Let's say I learnt some formulae to solve permutations and combinations. I am trying to learn it by myself, when I think I've already learnt the principle and I can make use of inductive thinking to cope with all the problems about both issues, I find problems I can't solve, and when I go back from scratch and see how someone else solved the problems I only find different gaps I had not filled in my mind to understand the complexity of these topics and I think I build a more complex understanding, and again fail with problems I think at this stage I would be able to solve. I conclude saying I need to work more on that, but I've thought I had learnt something so I did not realize about my limitations till I was challenged.
Hi Cesar,
Your question and posting -- To what extend do we understand our own limitations to self-regulate our learning -- is an interesting question.
You say that you have learnt some formulae and algorithms to solve permutations and combinations problems. You go on and say that when you try to learn it by yourself and think that you have already learnt the principle and can make use of inductive thinking to cope with the problems about both issues (i.e., permutations and combinations), you find problems you can't solve. Thus, you begin to understand your own limitations to self-regulate your learning
This example (i.e., to solve permutations and combinations problems by appealing to learned formulae or rules of thumb so as to solve a certain intellectual problem) shows that it is important to distinguish between rote learning, memorized answers and answers based on learned formulae and algorithms, and self-generated answers, operational learning and understanding. If the latter is the case (i.e., operational, self- regulated learning and understating), then it is likely that you really master the concepts of permutations and combinations, and hence, you are capable of solving permutations and combinations problems whatever. If the former is the case (i.e., to solve permutations and combinations problems via learned formulae and algorithms), then it is likely that you find some permutations and combinations problems you cannot solve however much you can make use of inductive reasoning or thinking to cope with these -- or similar to these – problems.
In what follows, I present a Piagetian problem that shows that to solve a formal operational problem (e.g., a proportionality problem) via operational understanding or self-generated learning is quite different from solving such problem by applying a learnt formula or algorithm.
Suppose that you have 2 bags, say, bag A and bag B. In bag A, there are 2 white pearls (favorable cases) and 3 black ones (unfavorable cases). In bag B, there are 3 white pearls (favorable cases) and 4 black ones (unfavorable cases). During the experiment, both the bags and their respective beards remain visible so as to not transform the problem into a memory problem.
Here it goes the problem: “Is it more likely, at a first trail, to take away a white beard from A, from B, or is it equally likely to take away a favorable case from either A or B and why?”
You give a “correct answer and justification” when you say that it is more likely, at a first trial, to take away a while beard from B than from A because when you divide the favorable cases by the possible cases (i.e., 2/5, bag A; 3/7, bag B) you get a greater ratio in B than in A. In other words, you solved the above mentioned problem by applying a learned formula or algorithm. Does this mean that you have an operational, self-generated understanding of the proportionality concept? Not necessarily, just because you solved the problem by applying a learned formula or algorithm.
Consider now the following response and justification: “It is equally likely to take away a white beard from either A or be B because when one first compares the white beards (favorable cases) with the black beards (unfavorable cases) either in bag A or bag B (two intra-comparisons), and then one compares the number of black beards contained in bag A to the number of such beards in bag B (one inter-comparison), for example, one can see that, in both bags, there is more one black beard than white beards (2 and 3, bag A; 3 and 4, bag B, respectively).
Needless to say, this answer is incorrect because it is more likely to take away a white beard from B than from A. Although incorrect, such answer is, in Piagetian terms, a formal operational and self-generated answer and does not relies on a learnt formula or algorithm.
Imagine that you give the following correct answer and justification: “It is more likely, at a first trial, to take away a white beard from B than from A. In fact, it is true that, in both bags, there are more one black beard than white beards. However, it is more likely to find a favorable case in B than in A because I understand and, so to say, discover that the more a proportion increases the more it tends to be equal to 1, as is the case, for example, of 100/100, or 1000/1000.
I think that the more we find problems that we can solve on certain occasions and situations, but not on others, the more we understand our own limitations to self-regulate our learning.
Of course, it is possible to accelerate the acquisition (more than the understanding) of certain concepts, like permutations, combinations, and the like. Paraphrasing Piaget, I would the following to ask: (1) is such acceleration beneficial or rather detrimental to the individual’s development and education? For Piaget, whenever one prematurely teaches a given individual something that s/he could have discovered by himself/herself, that individual remains deprived of understating it completely; (2) is this learning through acceleration lasting?; (3) how much generalization is possible?; and (4) what was the individual’s operational level or stage of development before a given learning experience and what more complex structures has this learning succeeded in achievement? (See, for instance, Piaget, J. (1964). Development and learning. In R. Ripple & U. Rockcastle (Eds), Piaget rediscovered (pp.7-20). Ithaca, NY: Cornell University Press).
Thank you Friend,
Most interesting example to follow.
In my capacities as : me, teacher, mother, provide, leader, Researcher, I realised to realise my limitation, taking challenge,
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