A-level Teaching, Uncategorized

Groundhog Day and planning to fail

Some mathematical proofs are essentially magic, aren’t they?  “OK, there’s nothing up my sleeve, right?  Now watch what happens when I multiply by \frac{1-cos(x)}{1-cos(x)}!”  “Stay with me now: we’re going to take logs for reasons that will become clear later, then substitute tan(2t) for x.”

Students say “I understand what you did, but how did you know it would work?  I would never be able to come up with that.”  This is a big problem with the way we teach maths, especially at the more advanced level.  We want students to develop into resilient problem solvers who learn from their mistakes, but most of the time all they see modelled from their teachers is carefully crafted examples, pruned and polished until they are as efficient and sanitized as we can make them, with no hint of the rough-n-ready manner in which they were born.

When’s the last time you stood up in front of a class of students and tried to tackle a problem that you didn’t already know how to solve?  Not just through inadequate subject knowledge or bad planning, but in a deliberate attempt to model failure and how to deal with it for our students.  They already know what success looks like, and half the time we make it look so effortless they get even more demoralised when their own experience doesn’t match up.  Don’t they deserve to watch us flounder, sweat, get stuck and be confused once in a while, too?

Even if we occasionally take a punt at an unfamiliar question in class, I’m guessing for most topics we can see how the problem will play out before we start (or at the very least, its general trajectory), which doesn’t quite fit the bill.  To get well and truly stuck, you either have to be testing the boundary conditions of your subject knowledge comfort zone (Further Maths A-level has a few areas like this for me, still), or tackle something like a UKMT maths challenge problem, or something from the excellent Integral Maths Ritangle competition.

Solving a maths problem to me feels like the film Groundhog Day.  The first time you tackle it, it’s a mess.  You may not get the solution at all, or even close to it.  Perhaps when you come back to it, having reflected on the dead-ends and false trails, you can do better.  Define your variables, because that tripped you up last time.  Clarify your assumptions, and tidy up your notation because those simultaneous equations get messy fast.  A couple more attempts, though, and you can solve it in your sleep.  You even spot shortcuts that would never have occurred to you the first time around.  And when you miraculously pull the perfect substitution out of your hat the next day in class, and they whisper “It worked!  But how did he know to do that??” it’ll look like magic when really it was good old-fashioned hind-sight.  A colleague shared a geometrical problem with me that he said could be done in your head “after you know the answer”.  Which makes perfect sense to mathematicians: solving a problem doesn’t just give you a solution, but an insight into the entire scenario.

When I’m feeling especially reckless, I bring my selected unsolved questions into the classroom.  Narrating my thought processes, and letting the whiteboard fill up in the half-structured half-chaotic manner that back-of-the-envelope problems do, I can show students what it looks like for me to struggle with an unfamiliar problem.  They see the wild goose chases, the uncertainty, the dubious assumptions and second-guessing.  They see me get stuck, and take stock of what I’ve tried and how it helped, if at all.  They see me back-track, and start again from scratch, and get it wrong, and hunt for that dropped minus sign, and look for ways to verify my solution.  And with any luck, in the post-mortem, we’ll spot that elusive, elegant pathway through the maze which is often only visible from the end.

How we learn, as shared by @magicalmaths
A-level Teaching

Why teaching A-level is the best preparation for teaching A-level

I thought I understood A-level Maths until I started teaching it.  As @magicalmaths tweeted the other day, we learn some of what we see or read and rather more of what we discuss or experience for ourselves, but none of it compares to how well we learn by teaching others.  If you’re looking for the ideal training course to equip you to teach A-level Maths, you need to go back to the A-level Maths classroom, but this time stand at the front.

How we learn, as shared by @magicalmaths

I was very proud of my own A-level results, but have become increasingly convinced that there is a yawning gulf between being able to get the best grade at A-level and being able to teach others to do so.

It’s now more than 5 years since I started teaching and the main difference between me now and NQT me (other than his gloriously unkempt facial hair and his lack of a wife (correlation/causation?)) is the fact that now I’m starting to realise just how much I don’t know.  If I had to quantify it (and I am a maths teacher, so I do), I’d say my level of understanding of A-level mathematics has more than doubled since I started teaching it.  And by far the most effective tool for increasing my understanding has been actually having to teach it.  Tough luck on the first class or two (don’t worry – unless you’re really clueless they’ll cope just fine), but by the time you’re teaching a topic for the second or third time you’ll know it inside out, back to front or under any other geometric transformation you can imagine (180 degree rotation is a useful one for correcting student errors from the front).  When teaching a topic, knowing what you know and what you don’t is crucial.  You’ll probably be somewhere along this scale with any given topic:

1. Grasp the key idea.

2. Answer simple questions.

3. Answer more complex questions.

4. Interpret/analyse the general form of solutions.

5. Map out a path to the solution without having to solve the problem.

6. Make up your own questions (including all relevant (and no unnecessary) information.)

7. Identify the potential pitfalls for a problem.

8. Grasp the key idea that extends/generalises this one.

For example, when learning about projectile motion in Mechanics 1:

1. Understand how horizontal and vertical motion can be treated separately.

2. Calculate horizontal range from a given initial speed and angle.

3. Calculate the possible launch angles for a given speed, horizontal range and initial vertical height.

4. Explain qualitatively why doubling the speed quadruples the horizontal range, and why 45 degrees gives the maximum range.

5. Predict the quadratic equation from vertical motion, the linear one from horizontal motion and how the key variable (time) connects the two.

6. Accurateley estimate the speed of a bullet / arrow / cricket ball and form a question with enough unknowns (but not too many).

7. Warn students to look out for negative displacement vertically, remind them that the initial velocity is not a force and therefore horizontal speed doesn’t change.

8. Describe the modifications necessary to our model should we choose to take into account air resistance.

Trebuchet Projectiles

And solving equations of the form a sin(x) + b cos (x) = c for Core 4:

1. Recognise how the sin(A+B) or cos(A+B) formulae can be rearranged to get a single function.

2. Solve questions where you’re told whether it’s sin or cos, and whether it’s + or -.

3. Rearrange and simplify trig equations in a variety of forms to get a single trig function in the equation.

4. Describe how the Rcos(x+a) format relates to the original functions through graphical transformations.

5. Consider the signs of the sin and cos components to determine whether you need Rcos(x+a), Rcos(x-a), Rsin(x+a) or Rsin(x-a), and what happens if you choose the ‘wrong’ one.

6. Show how interference of two waves simply generates another wave, each wave being a transformed trig function.

7. Remind students that, depending on their choice of function, the primary solution for the angle may not be the correct one, and how to find the right one.

8. Consider how graphs of functions may be modified in other ways than adding.  Does multiplying two trig functions give a trig function?  Are there similar patterns for tan(x)?

In the last few weeks I’ve had some moments where I’ve glanced at a student’s work and leapt straight in at point 7 identifying their misconception with almost uncanny accuracy in seconds.  But I’ve also jumped to point 6 in class without realising my understanding was still at a 3, wasting 45 minutes of lesson time getting things wrong on the board.

If you’ve only just begun to teach A-level, don’t be put off by how well another teacher may seem to understand a topic.  It’ll come; you just have to be prepared to get stuck in and teach it for a bit first!  I have an ever-increasing list of topics I want to ‘really get my head around’, but in the meantime, knowing my limitations enables me to plan properly and get support from more experienced teachers when I know I need it.