1. Positive Algebra From arithmetic to algebra Jaap den Hertog Freudenthal Instituut Universiteit Utrecht J.denhertog@fi.uu.nl

4. “ I used to be good at arithemetic, but now I don’t understand anything anymore.”  Counting in primary school grows into advanced and more sophisticated counting  You cannot maintain what you never learned  When do you use your calculator?

5. Continuous learning trajectories  To introduce negative numbers and to use them  Knowledge about fractions as a preparation to working with algebraic expressions  Rules, patterns, structures

6. 27 – 38 = ….?

7. 5 × -3 = -15-1 × -3 = 3 4 × -3 = -12-2 × -3 = 6 3 × -3 = -9always 3 more 2 × -3 = -6 1 × -3 = -3 0 × -3 = 0 A pattern

8. What is the power of algebra?  Reasoning and generalizing: is it always?  Are you sure? Is it certain?  Not only knowledge of (f.e. number system) but also knowledge about  Development of thinking models

9. A continous learning trajectory  Developing a fraction language  Reasoned divide  Perform operations within the context  To relate ‘Part of’ to multiplication  Towards the development of routine procedures  Fractions on the number line  And what is next …?

10. Two thirds of 4500 2/3 times 4500 × 4500

11. A learning process and struggles  π/4; 1/4π; π ÷ 4; they are all the same, but different  Add up the same number with the nominator and the denonminator  You divide a number and the result is larger. Why?  Add up the nominators and the denominators. Is the new fraction bigger or smaller than the sum of the fractions?  Is there a smallest fraction greater than zero?  How is the number system extended?

12. A square of 1 bij1. Write the area of each piece as a fraction and add up.

13. When is formal arithmetic with letter fractions introduced?  For which students is it important?  In which grade do we start?  What are the preparations for the students?

14. Which formula is equivalent with…

16. Are there more examples? Is there a formula?

17. Simplify fractions

18. Reasoning with formulas  Adjust / prepare formulas yourself  Discus the effect of changes in variables and / or numbers

19. Recommended maximum heart rate For years, the following formula was used: Maximum heart rate = 220 – age  Who has a higher maximum heart rate, someone in your class or one of the teachers?

20. Recommended heart rate Recently the formula has been changed Maximum heart rate = 208 - (0.7 x age) What are the consequences of using this formula: is your heart rate higher or lower than the recommended rate?

21. Summary  Continuous learning trajectories from Primary school and Secondary school  Introducing negative numbers in primary school, but the formal operations in secondary school  Fractions are not “ready” after the primary school  Fractions in secondary school  Do not avoid fractions in secondary education, but also include letters  Learning processes in developing and adapting formulas