Thanks for your interest in my article. I certainly believe there's a place for rote, as your examples highlight. But to apply their knowledge of number in the real world, students need to combine their procedural skills with an understanding of how numbers behave and relate to one another. Indeed, understanding and context aid memory even if that is the ultimate goal. As does engagement - which rote learning, as practiced in most schools, largely dispenses with.
The revised title is equally misleading. Problem solving is so universal one could argue it is the defining human trait. Our approach to problem solving on the other hand is heavily influenced by culture and personality.
Great piece; I think we can all relate to Gmail moments.
Worth noting that what delights users initially will likely be taken for granted over time (see the Kano model, for example). An exciting challenge for Product Managers is to constantly dig out these unknown needs you mention.
It's important to note the distinction between breezing through course material and breezing through an exam.
I have taught plenty of students who possessed a deep understanding of mathematics but could not perform within the confines of a structured exam. They may be berated for performance or even lack of preparation, but given that problem-solving is the main criteria for admissions this surely reflects the limitation of the measuring tool.
Similarly, many students who breeze through A Level exams may have thought they'd mastered topics like calculus, but it turns out the exams only called on their computational fluency. The moment they are pushed conceptually and asked to formulate their thinking with the rigour that undergraduate mathematics, they come unstuck.
Very fair question - alas, I did not have the authority to change the system (and, for selfish reasons, would not want to as my exam scores flattered my actual ability).
Worth noting that written exams do not feature at all in research. There's a recognition at this level that mathematical understanding can not be captured through blunt testing instruments.
Do you think you're confusing existing "bad exams" with all possible exams?
I don't recall ever sitting an exam with 10 questions where if you got 2 of them out you'd come first. You'd mark such a "hard" paper based on "this is a reasonable avenue of exploration", "this is a good idea that won't work", "this approach will prove ultimately futile but given the student hasn't seen anything like it before it's definitely worth some points."
Why are you trying to achieve just that in an interview?
Research isn't done in interviews any more than research is done by sitting exams. However sitting down and exploring ideas with a pen and paper by yourself is likely closer than "creating an impression" in an interview. An interview where you, the interviewers, are saddled with the irksomeness of having to roughly account and adjust for your bias as much as you can be aware of it (does she really look like a mathematician? I didn't like sportsman Ben's approach as much as bespectacled whoever - is the impression based on the maths alone or is the form of it's presentation fairly important).
The huge advantage of grading an exam is you can exclude all of that gumf we all carry (with differing expressions of it) by simply not knowing who wrote it. Not their name or anything.
The other big win is that if you publish the blessed thing you can make such exams more normal. Budding mathematicians
might practise such skills and get better at them, younger.
As it is you're selecting for the children of mathematicians who get such practise at home and excluding the very smart who honed their exam skills alone because that's all they encountered.
I also got very flattering scores on mathematics exams but I wouldn't mistake that for anything beyond my ability to act the student performing seal at that particular game. Any exam where it is remotely possible to get 100% does not test anything like creativity - creativity is the thing that is most important, no?
"Moonshots in Education" is a book full of case studies from all around the world. Implementations are as diverse as the conditions of learning environments. Intelligent tutors may be fully integrated, squeezed on the sidelines, or somewhere in between.
