The final assessment

The list of concepts at the end

In the final version of the list of 59 concepts Anne indicates for six of the concepts that they were unknown to her at the start of the course (anchor point, cognitive network, own construction, informal procedures, narrative to a problem, time table method), but that she can now – at the end of the course – tell a teaching narrative in which that concept has

Theory-enriched practical knowledge in mathematics teacher education

meaning. These narratives are for the most part derived from all four categories (own practice, ‘The Guide,’ literature or the introductions and discussions during meetings at the Pabo). In addition, she indicates for three other concepts that they have been further broadened during the course (structured material, strategy, multiplication). This is the case particularly for ‘multiplication’ in general. ‘Proofs’ for these concepts having made a lasting impression on Anne come from the triangulation of data as described in the previous sections, but also from reporting on activities in her teaching practice and during individual study. For instance, she reports comprehensively on a study of eight students in grade 2, where she maps the cognitive (times-table) network of the children (appendix 4), describes the learning trajectory for multiplication in the mathematics textbook series ‘Wis en Reken’ and interviews the teacher on learning and teaching the tables. All activities are related to the learning question she chose: “What is the connection between strategies that are offered and strategies used by the children?” In the analyses and reflections she makes meaningful use of theoretical concepts and makes connections between concepts.

Narratives for selected concepts, the practical study and the personal learning question

The two stories she has to write within the framework of the final assessment for two concepts (appendix 5) are yet another proof of her competence to meaningfully clarify concepts from her own practical experiences (nrs. 1-3, 10). Anne chooses the concepts ‘anchor point’ and ‘strategy.’ In both cases she provides a clear argument, successfully integrating eight concepts into each of them.

In addition, Anne performs an extensive study in her practice school about the knowledge children have of tables.

As is the case with the two narratives of practice, these activities are placed in the context of her own learning question. She supports her findings with the examples, schematic representations of the children’s table networks (appendix 4) and theory-laden notes. Even in the summary of her findings (see below) she uses many theoretical concepts.

My findings. There are several things I noticed during the interviews. - The strategy the children add to the table is counting with jumps.

- The supporting problem [anchor point; w.o.] and counting with jumps are used often by these children. These strategies may be useful as well for solving other problems, but can also be hard to use.

- Some of the children knew they had to make jumps, but were uncertain about how to determine the size of the jumps.

- An incorrect supporting problem was used a number of times. - I also found that several strategies were used in one problem. - Children do have a preference, which differs per child. - Doubling and halving are not used much.

- A number of children have automated the problems that were looked at, others find it difficult to apply the strategies.

In the reflection on an interview with the teacher of the class where she conducted the above-mentioned study, she considers the teacher’s actions while using theory meaningfully and in mutual connection(nrs. 3, 15).

As a result of her study in the framework of her teaching question she examines in detail the character of the connection between the strategies on offer and their use by the children.

(...) Particularly the supporting problem [anchor point; w.o.] and the reversing [commutative; w.o.] strategy are important, in addition this class uses counting with jumps (repeated addition) to ultimately reach the answer. Though the teacher follows the teachers’ manual, she does place her own accents. The children follow the teacher, but also use their own ways and names. The various strategies do turn out to be somewhat complicated for some children. They find it difficult to choose the right supporting problem or the right size of the problem (…). Strategies are needed to automate and memorize the problems. You can also reverse this. That memorised problems are needed for the strategies. Think of the supporting problems. They can calculate new problems through problems they already know. Strategies, automating and memorizing are inextricably linked. The use of strategies is not limited to multiplication, but occurs in all other areas of mathematics education. Another reason to offer strategies is the opportunity for checks. In practice you encounter children who have memorised problems wrong. By calculating problems using the strategies you can check the answers. Provided the strategies are used correctly, which is sometimes difficult for the weaker students.

In her final conclusion Anne emphasises the importance of developing strategies in students, but she also asks to what extent weaker students will profit from being offered a variety of strategies if that offer is not accompanied by sufficient individual attention. Here, Anne shows that she possesses a network of meaningful relations between theoretical concepts (nr. 15).

Reflective note for the multimedia teaching situation (the ‘suitcase full of balls’)

Anne writes – as do her student peers – a reflective note (A4) to finish the course. Her note (appendix 6) consists of 628 words and seven meaningful units (section 4.2.5; table 4.2). It turns out that Anne use more than one theoretical concept in six units and also meaningfully connects these concepts. She also touches upon explanatory reasonings of what happens in the situation in all units. Nowhere does she limit herself to factual describing of occurrences or interpreting the situation without providing some founding for her opinion. In three units she goes beyond explaining or clarifying what happens and she in fact responds to the situation by reflecting on the teacher’s actions and by considering alternatives for the approach she observes. This happens for example in unit 6 (see the quote below), where she differentiates between shortened counting as an activity and as a strategy. This is an increase in level in her thinking and reasoning about the situation.

Theory-enriched practical knowledge in mathematics teacher education

(...) Afterwards they count together in the same way that the class started.

You might ask whether this is the correct approach. If you want to work from the children’s view, it’s good to first follow the children, the problem can be stated afterwards. The emphasis will be only on shortened counting as an activity, not as a strategy.

If you want to offer shortened counting as a strategy for multiplication, you must start with the times problem. Then you can use shortened counting to solve it, with jumps of 5. I think this was her [the teacher’s; w.o.] goal, and she achieved it. It’s good she offers this for large numbers. You avoid that children know the answer immediately. Then the strategy wouldn’t come across so well.

She starts from the actual, observed situation, and questions (nr. 6) what the consequences of different approaches might be. Yet she still tries to think along with the goal that the teacher presumably (nr. 4) has in mind. She reaches a concluding consideration through if-then and cause-effect reasoning.