Influence of Government and Policy Makers on UK Classrooms

“Mathematics educators have often emphasized reasoning as one of the primary goals of

learning mathematics.” (Hwang et al., 2017)

The aim of this essay is to analyse the empirical literature which discusses the current government and policy maker’s influence on UK classrooms which suggest that introducing East Asian pedagogical practices would improve mathematical achievement and numerical reasoning to raise standards towards levels seen in renowned internationally successful nations.

 In 2012, the Programme for International Student Assessment (PISA) study by the Organisation for Economic Co-operation and Development (OECD)’s Programme tested over half a million children from over 65 regions, countries and economies around the globe, with a focus on mathematical attainment. Mathematical performance, for PISA, was formulated to measure the ‘mathematical literacy of 15-year-olds to employ and interpret mathematics in a variety of contexts to describe, predict and explain phenomena, recognising the role that mathematics plays in the world.’ (OECD, 2018). To be successful on the PISA test, students must be able to reason numerically and use ‘mathematical concepts, procedures, facts and tools to describe, explain and predict phenomena’. (OECD, 2016). The 2012 study illustrated findings showing that children from Shanghai and Singapore were the top performers in mathematics, with the equivalent of nearly three years of schooling above most other counties being displayed by children from Shanghai. Whilst there were many other East Asian nations also in the highest performing group, three of our European Counterparts were also there: Liechtenstein, Switzerland and the Netherlands. Results indicate that ‘23% of students in OECD countries, and 32% overall, failed to master the simplest maths problems’. (OECD, 2014). Almost half a million children took part in the PISA assessments in 2015, these children represent about 28 million 15-year-olds in the schools across the 72 participating nations, economies and countries taking part. (OECD, 2018). These findings and conclusions show that four countries within Asia continue to ‘outperform all other countries/economies in mathematics’ (OECD 2016). The first PISA results (OECD, 2014) surprisingly claim that ‘only 20% of the students in OECD countries frequently encounter mathematics problems that are set in real-world context and where argumentation skills are demanded’.

Another international-scale study, the Trends in International Mathematics and Science Study (TIMSS) was designed to provide a ‘perspective’ on international teaching and learning in mathematics and science ‘designed to inform educational policy and practice’, was carried out in 2011. Singapore, Korea, Hong Kong, Taiwan and Japan were ranked in the top 5 respectively for 10-11 year olds (Mullis et al., 2012); the 2015 study showed that those same nations remained the most successful; with the remainder of the world remaining at least 23 points behind (the same margin as in 2011). The purpose in TIMSS’ aims are to ‘understand how mathematics curricula should be improved and how to improve students’ mathematics achievements’ (Mullis et al., 2012) through testing the numerical reasoning skills of children on an ‘international scale’. Furthermore, in 2015, results indicate that only 6% reached an advanced mathematical level where they are able to ‘apply understanding and knowledge in a variety of relatively complex situations and explain their reasoning’. In ‘Singapore, Hong Kong SAR and Korea 41-50% of pupils achieved the advanced benchmark, but 10% or fewer did in 34 of the 49 countries that took part’ at age 10-11 years. (Mullis et al., 2015). These results have been discussed, analysed and sensationalised; making national and international headlines on numerous occasions – educators and policy makers in the UK create the impression that it is surprising, demoralising and depressing that we are unable to challenge the mathematical abilities of the most successful countries. These reports of international success, however, should be used to motivate other nations and demonstrate the capabilities of children when given the ‘optimal conditions to succeed’. Mullis et al (20012) states that ‘they demonstrate that our children’s current achievement is not the best they can do; they can achieve much more’.

There have been many studies on the impact of East Asian methods due to their successful results in PISA and TIMSS surveys. For their theoretical study, Jerrim & Choi (2014) aimed to develop a ‘better understanding of how children’s performance on internationally standardised math tests changes between ages 10 and 16’, through comparisons between the experiences of English children compared to the highly successfully performing East Asian jurisdictions (Japan, Singapore, Taiwan and Hong Kong); discussing reasons why children from such countries are, on average, more than one school year ahead of their Western peers (Jerrim and Choi 2014). Fundamental limitations in this study are expressed at the outset in that ideally, longitudinal data would be available to analyse findings in this study, however, the researchers’ use of cross-sectional data ‘repeated cross-sectional data, where samples have been collected from the same, or very similar, cohorts of school children at various points in time’. Jerrim & Choi create a credible study by discussing limitations to their empirical study in that there are some conceptual differences in the skills being measured between PISA and TIMSS data collection, however, they question whether the slight difference in focus is of substantive importance. (Detail findings and methods – focus on rigour and credibility)

Conversely, Jerrim and Vignoles (2015) studied two of the most frequently asked questions by education policymakers ‘What drives East Asian educational success?’ and ‘What can we do to catch up’? Their paper attempts to provide some robust evidence to begin to fill this important gap in the literature. Specifically, it provides evidence as to how introducing a particular East Asian inspired teaching method into a Western schooling system influences children’s mathematics test scores. (Detail findings and methods – focus on rigour and credibility)

Boyd and Ash (2018) investigate teachers’ beliefs during implementation of a textbook-based Singaporean mastery approach to teaching mathematics. (Detail findings and methods – focus on rigour and credibility)

There has been a vast amount of study into different approaches that could be adopted to improve mathematical reasoning within the primary classroom. Key findings of Nunes et al‘s research study found that ‘mathematical reasoning, even more so than children’s knowledge of arithmetic, is important for children’s later achievement in mathematics’. Mathematical reasoning and knowledge of arithmetic (assessed at age 8-9 years) make ‘independent contributions to children’s achievement in mathematics in KS2 and 3’ (Nunes et al.,2009). While both are important, Nunes et al. (2009, p.1) claim that mathematical reasoning is more important than knowledge of arithmetic for achievement in KS2 and 3. Mathematical reasoning has been defined in a number of ways (Bolton, 2017; Erdem & Gürbüz, 2015; Holton, Stacey & Fitzsimmons, 2012; Herbert et al, 2015): though researchers generally accept that mathematical reasoning involves critical thinking; focus on mathematical relationships; drawing inferences; involving in-depth discussion; metacognitive explanation; argumentation; justification of solutions; and reflection on the strategies and methodology applied in the process. Corollary, researchers commonly agree that reasoning and proof form the foundation of mathematical understanding. This large longitudinal study survey of children investigated children’s progress at different stages through primary and early secondary school. Findings discuss the advantages of heterogenous groupings in place of more traditional homogenous (ability) groupings which ‘in Primary school improves the mathematical reasoning of children in the top ability group, but the effect is small. It hinders the progress of children in the other groups.’ Claiming that children’s self-confidence in the subject has a small but significant impact on attainment in mathematics reasoning.

The aims of Herbert et al. (2015)’s research study was to create a framework of teachers’ perceptions towards and of mathematical reasoning which would enable the tracking of teacher perceptions (which in turn could then be utilised in further study to adopt professional learning and enhance pedagogical application). Herbert et al. claim that the framework produced can provide a ‘vehicle to assess teachers’ awareness of aspects of reasoning’; facilitating this tracking tool could both evaluate professional learning requirements and assess the impact of this learning. Furthermore, claiming the framework capable of ‘maximising the change in teachers’ perceptions of mathematical reasoning.’ A pragmatic philosophy appears to have been adopted when selecting this phenomenology paradigm; to investigate the participants’ perceptions of the concept of investigation as opposed to the researchers’ bias, experience or predetermined ideas. (Describing learning as a change in the way a student conceives the object of learning (Booth, 1997; Ramsden, 1988)). Herbert et al. (2015)’s investigation clearly followed a robust and rigorous process underpinning the importance of eliminating bias in the phenomenology paradigm; illustrating that ‘phenomenography to be an effective methodology to provide evidence for establishing this framework’. Limitations into generalising this study included the socio-cultural features of the schools participating; limited teaching experience in some participants. Additional limiting factors are highlighted as teachers’ lack of vocabulary to discuss mathematical reasoning. However, the framework successfully developed creates opportunity for further research to determine the effectiveness of professional learning on participant’s perceptions of mathematical research.

From an opposing perspective and approach to improving attainment of mathematical reasoning; the research study of Gürbüz & Erdem (2016) uses correlation analysis to determine a relationship between mental computation and mathematical reasoning in primary aged children. An explorative approach appears to be used to facilitate the quantitative research methods allocated to the investigation; utilising a correlation model. This study focused on 118 primary aged children nominated using random selection techniques to identify participants from ‘low and middle socioeconomic areas in a city in Turkey’. An analytical survey and Inferential analysis was used allowing generalisations to be formed. Correlation analysis found a ‘relationship between mental computation and mathematical reason of the students involved in the study’. In literature, only correlation values of 0.65 or higher in education research will show that it represents the correlation correctly and will allow individual predictions that are reasonably accurate for generalisation purposes. This study found a 0.654 correlation that there is a ‘significant’ and ‘highly positive relationship’ between students’ mental computation and mathematical reasoning’. This robust and credible study used clearly defined parameters to analyse and discuss findings. Conclusions identified a highly positive relationship between mental computation and mathematical reasoning. To move from explorative to more transformative research, Gürbüz and Erdem (2016) suggest further study to investigate the relationship between mental computation and mathematical reasoning qualitatively; examining the impact of addressing mathematical misconceptions, perceptions and ability on mathematical reasoning through more constructivist and transformative descriptive analysis.

With regards to policy-makers making decisions on adopting East Asian pedagogies and teaching formulas, Jerrim & Choi (2014) state that East Asian children vastly out-perform their English peers even when they have been through the English schooling system.[1]. A claim that is justified by indicating that perhaps it is generally high expectations along with cultural and community ethos that contribute to high levels of achievement, which in the UK cannot be a short-term focus seeing as it is notoriously difficult to modify people’s attitudes and beliefs ‘. Bray (2003) also highlights issues with this kind of cultural expectation which can result in ‘pressure which students (physical and psychological) and parents (financial) must put up with’. However, is this the kind of pressure which could allow our children to achieve at the superior level in PISA and TIMMS similar to that of our East Asian counterparts to ensure financial prosperity and long-term economic success?

Some studies have found that many primary teachers are not confident in defining reasoning (Loong et al., 2013). Teaching methods, attitudes and perceptions in an education system along with teacher subject knowledge, pedagogical application and professional development are paramount in improving achievement in mathematical reasoning in primary classrooms. Herbert et al.’s research could be used as a basis to create a tracking tool to assess perceptions and to improve professional development outcomes as a new curriculum initiative is introduced.

Developing classroom dynamics including specific grouping and encouraging confidence in mathematical ability can improve children’s achievement in mathematical reasoning (Nunes et al, 2009). Providing sufficient opportunities to engage in mental computational mathematics is important to ensure children’s mathematical knowledge can be applied to other contexts (Gürbüz & Erdem, 2015). This mental computation forms an important component of the method in which Eastern Asian countries apply their mathematical teaching methods with significant success internationally as seen in the PISA and TIMSS results.

Even when policies and teaching methods have been proven to be effective in East Asia, culture and context potentially limit the extent to which such initiatives can be successfully transferred to other countries (Cowen, 2006,). Family and social commitment to education is also reflected in the large number of weekly hours East Asian students spend in self-study activities (Jerrim & Choi, 2014) and, as Zhu and Leung (2011) argue, the ‘great impact extrinsic motivation has on their mathematics test performance (much more so than their Western peers)’. However, here the issue of causality exists. (Jerrim and Vignoles, 2015) Indeed, it appears that instead of at secondary level that mathematical reasoning develops fully, it appears that East Asian countries that top the PISA and TIMSS survey appear to ’pull ahead’ before age 10 and then maintain the mathematical superiority which exists between them and other countries globally. Furthermore, these children display a ‘situated mathematical mindset’: a belief held in varied ways by teachers and children, that the more you practice at the edge of your current attainment level in maths, the more intelligent you will become as a mathematician (Boaler, 2016).

Whichever, teaching method is adopted, researchers generally agree on one vital component for mathematical and numerical reasoning success – mastery is paramount. ‘Learning for Mastery (LFM) is a group-based, teacher-paced approach to mastery instruction wherein students learn, for the most part, cooperatively with their classmates’ (Block and Burns, 1976). Traditionally, in the UK classroom, approaches to differentiation commonly involve some children being identified as ‘mathematically weak’ and are taught a reduced curriculum with simplified mathematical work to carry out, possibly in heterogenous ability classes but generally in same-ability groupings; whilst others are identified as ‘mathematically able’ and given further challenges and extension tasks, or more simply moved to the following years’ skills. This approach has been adopted with the best of intentions: to offer additional support to those encountering difficulties with understanding mathematical concepts, with a view to ensuring competence of key concepts within mathematics. Nonetheless, this can only have negative connotations for these children surrounding their mathematical ability, the perception of their own mathematical ability, their motivation and mindset surrounding the subject and their future numerical application. In the light of international evidence from high performing jurisdictions in the Far East, and ‘mindset’ research (Hattie, 2012), mastery appears to be the most common form of pedagogy within those nations who boast the highest levels of success. Most modern mastery applications stem from the word of Benjamin Bloom (1971, 1976, 1984), who discussed how teachers might adapt their pedagogy to improve learning in classrooms. Bloom suggested ‘if teachers could provide the necessary time and appropriate learning conditions, nearly all students could reach a high level of achievement’ Guskey, 2010). ‘The notion that Singaporean teachers place more emphasis on whole class mastery of concepts is supported by the Teaching and Learning International Survey (Micklewright et al 2014)’ (Jerrim and Vignoles, 2015). The class simply do not move on until every member of the class has acquired mastery of each concept. More able children investigate the aspect in more depth, whilst the teacher focuses on those children who need more support in achieving the mastery. Additional support in terms of parental partnerships and after-school tutoring is also commonplace for these children in order that they can also become fluent in the fundamentals of mathematics along with their classmates. Furthermore, the’ Singapore system concentrates more on developing problem-solving skills rather than mental arithmetic’ (Cohen, 2017). Indeed, the English National Curriculum for Mathematics states that ‘… decisions about when to progress should always be based on the security of pupils’ understanding and their readiness to progress to the next stage. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent with earlier material should consolidate their understanding, including through additional practice, before moving on’ (Department of Education, 2014). In many mathematical schemes of work in the UK, there are time constraints to different areas of the curriculum to ensure adequate ‘coverage’ – these schemes create constraints on the time for which each concept is taught. In Wales, one scheme of work indicates that one week at the start of each year should be dedicated to place value, followed by one week of addition, subtraction, multiplication and division respectively – arguably the most important cornerstones of mathematics, without an in-depth knowledge of each, children could not possibly hope to compete on an international scale with those children from our East Asian counterparts. Researchers studying the importance of learning for mastery (Jerrim and Vignoles, 2015; Guskey, 2010; Mercer, 2006; Mevarech, 2015) would strongly disagree with this padagogy, agreeing with a 2012 publication by the independent Advisory Committee for Mathematics Education (ACME) which advocates ‘depth in place of acceleration’ and which states that ‘users of mathematics should experience a deep, rich, rigorous and challenging mathematics education, rather than being accelerated through the school curriculum’.

It could be suggested that although elements of East Asian pedagogy can be learned from and are generally agreed to be highly effective, it cannot be proven unequivocally that they could be implemented successfully within any other educational setting to equalise mathematical performance with East Asian nations without also (impossibly) committing to cultural and historical mirroring. Instead, a focus and commitment of changing the methods and pedagogy through which mathematics is taught in the UK appears to be a more realistic target. Ensuring mastery in the fundamental mathematical areas of place value; addition and subtraction, multiplication and division; their inter-relationships; and the reasoning and application of methods and strategies to solve calculations and numerical problems, should instead be the focus to improving numerical reasoning and mathematical attainment within the primary classroom.

References:

• Advisory Committee on Mathematics Education (2012). Raising the bar: developing able
• young mathematicians. [Online]
• Available at: http://www.acme-uk.org/media/10498/raisingthebar.pdf
• [Accessed 20 November 2018]
• Akerlind, G. (2005). Learning about phenomenology: Interviewing, data analysis and the qualitive research paradigm. In J.A. Bowden, & P. Green (Eds.). Doing Developmental phenomenography (pp.74-100). Melbourne: RMIT University Press.
• Bloom, B. S. (1971). Mastery learning. In J. H. Block (Ed.), Mastery learning: Theory and practice (pp. 47–63). New York: Holt, Rinehart & Winston
• Bloom, B. S. (1974). Time and learning. American Psychologist, 29(9), 682-688.
• Bolton, T., (2017). Oxford Education Blog. [Online]
  Available at: https://educationblog.oup.com/primary/5-ways-to-improve-mathematical-reasoning
  [Accessed 30 September 2018].
• Booth, S. (1997). On phenomenography, learning and teaching. Higher Education Research and Development, 16(2), 135-138
• Cohen, D. (2017). How the Child’s Mind Develops. (3^rd Ed.) Taylor & Francis.
• Cope, C. (2002). Educationally critical aspects of the concept of an information system. Informing Science, 5(2), 67-78.
• Cowen, R. (2006). Acting Comparatively upon the Educational World: Puzzles and Possibilities. Oxford Review of Education 32 (5): 561–573
• Department for Education (England). (2014). Statutory guidance: National curriculum in England: mathematics programmes of study. England. [Online]
• Available at: https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study/national-curriculum-in-england-mathematics-programmes-of-study [Accessed 20 November 2018]
• Giorgi, A. (2012). The descriptive phenomenological psychological method. Journal of Phenomenological psychology, 43(1), 3-12.
• Gürbüz, R. & Erdem, E. (2016). Relationship between mental computation and mathematical reasoning. Student Learning, Childhood Voices. Cogent Education, 3: 1212683
• Guskey, T. (2010). Lessons of Mastery Learning. Educational leadership: journal of the Department of Supervision and Curriculum Development, N.E.A. 68. 52-57.
• Hattie, J. (2012). Visible Learning for Teachers. (1^st Ed.) Routledge.
• Herbert, S., Valeb, C., Bragg, L.A., Loong, E., Widjajab, W. (2015). A framework for primary teachers’ perceptions of mathematical reasoning. International Journal of Educational Research 74, 26–37.
• Holton, D., Stacey, K., & FitzSimons, G. (2012). Reasoning: A Dog’s Tale. Australian Mathematics Teacher, 68(3), 22–26.
• Hwang, J., Runnalls, C., Bhansali, S., Navaandamba, K. & Choi, K.M. (2017) “Can I do well in mathematics reasoning?” Comparing US and Finnish students’ attitude and reasoning via TIMSS 2011, Educational Research and Evaluation, 23:7-8, 328-348, DOI:10.1080/13803611.2018.1500293
• Jerrim, J. & Choi. Á. (2014). The mathematics skills of school children: how does the UK compare to the high performing East Asian nations? Journal of Education Policy 29(3): 349-76.
• Jerrim, J. & Vignoles, A. (2015). The causal effect of East Asian ‘mastery’ teaching methods on English children’s mathematics skills? UCL Institute of Education. Department of Quantitative Social Science Working Paper No. 15-05.
• Kaur, B. & Yeap, B.H. (2009). Pathways to reasoning and communication in the primary school mathematics classroom. Singapore: National Institute of Education.
• Kaur, B. (2012). Some “What” Strategies that Advance Reasoning and Communication in Primary Mathematics Classrooms. In B. Kaur & T.L. Toh (Eds.), Reasoning, communication and connections in mathematics: Yearbook 2012 (p.75-88), Association of Mathematics. World Scientific Publishing Co. Pte. Ltd.
• Kaur, B., Yeap, B.H., & Kapur, M. (2009). Mathematical problem solving. Singapore: World Scientific.
• Kosyvas, G. (2016) Levels of arithmetic reasoning in solving an open-ended problem, International Journal of Mathematical Education in Science and Technology, 47:3, 356-372, DOI:10.1080/0020739X.2015.1072880
• Loong, E., Vale, C., Bragg, L., & Herbert, S. (2013). Primary school teachers’ perceptions of mathematical reasoning. Mathematics education: Yesterday, today and tomorrow. Proceedings of the Thirty-Sixth Annual Conference of the Mathematics Education Research Group of Australasia. Melbourne: MERGA466–472.
• Mercer, D. (1986) Mastery Learning, British Journal of In-Service Education, 12:2, 115-118
• Mevarech, Z, R. (2015) The Effects of Cooperative Mastery Learning Strategies on Mathematics Achievement, The Journal of Educational Research, 78:6, 372-377
• Micklewright, John; John Jerrim, Anna Vignoles, Andrew Jenkins, Rebecca Allen, Sonia Ilie, Elodie Bellarbre, Fabian Barrera and Christopher Hein. 2014. ‘Teachers in England’s secondary schools: Evidence from TALIS 2013.’ Department for Education Research Report DFE-RR302. Accessed 04/11/2018 from https://www.gov.uk/government/publications/teachers-in-secondary-schools-evidence-from-talis-2013
• Mullis, I.V.S., Martin, M.O., Foy, P. & Arora, A. (2012). TIMMS 2011 International Results in Mathematics. Retrieved from Boston College, TIMSS & PIRLS International Study Center website: https://timssandpirls.bc.edu/timss2011/downloads/T11_IR_Mathematics_FullBook.pdf
• Mullis, I.V.S., Martin, M.O., Foy, P., & Hooper, M. (2016). TIMSS 2015 International Results in Mathematics. Retrieved from Boston College, TIMSS & PIRLS International Study Center website: http://timssandpirls.bc.edu/timss2015/international-results/
• Nunes, T., Bryant, P., Sylva, K. & Barros, R., (2009). Development of Maths Capabilities and Confidence in Primary School, Oxford: Department of Education, University of Oxford.
• OECD (2014), PISA 2012 Results: What Students Know and Can Do (Volume I, Revised edition, February 2014): Student Performance in Mathematics, Reading and Science, PISA, OECD Publishing, Paris.
• OECD (2016), PISA 2015 Results (Volume I): Excellence and Equity in Education, PISA, OECD Publishing, Paris, https://doi.org/10.1787/9789264266490-en.
• OECD (2018), “Mathematics performance (PISA)” (indicator), https://doi.org/10.1787/04711c74-en(accessed on 02 November 2018).
• Ramsden, P. (1988). Studying learning: Improving teaching. In P. Ramsden (Ed.), Improving learning: new perspectives (pp. 13–31). London: Kogan Page.
• Starks, H., & Brown Trinidad, S. (2007). Choose your method: A comparison of phenomenology, discourse analysis, and grounded theory. Qualitative health research, 17(10), 1372-1380.
• Zhu, Y., & F. Leung. (2011). Motivation and Achievement: Is There an East Asian Model? International Journal of Science and Mathematics Education 9 (5): 1189–1212.
• Wong, K.Y., Lee, P.Y., Kaur, B., Foong, P.Y., Ng, S.F. (2009). Mathematics education: The Singapore journey. World Scientific Publishing, Singapore.
• Boyd, P. Ash, A. (2018). Mastery mathematics: Changing teacher beliefs around in-class grouping and mindset. Teaching and Teacher Education 75 (2018) 214-223.
• Yew Hoong, L., Weng Kin, H., & Lu Pien, C. (2015). Concrete-Pictorial-Abstract: Surveying its origins and charting its future. Mathematics Educator, 16(1), 1e19.
• Boaler, J. (2016). Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching. San Francisco: JosseyBass

[1] In 2011, 78.5% (76.6% in 2015) of Chinese children achieved five or more A*–C grades including Maths and English. This compares to a national average of 58.2% (57.1% in 2015); 61.8% (61.1% in 2015) of Asian pupils achieved five or more A*–C grades including Maths and English https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/219306/sfr03_2012_001.pdf; https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/494073/SFR01_2016.pdf