Close your eyes for a moment and think of Sinθ. OK, so one of three things will have just happened. You may have simply ignored the request and carried on reading like some kind of popular science reading maverick - if so, well done, you've shown real integrity. The remaining spineless creatures fall into two groups. For the first it will have conjured up supressed memories of endless school maths classes with their stale, out of date teachers and text books which lost interest years ago. After a few seconds drool started dripping out of the corner of your chops and you fell into the deepest sleep you've had in weeks. You've just woken to find that 6 hours have now passed since you began reading this article. Welcome back. The third group, which I imagine is quite a small proportion, closed their eyes and successfully thought about the fixed relationship between a right-angled triangle's angles and the ratio of it's sides.
So how can it be that we all spent countless hours diligently learning trigonometry at school but most of us have little understanding of what Sine, Cosine or Tangent actually mean? One reason lies in the way maths is taught. Understandably the focus is on passing exams, and this means providing the tools to answer the questions. For trigonometry a mnemonic such as SOH-CAH-TOA seems like the perfect tool - work out which of the values you're been given in the question, turn the handle and Ta Da! Out pops an answer, exam passed, phew! But what has actually been learnt? Nothing more than how to solve the question.
It's a simple fact that ideas are easier to absorb when we're able to stitch them into the existing fabric of our knowledge - if you want to know how to subtract 1/16th from 3/8ths then look no further than your nearest teenage stoner.
The efficacy of a tool seems to be the justification for its use and no real time is spent understanding why it works or where it comes from leaving the student with a head full of information and no actual knowledge. The physicist Richard Feynmann makes this point in an interview by describing how when he was a child he asked his father why a ball placed on top of his toy trailer stayed stationary when the trailer was pulled from under it. His father told him "it's called inertia, however nobody knows what inertia actually is". Feynmann points out that his father appreciated the difference between knowing what something is called and what something actually is. Without a fundamental understanding of where an idea or tool comes from then you're deprived of the foundation needed to build your knowledge upon - like trying to grow the foliage of a tree which has no roots.
In Carl Sagan's bestselling novel The Demon Haunted World he explains the importance of providing children with a baloney detector kit - which he describes as a device for fortifying oneself against religious zealotry, propaganda and all other forms of deception. The kit including 9 pieces of guidance such as: 'Wherever possible there must be independent confirmation of the facts and 'Arguments from authority carry little weight'. Learning by Rote is the antithesis of Sagan's advice and teaches students to absorb information blindly and without question.
It's a simple fact that ideas are easier to absorb when we 're able to stitch them into the existing fabric of our knowledge - if you want to know how to subtract 1/16th from 3/8ths then look no further than your nearest teenage stoner. As a teacher leads a child further into the rabbit hole of mathematics the concepts become ever more abstract meaning their ability to relate back to the world they know diminishes. Eventually they focus on learning by rote and with it lose everything interesting about maths. For instance they may understand that Pi is the ratio between a circle's circumference and it's diameter or maybe just see it as a number and a button on a calculator. Either way this misses everything that's interesting - Why is the ratio the same for every circle? Did we invent this number or discover it? If the number is infinitely long (irrational) then does any sequence of numbers you can think of appear somewhere in Pi?
This is not just about making children better at maths, it's about tapping into the same wonder and curiosity about our world which led mathematicians and scientists to make the discoveries in the first place. This takes us beyond being calculators regurgitating figures and ignites what it means to be human. Sorry I forgot to mention, you can open your eyes now.