It might be Friday afternoon with an often challenging class. Additionally there may be rainy and windy weather forecast – which, as every teacher knows, is the worst possible combination.

On top of that, the content is something students always find challenging. And, to be completely honest, it’s not even the teacher’s favourite topic.

Chances are, this teacher will spend their Thursday evening doing lots and lots of preparation. Despite what some politicians may say, it’s extremely rare to find a lazy teacher. Teachers will habitually expend vast quantities of their supposedly free time on getting ready for school the following day.

And thus, our teacher will stay up into the early hours, scouring the internet for the best resources they can find. They’ll be thinking about, re-thinking (and perhaps overthinking) what they’ll be doing, minute to minute.

This teacher will plan, and then re-plan, and then tweak and improve until they eventually have an all-singing, all-dancing lesson ready to meet the next day’s challenges. One that’s sure to make the lesson go smoothly and facilitate a positive, rather than negative learning experience.

Running on empty

What transpires the following day is an order of events that many of us will have gone through ourselves. The teacher wakes up tired and grumpy from lack of sleep. Having spent their precious evening (and a hefty chunk of sleep time) hard at work on their preparation, they’re simply not operating at their best come the following afternoon. Yes, the lesson is fully prepared. The teacher is not.

Oh, they know their stuff and they’ve done their homework, all right – but that’s precisely the problem. An exhausted teacher is never best placed for handling a challenging lesson or class.

Our teacher finds it hard to think quickly. Their judgment calls aren’t as good as they might be. They’re slow to respond to difficulties, and the lesson ends up embodying all of the teacher’s worst fears.

Teacher wellbeing matters too

Teachers can be very selfless people. They know perfectly well that they could earn more money for less effort engaged in some other profession. But they care about young people and believe that education matters.

Teachers want students to get the best possible start in life. To that end they’re committed to giving something back by playing their part in that. Where the children are concerned, it can sometimes seem as though no sacrifice is too much. As we hear so often, children only get one chance at their education,.

No teacher wants to be a hypocrite, challenging students during the day about the effort they’re putting in and demanding punctual returns of homework, only to then not do those same things themselves as part of their job. Schools exist for students, not for teachers – ergo, we must put students at the centre of everything.

Yet while some of those sentiments might be true, they can combine to create a toxic work environment for teachers. Of course, schools should put students and their learning at the centre of all that they do. But teachers matter too.

Schools are more than just learning environments for students. They’re also workplaces for both teaching and non-teaching staff, and their levels of wellbeing matter as well.

What’s unhelpful is seeing ‘teacher wellbeing’ presented as being in competition with students’ best interests. The question of ‘Who matters more – the students or the teachers?’ presents a false choice, since exhausted, demoralised teachers are never going to be in students’ best interests either.

Preparation in the round

In a sense, there was very much a lack of preparation ahead of that difficult lesson. The paperwork may have all been in order, and the lesson’s technicalities expertly considered – but conspicuously absent was any form of emotional preparation on the part of the teacher.

Teaching is an emotionally demanding, often draining occupation and virtually impossible to do well when running on empty. It’s not a selfish act for a teacher to prioritise their own wellbeing and sanity. Rather, we should view it as something that is beneficial for everyone.

Preparing ‘the teacher’ is just as important – perhaps even more so – than what we might traditionally view as preparing ‘the lesson’.

In practice, this ‘teacher preparation’ might involve visiting the gym, or relaxing with family and friends. Far from being trivial, spending time in this way contributes to important emotional preparation for the demands of the day ahead.

In an ideal world, all teachers would be able to complete the entirety of their paperwork by the end of the day. They’d then be able to head home and enjoy an evening of entertainment and/or restful social downtime.

Unrealistic aspiration

In reality, this can seem an unrealistic aspiration for many, calling for some hard choices. Is it wise to go in with a ‘good enough’ lesson plan, after an evening spent relaxing and an early night? Or better to produce a stellar lesson plan that’s then delivered by a stressed and worn-out teacher the following day?

There may be no right or wrong answers here – but seeing emotional preparation as being of equal importance to content preparation might at least start to redress a balance that’s tipped too far in one direction.

If we continue to prioritise short-term goals (the quality of the next day’s lesson) over long-term teacher wellbeing, then we shouldn’t be surprised if the rate of teachers leaving the profession continues to rise, to the point that it becomes no longer possible to provide ‘schooling’ in the sense we’ve become accustomed to.

Teacher wellbeing and common sense

The process of teaching draws on a complex package of skills and requirements. Teachers are never just preparing for ‘a lesson’ or even several lessons; we’re preparing to be around young people, with all the challenges and opportunities this presents.

Depending on the subject, some teachers may find lesson content emotionally charged and draining to teach. In some instances, we might need to work at being in an emotionally healthy place ourselves before being able to do a good job of that.

More broadly, however, we simply don’t know when a student will suddenly come to us with a personal question or problem they want to talk about. We’ll often say that teachers should always be available to be approached about anything – but do we perform the requisite work/rest on ourselves in order to prepare us for that?

Being as well-rested and refreshed as possible (even if that means making hard compromises with respect to lesson preparedness) will stand us in good stead to be maximally useful.

Even if it’s just taking some common sense steps to look after our own emotional wellbeing, we’ll be better role models and more supportive and helpful adults when students approach us – for whatever reason that may be.

Colin Foster (@colinfoster77) is a Reader in Mathematics Education in the Department of Mathematics Education at Loughborough University and has written numerous books and articles for mathematics teachers; for more information, visit foster77.co.uk

The post Teacher wellbeing – Students need good content AND rested teachers appeared first on Teachwire.

Look at the four pairs of simultaneous equations below:

Which ones could you solve by adding the equations together? Which ones could you solve by subtracting one equation from the other? Students may be unsure and not know how to decide.

The solution

Don’t worry about solving the equations just yet. All I want you to do is simply add together each pair of equations. And also subtract each pair of equations. See what you get.

By adding, students should obtain the following:

And by subtracting the second equation from the first equation:

Students may not bother to write the 0𝑥 and 0𝑦 where there are no 𝑥 and 𝑦 terms, and this is fine.

They may make errors, particularly when subtracting the negative terms – so for the subtractions, they may end up with the wrong answers shown in red below:

Writing out the difficult subtractions explicitly may help:

When does adding eliminate an unknown?
This happens when two terms are equal in magnitude, but of opposite sign
(e.g., 2𝑦 and –2𝑦).

When does subtracting eliminate an unknown?
This happens when two terms are equal in magnitude, and of the same sign
(e.g., –3𝑦 and –3𝑦).

Sometimes, it can help if students remember the following: When the Signs are the Same you Subtract.

Can you find a pair of equations where either adding or subtracting will lead to elimination of one of the unknowns?

An example would be 3𝑥 + 2𝑦 = 11 and 3𝑥 – 2𝑦 = 7.

The solution to all of these pairs of equations is 𝑥 = 3, 𝑦 = 1.

Checking for understanding

To assess students’ understanding, ask them to create four pairs of simultaneous equations of their own, two of which can be solved by adding the equations, and two of which can be solved by subtracting the equations.

They should label clearly which are which.

Colin Foster (@colinfoster77) is a Reader in Mathematics Education in the Department of Mathematics Education at Loughborough University and has written numerous books and articles for mathematics teachers; for more information, visit foster77.co.uk

The post Simultaneous equations – When to add and when to subtract appeared first on Teachwire.

They’ll likely be told something along the lines of, ‘Oh, you should go and see Mr Smith – his behaviour management is fantastic!’ Is this good advice?

In the event, our teacher makes arrangements with Mr Smith to watch him teach his most difficult class – which coincidentally, happens to be the very same class our teacher has been struggling with. Perfect! So she sits at the back of the classroom, ready to watch this ‘master teacher’ do his thing, pen poised to note down all those novel strategies, clever comebacks and aspects of body language she can later imitate back in her own classroom.

And… nothing happens.

The students all behave impeccably. Our observing teacher is impressed with Mr Smith’s skill, certainly – but what has she actually learnt? Not much. “He didn’t seem to do anything – they just behaved!” she says on her return to the staffroom.

Study the non-experts

It may sound counter-intuitive, but it’s actually hard to learn from the very best. If you want to be amazed and impressed by a magic trick, go watch the best magician you can afford tickets for. If you want to learn how to be a magician yourself, get inside those tricks and understand how they’re done, observing the best magician on the circuit will teach you little that will help you imitate them.

In fact, you’ll learn far more from seeing a less expert performance, even one in which mistakes are made. Catching sight of a playing card disappearing into the magician’s pocket, or the tip of a handkerchief poking out from behind their hand will instantly clue you in on how you might go about imitating – and ultimately even improving on – the performance you’re seeing.

If you want to learn the art of magic, it’s the second-rate magician who’ll be much more useful to you.

I would venture that something similar applies to observations of colleagues within school. To learn good behaviour management you have to see good behaviour management – not just good behaviour.

A crack in everything

You need to see things working and things going wrong, and what the colleague does in response.

You need to see problems bubbling up and being cleverly averted, as well as overt instances of bad behaviour being addressed appropriately. You need to see the cracks. As Leonard Cohen once wrote, ‘There is a crack in everything. That’s how the light gets in.’

You may well want to see someone who’s advanced further along the teaching path than yourself, but it’s perhaps best to not seek out someone who feels they’ve reached the end of the path and has sat down for a nice drink. Watching a colleague who’s still struggling a bit – though hopefully

succeeding to some degree – with a difficult class will give you lots more to think about, compared to watching colleagues whose attention isn’t on behaviour because (at least for the moment) it doesn’t need to be.

Timing is everything

Of course, those ‘superhuman superteachers’ who never put a foot wrong are mythical. All teachers struggle from time to time. When someone appears to be light years ahead of us, it might just be that we’re out of sync with their timing.

There’s a good chance that Mr Smith experienced struggles of his own with that class during the first few weeks of term, in the process of establishing routines and gaining respect. That would have been interesting to watch, for sure.

But now, with that behind him, perhaps all he has to do is maintain what he’s set up via small actions that won’t be immediately apparent to an outside observer. The risk of visiting his lessons now is that the observing teacher may end up honing in on tiny, relatively insignificant aspects of Mr Smith’s behaviour and try duplicating these in her own practice, when in all likelihood they’ll have little effect.

Those behavioural elements aren’t what make his lessons successful, but they’re the only things she can see in his lessons that she can take away with her. And even then, she may well come away feeling demoralised about how far below the perceived standard she feels she is.

Works in progress

It takes confidence to open up your classroom to colleagues when you know you’re far from

‘perfecting’ your craft – but it’s precisely the messy, ‘work in progress’ phase of classroom teaching that’s most helpful for others to see.

By contrast, watching the ‘finished product’ can be quite uninformative. An observer benefits from seeing something working, but also from seeing it go wrong, and then trying to tell why things went differently each time.

We need to let colleagues see us with all our imperfections, warts and all. In a way, this can take the pressure off those being observed. Rather than desperately hoping that everything will go 100% smoothly, and that you’ll give an ‘impressive’ performance, you can instead just aspire towards something useful emerging from the lesson.

This will allow your focus to be placed firmly on the students you’re teaching – where it should be – and not worry about how you’re coming across to observers.

Teachers will sometimes say to a colleague, self- deprecatingly, ‘You can come and watch if you want, but it won’t be amazing.’ That, though, is the point – an observer shouldn’t be looking for ‘amazing’. There’s little value to be had in being ‘dazzled’ by a star performance.

Learn, don’t imitate

Instead, as mutually supportive colleagues, we should want to learn from both our successes and difficulties. The important thing is for observers to witness something that will allow them to reflect on their own practice, and prompt them to consider aspects of their own methods they might approach differently.

Merely ‘copying and pasting’ another colleague’s techniques, however tried and tested they might be, is unlikely to work. All teachers are different, and we all need to find different ways to succeed in our different contexts.

That doesn’t mean we have nothing to share, however – far from it. Seeing what our colleagues down the corridor are doing will give us opportunities to consider different options. While observing, we have the time and space to watch the students and gain a different perspective on the classroom through their eyes.

Above all, we have the chance to learn from any mistakes made – whether they be ours or our colleagues’.

Colin Foster (@colinfoster77) is a Reader in Mathematics Education in the Department of Mathematics Education at Loughborough University and has written numerous books and articles for mathematics teachers; for more information, visit foster77.co.uk

The post Professional development – Why teachers SHOULD NOT learn from ‘the best’ appeared first on Teachwire.

Imagine that you ring up and order a 20-inch pizza to be delivered. When they arrive, they say they are sorry, they don’t have any 20-inch pizzas, but they have brought you two 10-inch pizzas instead. Is that OK?

Students may worry about things like the amount of crust, the shape of the slices, or how the pizzas would be shared out, but not realise that in fact two 10-inch pizzas contain only half as much pizza as one 20-inch pizza!

The solution

Rather than explaining the point, just display the image below, without any words, and ask: What does this have to do with the pizza question?

If the diameter of the blue disc is 20 inches, then the diameters of the red discs will be 10 inches. The total area of the two red discs is clearly much less than the area of the blue disc.

Since the area of a disc is 𝜋𝑟², where 𝑟 is the radius, Area ∝ 𝑟², or, since radius ∝ diameter, 𝑑, Area ∝ 𝑑². This means that twice as big a diameter gives a disc with four times as much area. The two red discs in total cover only half of the area of the blue disc!

Another way to think about it is to imagine stretching one of the red discs until it is twice as wide, but without changing its height.

It now becomes an ellipse, with twice the area of the original red disc. This means that the area of the ellipse is equal to the sum of the areas of the two red discs.

This red area is half of the area of the blue disc, because the red ellipse would have to be stretched vertically by another scale factor of 2 for it to become congruent to the blue disc. This would take another doubling of its area to match the area of the blue disc:

Students may think this is just a weird thing about circles, but it is actually completely general. Whenever any 2D shape is enlarged, for the enlarged version to remain similar to the original shape, it must be stretched by the same factor 𝑘 both horizontally and vertically.

Each of these stretches increases the area by a factor of 𝑘, so overall the area will increase by a factor of 𝑘².

Checking for understanding

To assess students’ understanding, ask them to imagine scaling up a sphere in 3D. What would happen to the volume?

Multiplying the radius (or diameter) of a sphere by a factor of 𝑘 will require three stretches this time, in order for the new shape to be spherical (i.e., similar to the original sphere). The three stretches, each with scale factor 𝑘, in all three perpendicular directions, will scale up the volume by a factor of 𝑘³.

Colin Foster (@colinfoster77) is a Reader in Mathematics Education in the Department of Mathematics Education at Loughborough University and has written numerous books and articles for mathematics teachers; for more information, visit foster77.co.uk

The post Area and volume – What happens to the area of a disc when the disc is enlarged? appeared first on Teachwire.

The difficulty

Fill in the missing numbers in the table below:

Students may be confused about what ‘significant figures’ means and round to 1, 2 and 3 decimal places instead.

Or they may be confused about whether to count zeroes or not, so they might think that 30.5449 to 2 significant figures is 30.5.

Or they may use incorrect ‘successive’ rounding when working out 30.5449 to 4 significant figures, rounding 30.5449 first to 30.545 and then rounding this rounded value to 30.55.

The solution

For each of these numbers, say which column the most significant digit is in.

30.5449
0.030 544 9
3054.49
5.3449
0.53994
534.99

This could be a quickfire activity, with flashcards and mini-whiteboards. Each time, all students need do is identify the column (ones, tens, tenths, thousandths, etc.) containing the digit worth the most.

It will always be the first non-zero column from the left. Students might incorrectly think that it is the largest digit – e.g., the 9 in 30.5449, even though this is actually worth the least.

To round to 1 significant figure, we round to this column. So, to round 30.5449 to 1 significant figure, we round to the nearest 10. To round 0.0305449 to 1 significant figure, we round to the nearest 0.01. To round 3054.49 to 1 significant figure, we round to the nearest 1000.

Write down 10 more statements like these, making the numbers as varied as you can.

Students can be allowed to abbreviate their sentences to avoid repetition, or you can just ask them to list the two numbers for each statement. Alternatively, you can download a printable ‘Significant figures’ task sheet from the link above.

Now look at what happens for 2 significant figures. To round 30.5449 to 2 significant figures, we round to the nearest 1. To round 0.0305449 to 2 significant figures, we round to the nearest 0.001. To round 3054.49 to 2 significant figures, we round to the nearest 100.

Students might need reminding that ‘rounding to the nearest 0.1’ or ‘rounding to the nearest tenth’ is the same as ‘rounding to 1 decimal place’. Write 10 more statements like these, again varying the numbers and using several different numbers of significant figures.

Now complete the table below. The answers are included here in red – a blank table can be found in the aforementioned task sheet.

Here, we have the number 247 406. Why would it be a problem If we change that number to 247 401?

This may be hard for students to answer, but the transformation 247 406 → 247 400 must be rounding to the nearest 100. However, 247 401 → 247 400 could be rounding to the nearest 100 or the nearest 10, leaving two possible answers.

Checking for understanding

To assess students’ understanding, ask them to create a table like the one above for their partner. Swap, complete, and then swap back and check.

Colin Foster (@colinfoster77) is a Reader in Mathematics Education in the Department of Mathematics Education at Loughborough University and has written numerous books and articles for mathematics teachers; for more information, visit foster77.co.uk

The post Significant figures – Help your students master the art of rounding numbers appeared first on Teachwire.

After all, human beings are complex creatures. The process of learning anything complicated can quickly become a dense, involved journey full of twists and turns.

We can’t simply and directly control another human being’s thinking and shape it in exactly the way we desire – which, on balance, is surely a good thing for preserving our sense of freedom and agency!

Is a curriculum linear?

We therefore shouldn’t try to operate as though learning is simply a straightforward process of incremental progress – like building a wall, where one brick is placed atop another, and then another, until the desired endpoint is reached. Learning is much more problematic than that.

There may be times when we have to go back and unpick something that we thought our students had already learned. In the process of doing so, it might seem that we’re not making much headway, yet taking this step could prove essential for our students’ long-term benefit. Teaching can sometimes feel like taking one step forward and two steps back, but that doesn’t necessarily mean we’re doing anything wrong.

I tend to agree that this is often the case, but I also believe that the ‘learning is linear’ mantra can sometimes have a disempowering effect when it comes to teachers’ planning. If the process of learning really is as complex and uncontrollable as it seems, then what can teachers possibly seek to do, other than turn up in the classroom, hope for the best and attempt to deal with whatever occurs?

What’s the point of carefully planning and sequencing the details of your lesson or curriculum if ‘learning isn’t linear’? Should we instead simply offer students a wide variety of ‘rich’ experiences, and trust them to make whatever sense they can out of what we provide them with?

Taking account of differences

I personally think that teachers are capable of achieving much more than this. Regardless of whether learning is indeed linear or not, one thing that definitely is linear is time. However ‘non-linear’ a given student’s experiences of a subject may be, their lessons are guaranteed to come at them one after another, in a specific order and sequence.

Students may well take different paths before arriving at a deep understanding of the subject, but each of those winding paths will necessarily unfold through time, with one event happening after another. It therefore seems to me that we don’t need to be apologetic about offering a ‘linear’ curriculum, providing we recognise that students’ movement through it may involve some circling back and revisiting of areas at a later date. As Pete Griffin observed, “Teaching takes place in time; learning takes place over time.”*

No teacher with more than five minutes of experience will naively assume that students will immediately understand and retain everything that they’re taught, with instant success the first time they encounter any given topic. The point of regular formative assessment is to try and remain in touch with where students are at each moment in time, so that we can support them as effectively as possible in moving forwards.

A one-size-fits-all approach clearly won’t make sense if it fails to respond to what the teacher sees in front of them – but I would also advise against going too far in the opposite direction, by exaggerating the differences between our students.

Avoiding linearity

One of the major ways in which students differ is in terms of their prior knowledge. This doesn’t necessarily mean that they’re on different ‘tracks’, and therefore need to journey through different ‘curricula’ – they might simply be at different positions along similar trajectories.

There might need to be more intensive support for some students, to help accelerate them along and build securely on what they know, but this needn’t amount to them taking a substantially different path. In fact, there are likely many more similarities between most students’ trajectories through any given subject than differences.

Postmodernists tend to talk negatively of ‘linear thought’, as if it’s something that’s old-fashioned and outdated. Instead, they talk highly of ‘non-linear’ thinking that’s based more on intuition and subjectivity. Yet I think it’s a serious obstacle to good curriculum design if we allow ourselves to conceive of ‘linear’ as always being a bad thing.

Carefully plotting out the knowledge students need to develop at different points, while taking serious account of the necessary prerequisites, is just sensible curriculum planning, and still seems like the best bet for ensuring that as many students as possible can progress successfully in their learning.

Of course, there can be no guarantees that Student X will definitely learn Content Y on Day Z, and never need to go back and look at it again. The whole idea of retrieval practice is built on the notion of forgetting. As teachers, we’re aware of how important it is that we return to previously-covered ideas so as to reinforce them and build on them.

Yet at the same time, it’s necessary for us retain a clear sense of the big-picture learning path we want students to take, if we’re to effectively plan for students’ progression.

Keep to the path

It might not be as catchy as ‘learning isn’t linear’, but ‘learning isn’t monotonic’ might better capture the intended point.

‘Monotonic’ progress is unidirectional – for example, always getting better each day. That might seem like an unrealistic expectation at first glance, but while there will inevitably be ups and downs as students move through the curriculum, zooming out should reveal a general progression along an intended path.

Planning for this doesn’t equate to making simplistic assumptions around ‘linearity’. It simply acknowledges that we have a clear goal in sight for our students, and that we’ve carefully thought through how best to help them get there.

When this is backed by effective formative assessment, we can rapidly identify where particular problems might arise for certain students along the way. That’s the point at which we can take the opportunity to circle back, revisit things and intervene.

Yet the overall picture is one of progress – in a linear direction, along a path we’ve planned for and prescribed as carefully as we can. That’s what we need to aim for.

*Griffin, P. (1989). Teaching takes place in time, learning takes place over time. Mathematics Teaching, 126, 12–13

Colin Foster (@colinfoster77) is a Reader in Mathematics Education in the Department of Mathematics Education at Loughborough University and has written numerous books and articles for mathematics teachers; for more information, visit foster77.co.uk

The post Learner progress – Don’t apologise for offering a ‘linear curriculum’ appeared first on Teachwire.

The difficulty – adding and subtracting surds

Look at these statements. Are they true or false? Why?

√12 + √3 = √15
√12 – √3 = √9
√12 × √3 = √36
√12 ÷ √3 = √4

Students may need calculators to be sure. Three of the right-hand sides are square roots of perfect squares, so students may recognise these integers (√36 = 6, √9 = 3, √4 = 2).

The multiplication and the division are correct, but the addition and the subtraction aren’t.

The solution

How can we be sure whether these are true or false?

We can show that √12 × √3 must equal √36, because squaring each of these expressions gives us the same value:

( √12 √3 )( √12 √3 ) =? √36 √36
( √12 √12 )( √3 √3 ) =? √36 √36
12 × 3 =? 36

After squaring, the left-hand side equals the right-hand side, meaning that √12 × √3 = √36.

Adding surds with a calculator

If we try this with the addition, we get a problem:

√12 + √3 =? √15
( √12 + √3 ) 2 =? √15 √15
( √12 ) 2 + 2√12√3 + ( √3 ) 2 =? √15√15
12 + 2√12√3 + 3 =? 15

So, we can see not only that these are not equal but that the left-hand side in a case like this is always going to be bigger (because of the extra 2√12√3 term, which must be positive).

o, √12 + √3 > √12 + 3, not √12 + √3 = √12 + 3.

Square rooting is sub-additive, which means that, unless either a or b is zero, √a + b < √a + √b.

The radical symbol √ does not behave like multiplication – e.g. something like 3(a + b) = 3a + 3b).

Square rooting is not distributive over addition like multiplication is.

However, we can simplify √12 + √3, by using what we have seen about multiplication of surds.

The number 12 has a square factor (4), and so we can write √12 = √4√3, and because 4 is a square number, √12 is equal to 2√3.

So, √12 + √3 = 2√3 + √3 = 3√3.

(This last step is just ‘counting in √3s’ and is analogous to collecting like terms.)

Students should check this on their calculators. We can simplify additions and subtractions of surds in this way whenever there is a square number that is a factor of the number being square rooted.

What would √12 − √3 be equal to?

This time, √12 − √3 = 2√3 − √3 = √3. It may look strange to write √12 − √3 = √3 (students may think it should be √6 − √3 = √3), but it is correct.

Checking for understanding

To assess students’ understanding, ask them to find as many surd expressions as they can that are equal to 3√5. For example, they could start with 4√5 − √5 and convert this to √80 − √5.

There are many possibilities, and generating lots of these is an excellent way to practise using these ideas. They could try to make the 3√5 as concealed as possible, and make some that look as though they might be equal to 3√5 but aren’t!

Colin Foster (@colinfoster77) is a Reader in Mathematics Education in the Department of Mathematics Education at Loughborough University and has written numerous books and articles for mathematics teachers; for more information, visit foster77.co.uk

The post Adding surds – What students get wrong about expressions with square roots appeared first on Teachwire.

It might seem like a natural thing to try and do, but it’s a trap that benefits neither teacher nor students, however well-intentioned it might be. Attempts at forced humour and slang will likely come across as awkward and embarrassing to ‘old hands’, though it should be noted that the young teachers in question are still attempting to build bridges.

The reason they’re doing it is to try and be approachable and accessible to their students, and ultimately have a positive influence on their students’ lives, which is difficult to argue with – but might there be a way of achieving these ends more effectively? Is it always wrong for teachers to want to be liked?

A natural impulse

At some level, almost everybody wants to be liked. Only sociopaths have no interest at all in what other people think of them, nor any desire to please anyone else. Given that it’s a caring profession, teaching isn’t exactly replete with sociopaths. Indeed, one would hope that those entering the profession do so at least partly because they have some degree of liking for young people, and wish to form positive, professional relationships with them.

While teaching involves interacting with all kinds of people – colleagues, senior managers, parents, etc. – teachers will typically spend the largest portion of their working day ‘alone’ in rooms full of students. In this sense, it would be surprising if teachers didn’t, at least on some level, want to be liked.

But often, this very natural, human feature is portrayed as a problem – a professional character flaw, even. Teachers, particularly those at the start of their careers, will often be told that if they’re at all concerned with being liked, they’ll be less effective at their job.

The reasoning goes that ‘wanting to be liked’ makes a teacher more reluctant to push their students to excel academically, and less likely to challenge them when their behaviour falls short – neither of which are ultimately in the students’ best interests. Moreover, worrying about being popular leads to low standards and expectations, so consequently, teachers are advised to simply not concern themselves with whether their students like them or not.

Instead, they should be as indifferent to this as they can possibly be, because it risks placing entirely the wrong focus on the job they do. The teacher is a professional who, just like doctors, police officers, elected representatives, business leaders or any other professionals, may not always be popular among those affected by their decisions. They will
often have to make choices that satisfy some higher, longer-term purpose, rather than merely pleasing the people around them in the moment.

Experiencing and remembering

And yet, I see this line of reasoning as an unworkable, and ultimately counterproductive approach, since it demands that teachers suppress their emotions in a job that’s often emotionally demanding. It’s impossible to truly care for someone in a strictly unidirectional manner without any concern for how they might feel about you. The kind of advice outlined above will merely serve to make the most caring teachers feel guilty for caring, and perhaps push some of them out of the profession altogether.

I believe a much better solution can be had from applying the insights of Nobel-prizewinning psychologist, Daniel Kahneman*, who draws contrasts between the two categories of ‘the experiencing self’ and ‘the remembering self’. The experiencing self is what you experience at the time, in the moment, whereas the remembering self is how you think back on an experience, from some unspecified point in the future.

What’s important is that these two ‘selves’ frequently aren’t the same, and can often have quite different – perhaps even diametrically opposing – aims.

Applying this to the problem of wanting to be liked, we could try focusing on the students you teach now as they will be some time in the future – say, five or so years from now. When they think back on their school days, you as their teacher and the circumstances of today, this week or this year, how will they feel about it then? Will they like you then?

Sometimes, students who have left school will pop back in to say hello to their ‘old’ teachers. Imagine if such a student were to say, ‘I actually never liked you when you taught me, because you were always saying I could do better, and you were always pushing me to improve my work and picking me up on things when I wasn’t trying. And that was really annoying. But I’m really glad you did, because I learned a lot in the end.’

Future perspective

In this case, the former student’s experiencing self didn’t like the teacher, but the remembering self did. Students are like adults, perhaps even more so, in that they don’t always possess the maturity required to discern what’s in their best interests – but that doesn’t have to mean we should actively not concern ourselves with what they think or want.
Instead, we can mentally fast forward to their future selves, and do our best to try and listen to what they might wish to say to us from the future.

Of course, this mustn’t become an excuse for ignoring everything that happens in the here and now, in the hope that it will be ‘worth it’ for some unspecified future life (‘No pain, no gain’). It matters that students get to feel safe and cared for at school, and enjoy their schooldays. It matters that their teachers feel able to present as human and approachable people who can be turned to in a crisis.

However, even when such situations present themselves, it still makes sense to sometimes think of the students we’re teaching now in terms of the people they will be as future adults, however hard that may seem at times. We should try and ask ourselves from time to time what we think they might have to say about us then, from that future perspective.

Perhaps wanting their future selves to think highly of us might help us to recalibrate what’s best for them in the here and now.

*Kahneman, D., & Riis, J. (2005) ‘Living, and thinking about it: Two perspectives on life’ from F.A. Huppert, N. Baylis, & B. Keverne (eds.), The science of well-being Oxford University Press

Colin Foster (@colinfoster77) is a Reader in Mathematics Education in the Department of Mathematics Education at Loughborough University, and has written numerous books and articles for mathematics teachers; for more information, visit foster77.co.uk

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Surely only the most socially inept and insensitive colleague would respond with the classic, ‘Well, they’re fine with me!’ – because what could that possibly achieve, other than make your colleague feel small?

Teachers want to be supportive colleagues, and will frequently recognise that ‘we’ve all been there’. This colleague has obviously had a very stressful lesson, is feeling low and needs some affirmation and support. It follows that we’d want to build them up, not knock them down – but what would you actually say? And how can you prevent your ‘supportive’ response from simply making them feel even more hopeless?

Blaming students

I find that supportive colleagues will tend to say things like, ‘Oh, that group! They’re exactly the same with me – they take no notice of anything.’ This seems like a nice thing to say. Perhaps the colleague no longer feels quite so much like a failure. It isn’t just them, because others have evidently experienced similar issues.

But of course, the natural implication of ‘It’s not you’ is ‘It’s the students’. And so the conversation will often descend from there into demonisation of ‘these students’. ‘What is it with kids like that? They have no respect for school, or for any authority whatsoever. Their parents are just as bad – they take no interest,’ and so on.

Is this OK? What if the colleague had entered the staffroom saying something more extreme, like ‘Those Y9s are vile!’ Would you challenge that way of talking about students? Or would doing so make you ‘unsupportive’, as if you don’t understand how hard teaching can be?

These kinds of ‘supportive’ remarks (‘Oh, I know! Tell me about it!’) can quickly gain momentum in a friendly staffroom. Everyone’s keen to show their wounds and how agreeable they are. That may make the colleague feel a little better in the short term, but fundamentally, I think it ends up being disempowering.

The message ‘It’s not about you’ is hard to separate from ‘There’s nothing you can do about it’. If those students are ‘just like that’, and it has nothing to do with your teaching, then what hope is there? You may as well find another school – or another career.

A safe space

People will often defend ‘letting off steam’ in the staffroom, and say that conversations around what I’ve called ‘demonising’ students are somehow therapeutic. The argument goes that the staffroom should be a ‘safe space’, and that so long as such conversations take place out of earshot of students or visitors, it shouldn’t be policed with regard to what’s said.

I’m not so sure. I’m no professional counsellor, but when I completed a short counselling course many years ago, it challenged my presuppositions. Previously, my assumption around counselling was that it was about being nice and patient, and accepting whatever the client said. It involved saying ‘Poor you’, and giving the person time to talk about their feelings so that they felt ‘listened to’ and affirmed.

While I’ve no doubt that at least some of those things can be important for a good counselling relationship, I also discovered that experienced counsellors are actually often quite ‘tough’ with their clients – even in ways that could seem cringeworthy to outside observers.

Mentors learn to be highly skilful at challenging people to make changes and take responsibility for their actions. This isn’t about blaming anyone – the point is that even when something is in no way your fault, that doesn’t mean you’re completely powerless to do anything about what happens next. Yet holding those two things together isn’t easy.

Rethinking ‘supportiveness’

I’m not saying we should all start acting like amateur counsellors, as that could do more harm than good. I do, however, wonder if the prevailing sense of what it means to be ‘supportive’ needs a rethink.

Many years ago, I was in a staffroom when a colleague entered saying something along the lines of ‘Those Y8s won’t do a single thing I say’. The collective reaction was to nod along sadly – but then one teacher glanced up from her marking, and shocked the room by looking straight at the colleague and saying, “You have to make them”.

What an awful thing to say! The atmosphere of the room instantly changed, and the colleague who had entered walked off, clearly annoyed, saying “Well, how am I supposed to do that?” It seemed at the time like a textbook example of a highly inappropriate response.

Many years later, when the ‘Won’t do a single thing I say’ teacher was retiring, he cited the other colleague by name – though not in relation to the incident I’d observed – as being the best and most supportive colleague he’d
met in his career.

Perhaps the comment was therefore made within the context of a trusting relationship, the marking colleague with the ‘inappropriate’ response having, in a sense, ‘earned’ the right to say what she did? Or maybe it was a blip in an otherwise more ‘supportive’ set of encounters?

In any case, it made me wonder whether we’re often more concerned with saying something that will signal us as being ‘supportive’ and make us look good, rather than saying something that may actually be helpful to the other person in the long run.

Time and tone

To be clear, I’m not advocating saying deliberately tough things to your colleagues in the staffroom. Indeed, the most useful conversations usually need to be held in private, rather than in front of everyone, and are perhaps best had later on, once the ‘heat’ of the moment has died away.

Yet even then, do we teachers ever share those tough messages? Or are we more minded to simply stick to nice, safe things that make us seem virtuous, but are ultimately unhelpful – even counterproductive? With our supposedly kind remarks, are we sometimes just enabling people towards defeatism, inaction and hopelessness?

Taking responsibility for dealing with things that aren’t your fault seems like an important aspect of being a professional. We must find ways of talking about strategies for handling challenging situations and learn from one another – while accepting that none of us are perfect, and that we all have struggles. Often quite similar ones.

Pointing out that there might be something we can do to address a particular challenge doesn’t amount to blaming us for causing it. Indeed, it might even be what ultimately sets us on the path to improving the situation.

Colin Foster (@colinfoster77) is a Reader in Mathematics Education in the Department of Mathematics Education at Loughborough University. He has written many books and articles for mathematics teachers. foster77.co.uk

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In the real world, we need each other. One of the most important things we might hope our students learn in school is how to collaborate and communicate effectively with others, since it’s an absolute necessity for a well-functioning society. We also know that employers are actively looking for recruits possessing the kind of soft skills that enable them to do this – hence, lessons will often involve students learning in groups.

‘Learned helplessness’

But does that argument actually make sense? The are many aspects of ‘real life’ that aren’t just less relevant in a learning context, but sometimes the exact opposite of what’s needed.

For instance, in the real world we tend to ‘play to people’s strengths’. When young children are baking a cake together, you might hear something like, ‘You’re good at cracking eggs – you do that. I’m good at weighing – I’ll weigh out
the flour.’

This is good on many levels. It encourages a positive, can-do attitude and gives the children a means of celebrating their skills. Having been made aware of their different strengths, they can work together as a team to get the job done more efficiently. Any employer would be pleased with that!

But from a learning point of view, how will Child 2 ever get good at cracking eggs if Child 1 always does it? How will Child 1 ever get good at weighing if they always rely on Child 2 to do it for them?

This kind of ‘learned helplessness’ can be insidious. There’s no deliberate effort to deskill anyone, but that’s what ultimately happens. Children get classed as ‘good egg crackers’ or ‘good weighers’, and become increasingly dependent on others to perform certain tasks.

In a work context, an employer may well not care about the individuals assigned to a task. As long as they’re productive cogs in the machine, they’re just a means to an end. If an employer is unlikely to ever redeploy its staff to different roles, it might not be bothered about whether they know anything beyond what it needs
them to do. Specialisation equals efficiency.

Thinking differently

In education, however, it’s all about the individuals. In education, work is simply a means to an end. The motivation at play isn’t that there are pages of exercises that need to be completed, or essays that need to be written – those will all end up in the bin eventually. The point is what’s learned in the process.

When our priority is learning, we need to think very differently. The group work that enables tasks to be completed more efficiently often isn’t all that helpful. For learning to occur, things need to be slowed down and everyone needs to get their chance.

Simply playing to each person’s existing strengths serves to imprison learners in small pockets of capability, and prevent them from developing important skills they might lack.

It misdirects practice to where it’s least needed, placing the need to ‘learn things’ firmly at odds with the desire to ‘get things done more efficiently’.

Prioritising learning

The main challenge of having students collaborate in the classroom is to avoid the left-hand column of the diagram below and promote the one on the right. And that’s hard.

To learn, we have to focus on our weaknesses. Doing this can be difficult and slow, and may well frustrate other members of the group if they have a ‘doing’, rather than a ‘learning’ focus.

During a pair work activity, I once heard a learner say, “It will be faster if you just do it and I just watch.” Any teacher would be alarmed by this, but in a sense, that learner may actually have been right. If the faster child stopped to explain what they were doing, this would have slowed them down.

If the ‘job’ is to complete the task, and if that’s what the teacher is rewarding (‘Well done! Group A is finished already!’), then we shouldn’t be surprised when children find ways of doing precisely that which serves to prevent learning. We should remember that in non-learning situations, this may indeed be the perfectly sensible thing to do.

Groups or no groups?

To some people, this is a fatal problem with group work, and one of the reasons why they believe it to be incompatible with learning, but I don’t think that’s necessarily the case.

I think that unplanned or poorly-planned group work is the problem. Too often, students will begin lessons working individually, and gradually morph into collaboration as they start to approach harder questions. Sometimes, the culture within a classroom can be that it’s always okay to ‘work together’. If students politely ask to do so, then surely only the meanest teacher would say no.

The usual concern that arises here is that the students will then waste time ‘off task’ and fail to complete it, but the concern I have is almost the opposite – that by working together, the students will be too successful. They may get on too well, too quickly, beyond what either of them might do unaided.

This means that even in a best-case scenario for pair work, you’ll potentially end up with two partially-skilled, complementary people who may ‘work well together’, but are unable to perform well individually – and that’s in neither learner’s best interests.

Mind the gaps

With groups larger than two, teachers will tend to complain even more vocally about group work, chiefly in terms of freeloading and timewasting. Giving each group member a role (which might rotate over time) and making them individually accountable for the entire product of the group can be helpful. Even then, you still risk each person learning only part of what you’re trying to teach.

Learning to perform effectively as a group can’t help but create gaps that go overlooked, if only because someone else will be covering them. Like it or not, even outwardly ‘successful’ group work that sees students successfully completing their tasks will ultimately lull students into a false sense of security.

Colin Foster (@colinfoster77) is a Reader in Mathematics Education in the Department of Mathematics Education at Loughborough University and has written many books and articles for mathematics teachers; for more information, visit foster77.co.uk

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