Pupils’ over-use of proportionality

1. How numbers may change solutions … Pupils’ over-use of proportionality Dirk De Bock Wim Van Dooren Marleen Evers Lieven Verschaffel Center for Instructional Psychology and Technology Catholic University of Leuven Belgium

2. SECONDARY SCHOOL PUPILS’ILLUSION OF LINEARITY OUTLINE Introduction  « over-use of proportionality »? Empirical and theoretical background  previous study on arithmetic problems  how numbers may change solutions A new empirical study  method  main results Conclusions and discussion

3. Over-use of proportionality Students’ tendency to treat every numerical relation between numbers as if it were linear (or proportional) (see, e.g., Freudenthal, 1973; Rouche, 1989; …)

4. Stacey (ESM, 1989): To make a ladder with 2 rungs, I need 8 matches. How many matchesdo I need to make a ladder with 10 rungs?  Most frequent error: 40 matches Examples

5. Examples Aristotle: “An object which is 10 times as heavy as another object, will reach the ground 10 times as fast as that other object”

6. SECONDARY SCHOOL PUPILS’ILLUSION OF LINEARITY OUTLINE Introduction  « over-use of proportionality »? Empirical and theoretical background  previous study on arithmetic problems  how numbers may change solutions A new empirical study  method  main results Conclusions and discussion

7. Previous study • Missing-value arithmetic problems • Evolution of over-use of proportionality from 3rd to 8th grade • (cf. Van Dooren et al., Cognition and Instruction, 2005)  already present in 3rd grade  considerable increase until 6th grade (~ emerging proportional reasoning skills)  decrease afterwards

8. Previous study “CONSTANT” PROBLEM A group of 5 musicians plays a piece of music in 10 minutes. Another group of 35 musicians will play the same piece of music. How long will it take this group to play it?

9. Previous study “ADDITIVE” PROBLEM Ellen and Kim are running around a track. They run equally fast, but Ellen started later. When Ellen has run 5 rounds, Kim has run 15 rounds. When Ellen has run 30 rounds, how many rounds has Kim run?

10. Previous study “AFFINE” PROBLEM f(x) = ax + b The locomotive of a train is 12 m long. If there are 4 carriages connected to the locomotive, the train is 52 m long. If there would be 8 carriages behind the locomotive, how long would the train be?

11. But … Numbers may change solutions! Literature on (acquisition of) proportional reasoning  Frequently reported error: additive reasoning  Esp. with « messy » numbers/non-integer ratios a mixture with 20 kg sugar for 100 l water tastes equally sweet as a mixture with 60 kg sugar for 140 l water a mixture with 21 kg sugar for 95 l water tastes equally sweet as a mixture with 23 kg sugar for 97 l water 20 + 40100+ 40 (e.g., Noelting, 1980, Hart, 1984, Karplus, Pulos, & Stage, 1983)

12. So numbers change solutions! • Integer ratios facilitate proportional reasoning to • proportional problems • What with proportional reasoning to NON- • proportional problems? •  Earlier studies: always integer ratios •  « It remains a question for further research • whether an approach with non-seductive • numbers will prevent children from making the multiplication error » • (Linchevski et al., 1998)

13. SECONDARY SCHOOL PUPILS’ILLUSION OF LINEARITY OUTLINE Introduction  « over-use of proportionality »? Empirical and theoretical background  previous study on arithmetic problems  how numbers may change solutions A new empirical study  method  main results Conclusions and discussion

14. Method • 508 students, 4th, 5th and 6th grade Test with 8 word problems: Non-proportional • Nature of numbers in problems was manipulated

15. Ellen and Kim are running around a track. They run equally fast, but Ellen started later. When Ellen has run 16 rounds, Kim has run 32 rounds. When Ellen has run 48 rounds, how many rounds has Kim run? x2 16 32 x3 II-version 48 ? Method • Manipulation of numbers:

16. Ellen and Kim are running around a track. They run equally fast, but Ellen started later. When Ellen has run 16 rounds, Kim has run 24 rounds. When Ellen has run 36 rounds, how many rounds has Kim run? x1.5 16 24 x2.25 NN-version 36 ? Method • Manipulation of numbers:  One version at random for each student

17. Results PROPORTIONAL PROBLEMS II version NN version

18. Results ADDITIVE PROBLEMS II version NN version

19. Results CONSTANT PROBLEMS II version NN version

20. Results AFFINE PROBLEMS II version NN version

21. SECONDARY SCHOOL PUPILS’ILLUSION OF LINEARITY OUTLINE Introduction  « over-use of proportionality »? Empirical and theoretical background  previous study on arithmetic problems  how numbers may change solutions A new empirical study  method  main results Conclusions and discussion

22. Conclusions • Hypotheses generally confirmed • Proportional problems - Integer ratios facilitate correct reasoning  confirmation of other findings in literature - Older students less “hindered” by non-integer ratios

23. Conclusions • Hypotheses generally confirmed • Non-proportional problems - Integer ratios also facilitateover-use of proportional reasoning - Additive problems:  non-integer ratios cause more correct answers - Other non-proportional problems:  non-integer ratios cause more other errors - Older students ‘benefit’/‘suffer’ less from non-integer ratios

24. Discussion Theoretical implication • Not only key words or problem formulations • Also the (combination of) numbers in a problem can be associated with a solution method • Association interacts with students’ prior knowledge

25. Discussion Methodological implication • Assessing over-use of proportionality using items with integer ratios may strengthen the effect • Only / especially for younger students

26. Discussion Practical implication • During classroom teaching of proportionality: • Explicitly discuss (validity of) criteria students choose to apply proportional methods • Take care to use variety of examples, not sharing same superficial task characteristics