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

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