Originally posted elsewhere:
“An exercise is a question that tests the student’s mastery of a narrowly focused technique, usually one that was recently “covered.” Exercises may be hard or easy, but they are never puzzling, for it is always immediately clear how to proceed. Getting the solution may involve hairy technical work, but the path towards solution is always apparent. In contrast, a problem is a question that cannot be answered immediately. Problems are often open-ended, paradoxical, and sometimes unsolvable, and require investigation before one can come close to a solution. Problems and problem solving are at the heart of mathematics. Research mathematicians do nothing but open-ended problem solving. In industry, being able to solve a poorly defined problem is much more important to an employer than being able to, say, invert a matrix. A computer can do the latter, but not the former.”
Paul Zeitz, The Art and Craft of Problem Solving
Imagine a school for surgery. Gone are classes in anatomy or repetitive drill with models. Instead, surgeons are trained in authentic contexts. They learn anatomy just-in-time such as when they are elbow deep in somebody’s abdomen. If this sounds like a bad idea then it is because we readily recognise the difference between an expert surgeon and a surgical student. Those studying surgery need to master a lot of knowledge and skill before they get anywhere near an authentic problem; a real-life human.
In reality, medical students spend a lot of time cutting-up cadavers. They also spend time in lectures and studying from books. A friend of mine had an anatomy chart on the back of the toilet door. And she would memorise it like a parrot. Was this ‘rote’? She certainly memorised things but she also understood them.
Only when such knowledge is robust do students start to integrate it with other skills in an attempt to solve problems. Even the vogue for ‘problem based learning’ for the training of diagnosis starts a good way up the ladder with a huge amount of biological knowledge already assumed.
Unfortunately, this clear distinction between experts and novices fades when life-and-death issues no longer hold it in stark relief. Many assume that the best way to teach mathematics is to require students to behave like professional mathematicians; solving open-ended problems. The best way to learn science is therefore to conduct scientific investigations just like real-life scientists do and, of course, students of history should be ploughing through primary sources.
However, in these contexts, our budding students are likely to learn less. They may become cognitively overloaded, with too much information to process. They may simply be exposed to a problem sub-type so infrequently that they don’t effectively learn from it or they fail to link it to other examples (If you want to improve your golf drive, you don’t play nine holes of golf; you go to the driving range and hit fifty or sixty drives). If you want to train future experts then you need to isolate the various components of that expertise, train those components and then systematically bring them together through a carefully chosen set of contexts that illustrate the deep structure of the concepts or problems.
But there is a persistent clamour for ever more authentic learning. Is your authentic learning not working? Then it must not be authentic enough. And it is a favorite trope of amateurs with an opinion.
It comes from a kind of romanticism. We look at, for instance, mathematical drill in class, we note that this is not always enjoyable for students and we proclaim, “But that’s not what real research mathematicians do!” We yearn for a shortcut so that students may appreciate the beauty of the subject without the grind.
But no such magical techniques exists. I am sure that expert players of the harp derive enormous pleasure from it. Yet, I am not going to replicate that pleasure by just plinking about on a harp myself. I will know that the sounds I make are inexpert and this is unlikely to be motivating. Instead, if I wish to experience the pleasure then I must defer this and undergo the short-term pain of practise – drill, scales and the like.
Maths, science and anything else worth learning are no different. Expertise does not come from simply copying what experts do.