On Jun 29, 2014, at 7:23 AM, GS Chandy <firstname.lastname@example.org> wrote:
> I have long been struck by your emphasis on 'algebra' as [possibly] representing the 'initiation', as it were, of 'mathematical thinking' by a learner. This is, I believe, often a characteristic error made by those who 'came to math' via engineering. (I myself am one such, and for quite some time I had believed that 'mathematical thinking' really started with algebra).
When I say algebra I mean algebraic reasoning and formal thinking. Algebra tends to be the first time in grade school that students are exposed to mathematical reasoning and formal thinking. Assuming of course that the curriculum is authentic.
> As a matter of fact, I believe there is much sound and tested scientific evidence for the view that true 'mathematical thinking' is actually seen in/ demonstrated by a learner when he/she studies geometry, or patterns perceived in nature, or symmetries seen in many of the objects (natural and humanmade) around him, or even when he/she studies his/her fingers with some profound questions in the mind. I believe - but am unable to provide clear-cut evidence - that 'mathematical thinking' may well be demoinstrated by human children almost from the time they are born, when they open their eyes, and so on.
First, when I say *mathematics* I mean mathematical reasoning and formal thinking, which, when it does occur in a student it doesn?t occur till adolescence, except at the extreme tail end of the distribution. I don?t have a problem with conjecturing what might be some of the primitive ingredients of mathematical reasoning and formal thinking, except when the theory says that the ingredients are things that everyone has. This is what I mean about claiming that the sky is orange. If you create a theory that predicts that the sky is orange, yet the sky is blue, then there is something wrong with your reasoning because it lead to an incorrect conclusion. Likewise, if you claim that everyone or almost everyone has the ingredients for mathematical reasoning and formal thinking, yet only a small percentage actually attain theses things, then your reasoning is flawed.
The only way to validate any theory that everyone or almost everyone has the ingredients for mathematical reasoning and formal thinking is to attain mathematical reasoning and formal thinking in everyone or almost everyone. And I would say that nothing in the last 50 years of education has been tried more. And nothing, sadly, has had less results.
In science you study the data and then create models consistent with that data. Reformists, by their very nature, don?t work by that method. They pick what they want the result to be first and then invent science to support it. This is where they generally run into very serious problems. They get very loose (strange) with their chain of reasoning. The data says that few students attain mathematical reasoning and formal thinking. I am ok with reformists thinking that they can change this, but they have to recognize that when they say they can changes this, they are going against the data and science and the only way they can change this is to go all the way and attain mathematical reasoning and formal thinking in everyone or in almost everyone, or even in just a significantly greater percentage of everyone.
Does that make sense? In other words, there is nothing wrong in thinking that the data is wrong, but the only way to prove it is to change the data. You have to make everyone, or almost everyone attain mathematical reasoning and formal thinking. Proving that you think the data is wrong or proving why you think the data is wrong, doesn?t prove that the data is wrong. You have to change the data itself.