 weebecka: I think part of the problem is people assume I'm not listening so they don't bother to work at it.
I am not sure about that. I think part of the problem is that you don't seem to realise that people have thought about a priori and a posteriori knowledge in maths, whether or not it is objective or subjective knowledge ( it is pretty standard fare for the IB ) but that it has little relevance if the students cannot add two numbers together. Now it may be very well discussing what is, in reality, 7 x 8 or how we can describe a n-dimensional space using algebraic structures, but it doesn't get the students anywhere. In my mind all it is doing is purposefully avoiding the hard grind of actually teaching some fundamental building blocks of arithmetic, in order to pretend or even play at teaching mathematics. I do not teach in a problem school, and I never have. The type of schools I teach in consider a problem student as one who cannot obtain an A or A* at IGCSE ( even that is not really true - my school does not even sit any external exam until the students are in their final year before university ). Class sizes are seldom in double figures and resources are anything you want. Even in this type of setting, the students learning the fundamentals of arithmetic and algebraic manipulation is vital. Without it they cannot do anything, learn anything or express themselves in any shape or form mathematically. All this talk about students creating theorems for themselves and discovering mathematics is frankly a pipe dream. An unfortunate relic of some 70's ideal still holding sway in teacher training colleges. Even if you take the most elite mathematical hothouses the English public school system has to offer, the US advocates of the Harkness system, etc. etc. you still require a great deal of guidance from a teacher. If at the end of the day the students cannot do any basic arithmetic nor can they perform rudimentary algebra, then you, as a teacher, have failed. You may blame the failure on extenuating circumstances, on poverty, on poor social conditions, on a poor primary education but it does not excuse the fact that a student under your care has failed to learn basic mathematics. Each time a student leaves my classroom after two years of being in there without passing their mathematics IB exam at whatever level it may be at then I have failed them as a teacher. Do I believe all students are capable of learning? Yes, fundamentally all are capable of learning. Do I believe all are wanting to learn? No, most definitely not, no matter what excitement you may bring into your classes.
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