re: TERC on Establishing Truth in Geometry
Oh really .... I don't know where to begin!!!!
"establishing the validity of ideas is critical to mathematics"
I know straight away that my blood pressure will need to be checked by the time I get to the end of this!!!
But wait, there's more!!!
"Most mathematics instruction and textbooks, however, lead us to believe that mathematicians make use only of formal proof -- logical, deductive reasoning based on axioms."
Of course, that should read "Most mathematics instruction and textbooks AND ALL MATHEMATICIANS ..."
"evidence for its validity in the form of a proof"
By this stage, it's pretty obvious to me that the author isn't a mathematician. "Validity in the form of a proof" ..... what other type of validity is there??
Of course, the trained mathematician should have their 'proof by obfuscation' alarm bells ringing by now.
"For a mathematician, often this internal testing can take the form of proof as one attempts to perform the socially accepted criticism of one's argument."
Is this even English? [ed.: I've been asking myself the same question. When I finally learn how to diagram sentences, I'll be able to answer it.]
"However, does proof convince students? Do they see it as a way to establish the validity of their ideas or, as Hanna (1989) suggests, as a set of formal rules unconnected to their personal mathematical activity?"
They'd better see it as (ahem) "a way to establish the validity of their ideas" or their teacher hasn't really communicated the difference between Science and Mathematics too clearly.
Worthy of mention is the desire to "convince students" .... you may accuse me of semantic nitpicking here, but it's important!!
"Ironically, the most effective path to engendering meaningful use of proof in secondary school geometry is to avoid formal proof for much of students' work."
Which roughly translates as "In order to save the village we had to destroy it!"
I think Melanie Philips (British author) summed it up best in her book 'All Must Have Prizes':
"A fundamental shift in emphasis from knowledge transmitted by the teacher to skills and process 'discovered' by the child has undermined the fundamental premises of mathematics itself. The absolutes of exactness and proof on which the subject is based have been replaced by approximation, guesswork and context."
Melanie Phillips, All Must Have Prizes