2.
GRD, met Booijmans

4. 1985

5. Design heuristics of RME
Guided reinvention
Didactical phenomenology
Emergent modelling
Heuristics - not a cookbook, general concepts, FI-design culture

6. Guided reinvention Reinventing mathematics
Guided by the teacher and by carefully chosen activities
Class as a community

7. Area as an example
Grade 5, ze wisten allemaal lxb, maar snapten er niks van. We besloten maar eens vanaf het begin te beginnen.

8. Designing a lesson series
Analysis: what is area and what makes area a difficult concept for students? (didactical phenomenology)
What problems can we use to raise discussions about the mathematics of area? (guided reinvention)
Design a ‘hypothetical learning trajectory’ (HLT) for the whole series and for each lesson

9. What is ‘understanding area’?

10. allebei 16

11. 60

12. 60 en 58

19. What makes area a difficult concept?
Geef ik ze 7 minuten voor

20. What makes area difficult?

21. Measuring area, main ideas ...
Area refers to the size of a surface
Area is not bound to a certain form; also non-rectangular surfaces have an area.
Area can be measured with different measures, but if you want to compare areas you should use the same measure
Choose a measure that connects all along the borders. With a measure that is round, for example, you cannot measure area properly
If you want to know how often a certain measure fits, you can often use multiplication

22. ... main ideas ...
If you want to multiply, it is handy to use a square form as the measure, as this avoids the problem of the measures laying in different directions
To determine how many times a measure fits in a certain direction, you can measure with a strip. This means that you can calculate area from known lengths.
It is useful if everybody measures area with the same, standard units
Just like when we measure length in standard units (mm, cm, dm, m, and so on), it is possible to convert a given area from one standard unit into another standard unit.
If you double both the length and the width of a surface, the area becomes four times as big, not twice as big.

23. Measuring area by multiplication

24. Why area is difficult 1
If you want to know how often a certain measure fits, you can often use multiplication.
‘area = length x width’
Students do not recognize multiplication in a task on area
Reinvent (once again) the idea of computing area by multiplication

25. Measuring area by length
Plaatje anders

26. Why area is difficult 2
To determine how many times a measure fits in a certain direction, you can measure with a strip. This means that you can calculate area from known lengths.
We should help students to understand how area can be computed by measuring length
Reinvent the idea of using length to measure the number of squares

27. 2 lessons: measuring area with ...
pages A4
square pieces of paper
large, small pieces
circles
triangles

28. How many pages fit on a table?

29. How many pages fit on a table?
Context? Story?
HLT (hypothetical learning trajectory) for this lesson:
What will happen?
How can we guide learning?

30. What will happen? How many pages A4 fit on the 4 small tables?
halvetafel

32. What did happen? 4 tables? multiplication? overlapping
pages in different directions
Stuff for discussions!

33. Discussing multiplication
“Nikki and Arthur made a discovery”

34. dat ene blaadje dubbel tellen

35. Misunderstanding?
dat ene blaadje dubbel tellen

36. Discussions about multiplication overlapping the shape of the unit
large vs. small units

37. From squares to ‘length x width’
How would you help students with this step?

38. From squares to ‘length x width’

39. filmpje
Ook nog even terug naar het begin, waar kind afpast, covers

40. Contexts
It should be a real problem for the students (1. not too easy, 2. genuine)
Using their out-of-school knowledge
Motivating
Evoking the right discussions
Sometimes a first step in modelling

43. From context to model

44. Models, modelling
First: context-specific models, informal solutions: ‘model of’ a certain situation
Later: object-like character: ‘model for’ more formal mathematical reasoning

45. Useful problems? Indonesian master students
Task: design a set of problems on area, for students grade 5 who know the formula, but do not understand area

46. Useful problems?
In many Indonesian schools the pupils take off their shoes before they enter the classroom. How large should a shoe rack be for the class?

47. Useful problems?
The pupils are given a drawing of nine rice fields. Which one is the largest?

48. Useful problems?
Mineral water is sold in conical cups. How can you arrange them in such a way that as many as possible fit on the table?

49. Useful problems?
The pupils are asked to design paper boxes for 24 cubes of 1cm x 1cm x1cm. How much paper do you need?

50. HLT, hypothetical learning trajectory
Predict how students will solve the problem
AND give suggestions of how the teacher should react

51. Covering with A4, squares, triangles, circles
Covering, predict what children will do:
overlapping
borders
circles
different orientation of pages
Mistakes are welcome – for discussions!

52. Covering with A4, squares, triangles, circles
Multiplication
counting one by one
Discussion: what units wil make counting easy?

54. This week
Design a series of lessons
Learn form each other
Have fun