TUBERIAS DE FUNDICION, ACCESORIOS, VALVULAS Y VENTOSAS

A ‘theoretical concept’ is defined as a concept from a list of 59 general pedagogical concepts and pedagogical content knowledge concepts which is part of the learning environment of the student teachers (appendix 2A,B).

Notions of theoretical concepts and lay concepts

A notion of a concept is seen as a synonym or a description that within the given context lends the same meaning as the ‘mother concept’ – which is always the concept mentioned first – in the list of 59 concepts. These are words or expressions that occur in the descriptions of these same ‘mother concepts’ from the register of The Guide. A theoretical concept or the notion of it manifests as factual information in a text, not as an interpretation of what a student might have been thinking.

It does happen that theoretical concepts are referred to by a name that is the same as concepts that occur in daily use, for instance the concept ‘multiplication.’ If no difference exists between use by students and use by lay people, it will not be considered as a theoretical concept. That difference will be expressed when there is a clear pedagogical surplus value, for instance when the concept is used in relation to another concept within the context of the given teaching situation or if the concept is

Theory-enriched practical knowledge in mathematics teacher education

used in a more contemplative sense41_{. In the sentence: “Fariet is doing multiplications,”}

‘multiplication’ is taken as a lay concept if no further connection is made or explanation given, while in the sentence: “Fariet uses smart multiplication with tens,” ‘multiplication’ is seen as a theoretical (pedagogical content) concept.

On the other hand there are also words that are not identical to one of the 59 ‘mother concepts,’ but that can have the same meaning. Sometimes they have the character of a ‘lay concept.’ An example is the phrase ‘make visible’ with the mother concept ‘visualizing.’ These concepts are scored if they occur in the description of the mother concept in the register of The Guide; they are also included in the score list together with the mother concepts. For these concepts it is also the case that they are only scored if their use within their context lends a meaning that is equivalent to the mother concept. Characteristics level 1: no visible use for theory

No visible and relevant use of theoretical concepts is observed; at most there is relevant use of notions of theoretical concepts.

The use of irrelevant theoretical knowledge occurs in case of incorrect or improbable statements42 _{or intuitive judgments in which theoretical knowledge has no meaning and}

has only been ‘mentioned.’

Characteristics level 2: reproductive or mechanical use for theory

Visible and relevant use of a theoretical concept can be seen in a sentence or in a cluster of sentences.

Where two or more theoretical concepts or theoretical notions are being used, there is no visible insight from the student teacher into the coherence between those concepts or notions of concepts. No use of relative language is observed, either on its own or in combination with demonstrative language.

Mainly reproduction of theory takes place.

Judging with the benefit of a theoretical concept occurs on the basis of simple reasoning. Characteristics level 3: integrating and synthesizing theory

A visible and relevant use of two or more theoretical concepts is observed, with visible insight by the student teacher into the coherence between those concepts or notions of concepts.

Judgments and conclusions are made with the benefit of theoretical concepts on the basis of logical reasoning (if... then implications, use of arguments, (re)considering, making relationships, generalizing), among other things with reference to literature. Sometimes a student’s ‘own theory’ is formulated and founded; reconstruction of theory takes place. In section 3.9 the concept of ‘theory-enriched practical knowledge’ (EPK) was introduced as a derivation of the concept of ‘practical knowledge.’ Within the framework of this study, level 3 of the use of theory is seen as an important indicator for theoretical enrichment of practical knowledge.

Table 5.2 gives an overview of the twelve categories for the nature (horizontal) and the level (vertical) of theory use by students. Table 5.3 shows examples of each of these categories and table 5.4 describes some examples of doubtful cases encountered by experts when scoring the meaningful units.

Table 5.2 Reflection Analysis Tool. Brief description of the twelve score combinations A Factual description facts: who, what, where, how B Interpreting For instance opinion or conclusion without foundation C Explaining For instance ‘explaining why’ D Responding, gearing to For instance, anticipation, continuation or alternative design, meta-cognitive reactions Level 1 A1 Factual description of events without use of theoretical concepts. B1 Interpretation of events without use of theoretical concepts. C1 Explanation of events without use of theoretical concepts. D1 Description, alternative event, continuation or meta- cognition without use of theoretical concepts. Level 2 A2 Factual description of events using one or more theoretical concepts without mutual connection. B2 Interpretation of events using one or more theoretical concepts without mutual connection. C2 Explanation of events using one or more theoretical concepts without mutual connection. D2 Description, alternative event, continuation or meta- cognition using one or more theoretical concepts without mutual connection. Level 3 A3 Factual description of events using one or more theoretical concepts with a meaningful connection. B3 Interpretation of events using one or more theoretical concepts with a meaningful connection. C3 Explanation of events using one or more theoretical concepts with a meaningful connection. D3 Description, alternative event, continuation or meta- cognition using one or more theoretical concepts with a meaningful connection.

The units (table 5.3) have been borrowed from reflective notes in the final student teachers’ assessment (appendix 12), which was the closure of the course within the framework of the research.

Theory-enriched practical knowledge in mathematics teacher education

Table 5.3. Examples of the combinations A1 up to D3 Score

combination

Example of the meaningful unit

A1 At the front of the class there is a suitcase with tennis balls.

Explanation: This is an actual reproduction of the situation. No theoretical concept is used.

B1 Before the teacher counted the balls in the suitcase together with the children, she probably told them the exciting story of how the suitcase got into the classroom. You can see that the children are very involved in this lesson.

Explanation: The first sentence is an interpretation of what might have happened before the observed situation; the word 'probably' is an indication, just like the expression ‘very involved’ in the second sentence. The word ‘exciting’ in the first sentence shows a ‘weaker’ signal.

No theoretical concept is used.

C1 It is precisely the choice for a large quantity of balls that evokes students’ thinking. The large number of balls makes it less likely that they will just count them.

Explanation: It is indicated why teacher Minke sets the students thinking. No theoretical concept is used.

D1 Placing the cylinders with balls might have been done at a slower pace, which could give space for doing arithmetic in between; this is how I would do it in any case.

Explanation: In the reflection the student teacher gears towards concepts of a possible alternative for the teacher’s approach in the observed situation. No theory is used.

A2 The suitcase with balls that was put down by ‘Black Piet’ is used by Minke as a reason to count (in a structured way) with the children. The fragment starts at the moment that the balls are snatched away and are put in transparent cylinders.

Explanation: It is a factual reproduction of a situation, in which one theoretical concept (structured counting) is used.

B2 The children count once more up to 100 in the same way (strategy) Fariet did. Minke indicates that Fariets’ way of thinking makes sense; that response will reinforce his self-confidence.

Explanation: The second sentence points towards an interpretation of the situation; one theoretical concept (strategy) is used. The final clause can be seen as a notion of the concept ‘pedagogical climate.’

C2 Minke is working with the whole group. Counting together with jumps carries the danger that not everybody participates in the activity. I can see that with two children who are doing different things while the class is counting.

Explanation: The student teacher postulates a ‘thesis’ and an associated ‘proof’ for it.

Theoretical concepts are used (group teaching, to count with jumps); however, those concepts do not have a coherent meaning that is relevant for the third level of using theory.

D2 Hereafter, it could be possible to let make the students a table network, starting with the sum 20 x 5 = 100; hang it up and discuss it.

Explanation: The student teacher gears to the given situation in terms of a possible continuation on the observed activities.

One theoretical concept (table network) is used.

A3 By moving the cylinders the teacher makes another grid model. Now there is a rectangle of 10 x 10. Next, she let the students give meaning to the new model, working with doubling and halving in a very concrete way. She emphasizes that the multiplication is/sounds different, but that the answer remains the same. She writes the new multiplication on the blackboard as well and again connects the concrete and the abstract sum.

Explanation: This is a factual reproduction of three successive events. Three pedagogical concepts (grid model, rectangle model and doubling and halving) are used coherently.

B3 Fariet gives a handy solution for 13 x 5. He immediately thinks of the

multiplication that is really represented by the 13 cylinders. So far his class has only done the tables up to 10 x 5 (I assume), but he already understands how to calculate the five times table above 10 x 5.

Explanation: The words and expressions ‘handy,’ ‘he immediately thinks of,’ ‘I assume’ and ‘he already understands ... above 10x5,’ indicate an interpretation of the situation.

The concepts ‘multiplication 13x5,’ the ‘13 cylinders’ (notion of material) and the ‘tables up to 10 x’ are used coherently.

C3 The class already comes up with 2 x 5 followed by 3 x 5. Because she visualises the five times table for the children, they can also tell a story to accompany a problem. 1 x 5 will be possible to see as 1 tube times 5 balls. She also makes a connection between concrete material and a grid model. At one point Clayton is counting 10 x 5, the teacher confirms this for the class. In fact a transition is being made here from multiplication by counting to structured multiplication.

Explanation: the whole text has the character of an explanatory description, with the words ‘because,’ ‘also’ and ‘in fact’ functioning among other things as signal words. Seven concepts are used in connection (five times table, visualises, story to accompany a problem, concrete material, grid model, multiplication by counting and structured multiplication).

D3 Do the children really see the tens in the rectangle model? The teacher could have asked on with Fariet: “Fariet, how do you see the 10, 20...? Can you tell me or point it out, Fariet?”

Explanation: The student teacher anticipates the situation in terms of a possible alternative for the teacher’s approach. The concepts ‘tens,’ 'really see’ (notion of structure), ‘rectangle model’ and ‘asking on’ are used coherently.

In some cases there was some doubt about the score combination for a unit. This doubt was expressed in differences between expert scores or by both experts finding at first that a unit qualified for two or three scores.

Theory-enriched practical knowledge in mathematics teacher education

Below (table 5.4) some examples of such doubtful cases are described, together with the considerations that led to an unequivocal score combination.

Table 5.4 Examples of doubtful cases of scoring units Example 1

Unit In this fragment we see teacher Minke, teaching grade 2. Using a splendid

context, ‘The trunk full of balls,’ she teaches learning to multiply.

Doubt between A2 and B2

Considerations by the experts

The first sentence has an A-character. However, the expressions ‘splendid context’ and ‘learning to multiply’ in the second sentence indicate interpretation (B). It is not clear how ‘learning’ is taken here: learning to multiply in general? Has the first introduction been meant, the conceptual attainment of learning to multiply? Because of the link to the ‘trunk full of balls,’ the application of the concept ‘context’ can be considered as meaningful use of a theoretical concept. Level 3 is not reached: no meaningful relationship has been made between theoretical concepts.

Conclusion by the experts

B2

Example 2

Unit At first, you would think that this fragment belongs to the introductory stage

of learning to multiply, because it starts out with a context (with the balls), in which the five times table of multiplying is hidden. One can watch the children counting smoothly by fives and even making a link between the table of 5 and 10, doubling, halving and, counting with jumps on the number line. Soon after this, strategies are used and links are made. Therefore, you could also say: these activities belong to the memorizing stage of learning to multiply. This is strange: contexts, materials, but nevertheless the

memorizing stage (or not?).

Doubt between C3 and D3

Considerations of the experts

The first part of the text reflects a C-character: the assumption that the fragment belongs to the introductory stage of learning to multiply, is founded with an argument. On the other hand the expression ‘at first you would think’ has a fairly subjective, non-neutral character, which is typical for D (and for B). The same could be said about the sentence ‘Therefore, you could also say: these activities belong to the memorizing stage of learning to multiply.’ In the last sentence the author of this unit asks himself a question, as a result of the situation; that underpins the choice for score D.

Concerning the level of theory, it is obvious that this text characterises level 3, showing meaningful relations between several theoretical concepts, namely some stages of learning to multiply, materials, context, the five times table, counting with jumps and doubling and halving.

Conclusion by the experts