Improving the teaching of mathematics in the United States by using formative approaches
Submitting InstitutionUniversity of Nottingham
Unit of AssessmentEducation
Summary Impact TypeSocietal
Research Subject Area(s)
Education: Curriculum and Pedagogy, Specialist Studies In Education
Summary of the impact
A design research programme in mathematics education by The University of
Nottingham has been taken up by two powerful US change agents — the Bill
and Melinda Gates Foundation and the National Council of Supervisors of
Mathematics — as a key element in their strategies to improve the quality
of teaching and learning in secondary mathematics classrooms across the
Beginning with small-scale design research on diagnostic teaching in
mathematics, effective principles for the design of lessons were developed
to enable teachers to adapt to students' learning needs. These principles
were then engineered into robust products and processes through
systematic, closely observed classroom trials.
Design research takes a formative approach, using educational theory and
ongoing feedback from end-users to systematically design and refine
innovative products and processes (`tools'). This is an iterative process,
in which both the tools and the theories are refined . The
impact of this work is built on a sustained, coherent programme of design
research in mathematics education.
Building on earlier work at The University of Nottingham led by Professor
Hugh Burkhardt (emeritus professor since 1996), Malcolm Swan has explored
the advantages to learning of promoting active classrooms where students
teach their peers or devise their own assessments. For example, a funded
study into diagnostic teaching in Further Education (FE) (1995-1997)
showed that collaborative discussion materials can be highly effective
when used appropriately, even with low attaining students. The study also
offered insights into the ways in which teachers' beliefs (about
mathematics, teaching and learning) affect the ways in which they use
teaching materials and, conversely, the ways in which the materials affect
beliefs and practices .
This programme of work led Swan and the team to the conclusion that in
order to achieve large-scale impact, the research needed to focus more
explicitly on developing replicable models for teacher learning and
on-going professional development (PD). Design principles for teacher
learning were developed through structured reflection on a sequence of
carefully-planned teaching experiences. These principles were refined
through work on the government-funded Learning Mathematics through
Discussion and Reflection project (2000-2002). The resulting PD
toolkit, which supported the diagnostic teaching of algebra, was published
by the Department for Education and Science (DfES) and distributed to all
FE colleges. This led to a research project on the effects of implementing
collaborative approaches to learning mathematics in 40 GCSE retake
classes. Swan's book Collaborative Learning in Mathematics ),
which summarises this project, was described as "an outstanding piece of
research; it is up there with the best applied educational research in the
UK, and the best international mathematical education research" by
Margaret Brown (Education Sub-Panel Chair, RAE 2008). Alan Schoenfeld,
ex-president of the American Educational Research Association, and
Elizabeth and Edward Conner Professor of Education, University of
California at Berkeley commented:
I think Swan's work is exemplary and that it should be widely known. By
way of preliminaries, I note that the challenge he took on is significant:
having a real impact on teacher's beliefs and practices is extremely
difficult. Hence any clear signs of positive impact are notable in
themselves. But in this case, the impact is just the tip of the iceberg.
The DfES then commissioned the design of a multimedia PD resource, Improving
Learning in Mathematics [4, 5], for teachers working across
the 11-18 age range. The material was trialled in 90 colleges before being
distributed to all English FE colleges and secondary schools. OfSTED
described this resource — commonly known as `the Standards Unit box' — as
offering "highly successful approaches to teaching, learning and
professional development" (OfSTED, 2006) and the Royal Society's Advisory
Committee on Mathematics Education called it "exemplary" (ACME, 2006).
This design work for professional development continued through the
development of the Bowland Professional Development Modules which
incorporated further influential research on formative assessment and
problem solving . Bowland Maths was an award finalist at the
2010 British Educational Training and Technology Show and has been used by
60% of schools in England as well as being the foundation for further
collaborative work funded by the European Commission's framework programme
References to the research
1. Burkhardt, H. (2006). From design research to large-scale impact:
Engineering research in education. In J. Van den Akker, K.
Gravemeijer, S. McKenney, and N. Nieveen (Eds.), Educational design
research. London: Routledge. Available on request.
3. Swan, M. (2006). Collaborative Learning in Mathematics: A
Challenge to our Beliefs and Practices. London: National Institute
for Advanced and Continuing Education (NIACE) for the National Research
and Development Centre for Adult Literacy and Numeracy (NRDC). (350 pages
+ CD Rom showing professional development resources and research
instruments). Available on request.
5. Swain, J. and Swan, M. (2009). Teachers' attempts to integrate
research-based principles into the teaching of numeracy with post-16
learners. Research in Post-Compulsory Education. 14, 1, 75-92. (DOI:
10.1080/13596740902717424 — returned in REF 2).
6. Swan, M., Pead, D., Doorman, M. and Mooldijk, A. (2013). Designing and
using professional development resources for inquiry-based learning. ZDM:
The International Journal on Mathematics Education. (DOI:
10.1007/s1185801305208 — returned in REF 2).
• Department for Education and Skills (2004-05), Development of
Mathematics Teaching and Learning £108 000 (Swan PI).
• National Research and Development Centre (2005-07), Maths4Life £38 500
• Bowland Charitable Trust (2007-13) Bowland Maths (four linked projects)
£492 000 (Swan PI)
• European Commission (2010-13), Promoting Inquiry in Mathematics and
Science Education across Europe. £2 750 000 (Wake and Swan Co-Is; Prof
Katja Maass, Freiburg- PI).
Details of the impact
Design research produced by the CRME, under the leadership of Professor
Malcolm Swan (promoted in 2009) has had a substantial influence on the
strategies employed by the largest charitable organisation in the world,
The Bill & Melinda Gates Foundation, and on the US National Council of
Supervisors of Mathematics (NCSM). The Foundation's goal is to vastly
increase the number of `college and career ready' US students graduating
from high schools. The NCSM is seeking to improve the quality of
mathematics teaching and learning in US secondary schools.
In the US, individual states have autonomy in education, so national
improvement strategies are voluntary. Prompted by President Obama, the
Gates Foundation funded the development of the Common Core State Standards
in Mathematics (CCSSM or Standards). These Standards, which have now been
adopted by 46 States, emphasise `mathematical practices', i.e. the
processes of doing mathematics. The Gates Foundation recognised the
strategic challenge of finding a scalable implementation strategy to make
these mathematical practices a reality in US classrooms. The Foundation
aimed to meet this challenge by commissioning formative assessment tasks
supported by teaching materials. This was the core of The Gates
Foundation's improvement strategy.
In 2009, in recognition of CRME's longstanding experience and expertise
in design-research, the Gates Foundation commissioned a large-scale
programme to design and trial lessons called `Classroom Challenges' (CCs)
[A]. These lessons were intended to improve the teaching of concept
development and non-routine problem solving. The principles of diagnostic
teaching and formative assessment are key design features of these CC
lessons. The CCs are now central to the Gates Foundation's implementation
strategy; they have also been adopted by many of the Foundation's
Vicki Phillips, Director, College Ready Education at the Gates Foundation
The Foundation saw formative assessment as central to its support for the
country's improvement strategy in mathematics. [CRME's] research in this
field, and the excellent materials developed from it, were well known.
They epitomize the `fewer, clearer, higher standards' of the Common Core.
Our strategy covers school systems across the country. The response from
teachers to these Classroom Challenges has been enthusiastic, remarkably
so since they take most teachers into new terrain, mathematically and
The remarkable response to these materials is evidenced by an average of
76,000 downloads of CCs a month during 2012-13 [C]. This is in
addition to downloads from other mirror sites (e.g., the Association for
Supervision and Curriculum Development).
At school district level, the CCs are now being used to help teachers
interpret the Standards. Valerie Mills, the President-Elect of the
National Council of Supervisors of Mathematics (NCSM), explains:
In K-12 districts [the CCs] are being used as the basis of study groups
and other forms of professional learning. In universities, they are being
used in mathematics methods courses to help pre-service teachers become
acquainted with the CCSSM. In both settings they are helping educators
think more carefully about what it means and what it looks like to teach
for understanding, to really `do' mathematics, and to become proficient in
the Standards for Mathematical Practice. The Classroom Challenges are
vivid instantiations of the CCSS and have been powerful communication
tools for us as we work with the 20,000 teachers in our direct service
area and the more than 100,000 teachers across our State. [D]
In 2012, the NCSM recognised the power of the CC lessons to improve the
teaching of mathematical practices and so decided to base its nationwide
professional development strategy for the next two years on formative
assessment, built around the CCs. From July 2013, NCSM began a series of
three-day `Leadership Academies' using the CCs. These Leadership Academies
each serve several hundred mathematics education supervisors from
districts and schools across the country. Valerie Mills explains that
the...resources and the experiences in the Summer Leadership Academy will
enable participating leaders to begin to increase the use of formative
assessment strategies in their districts, shifting teaching practices and
improving student outcomes.
Furthermore, she describes the resources as "sturdy hinges on
which to open doors to more effective instruction".
The CC materials are designed to support the original research findings
that formative assessment produces more robust long-term learning of
concepts and skills, and more independent students who can use their
mathematics to tackle non-routine problems effectively and with
Pearson, one of the largest educational publishers in the world, is now
developing the first tablet-based full curriculum for Mathematics.
Becoming aware of the CCs, it has now licensed them for inclusion in this
online course, and is working with CRME on their adaptation for this new
Sources to corroborate the impact
A. Chair, Common Core State Standards Mathematics Writing Group
B. Director, College Ready Education, The Bill & Melinda Gates
C. Data from project website: http://map.mathshell.org/materials/index.php.
Download statistics available on request.
D. President-Elect, National Council of Supervisors of Mathematics.