Werner Blum had the task of firing up a homebound audience with the title "Quality Teaching of Mathematical Modeling - What Do We Know, What Can We Do?" He gave a few starting examples of modelling: Hassel pick-axe - how tall would a giant have to be to fit the pick-axe? And: is it worthwhile to go down to town to buy a t-shirt where it is cheaper? He used such examples to go through familiar steps of modelling processes.

Modelling competency is at the heart of PISA. Modelling is a cognitively demanding activity, because it involves several competencies, both mathematical and non-mathematical (including ethical considerations). Each step in the modelling process is a potential cognitive barrier for the students: 1. understanding the situation and constructing a situation model. Students have learned that they can solve problems without considering the words. "suspension of sense-making". "an orchestra needs 40 minutes to play Beethovens 6th symphony. How much for 9th?" 2. Simplifying and structuring - students are loath to take decisions on their own, for instance to round off numbers. 6. Validating - is seen as the teachers' job Students tend not to transfer. Which makes it doubtful if there is such a thing as general modelling competency.

He listed four kinds of justifications for modelling: pragmatic, formative, cultural, psychological. Based on these justifications, several "perspectives" of modelling can be seen, when combined with characterizations such as "authentic" or "mathematically rich" - I will not go into that here. But it is an interesting way of structuring the discussion, and I wonder if a similar framework would be useful in the field of HPM as well (and of course it could be that such a framework has already been proposed in articles I haven't read).

Finally, Blum gave ten rules for teaching modelling (based on empirical evidence): 1. Effective and learner-oriented classroom necessary 2. Activate learners cognitively also necessary 3. Activate learners meta-cognitively also necessary 4. Variety of suitable examples (real world contexts and mathematical contexts and topics) - transfer cannot be expected. Real world contexts help reducing the "suspension of sense-making. 5. Teachers ought to encourage individual solutions (they tend to favour their own solution) 6. Competencies evolve in long-term learning processes. Repeating and practicing is necessary 7. Assessment must reflect the aims of modelling appropriately 8. Parallell development of competencies and beliefs and attitudes 9. Digital technologies can be powerful tools: experiments, investigations, simulations, visualisations or calculations 10. Mathematical modelling can be learned by students supposed there is quality teaching.

Dongchen Zhao presented, on behalf of himself and Yunpeng Ma, a paper with the title "An analysis of the characteristics and strategies of the excellent teachers in mathematics lessons in primary school". There was a curriculum reform in China in 2001, fully implemented from 2005. The previous one was from 1992. Zhao gave some short comments on the 1992 curriculum and then gave an introduction to the new curriculum.

The lessons analysed in the project were prize-winning lessons from the National Contest in Exemplary Lessons. Unsurprisingly, the prize-winning lessons were found to comply with the new curriculum (which might be why they won prizes in the first place). He gave examples of how this was done. More interesting, perhaps, is other findings when looking at these videos. All lessons were dominated by public interaction (again not surprising, I guess, because it must be difficult to make an impressive video of students working individually, for instance). For us, the most interesting finding is perhaps that lots of the student speaking was done in chorus - up to 66 percent in one of the lessons... This seems to suggest that the teachers didn't often ask students to bring forward their way of thinking, but rather were asking rhetorical questions with only one valid answer. Moreover, students rarely raised questions by themselves - the teacher did most of the asking. It is interesting that lessons that are judged as good lessons in some respects, have such worrying characteristics in other respects.

I am thinking that such a lesson contest could be a cool idea also in Norway - not least because it would have teaching experts discuss - in concrete cases - what constitutes good teaching. After all, our discussions are so often concerning only hypothetical situations, and not real-life lessons with all their quirks.

This ended the ICME12. I hope to be back at ICME13 in Hamburg in July of 2016. But first, it is the HPM conference in Daejeon beginning on Monday...

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