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Been looking for something like this for ages. Great stuff.

Now you're just being pedantic.

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.

That's fair enough - was not intending to mislead and hopefully the amended title is more reflective of the piece.

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.

"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.