Learning from Others

1. Learning from Others Paul Andrews Wednesday 11 April, 2012

2. Why is learning from others important? • In 1994, while working at the MMU, I was able to visit Budapest to observe mathematics being taught in that city. • Hungary has a long and well-established tradition of mathematics teaching excellence and I was immediately captivated. • The picture shows an isosceles triangle with an area of 9 units squared. • The triangle has been drawn with each vertex on an integer grid point. • One of the vertices is at the point (3, 1). • How many different triangles satisfy these conditions?

3. Is interesting • Offers new lenses for analysing my own classroom • Challenges my assumptions • Makes the strange familiar and the familiar strange • Facilitates an evaluation of adaptive potential • Helps me understand how culture impacts in hidden ways on what I do • Allows a system to evaluate its effectiveness Investigating how others do things

4. Framing today’s talk • The new (not so new now!) government has made a big issue of listening to stake-holders as preparation for curricular reform. • It has also indicated, through Tim Oates’ report, that it is looking at high achieving jurisdictions for curricular steers • In this talk I hope to outline the need for such an approach, while highlighting both potentials and pitfalls of attempts to learn from others • In so doing I will draw on my experiences of visiting European mathematics classrooms, something I have been doing for almost twenty years.

5. Is there a problem With English (UK?) mathematics teaching?

6. Yes, I think there is and it’s long lasting • In the 1990s, at the height of the SMP’s 5-16 scheme’s popularity, I was working in teacher education at the MMU. • Such lessons were described by a foreign observer thus: They start with the teacher completing various administrative matters including a register. Students would then work on their own while the teacher circulated the room busily helping students with difficulties and recording their latest booklet test marks. Finally, a few minutes before the bell, teachers would ask students to pack away (Leung, 1995). • An exemplary lesson in Rochdale • A student learning ratio

7. A (post-SMP) outsider’s perspective • English mathematics privileges pragmatism over theory. • A spiral curriculum enables topics, taught over a small number of lessons, to be introduced at an elementary level, picked up again later and taught independently of any sense of structure. • New concepts or methods are typically given as information or in the style of a recipe. • Proof is rare with theorems warranted by experiment, example or teacher assertion. • Standard algorithms are of minor importance with priority given to students’ own solutions. • The use of precise mathematical language is unimportant. (Kaiser, 2002; Kaiser et al., 2006)

8. The typical English text book • Comprise shorter chapters and fewer words and examples per page than text books in France or Germany. • Incorporates little explanatory text, although there may be worked examples to overcome the need for teacher mediation, • Place little emphasis on technical vocabulary. • Comprises exercises, typically of questions similar to the worked example, little cognitive demand and little obvious scope for extension • Rarely places mathematics in context. (Haggarty and Pepin, 2002; Pepin and Haggarty, 2001)

9. The typical French textbook • Every chapter is divided into three parts - activités; l’essentiel; exercices • Activités are small investigations, practical or cognitive activities intended to introduce pupils to a notion. • L’essentielcorresponds to the compulsory part to be taught to and understood, by all students. It is often referred to as the coursand is presented in words and worked examples that the teacher exploits. • Exercises are typically graduated in order of difficulty. • French text books are written by teams of colleagues connected to an IREM - Institut de Recherche sur l'Enseignement Mathematique. Typically, IREM teams comprise academicsand teachers who systematically integrate research in mathematical didactics. (Pepin and Haggarty, 2001)

10. English teachers’ beliefs about mathematics Hungarian teachers’ beliefs about mathematics (Andrews, 2007)

11. English teachers’ professional goals (Andrews, 2010)

12. Hungarian teachers’ professional goals

13. Another Hungarian problem • Grade 5 students were invited to order the above from smallest to largest. • The solution was managed collaboratively, involved at least ten volunteers and lasted several minutes. • The first contributor wrote a 1 against 0,405, the second a 2 against 0,45 and so on until the list was completed. • All contributors articulated their thinking: the first child said that all the numbers had a four in the tenths position, but only one had a zero in the hundredths. • (Andrews, 2003)

14. What and where are the alternatives? Looking at other systems

15. Does PISA help us? • The OECD’s programme of international student assessment (PISA) undertakes multinational assessments of how well “education systems prepare students for real-life situations” (OECD, 2009: 9). • Each cycle assesses 15 year-old students’ application of subject knowledge – mother tongue, mathematics and science - to authentic or real-world situations (Adams, 2003). • The outcomes have been viewed as benchmarks against which educational policies and practices have been evaluated, not least from the perspective of education as the lynchpin of economic success. Indeed, “A modest goal of having all OECD countries boost their average PISA scores by 25 points over the next 20 years… implies an aggregate gain of OECD GDP of USD 115 trillion over the lifetime of the generation born in 2010... Bringing all countries up to the average performance of Finland, OECD’s best performing education system in PISA, would result in gains in the order of USD 260 trillion” (OECD, 2010: 6)

16. The Finnish PISA Phenomenon? Such has been the interest in this small country’s success that envoys from all around the world have visited Finland to uncover the story behind the success (Laukkanen, 2008).

17. How the Finns explain their success 1 • The comprehensive school system is based on equity for all, irrespective of gender, social status or ethnicity, and a compulsory nine year basic curriculum (Laukkanen, 2008; Sahlberg, 2011; Välijärvi, 2004 ). • Students, who are neither tracked (Antikainen, 2006; Reinikainen, 2012) nor streamed (Halinen and Järvinen, 2008; Lie et al., 2003), are taught in schools typically construed as learning and caring communities (Aho et al., 2006; Sahlberg, 2007). • The right to choose the school their children attend has little influence on parents’ decision making (Poikolainen, 2012). • Finland achieved the lowest PISA-related between school variation (Halinen and Järvinen, 2008; Liang, 2010; Reinikainen, 2012; Schleicher, 2009).

18. How the Finns explain their success 2 • Integrated SEN, involving around 1 in 6 students and requiring no formal statement of need, begins when difficulties arise and has both reduced the stigma of special needs and promoted inclusion (Halinen and Järvinen 2008; Savolainen 2009). • Primary years SEN provision is typically focused on supporting pupils’ mother tongue and basic mathematical skills (Hausstätter and Takala, 2011; Kivirauma and Ruoho 2007). • Thus, since PISA and Finnish special education both focus on the same areas “it seems plausible that the special educational system in this country plays a positive role in relation to PISA” (Hausstätter and Takala 2011: 276), not least because Finnish students’ mathematical word problem competence is a function of their reading competence (Vilenius-Tuohimaa, et al., 2008).

19. How the Finns explain their success 3 • Teaching is a popular career choice among school leavers, even though fewer 20% of applicants are successful (Laukkanen 2008; Niemi & Jakku-Sihvonen, 2006). • Finnish teachers are talented and continue to enjoy high public esteem (Sahlberg, 2007, 2011; Simola2005; Tuovinen, 2008), being considered professionals who know what is best for their students (Aho et al 2006). • A master’s degree, requiring 4 to 5 years to complete, is an essential prerequisite (Antikainen, 2006; Laukkanen, 2008; Jyrhämä et al., 2008; Niemi & Jakku-Sihvonen, 2006) that ensures “an academically high standard of education for prospective teachers” (Niemi, 2012: 29) and have been well received by teachers who saw them as enhancing their professional status (Jyrhämä et al., 2008).

20. An example of Finnish teaching • This clip derives from the first lesson on percentages taught to a grade 5 class. • The teacher was considered locally as effective, not least because, teachers working in partnership with their local universities’ teacher education department would have had “to prove they are competent to work with trainee teachers” (Sahlberg 2011, 36).

21. Perspectives on Finnish teaching • Finnish teaching is a teacher-dominated practice that has changed little in fifty years and created an intelligence and emotional wasteland (Carlgren et al., 2006). • A mid-1980s commission of enquiry advocated, inter alia, a “diversification of teaching methods” alongside a shift of emphasis from “routine skills onto development of thinking” (Kupari 2004, p.11). • However, a government commissioned external review found a conservative workforce, uncertain how to adapt to expected changes, continuing to teach as it always had (Norris et al., 1996). • More recently, a prime ministerial initiative, aimed at improving mathematics teaching through substantial in-service opportunities for teachers, seems to have resulted in little change (Savola, 2010), particularly with respect to systemic expectations of mathematical problem solving (Pehkonen, 2009).

22. Finnish PISA success: at what cost? • Earlier deductive approaches to mathematics have been replaced by procedural approaches that have marginalised logical thinking, elegance, structure and proof (Malaty, 2010). • Finnish performance (520) on TIMSS 1999, the sole TIMSS in which Finland participated, was average for European nations and masked disappointingly low scores on algebra (498) and geometry (494) • This reflects a decline in the mathematical knowledge necessary for students to continue to higher education (Astala et al., 2006; Tarvainen and Kivelä, 2006). And…,

23. Alternative explanations for Finnish PISA success • A strong Finnish identity grew from successive periods of Swedish and Russian colonialism lasting from the mid-thirteenth century until independence in 1917 (Niemi, 2012). • For more than four hundred years, reading competence was a prerequisite for receiving Lutheran sacraments. Failure in the public examination, or kinkerit, meant a denial of permission to marry with the consequence that Finns have, for centuries, been raised in a culture of high expectations not only with respect to learning but also personal responsibility (Linnakylä, 2002). • This created a community with a “strong appreciation for education” (Halinen and Järvinen 2008: 80); reading is valued so highly that the Finnish library network is among the densest in the world, with Finns borrowing more books than anyone else (Sahlberg, 2007). • Such traditions may explain why there is essentially no illiterate underclass in Finnish society.

24. My perspectives on Finnish teaching • My analyses, based on video data collected during the period 2003-2005, indicate that teachers work within a tradition of implicit didactics – they offer definitions and model solution strategies in ways that expect students to infer meaning. • Teachers rarely, if ever, seek or offer clarification during public discourse. Students are left to infer meaning from such exchanges. • Students are encouraged, implicitly and explicitly, to make extensive notes. Teachers write extensively on the board, typically very slowly in capital letters. • There are various indications that students are expected to discuss their notes and sense-making at home. (Andrews, 2011a; Andrews, 2012; Andrews et al., 2012)

25. Comparing PISA with TIMSS.…

26. Looking at Flanders Is it a more appropriate site for investigation?

27. Perspectives on Flanders • Although nominally comprehensive, Flanders operates a differentiated school system according to whether they are public or Catholic (Opdenakker et al. 2002: 405). • With respect to secondary education, the core, including mathematics, is the same for everyone. However, students elect to follow vocationally oriented, humanities oriented or classically oriented tracks. There is a general understanding that these form an academic hierarchy (Op ’t Eynde et al., 2006). • Research shows that school type and track influence significantly the mathematical learning of Flemish students (Brutsaert, 1998; Opdenakker et al., 2002; Pugh and Telhaj, 2007; Pustjens et al., 2007) • Thus, in relation to the UK, Flanders may be more appropriate research site, particularly from the perspective of its multicultural demographic.

28. A Flemish perspective on percentages • These clips show a teacher working with a grade 5 class on percentages. The lesson had started with some percentage-related artefacts brought from home. • As with the Finnish teacher above, she was considered locally as effective, having worked with a nearby university on a curriculum development project focused on introducing ‘realistic’ approaches to mathematics teaching. • The English teacher did something entirely different

29. Observational research shows • Flemish teachers exploit high level questioning, sharing of student ideas and explanation, augmented by their coaching - using closed questions, hints, prompts, or feedback to facilitate students’ completion of tasks – and activating their students’ prior knowledge. They pay little attention to motivating their students, within class differentiation or activities focused on individual exploration (Andrews, 2009a) • They privilege the development of their students’ conceptual knowledge above procedural knowledge, although both are emphasised frequently. They engage students in public reasoning and highlight structural links and issues of elegance. They rarely offer students genuine opportunities to problem solve (Andrews, 2009b).

30. eliciting general principles What can we learn from classroom observation studies?

31. Looking for general principles • In the following clips I show how three teachers, Sami from Finland, Pauline from Flanders and Emese from Hungary, approached the teaching of linear equations. • In all cases, the three teachers had engaged their students in some preliminary activities focused on the use of mental strategies, essentially arithmetical, to solve equations with the unknown on one side. • The clips show their introductions to the formal procedures the teachers intended their students to follow subsequently.

32. General principles? • All teachers exploited algebraic equations, knowing that intuitive approaches would fail. All invoked the balance and all annotated their solutions similarly. • Sami, despite his animated introduction, made little use of the balance. Despite implicit acceptance of his student’s subtraction of x, subsequent actions indicated that he had a clear view, which students had to infer, as to what was acceptable. • The solution to Pauline’s very complex equation, which comprised a single allusion to the balance, was driven by many questions, highly stylised and invoked several familiar concepts to highlight inter- and intra- topic links. • Emese exploited a ‘realistic’ word problem to derive her equation. She sustained the balance throughout, made explicit the relationship between sketches and symbolic representations and questioned constantly. • Both Sami and Pauline operated in exclusively mathematical worlds although it was Pauline and Emese who included checks at the conclusions of their expositions. (Andrews, 2011b; Andrews and Sayers, 2012)

33. A Japanese problem The picture below shows the boundary, EFG, between two fields. Each field is owned by a different farmer and both agree that their lives would be made easier if the boundary were straight. Where might the straight boundary be drawn in order to preserve the areas of both fields?

35. Back to Hungary (thanks to Jenni Back) • A grade 1 class was working on number sequences. Klara, wrote the following on the chalkboard 3 7 6 10 _ _ _ _ _ • Klaraasked her children to read the numbers before asking them to identify the rule for the sequence. Students volunteered that the first jump was add four, the second was subtract one, add four and so on. The class then predicted, and Klara wrote in the spaces, that the next numbers in the sequence were 9, 13, 12, 16 and 15. • This was followed a series of questions, each pertaining to one of the numbers of the sequence. For example, I am thinking of the largest one-digit number , yielded nine, at which point she wrote I above the number. Other questions related to the value of a particular Cuisenaire rod, and concepts like more than, two digit number, double, the sum of the digits, smaller neighbour, bigger neighbour, the difference of the digits.Each correct answer prompted Klara to write a letter above it the corresponding number.

36. It continued.... • Finally, with all other numbers having been accompanied by a letter, Klara asked her students to suggest properties for 10. Their propositions included • It is the bigger neighbour of 9. • It is the smallest 2-digit number. • It is the smaller neighbour of 11. • The sum of its digits is 1. • It’s an even number. • It is the sum of the 1 and 9. • Klara: Yes, it is the sum of the 1 and 9… and who knows the letter in my hand? • Chorus: SZ • Klara: Yes, so where are we going today? • Chorus: Bábszínház

37. And just to reinforce why not Finland • The following clip comes from a grade 6 lesson on polygons, taught by a teacher also considered as effective. • Finland, I believe, highlights well the dual role of social equity and high expectations in educational achievement. • It has, I believe, little to offer by way of adaptive potential. • I turn, finally, to what I believe research on others can tell us about a good mathematics curriculum and how it is presented to children.

38. A ‘good’ mathematics curriculum expects • Mathematics to be difficult • Mathematics to be problem-solving activity • Problems to exemplify generality and problematise particularity • Teachers to develop rather than state concepts and procedures • Teachers to make connections within and between topics • Students to engage with proof and justification • Teachers to encourage mathematical vocabulary • Applications to be subordinated to mathematics itself • Mathematical ideas to be revisited constantly within the problems offered

39. And pedagogy? • Teachers expect to teach each class as a unit – within class differentiation is an alien concept • Teachers talk, or manage the talk of others, for the majority of a lesson • Students are expected to operate in a public domain • Students spend little time working alone • Routine exercises are few - teachers present a few carefully selected and challenging variational problems • Homework is frequent but short and bridges successive lessons.

40. Finally, teachers work within three curricula • I have suggested (Andrews, 2011c) that teachers work within an intended curriculum reflecting the knowledge and skills privileged by the system in which they operate. • They work within a received curriculum, amenable only to inference, reflecting the hidden and culturally derived beliefs and practices teachers acquire by dint of being who they are. • Finally, they work within an idealised curriculum, which is articulable, reflecting individual teachers’ personal and experientially informed beliefs. • Comparative research can influence how a system constructs its intended curriculum. It can even influence the idealised curricula of individuals. However, changing the received curriculum will require substantial investment and political commitment.

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