Whilst marking examination papers, I was chatting with a colleague, an experienced Maths Methods and Specialist teacher. We were looking at a couple of students who intend to do Methods next year, but currently fall well short on what I would consider very basic things - factorising quadratics, learning (or being able to find) exact values for trig functions, that kind of thing.
I firmly believe that we should teach for understanding. But at the same time there is a certain amount of spade work to be done. Students need to learn certain facts and processes - ideally this will be done with full understanding, but in reality the understanding may come some time after these processes, and sometimes as a result of using them and becoming more familiar with them.
But in maths, if a Methods student does some homework and, let's say, can't factorise x2-9, or doesn't know what sin(30o) is, or when told to differentiate x2 from first principles and given half a page in which to do it, just writes down f'(x)=2x, then that student hasn't done the spade work. He hasn't bothered to go away and commit some facts to memory, or to review a process adequately until he understands.
In these cases, my reaction will usually be something like "This first one is a difference of two squares. Remember that?" "Oh yeah, right!" "Ah well, next time...". Or, when solving a trig equation, I'll find myself more or less repeating what I said in the first place. "This tells you the acute angle, and the negative value tells you it's in these quadrants." This doesn't need me to explain again, it needs the student to look over their notes and follow the process demonstrated again and again until they at least remember (and, ideally, understand why...)
But my colleague suggested that not learning these things is like an English student writing an essay on Macbeth without having seen or read the play. Such an essay would, of course, be pretty rubbish. And the English teacher would not hesitate to say "You haven't read the book, and what you have written is hence a pile of rubbish. Go away, get it done properly, and don't waste my time like this again."
In maths, however, we have a tendency to fall away from this, and reduce our expectations of the students. "Oh, you don't know how to do this? Don't worry, I'll explain it to you again." It's like the English teacher reading out the book to a student who didn't go away and read it.
Perhaps we do this because of the culture of failure around mathematics. People say "Maths is hard, don't expect to be able to do it." When really they should say "Maths is hard, so you'd better work really hard if you want to master it."
But I feel the pressure to let students get away with this kind of behaviour because, if I don't, I worry that the students will go away thinking that I don't care, or that the reason they can't do this must because I didn't explain the concept properly in the first place. But on reflection, this habit is actually quite damaging to the students - setting such low expectations, it's not surprising that they get into these dependent habits. A lot of this is reflected in Dweck's Self Theories, how when we think we are helping, we are actually making things worse.
The challenge for me here is to set these high expectations, but be really clear to students what I expect, and why. If I just tell them "You haven't bothered", I can expect to be fielding calls from parents that afternoon. For a lot of students, they will have never been expected to sit and struggle with a piece of maths, and they need to know how to do this productively.
To this end, I'm also looking forward to trying a bit of 'flipped classroom' structure, whereby for homework I might get students to watch a video explaining something, and expect them to achieve mastery (up to a certain, realistic level) by playing the video over and over, practising at home, and having the expectation that they will arrive prepared, and not be able to say "I watched it once, and it was hard, so can you explain it to me again now?"
I'll be teaching an intense course for Year 9 students for the last two weeks of this term, and I'm hoping to try this technique during this time. We'll see how it goes...