Fátima Mendes Escola Superior de Educação Instituto Politécnico de Setúbal Portugal

1. Developing Multiplication Fátima Mendes Escola Superior de Educação Instituto Politécnico de Setúbal Portugal

2. The project and the main objective • The Project • Developing number sense: curricular demands and perspectives (DSN) • Was developed from January 2005 to December 2007 • Main objective • To study number sense development with children from 5 to 11 years old. • The team worked in two intertwined characteristics • curricular development • developed, experimented in several classrooms and reformulated task and task chains • educational research level • studied the way children develop number sense in problem solving contexts Fátima Mendes, ESE de Setúbal

3. The expression number sense • Knowledge and facility with numbers, witch includes multiple representations of numbers, recognizing the relative and absolute magnitudes of numbers, composing and decomposing numbers and selecting and using benchmarks. • Knowledge and facility with operations, witch includes the understanding of the effects of operations on numbers, the understanding and the use of the operations properties and their relationships. • Applying knowledge of and facility with numbers and operations to computational settings, which includes the understanding to make connections between the context of a situation and the computation procedures, requiring knowledge of multiple computational strategies. McIntosh,Reys & Reys (1992) Fátima Mendes, ESE de Setúbal

4. The task chain – Forming groups, The loft wall and Chewing gums • This paper focus on a case study that analysed a 2nd grade classroom • This chain had 3 tasks with the objective of : • developing the understanding on multiplication • the use of different calculus strategies related with multiplication. • The chosen contexts • Are familiar to children • Try to facilitate the understanding of the multiplicative structures • Try to facilitate the informal use of the multiplication properties. • This option follows Treffers & Buys’ (2001) ideas • about relevant contexts • about the way they see the learning trajectory of multiplication – from solutions specific to a context to a more generalizable solution grounded on models. Fátima Mendes, ESE de Setúbal

5. The task chain – Forming groups, The loft wall and Chewing gums Fátima Mendes, ESE de Setúbal

6. The task Forming groups • Task 1 - Forming groups • The teacher has 128 sheets. • She wants forming groups of 30 sheets. • How many groups can she form? Explain your thinking. • And if she wants forming packages of 10 sheets? • And if she wants forming packages of 20 sheets? Fátima Mendes, ESE de Setúbal

7. The task chain – Forming groups, The loft wall and Chewing gums Fátima Mendes, ESE de Setúbal

8. The task The loft wall • Task 2 – The loft wall • Sara’s father wants to catch a loft wall with bookcases. • The height of each bookcase is 42 cm. • Sara’s father has been able to heap up 4 bookcases, up to the roof. Which is the height of the wall? • And if the height of each bookcase is 21 cm? • How many bookcases he needs for the same wall? • - Sara’s father has experimented to put the bookcases with 21 cm of length, side by side, and he can put 9. Which is the length of the wall? Fátima Mendes, ESE de Setúbal

9. The task chain – Forming groups, The loft wall and Chewing gums Fátima Mendes, ESE de Setúbal

10. The task Chewing gums • Task 3 - Chewing gums • Observe the prices of the Chewing gums • If you want to buy 5 chewing gums, how much is it? • And how much is 7 chewing gums? • And how much is 10 chewing gums? Fátima Mendes, ESE de Setúbal

11. The exploration of the task chain in the classroom • The tasks were explored in a 2nd grade class with 19 pupils • The lessons were organized in three parts • First one - the teacher presented the tasks and clarified the doubts posed by pupils • Second one - pupils worked alone or in groups on the task. • Third part - the teacher organized an all class discussion • - different strategies and procedures were shared and analyzed • - children explain their ways of thinking and understand the explanations of the colleagues Fátima Mendes, ESE de Setúbal

12. The task Forming groups The procedures that children used to solve the task Forming groups • In the first part of task, we identified four different • strategies: • The understanding and the use of 30 as a group, but • without being able to add the several groups of 30, • not understanding when to stop – Íris’s strategy • The use of repeated addition of 30, counting • the number of times that the 30 was repeated • – Sara’s strategy (Figure 2) Figure 1 - Íris’s strategy Figure 2 - Sara’s strategy Fátima Mendes, ESE de Setúbal

13. The task Forming groupsThe procedures that children used • In the first part of task, we identified four different • strategies: • Counting jumps of 30 and related the counting • with the results 30, 60, 90, 120. • We can see a multiplicative structure based on • proportional reasoning – Hugo’s strategy (Figure 3). • The use of division as inverse of multiplication • thinking that with 30 sheets we can make 1 group, • with 60, 2 groups… - João Pedro’s strategy (Figure 4) Figure 3 - Hugo’s strategy Figure 4 - João Pedro’s strategy Fátima Mendes, ESE de Setúbal

14. The task Forming groupsThe discussion of the others questions • The teacher asked And if we had form packages of 10 sheets? • One pupil answered immediately: • Pupil: 12, because 10x12 =120 • After organizing written registrations of the question – And if we had form packages of 20 sheets? Gonçalo answered: Gonçalo: Because it is the double. Elvira: The double of what? Yes, there is a double relation. Carolina? Carolina: Because it is half. Because 6+6=12. Elvira: And how does it relate with sheets of paper? Hugo: Because the double of 10 is 20. Elvira: Can someone explain this better? Hugo: Because 20 is the double of 10 and 12 is the half of 6. (reformulated…) Because 12 is the double of 6 and 10 is the double of 20. Fátima Mendes, ESE de Setúbal

15. The task Forming groupsThe discussion of the others questions • The teacher asked for others strategies • Elvira: Others strategies? Gonçalo? • Gonçalo: 5x20=100 and 1x20=20 and 100+20 =120 • The procedures that children used to calculate 6x20 • The product 12x10 to solve 6x20, using the relation half/double. uses another. • The distributive property, he calculates 5x20 and 1x20 (known facts) adding the partials products (Gonçalo strategy's). Fátima Mendes, ESE de Setúbal

16. Final Conclusions • The analysis of the different processes used by the children • some of them seem to understand the effects of the addition and multiplication procedures and their relations; • two children that even seemed to understand the relation between the multiplication and division procedures, even though in a modestly structured form • As is mentioned by Beishuizen (2003), an entire work based on numbers and their relations further helps pupils with their understanding rather than the premature introduction of the algorithms. Fátima Mendes, ESE de Setúbal

17. Final Conclusions • The analysis of the different processes used by the children • almost everybody else seems to have understood that 10x12=12x10=120 • some of them seem to think using more organized procedures, utilizing the doubles and the halves relating directly to the distributive property of the multiplication towards the addition. • There seems to be an indication of some knowledge and facility in what concerns the multiplication procedure and its properties. Fátima Mendes, ESE de Setúbal

18. Final Conclusions • The analysis of the different processes used by the children • At least one of the pupils used known products and the 10 factor. • Some of them seem to think using more organized procedures, utilizing the doubles and the halves relating directly to the distributive property of the multiplication towards the addition. • Almost every pupil seemed to be aware of the existence of multiple strategies, which was aided by the discussion and the confrontation between the different procedures. • It was clear from the discussion, apart from the pupils’ seizing of the multiplicative strategies and a growing understanding of the relations between the numbers, that there was a greater relation between the processes used and their verbalization. Fátima Mendes, ESE de Setúbal

19. Final Conclusions • The analysis of the different processes used by the children • By analysing the entire process developed throughout this task, we can say that there is some development in terms of the use by some of the pupils of strategies which are more structured and formal. • We seem to be able to say that, the mentioned chain, and particularly the task illustrated in this paper contributed for the children’s development of their number sense, in aspects concerning : • the knowledge and facility with the numbers and the addition and multiplication procedures, as well as with • the applying knowledge of and facility with numbers and operations in various calculus situations. Fátima Mendes, ESE de Setúbal

20. References • Beishuizen, M. (2003). • Beishuizen, M., Gravemeijer, K.P.E. & E.C.D.M. van Liesthout (Eds.). (1997) • Brocardo, J. (2006) • Cebola, G. (2002). • Fosnot, C., & Dolk, M. (2001) • Heuvel-Panhuizen, M. (2001). • Mcintosh, A., Reys, B. J., & Reys, R. E. (1992) • Mendes, F., & Delgado, C. (2008) • Simon, M. (1995) • Simon, M., & Tzur, R. (2004) • Treffers, A., & Buys, K. (2001) Fátima Mendes, ESE de Setúbal

21. Developing Multiplication Fátima Mendes Escola Superior de Educação Instituto Politécnico de Setúbal Portugal