The question is not so much what kind of knowledge may be used when a teacher generates an example (say in a math class) but what kind of knowledge needs to be used and combined in order to create meaningful examples with lasting teaching effects. We know from research some major factors:

·         Individually generated information clearly supports both acceptance and recall of information. The effects are enhanced, if individual differences in usability ratings are taken into consideration.

·         People tend to favor information with a high personal relevance and entertaining values.

·         Better understanding needs deeper processing. But there is a tendency to understand entertaining media ‘at a glance’ resulting in a simple representation that is not linked to verbal information.

·         Deeper processing can be elicited by teaching appropriate learning techniques.

·         Supporting understanding demands the construction of semantically connected pieces of text and pictures, activating appropriate pre-knowledge, providing learning strategies for multimedia, and changes of media and learning perspectives to support the construction of comprehensive mental models.

·         Research support the consideration of individual differences in abilities and interests in order to enhance the understanding processes.

From a teacher’s perspectives this means that not only subject matter knowledge is needed (e.g. about mathematics) but also on general didactics (e.g. how to generate learning perspectives), special didactics (e.g. math didactics), general knowledge about individual differences (e.g. the visual abilities needed in geometry), and specific knowledge of interests, experiences, and pre-knowlegde of the students in the class. I have attached a brief review article and some research we did about these issues.

When a teacher selects and uses examples, he does that for clearer understanding of the subject matter.  It also shows the expertise of  the teacher"s knowledge of  the subject. When examples are used, it makes it easier for slow learners to catch up the lesson and can also meets the diverse students needs in the classroom be it in any subject.

Your question seems to be revolving around two concepts to me. The first is essentially Bloom's Taxonomy, the other is the styles of learning.

When using examples, particularly the maths examples you suggest, it would fall into any number of the initial five stages under the Cognitive Domain: remembering, understanding, applying, analyzing, or evaluating, depending on the pedagogical function of the example.

In terms of styles of learning, this would coincide with the Logical domain, which emphasizes logic and reasoning over other approaches (such as visual/spatial, auditory, physical, or verbal, to name a few others).

While the teacher may be demonstrating their own knowledge or understanding of their subject/materials, from a learning perspective the teacher is not doing this to emphasize their own knowledge so much as to extend knowledge to others. To use examples effectively, the teacher must have a knowledge of the subject, but the purpose of using examples has less to do with reflecting Subject Matter Knowledge/Pedagogical Content Knowledge and more with the modes of teaching their subject.

While you can argue that by using examples to convey a lesson using logic-based approaches to learning is a component of pedagogical content knowledge, PCK and SMK both rely on a conscious awareness of the content; I know people who teach and have no pedagogical understanding of what they do - they just do what worked for them as a student without conscious thought.

Just my take.

Cheers,

Rob

In my opinion, it is important to distinguish between two different types of examples: prototypical (1.3 as a decimal number, or and equilateral triangle with a horizontal side as example of triangle) and nonprototypical (any infinite decimal as 1.33 ... as a decimal, or an integer number as a fraction).

This distinction is related to advanced mathematics didactical issues as the use of ostensive definitions, teacher management of students’ processes of generalization, teacher's knowledge of student thinking and its development (Van Hiele levels, for example).

I think that not prototypical examples are the most appropriate to study and deepen teacher’s knowledge.

The problem with examples is that they do not always produce the general learning they are supposed to.  

(1) Students may (and do) attach importance to the wrong or irrelevant parts of the example and faithfully apply this to their application of the subject.  They may always include a statement which was relevant to the example, but not in general, or they may use a procedure which applies to a particular category for all subjects.

(2) Students may only be able to deal with exercises in presented in a very similar form.  This is tacitly used in higher education where apparently high level problems are dealt with at length in tutorials, and then exactly the same form is used in exams.  (In extreme cases, the same question but different numbers for the calculation.)  This gives a false impression of the actual competence of the student. but impresses inspectors and external examiners.

(3) Students may treat the example as the only application.

Thank you very much everyone for your contributions ; They are very enlightening.

Our focus is the knowledge of math teacher, and our research is to identify knowledge ( within Subject Matter Knowledge and Pedagogical Content Knowledge ) are necessary for effective teaching . Examples spaces ( see Mason and Watson) in relation to a specific part of that content knowledge. But there are different in nature the mathematical knowledge on the examples in relation to the phenomena of the emerging concept that knowledge of the role of syntax counterexample in mathematics or knowledge on how to choose and sequence examples for teaching specific content mathematical. These different types of knowledge regarding the examples I wish it were the focus of debate.

The particular area of mathematics teaching that I know about is teaching the subject to engineers at university (at two universities myself and by discussion with other academics).  The most prestigious institutions only accept students with a high grade (often A as minimum) in mathematics.  Others may accept students with no advanced qualification.  I would say that the teaching challenges are different.  To roughly quote a friend "I have a class of students who are only in University X because they could not get into universities A, B, C......  They are doing engineering because they could not get into management and business studies.  They are only doing maths because it is a required class.  They have no interest and only want to pass with as little effort as possible.  That is a real teaching challenge."

For any subject, the two key factors are what the student knows already (including what is known, but is actually a misunderstanding that has to be undone) and the objectives of the course.

It is notable that "the maths problem" in engineering has been discussed many times in the UK.  You can find books and papers by a search of Google Scholar, e.g.:

http://www.hull.ac.uk/engprogress/Prog3Papers/Edward.pdf

I have several times been asked to referee papers for journals or conferences which boast that an engineering department has solved "the maths problem" by taking the teaching away from the maths department.

Some common problems appear to be:

(1) Teaching engineers is supposed to be a simple task which can be assigned to any mathematician, even a junior one without any preparation.  Thus the same module may be taught by a different person every year, some good, some poor.

(2) Mathematicians assume that maths is worthwhile and interesting for its own sake.  The above document notes that school students may be able to do algebra but not know why. 

(3) Conversely they try to be helpful by (a) assuming that all engineers are the same, so applying maths to the oscillation in an electronic circuit will equally inspire civil and chemical engineers (b) Giving examples at an inappropriate level of the course.

(4) To get the maximum benefit on what is called the "Full-Time Teaching Equivalent" - i.e. service teaching all the engineering students for 10% of the year is counted as the same as teaching 10% of the numbers as full time - they are put in the largest classes possible.  (See also 3(a).) So-called tutorials or examples classes may have just a few postgraduates in a room with a lot of students so that most students do not get even a few minutes direct help.

There are some issues on which I would think from the perspective of knowledge that shows a teacher when you select or use examples.

To investigate about the space of examples of a teacher, in a specific mathematical topic, could provide us information about the level of understanding of the teacher about that topic. Thus, the choice of a collection examples that show, in each case, a key feature of the concept, overshadowing the other or others allowing certain relationships between them, can give us information on the teacher's knowledge about the characteristics and their relationships, representing knowledge on the subject (SMK); but also it gives us information about the progression in teaching that concept or topic by the teacher that in some cases, it may be conditional on their knowledge of learning difficulties of their students on the same topic, which is pedagogical content knowledge (PCK). The identification of these different types of knowledge and their relations can be useful in the initial training of mathematics teachers.