Learning Studies in Sweden

1. Learning Studies in Sweden

2. Short division

3. Öjersjö • Öjersjö is a small village just outside Göteborg on the west coast of Sweden. There are 700 students at our school in the age of 6 to 16 years Göteborg

4. How it all started… • In Öjersjö we have worked with Learning Study for six years. In the beginning the University acted as tutors, but now we manage on our own

5. Ourstudy • This year our case study focuses on our experiences from practising Learning Study in Mathematics on the area of algorithms for division. Short division

6. Ourstudy • A tutor from our own school, not from the University • Four teachers • Four classes, grade 6-7 (age 12 – 13)

7. The learningobject • Understanding the method and knowing how to use it • Knowing when to use it

8. The critical aspects Too much respect for the method • Understanding the number system • Which is the largest and the smallest of the ”memory numbers” ? • Analysing the relationship to multiplication • The direction

9. Method

10. Lesson 1 • Focusing on the direction. Comparison of the four fundamental rules of arithmetic 3 starts from the right ( ”backwards”) Only 1 starts from the left 836 836 836 836 =* 2+ 212- 212 2 • When is division useful?

11. Results after lesson 1 • The direction seems to be no problem at all • The students understand division with the ”containerthinking”, not only ”dividing in parts” • However, the value of each number, depending on its position is still unclear • How many tens are two hundreds? Results table: Short division

12. Changes in lesson 2 • More student activities • Focus on the value of each number (the number system)

13. Lesson 2 • We wrote three errors on the board and let the students investigate the problems in group discussions 535 = 101 535 = 17 535 = 161 5 5 5 • The number system, different values for different numbers • In how many ways can you explain 328?

14. Results after lesson 2 • They understand the number system much better • The students still, however, have difficulties with the ”memory digit” algorithm Results table: Short division

15. Changes in lesson 3 • In how many ways can you say 328 SEK? • The relationship between division and multiplication • More analysis of the method

16. Lesson 3 • The relationship between division and multiplication. • The division 328 = 63 4does not correspond, because 63 ∙ 4 ≠ 328 • Going through the method again, explaining the algorithm • The meaning of the “memory digit”

17. Results after lesson 3 • The Swedish money system is useful for understanding the number system if you use the right values, i.e. 100, 10 and 1! • The students gained an increased understanding of the division algorithm. Results table: Short division

18. Changes for lesson 4 • Student activities: throw the dice to decide numerator and denominator • More focus on the number system

19. Lesson 4 • In what ways can you express 5 3 2 8 (by using various thousands , hundreds , tens and units) • The students solved random tasks • We used dices (0-9) and wrote the digits in the squares

20. Results • The students understand and can use the algorithm with a ”memory digit” Results table: Short division

21. Difficulties during the study • Problems to see what to focus on (unexperienced teachers…) • Lack of time. Result: We did not all agreed about how to take the next step before we took it • The vaccination programme disturbing lesson 2

22. Conclusions • Short division is considered difficult by many students and their parents, and therefore they show more respect for the method than they really should • Understanding the number system is the key • The significance of the numbersystem for understanding values. • The relation between numerator, denominator and quotient. • Looking at division in two ways: ”dividing” and ”containing” helps to understand the method and when it is useful

23. The teacherexperience – over the last years Teachers try to work together on the critical aspects of an area - taking part of any previous Learning Study in the area We carefully define the goals of each area as precise as possible Clearer objectives and expectations for the students Work with flexible student groups - sometimes grouped by level of ability Greater variety of working methods and examinations

24. The student experience – over the last years • More enthusiastic teachers • Positive experience with students grouped by level of ability • Positive experience from having several teachers involved in appraisal of exams • More variety in teaching • The students are motivated and stimulated - every lesson is important

25. The student resultshavebeensubstantiallyimproved over the last years Source: Department of school (2004-2007). Jens Gerhardsson and Tuula Maunula (2008)

26. Recommendations for further studies: • It is important that the variation theory is equally well known by most of the group members • Try to do more of the planning in the beginning of the study • Don’t make too many changes at a time • Good circumstances around the lessons in the study are important to achieve the most from the study

27. Marianne Burenius Lena Dahlen Jenny Ljungberg Tobias Sundin Johanna Wallinder