I am an old-fashioned sort of guy in many ways. For one, I have never embraced the general, modern concept of 'entertainment' and 'fun' as a foundation for life or education. I find much of the material that is presented these days in the interest of making learning 'fun', too much like fairy floss—fluffy and sweet, but when you bite into it, there's nothing really there. Where ever possible, life and learning should be enjoyable, but this is as much about attitude as it is about individual activities. There will always be things in life that have to be done, or tasks that have to be mastered, which are not necessarily enjoyable and certainly not a load of 'fun'. Ultimate mastery of many of these tasks, however, will invariably be very satisfying.
I have never found maths 'fun'. I have found some of the 'higher concepts' that I encountered during my university studies fascinating, and even 'neat', but my mastery of the subject (I am no maths wizard, but my undergraduate degree included an 'honours level' maths sub-major) has involved a lot of hard work. I do not recall ever having encountered anyone who has mastered mathematics in the course of 'having fun'. The more gifted mathematicians I have known certainly had fun in the course of exploring the subject, but I would suggest that this sort of 'fun' is more akin to the personal satisfaction that one derives from successfully meeting a challenge.
Certainly, it would seem that the sooner one masters the basics of mathematics, the more enjoyable is the subsequent learning process. Without a strong foundation, more advanced mathematics can certainly become a hard slog, if not impossible. To this end, my attitude from the outset was one of mastery of the fundamentals of addition and multiplication by rote learning of tables. I know this is not a popular approach in 'progressive' circles, because it it seen as a tedious bore. I, for one, however, never found learning tables a tedious bore, and never have I encountered anyone who learned their tables this way who does not think that this aspect of their education has stood by them for their entire life.
My fundamental message in teaching maths has been "Knuckle down, do the hard yards now, and you will lay a foundation that will support any path you ultimately choose to follow." At every opportunity, I have tried to illustrate how useful mathematics can be, and more particularly how important it is to have your mathematics 'on call' (i.e. learned, memorised, and in your head).
Barb didn't have the same conviction that I had when she started teaching Steve maths. She went through a number of the 'modern' approaches before agreeing that there was no substitute for just 'knuckling down' and learning multiplication tables. She also used flash cards to help develop practical competence with the basic operations.
Barb used the Saxon Maths series of texts during Steve's primary years, and we know several families that have used these texts successfully. When I took over Steve's maths education in Year 7, however, it was apparent that he was just coasting through this material, relying on the way the material was presented to learn, rather than making any specific effort. This may sound really neat to some, but it was not at all 'challenging'. When it was apparent that this was indeed the case, we switched to using the Understanding Maths series from Accelerated Maths Learning. These books require the student to make much more of an effort, to be more actively involved in the learning process. The Saxon texts may not have been the real problem here, but I felt that a change of approach was required to bring about a necessary change in attitude towards actually mastering mathematical skills, rather than completing exercises.
I also checked the NSW Board of Studies Mathematics syllabus to see what might have been expected of a Year 10 student. I felt that this was a lot more helpful than the Science syllabus in that it was a lot more specific about what needed to be learned.
After some initial push-back (Steve was very comfortable just cruising along...), the Understanding Maths series was well received by our student. As I mention in my overview of our Science curriculum, my ultimate aim was to present mathematics in the broader context of 'natural philosophy'. In one sense, this is just presenting maths in a context in which it is commonly used. To this end, I didn't necessarily follow the sequence of chapters as they were presented. For example, one of the first subjects I taught in Year 7 was logarithms. Since this was pretty much the way every student started maths in high school before calculators came on the scene, I was not put off by the fact that this was an 'extension' subject at the end of the Advanced Year 9-10 text. To me, this was an essential concept to grasp before using scientific notation, the foundation of much of the mathematics used in science.
Thereafter, I identified the fundamental mathematical skills, required at various points to support the science curriculum that I had planned, and developed these individual skills more or less using the sequence presented in the text. When covering mathematically intense areas of physics and chemistry, maths lessons were devoted entirely to developing the techniques being used at the time. In this way, we covered the entire syllabus, as presented in the Understanding Maths texts, in under three years, with the extra year essentially devoted to the application of the relevant mathematical techniques in the other sciences. Having said this, I am not entirely happy with the progress we made with the maths we covered. It is pretty clear to me, however, that it is not understanding that is lacking. Rather, it is the skill that comes with conscientious application that is in need of a little more development.
During Year 10, we encountered the mathematics enrichment program developed by the Australian Mathematics Trust. This program contains some excellent material, that in our case at least has helped to encourage thinking about different approaches to problem solving.
One of the problems with which I have struggled, to some extent, is presenting mathematics as more than a mechanical exercise, while still building the mechanical foundation that I believe is necessary for the practical application of 'the craft' of mathematics. In my own case, mathematics has always been primarily a tool. The 'neat' aspects of mathematics have really only ever been a diversion, even though an enjoyable one. Problems relating to symmetry and topology were very much a part of the process of molecular design that underpinned my early life in chemical research and were always challenging and quite enjoyable intellectual exercises. As such, I had not hitherto given a great deal of thought to how one might impart this level of appreciation for mathematics on an unwilling student. While we have only recently encountered the AMT materials, they appear to be dealing with this issue quite effectively.