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.