is something that I don't feel most GCSE textbooks have got right. Most exercises are presented with a few examples at the top, then a large chunk of problems, requiring teachers to carefully guide pupils through the exercises and making independent learning quite difficult. The Edexcel series of A Level textbooks on the other hand have excellent worked examples with full explanations, and often include pointers in exercises referring pupils to the specific example that might be useful in that case - it would be good to see more of this earlier on in pupils' education.

3. Combine graphics with verbal descriptions
4. Connect abstract and concrete representations
I jotted down a couple of key points from this linked pair of recommendations; firstly, Mike suggested that graphic representations should be accompanied by verbal descriptions rather than lots of written text, keeping the image clean and allowing for discussion. Secondly, Mike pointed to evidence that ​the thinking that understanding develops from concrete to abstract is flawed; young children are capable of engaging in abstract mathematical thinking (a point that tied in nicely with my third workshop choice, "A Golden Age?" from@dannytybrown), while adults and more advanced learners don't get rid of concrete representations completely.  I've seen evidence of this in my secondary practice (see this blog on completing the square), and Mike highlighted an example from his experiences as a university student when tackling higher-level concepts - despite being able to follow an abstract procedure, he had limited understanding compared with peers who could ground these concepts in concrete imagery.

5. Quizzes
Mike described the American process of setting "non-threatening" or low-stakes quizzes, both as a pre-assessment at the start of a unit and to re-expose pupils to concepts later on. Pre-assessments seem to be more a part of standard practice than they used to be, so it was interesting to see during Mike's reveal of the evidence levels later on that there was low evidence that this actually increases pupils' attainment. It was, however, interesting to see that quizzes as a means to re-expose pupils to content did have significant impact.

6. Ask deep explanatory questions
Mike used this as a springboard for his discussion about problem-solving and reasoning. Without deep questions, pupils only really experience a shallow, procedural version of mathematics. He highlighted the idea of posing a question at the end of the lesson and allowing pupils to "sleep on it", with the suggestion that the brain will unconsciously continue to work on the problem.

Mike then discussed a few points around problem-solving and reasoning. This quote from ​Development of Maths Capabilities and Confidence in Primary School (DSCF-RR118) seemed to strike a chord with many people:

This is further elaborated in the document as follows:

Mike highlighted the implication that skills in arithmetic are not a predictor of problem-solving skills (and vice versa), then drew the distinction between direct and indirect objects of learning. The direct objects of learning are the topics we teach, and what pupils are most likely to identify when discussing their mathematics - "we did fractions this week". The indirect objects (can't be taught directly) break down into a list of three proficiencies:

1. Fluency
2. Problem-solving
3. Reasoning

​Mike then discussed the importance of ensuring that tasks planned to develop problem-solving capabilities don't "decline" into procedural or fluency-based tasks and that teachers should be aware of "taking thinking away" from pupils by immediately diving in to help when pupils get stuck. He offered the suggestion that slow thinking is accompanied by emotional discomfort and tension, and that we're almost hard-wired as teachers to remove elements of discomfort and tension from our classrooms - in removing this tension, we also remove the problem-solving element and the task declines to a more procedural activity. Pupils should be given long enough to think carefully about the problem and be allowed to struggle - but crucially, should not be allowed to think unsuccessfully for long enough that they get fed-up and disheartened with the task.

We should also think carefully about the sorts of problems we select for our pupils; this is what Mike called the ABC of problems.

1. Authentic problems
These are problems that pupils could conceivably encounter in their current circumstances - Mike offered the example of "plan a school disco". The issue here is that authentic problems require authentic solutions, and realistically, we tend to "find a way around" such problems without engaging in much deep mathematical thinking. 

2. Believable problems
​These are problems that pupils could believably encounter at some point in their lives. Mike highlighted the issue with predicting such scenarios, and what's believable or relevant for one pupil may not apply to others in the class; constructing such problems is very difficult.

3. Curious challenges
I loved the terminology used here, and immediately thought of Don Steward's work. Rather than invent contexts and problems which may not be relevant to pupils, we look instead to provide them with challenges that hook them in - these may or may not be related to a "real-life" scenario. Mike offered a particularly excellent example that I'm looking forward to using with pupils:

Mike also offered a couple of rubrics for planning and developing lessons to enhance problem-solving skills. The first of these was around the teacher's duties during such activities:

1. Choose the tasks - we use our expertise to select tasks as above;
2. Set them up - give the tasks a narrative or story;
3. Orchestrate discussion of solutions - this is where the teacher should be most active during a problem-solving task, and focus should be on the mathematical content behind the problem, rather than just hearing voices. The teacher's job is to look for the mathematically significant answers and draw these out.

The second plan offered was how to get pupils to engage with this discussion process. Mike cited a study looking at pupils' reactions to discussion in upper primary - pupils thought it very important to have their say on the mathematics they had discovered, but frequently did not see the importance of listening to others. Suggestions included:

1. Repeat and revoide - ask a pupil to repeat what another has said;
2. Rephrase - ask a pupil to explain another's answer in their own/different words;
3. Build on - ask a pupil to add to a point raised by another;
4. Agree/disagree - ask a pupil their opinion and give an explanation.

Following Mike's address, we all engaged in the usual "speed-dating" activity to share resources. I'd brought along a simple idea I nicked from a colleague last year - a double-sided plastic mat with single integer values. It's a gimmick, but an engaging one - I've used it for quick-fire questions and ask pupils to fold their mats to show the correct answer - the kids seem to love it! For example, you could ask pupils to fold and show you a square number:

Vicky shared a great idea for a hands-on approach to introducing angles in parallel lines; she described how the DT department has constructed a solid model of the parallel lines diagram from coloured Perspex, with two angle shapes tha slotted in. Pupils then used this to discover where angles were the same and developed rules from this.

Kate (@KateTarno) shared her department's idea of attaching clear, raised Perspex to the tops of pupils' desks. These then functioned like mini-whiteboard paint, and pupils were encouraged to use these for working out. She also explained how worksheets could be slotted underneath the Perspex, both saving on reprographics costs and providing an easy way for pupils to alter their work when working on diagrams.

Unfortunately I didn't get my third date's name and have subsequently lost the paper resource, but she shared a tangram-type template including most quadrilaterals, great for making 2D shape properties a little more interesting.

My workshop choices for the rest of the day were as follows:

1. Avoiding Misleading Assumptions (Peter Mattock​)
2. 10 Things You Should Know About the New Maths A Levels (Christine Andrews, Andrew Taylor and Gary Wing)
3. A Golden Age? (Danny Brown)
4. More Interactive Problem-Solving Models (Matt Dunbar)

Individual blogs on each of these will be coming over the next week or so!

While the Internet is possibly the best thing that's happened to teaching and education in the last thirty years (think TES resources, Twitter, independent sites, quick communication channels), it also comes with a minefield of copyright issues. A couple of weeks before I left my last job, an email went round to all staff reminding them of copyright issues with downloading and using images from the Internet - and I used to be as guilty of this as the next person. When planning a lesson, it's very easy to just Google and pull in an image from anywhere to jazz up a PowerPoint - accessing images from all over the world means that you can easily illustrate a point about real-life applications of the mathematics pupils are studying. However, with the prevalence of online resource sharing, it's more important than ever to make sure that images used and redistributed are actually done so legally.

We started with a quick refresher, then we had a go at the first activity here, comparing one rod to another. We looked at links both ways, so wrote 6w = d first, then w = 1/6 d. It took a lot of annotating and re-explaining for them to completely get the idea, and I think I need some more problems along these lines to make sure they are really confident. The links to later work on solving equations and inverse operations are really clear though!
​
Next, we had a look at situations where the two Cuisenaire expressions weren't the same length. I wanted them to focus on just comparing part to whole here, so we only really talked about y = 5/6d, but at some point we'll go back and explore d = 6/5y further. This links so well to mixed numbers and improper fractions, as well as later work using proportional reasoning and scale factors.

We then had a go at the second task (scroll down on previous link), getting them to write statements to compare different sizes. I need to develop this further, because the level of difficulty ramped up too much for some of them. The couple who got it straight away were happy to continue to find their own expressions, and developed some quite complex ideas!
I'm definitely adding Cuisenaire rods and the NRich environment to my repertoire - the pupils are finding them really engaging and they're a good stepping stone on the way through algebra tiles to abstract representation. Just got to keep tweaking these lessons and make some more resources now!

A question that one pupil had was about time - namely whether the analogue 12-hour clock was Imperial and the digital metric. His reasoning was that, similarly to feet, inches and ounces, the analogue clock was "old-fashioned". I put him straight, but it did get me thinking about the idea of decimal (base 10) time - it's always an interesting discussion when you're teaching factors and multiples to ask pupils to think about why we use base 12 for a clock and why there are 60 minutes in an hour.
​
To satisfy my curiosity, I went Googling and read a very interesting article that confirms that decimal time was considered and briefly summarises the principles behind it. If you're teaching units of measurement soon, it's worth a read.
​
(Image: By shakko - Own work, CC BY-SA 3.0, commons.wikimedia.org/w/index....)

So the restaurant I'm at has red and green cards,  but no amber...

Yeah, tonight's is a bit of a placeholder post, as I've been busy, followed by asleep (post-half term lurgy), followed by actually out of the house for a friend's birthday meal.
​
Sometimes switching off is the best thing to do! I'll make the last three posts particularly good to make up!

Putting the kettle on.
Reading random education news articles, getting cross and closing them halfway through.
Organising several appointments with estate agents, solicitors and dentists.
Checking my emails.
Ringing an old colleague for a chat.
Actually doing some washing up even though we still have clean plates.
Scrolling through the #mathschat hashtag on Twitter.
Typing the first few words of a blog, then deleting them again.
Investigating "interesting mathematical dates" - there's nothing great this year, but I enjoyed 08/02/16. 04/04/16 is quite pleasant too.
Napping - because I'm always tired at the moment.
Annoyed that I can't be bothered to make a new resource to blog about.
Tea-drinking.
Idling away the time by watching a couple of episodes of "The 100" - enjoying very much.
Not doing anything even slightly meaningful with my evening!
Glad that I've somehow managed to turn my brain mush into something resembling a blog post.

(Normal service resuming tomorrow)

​Image credit: By Alexandre Normand from San Francisco, United States - Procrastination (No Wall Uncovered VII), CC BY 2.0, commons.wikimedia.org/w/index....

I prompted them towards writing down "sentences" for the blocks we built together - so "brown + red = orange". Once I'd written a couple out in full, someone suggested we just use initials, so this became t + r = o (brown is "tan" in Cuisenaire language - yeah, that threw me too).

​I gave them some square paper and got them to build some more expressions equal to one orange (i.e. 10). They would have happily done this all lesson if I'd let them!

A few pupils had gone for 10 white blocks make one orange, but we had a big mixture of notation, including 10(w), 10 x w, w x 10 and just 10w. Interestingly, no-one wrote down any incorrect notation when simplifying (like writing w to the power 10, which normally happens if I do algebra without manipulatives). We quickly got to grips simplifying the rest of their expressions too.

​Next up, we had a go at the task I'd written (here) - they found this really straightforward, but weirdly started introducing 1s in - so writing 2g + 1r instead of 2g + r. We had a quick chat about that.

A couple of really interesting things came up here. Firstly, the four-part equation (5th line down) seemed really natural to most of them, and they were comfortable with the fact that if A = B and B = C then A = C and so on. Secondly (and it's boxed because we talked about it quite a lot), the last two equations came from the 1st and 4th rod pictures - there's so much scope here for further work on equations!
​
(Header image: By Celcom, CC BY 3.0, commons.wikimedia.org/w/index....)

It turns out it's not quite as simple as I thought; there's definitely plenty of scope for practice using factors, multiples and systematic working with this one.
​
​Essentially, it comes down to:

In cases where the length and width are co-prime (have a highest common factor of 1), it's quite easy to spot.

It's interesting to explore the links between the 3x2 grid and the 6x4 grid. Multiplying the length and width by 2 has the effect of multiplying both the HCF and the LCM by 2. If pupils connect that grid size = HCF x LCM, there's an interesting discussion to be had here around length and area scale factors.

I also really like the connection to prime factors. It's easiest to see on a bigger grid - we can split the cut on a 12x9 into 3 lots of a 4x3 cut, creating a building block for other sizes - so you could ask pupils how many crossings in a 16x12, or 20x15, or just get them to find other grid sizes that fit with this "family".
​
If you examine the 4x3 block further, you can see that it's made of two 2x3 blocks, one a 180 degree rotation of the other. We can view the 2x3 as a base building block - we put two together to give the 4x3 block, then fit 3 of these along the length of the rectangle to complete a side length of 12 (see the diagram, it's much easier on there!). This gives a lovely visualisation of the product of primes for the side length of each shape.

Primary, secondary, college or uni - which was your favourite?

Sixth form was fantastic, because I spent most of my time doing Maths, Physics and German - three of my favourites by the time I finished Year 11. I really struggled to pick my A Levels, but I'm happy with the choices I made now. University was my favourite, though, because I dropped German and picked up Philosophy in my second year; bizarre as it sounds, I really enjoyed going to lectures and coming out not having a clue about what we'd done!

Which of your teachers do you still admire?

​My maths teacher in Year 7 still sticks in my mind - he did nothing but algebra for an entire year, and most of us could solve quadratics by the end of the winter term. He was proper old-school, and we worked in complete silence all lesson - I really enjoyed just being able to get on with my work!

My GCSE maths teacher was a big influence in deciding to become a teacher; as a class, we'd not had a lot of success with good teachers, and we'd played up something rotten for his predecessor. He had a fab dry sense of humour and really knew his subject, and encouraged me to do Maths at A Level.

Which lessons do you still remember?

Triple Science lessons at Secondary were great fun - Chemistry was a bit of a riot (my teacher once ate glue when we asked why it said "non-toxic" on the side), and our Physics teacher used to happily tell us we were all going to fail completely if we didn't sort our acts out.

About seven of us decided to pick Latin as a GCSE option, and we were taught in the smallest classroom ever (essentially a cupboard) by the most batty but fantastically intellectual teacher. She was a real stickler for old-fashioned schooling, and we got so much homework every week.

What are the main differences between the classrooms you were taught in and those you work in now?

They all have interactive whiteboards - I remember the first one appearing in one of the maths rooms while we were in sixth form, and it was a guarded treasure!

Technology is great in the classroom - a few examples I can think of include my mini-webcam, which I use to photograph and display pupils' work, my interactive whiteboard and the maths tools on Promethean Studio, which are great for demonstrating constructions, and the growth of excellent revision sites like Hegarty Maths and YouTube tutorials, which mean that pupils can access a teacher's help outside of lessons. Then there's the fact that I can communicate with and get teaching ideas from the community of maths teachers on Twitter, via Staffrm and the TES resources section - it's much easier to share resources now that I imagine it was twenty years ago.

Despite all this, I still use the "normal" whiteboard in my classroom every day. I tend to create a PowerPoint/Flipchart for most lessons I teach, but these are very basic and used to scaffold the sequence of activities for the lesson. While there are some beautifully designed electronic resources out there, I steer away from anything that's very step-by-step, because I feel like it interrupts the flow of the lesson - it becomes more about me delivering a lecture than responding to what pupils are saying or thinking. Most of the time, I pre-write any examples I'm going to use, so I can choose them carefully to sequence what I want pupils to discover, but I don't pre-write solutions - that's done by hand, either annotating on top of an electronic resource or on the standard whiteboard I have by my electronic version.

I'm the same with books - a couple of years ago, a friend gave me his old Kindle with over a hundred books already loaded on. I've picked it up once and really didn't get on with it - I found the experience of not being able to turn the pages of the book really offputting. Since then, I must have bought and read another thirty-odd paper books, all of which require storage space (see small selection in header photo!). The advantages of electronic books are apparent, but I still can't quite take that part of my life completely online either.
​
It'll be interesting to see how technology and education develop together in the future, but I'm still convinced that some learning has to stay rooted in a physical environment.