https://www.youtube.com/watch?v=7EUdhbC4ClY GRD, met Booijmans

1985

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

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

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

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

What is ‘understanding area’?

allebei 16

60

60 en 58

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

What makes area difficult?

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

... 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.

Measuring area by multiplication

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

Measuring area by length Plaatje anders

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

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

How many pages fit on a table?

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

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

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

Discussing multiplication “Nikki and Arthur made a discovery”

dat ene blaadje dubbel tellen

Misunderstanding? dat ene blaadje dubbel tellen

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

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

From squares to ‘length x width’

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

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

From context to model

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

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

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?

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

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?

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

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

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

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

www.fisme.science.uu.nl/en/impome/

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