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Development of concept of division – from intuitive models to division of fractions

35th ANNUAL CONFERENCE OF THE ATEE Budapest, august 26th – 31st, 2010. Development of concept of division – from intuitive models to division of fractions. Maja Cindrić, Department for Teacher Education, University of Zadar, Zadar, Croatia mcindric@unizd.hr

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Development of concept of division – from intuitive models to division of fractions

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  1. 35th ANNUAL CONFERENCE OF THE ATEE Budapest, august 26th – 31st, 2010. Development of concept of division – from intuitive models to division of fractions Maja Cindrić, Department for Teacher Education, University of Zadar, Zadar, Croatia mcindric@unizd.hr Irena Mišurac Zorica, Department for Teacher Education, University of Split, Croatia irenavz@ffst.hr

  2. Kilpatrick and others (2001) : conceptual understanding procedural fluency Strategic competence Adaptive reasoning Productive disposition Figure1. : Interwined Strands of Proficiency, from Adding it up What does we want for our children to acquire by learning mathematics ?

  3. What does we want for our children to acquire by learning mathematics ? • Balance between conceptual understanding and procedural skills • ability to use a flexible application of knowledge learned in appropriate situations • combination of knowing the facts, knowledge of procedures and conceptual understanding • Students who memorized facts and procedures without conceptual understanding often are not sure when and how to use it so they know their knowledge is very fragile (Bransford and others 1998)

  4. What does we want for our children to acquire by learning mathematics ? • Well connected and conceptually grounded ideas simply can be use in new situations (Skemp 1976) • practice algorithms in mathematics, without conceptual understanding are often quickly forgotten or remembered incorrectly • understanding of the concepts, without fluency in the performance of algorithms, may present an obstacle in solving problems

  5. Contemporary mathematics curricula • emphasizes the optimal balance between the development of conceptual and procedural knowledge • many teachers are influenced by traditional teaching, which emphasizes practicing algorithms • teachers are aware of contemporary ideas, but do not feel confident to change the way teaching … • …or they don’t know how ? • to be sure we conducted research on understanding the concept of division

  6. We asked ourselves : • what division means for children, students and mathematics teachers • What means to develop conceptual knowledge of division

  7. Children's intuitive knowledge of division • even young children, can solve many different types of problem-solving tasks with direct modeling of problem situations in the task (Carpenter, Ansell, Franke, Fennema, Weisbeck, 1993) • Children are insistent in using intuitive knowledge despite the traditional methods explained by teacher

  8. Children's intuitive knowledge of division • Mama sent Lucija to the store and gave her 100 kuna, to buy two cakes, and 3 packages of eggs. Each cake cost 15 kunas, a package of eggs, 11 kunas. Lucija wanted to buy a chocolate egg, which cost 3 kunas. Mom told her that with the rest money can buy what ever she want. How many chocolate eggs can Lucija buy?

  9. Children's intuitive knowledge of division • all basic arithmetic operations are associated with the unconscious primitive intuitive model, which mediates in search of arithmetic operations needed to solve a mathematical problem (Fishbein 1985.) • two intuitive models that children use when the situation requires a division problem : • partitive model and measurement (quotative) model

  10. Children's intuitive knowledge of division • On the table are 12 apples. I want to put apples into three baskets, so that contains the same amount of apple. How many apples are in each basket? "- Partitive division • On the table are 12 apples. I want to put apples in the basket, so that in each basket contains three apples. How many baskets will be filled with apples? "- Measurment division

  11. Children's intuitive knowledge of division • Fishbein and others argue that intuitive models can impede, discourage or even prevent a child to solve mathematical problem • For 12 : 3 child is said : Grandfather has a 12 cookies and 3 grandchildren. How much cookies will each grandchild get? • For 3.21 : 0.75 child is said : Grandfather has a 3.21 cookies and 0.75 grandchildren. How much cookies will each grandchild get?

  12. Definition of concept by Gerard Vergnaud • Concept is three-uple of three sets : C = (S,I,R) • S: the set of situations that make the concept useful and meaningful • I: the set of operational invariants that can be used by individuals to deal with these situations • R: the set of symbolic representations, linguistic, graphic or gestural that can be used to represent invariants, situations and procedures.

  13. Problems situations for division of whole numbers invented by elementary school children • for the study consisted of 135 elementary school children from 8 till 10 years old • to write three problem tasks for which solutions will contain numerical expressions 12: 3, 45: 3 and 72: 12 • How many children know to write a correct problematic situation? • If the problem situation is the exact, a which classes of situation students choose and which division model • If the child does not choose an adequate problematic situation, where he/she make mistakes

  14. Problems situations for division of whole numbers invented by elementary school children

  15. Problems situations for division of fractions invented by elementary school children • subjects for the study consisted of 241 elementary school children in 6th grade • Students are asked to write one problem task for which solutions will contain numerical expressions 12: .

  16. Problems situations for division of fractions invented by elementary school mathematics teachers • subjects for the study consisted of 122 elementary school mathematics teachers • Teachers from different schools in the southern Croatia (Split and Zadar County) • teacher was required to write three problem situations for three different division: 12: 3, 12: , : .

  17. Problems situations for division of fractions invented by elementary school mathematics teachers

  18. Problems situations for Division of Fractions invented by students from teacher studies • Subjects for this study consisted of 173 prospective teachers from University of Zadar and University of Split

  19. Thank you !

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