There is a narrative in pretty much everything - and the learning journey of a student in Maths class is no exception. The 'relationship' that students come to have with mathematics is to some extent determined by how they can relate to this narrative. Mathematician Dr. Vicky Neale put it very eloquently on an episode of The Infinite Monkey Cage:
"One of the reasons that people switch off [from Maths] is when they stop being allowed to play around with mathematical ideas. [...] Maths is about playing around and seeing what you can dicover and the longer people have the opportunity to do that at school the longer they are doing real mathematics." (Dr. Vicky Neale, The Infinite Monkey Cage, Series 10 Episode 1, 07:50)Perhaps sometimes statements we use in the Maths classroom should be preceded by a "Spoiler Alert."
By presenting a necessary 'must-be' maths fact like this to students before they have had the chance to 'play around' and potentially see it for themselves, we are robbing them of the narrative of discovery that characterises what the subject is all about and what so many mathematicians (amateur and professional) enjoy so much about it. Ideally, this fact should be the end point of a narrative of discovery for every student - telling students the fact before they have had a chance to explore the idea for themselves is nothing short of a Mathematical Spoiler.
(In his empassioned essay, Paul Lockhart describes students exploring this mathematical idea in a natural and very mathematical way and coming up with their own arguments. (p20))
Indeed, if students are fed a diet of too many transmitted mathematical facts that they are expected to accept without the chance to 'play around' and see for themselves, it is likely that the sense of enjoyment, wonderment and curiosity that mathematics has the potential to induce will be dulled. If students passively receive facts, it is no wonder many children, and subsequently adults come to believe that Maths is not for them - that they are somehow 'not a maths person.' For students to feel confident and interested in Maths, they must (as Dr. Vicky Neale put it in the above quote) have the opportunity to play around and see what they can discover.
I am not advocating a classroom of students working aimlessly playing around with mathematics, but carefully guided explorations that enable students to see necessary 'must-be' maths facts for themselves. Yesterday I wrote about my class discovering Pythagoras' Theorem through a structured activity - I certainly do not believe the wonderment and curiosity some students experienced would have been the same if I had merely presented the algebraic rule to them. Thus the end result (in this case a² + b² = c²) would serve as a mathematical spoiler for students' journey of discovery.
So although it is our job as teachers to impart knowledge, it is equally important that we think carefully about how to craft the journey so as to ensure students feel the joy of 'playing with mathematics', the joy of discovery and the further curiosity this arouses. Constantly presenting facts to students that they could discover for themselves will serve as Mathematical Spoilers that will disengage them and may lead them to the conclusion that maths is simply not for them.