In mathematics education, teachers selects and uses examples; these examples shows us some aspects of their mathematical knowledge and pedagogical content knowledge.Our interest is to identify these aspects of knowledge.
I'm not sure what exactly do you mean by exemplification. My current research program addresses this in detail. Your idea is a good one , although I am hoping that k-12 educators are more than well-equipped to teach possibly any of those courses (I am unfortunately finding that it is not the case :-(
If i understand the question
When the learners are facing a difficulty, teachers use examples, this approach can help learners to concentrate on understanding.
By analyzing the examples used by the teachers we notice that they mobilize (transport) their knowledge of a context (current course) towards other less complicated. The learners interact better with the supplied examples.
This analysis (examples) showed us that there are three levels of knowledge structuring: knowledge of the domain, and the nature of the element of the domain, and the concepts.
if it is the case, may be this paper can help you : Knowledge structuring for learning by level
http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=7358392&newsearch=true&queryText=Knowledge%20structuring%20for%20learning%20by%20level
I agree with George with a reminder that this is true across all disciplines. A suitable example (or experiment) is the most graphic and productive learning.
Yes George, carefully chosen examples (experiments) are pointers on the way via "professional life". They can become the starting points for their own thoughts.
Although it does not have to be conceived in this way, Pedagogical Content Knowledge often focuses on the way the teacher organises the material to be taught - the order of presentation, analogies and examples used, and so on. I do not wish to diminish the importance of this. Acquiring disciplinary concepts and understanding is important and, contrary to some thinkers, I believe can be effectively acquired by students when the teacher has organised the material well and directly teaches it. However, concepts and understanding are not all that students need to develop. There are also disciplinary ways of thinking. Although some writers on PCK recognise this, disciplinary ways of thinking are not always explicit in concepts of PCK. We suggested a partner concept to PCK - Pedagogical Process Knowledge (PPK) to try and make this aspect of teaching more explicit in the context of science education ( https://www.researchgate.net/publication/242220122_Adding_Pedagogical_Process_Knowledge_to_Pedagogical_Content_Knowledge_Teachers%27_Professional_Learning_and_Theories_of_Practice_In_Science_Education )
Processes are, perhaps, even more to the fore in mathematics - if my days as a student are still typical, one was introduced to some mathematical concepts and then given questions to apply them to. However, the latter process was not always well supported, so that they seemed like mechanical hoops to go through with little purpose. So a focus on how teachers support the processes involved in practising mathematics might be even more important. As seems implicit in George's and Josef's contributions, that might be more than giving examples but somehow better helping students to use them in developing mathematical processes.
Article Adding Pedagogical Process Knowledge to Pedagogical Content ...
Sorry, George. In my opinion your answer is correct, but also is obvious.
There are large parts of our courses which cannot be understood by the most of the students, without the associated supporting examples. For example, as an appropriate example of applications of the contraction principle, I recall the approximation of the solutions of some equations which cannot be solved exactly, by means of successive approximations method. Another supporting example related to the same theorem is deducing the proof of the existence and uniqueness theorem of the solution of the Cauchy problem for differential equations by means of contraction principle. The implicit function theorem itself can be proved using the same contraction principle. On the other hand, the important particular cases of the implicit function theorem cannot be well understood without some corresponding numerical examples. Even the evaluation of the error in computing the sums of some series giving numbers such as e, ln(2), pi, etc., are based on some numerical evaluations, based on approximation theory.
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