Order of Operations

Order of operations used to get right on my wick. I used to hate teaching it.

Mainly because I did it really badly. Just by showing it and going ‘but it’s a convention! It’s needed for other stuff!’

Pedagogically I was all fur coat, no knickers.

The problem with the order of operations isn’t the operations themselves, but it’s in their reading of the maths in the first place. Naturally, students read from left to right, but really, those more experienced will look at the problem as a whole and prioritise their attention to the bits they’re going to do first.

I liken this to looking at a photograph. We don’t look at the top left corner of an image and examine each pixel to the top right, repeating for each row.

I demonstrate this with a photograph of Mont Blanc.

in2white1

A screenshot of a section of the largest photograph in the world, linked below

The Largest Photograph In The World
The about section is fascinating as well, and well worth pointing students in that direction for those so inclined.

Zooming right in on a bit of the sky in the full screen version of the image, I slowly scroll right for a bit until I’ve made my point, then go back, nudge down, and start scrolling right again as if I were ‘reading’ the photograph like I would read a text.

in2white2

A zoomed in patch of sky. Not that you’d know

This is clearly a daft way to look at a picture, so we have to zoom out for the full effect.

We notice the mountains first. We look at the big stuff for the overall feel of what’s going on and then we look for more detail, such as the little cable car.

This is how we ‘read’ maths. Big stuff first, then details.

I use GEMA as a reformed BIDMASer.

Group
Exponent
Multiplicative
Additive

The groupings of either brackets or fraction lines etc. serve as highlighters in my increasingly tenuous mixed metaphor. (Or like a little man on the mountain with a big sign going ‘Oi look at me’ dragging your focus to him first.)

I show the following calculations, starting from the bottom:

33
3×3
3+3

We have addition, then repeated addition, then repeated multiplication, all with two threes. These are all linked by an operation, giving the values 27, 9 and 3 respectively, so in order of size we get the EMA in GEMA. This is contingent on students recognising the links between multiplication and division, and addition and subtraction.

Even then, this seems a bit flimsy when we consider mixtures of operations, for example two threes related in division has a lower value than two threes related by addition. Likewise if we had numbers less than one, the order of size could be different, and that’s without introducing more complex functions. (If you took inverses of the three examples used above though we’d still get the cube root of three > three divided by three > three subtract three.)

The three examples above aren’t meant to be a catch-all explanation of what’s happening, but a way in for students to gain familiarity with the convention.

Students also become more familiar with reading mathematics, and its difference to reading language, which is a valuable habit for geometry, proof and general revision later on in their maths learning.

I later give examples of how I would answer something more complex like the example below, dealing with negatives, a fraction line and a vinculum. This also starts to embed doing something with the quadratic formula instead of painstakingly transcribing it into a calculator, which makes me dry heave.

20170407_212654

Even without variables, there’s a lot to get confused with here if you’ve only a superficial fluency in your order of operations.

The aim with order of operations is to build up to procedural fluency, as it isn’t a mathematical concept, but a part of mathematical literacy.

Without a thorough grounding in the order of operations, students are going to find it gets in the way of doing maths, much like negative numbers and the foundations of algebra.

I don’t know if this is the optimal way to introduce the order of operations, as it depends on a lot of prior knowledge about the nature of operations. I’m also wary of introducing more metaphor when maths is heavily abstract as it is, but I have found this way of introducing it to be effective.

Advertisements

Teachers Are A Finite Resource

During observation feedback I was once given a list of things I hadn’t done that would have demonstrated ‘rapid and sustained’ progress. Thankfully, another member of my department stepped in and asked what I should prioritise, as to fit that in I would have had abandon something else.

Everything in the lesson was valuable so instead it was suggested I could mark every book after every lesson to cut down on AfL time to hone in on ‘rapid and sustained’ progress.

(‘Rapid and sustained’ always reminds me of the violent aftermath when I drank quite a lot of tap water in Asia. I spent two days in the foetal position on my bathroom floor.)

This suggestion is seductive. Lessons have finite time and schools have finite money, but marking every book every day affects neither of those things, so crack on.

When there’s no ‘extra’ (and that’s before getting on to actually having less), it’s teachers’ time and conscience picking up the slack. More meaningless data, more detailed feedback, more personalised learning. Even Sisyphus’ boulder rolled back down the hill sometimes.

It isn’t sustainable (“well it’s not about you, it’s about the children”), and more exhausted teachers is leading to more burnout, more quitting and more crap teaching as teachers teach to survive the hour/day/week.

It was reported this week how much unpaid overtime teachers do, but the hours aren’t the problem. It’s the intensity. At university I was a supervisor at a branch of that big green coffee shop, running the whole of Saturday from 7am-7pm. Being on my feet for twelve hours with nobody to cover my break was less tiring than teaching five hours straight.

Planning lessons is intellectually hard work (and bloody enjoyable), and classroom teaching requires a level of alertness, constant reaction and adaptability more so than a lot of other jobs.

Plenty of professions work upwards of 60 hours a week, but if they need to pee they can go to the toilet. If you’ve got kids coming to see you at break and lunch for detentions, or for help, or if you’re on duty you can’t. That parched throat stays parched.

Friends I know work in theatre and they say eight shows a week isn’t sustainable over a long period of time. There’s no proper time to rest. Two-show days with a matinee and evening performance are endurance trials, where they’re on full alert for a high-period of time. Similarly, we’ve a family history of truck drivers and they need to be well-rested and alert, and their employment is set up in a way that prioritises that.

Even with a full timetable teaching would be sustainable if teachers had supportive systems in place, and if after a day of teaching you could talk to another adult about something other than a new admin task to be done, some spreadsheet, the anxiety of where we’re at with the new GCSE or how you’re not keeping up with Kafka’s marking policy.

Yes the holidays are brilliant, and consistently I’d get a holiday illness for the first few days, unable to really do anything other than sleep. Then there’s time to catch up on the marking, be a bit of a human being and then get back to it (although there are the holiday revision sessions to get through).

I only teach the occasional whole-class lesson in my current role, and I’d forgot how much energy it needs. It’s only been two months since I was a classroom teacher on a full timetable but I had completely forgotten the gyroscopic whirlwind it creates. I now have the mental energy to actually think about teaching maths properly, which should be a large part of an early career teacher’s job, not an afterthought.

Professionalism in education can be seen as: having your paperwork up-to-date instead of engaging with effective CPD; doing six million things by decree instead of pesky moaning, questioning impact or discussing the best way to teach your subject; evidencing impact instead of having any.

Not having my seating plans in the ‘consistent format’ didn’t make me feel less of a professional, but I did when discussing the outcomes of our graded book scrutinies.

That’s before even going into multi-pen marking. As far as resourcing goes, it cannot be justifiable for ten people on upwards of £22k/yr to spend rare collaborative time wanging on about the colour of a bic.

The rapid and sustained stuff that takes teachers’ time, focus and energy isn’t helping them to help others. I’m now paid hourly, so it becomes very explicit what I’m spending my time doing in school. ‘If you could…’ soon disappears when it’s billable. I don’t need to go to meetings, so I don’t go to meetings.

Teachers are a finite resource and all the platitudes in the world won’t change that.

Methods: Solving Equations

To each his own. All roads lead to Mecca. There’s more than one way to skin a cat. We’ve got plenty of ways to describe people being able to find their own path to the same place, and the path of least resistance to a solution is something massively valued in mathematics. Elegance and efficiency are worthwhile aims, but not all methods support progress to higher levels of maths.

My new role as a school-based tutor means I do lot more close reading of student maths as it happens, and on the whole I’ve been trying to let students use their own methods to cut down confusion. No more ‘whack it in a box’ for anything to do with proportionality and multiplicative reasoning unless the student’s got nowhere to start.

20170216_112744

The Standards Unit box is a go-to for conversions, percentages, etc.

I used to think it doesn’t matter what they do if it ‘makes sense to them’. Teach them 3 ways to do division and let them pick the method that they like, understand or feel most comfortable with.

I’m not convinced anymore. We should be teaching the methods that allow students to access as much of the curriculum as possible.

All teachers have pedagogic opinions on what’s more effective in maths. I love a grid for most things and I find chunking heinous. I think discussing these things with people who don’t agree is one of the most interesting and intellectually stimulating aspects of being a maths teacher. Others disagree. There’s a lot to argue about for a subject that’s often said to be unique in virtue of its definite answers.

So equations.

3x+6=18

The language used varies from teacher to teacher. We could ‘balance’. We could ‘get x on its own’. We could ask ‘what’s 1 x?’.

Some people describe processes I’m convinced come from a random word generator, like ‘float and ping’, ‘crossing the river’ or ‘gouging the onion’. (That last one is a lie)

One of my tutees solves equations like this:

student-method

They start with the unknown, then follow BIDMAS (another thing for another day) until they get to the ‘answer’ of 18. Then they work backwards with inverse operations to find the value of the unknown.

I thought this was brilliant. It was accessible. It linked with something they already knew with the BIDMAS and the student felt confident with it. It feels a bit like dividing fractions with ‘keep, flip, change’, but I’m there to get them through an exam (again, another thing for another day).

Then it dawned on me that this wasn’t going to work for much beyond the most simple one-step equations. This method is not going to support progress in linear equations, never mind beyond.

3=30-3x

Aside from the cacophony of threes, the negative coefficient can become problematic, as does what to do next in my tutee’s method.

330-3x

Much less obvious for the student to unpick.

This BIDMASsy arrows thing could still work if you rewrite the equation from 3=30-3x to 3=-3x+30.

rearrange-3-3x30

But 30-3x being equivalent to -3x+30 is a big leap for lots of kids. I seem to spend my life reminding kids subtraction isn’t commutative, and now I’m saying that thing with a negative symbol can be written either way round?

The only way I approach this is to think of collecting terms, as below, but then this depends on students being able to navigate the links between negatives and subtractions, which for weaker students adds another dimension of difficulty to solving equations instead of fitting in with what they already know.

numbers-as-like-terms

I’m not even going to start on students ending up with (7+4)-(3+6) from BIDMAS

The other way to rewrite the equation would allow the student to crack on with their arrows too:

rearrange-33x30

None of this is wrong per se, as it would enable the student to solve the equation, but learning BIDMASsy arrows for one thing, and then “adding/subtracting unknowns to both sides” is using two different methods for one question. It feels clunky and all over the place. “If it’s negative you have to add to both sides before your start” is only a whisker away from “gouge the onion” in terms of meaning if it doesn’t fit into established understanding.

Students learning methods for every case is ineffective and makes maths more difficult in the long run, as it downplays any flexibility of thought and we end up with hundreds of baseless ‘rules’ with no context.

When I teach solving equations I have post-it notes on a metre stick and do lots of flailing if we don’t apply the same operation and magnitude to each side.

I’m calling it ‘Slapstick for Learning’, and it’s coming to an INSET near you soon.

I want them to be focusing on the truth of the equality instead of what the unknown is. When we first introduce solving equations students can often solve them by inspection.

2x+5=11
“Yeah but it’s 3 though”

I don’t think this is being able to solve an equation though. It isn’t enough. They can solve that equation.

By focusing on the equality,”Is this still true?” becomes the question rather than “what do I do?” or “What’s x?”

4

Is it still true?

7x

Is it still true?

This way, students can practise some substitution, and then when we get to solving inequalities and we need to know if our sign needs changing it’s not another inaccessible idea. It’s all linked.

By manipulating unknowns on both sides at this stage students will be more prepared for unknowns on both sides, or even to solve minefields like:

19=4-3x

In this case, the negative coefficient of x, the negative solution and sometimes even the fact the 19 is on the left side of the equals sign can cause students to stop dead in their tracks and we haven’t even got to equations with unknowns on both sides of the equality yet.

If we know we can do things to that equation that keeps the equality intact, students can now focus on what option would be useful. By framing it this way they can start to have a go at manipulating the equation to see where they get to, and when they might need to go in a different direction, but this too relies on prior knowledge, by being comfortable with algebraic notation and the effects of operations on expressions.

In the first equation, 3x+6=18, I can discuss with students whether or not they would factorise first and divide, when would they add the constant, when wouldn’t they, give me an example of an equation where it would make sense to factorise first, give me an example where it wouldn’t, etc. By approaching teaching equations in this way, students can begin to reflect on what they could do when they get an equation, which is the first step towards what they will do. This is problem solving.

Both fluency in manipulating algebra and the standards unit box diagram allow you to tackle problems across a wide-range of content in algebra and proportionality.

When we talk about students being flexible in their approach to solving equations, I take that to mean their algebraic manipulation. They can preserve the equality of the equation to get to where they need to be and to discern the most efficient way, even at the level of simple two-step equations like 3x+6=18.

That principle of equality becomes essential for solving harder equations and if students are secure with it, solving inequalities and changing the subject becomes an extension of their previous learning rather than ‘another topic’.

In the long run, that’s better for students’ problem solving skills, self-efficacy and resilience. Learning 3 superficial methods and picking one isn’t.

problem-solving

“We’re now going to have a lesson on problem solving.” (Image source)

Questions that increase in difficulty one niggle at a time are great for this, and Dave Taylor‘s website, increasingly difficult questions, is as much a planning tool for honing explanations and concept-building as it is a bank of quality exercises.

I’m sure there’s lots of things I teach in a way that allows students to answer questions and solve problems at the level of challenge they’re at then, but might cause problems later on, particularly at AS/A2 level.

Maths teachers of year 11s up and down the country are getting tactical on what they do between now and exams. When is it alright for a quick and dirty method? I want to say never, but realistically I think sometimes there’s a trade-off for time.

Of course, two days later I taught a paired session and I completely changed my mind on this when it came to expanding brackets, but that too is another thing for another day.

Relevance

The George Monbiot article in the Guardian today argues for a relevant curriculum to the 21st century that’s against routine tasks and favours exploration rather than indoctrination. It’s factual inaccuracies on the state of education and it’s solutions are something more articulate and knowledgeable minds can hash out more clearly, as the comments on it attest.

Has anyone actually had an education like it’s described there? All factory (eh?) and everyone getting the same thing? I certainly didn’t in the mid-noughties at school.

But I did at Taekwondo.

As a fat gobshite nerd, I definitely had no business being there. It was utterly irrelevant to me, my life and my background.

I began learning Taekwondo because our posh neighbours went and it meant I had to stop playing out three times in the week while they put on their white suits and hit things. I wanted to go because I thought I’d become a power ranger without the zords.

When I got there, it was walking up and down a room doing the same thing for a lot of the lessons. Literally marching up and down a hallway doing kick after kick. Make it better. Make it more precise. Make it faster. Make it sharper. You can’t learn that fancy spinning kick until the simple kick is perfect. Not good. Perfect.

I think about it a lot when I’m in the classroom. “They’ve got it move on.” No, that’s not enough. Have they got it well enough to underpin what comes next? “If they can do three they understand.” I don’t consider three a warm-up.

I did this for eight years, and passed my black belt exams at 18. I am fiercely proud of that, more so than any academic qualifications. The black belt was harder. Academic stuff? Well, I’d be reading anyway so I might as well get something out of it. The black belt? Discipline, focus and challenge.

If I were to spend my teenage years following my dreams I’d have still gone, but I would have stopped pretty quickly. If I had someone telling me repeated practice was damaging, or saying I ought to be learning something more relevant, I wouldn’t have stayed there.

Why does it have to be a traditional Korean martial art? What use is that? What situation in life calls for a perfectly executed ap chagi?

Did it lead to a crushing of my precious adolescent creativity? I don’t know. Every person I know seems to have a different understanding of what creativity actually is, but I managed to do alright in fights, even without my glasses on. When someone’s foot is launching itself at your head, you’re grateful for the practice. I’m not sure if not getting battered is seen as applying a set of techniques in a new context ‘creatively’ or not, but it was the most formative experience of my adolescence.

As a child, my curriculum would have been pogs and Pokemon cards if I’d have had anything to do with it. If I’d had a ‘relevant curriculum’ I might have agreed with my parents who said ‘people like us don’t go to university’, but I had teachers and instructors who were too busy teaching their subjects to worry about their relevance. We were taught about Balinese dance, why Hemingway’s short stories are so effective and universal truths about triangles. We were given cultural capital.

Students are in about five hours of lessons a day. The rest can be used to learn things that are relevant. For those five, students should be learning things they won’t get anywhere else.

Any call for a ‘relevant’ education that students can be ‘engaged’ with is a call for our kids to know their place and accept the limits of their worldview.

Notation

“Why are you teaching formal notation to year 7 anyway? When you’re problem solving do you actually use it? Why should our students?”

I saw the teacher from the classroom next door grin at me, goading me into responding. This comment came from a consultant during our department briefing while I was discussing what I was putting on the end of unit assessment.

If biting your tongue was an Olympic sport I’d be making the podium in Rio. We’ve had consultants galore this past 10 months. I have a draft folder full of bitter ripostes and wry lists of utterly daft advice.

I’ve nothing against consultants in theory, but I’d rather take my advice from active practitioners. There’s often a disconnect between what’s desirable and what’s possible for a full-time main-scale classroom teacher. Or as I unprofessionally put it: “You could say it’s ‘best practice’ to go t’moon duck, dunna make it possible. Get me rocket out shall I for lesson 3?”

To my mind, mathematical notation and formal solutions are how we communicate mathematics. One of our TAs is just as militant about formal arguments as I am. Use implication arrows, line up your equals signs, substitute variables in formulae for unknowns in the question.

It seems overly fussy for examples like the one below:

straight line eg 1

A simple angle question…

straight line eg 1 notation

… with a solution written formally.

I was honest with my mixed attainment class about why we were doing this upfront. “Yes, you might be able to do this by inspection (my inner pedant can’t stand ‘I did it by looking’ or ‘I know it’), but actually, if we get into this good habit, it’s going to make things like example 2 easier later on in the lesson.”

eg 2

Example 2

In truth, the reason I labour so much at this notation isn’t to answer questions like example 2. Linking algebra to calculations and geometry at any level is worthwhile. While this looks extremely procedural, it actually aids fluency when we get to the higher concepts in maths, like a musician’s scales or an athlete’s general fitness.

Teaching geometry through an algebraic layout has made Pythagoras much easier for my lower attaining key stage 4 classes to access. None of this ‘long side add, shorter side subtract’. It also subtly shifts the focus in solving equations from finding the unknown to preserving the truth of the equals sign, which is a vital change in mindset when students start manipulating algebra later on.

The emphasis on formal notation went through the entire unit, from simple angle questions to properties of polygons and angles in parallel lines.

Do I use formal notation when I’m solving a problem? Yes and no.

Sometimes I do. If I want someone else to read it? Too right I do. Mathematical notation is glorious in its economy when expressing ideas. My exam classes know full well their exams get marked by examiners, not detectives. Why not embed this early?

Sometimes I don’t. But I’m a maths teacher. I don’t teach maths teachers, I teach children. I want to see what they’re doing. I want to see where they go wrong. It’s impossible to ‘see’ thinking, but I can infer much more from formal notation than from different odds and ends all over 30 different pages.

Below are the documents for the first lesson of the unit, which ran through a lot of the basics of formal notation.

Slides and Notes – Using Formal Notation
Worksheet – Using Formal Notation

Everyone in the department taught this lesson to their class, and initially, I wondered if it would be too dry, but I underestimated our kids. They properly got stuck in.

My amazing TA wasn’t surprised though. “Kids don’t like faffing, they like knowing stuff.”

I’d take her advice over a consultant any day.

Winding My Neck In

When I got in yesterday from spending the day with friends I got stuck in a twitter spiral. I couldn’t stop reading each hot take on the Michaela story that had been in the news about lunches. I was instantly reminded of two other incidents that I’m describing incredibly vaguely as they’re not really my stories to tell.

Incident 1: A friend of mine works in property. His company was dragged to dirt in the newspapers for evicting a family just before Christmas. Cue outrage. “The company wasn’t available for comment” was seen as evidence of them not being able to justify what they had done. What had actually happened was a couple had been evicted from their business premises (not their home) after 6 months of missing rent and a refusal to engage with the company in how they could be supported. The company wasn’t available for comment because they’re incredibly small and incredibly busy. A small firm specialising in sustainable and ethical development is now an evil money-grabbing shadow organisation calling for a return to workhouses.

Incident 2: A different friend is quite prominent within a niche media circle. They get brutally attacked. Their silence on the matter across social media is an indictment of its fabrication because if the truth is on your side you’ve nothing to hide. They then address the matter on social media and this is an indictment of its fabrication because you’re clearly just oversharing for the attention. Death threats and abuse follow them and their family. The bulk of this abuse is from anti-bullying campaigners, who feel like their cause has been undermined. I struggle to move under the weight of the irony.

I can think of other examples of where I’ve been in similar situations with much lower stakes, and a lot of them are from within teaching. For a profession, we spend a lot of time slinging mud and then moaning that the place is ditched. I find myself doing it too, and it’s childish.

This need for us all to go full Jessica Fletcher on everything is wearing.

“If it’s [insert worst case scenario], then I am appalled and call for [sacking/putting in the stocks/some form of flagellation]”. This kind of comment doesn’t get enough flack as the most tedious thing on social media. We trot out axioms as badges of honour, usually related to personal experience as if that settles a matter.

Do I have an opinion on the Michaela lunch stuff? Yes.
Is it valid? Not really.

Yes, to know the ins and outs of the situation would be really interesting, but even with a forensic rundown of events, I’d still be basing my opinion on my personal experiences of the school and my ideas of what school socialisation should be. Just like how my friends are cynical money-grabbing attention addicts to those who feel comforted to take a moral high ground regardless of truth.

Any situation involving people is difficult, messy and nuanced. Reducing all that to a binary right or wrong cheapens us. If teachers can’t deal with each other from a starting point of charity in public, how will our students?

I’m taking my Columbo trench coat off, winding my neck in and putting the bloody kettle on.

 

Why Teach? A Self-indulgence.

There are problems with recruitment and retention.

It isn’t clear how these will be addressed in the foreseeable future, and ‘that’ advert, with its panning shots and relaxed teachers, seems to have ruffled feathers, which I get, but given how difficult schools are finding it to recruit, moaning about an advert feels like rearranging deckchairs on the Titanic.

This half term’s been quite challenging. There’s no breathing space in my days now my NQT time’s gone, and a maximum timetable is like a weekly Herculean task.

In my training I didn’t cry once. I cried three times in one day this last half term, and I’m not entirely sure why.

Other recently qualified teachers, both on twitter and in real life, seem to be looking around wondering what happened. Friends, family and even colleagues (!) have questioned why I’m still a teacher.

It’s unanimous. “Get out.”

Like the Amityville Horror, but with gluesticks.

*whispers*

*whispers* “Put… the lid… back on… or it’ll… dry… out…”

Student perceptions of teaching aren’t much better.

“Do you need GCSEs to be a teacher?”
“You got all As and A*s at GCSE sir? You could have done ANYTHING.”

Capture

So, given just how often I seem to be justifying the choice in career to loved ones, I’m going to be a bit self-indulgent and lay out why I decided to start teaching.

NB: There seems to be a culturally pervasive idea that ‘a teacher’ is a type of person, rather than a description of their work. IMHO you’re not born a teacher; it’s actually a lifestyle choice. I don’t think we have a monopoly on socially conscious work.

When I first went to university I was going to study composition and the piano, then write music for a living. This lasted until I realised I’d have to practise 4 hours a day.

In the end I chose to study philosophy, and took electives in anthropology, music (non-performance courses) and theology. Educationally, I’d never been more satiated. I studied enough philosophy to get a single designation degree, and then studied more theology alongside it. I just liked learning stuff.

I’d never considered teaching. I taught piano for a bit during sixth form and I didn’t feel that ‘heal the world’ vibe other friends had. It was just less greasy than the chippy, and paid better.

I first considered maths during logic classes. I loved philosophical logic. It was like solving simple equations. My tutor wasn’t keen on my comparison.

“Dan don’t say that. It puts people off. We prefer to say it’s like word puzzles. People hate maths.”

I was livid. It hadn’t occurred me that saying something was ‘like an equation’ would put people off a subject.  I’d always loved maths. I did A level and found it really interesting, but I was always a ‘performing arts’ kid. Nobody else in the family had been to university before, and if I was going, I’d be going to do music and/or drama.

I visited a school and quite liked it, so I applied for a deferred PGDE in maths so I could teach English abroad for six months and then spend six months on the old maths enhancement course to sharpen my subject knowledge.

Both experiences made me want to teach even more. I loved my training, and I love the job, even though I sort of fell into it. Face first.

Teaching is the most intellectually stimulating job I can think of. In my four years of philosophy, there were no answers, just more refined questions. Then suddenly, I’m teaching mathematics and I can show you proofs.

You can find the area of a parallelogram by multiplying the length of the base by the length of the perpendicular height, but don’t take my word it, there’s reasoning you can follow to see why it works.

You’ve only got to look at the comments section of a website to see how even the simplest ideas can be rowed with, but you can’t argue with the area of a parallelogram.

It’s an irrefutable truth about the world that’s accessible to children.

I don’t have the vocabulary to explain how much that makes me fizz.

You also get to see how people learn day in, day out.  Strictly speaking I suppose you don’t as we can’t ‘see’ learning as such, but if you’re dead nosy (*surreptitiously raises hand*), you get to hear their ideas and shape how they change, which yes, might be part of a socially conscious idea to improve the world, but it’s also fascinating.

“What do you think it’d be like to live in four dimensions?”
“I don’t get fractions. I’ve never got them. I never will.”
“Are squares rectangles?”
“What’s the exterior angle of a concave polygon then?”

As well as that, working with young people, you see the whole gamut of human experience. Within 20 minutes a student’s story can break your heart, whilst someone else has made you howl with laughter. Their energy is a challenge, as is their lethargy.

We’re at the end of the half term break and I’m a bit more rested, so the bags under my eyes have receded back behind my rose-tinted glasses somewhat, but I stand by teaching as an attractive career choice.

And let’s be right, we’re teachers. We know this. We know the job has the potential for needing to beat people away with sticks.

If the pay and the ‘making a difference’ doesn’t sway it for you, then how about the variety, the life you see, the community you join and the intellectual stimulation of it all?

And it’s still less greasy than working in the chippy.

Supercalifragilistic First Time I Taught Quotient

When trying to express oneself it’s frankly quite absurd,
To leaf through lengthy lexicon, to find the perfect word.
A little spontaneity keeps conversation keen,
You need to find, a way to say, precisely what you mean.

Mary Poppins

I’m a bit of a pedant. Irritatingly so. In my defence I’m getting better. I no longer correct apostrophes in text messages from family and friends, or quibble over ‘there is’ being ‘there are’, but in my classroom, be precise.

6+ 1 = 7 + 2 = 9

photo credit: pedro via photopin (license)

Photo credit: pedro via photopin (license)

I correct ‘I done…’ to ‘I did…’ when my classes give explanations, and like a good worker bee I put keywords up on the board. Sometimes I put keywords up on the board. I’m not that consistent with it, as it’s normally an afterthought. It’s ‘best practice’ to highlight keywords with students. It’s ‘literacy across the curriculum’.

It isn’t.

The way I was slapping some maths words on the board at the start of a lesson, never to be seen or heard of again, does nothing to improve literacy. The only students getting to see them are those that get the date and title down in their books pretty sharpish anyway.

I was the Sunday Christian of literacy.

Two of my key stage three classes were learning decimal division last week, and I decided to change my approach. I taught them the terms ‘dividend’, ‘divisor’ and ‘quotient’. Actually taught them.

Capture1

With a bit of questioning on mini whiteboards I got students to identify the dividend, the divisor and the quotient. The example in the attached slides  progress through some question prompts for students. (found here: Dividend, Divisor & Quotient)

Capture2

In the last slide, I first asked students to tell me the quotient and remainder of the division calculation. I then used the same slide and asked them to convert the improper fraction into a mixed number. By changing nothing but the language I used in the question, students explicitly saw the link between fractions and division in a new way, and linked that to finding a mixed number.

I think there’s lots more to build upon with this, to the point where this kind of literacy opens up access to the subject in a much wider way than ‘students can read and understand the questions’.

Knowing a mixed number is the quotient and remainder of a division instead of ‘how many times the bottom goes into the top and write that big and then the rest write over the bottom again next to it’ is difference between knowing maths and knowing rammel.

In every assembly on bullying students are told how powerful the language we use can be, but I rarely take the time to explicitly teach the language of my subject, and then insist on its use in the maths classroom.

Those words are just as powerful, and it’s about time I taught them properly.

Using the Quadratic Formula: Homework

In year 11 at the moment we’re recapping some key algebra skills.
As a homework here are some collated exam questions.

Part 1 deals with simple substituting and solving:
Using the Quadratic Formula HW Part 1

Quadratic Formula HW Part 1

Part 2 involves setting up the quadratics from different context and are broadly differentiated with R being the simplest, followed by A and then G.
Using the Quadratic Formula HW Part 2 A
Using the Quadratic Formula HW Part 2 G
Using the Quadratic Formula HW Part 2 R

R: The compound shape is split up already

R: The compound shape is split up already

A: Using the area of a trapezium to set up the quadratic

A: Using the area of a trapezium to set up the quadratic

G: Setting up a quadratic using simultaneous equations and compound area

G: Setting up a quadratic using simultaneous equations and compound area

All sections come with a success criteria for clear feedback on particular skills.

Hey Teacher, Leave Them Kids Alone

Induction’s been officially passed, new term INSETs are done and dusted, and it feels like the holidays were an age away already. September’s here, and it’s time to get back into the swing of teaching. The timetable’s somewhat fuller, and there are less names to learn this time round, but it’s still equally nerve-wracking and exciting to be back at school.

When it comes to new year resolutions, mine are to stop over-planning, calming down a bit with marking and then making sure I leave work earlier than last year.

It seems counter-productive to actively try and do less this year, but it’s not. Last year, there were lessons where I was doing more maths than my students. That’s definitely counter-productive.

My new year 9 class were in their third lesson with me last week, (multiplying fractions) and I set them a textbook exercise. They were working in silence, and I gave them the chance to get their teeth into it, to focus. Not having some gre’t lummox faffing about asking you questions and confusing you for 15 minutes whilst you got to get your head down and practised meant that when we stopped, I could really start to question what they had understood.

“Why was question 4 different to question 5?”, “Which question was harder, question 8 or question 14? Why?”, “Write a question you don’t think you could solve this way, and suggest why not.”

If you’ve got no hook to hang your hat on, you’re just chucking questions around.

The breathing space for them allowed them to focus on what they were doing instead of what I was doing. The atmosphere was calm and the students were really proud with what they had achieved, which in terms of building confidence might allow them the security to attempt more involved problems applying their skills.

When it comes to building in routines and getting students to really hone a skill, unsolicited silence as they concentrated on getting through the task was a wonderful surprise.

If students aren’t working on a task long enough to run into a problem, they certainly aren’t going to have the time to dig themselves out of it.

I don’t need to micro-manage my students every second in my classroom and write it down, I don’t need to slave over every single thing my students write in their books either.

This year, I’m going big on routines, and sometimes that routine will involve doing a lot of similar questions quietly.