Teaching Musings

How did you get into teaching?

I once upon a time was a student at the school I presently teach at, Tiffin School, before embarking on my 4-year degree at Oxford University, where I was at Worcester College. My 4th year dissertation was rather fitting given my current profession, using machine learning techniques (basically statistics) and a whole load of semantic analysis to automatically surmise mark schemes from actual GCSE Biology textual student answers, courtesy of Edexcel.

Upon leaving university I spent a year working for an investment bank in Canary Wharf, coding trading algorithms for Bond Traders, which sounds way more interesting than it actually was. It wasn't very long until I found this line of work somewhat unedifying (the classic "disillusioned city worker"!), and when my old project supervisor let me know of a new project for which I could be a PhD student, or to use the Oxford vernacular, 'DPhil student', I leapt on the opportunity. During my PhD I became extensively involved in teaching, both in delivering classes and giving tutorials to some visting Princeton students at my college. It was hard work, but I really enjoyed working with students and putting together resources. In the last year of my PhD, when considering my future options, I contacted my old school (Tiffin) so see if I could spend a week of holiday there to observe some lessons, and perhaps teach a few too; a request gratefully granted. I loved it, and the school offered me a GTP place (now rebranded as 'School Direct'). I chose GTP over a PGCE because I was quite keen to have my own classes from the get go and due to my prior university teaching experience. Teaching is very much a profession that I chose rather than fell into; I left Oxford with fond memories, but excited about the job that lay ahead. In my 4 years teaching I have not regretted the move from either banking or academia to teaching for a second, and highly recommend it as a career! I would be delighted to discuss with anyone interested.

How do you go about planning a lesson?

I generally think about the following factors when putting together a lesson.

  1. Exposition: How well does exposition of new material enable students to see the whole picture? Is it too prescriptive of a particular method? Or in the opposite extreme, does the reliance on weaker students reasoning through every step from scratch mean they don't have a clear sequence of steps they can follow through? In general, I try to reduce theory to its basic essence that tries to minimise what students have to 'remember'. Take for example expanding two brackets; a method such as 'FOIL' (First-Outside-Inside-Last) might seem like a sensible method to teach for weaker students, but personal opinion is that it does not build understanding, and causes problems as soon as they encounter more than two items in one of the brackets. Meanwhile, the idea of "multiplying each item in the first bracket by each in the second" is not prescriptive of the order of items in the expansion, generifies to multiple items (and even multiple brackets) and most importantly, minimises what students actually have to 'remember'. Another SK4TL gripe of mine is using the '0th term' when finding the constant term in the nth term formula of a linear sequence. Suppose we had the sequence 2, 5, 8, 11, ... Students could identify the '0th term' is -1 and hence get the formula 3n - 1. But would it not be much better if students could appreciate that 3n would give the sequence 3, 6, 9, ... and think of the -1 as a 'correction' to get to the desired sequence? It builds understanding of how the 3n term functions in their formula, rather than following a blind series of steps to get a formula. This principle of 'adjustment' also nicely extends to other types of sequence, e.g. quadratic.

    In addition to trying to teach in a way that promotes understanding, I sometimes like to think about how students 'internalise' concepts. This might be how students visualise a concept (e.g. for addition/subtraction involving negative numbers, picturing movements on a number line), but more generally how they 'see' the maths. With weaker students I've seen them being able to arrange y = x + 2 to y - 2 = x without difficulty, but somehow y = x + 2 to y - x = 2 presents more difficulty (possibly considering a division by x). They can't 'see' the syntactic structure of x + 2, i.e. as + being a binary operator with two (commutative) arguments, and instead internalise the expression as x and +2.
    There's also much to be said about the extent to which questioning of students is used during exposition, which I will not discuss here!

  2. Structure: Many of my early lessons were plagued with bad structure (and still often is!). Looking back at some of my older materials, there wasn't always enough opportunities to assess student's understanding before they embarked on independent exercises, other than particular students targeted through discussion; this is why you'll now see 'Test Your Understanding' slides consistently through my slides just before a main exercise. My other common problem was/is to give insufficient time to the main exercises after exposition/mini-exercises. When I update resources, it is usually because I'm trying to cover too much theory in order to have a sufficient mix of content in the exercise.
    I am otherwise very nonprescriptive on lesson structure and think the three-part lesson is a helpful heuristic but is certainly not something I always adhere to.

    In addition to how to structure a particular lesson, I often spend more time thinking about how to split a topic up over multiple lessons, and how best to scaffold. For straight line equations for example (we first teach this in Year 8), I always start with a lesson establishing the relationship between a line and its equation (with some plotting), then one or two lessons on gradient, and only then progress on to the full y = mx + c. It's often through teaching the topic a number of times (and getting it wrong!) that I get a sense of the best ordering, and even 4 years into school teaching I'm constantly having to evolve how I do things.

  3. Assessment: An obvious one this! Check out for example the idea of 'coloured cards' on the Lesson Ideas page. If I was to highlight the single biggest risk in my own lessons when going into 'teaching auto-pilot', I would say it's allowing the weakest students to slip under the radar and not having a full sense of what they understand at the end of the lesson. I find I have to make a concerted effort!

  4. Differentiation: Note to teacher trainees - it is a myth that the only way to explicitly demonstrate differentiation in a lesson is to have separate activities for different ability learners. I've never been a fan of the "Some learners will, most learners will, all learners will..." model (partly because it can often be so contrived), instead trying to make all core content as accessible as possible, with some 'stretch' objectives for more able students. For the 'typical' lesson, I think it's mostly a case of having an awareness of learners strengths/weaknesses, ensuring earlier questions of activities/exercises are accessible to lower attainers, targeting questions appropriately and dedicating more individual time helping individuals struggling. With regards to 'extension' questions, try to avoid too 'open' questions. Extension does not equal 'more ponderous questions', and even able learners like some clear direction. UKMT problems are particularly perfect in this respect, particularly if you can find questions relevant to the topic.
    Curiously I've found how I 'market' such extension questions makes a huge difference to students' willingness to engage in them. When I started teaching I called extension questions 'Conundrums', and students took little interest. Then I instead used a N symbol as the question numbers ('N' in Wingdings!), referring to them as 'killer questions'. Result: opposite problem of having to ensure able students have done some 'core' questions before wanting to launch themselves into them straight away!

More on 'internalising' maths

I mentioned above how students' understanding of mathematics and their ability to further apply it, is maximised when they are able to effectively 'internalise' it, i.e. picture it. Internalising also helps students do certain aspects of a method (or all of it) mentally, in most cases reducing the risk of error. Here's a list of examples, some more obvious than others:

  • Negative Numbers: Picturing the number line, and the resulting movements when adding or subtracting numbers.
  • Algebraic expressions: Repeated verbal use of phrases like "3 lots of x2" when referring to "3x2" to both (a) avoid students doing crazy stuff when they try to add said terms and (b) avoid substituting a value incorrectly because they square last rather than first.
  • Linear Sequences: Once students have initially identified the kn term (say 3n), they can literally picture the 3 times time table superimposed on the original sequence, thus being able to mentally see the adjustment required.
  • Fraction/Decimal/Percentage Equivalences: If students think 1/5 is 50% or 1/4 is 40% then they don't have a mental appreciation of the magnitude of numbers. They need to appreciate that 1/5 is closer to 0 than 1, while 50% is halfway, and thus have a sense that such an answer is not sensible. Fractions can obviously be pictorially represented; students might struggle with "what is 1 divided by 1/5" until you phrase it like "how many fifths of a pizza are in a whole pizza".
  • Percentages: Similarly, students need to build an appreciation that say 1.1 is just more than 1, and thus multiplying by it just slightly increases it (subsequently making the mental jump that this would be a 10% increase). A full appreciation of this avoids them having to write out a first step where they do 100 plus or minus the percentage change, and be less likely to confuse different methods within percentages (e.g. calculating a value after a percentage change and finding the percentage one amount is of another).
  • Angles: Students often find it hard to picture where the alternate angles are in a diagram when there are more than 3 lines. Identifying the line connecting two parallel lines, the looking at the two ends helps students to see this Z better.
  • Solving Equations: The age-old idea of balancing scales.
  • Changing the subject: The idea that they are peeling off layers around the subject by undoing them, thus having a better appreciation of the order of steps.
  • Straight Line Equations: A constant mental note in the back of their mind of the correspondence between points on the line and the line's equation. When finding gradient using "change in y over change in x", mentally looking at the increase/decrease to the numbers rather than subtracting the two (which makes it error prone when the y values are say -2 and -5).
  • Trigonometry: I like to emphasise the role of sin/cos/tan as 'functions' (or 'number machines' at the first stage they're taught) which inputs an angle and outputs the ratio between the appropriate pairs of sides. Students therefore have a better appreciation that these functions are always evaluated on an angle. I don't see the process of finding an unknown angle as different from the forward process, more that students have the inverse trig functions added to their arsenal of 'changing the subject' skills. So with say sin(theta) = 3/4, they can make theta the subject by applying inverse sin to both sides of the equation.

And so on...

Creating Teaching Resources

When creating a set of resources of a new topic, my stages are:

  1. I always first check (a) the specification (b) if GCSE/A Level, look at a large variety of exam questions and compile where necessary (c) textbooks, (d) sometimes the TES website and (e) my UKMT database for Maths Challenge/Olympiad questions relevant to the topic (UKMT now has their own free tool to compile questions by topic). This gives me a full sense of what I want to cover.
  2. As above, I then carefully think about the mathematical story I wish to tell, giving particular attention to the best place to start when first introducing the topic, whether a motivating question, or a simplified question that they can use existing knowledge to solve. I consider both how the topic is segmented into lessons and the best ordering of these, but also within a segment of exposition, how best to gradually scaffold the learning.
  3. I consider how much material I want to cover before I have a 'mini-exercise' to test a particular concept.
  4. For exercises, it's vitally important you provide answers if you want others to use them! I put all answers within my slides under clickable boxes. This means I never have to fumble around looking for answer books.

13 Tips for Teacher Trainees

  1. Don't underestimate quite how big the difference between your 'subject knowledge' and 'subject knowledge for teaching' is. I naively started teaching thinking that my moderately decent subject knowledge meant that I could concentrate on the less mathematical aspects of teaching. In hindsight, I can categorically say that I WAS COMPLETELY WRONG. Do you have a good understanding how different learners internalise different concepts in maths? Are you aware of the variety of different ways a topic can be taught and the potential pitfalls of each? Are you aware of the various minutiae of mark schemes? Are you aware of common misconceptions that students might have and how similar topics in maths might contribute to these? Do you know how to structure theory in a larger meatier topic that best scaffolds understanding?
    One of my first responsibilities for an 'exam class' was taking on a Year 11 bottom set in my NQT year. While they ended up with respectable results, seeing the breakdown of their loss of marks in their actual GCSE exams was a highly educational experience for me, in terms of realising where I had neglected certain subtle parts of the specification, avoided certain emphases, or just plain taught some topics badly the first time around (e.g. non-right angled triangles) which I had to rectify come revision time. For some students, some of whom where 1 or 2 marks off the A* threshold, my SK4L blunders undoutedly made the different between one grade and another. Know your official specification: don't rely just on your department's scheme of work and past paper questions you've seen come up. 4 years into teaching I still feel I'm learning an enormous amount about SK4T, and benefit from regular conversations (and sometimes debates!) with colleagues in this respect.
  2. The most important aspect of your lesson is the extent to which students are learning. Experiment with various activities and such, but reflect on how these tell the mathematical story of the topic you're teaching. Choose activities based on their teaching merits rather than nature of the activity itself.
  3. Don't try to cover too much in your lesson. I think the danger (based on my own experience, and observing others) is generally covering too much rather than too little. Having to rush later portions of your lesson is always a bad thing.
  4. Don't talk too much. The biggest risk of my earlier lessons (and to be honest, I still have to be conscious of today) is to spend far too much time in the exposition/dialogue part of the lesson, leaving too little time for students to do independent exercises. This is quite frequently due to covering too much (as above), and sometimes because students ask you too many questions!
  5. Use 'positive' behaviour management: I'm no behaviour management expert, but I've found where the most disillusioned or disruptive students have been on task is where I've used praise to reinforce good work ethic/behaviour. It's been less effective where there's a sense of 'battling' against the student (the latter which is sometimes easier said than done to avoid). A few ideas that I've found helpful:
    • I have a 'GB' list (GB standing for 'Good Boy') on the board for students who work well during exercises or make a particularly good contribution to class discussion. This usually entails a 'merit' at the end of the lesson if they remain on the list. While 'Good Boy' is intentionally jokingly patronising in the wording, students love having their efforts recognised. And I've found students, whose attention is liable to wander, saying things like "Look sir, I'm working well!". I've even extended its use to A Level students in an 'ironic' fashion. Except not so ironic apparently, because they love being on the Good Boy list.
    • I've found the phrase "I see [X] is working well", sometimes followed up with "[Y] is working less well", while circulating the classroom, often helps.
    • My sanction system within lesson is to have a warning for first offence (on a 'W' list on the board) followed by detention at lunchtime. Warnings might be for talking when I'm talking, leaning back on their chair, slacking during independent exercises, etc. I often write a name on the 'W' list without any verbal recognition, and continue with the lesson; I find this draws less attention to negative behaviour and maintains the flow of the lesson.
    • ALWAYS follow up on sanctions. If you've told a student they've got a demerit then don't issue it, or they don't turn up to a detention and you don't notice, your authority is immediately undermined.
  6. Try out new styles of activities and have fun, but you may want to play it safe in formally observed lessons.. At the end of my teacher training year, where I was visited by an external assessor, I had to teach two lessons. One was to my mixed ability Year 8 class on 'combinations', and the other my top set Year 9 class on using a completed square and roots to sketch a quadratic. My first was much more adventurous, where after some initial exposition, students had to move about the classroom in pairs to solve different puzzles (each puzzle led to another stuck up in a different part of the classroom). In between that lesson and the Year 9 one, my internal mentor had to meet with the external mentor to discuss my general progress throughout the year, where the external mentor also fed back on my first lesson. Immediately before my second lesson my internal mentor briefly gave me a private 'pep talk' to relay the feedback from the assessor: that the activity format had some flaws and there were certain teaching skills he still wanted to see on show. Prior to the second lesson I had previously been more worried about the second lesson in that it was too 'straight down the line', based partly on (in hindsight, questionable) advice by my external mentor that my lessons had to be episodic, involve a variety of different types of activities, and so on. Thankfully the assessor loved my second lesson, where the more traditional lesson format had allowed me to concentrate on core teaching skills. Assessors don't necessarily want an all-singing all-dancing lesson; they want to see that you can do the 'basics' well.
  7. Imagine that a observer was hovering around your weakest students in each lesson: What would they say about how they've engaged in your lesson? I often have to make a conscious check that their understanding has been checked at least by the end of the lesson.
  8. Learn your students' names as soon as possible: I print photos out from the school database, and if I'm feeling adventurous, arrange these on to a printed seating plan as an extra aide memoire. I find that (a) this improves my behaviour management if I can refer to individuals by name and (b) I am less likely to avoid choosing particular students for questioning because I can't remember their name.
  9. Be receptive to feedback: While I only have some experience of working with teaching trainees, I've heard lots of anecdotes of trainees through the years of trainees being adverse to feedback. And I certainly know that I was on occasions too defensive of my teaching during my training year. There's two possibilities when you receive a piece of feedback:
    • They're right: Some feedback you can only appreciate in hindsight. I remember my external mentor once critiquing the way I had explained completing the square to the class, and suggested the phrase "throwing it away" in reference to getting cancelling out the square of the second term in the square. I didn't really get it at the time, responding that I thought that was exactly how I had taught it. In hindsight I have a much greater appreciation that subtle changes in the language you use can make an enormous difference to the clarity of your exposition. He was right.
    • They're wrong: Sometimes teachers have completely different opinion on aspects of teaching. But most of the time, when you disagree with a piece of feedback, I think it's just better to swallow your pride and nod; maintaining your professional relationships with staff is important. Feel free to justify why you made a particular teaching decision, but don't overdo it, and be mindful of the manner in which you do so. In hindsight you might realise they're right. And they might be wrong, and you can do it your own way next year.
  10. Keep your files well organised: There's two ways to organise your files: (a) By year group, then by strand (Algebra, SSM, DH, Number), or (b) by strand only. I use the former as we often reteach certain topics in later years, but with a slightly different angle. Ensure all your resources for a topic are in a single directory, and if you need it in multiple places, create shortcuts. If you're working on slides from home, and gradually updating resources, it's incredibly important you have a designated place for your 'master copies', which you can guarantee will have the most up-to-date version (and upload straight to this location if you work on a copy on any other device). For me this is my local drive on my school computer (I have to propagate any changes to the teacher shared area, this website and TES!). I have a "Last Updated" box on my title slides in case I find two copies of the same resource.
  11. Send a nice email to parents if a student makes significant progress: Students love this recognition, parents like hearing something nice about their child, and any improvements are likely to be more enduring.
  12. Make sure students always have something to do: I tend to have 'backup' activities, not necessarily related to the topic at hand, which I can whip out in case I run out of questions (or if we're going through a test and they got little wrong, etc). Maths Challenge papers are good! If you have mini-exercises before the main independent exercises, ensure there's at least a second question so no one is waiting around (for my own resources, I'm referring to 'Check Your Understanding' slides).
  13. Don't give up! Sadly a number of teacher trainees have bad experiences and drop out before the year is through. If you're finding it tough, just keep in mind: (a) Your negative experience might be with the school you're at rather than teaching itself. (b) While you'll obviously have a greater number of lessons in your NQT, I actually found it easier, without the pressure of regular observations, writing lessons plans, collecting evidence and writing essays. (c) If your concerns are workload related, feel comforted to know that most others are in the same boat! While my workload has never decreased (although I make a lot of work for myself!), it feels more manageable as you become a more confident classroom practitioner and you build up a bank of resources to use.

10 Technical Tips for Creating Teaching Resources

While considerations such as mathematical content and the manner in which this is disseminated to students is absolutely key, I do think that the design, presentation and general professionalism of teaching resources can often be put by the wayside. Here's a few tips that I've found helpful in creating my own resources:
  1. I would personally avoid using proprietry software (such as Smartbook) to create your software. This causes problems in that (a) students will find it harder to access your slides should you make them available and (b) different schools use different software, and thus changing schools yourself may make all your resources void, not to mention that other teachers may not be able to make use of your resources. Powerpoint files are supported by all devices and there are a number of free viewers for non-Microsoft platforms. Simple as.
  2. Get adept at using the equation editing tools in Office. I admit I'm slightly fastidious when it comes to mathematical typography, including algebraic x's written as 'x', fractions written with a diagonal slash, e.g. â…“ (which leads to students encountering problems at A Level), etc. The biggest time saver in my entire teaching career? Alt + Equals. When in Word/Powerpoint/Publisher, this enters equation mode. You can type a fraction for example using a/b followed by space to complete the structure, and use bracketing to keep things together, e.g. a/(b + c) (brackets will automatically be removed). You can get many mathematical symbols by prefixing them with backslash, e.g. \pi (followed by space), \sigma and \Sigma for lower and upper case sigma, and so on. You can use x^(2+z) to get x2+z and use underscore for subscripting. Try for example: \Sigma_(k=0)^n (k^2) and x \in \doubleR. You can press the Right arrow when at the end of your equation to exit equation mode. I have a fuller guide/cheat sheet here: here
  3. If you're putting complicated diagrams in Microsoft Word, it's probably better to do the diagrams instead in Powerpoint, press the Print Screen button, open up Microsoft Paint (or your favourite graphics software!), Ctrl + V to paste, select the region of the diagram you want, Ctrl + C, then paste into Word. Sounds like a lot of steps, but I do it in seconds. Diagrams are a bit of a nightmare in Word, tend to render differently when viewed on different devices, parts of your diagram magically seem to get separated from the rest, all goes to hell if you column-ise your page, and so on. I've found switching to images saved me a lot of pain, and I simply maintain a separate Powerpoint file for diagrams should I need to modify them.
  4. Establish your own design aesthetic in your slides, if you intend to regularly share your resources with others. This might be a colour scheme, a distinctive title slide, etc. Such 'branding' might seem superfluous, but I do think there's something to be said for establishing your identity as a content producer. Have a template Powerpoint that you can copy when you create a new resource.
  5. Use the 'Curve' tool within the Shapes submenu within Powerpoint to draw angles, not the 'arc' tool.
  6. In Powerpoint you can set animations to be triggered by clicks to objects. I use this for example for my 'Click for Animation' buttons and all the revealable green question marks. Once you've added an animation to an object, open the 'Animation Pane', right click the relevant animation, select 'Timing' and then 'Triggers'. You can then select the object which triggers your animation upon clicking.
  7. To quickly copy an item, hold the Ctrl item when you drag the item. Holding the Alt key when drawing lines avoids the line 'locking' to other objects.
  8. Avoid animations which may unnecessarily distract students and are not relevant to the content, e.g. slides transition animations, unsubtle object animations ('fade' and 'wipe' should be your staple!).
  9. You can assign a 'link' to an object to another slide in your presentation. Suppose you want and first slide with different question categories, where clicking each moves the user to the relevant slide. Simply right click and object and select 'Hyperlink', followed by 'Place in this document'.
  10. Comic Sans is evil and must be stopped at all costs.

Teaching Methods I'm Not Fond Of

I've created a series of posters for teachers here.

What is your strongest and your weakest area of mathematics? What areas of mathematics do you find most interesting?

Maths Olympiad-wise, I'd say my strongest area is Combinatorics (i.e. counting combinations of structures and objects), and my weakest questions the horrid ones you find at the end of BMO papers which combine inequalities with geometry. I find Number Theory the most interesting field of mathematics, which studies the integers (whole numbers), such as primes and so on. It's particularly fascinating how in order to reason about the integers, we have to consider real and complex numbers, a sub-branch known as Analytic Number Theory. Perhaps my favourite proof in the entirety of mathematics is the proof that the probability that two randomly chosen positive integers are 'coprime' (i.e. share no factors other than 1) is 6/π2. While the proof is moderately simple, it touches upon a number of areas, such as the relationship between the product of all the primes and a summation involving all the positive integers, and remarkably involves the constant π, which you would ordinarily associate with circles.