On teaching students to show their working

Karl F Gauss (1777 – 1855) “I have had my solutions for a long time, but I do not yet know how I am to arrive at them.”¹

I have struggled with the problem of persuading the talented students to show their thinking in any detail – the greater the gift, the less likely they are to want to explain the details.  After all, if it is blatantly obvious, who wants to discuss why it is so?

“Because I say so!” is a non-starter – too many of them can turn mulish.  The argument that it helps one to find one’s errors doesn’t work – they, having genius egos and little experience, do not really believe that they can be in error.

In literacy and other communication-loaded subjects, the argument that the details are needed to help the audience understand one’s thoughts may help.  Unfortunately, if the student feels that the teacher knows the details already, ve may logically respond that details are therefore not required.   More powerful for the score-motivated, however, was the clear opening statement that the person assessing one’s work will have a mental checklist of “things to give points for”, and that including those statements will get one points.  This was particularly important for a student, asked to discuss  a complex piece of writing, who skipped all the obvious bits (the teacher’s focus, of course!) to discuss subtleties in the subtext .  However, unless the teacher has written a very careful programme, has a very clear marking key linked to the topic, and has provided examples of pieces showing discussions of  “obvious bits” ,  and the student has attended, this leads to the lesson being that the teacher wants the student to guess what the teacher wants to see –

Now,  here is a win/win situation: that “unless” sounds like good teaching practice,   and the latter outcome reflects the reality of many test and workplace experiences .²

But then, there are those with much more internal motivations.   One approach for them which  I enjoyed involves a little history and psychology:

Many experts find they “know” but can’t easily explain why – like the soldier in Afghanistan who believes there is a trap, but can only say that “that street gives me a cold feeling” – not much help when training a rookie.   I knew a mathematician who had spent three years working on proving that two different areas of extreme geometry which were using different theorems could be reduced to one set of theorems – he “knew” they were linked, but had to prove it using both sets of theorems.  In higher maths, I myself often knew the answer to a problem and then had to work to choose the approach to showing that it was so.   

Even the great scientists and mathematicians experience this:  thus, the Gauss quote: Karl F Gauss (1777 – 1855) “I have had my solutions for a long time, but I do not yet know how I am to arrive at them.”¹ 

If all these people know things are so, then it is not surprising  that some  students will start to know also.  But how does it come to be so? 

When   one is learning to read, one remembers the shape-sound match, then sounds-out the letters, then links the sounds to find the word, and try to remember that one while reading the next word.   At each step, neurons are firing chemical messages and stimulating each other to grow and link across brain structures.  Later, the shape-sound match is known: neurons have made linkages, so the visual system gets the verbal system for the letter-sound doing its job below consciousness.   Then the sets of letters in each word start having neural networks, so the whole word is known.  Later, groups of words are linked, so that if a new word is made up of them, the set of activated networks means that the new word is known.  With much reading (and especially with good speed reading training) the networks may link so that entire sentences may be grasped by the expert neural systems rather than  read “word by word” by the conscious mind. 

In all aspects of learning, it comes to the same thing: neurons signalling and  linking and signalling.  What you practice is what you get good at.

If you practice just knowing, and you have made a faulty connection somewhere, that fault is strengthened.  That is the first reason to check the working consciously.

If you practice just knowing, you can unconsciously be using systems elsewhere in your brain – other expert systems can be involved without your knowledge.  This often happens where science uses mathematics, or any subject uses literacy.   Examining your processes can make surprising linkages available to make useful shortcuts – for example. having to remember only one of five physics formulae because they are all derived from one mathematically.  Making links where no-one else saw them is what gets people Nobel prizes.  That is a second reason to check the working consciously.

If you practice just knowing, you do not practice self-questioning.  To question one’s own inner processes is a valuable skill in later life, when more complex problems engage many neural networks – many adults spend long hours with psychologists, trying to figure why they make such bad decisions and how to improve their lives.    Start practicing with small things – like adding three-figure numbers – and develop the skills you will need for the big things later.  That is a third reason to check the working consciously.

Practicing just knowing means you cannot be sure that you have really grasped the whole of the problem.  In a hurried situation it is useful to be sure enough to gamble on, but where there is time –  conscious checking allows certainty.  That confidence is a fourth reason to check the working consciously.

If you practice just knowing, you can be manipulated by events elsewhere in your brain.  If conscious checking gives an answer different from your knowledge, you have a starting point for finding out what you know (in the normal sense) which contradicts your conscious thinking.  Sometimes, there is a fault which can derail your life (see the psychologist’s work, as above.)  And, from experts in fields from medicine to farming, there is another case:  if you know the area really well, your gut feeling is often right – and finding out why is the path to excellence.    That is a last reason to check the working consciously.

Finally, the more complex the problem, the more useful it is to do the working in writing and in a careful format, so that each step can be checked over again.  Doing this gets easier with practice – so practice doing it in writing, neatly, and in the standard form for the subject.

 Go, and justify your answers from now on.  Or I will go over this again.

___________________________________________________________________

1.  K.F.Guauss, cited by Paul Williams (2010) p.338.  In Afterword in Sturgeon, N. (Ed.) Case and the Dreamer (pp.327-354)
Berkeley: North Atlantic Press

2. A teacher’s view of this fact of life, and advice on teaching students to live with it:  Mark Lopez “The Little Black School Book ” Vols. 1 and 2, 2008 and 2011, Connor Court Australia.

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