I’ll be upfront and say that this is how I think we should teach maths:

- The teacher should chart a course through the subject, explaining key and threshold concepts along the way. Students may not ‘get’ everything at the first attempt but as Fields Medallist, Timothy Gowers says: “
*It is quite possible to use mathematical concepts correctly without being able to say exactly what they mean. This might sound like a bad idea, but the use is often easier to teach and a deeper understanding of the meaning often follows of its own accord*”. So, once the student commits the basicof mathematics to his or her long term memory (I admit that motivating students is a challenge here and that is where the skill of the teacher comes in), they will be able to progress on to the next level and become proficient at the*language**doing*of mathematics, building confidence and hopefully some enthusiasm along the way. In time, they might return to those rules that they didn’t quite understand and look at them in a new light and with new insight. (I’ve done this many times myself and had to think very hard recently about what raising a number to an irrational power actually*means*. And that’s 32 years after getting my engineering degree.) - The students’ understanding (which is really only another form of memory – what else could it be?) of mathematics and their ability to think critically and even creatively is built up by challenging them with lots and lots of problems of ever increasing difficulty and ever increasing breadth, drawing on more and more of their prior learning. The better posed the problems are, the more the student will learn: too hard, or if students simply don’t have the tools to tackle the problem, and they will switch off; too easy and they will be unstimulated and get bored. And, the more interesting these problems, the better. Ideally, you don’t want the student to ask “so what?” after spending half an hour getting an answer. (People who really love maths never ask the “so what?” question.) If the student is fully committed and studies independently using scientifically sound study methods, they have every chance of becoming proficient at mathematics. But the key in all of this is that the teacher is knowledgeable, motivating, encouraging and supportive. Mathematics, after all, is fundamentally abstract. There is no getting away from that fact and most mathematics (as opposed to arithmetic, perhaps) can be classified as biologically secondary knowledge so learning it doesn’t come easily to the majority.
- Of course, you can’t talk about teaching without talking about assessment and if we want our students to become critical thinkers then we need to adopt methods of assessment that
*require*them to think critically. In effect this means that when it comes to assessment, students should expect the unexpected. However, there is a belief abroad, at least in Ireland, that the unexpected is unfair so a change in mindset is needed; from students, teachers and parents.

In the last couple of decades there has been a steady shift away from this approach, which could be classified as ‘traditional’, towards more ‘progressive’ approaches of which inquiry/discovery-based methods are a significant part. Canada, especially the state of Ontario, has gone down the discovery route and there are those who would suggest that this has been a disaster.

By and large, discovery methods, which have an air of plausibility about them, are seen as inherently good, although there has been something of a backlash in recent years, sparked in many cases by the obvious fact that many students, even third level students, seem to lack basic skills, especially in mathematics. This seems to be a worldwide problem (except perhaps in Asia) but was not always the case.

The basic idea behind inquiry/discovery methods is that by tackling ‘real world’ problems, “constructing their own knowledge” in the process, students develop a deeper understanding of the mathematics required to solve those problems. (Presumably the belief is that knowldege and skills acquired by solving real world problems transfers to other problems so, in a sense, discovery learning eventually ‘eats itself’.) Apart from the fact that ‘understanding’ is a difficult concept to define, the precise role of teacher guidance in inquiry based methods is often hard to nail down. It seems to vary from almost no guidance to so much guidance (‘enhanced inquiry’) that is difficult to see how this form of teaching is any different from traditional methods.

Now I’m just an engineer who is not an expert in pedagogy but the impression I get from reading the education literature is that, *at the* *very best*, methods of learning that are not teacher-led are unproven. Indeed, there is absolutely no consensus as to how best to teach, not only mathematics, but almost every single subject. So despite what camp you are in, traditional or progressive, you should tread very carefully when you get into the realm of educational policy and curriculum reform. And if you are a ‘progressive’, you need to be extra careful because the extraordinarily creative and innovative 20^{th} century, and the first decade of the 21st, was built on the foundations of traditional education and traditional methods should be discarded only in the face of ** extraordinary** evidence. To me, that is the only intellectually honest way to proceed. Otherwise, you will just end up doing large scale experiments on children. This is what we have done in Ireland with the Project Maths initiative which, despite claims that it is evidence-based, has sparked a large degree of concern amongst genuine and thoughtful educators, especially at third level. These are people with no axe to grind but who are leaving exam board meetings depressed at the extent of failure in mathematics modules. Many bemoan the lack of very basic skills shown by second and third year college students. The general feeling amonsgt third level lecturers is that whatever way students are learning mathematics at school, they are not acquiring the basic ‘grammar’ of the mathematical language to the extent that this grammar is imprinted in their long term memory, freeing up their short term memory for all of that critical and creative thinking that we want them to do. It is very hard to see how discovery methods will fix this problem.

But the existence of pedagogical camps is a problem because very often people see what they want to see when they examine research. Confirmation bias and motivated reasoning are rampant. A really good example of this was a recent article by renowned maths educator, Jo Boaler, in which she claims that some recent PISA results could be interpreted as showing the superiority of inquiry methods over more traditional methods in which learning basic maths facts and rules ‘off by heart’ is a key component. (She singled out Ireland for particular criticism despite the fact that our PISA ranking is higher than many countries of whose methods she approves.) Her article has been thoroughly demolished by Greg Ashman, an English (i.e. from England) teacher and blogger based in Australia. (I had previously raised my own doubts about her interpretation of the data.) I don’t know a huge amount about Boaler but I plan to read her work in more detail so that I can arrive at some sort of informed opinion. I have to say, though, that her response to the PISA data worries me. It seemed to me to be ideological.

The signs are, what with the huge emphasis on fairly nebulous skills as opposed to knowledge in the new Junior Cycle, that the Irish education system is going to go down the ‘progressive’ route and my view is that this will be disastrous, especially if there is a large emphasis on discovery methods. I think we need to remember Newton’s famous statement: “If I have seen further, it is by standing on the shoulders of giants”.

**Further Reading**

- The case against minimal guidance

http://cogtech.usc.edu/publications/kirschner_Sweller_Clark.pdf

- Being intellectually honest in education

https://www.amazon.com/What-Everything-About-Education-Wrong/dp/1845909631

- The case for fully guided instruction

http://www.aft.org/sites/default/files/periodicals/Clark.pdf

- Why knowledge is important for critical thinking

http://www.aft.org/sites/default/files/periodicals/Crit_Thinking.pdf

- Why learning basic maths facts is important

http://www.aft.org/sites/default/files/periodicals/willingham.pdf

- The cognitive niche – or why human beings might find maths hard! (See the section on “Emergence of Science and Other Abstract Endeavors”)

http://www.pnas.org/content/107/Supplement_2/8993.full

Pingback: Ninth Level Ireland » Blog Archive » How should we teach maths?

Greg I must say I have been influenced by your writing on this recently but probably moreso by the disaster my daughter went through in the leaving cert 3 years ago when project maths came in. (and she went on to study engineering) . I had thought that my own command of maths was based on problem solving but on thinking back it was probably based on solving lots of theoretical problems from the Christian Brothers book in the seventies followed by practical problems in college when I had a good grasp of the techniques. Having said that, we have to improve the teaching of traditional mathematical techniques and I think what you have described may well be done better by computers than by teachers: “The students’ understanding …. of mathematics and their ability to think critically and even creatively is built up by challenging them with lots and lots of problems of ever increasing difficulty and ever increasing breadth, drawing on more and more of their prior learning. ” This needs to be personalised for every student which is too much effort for teachers. Adaptive systems (such as used by the Khan Academy), particularly AI based ones with access to large data on previous student behaviour, that can accurately evaluate the current position of a student and select progressively more difficult challenges, are likely to perform better than most teachers, even the better ones. However, this idea may cause some cognitive dissonance” in those researchers whose bias is against automation of the process of teaching.

Interesting. One of the things I notice in third level these days is the huge variation in ability within a single class. Some personalisation would be really useful.

An interesting article Greg.

I am not a maths teacher but I and many others share your concerns about the direction maths teaching, and for that matter the teaching of many subjects, is going in Ireland.

An interesting article I found on project maths is from Aidan Roantree, senior maths teacher at the Institute of Education. My own daughter is doing the leaving certificate this year and attends grinds at the Institute. She feels he is a fantastic teacher and I therefore think his views are worth listening to.

http://www.irishtimes.com/sponsored/exam-times/maths/project-maths-a-teacher-s-perspective-1.1663014

Yes, I read that article and it is reassuring for someone like me (we’ve no teacher training in third level!) to hear an experienced teacher saying these things.

Enjoyed your post about math and gave you some publicity. https://www.facebook.com/fixontmath/

Tks Teresa!

Greg