1. THE PISA MATHEMATICS RESULTS IN CONTEXT Elizabeth Oldham Trinity College, Dublin Second National PISA Symposium 6 April 2005

2. Outline Historical background –Realistic Mathematics Education v. “modern mathematics” Current relevant issues in Ireland The PISA 2003 Irish mathematics results –Overall results –Performance on subscales and individual items Implications and possibilities

3. Historical Background Origins of Realistic Mathematics Education –A (Freudenthal’s!) reaction to “modern mathematics” The “modern mathematics” movement –Mathematics of the “Bourbaki group” (from France) Abstract (sets & structures) – very “pure,” no contexts Rigorous, logical, with precise terminology Vertical, not horizontal, mathematisation emphasised –School mathematics was dated – mismatched to this Hence, a “top down” and “mathematical” influence… … not a pedagogical one

4. Influence in Ireland –The 1960s second level syllabuses “bought into” modern mathematics … and continued to do so in the early 1970s revision –The 1971 Primary curriculum was less affected Developments –The legacy at second level persists, albeit diluted … with some indications of change in exams.; No chance since the 1960s for a basic critique!!! –There is more focus on horizontal mathematising in the revised Primary Curriculum … which emphasises problem-solving

5. Current Relevant Issues in Ireland Dissatisfaction with achievement; e.g. see –Reactions to Ordinary LC performance, 2001 on –Dropout/difficulty at third level Culture of maths. teaching and learning –Not a mathematising culture … especially for lower achievers? –Focus on teaching towards predictable examinations (Elwood & Carlisle, 2003) … and associated didactical contract: “This technique will be used in Paper II, Question 6, part (b) (iii)”

6. In fact –“Three-part” questions in the exams. were intended to include applied / problem-solving “part (c)”s … ensuring but restricting the “problem” aspects However, teachers and students maybe try to cover all possibilities as isolated example types… … increasing the “content” and not achieving the “process” –The JC Guidelines (pp. 91, 96-100) emphasise the importance of the other objectives … and lay the groundwork for their eventual assessment? [Many other factors – “outside the scope”]

7. The PISA 2003 Irish Maths. Results The reports contain –Descriptive data on achievement Ranking of countries by mean score Country means and standard deviations Scores at key percentiles Percentage of students at “proficiency levels” Subscale scores –Further analyses Achievement differences between schools Associations with background variables … etc., etc., etc.

8. Ranks tell one almost nothing until one knows the context … … but in this case we have been given some context … … and will find more Note

9. Overall results “The story as before” (i.e. in 2000)

10. Ranks and means Note –OECD mean is 500, SD is 100 Ranks –2000: Ireland was 15 th out of 27 –2003: Ireland is 17 th out of 29 / 20 th out of 40 Means –2000: Ireland’s was 502.9 –2003: Ireland’s is 502.8 … effectively on the OECD mean … both times “Could do better?”

11. Country comparisons in 2003 –Countries scoring significantly higher than Ireland include Pacific Rim (as always!) The Netherlands (they should, shouldn’t they?) Finland (the success story of PISA 2003...) –Countries scoring at the same level include France Germany –Countries scoring significantly lower include Hungary (“success” in earlier TIMSS study) USA

12. Distribution Standard deviations –2000: Ireland’s was 83.6 –2003: Ireland’s is 85.3 … in each case, one of the lowest Scores at key percentiles –Not a great “tail” … i.e. comparatively few low scorers… –… but not a great “head” either “Proficiency levels” (see handout) Empirically determined, but can be associated with skills –Again, Irish scores “bunch” (see graph below) Rather homogeneous system

14. Subscale scores Recall the four subscales –Uncertainty Mainly statistics and probability –Change & Relationships Algebra and functions, but other areas also –Quantity Number, applied arithmetic and measure –Space & Shape Not so much formal geometry as measure etc. Also recall the competency clusters –Reproduction, Connections, Reflection Some items “released” for inspection

15. Uncertainty –Mean score (517) significantly above the OECD mean (502) Released items (see handout – more on web) –“Robberies” (Connections, level 6 for full credit) Statistics, close to the syllabus, but hard; better than OECD mean with regard to partial credit –“Earthquakes” (Reflection, level 4) Probability; outside the syllabus but above OECD mean In our culture? Chancers ? Probability aspect missing from many syllabuses No learned helplessness ?

16. Change & Relationships –Above the OECD mean (506 versus 499) Released items –“Internet relay chat” qus. 1 (Connections, level 3) & 2 (Reflection, level 5) Qu. 2 tests what? … we scored well above the OECD mean –“Walking” (Reproduction, level 5) Algebra; on at least the Higher syllabus, but not an emphasised “routine” – below-average score

17. Quantity –502 in Ireland; OECD mean 501 Released items –“Exchange rate” qus. 1 & 2 (Reproduction, levels 1 & 2) & 3 (Reflection, level 4) Applied arithmetic; illustrates Irish scores “bunching” –“Skateboard” qus. 1 (Reproduction, level 3 for full credit) & 2 (Reproduction, level 4) Qu. 2 is on the LC syllabus… … hence, not reproduction for us, and relatively hard Archetypal PISA: picture, context knowledge probably helpful...

18. Space & Shape –476 compared to OECD 496; significantly lower Released items include –“Carpenter” (Connections, level 6) Measure / geometry; hard, especially for us! –“Number cubes” (Connections, level 3) Outside the syllabus; below OECD mean score Curricular clash!

19. Implications and Possibilities We do not know all “raw” scores / item facilities –We are talking relative performance Overall picture: strengths and weaknesses –The low scorers do get off the ground (relative to such in other countries)…. … we do present most students with more than (say) social arithmetic sums? … on unfamiliar items, they had not learned to be helpless? … we have fewer “minorities”? –Our higher scorers “could do better” … the culture of mathematics teaching and learning?!!!

20. Possibilities for change Difficult, but…. The French connection A vignette based on visiting student-teachers

21. Curricular mismatch with PISA is not an excuse... … but neither is it an automatic recipe for change A challenge!

22. PISA / RME: horizontal and vertical components Ireland / Bourbaki: vertical component Where do we want to go?