Incentives in the learning environment

We next discuss what the first, simple version of MILE offered the two student teachers and their coach/researcher, and focus on the question: What makes MILE educative? We probe to answer that question by elaborating main incentives in the learning environment.

Search words, search questions, and learning and investigation questions.

In their investigations in MILE, the two student teachers were limited by the simple search engine and their own search questions. In the first stage they often used ‘Telling stories of grade 2’ (Oonk, 1997) as a source for finding search words (e.g., mental action, ‘playing Dumb August’26_{, ‘egg box’}27_{). To some extent the ‘The World in}

numbers’ textbook and the teacher’s guide served the same function. To prepare for meetings, they used the textbook to plan the same lesson as the lesson taught by the MILE teacher, and then observed that lesson. This made the subsequent discussion more reflective than it had been previously. They raised questions such as: Did I devote sufficient time in my lesson to dividing numbers? Did I leave too much to the children? Can they actually estimate prices?

After seeing how the video teacher approached the lesson, they would draw comparisons and perhaps revise their own lesson plans.

Revealing practical knowledge. The student teachers were impressed by what they saw in MILE. The discourse often ended in personal analyses from different points of view. They called upon mathematics aspects but also took pedagogical and content pedagogical positions. From the beginning they were convinced that they could learn a lot from MILE teachers. Dieneke explained: “What are the ‘good questions’ that Minke asks? I noted how well she formulated an exercise: ‘If you have thought and drawn one way, try to think of another one.’ I would not mind hearing this remark a thousand times so I can imprint it in my mind. Instead of a compulsory exercise such as ‘Give at least 3 solutions,’ Minke shows she appreciates one solution and she encourages the pupils to think further. In my opinion, her choice of words is a clear example of good teaching.” However, Dieneke did not accept Minke’s practical knowledge unquestioningly. She analyzed, interpreted, and put forward arguments to substantiate what she believed she had observed. In doing so, she touched on the practical knowledge that appeared to guide Minke’s actions (in effect, unpacking this practical knowledge). In addition, she recognized ‘usable material’ even in the routines of the daily classroom activities. For example, Dieneke learned that Willie – Minke’s peer teacher – could quickly quiet the children and get their attention by remarking: “Everyone turn the calculator in your head on.”

From the beginning Hayet philosophised about the additional value of MILE as compared with teaching practice or lectures, and she also recognized the practical

knowledge of the expert teachers in MILE. She was especially impressed by the interviews, in which the teachers told about their plans for the next lesson: “Of the three types of video (transfer, interview, lesson), the interview made the most impression on me. It was like looking into the head of the teacher and finding out secret information. Going through a lesson step-by-step in this way is very practical and concrete. None of my tutors ever did this for me.”

Theory from practice. Searching for a direction to their investigation, Dieneke and Hayet got the idea to design a video for their peer first year student teachers, inspired by a lesson in which teacher Willie introduced the five times table. They became interested in the ways that Willie translated concrete material to the children, the children worked with that material, and the material precipitated mental action. By connecting to relevant theory, Dieneke and Hayet would make a statement about encouraging the rise from material to mental level in children’s learning. They formulated questions, made notes, then started theorizing: “We have made the following statement based on this video: ‘If the transition from concrete to mental action does not take place in sufficiently small and logical steps, the (material and mental) actions will remain separated from each other. The main objective is to couple these actions together (…).’ ”

Theory of practice. After the above mentioned discussion, the teacher educator wrote an extensive (electronic) annotation to make the student teachers aware of the distinction between a mechanistic ‘step-by-step’ approach and realistic didactics in which properly conceived ‘learning’ jumps in teaching are encouraged. He also focused on theoretical views on different levels of subject matter and learning processes applied to the structure of mathematics courses. The student teachers became very interested as they addressed these problems. Dieneke recognized the danger of misunderstanding rules from her own past education. In the next meeting, she revised a previous statement about a video fragment in which a pupil shows that thinking of egg boxes helps her interpret 43 as 40 (four egg boxes) + 3 (separate eggs): “This statement applies to small logical steps, raising pupils’ levels, and direct support. Support and raising pupils’ levels are similar concepts. The material is first used as a support. When pupils no longer need it and begin to construct mentally, their level of competence rises. Materials always help and support, provided you introduce them properly. If you do not do this, they are only extra ballast and lead to confusion. The fragment ‘Think of egg boxes’ is a good example of this since Minke had not referred to them before and ‘suddenly’ introduced them without moving through a sequence of small logical step towards them (…).”

The practical meaning of theory. The discussion about how to raise levels of thinking required student teachers to find relevant theoretical knowledge. They questioned the teacher educator and studied relevant articles and textbooks. They realized that MILE

Theory-enriched practical knowledge in mathematics teacher education

contains information (practical knowledge) that cannot be found in textbooks or teachers’ guides and compared the theory taught in lessons at college with the observed theory linked to the real-life situations in MILE. For example, by relating theory to practice, Hayet became aware that the terms concrete, abstract, mental, and formal initially put her on the wrong track: “In my opinion, ‘the abstract level’ means formal mathematics. ‘Mental actions with material’ contains the word mental, but does not belong on that abstract level. It should be placed on the concrete/material level. Actions with material are performed ‘in your head’ (it is not a physical action), but they are concrete, or in other words, conceivable and meaningful (…).”

The discourse as driving the learning process. The discourse about the practice in MILE was elaborated in corridor chats and e-mail communications. The student teachers were aware of the influence of the discourse on their cooperation and their learning processes. As Hayet put it: “I find it rather comical. If I had watched the video on my own, I would probably have missed the whole scene. The discussion that was brought about by that ‘small interesting incident’ is for me just like the didactics involved the most instructive part of the whole meeting. Another good thing about the discussions is that we start with critically thinking about what Minke does and why she does it. Is it part of her conscious strategy? And then the emphasis shifts to our own experiences and didactic considerations.”

One’s own practice as reflective. The practice in MILE engendered student teachers’ linkages to their own teaching practice. This happened in a natural way as they analyzed and compared the actions of the MILE teacher to their own experiences or anticipated future practice. Hayet compared her failed experience with ‘Playing Dumb August’ of her pupil Keltoum to the successful experience of MILE teacher Minke. Placing the ‘Dumb August’ approach in a wider perspective, she started formulating questions and hypotheses. She wondered about the relationship between one’s approach to ‘Dumb August’ and learners’ attitudes about making mistakes: “When you want to teach children to investigate mathematics (i.e., try out more than one strategy), it is important that you see making mistakes as part of the process. If you do not, you go straight back to mechanistic viewpoints: there is one way to solve a problem – the right way, mistakes are bad, and children who make mistakes are stupid. This MILE episode made me realize the strength, but also the possible dangers of the ‘Dumb August’ method.”

Final presentation. Both student teachers tell a lot of stories about their pioneering in MILE (Blikslager & De Bont, 1997; Oonk, 1999). In the closing presentation of their assignment, Hayet explained how her investigation in MILE had made her aware of what is behind theory in the lectures she hears in teacher education courses. She believes that MILE stories will help her keep in mind the connection between theory and practice.