Prior to the proliferation of the University system, the primary means of passing down knowledge was through apprenticeships. The mentor--usually someone currently practicing a craft or skill--would take someone on as an apprentice and through this relationship the mentor would receive help and labor and the apprentice would learn the craft. Many have argued that this was the most efficient and effective method for passing down information and teaching ever devised. Modern science has started to back that up.
In the 1980's and 90's, researches at the University of Parma, Italy were running tests on the macaque monkey and accidentally discovered what has come to be known as "mirror neurons". They noticed that the brain activity responsible for kinesthetic control not only showed up in the primates that were performing a particular function--say, grasping for some food--but also showed up in the primates watching the action being done. Essentially, the exact same neuron patterns fire when we watch someone else doing a task as the ones that fire when performing a task. Scientists have postulated that this neuron mirroring allows us to learn how to do something by observation. It is by careful observation of someone actually doing something and then doing it ourselves that we learn with the greatest comprehension and retention. So much for lecture. . .
The difficulty with teaching problem solving (mathematical, coding, or otherwise) is that the process is mental and therefore not overtly displayable. A keen mind may look at the work of another and discern his/her solution and even perhaps steps to solving the problem along the way, but this is through reasoning and deduction rather than mirroring actions in real time. The question becomes, is their a method whereby a mentor can teach the process of problem solving to a student?
How does one display the mental process of solving a problem? Clojure inventor Rich Rickey suggests that most problem solving happens at the unconscious level with the conscious mind playing skeptic to flush out its validity. Most of us have experienced unexplained bursts of insight when attempting to resolve something. But, can you teach people to be inspired?
Ultimately a problem is merely a question, and when broken down, a sum of many questions each with small answers discovered through previous knowledge and micro-inspirations. The core of solving new problems is solving old ones. Breaking problems into pieces and auxiliary problems by identifying the conditions, data, and unknowns circumscribes all problem solving. So on to teaching.
Consciously or subconsciously we are asking ourselves small questions throughout the process of solution finding. So to display this process to a student, the core tool is the socratic method. It is by asking successive questions to the student that we teach him/her how to ask themselves the key questions needed to solve all problems. By successively deconstructing a problem into simpler and simpler pieces and asking corollary questions we find a piece of the puzzle that they can solve. When you find that you've gotten down to the basest level and they can't answer the question you have ascertained a hole in the students knowledge that can then be filled through explanation and demonstration. The student now has an activity to mirror. If the process is continued often enough, the student will begin to ask themselves the questions without prompting.
There are many other parts and nuances to effectively teaching people how to solve problems. And like any complicated skill, it must be repeated by both student and teacher in order to gain proficiency. But at the core effectively teaching any skill requires the demonstration of both the seen and unseen parts. From this point we can proceed and add to our pedagogy.
The methodical application and practice of these basic principles will help a mentor teach his students problem solving skills rather than merely making them memorize steps to finish a puzzle. By teaching steps to a predetermined process we reduce the portability of the knowledge we are giving and ensure their long term dependents on our presence. By teaching the skills of solution finding we ensure that the student can then go about autonomously in a manner of self-teaching and solving.
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