Those who work in education usually have a number of anecdotes ready at hand. Often these describe someone's humorous or otherwise notable experience with a student or a colleague. The majority of these anecdotes do not pretend to be anything more than their face value. This changes, however, when they carry or evoke a message. Such anecdotes can then acquire a considerable value. Professor Hans Freudenthal was a world-renowned storyteller. Do any of us not know a few of his fine stories about his children or grandchildren? Take, for instance, the following anecdote about his grandson Bastiaan (Goffree, 1994).

"We're sitting at the dinner table. Bastiaan opposite his younger sister, father opposite mother, grandmother opposite grandfather. Suddenly, during dessert, Bastiaan says, holding up a spoon containing six blueberries, "we're this many". "Why?", I ask. "That's how I  see it", says Bastiaan, continuing: "two children, two grown-ups and two grandpa and grandma." (Perhaps the berries in the spoon were lying in the same die pattern as our seats around the table, but that I didn't see.)

This proved to be no coincidence. The next day, holding four strawberries in the palm of his hand, Bastiaan said, "This many live in our house." At that time, Bastiaan was still uncertain of how to use numbers and he absolutely refused to count, which is unusual at that age. Rather than having an understanding of numbers, in this story he displays something of a geometrical comprehension, which is perhaps normal at this age." Bastiaan (4;3) didn't count, but he recognized similarities between quantities by their geometrical structure. This is one of the many examples of the way in which Freudenthal, through an anecdote, demonstrated insights into children's learning processes. Anyone who has once heard or read such an anecdote will not soon forget either the story or its purpose. This is the strength of such stories: they are theoretically charged and they carry with them the kernel of ideas on learning theory and education. And this was precisely Freudenthal's objective with these anecdotes: to describe observations of unique learning instances that provide insight into learning processes - particularly into learning leaps or level increases.

Such characteristic 'classic examples' or paradigms can be of great import for those who are involved in education and who wish to continue to learn from their own practice. As Freudenthal (1980) states:

"One can learn more from a single paradigmatic instance than from a hundred irrelevant ones (...) Such an opportunity should be taken advantage of."
 

Freudenthal himself took optimal advantage of such opportunities, especially with respect to forming a link between theory and practice. And this particular link can be formed extremely well through concise, theoretically charged anecdotes. Paradigms have a natural connection to 'narrative ways of knowing'. They recount quality observations and provide them with a reflective note and a strong theoretical charge.

These anecdotes offer lucid insight into an exceptional phenomenon. Through this 'narrative way of knowing' (Gudmundsdottir, 1991), such anecdotes are seen as an excellent way of closing the gap between theory and practice. Anecdotes can help one understand how theoretical elements can be used in practice and they can also aid one in recognizing the theoretical elements in this same practice. The paradigms described here are such anecdotes.

Four examples

David and his teacher.
David: 15's odd and 1/2 is even.
Teacher: 15's odd and 1/2 is even? Is it?
David: Yes!
Teacher: Why is 1/2 even?
David: Because erm, 1/4 's odd and 1/2 must be even.
Teacher: Why is 1/4 odd?
David: Because it's only 3.
Teacher: What's only 3?
David: A 1/4
Teacher: A 1/4's only three?
David: That's what I did it in my division.

Anyone reading this for the first time will probably not have the faintest idea what is going on. But once the idea of models for fractions is introduced, the 'clock model' will spring to mind. This anecdote is used in Mathematics Education classes at institutes for teacher education in the Netherlands as a way of revealing in a nutshell how children form their own concepts, in this case with respect to fractions. It shows how children will sometimes design a 'mental model' all on their own to aid them in providing numbers with meaning or solving problems. Educators who carry such anecdotes in their theoretical baggage will be able to situate and guide children's learning processes in increasing breadth and depth.

An educator will use such a paradigm with a group of student teachers and purposefully allow the tension of incomprehension to build. The denouement will be revealed either when one of the students suddenly calls out, or when the educator draws a clock or a circle on the board.

Steven
Steven (5) had drawn a pond with a few ducks swimming in a line. The teacher said, "I see you have five ducks in your pond." Steven looked at his drawing in some confusion and replied, "That's not five, because there isn't one in the middle."
 

This anecdote clearly demonstrates how Steven's concept of the quantity 'five' was limited to the image of the (geometrical) structure of a die. It reveals - as did the story about Bastiaan - a significant facet of how young children develop the numerical concept. The anecdote shows how a misunderstanding can arise that may disturb further learning either temporarily or for a longer period of time. Thanks to certain examples, Steven had developed an image of the concept 'five' in which he was focusing on the wrong aspects. This anecdote, too, is often recounted. It is seen through more quickly, but evokes nonetheless a theoretical context. What does a kindergarten teacher do to develop such an image, and what can be done to prevent this? Directly linked to this example is the question of how such images arise. How does a child develop this and what can a teacher do to help?

Els
Els was an average student in arithmetic, and she could solve bare problems pretty well. One day she was presented with the following problem: 'Next door to me live a family with a father, mother and son. The son is fourteen years old. His father is four times as old as he is. How old is the father?' Els drew (fig. 1):

and she said: "that's added up twice."

And then (fig.2):

and she said: "and so that's four times."

This anecdote about Els illustrates the problems models sometimes can cause children, and shows how an ostensibly insightful model can prove much more mechanistic for some children then one would expect. (The case in question involved the intersection model, where the idea of addition lies in the background.) At the same time, it demonstrates how progressive schematization can be a natural approach. The educator can turn this problem into an educational one: what exactly are the educational implications of this anecdote? Based on this observation, what would you now decide to do and why?

Paul and Necmiye

In an interview situation, Paul and Necmiye were asked certain multiplication products. Paul was rather slow and didn't seem to know all the products by heart. Sometimes he could be seen counting on his fingers and would later report a strategy that did not seem to correspond to his behavior, although it was a good strategy. Necmiye knew all the multiplication products she was asked. Sometimes she repeated the multiplication: "oh, yeah, that's 6 times 3, urn, that's..." and then gave the answer. She mentioned no strategies; "I just know it", she stated frequently. The last problem asked was 12 x 6. Necmiye didn't know this one, nor was she able to figure it out. She remembered 11 x 6, but 12 x 6 proved too difficult. For Paul, on the other hand, this problem was no different from any of the others. This anecdote clearly displays the power of strategies and their advantage over mindless memorization. The educator can first play the part of the 'Nieuwe Media' videotape of this interview (Van Galen et al., 1989) in which Necmiye and Paul are first asked a number of products, and then Necmiye is asked the product of 12 x 6. This is a fine moment to stop the tape and have the students think about memorization and use of strategies. The discussion will change once the students have seen Paul solve 12 x 6.

What an educator can do with paradigms at an institute for teacher education

- Paradigms are illustrations of a theory. They help one better understand and situate the theory.

- Paradigms also help one situate one's own experiences with respect to certain theoretical ideas. They clarify one's own experience in practice.

- Paradigms reinforce theoretical ideas. Such anecdotes aid one in remembering and recalling the theory.

- Paradigms are therefore also a label for a theoretical idea. Thanks to the paradigm, we know exactly what is being discussed.

- Paradigms can also provide the occasion for a small investigation at one's own internship school. Do some kindergarten children here also believe that five always has to have a dot in the middle? Such investigations produce new anecdotes that can once again be used to comprehend portions of the theory.

- Paradigms can also be included on a test as a way of asking about certain parts of the theory.

A personal collection of paradigms

A fine collection of paradigms enables one to acquire a grip on and insight into one's own teaching practice and also enables one to improve its quality. All educators have such a repertoire of personal anecdotes. Although it should be noted that everyone has personal preferences, some anecdotes simply make a stronaer impression than others. In addition, anecdotes will gradually become one's personal property: something that was first read or heard will be combined with one's own experience and thereby be come a new, personal, story. Each educator should possess a variegated collection of anecdotes, particularly for sections of the theory which cause more difficulty or are less directly appealing to the students. In addition to orally recounted anecdotes, the educator's repertoire may also consist of audio or video fragments of class observations, fragments from the educator's of the students' journal, or passages from a book.

The collection

It is clear that many sources can be used to build a personal repertoire. Anyone in search of fine paradigms should take a look at:
- The primary school.
- Student journals.
- Anecdotes of colleagues.
- Professional literature.
- Research literature.
- Video clips Educational television programs.
 

Collecting paradigms is not all that easy. After all, not every educational anecdote is necessarily a paradigm. They only become paradigmatic when the students are able without difficulty to discover the theory within the story. Good paradigms reveal instantaneously what is going on, contain little superfluous material and are easily remembered. In order for a paradigm to become a label for a section of the theory, it must also stand for such an element and be able, later, to easily evoke the theory.

6 Designing education

Since the new institutes for teacher education were established in the late nineteen-seventies, a distinction has been made between the various professional roles of a primary school teacher. The following roles are listed in the Model of an Educational Curriculum (MOLP) (SLO, 1982): children's mentor, educator, developer, innovator, and discussion partner (MOLP, p. 39). Since the appearance of the MOLP, there has been the occasional tinkering and, occasionally, the role of educator has disappeared from view. But nowhere is a teacher regarded as a professional problem solver, and for good reason. Learning through problem solving is not an unfamiliar concept in mathematics education. Problem-oriented mathematics education is held in high regard and compares favorably with so-called 'task-oriented' mathematics education. Mathematics is knowledge (and skill and disposition!) that is pre-eminently suitable for solving certain problems. Even the manner used to solve problems has a mathematical character.

The process begins with problem identification: a problem is identified or a given situation is problematized, in order for one to get a handle on it. At the same time, certain knowledge is actualized and experiences from previous problem situations are recalled. This is followed by the phase of problem analysis: an attempt is made to structure the problem and ideas arise for an initial approach. Sometimes the situation is so transparent that a plan of approach can be designed directly. With complex problems, however, that admit no algorithmic solution, a heuristic (searching) approach will have to be taken. Extremely obstinate problems also exist, of course, which defy any and all repertoires of heuristics. These are called 'wicked problems', and here a search must be undertaken for an entirely different path of approach. Available knowledge and skills are used while tackling a problem, and it is necessary at times to expand one's collected knowledge, whether through the aid of others or through one's own discoveries. Here, the learning instances during the learning process are explicitly revealed. If the focus is primarily on learning, it is critical that one or more reflective moments be included in the process.

For certain professions or areas within these professions, it has proved possible to acquire expertise through solving problems. In its original form, this was termed 'problem guided' learning (Schmidt & Bouhuijs, 1980). More recently, 'Probleem Gestuurd Onderwijs' (PGO) or 'Problem Guided Education' has become a popular term in the Netherlands. Since the late nineteen-seventies, the University of Limburg has designed its medical studies according to this approach. The students learn the profession of medical practitioner through solving (medical and related) problems. This takes place in educational study groups of approximately twelve students, who work according to a set approach to problem solving. In this approach, the primary focus is on learning, while problem solving and the problem solving process are subordinate. In PGO circles, the high motivation of these students is spoken highly of, as well as the fact that they learn to apply their knowledge to problems in actual practice and that new knowledge is acquired in direct relation to practice. This method works wonderfully in the medical world, where diagnosis is at the heart of a doctor's work; diagnosing is, after all, solving problems. A similar approach can be seen in the law department at the University of Maastricht, where the education was also designed according to the PGO model.

How does this apply to the teaching profession?

A teacher's main function in the classroom is not that of problem solver: not when preparing the lesson, not when choosing a textbook, nor when helping a slow arithmetic student. Perhaps we could classify making a diagnosis as problem solving. But a teacher's approach - which involves catering to a wide range of educational needs - does not have the characteristics of the problem solving approach described above. Catering to a wide range of educational needs (and many other activities on behalf of students) is a matter of observing, inventing questions, designing problems and accompanying tips and explanations, thinking up ideas, talking with students, encouraging the students to reflect on their own activities, etcetera. In the case of adaptive education (in which the teacher takes advantage of the differences between students), a similar situation is found - only here it is not limited to a student. Evidently, a teacher is an educational designer rather than a problem solver. This should be kept in mind if one is planning to organize (sections) of an institute for teacher education according to the PGO model. This case therefore only concerns motivation and learning applications.

The focus in this chapter is on educational design. Student teachers must learn to design education themselves, and they can learn to do so in the following ways: by observing how their own education was designed for them; by - as professional practitioners of reflection - designing education together with an expert at their institute; by becoming familiar with what professional educational designers have created; by watching education being designed at their internship school and by questioning the designers; by trying out their own creations in educational practice.