Sunday, 18 November 2012

Trialling the Flipped Classroom

As we begin the new 2013 classes, I am going to be trying out some flipped classroom techniques with the new Year 12 Methods class.

For homework, they will be expected to watch a video explaining a concept (in the first case, the chain rule), get some basic skill practice, and then have some larger questions which they will be expected to discuss in class.

It is important that the students have procedural fluency in the chain rule. But I am starting the video by hopefully creating some need as to why it is important, and finishing with some challenging questions which should get them to think a bit, and create confidence in their own abilities if they are successful (e.g. by differentiating sin(2x) having never even seen differentiation of trig functions before). This is following advice from Dan Meyer's blog, and hopefully will address criticisms of the Khan Academy.*

This relates to my last post about how we perhaps don't foster enough independence in our students. I want them to feel responsible for going away and watching the video as many times as they need to in order to understand, with the expectation that they don't need the teacher holding their hands at every stage.

In doing this I do want to be realistic about the workload it might create. It would be unrealistic of me to spend hours and hours on top of my normal planning time. Which is why my screencast is done in one take, and can be expected to contain the odd mistake and the odd word fumble (though I think I have corrected myself).

The course will appear here. Comments/feedback welcome and appreciated.

* Reading tweets and blogs, there is a fair bit of criticism of TKA. Whilst such criticisms are often valid, they're not damning, and it's important to give credit for the mission statement of providing a free quality education for all.

So, I watched TKA's video on the chain rule, after I made my video, to see how they compared. In order to create a sense of challenge in students, both videos begin with "Nobody wants to expand (2x-5)7 if they can avoid it."

After that, we diverged. TKA taught by examples, using the procedure "differentiate outside the function, then inside, then multiply the two answers." (my paraphrasing.) I started from a more 'first principles' approach, using more formal notation. My feeling is that the latter is better for accuracy, particularly when you get e.g. chain rule within a chain rule, maybe within a product rule.

Watching TKA also made me quite conscious of the language used. "I will show you some examples before I tell you the rule" (from TKA) is quite teacher centred, and doesn't create in students the expectation that they can figure a lot of this out for themselves. But then, TKA videos are not designed specifically to be followed up by a teacher. Not to mention the fact that so far I have only watched the one video.

Wednesday, 14 November 2012

Do maths teachers set lower standards than other teachers?

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...

Monday, 12 November 2012

ABC's "The Future of Maths" - Some reflections

In The Future of Maths,  Conrad Wolfram argues against the teaching of e.g. long division algorithms, because these days we have computers to do this for us. Instead he wants to see maths used to answer real world problems which often involve modelling - e.g. are house prices in UK and Australia connected? What is the risk associated with a particular course of action?

I agree that teaching algorithms without understanding is fairly pointless. It used to be the case that learners of mathematics would be expected to do large numbers of long divisions, because what mattered is not that they understood what it meant to divide one number by another, or roughly how this process worked, but that they could produce the outcome required, i.e. the right answer. It is fair to argue that being able to repeatedly do these kinds of calculations quickly and accurately is less important than it once was.

However this does not mean we shouldn't teach it at all. We should teach long division so that learners have an understanding of what it means to divide one number by another. The meaning behind the algorithm requires and develops an understanding of place value, for example. And it gives learners a sense of roughly what the answer might be - in every day life we need to be able to work out roughly how to perform divisions mentally.

Wolfram argues that we should spend more time on modelling and problem solving. However this seems to be making the mistake outlined in Kirschner, Sweller and Clark (2006), which is that novice learners work the same way as expert practitioners. Yes, complex modelling and problem solving is a noble and useful goal, but we're not going to get people there without understanding and fluency of more basic skills.

I'm also uncomfortable with a recurring theme in this programme, mentioned by Wolfram and Keith Devlin, that we should only teach things which are explicitly useful in real life. If you want to make this a requirement of education, then you would want to scrap art, music, poetry, many parts of science (how many of us need to know about xylem and phloem in our lives?) and more. It might sound a bit pretentious but I'm of the opinion that learning mathematics, along with other subjects, is valuable and reward in itself, and that it teaches an intellectual rigour that can be applied to more concrete subjects as required.

Devlin made a similar point that being able to do arithmetic is less important as having a number sense. This sounds reasonable, though I wasn't sure what he means by a number sense. I take it to understand (as above) what it means to divide, or multiply, or work with fractions, and know when you would want to do these various operations. His discussion of using algebraic reasoning in spreadsheets reminded me of an ATM resource where kids would construct spreadsheets to e.g. model mobile phone tariffs, and having tried these, I found that they were effective in getting across ideas of variables in a slightly more meaningful way than normal algebra exercises. 

I still maintain that, in order to get this 'number sense', you need to have some experience and confidence in doing numerical operations, but that this is a means to an end.

I was interested in something Terence Tao touched upon, i.e. collaborative work in mathematics. He wasn't sure how this would be used in the classroom, but evidence shows that communicating and explaining mathematics to others helps both the explainer and explainee. I am keen to encourage collaboration in classrooms, though whenever it has been the explicit focus of the lesson, it often takes over the mathematics. I remember lessons as a less experienced teacher where I would have kids make a powerpoint and explain something to the rest of the class - more thinking was done about the powerpoint and the non-mathematical aspects of the task. In working with Moodle, one of the visions for this model of education is a kind of inquiry-based learning, where kids would logon and exchange ideas on how to solve a particular problem. There are practical issues with doing this in mathematics (where you can't express ideas easily using a keyboard) and there is a danger of cognitive overload (Kirschner et al) and of the focus on the medium overtaking the focus on the mathematics. We need to teach collaboration progressively like any other skill, and not assume that novice learners will be as competent as expert practitioners.

From this programme, I will consider again how I can get more collaboration in my classroom, without falling into the trap of collaboration for the sake of collaboration. I found William's "Think Pair Share" construct useful for this. 

An ongoing aim for me is making sure that learners understand any algorithms they are doing - this has been a cultural challenge at times when learners are often more interested in passing exams than understanding the subject matter itself. Managing this kind of culture shift is a delicate and long term task.

Monday, 28 May 2012

Success Criteria in Maths

Our school have been looking lately at success criteria, and we were interested as to how they might be used in maths. Other departments use them more routinely and successfully than we do, so I observed an English teacher during a ‘success criteria’ part of a lesson to see if we can apply in maths what other departments are doing. I’ve written a fairly seminal reflection here that may guide discussion.

The learning intention was “Analyse the discourse and grammatical features of complex text.” Key words were identified, defined and broken down (e.g. lists of grammatical features given). Student input was sought, and reframed by the teacher. The question was asked “How will I know that my analysis is effective and worthwhile?” Kids spent about ten minutes discussing, in groups and as a whole class, how they would know. They came up with:

  •          I can do this analysis in different contexts
  •          Others can read my analysis and know about the text being analysed without having first read it
  •          My analysis will identify the audience/context of the text
  •          I can deconstruct complex text using my skills.

During the discussion there was a lot of recapping of previously learned material, e.g. mentions of implicit/explicit information in text, literary devices. I got the feeling that the kids were already fairly familiar with what needed to be done, which indicated that the learning intention cumulatively built upon previous lessons.

It certainly felt like an effective use of time, focusing the kids on exactly what needed to be done.

I wondered how this could be applied to the last maths I remember teaching before revision: “Sketch cubic graphs including all key features.” Modelling what was seen, I would ask the kids, “So what are the key features?” Shape. Intercepts. Turning points. Shape? Well, let’s see what the basic cubic looks like by trying some numbers in y=x^3. Intercepts? X=0, easy enough. Y=0, need to factorise. How?... This feels like it would have been more or less what I do anyway, but without referring to the parts as success criteria. Where in English there seems like a large pool of skills from which to draw, in maths it is far more rigidly defined – these are just the things you have to do to answer the question, and if you don’t do them, you won’t get to an answer. (And we would accept different routes to the same answer, and hence ‘process criteria’ wouldn’t be appropriate). In English, you might have what looks like an answer, but actually doesn’t do everything you would want.

So, discussing the learning intentions sounds like it gives a structure to the lesson and identifies what needs to be done, but I’m not sure I would make explicit success criteria. They seemed useful in English because, after producing a piece of text, some serious reflection needs to happen for the student to know they have done what they should have. In maths, once you have done it, it’s fairly obvious.

The structure here would not be appropriate for e.g. the first lesson where children see matrices. “Understand what matrices are and what operations can be performed with them.” What’s a matrix? Don’t know. What can we do with them? Don’t know. OK, listen carefully whilst I tell you… And it’s quite possible that there are English lessons where the content is too unfamiliar for the kids to discuss success criteria.

So at the moment I have no plans to get into success criteria as a routine part of my lessons. But I would be interested to know if other people have different experience of this.