Elephants, Butterflies and Moths in the Amazon Rainforest:

High Epistemic Quality for Equitable Learning in the Mathematics Classroom

Brian Hudson

TEPE 2016 Conference University of Malta

20th May 2016

Presentation informed by research outlined in two key research papers p  Hudson, B. (2015) Butterflies and Moths in the Amazon:

Developing Mathematical Thinking through the Rainforest, Education and Didactique, Vol. 9, Issue 2, 119 – 133. http://educationdidactique.revues.org/2322

p  Hudson, B., Henderson, S. and Hudson, A., (2015) Developing Mathematical Thinking in the Primary Classroom: Liberating Teachers and Students as Learners of Mathematics, Journal of Curriculum Studies, Vol. 47, Issue 3, 374-398. http://dx.doi.org/10.1080/00220272.2014.979233

Structure of the presentation p  Equitable Learning

p  Background context– the Developing Mathematical Thinking in the Primary Classroom (DMTPC) project

p  The Elephant in the Classroom

p  Butterflies & Moths (and spiders!) in the Amazon rainforest

p  Epistemic Quality

p  How high epistemic quality is a necessity for equitable learning in the mathematics classroom.

p  Whose interests are being served by the push towards high stakes testing and summative assessment?

Equitable learning p  UNICEF/UNESCO report on the Global Thematic Consultation in

the Post-2015 Development Agenda stresses education as a fundamental human right. The report calls for two main education specific goals to be addressed as part of the future development framework: equitable access and equitable quality education.

p  We define equitable learning (Hudson et al, 2016) as learning that produces educational justice (“Bildungsgerechtigkeit”), that enables students to overcome societal limitations of access to and success in education, fosters subject autonomy and allows for the development of participatory competences for life in society.

p  Hudson, B., Loquet, M., Meyer, M. and Wegner, A. (2016) Equitable Learning in France, the United Kingdom, and Germany – An Empirical Research Project on the Process of Schooling and Instruction in Europe, (Work in progress).

p  Sayed, Y. (2013) Envisioning Education in the post-2015 Development Agenda: Report of the Global Thematic Consultation on Education in the Post-2015 Development Agenda, UNICEF/UNESCO.

Developing Mathematical Thinking in the Primary Classroom (DMTPC) Project

p  Funded by the Scottish Government (2010-12)

p  Collaborative development of a Masters level course for teachers involving a technology enhanced blended learning approach.

p  Piloted with a group of 24 practising primary teachers from the local education authorities of Dundee, Fife, Angus and Perth & Kinross.

Rationale for the project p  Most mathematics lessons in Scotland still tend to feature some

form of teacher-led demonstration followed by children practising skills and procedures from a commercially produced scheme (SEED 2005).

p  These findings were confirmed by TIMSS (IEA 2008) which found that 72% of both P5 and S2 pupils were taught using a textbook as the primary resource compared to the international average of 65% and 60% respectively.

p  Scottish national surveys of achievement in 2009 and 2012 also reported that pupils using textbooks and working quietly on their own was the most common form of activity in mathematics classes in Scotland.

p  IEA (2008) Trends in Mathematics and Science Survey 2007 (Lynch School of Education, Boston College: International Association for the Evaluation of Educational Achievement).

p  Scottish Executive Education Department (2005) Assessment of Achievement Programme: Seventh

Survey of Mathematics 2004 (Edinburgh: Scottish Executive Education Department)

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Design of the course of study

19 September 2011 Online module opens 24 September 2011 Workshop 1 10:00 – 16:00 26 October 2011 Twilight session 1 16:30 – 19:30 7 December 2011 Twilight session 2 16:30 – 19:30 4 February 2012 Workshop 2 10:00 – 16:00 23 April 2012 Assignment submission

Outline structure: three key questions, key texts and an action research project p  Key questions

n  What is mathematics? n  What is mathematical thinking? n  What is good mathematics teaching?

p  Key texts n  Joe Boaler (2009) The Elephant in the Classroom n  John Mason et al. (2010) Thinking Mathematically – it’s

OK to get stuck!

p  Action research plan and project as the module assignment

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The Elephant in the Classroom

p  “I have called this book ‘The elephant in the classroom’ because there is often a very large elephant standing in the corner of maths classrooms. The elephant, or the common idea that is extremely harmful to children, is the belief that success in mathematics is a sign of general intelligence and that some people can do maths and some can’t.”

p  Jo Boaler

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Design research framework

Research Questions – Main Study

1.  What are the teachers’ perceptions concerning their levels of confidence and competence in relation to teaching mathematics?

2.  What are the teachers’ perceptions concerning their attitudes and beliefs in relation to mathematics as a subject?

3.  What are the teachers’ expectations of the impact on pupil learning arising from this course of study?

4.  How do these perceptions and expectations change as a result of participating in this course of study?

Methods and data sources – Main Study

p  Pre-trial survey of the teachers’ perceptions (n=26)

p  Pre-trial interviews with a sample of participants (n=4)

p  Post-trial interviews with a sample of participants (n=4)

p  Post-trial survey of the teachers’ perceptions (n=15)

p  Action research reports from teachers (n=10)

p  One action research report as the exemplar for this presentation

One teacher’s Action Research Project p  To what extent does

topic-based mathematics allow children to demonstrate their mathematical thinking?

n  To what extent do topic-based mathematical questions allow children to verbalise their thinking?

n  What effect does topic-based mathematics have on children’s levels of engagement?

Attack

Review

Entry

Ref: Mason, Burton and Stacey (2010)

Anna’s Action Research Project on “The Rainforest”

p  Primary 5/6 pupils

p  Time: 3 weeks

p  Measurement – mainly length and weight

p  First question: How could we measure these life-sized insects accurately?

Brazilian Huntsman spider

Questions for pupils to explore, analyse and record Four questions corresponding to Lessons 1 to 4: 1.  How could we measure these life-sized insects accurately?

2.  How could we mark out the different layers of the rainforest in our playground?

3.  Can you compare the length of the River Tay and the Amazon River?

4.  Is there a relationship between the weight of an animal and the layer it lives on in the rainforest?

Methods of data collection Data was collected in three ways:

n  Children were recorded informally during conversations with peers,

n  Quotes were taken during class feedback sessions, and

n  After the sessions children were informally asked to comment on the lesson and this feedback was recorded.

p  Also live observations were made and various parts of the activities were filmed to watch and analyse later.

Lesson 1: Measuring moths and spiders

Engagement with the activity

p  … all pupils were actively engaged. This part of the lesson clearly parallels the “Entry Phases” described by Mason et al. (2010) … (Anna)

p  During the measuring process, the children were asked to verbalise their thinking and demonstrate their measuring. Most children chose to write their measurements completely in millimetres only. One Primary 6 girl wrote short sentences to describe her measurements and when asked why … , she said, “It’s easier to see the numbers this way. It’s weird, they’re all (the spider’s legs) different lengths mostly.”.

Data Analysis of Lesson 3: Comparing the length of the River Tay and the Amazon

p  John (Year 5 boy) stated, “I

drew the Amazon and the River Tay on my piece of paper. I measured the paper and it was about 300 millimetres so we narrowed it down and got that every 5 centimetres was about 1000 kilometres. The River Tay is only 186 kilometres so it’s only that size” (Pointing to the part of their diagram labelled “River Tay”.)

Tracy’s intervention

p  The other children in the class were very interested in this and one Year 6 girl (Tracy) commented:

p  “The Tay is tiny compared to that, you could fit like, a hundred of the Tay into the Amazon!”

p  Anna notes how this comment was explored and extended leading to her question:

p  “How many times would the Tay fit into the Amazon River?”

Some reflections p  The discursive element of this

lesson proved to be a very effective tool to assess the pupils’ understanding and mathematical thinking. (Anna)

p  The question developed tremendously throughout the lesson due to their knowledge of the subject, and their ability to visualise the problems, the mathematics became accessible leading an evolution in mathematical thinking for all. (Hudson, 2015)

Epistemic quality p  This process of ‘mutation’ (Boaler, 2009) reflects the process of

didactic transposition, which changes the mathematical knowledge profoundly and which leads to the epistemic quality of the subject becoming degraded as it is transposed into school mathematics. Hudson et al. (2015) describe this mutated or degraded version of mathematics as mathematical fundamentalism and as being of low epistemic quality. It is characterised by an approach that presents the subject as infallible, authoritarian, dogmatic, absolutist, irrefutable and certain and which involves rule following of strict procedures and right or wrong answers.

p  In contrast high epistemic quality is characterised by an approach which presents mathematics as fallible, refutable and uncertain and which promotes critical thinking, creative reasoning, the generation of multiple solutions and of learning from errors and mistakes. (Hudson et al., 2015)

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The spectrum of epistemic quality in mathematics from low to high

Low epistemic quality p  Infallible and authoritarian p  Dogmatic and absolutist p  Irrefutable and certain p  Strict procedures p  Rule following p  Getting right and wrong answers p  Boring and de-motivating p  Inducing fear and anxiety p  Alienation from the subject itself p  Reinforced by excessive high

stakes testing and summative assessment

High epistemic quality p  Fallible and liberating p  Critical thinking, growth & change p  Refutable and uncertain p  Multiple solutions p  Creative reasoning p  Learning from errors and mistakes p  Engaging and motivating p  Enjoyable and fulfilling p  A creative human activity p  Supported by diagnostic feedback

through formative assessment for learning

High Epistemic Quality for Equitable Learning p  The findings highlight ways in which the ‘framing’ (Bernstein,

2000) of particular aspects of the traditional curriculum had an oppressive impact on learners in the ways that suppressed creativity and limited the exercise of learner autonomy by both teachers and pupils.

p  The weaker framing of Curriculum for Excellence shifted the locus of control over the selection, sequencing and pacing of what counts as legitimate knowledge towards these teachers.

p  Teachers’ own experience as learners of mathematics highlights the impact of the strong framing over the criteria for the formal assessment system, especially at secondary school level.

p  On-going challenge for continuing reform is the alignment of criteria for evaluating or assessing of the formal assessment system with the aims and purposes of the formal curriculum.

Whose interests are being served by the push towards high stakes testing and summative assessment?

Not those of children who are eager to explore the world and learn!

Thank you for your attention