Research by Nunes et al (2009) identified mathematical reasoning as the most important factor in a pupil's success in mathematics. Yet Ofsted's findings (Jones, 2015) show it is the least well-developed of the three national curriculum aims. Explore the reasons for this and how it could be addressed.
The 3 main aims of the
national curriculum are: reasoning, fluency and application- this is the new
mastery strategy which was implemented in 2012. It was influenced by high
performing countries in east and southeast Asia such as: China, Singapore,
Japan and North Korea (NCETM, 2014). The mastery approach places an emphasis on
the depth of understanding as opposed to a breadth of knowledge (Morgan, 2015),
this will give pupils the opportunity to fully understand a concept and begin
reasoning mathematically before moving on. Furthermore, research suggests that
this approach is having a positive effect on pupils using mastery- for example,
data from the research program in 2012 (by the Programme for International
Student Assessment [PISA]) suggests that by age 15 students from the high
performing countries are on average up to three years ahead in maths compared
to 15-year olds in England (2014, p4).
In this paper, I aim to
explore one of the three key aims- reasoning. The national curriculum defines
reasoning as: 'following a line of enquiry, conjecturing relationships and
generalisations and developing an argument, justification or proof using
mathematical language' (National Curriculum, 2013, p88). Nunes et al have
advocated for reasoning being the most important factor for a pupil’s success
in mathematics (p3). It is also regarded by academics as the aspect in
mathematics which is at the 'core of the subject' (Haylock, 2014, p37-38). This
is further highlighted by NRICH (2014), who suggest it is reasoning that helps
children to use their mathematical skills, inviting them to gain a deeper
conceptual understanding- making connections and draw inferences (Witt, 2014,
p3). In short, reasoning is the 'glue' that helps to make mathematics make
sense (NRICH, 2014).
However, teachers often
struggle with teaching reasoning and how to use it to prove mathematical ideas
and vice versa with students struggling to learn it (TESS India, no date, p3).
This coincides with Ofsted's findings, that reasoning is the least 'well developed'
of the three-national curriculum aims (Jones, 2015). It could be suggested that
mathematical reasoning may be an element that cannot be taught with rules,
however, it can be fostered, prompted and stimulated within pupils through
various strategies. This, however, would be down to a teachers practice within
the classroom. Therefore, within this paper, I would like to investigate the
factors which may cause a lack of mathematical reasoning- and suggest how they
may be overcome.
One factor (which is
highlighted frequently by Ofsted and academics), is teaching children rules to
tackle calculations without letting pupils explore or discuss- to play an
active process in learning.
Atkinson (1992, p12)
highlights that mathematical reasoning is 'based on understanding', therefore
constantly verbalising the information, is suggested to give children a basic
understanding- also known as an instrumental understanding. An instrumental
understanding is when an individual learns the rules and applies them to particular
circumstances (Fisher, 2005, p171). However, the issue with this is that rules
are easily forgotten- thus it is seen as a shallow level of understanding which
can place pressure on a learners memory. This is exampled by Cockcroft (1982),
who investigated the mathematical competencies of a group of adults. Finding that, adults (who
left school with what was considered to be respectable test scores) 'appeared
to have only one method of tackling a given problem' (p8). It can be inferred,
from this research, that the subjects had an instrumental understanding of
mathematics- as the adults were trying to replicate the 'proper method' shown
to them to achieve the correct answer. However, had the adults gained a
relational understanding (this is where the individual knows the reason behind
the rules, therefore, they can think through and reconstruct the rules for
themselves), they may have continued to try other methods- as it was reported
subjects 'lacked the ability and confidence to attempt a different approach'
(p8).
Therefore, it is important
that we provide children with various opportunities to develop their reasoning
skills- this could be done by planning an enquiry. As highlighted by Atkinson
(1992, p13) 'maths with reason is rooted in action- learning through doing'.
Children would need to decide how they would first tackle the problem and would
then have to draw upon a range of mathematical skills to work at the enquiry
(Cotton, 2013, p29). Academics and Ofsted (2012) frequently stress the importance
of children playing an active rather than passive role in the classroom. I feel
that enquiries- such as the ones supplied on NRICH would achieve this. A
Strategy, which I would like to try in school, is placing an enquiry on a
working wall, on it, children can place their work up. I believe this would
help children to develop their reasoning skills as well as their problem
solving as it allows pupils to view model examples of work. They can then refer
to them when improving their own work, helping them to become succinct. The
class is working as a problem solving team. This notion is further supported by
the National Numeracy Strategy, a framework developed to raise standards in
schools, advocating high quality learning is a 'two-way process in which pupils
are expected to play an active part by answering questions, contributing points
to discussion and explaining and demonstrating their methods to the class'
(found in Myhill and Hopper, 2005, p50).
The next factor has been
suggested by an article from the National Centre for Excellence in the Teaching
of Mathematics (2013). They highlighted that too often teachers give children a
statement and we do not give them a chance to develop their mathematical
reasoning to prove it. 'Proof is a peculiarly mathematical way of reasoning'
(Haylock, 2014, p43). NRICH (2015) identify that children progress in their
reasoning when they go through the five stages- this starts at describing and
ends at proving.
A strategy that is commonly
suggested (Turner and McCullouch, 2005, Chapter 3 p9) is to allow children to
choose their own resources to figure out the problem- 'manipulatives can be
powerful tools to support sense making, mathematical thinking and reasoning'
(NRICH, 2014). Some children may prefer to see the problem visualized others
may prefer to experiment with the problem using written methods- as long as the
learners are trying to make sense of the mathematical concepts. As Delaney
(2001, cited in Thompson, 2010, p73) states 'there is no mathematics actually
in a resource' but rather 'the mathematics is brought to the resource by those
who interact with it or is developed by them as they use it to support or
challenge thinking'. Children need to represent their ideas and findings,
therefore using manipulatives or written down formulas relating to the problem
is effective to guide the children through the process and prove their
understanding (Herringer, 2013). Although, according to the research by
Schoenfeld (1987), even though children may be assisted in their learning of
concepts by the use of manipulative apparatus, they are often unable to apply
those same concepts to problem solving situations. Therefore, it is important
that teachers consistently model to use of resources to children.
My final factor, is the use of
questioning. As Stripp (2015) highlights, when Chinese teachers ask for
answers, they always ask follow up questions such as how? and why? Requiring
children to use both inductive reasoning- making conjectures and solving the
problem through convincing- and deductive reasoning- justifying and proving
their conjecture (Haylock, 2014, p42-43). This can be done with both peers and
teachers, challenging pupils to comprehend the maths taking place, encouraging
them to discuss their conjectures and underlying thoughts, helping pupils to
clarify their ideas and make mathematical connections so that they can
articulate their reasoning (Ge Feng et al, 2007, p382). This was apparent in
schools that took part in the China-England teacher exchange program. One child
is quoted saying 'I like the Chinese lessons, I need to think very hard because
I know Miss Lu will ask me to explain why and I will need to have a good
answer!' (Stripp, 2015). It could be suggested that the child understands the
engagement and depth required in Miss Lu's class as opposed to the other which
may not require as much reasoning from the pupil.
To reflect, there are a
variety of ways teachers can promote reasoning within the classroom.
Fortunately, whilst exploring these factors, it became apparent that whilst I
have been in schools I have observed practitioners stimulate and regularly
encourage reasoning from pupils. This paper has also inspired various ways in
which I hope to encourage the use of reasoning within the classroom- for example
the use of a working wall. Additionally another factor, which I did not discuss
but that I am aware of, is the use of mixed ability groups. In Nunes et al
(2009) research, they discovered that streaming groups only slightly helped the
higher ability but hindered the progress of others. Therefore, I will ensure
that whilst doing investigation and enquiry work the children will be mixed
ability. Overall, I believe that reasoning would be the hardest aim to teach,
however, as also highlighted by Nunes at al (2009), it will help children to
progress in their future mathematical development.
Bibliography
Askew, M. (2016) Transforming Primary Mathematics:
Understanding Classroom Tasks, Tools and Talk. London: Routledge.
Atkinson, S. (1992) Mathematics With Reason: Great Britain,
Hodder and Stoughton.
Beckley, P. Compton, A. Johnston, J. Marland, H. (2010)
Problem Solving, reasoning and Numeracy: London, UK, Continuum International
Publishing Group.
Bottle, G. (2007) Teaching Mathematics in the Primary
School. London: Continuum.
Cockcroft, W. H. (1982) Mathematics counts: report of the
Committee of Inquirt into the teaching of mathematics in schools. London: HMSO.
[Online] Available at: http://www.educationengland.org.uk/documents/Cockcroft/cockcroft1982.html
[Accessed on: 6th December]
Cotton, T. .(2013) Understanding and Teaching Primary
Mathematics 2 edn: Harlow, UK, Pearson Education LTD.
Cowan, P. (2006) Teaching Mathematics A Handbook for
Primary and Secondary Teaching: Abingdon, UK, Routledge.
Davis, A. GOudling, M. Suaggate, J. (2017) Mathematical
Knowledge for Primary Teachers, 5edn: Abingdon, UK, Routledge.
Fisher, R (2005) Teaching Children to Think. London:
Original Illistrations
Hansen, A. (2017) Children’s Errors in Mathematics: London,
UK, SAGE Publishing LTD
Haylock, D. Manning, R. (2014) Mathematics Explained for
Primary Teachers, 5edn: London, UK, SAGE Publishing LTD
Haylock, D. Cockburn, A. (2013) Understanding Mathematics
for Young Children 4 edn: London, UK, SAGE Publishing LTD.
Herringer, N. (2013) Making sense of mathematics through
reasoning. [online] Available at: http://www3plearning.com/making-sense-mathematics-reasoning-2/
[Accessed on: 5th December]
Jones, J. (2015) OFSTED: Mathematics mastery primary conference workshop
materials [PowerPoint Presentation]. [Online] Available at http://toolkit.mathematicsmastery.org/app/webroot/js/tiny_mce/plugins/moxiemanager/data/files/Ofsted.pdf.
[Accessed on: 30th
November]
Jones, J. (2015) Charlie's
Angles - guest blog by Jane Jones HMI, Ofsted. [online] Available at: https://www.ncetm.org.uk/resources/46034
[Accessed on: 3rd
December]
Morgan, D. (2015) Debbie
Morgan. NCETM Director for Primary: a presentation to teachers on teaching for
mastery in December 2015. [online] Available at: https://www.ncetm.org.uk/resources/48432
[Accessed on: 6th
December]
Myhill, D. Jones, S.
Hopper, R. (2006) Talking, Listening, Learning. Berkshire: Open University Press.
National Assiociation
of Mathematical Advisers (2015) Five myths of Mastery in Mathematics. [online]
Available at: http://www.nama.org.uk/Downloads/Five%20Myths%20about%20Mathematics%20Mastery.pdf
[Accessed on 4th
December]
National Centre for
Excellence in the Teaching of Mathematics. (2013) Maths to share- CPD for your
school. [online] Available at: https://www.ncetm.org.uk/resources/30913
[Accessed on 10th
December]
NRICH (2014) Reasoning:
Identifying Opportunities. [online] Available at: https://nrich.maths.org/10990
[Accessed on: 4th
December]
NRICH (2014) Reasoning:
the Journey from Novice to Expert. [online] Available at: https://nrich.maths.org/11336
[Accessed on: 5th
December]
NRICH (2015)
Haringey. [online] Available at: https://nrich.maths.org/10757
[Accessed on 7th
December]
NRICH (2013) Manipulatives
in the Primary Classroom. [online] Available at: https://nrich.maths.org/10461
[Accessed on: 7th
December]
Nunes, T., Bryant, P., Sylva, K. and Barros, R. (2009) Development of maths capabilities and
confidence in primary school. [Online]
Available at http://dera.ioe.ac.uk/11154/1/DCSF-RR118.pdf.
[Accessed on: 30th November]
Ofsted (2012)
Mathematics: made to measure. Available at: https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/417446/Mathematics_made_to_measure.pdf
[Accessed on: 4th
December]
Pepperell, S.
Hopkins, C. Gifford, S. Tallant, P. (2009) Matqhematics in Primary School: A
Sense of Pregression, 3edn: Abingdon, UK, Routledge
Programme for International Student Assessment (2014) PISA
2012 Results in Focus: What 15-year-olds know and what they can do with what
they know. [online] Available at: http://www.oecd.org/pisa/keyfindings/pisa-2012-results-overview.pdf
[Accessed on: 5th December]
Stripp, C. (2015) Charlie’s Angles, Math, How can we meet
the needs of all pupils without differentiation of lesson content? How can we
record progress without levels? [online] Available at: https://www.ncetm.org.uk/resources/46830
[Accessed on: 3rd December]
Stripp, C. (2015) Charlie’s Angles, Math, Mastery in
mathematics: What it is and why we should be doing it. [Online] Available at: https://www.ncetm.org.uk/resources/45776
[Accessed on: 3rd December]
Sangster, M (2016) Engaging Primary Children in
Mathematics: London, UK, Bloomsbury Academic Publishing plc.
Stoushall, E. (2017) Yes but Why? Teaching for
Understanding in Mathematics: London, UK, SAGE Publishing LTD
TESS India. Developing mathematical reasoning: mathematical
proof. [online] Available at: http://www.open.edu/openlearncreate/pluginfile.php/134971/mod_resource/content/4/SM02_AIE_Final.pdf
[Accessed on: 3rd December]
Thompson, I. (2010) Issues in Teaching Numeracy in Primary
Schools. 2nd edn. Berkshire: Open University Press.
Turner, S. (2013) Teaching Primary Children in Mathematics:
London, UK, Bloomsbury SAGE Publishing LTD.
Turner, S. McCullouch, J. (2004) Making Connections in
Primary Mathematics. Oxon: david Fulton Publishers
Witt, M (2014) Primary Mathematics for Trainee Teachers.
London: Learning Matters