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# Equations and Inequalities - Making mathematics accessible to all

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More than ever, students need to engage with mathematical concepts, think quantitatively and analytically, and communicate using mathematics. All these skills are central to a young person’s preparedness to tackle problems that arise at work and in life beyond the classroom. But the reality is that many students are not familiar with basic mathematics concepts and, at school, only practice routine tasks that do not improve their ability to think quantitatively and solve real-life, complex problems.

How can we break this pattern? This report, based on results from PISA 2012, shows that one way forward is to ensure that all students spend more “engaged” time learning core mathematics concepts and solving challenging mathematics tasks. The opportunity to learn mathematics content – the time students spend learning mathematics topics and practising maths tasks at school – can accurately predict mathematics literacy. Differences in students’ familiarity with mathematics concepts explain a substantial share of performance disparities in PISA between socio-economically advantaged and disadvantaged students. Widening access to mathematics content can raise average levels of achievement and, at the same time, reduce inequalities in education and in society at large.

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### Equations and Inequalities - Making mathematics accessible to all

1. 1. EQUATIONS AND INEQUALITIES: MAKING MATHEMATICS ACCESSIBLE TO ALL Andreas Schleicher Chiara Monticone Mario Piacentini
2. 2. WHY MATHEMATICS MATTERS FOR PEOPLE 2
3. 3. 0 10 20 30 40 50 60 70 80 Norway Japan Italy Flanders (Belgium) Korea SlovakRepublic France Denmark Estonia Netherlands Austria Spain Sweden Germany OECDaverage CzechRepublic Ireland Poland England/Norther nIreland(UK) Finland Australia Canada UnitedStates Use or calculate fractions or percentages Use simple algebra or formula Use advanced mathematics or statistics % Use of mathematics skills at work Source: Figure 1.2, OECD Survey of Adult Skills (PIAAC) (2012), Table 1.1a. 3
4. 4. 1.00 1.10 1.20 1.30 1.40 1.50 1.60 Is in good general health Is in the top quarter of earnings Has a job 4 Adults with good mathematics skills earn higher salaries Increase in the likelihood of the outcome related to an increase of one standard deviation in numeracy, OECD average (22 countries) Odds ratios Source: Figure 1.3 OECD Survey of Adult Skills (PIAAC) (2012), Table 1.2 Adults with higher numeracy (by 50 points) are 53% more likely to have high wages
5. 5. 0 50 100 150 200 250 300 350 Canada91 Portugal93 Tunisia26 Macao-China HongKong-China UnitedStates33 Mexico18 Iceland-10 NewZealand Australia6 Japan18 Italy19 Denmark18 Latvia10 RussianFederation15 Belgium21 Brazil Korea-33 OECDaverage13 Liechtenstein Spain34 Indonesia-23 Greece22 France Switzerland Thailand-18 Luxembourg4 Norway33 Poland-7 Germany14 Ireland CzechRepublic14 Sweden17 SlovakRepublic-18 Finland19 Turkey-28 Netherlands21 Austria-10 Uruguay-27 Hungary-13 2012 2003Minutes Time spent in mathematics classes has increased Source: Figure 1.6 In 2012, the average 15 year-old student in an OECD country spent 13 minutes more per week in mathematics classes than in 2003 5 Change in time spent in mathematics classes between 2012 and 2003
6. 6. LearningQuality of instruction Opportunity to learn Ability Perseverance Aptitude School learning has many facets Students’ characteristics Directly shaped by teachers/ schools / systems 6 Opportunity to learn refers to the content taught in the classroom and the time a student spends learning this content
7. 7. 7 Many students have never heard of basic mathematics concepts OECD average Source: Table 1.7 0 20 40 60 80 100 Vectors Arithmetic mean Linear equation Never heard the concept Heard the concept often/a few times Know well/understand the concept %
8. 8. Conditioning factors • Characteristics of the: • Student • Schools • Systems Opportunity to learn • Exposure to tasks • Familiarity with concepts • Time in class Outcomes • Mathematics performance • Attitudes towards mathematics Analytical framework of the report Source: Figure 1.1 8
9. 9. WHY ACCESS TO MATHEMATICS MATTERS AND HOW IT CAN BE MEASURED 9
10. 10. Applied mathematics Working out from a <train timetable> how long it would take to get from one place to another. Calculating how much more expensive a computer would be after adding tax. Calculating how many square metres of tiles you need to cover a floor. Understanding scientific tables presented in an article. Finding the actual distance between two places on a map with a 1:10,000 scale. Calculating the power consumption of an electronic appliance per week. Pure mathematics Solving an equation like: 6x2 + 5 = 29 Solving an equation like 2(x+3) = (x + 3)(x - 3) Solving an equation like: 3x+5=17 How PISA measures exposure to applied and pure mathematics 10
11. 11. R² = 0.05 -1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 Indexofexposuretoappliedmathematics Weak relationship between exposure to applied and pure mathematics OECDaverage Source: Figure 1.8 OECD average Index of exposure to pure mathematics Less exposure More exposure More exposure 11
12. 12. 0 1 2 3 4 Sweden Tunisia Iceland Argentina Luxembourg Brazil CostaRica Greece Switzerland Belgium UnitedKingdom Portugal Uruguay NewZealand Ireland Chile Thailand Australia Mexico France SlovakRepublic Liechtenstein Lithuania VietNam Colombia Poland Italy OECDaverage Indonesia Turkey Finland Malaysia Denmark Austria Peru CzechRepublic Kazakhstan Albania Qatar HongKong-China Latvia Israel Hungary Netherlands Serbia Spain Romania Bulgaria Estonia Montenegro Germany RussianFederation Canada Shanghai-China ChineseTaipei Croatia Slovenia UnitedStates Korea UnitedArabEmirates Jordan Japan Singapore Macao-China Mean index Large international differences in familiarity with algebra…. Source: Figure 1.7 Sweden Singapore Macao-China Never heard the concept Heard the concept once Heard the concept few times Often heard the concept Knows the concept well 0.83 2.85 3.04 12
13. 13. 0 1 2 3 4 Sweden Iceland Netherlands Ireland Germany Austria Switzerland Liechtenstein NewZealand Argentina Malaysia Denmark Brazil Indonesia Luxembourg Tunisia Lithuania Finland CostaRica SlovakRepublic Qatar Australia OECDaverage UnitedKingdom Thailand Chile Slovenia Uruguay Colombia Spain Israel CzechRepublic Peru Mexico HongKong-China Japan Montenegro UnitedStates Canada Portugal Poland Italy ChineseTaipei Bulgaria Korea Estonia Croatia Hungary France UnitedArabEmirates Jordan Macao-China Romania Turkey Kazakhstan RussianFederation Latvia Belgium VietNam Serbia Singapore Greece Albania Shanghai-China Mean index … and in familiarity with geometry Source: Figure 1.7 Sweden Singapore Shanghai- China Never heard the concept Heard the concept once Heard the concept few times Often heard the concept Knows the concept well 0.54 2.94 3.56 13
14. 14. VARIATIONS IN STUDENTS’ EXPOSURE TO AND FAMILIARITY WITH MATHEMATICS 14
15. 15. -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 NewZealand Portugal Brazil Qatar Luxembourg Tunisia Jordan Australia Sweden Belgium Denmark UnitedArabEmirates Colombia Argentina ChineseTaipei Chile CzechRepublic Turkey Netherlands Malaysia Canada SlovakRepublic Austria Indonesia Romania CostaRica Thailand Switzerland Uruguay Bulgaria Latvia Montenegro OECDaverage Serbia Israel France Greece Finland Peru Mexico Germany UnitedKingdom Norway Estonia UnitedStates Hungary Ireland Poland VietNam Japan Shanghai-China Iceland Lithuania Italy Croatia Kazakhstan Slovenia HongKong-China RussianFederation Spain Liechtenstein Singapore Macao-China Korea Indexofexposuretopuremathematics Bottom quarter (disadvantaged students) Second quarter Third quarter Top quarter (advantaged students) Exposure to pure mathematics increases with socio-economic status Source: Figure 2.5b 15
16. 16. 0 5 10 15 20 25 Estonia Malaysia HongKong-China Denmark Finland Jordan Mexico Tunisia VietNam Canada Sweden Israel Macao-China Kazakhstan Latvia Iceland Greece Qatar UnitedArabEmirates CostaRica Indonesia Argentina Poland RussianFederation NewZealand UnitedKingdom Ireland Lithuania Australia UnitedStates Peru Shanghai-China Colombia Montenegro Romania Spain CzechRepublic Luxembourg OECDaverage Singapore Bulgaria Uruguay Serbia Thailand Turkey Portugal Italy Japan Croatia Switzerland Brazil SlovakRepublic Chile Korea Netherlands ChineseTaipei Belgium Slovenia Germany Austria Hungary Liechtenstein Variation explained by students' socio-economic status Variation explained by students' socio-economic status and schools' socio-economic profile Socio-economic profile explains 9% of the variation in familiarity with mathematics % Source: Figure 2.2 Variation in familiarity with mathematics explained by socio-economic profile 16
17. 17. Boy Immigrant Did not attend Girl Non- immigrant Attended pre- primary -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 Gender Immigrant background Pre-primary education Index of familiarity with mathematics Girls, non-immigrants and students who attended pre-primary education are more familiar with mathematics Source: Table 2.10 Note: OECD averages are computed only for countries with available data. OECD average 17
18. 18. 0 2 4 6 8 10 12 14 16 18 20 0 20 40 60 80 100 Percentage of students in schools that engage in a given practice Systems with more selective schools give more unequal access to mathematicsAccesstomathematics More equal More unequal Sources: Figures 2.10, 11, 21 Transferring low-achieving students to another school R2 = 0.42 Considering academic performance for admission R2 = 0.31Considering residence for admission R2 =0.28 Variation in familiarity with mathematics explained by students' and schools' socio-economic profile, OECD average % 18 %
19. 19. -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50 Macao-China HongKong-China VietNam Greece Korea Tunisia Austria Brazil Indonesia CostaRica UnitedStates Estonia Mexico Bulgaria Chile Canada Italy Colombia Luxembourg Thailand Argentina Israel France CzechRepublic Netherlands Kazakhstan Portugal OECDaverage Singapore Qatar Uruguay Sweden Latvia UnitedArabEmirates Shanghai-China SlovakRepublic Slovenia Australia Ireland Germany RussianFederation Switzerland NewZealand ChineseTaipei Hungary Students in the last year of ISCED 2 Students in the first year of ISCED 3 Stronger relationship between familiarity and socio-economic status as students progress to upper secondary education Source: Figure 2.13 Change in familiarity with mathematics associated with one-unit increase in students’ socio-economic status Index change 19
20. 20. Earlier tracking associated with more unequal access to mathematics Accesstomathematics More equal More unequal Source: Figure 2.15 Australia New Zealand Poland United Kingdom Variation in familiarity with mathematics explained by students' and schools' socio-economic profile, OECD countries 20 Austria Belgium Sweden Chile Czech Republic DenmarkEstonia Canada Germany Greece Hungary Ireland Israel Italy Japan Korea Luxembourg Mexico Netherlands Iceland Portugal Slovak Republic Slovenia Spain Finland Switzerland Turkey United States OECD average R² = 0.54 0 5 10 15 20 25 9 10 11 12 13 14 15 16 17 Students' age at first tracking, system level % of the variation
21. 21. Students in vocational schools are more likely to be socio-economically and academically disadvantaged Source: Figure 2.16 Odds ratios Morelikelytobe disadvantagedorlessfamiliar Lesslikely Change in likelihood of having less familiarity with mathematics or being socio- economically disadvantaged associated with enrollment in vocational schools 21 0.00 1.00 2.00 3.00 4.00 5.00 6.00 Ireland Croatia Hungary Spain Korea Slovenia Serbia Netherlands Belgium Montenegro Italy Macao-China Portugal Greece OECDaverage ChineseTaipei Israel Japan Bulgaria SlovakRepublic France Germany Austria RussianFederation Chile Shanghai-China Uruguay UnitedKingdom Australia Turkey Argentina CzechRepublic Luxembourg Thailand Kazakhstan Malaysia Switzerland Indonesia CostaRica Mexico Colombia UnitedArabEmirates Being socio-economically disadvantaged Having less familiarity with mathematics
22. 22. -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 Austria Switzerland Germany Turkey HongKong-China Croatia Portugal Macao-China Greece France Chile Argentina Luxembourg Iceland Montenegro Latvia Australia Jordan RussianFederation Qatar Serbia Kazakhstan Lithuania Italy Singapore OECDaverage Netherlands Tunisia Belgium Mexico UnitedArabEmirates Uruguay CzechRepublic Canada Estonia Brazil Indonesia Romania Japan SlovakRepublic Finland Spain Denmark Malaysia Peru Sweden CostaRica Slovenia Colombia Poland Hungary ChineseTaipei Bulgaria Thailand VietNam Korea Shanghai-China UnitedStates Before accounting for gender, students' socio-economic status, and schools' socio-economic profile After accounting for gender, students' socio-economic status, and schools' socio-economic profile Weak relationship between ability grouping and familiarity with mathematics Source: Figure 2.18b Ability grouping has a small negative association with familiarity once characteristics of students and schools are taken into account Change in the index of familiarity with mathematics associated with ability grouping Index change 22
23. 23. -0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 helps students learn from mistakes lets students decide on their own procedures makes students reflect on the problem gives problems that require thinking for an extended time gives problems that can be solved in different ways gives problems with no immediate solution asks students to explain how they solved a problem Disadvantaged schools Advantaged schoolsIndex change The use of cognitive activation practices is associated with greater performance and familiarity in socio-economically advantaged schools than in disadvantaged ones Source: Figure 2.23b Change in the index of familiarity with mathematics associated with use of cognitive activation strategies, OECD average The teacher… Higher familiarity Lowerfamiliarity 23
24. 24. • Exposure to, and familiarity with, mathematics increase with socio-economic status, and • Vary by students gender, immigrant background, and pre- primary education …individual characterist ics • Grade repetition, schools’ selection mechanisms, and between-school tracking are associated with more unequal access to mathematics • Weak, negative relationship between ability grouping and familiarity with mathematics for the average student …how systems and schools sort and select students • Disadvantaged schools have a (slightly) lower student-to- teacher ratio, but mathematics teachers in disadvantaged schools tend to be less qualified • The use of cognitive activation practices is associated with greater performance and familiarity in socio-economically advantaged schools than in disadvantaged ones …teaching resources and practices Key messages: How access to mathematics varies by… 24
25. 25. EXPOSURE TO MATHEMATICS IN SCHOOL AND PERFORMANCE IN PISA 25
26. 26. -30 -20 -10 0 10 20 30 40 Shanghai-China ChineseTaipei Albania Japan Macao-China Korea Kazakhstan Malaysia RussianFederation Switzerland SlovakRepublic Indonesia Poland Latvia Singapore HongKong-China Thailand Liechtenstein Portugal Qatar Uruguay Bulgaria Romania Montenegro Slovenia Italy Peru CzechRepublic Jordan Mexico Serbia Denmark Argentina Luxembourg Hungary Chile Iceland VietNam OECDaverage Austria Turkey Tunisia Belgium France Germany Lithuania Spain Colombia Australia Canada Estonia NewZealand UnitedArabEmirates CostaRica Sweden Norway Brazil Croatia Finland Netherlands Greece Israel UnitedStates UnitedKingdom Ireland Score-point difference High-performing countries do relatively better on problems requiring knowledge of geometry and algebra Country's/economy's performance on the subscale is higher than on the overall mathematics scale Country's/economy's performance on the subscale is lower than on the overall mathematics scale Source: Figure 3.1 Relative performance on the “Space and Shape” sub-scale 26 Japan Chinese Tapei Shanghai-China Ireland
27. 27. -1.60 -1.40 -1.20 -1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 Uruguay Spain Hungary Sweden RussianFederation SlovakRepublic Portugal France OECDaverage NewZealand Denmark Australia CzechRepublic Belgium Finland 2012 2003Logit Performance on tasks with a focus on geometry deteriorated between 2003 and 2012 Source: Figure 3.3c Performance Change in performance on items in the space and shape sub-scale between 2003 and 2012 (countries where the change is significant) 27
28. 28. 420 440 460 480 500 520 540 Less than 2 hours Between 2 and 4 hours Between 4 and 6 hours More than 6 hours Mathematics Reading ScienceMean score Longer class time up to four hours per week is associated with a large improvement in mathematics performance Source: Figure 3.4 Hours per week: OECD average 28
29. 29. -0.15 -0.10 -0.05 0.00 0.05 0.10 420 430 440 450 460 470 480 490 500 510 520 Less than 2 Between 2 and 4 Between 4 and 6 More than 6 Indexofdisciplinaryclimate Mathematicsscore Mathematics Disciplinary ClimateMean score Hours per week: Instruction time above 6 hours a week is more frequent in classes with poor disciplinary climate Source: Figure 3.6 OECD average 29
30. 30. -10 -5 0 5 10 15 20 Ireland Iceland Macao-China Liechtenstein Norway Bulgaria NewZealand Greece CostaRica Sweden Albania Slovenia Thailand SlovakRepublic Estonia Brazil UnitedStates Spain Chile Hungary Qatar Finland Canada UnitedArabEmirates Portugal Peru UnitedKingdom Tunisia Luxembourg Mexico Kazakhstan RussianFederation Denmark OECDaverage Uruguay Argentina Switzerland Colombia Lithuania HongKong-China Israel Australia Germany Poland Belgium Jordan France Netherlands Indonesia Turkey Montenegro Italy Latvia Singapore Japan Malaysia Serbia VietNam CzechRepublic Austria Romania ChineseTaipei Korea Shanghai-China Croatia Score-point change The relationship between time and performance is much weaker after accounting for school characteristics 30 Croatia Shanghai-China Korea Chinese Tapei Romania Austria Italy Japan Indonesia Netherlands Turkey Malaysia Singapore Relationship between time and performance among students in the same school and grade Liechtenstein Chech Republic
31. 31. 420 440 460 480 500 520 540 First quintile Second quintile Third quintile Fourth quintile Fifth quintile Quintiles of exposure Applied mathematics Pure mathematics Exposure to pure mathematics is more strongly related to performance than exposure to applied mathematics Source: Figure 3.9 Mean score 31
32. 32. -40 -20 0 20 40 60 80 100 120 140 160 180 SlovakRepublic Shanghai-China Albania Serbia Slovenia Macao-China Romania Portugal Bulgaria Mexico Brazil Colombia Austria Tunisia Indonesia Estonia CostaRica Liechtenstein Croatia Australia Hungary Kazakhstan Israel Montenegro Denmark Ireland Argentina Canada Peru Thailand Germany Jordan NewZealand Uruguay Chile Latvia OECDaverage UnitedArabEmirates HongKong-China RussianFederation ChineseTaipei CzechRepublic Netherlands Turkey Italy UnitedKingdom Poland Lithuania UnitedStates Belgium Sweden Singapore France Finland Norway Switzerland Qatar Malaysia Spain Luxembourg VietNam Korea Score-point change Exposure to pure mathematics is related to higher performance, even after accounting for school characteristics Source: Figure 3.11 32 Relationship between exposure and performance among students in the same school
33. 33. Charts Q1 Revolving Door Q2 R² = 0.39 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 2.80 300 400 500 600 700 800 Drip Rate Q1 Arches Q2 Stronger association between familiarity with concepts and performance on more demanding tasks Drip Rate Q1 Effectoffamiliarity Higher positive effect Lower positive effect Source: Figure 3.12 Difficulty on the PISA scale Revolving Door Q2 33 Odds ratio
34. 34. Familiarity with pure mathematics is enough to solve procedural problems… Scenario: Nurses calculate the drip rate for infusions using the formula: 𝐷𝑟𝑖𝑝 𝑟𝑎𝑡𝑒 = 𝑑𝑣 60𝑛 d is the drop factor in drops per mL v is the volume in mL of the infusion n is the number of hours the infusion is required to run Question: Describe how the drip rate changes if n is doubled but the other variables do not change. Drip Rate Question 1 34
35. 35. Korea OECD average Indonesia Malaysia Qatar Shanghai-China Chinese-Taipei R² = 0.57 -3.50 -3.00 -2.50 -2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 2.00 -1.00 -0.50 0.00 0.50 1.00 1.50 LogitfortheitemDripRateQ1 Index of familiarity with mathematics BEFORE accounting for countries’ performance on all the other tasks Source: Figure 3.13 Familiarity with mathematics and performance on Drip Rate Question 1: Country-level relationship 35
36. 36. Familiarity with mathematics and performance on Drip Rate Question 1 - Country-level relationship Korea OECD average Netherlands Luxembourg Spain R² = 0.22 -1.60 -1.40 -1.20 -1.00 -0.80 -0.60 -0.40 -0.20 0.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00 AFTER accounting for countries’ performance on all the other tasks Index of familiarity with mathematics LogitfortheitemDripRateQ1 Source: Figure 3.13 36
37. 37. …but being familiar with mathematics content might not be enough to solve problems that require to reason mathematically Scenario: A revolving door includes three wings which rotate within a circular-shaped space and divide the space into three equal sectors. The two door openings (the dotted arcs in the diagram) are the same size. Possible air flow in this position 200 cm Question: What is the maximum arc length in centimetres (cm) that each door opening can have, so that air never flows freely between the entrance and the exit? Revolving Door Question 2 37
38. 38. Familiarity with mathematics and performance on Revolving Door Question 2 - Country-level relationship Korea OECD average Indonesia Malaysia Qatar Shanghai-China Chinese-Taipei R² = 0.18 -6.00 -5.00 -4.00 -3.00 -2.00 -1.00 0.00 -1.00 -0.50 0.00 0.50 1.00 1.50 BEFORE accounting for countries’ performance on all the other tasks Logitfortheitem RevolvingDoorQ2 Index of familiarity with mathematics Source: Figure 3.14 38
39. 39. Familiarity with mathematics and performance on Revolving Door Question 2 - Country-level relationship Korea OECD average Netherlands Luxembourg Spain R² = 0.02 -5.00 -4.50 -4.00 -3.50 -3.00 -2.50 -2.00 -1.50 -1.00 -0.50 0.00 -0.80 -0.30 0.20 0.70 Logitfortheitem RevolvingDoorQ2 Index of familiarity with mathematics Source: Figure 3.14 AFTER accounting for countries’ performance on all the other tasks 39
40. 40. -30 -20 -10 0 10 20 30 40 Macao-China HongKong-China Tunisia Malaysia Estonia Mexico Denmark Israel CostaRica VietNam Kazakhstan Argentina Latvia Greece Lithuania Shanghai-China Romania Finland Ireland UnitedArabEmirates Japan SlovakRepublic CzechRepublic Bulgaria RussianFederation NewZealand Poland Indonesia Sweden UnitedKingdom Uruguay Jordan Montenegro Canada Luxembourg Iceland Serbia Singapore OECDaverage Slovenia Qatar Peru Turkey Colombia Australia Italy ChineseTaipei France Netherlands Chile Spain Croatia Thailand Portugal Brazil UnitedStates Belgium Hungary Switzerland Germany Austria Korea % of score-point difference Familiarity with mathematics explains 19% of the socio-economic performance gap Source: Figure 3.15 Percentage of the score-point difference between advantaged and disadvantaged students explained by different familiarity with mathematics 40
41. 41. 0.30 0.40 0.50 0.60 0.70 0.80 0.90 Macao-China Estonia Thailand Indonesia Jordan Iceland Finland Mexico Serbia HongKong-China Argentina Greece Tunisia Norway Kazakhstan Croatia Korea Malaysia UnitedKingdom Romania Canada Turkey Germany Latvia Colombia Denmark CostaRica Netherlands UnitedStates Qatar Switzerland Sweden RussianFederation Lithuania CzechRepublic Italy Brazil Montenegro OECDaverage VietNam Ireland Slovenia Australia Japan Austria Singapore Poland NewZealand Portugal UnitedArabEmirates Spain Belgium Uruguay Luxembourg Hungary France Chile Shanghai-China Israel Bulgaria Peru ChineseTaipei SlovakRepublic The task has a scientific context The task has a personal contextOdds ratio 41 Socio-economically disadvantaged students perform better on “familiar” tasks Socio-economicgapinperformance Larger gap Smaller gap Source: Figure 4.18
42. 42. • Countries where students have higher familiarity with geometry and algebra perform better in all tasks and relatively better on tasks requiring geometry and algebra • Performance on tasks with a focus on geometry deteriorated between 2003 and 2012 Structure of curriculum • Increasing instruction time in mathematics beyond 6 hours a week has no clear relationship with performance. The relationship differs substantially across countries, and within countries according to the quality of the disciplinary climate in the classroom • Exposure to pure mathematics tasks (equations) is strongly related to performance • Exposure to and familiarity with mathematics concepts may not be sufficient for solving problems that require the ability to think and reason mathematically Amount/type of mathematics tasks and performance • Almost 20% of the performance gap of disadvantaged students is explained by their lower familiarity with mathematics concepts. • Disadvantaged students lag behind other students particularly in those complex tasks requiring modelling skills and the use of symbolic language. Socio- economic disadvantage and exposure to mathematics Key messages 42
43. 43. OPPORTUNITY TO LEARN AND STUDENTS’ ATTITUDES TOWARDS MATHEMATICS 43
44. 44. Less than half of students enjoy studying mathematics 0 10 20 30 40 50 60 70 80 90 Austria-4 Hungary SlovakRepublic-5 Finland4 Belgium-5 CzechRepublic Korea Japan5 Norway Netherlands Luxembourg Poland-4 Canada UnitedStates Sweden Ireland4 Spain OECDaverage NewZealand Latvia Australia3 Germany-4 France-5 Macao-China RussianFederation Portugal Italy Iceland10 Switzerland-4 Uruguay Greece8 Turkey-5 Mexico8 HongKong-China3 Liechtenstein Brazil-4 Denmark Tunisia-9 Thailand Indonesia5 2012 2003% Source: Figure 4.2 Percentage of students who agree with the statement I do mathematics because I enjoy it" 44 The difference between 2003 and 2012 is significant
45. 45. 45 Exposure to more complex mathematics is related to lower self-concept, among students of similar ability Mathematicsself-concept Source: Figure 4.7 -0.40 -0.30 -0.20 -0.10 0.00 0.10 0.20 0.30 0.40 Liechtenstein Indonesia Argentina Thailand Kazakhstan Austria Luxembourg Netherlands Tunisia Qatar Japan Romania Macao-China Belgium Germany Bulgaria SlovakRepublic Malaysia Brazil Switzerland HongKong-China Shanghai-China Latvia Uruguay VietNam Estonia CostaRica Greece Lithuania Mexico Israel Denmark RussianFederation Peru Sweden Colombia CzechRepublic Chile Montenegro OECDaverage NewZealand Ireland UnitedArabEmirates Albania UnitedKingdom Turkey Croatia UnitedStates Hungary Jordan Spain Finland France Slovenia Canada Italy Singapore Portugal Iceland Poland Australia Serbia ChineseTaipei Korea Before accounting for performance in mathematics After accounting for performance in mathematicsIndex change Change in students’ self-concept associated with 1 unit change in familiarity
46. 46. Exposure to more complex mathematics is also related to greater anxiety among low-performing students Source : Figure 4.8 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 Malaysia OECD average Czech Republic Bottom quarter by mathematics performance Top quarter by mathematics performanceIndex Change in students’ anxiety associated with a change in familiarity, by students' mathematics performance 46 MoreanxietyLessanxiety
47. 47. Students with hard-working friends are more motivated to learn, especially in schools where students are least familiar with mathematics 0.00 0.50 1.00 1.50 2.00 2.50 I am interested in the things I learn in mathematics I do mathematics because I enjoy it I look forward to mathematics lessons Making an effort is worthwhile for the work I want to do Mathematics is important for what I want to study later on Schools where students are more familiar with mathematics Schools where students are less familiar with mathematics Odds ratio Source: Figure 4.11 Change in the probability that students agree with each statement, associated with having friends who work hard on mathematics 47
48. 48. High-performing students whose parents do not like mathematics are more likely to feel helpless 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 OECD average France Top quarter of mathematics performance Bottom quarter of mathematics performance Odds ratio Source: Figure 4.14 Change in the probability that students feel helpless when doing mathematics problems associated with having parents who do not like mathematics Children whose parents dislike mathematics have higher anxiety 48
49. 49. Students whose teachers provide feedback or specify learning goals are more familiar with mathematics 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 The teacher gives different work to classmates who have difficulties learning and/or to those who can advance faster The teacher has us work in small groups to come up with joint solutions to a problem or task The teacher gives extra help when students need it The teacher continues teaching until the students understand The teacher asks questions that make us reflect on the problem The teacher gives problems that require us to think for an extended time Students more familiar with mathematics Students less familiar with mathematics Index change Source: Figure 4.16 Change in the index of mathematics self-concept associated with having mathematics teachers who provide feedback or specify learning goals in every or most lessons 49
50. 50. Teachers’ feedback practices have a different relationship with anxiety depending on students’ familiarity with mathematics Source: Figure 4.15 -0.10 0.00 0.10 The teacher gives me feedback on my strengths and weaknesses in mathematics The teacher tells us what is expected of us when we get a test, quiz or assignment Low familiarity students High familiarity studentsIndex change Change in mathematics anxiety associated with having teachers who engage in these practices, OECD average MoreanxietyLessanxiety 50
51. 51. Using a computer during mathematics lessons is associated with higher motivation for learning mathematics 0.00 0.10 0.20 0.30 0.40 0.50 Japan Iceland Denmark Macao-China Liechtenstein Serbia Mexico Spain Estonia Germany CzechRepublic Austria Belgium Shanghai-China RussianFederation HongKong-China Finland Switzerland Norway Singapore Croatia Uruguay Italy Ireland Netherlands OECDaverage CostaRica Australia Latvia Slovenia Chile Portugal Poland Turkey Sweden Hungary SlovakRepublic Korea Greece NewZealand Jordan ChineseTaipei Israel After accounting for students' and schools' characteristics Index change Source: Figure 4.17 Change in intrinsic motivation for mathematics associated with using a computer in mathematics class 51
52. 52. • Exposure to more complex mathematics concepts is associated with • lower self-concept and higher anxiety among low- performing students, and with • higher self-concept/lower anxiety among high-performing students Opportunity to learn and attitudes towards mathematics • Peers: Having hard-working friends can increase mathematics self-concept, but students can develop lower beliefs in their own ability when they compare themselves to higher-achieving peers • Parents may transfer their feelings about mathematics to their children, even high-performing ones • Teachers’ practices can have a different relationship with students self-concept and anxiety depending on students’ familiarity with mathematics Mediating factors Key messages 52
53. 53. WHAT DOES THIS MEAN FOR POLICY? 53
54. 54. Develop coherent standards Develop skills beyond knowledge Reduce the impact of tracking Support teachers of heterogenous classes Support positive attitudes Monitor Opportunity to Learn Develop coherent standards, frameworks and instruction material for all students How: • Cover core ideas more in depth • Increase connections between topics • Review textbooks and teaching material accordingly A policy framework to widen opportunities to learn A policy programme in 6 points In Singapore the mathematics framework covers a relatively small number of topics in depth, following a spiral organisation in which topics introduced in one grade are covered in later grades at a more advanced level
55. 55. Develop coherent standards Develop skills beyond knowledge Reduce the impact of tracking Support teachers of heterogeno us classes Support positive attitudes Monitor Opportunit y to Learn Help students acquire mathematical skills beyond content knowledge How: • Replace routine tasks with challenging, open problems • Develop specific training for teachers • Integrate problem-solving abilities into assessments 55 A policy framework to widen opportunities to learn A policy programme in 6 points Recent revisions of the mathematics curricula in England, Scotland, Korea and Singapore emphasise the development of problem- solving skills
56. 56. Develop coherent standards Develop skills beyond knowledge Reduce the impact of tracking Support teachers of heterogeno us classes Support positive attitudes Monitor Opportunit y to Learn Reduce the impact of tracking on equity in mathematics exposure How: • Consider possibilities to delay tracking • Improve quality and quantity of mathematics instruction in non- academic pathways • Allow students to change tracks 56 A policy framework to widen opportunities to learn A policy programme in 6 points Sweden and Finland reformed their education systems in the 1950-1970s: a later age at tracking reduced inequalities in outcomes later on. Also Germany and Poland reformed the tracking system to reduce the influence of socio- economic status on student achievement
57. 57. Develop coherent standards Develop skills beyond knowledge Reduce the impact of tracking Support teachers of heterogeno us classes Support positive attitudes Respon- sibility Learn how to handle heterogeneity in the classroom How: • Provide students with multiple opportunities to learn key concepts at different levels of difficulty • Adopt student-oriented practices such as flexible grouping or cooperative learning • Offer more individualized support to struggling students 57 A policy framework to widen opportunities to learn A policy programme in 6 points In Finland, half of children with special education needs are mainstreamed and assigned special teachers, rather than being in special schools.
58. 58. Develop coherent standards Develop skills beyond knowledge Reduce the impact of tracking Support teachers of heterogeno us classes Support positive attitudes Monitor Opportunit y to Learn Support positive attitudes towards mathematics through innovations in curriculum and teaching How: • Develop, use and share engaging tasks and learning tools (including IT-based) • Learn how to give effective feedbacks to struggling students • Engage parents 58 A policy framework to widen opportunities to learn A policy programme in 6 points The 2011 revisions of the mathematics curriculum in Korea has reduced curriculum content to give more time to engaging activities that would improve students’ motivation
59. 59. Develop coherent standards Develop skills beyond knowledge Reduce the impact of tracking Support teachers of heterogeno us classes Support positive attitudes Monitor Opportunity to Learn Monitor and analyse opportunity to learn How: • Collect and analyse data on the implemented curriculum both from teachers and students • Support multi-year research and curriculum-development programmes • Analyse data on mathematics teaching practices from video studies 59 A policy framework to widen opportunities to learn A policy programme in 6 pointsThe Teaching and Leaning International Survey (TALIS) study is piloting an international video study of teaching practices to provide insights into effective teaching practices
60. 60. THANK YOU 60