The consideration of teacher education from a global perspective foregrounds the challenge of inequality as a core challenge for contemporary societies and for educational systems. The crucial role of education in relation to this challenge is highlighted in the UNICEF/UNESCO report on the Global Thematic Consultation in the Post-2015 Development Agenda, which 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. Accordingly this paper considers the relation between quality and learning and in particular that between epistemic quality and equitable learning. The work of Jo Boaler is especially relevant to the former in relation to her proposition about ‘the elephant in the mathematics classroom’. Of particular significance is her argument that in many maths classrooms a very narrow subject is taught to children, that is nothing like the maths of the world or the maths that mathematicians use. In our recent study on developing mathematical thinking we present this as an issue of epistemic quality (Hudson et al., 2015). High epistemic quality involves mathematics as fallible, refutable and uncertain, critical thinking, creative reasoning, multiple solutions and learning from errors and mistakes. In contrast low epistemic quality is characterised by mathematics as infallible, authoritarian, dogmatic, absolutist, irrefutable, certain, rule following of strict procedures and right or wrong answers. Additionally we consider how a thematic approach through the study of butterflies and moths in the Amazon rainforest resulted in mathematics becoming more accessible for all (Hudson, 2015). Such accessibility is central to equitable learning, which is seen as learning that produces educational justice (“Bildungsgerechtigkeit”). The paper concludes by considering how to redress the extent to which educational systems, and also everyday teaching practices and classroom interaction, reproduce inequality.

1. 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

3. 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

4. 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?

5. 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.

6. 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.

7. 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)

8. 8
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

9. 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
9

10. 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

11. 11
Design research framework

12. 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?

13. 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

14. 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)

15. 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

16. 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?

17. 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.

18. Lesson 1: Measuring moths and spiders

19. 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.”.

20. 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”.)

21. 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?”

22. 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)

23. 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)
23

24. 24
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

25. 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.

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

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

28. Thank you for your attention