Authentic learning

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.

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7 Comments on “Authentic learning”

  1. Hi Greg, isn’t the pursuit of an open-ended problem the way into finding and discovering a determination to master relevant exercises? My feeling is that too few people are giving themselves permission to go after big, real, open-ended problems because they’ve been led for too long through exercises that might one day mean they are in a position to solve other people’s problems. My education left me primed to be a ‘scientist’ in an important technical industry; but doing that work was never my problem, so I left. Other people in the same position can feel too scared or trapped so they don’t leave and become miserable, or become fascinated by position, promotions and benefit packages. I’d like to see more people going after their own open-ended problems… and I’d completely trust those people to find any help they need in the exercises they need to master to get closer to their own solution.

  2. Leah, A “just in time” approach to learning is what you are suggesting. I.e., present student with open-ended problem, and student will then have a need to learn what is necessary to solve it. That is not an efficient way to learn any more than you teach someone to swim by dumping the person in the deep end of the pool and yelling instructions from the side on how to swim. Ostensibly the person dumped in the pool needs that knowledge, but the “just in time” approach is going to result in one of two outcomes: 1) drowning or 2) he reaches the other side and says “I don’t know how I did that but I never want to do that again”.

    Support has to be developed incrementally with mastery of the foundational principles that allow application to problem solving. Open ended problems work best with those who have expertise.

    See: http://www.ams.org/notices/201310/rnoti-p1340.pdf and http://www.educationnews.org/k-12-schools/fran-henderson-pingry-and-me-a-tale-of-problems-vs-exercises/.

  3. Hi Barry, thanks – and I agree with what you say; it would be crazy to line a random bunch of kids up and dump them into the deep end of a pool. I’m also saying it’s …crazy… to take those same kids and move them collectively through swim exercise after swim exercise in the hope that some of them, one-day, will come to the problem that e.g. no one has yet swim X meters in under this time and then to decide to solve that problem. Most of them will be technically able to do it, but what if most don’t care? What if just being able to swim enough to have fun in the shallow end was all most of them wanted to do, as far a swimming goes. Or some wanted to dance in the water. And in all that time dedicated to speed-swimming exercises, they’ve got the message that the things they really do care about are less important than swimming quickly? So, what I’m asking is that teachers see that students come with their own open-ended problems (no presentation of problems required; inspiration yes, sharing of interest yes.. but no need to phrase an open-ended problem for a student to solve). If what a teacher has to teach -those exercises- is what a student needs to know, then we’ll have a genuine teaching/learning relationship going on. I believe the world can do without all the other exercises that happen without the student consent or engagement.

    • Don’t quite agree. Purpose of education is to prepare students. We don’t have a crystal ball. I don’t know which of my algebra students will want to pursue a STEM path. But algebra is basic for STEM, so I teach it as it should be taught. Non-routine problems are fine, but you start with routine and go from there. Most problems in STEM are going to be variants of basic problem types. There are those who go into research and spend years working on unknown problems, true. Different levels of expertise exist.

      See http://www.ams.org/notices/201310/rnoti-p1340.pdf

    • Roger Curtis says:

      One can only hope and history suggests that we don’t have to hope too hard that somewhere along the line the spark will catch. It might be because of a comment or observation made. It may stem from a connection made, a challenge issued by a fellow student or a simple day-dream. Give kids credit and stop worrying.


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