@2018 S Lomax and D Bowman (@maxthemaths @maths4ukplc)
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In the mathematics classroom challenge has four elements. Two of these reflect the mindsets of teachers, parents and children with regard to learning mathematics and two reflect the pedagogical challenges.
1) The challenge that every teacher believes all children are capable of learning mathematics
2) The challenge that all children have positive learning (academic) mindsets when thinking about mathematics.
3) The planning and design of questions required to challenge conceptual understanding
4) The planning and design of questions required to challenge mathematical thinking.
Mindset Challenge
In 1956 George Polya in his book How to Solve It wrote
“Mathematics has the dubious honor of being the least popular subject in the curriculum… Future teachers pass through the elementary school learning to detest mathematics … They return to the elementary school to teach a new generation to detest it “
This negative attitude towards learning mathematics continues. In Professor Adrian Smith's 2017 review of Post 16 Mathematics he wrote:
“Negative attitudes towards mathematics generally are a cause for concern. Gender has a heavy influence on mathematics participation, reflecting entrenched cultural attitudes towards mathematics”
Consequently the first challenge a teacher of mathematics faces is to change these negative attitudes that children bring into lessons. To assist this process, teachers need to be aware of the four learning (academic) mindsets identified in psychological studies (Farrington et al 2012). These are
i) Belonging
ii) Purpose
iii) Self Efficacy
iv) Growth
For children to progress in mathematics it is essential that both teacher and students have a positive response to all four mindsets. Although all are equally important but belonging is a particular issue in mathematics lessons. The majority of secondary schools stream or set children which implicitly gives each child a message of what they can achieve. Those in top stream/set believe they “belong” in the class where everything is possible whereas in lower streams the children “belong” with others not expected to succeed. This is a significant challenge. Even if a school is unlikely to move from streaming / setting (based on prior attainment), it is possible to change pupil’s minds of what they are capable of doing. Where they are now is a result of their opportunities and performance to date and as Carol Dweck comments in Mindset (2006):
“Remember, test scores and measures of achievement tell you where a student is, but they don’t tell you where a student could end up”.
The teacher must believe it is possible for all children to learn school mathematics. Although recent advances in England is changing the perceptions of who can learn mathematics, it is probably the case that too many teachers still label children as capable or not capable of achieving in mathematics. This attitude has to change throughout the school. This is possibly a bigger challenge than the actual teaching of the subject content!
How a child’s view of what they can achieve in mathematics is exemplified by this comment
“Thank you for all you have done to teach me and encourage me. Thank you for showing me there is a point and most importantly that I can do it. I have really enjoyed maths this year and can’t wait to be doing more next year! I never thought I would be saying this after doing my GCSE exams but it’s true and I can’t thank you enough for changing my view and making me believe in myself.”
Pedagogical Challenge
‘The National Curriculum for Mathematics (2014) aims to ensure that all pupils:
· become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately
· reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
· can solve problems by applying their mathematics to a variety of routine and nonroutine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.’
The mathematics classroom needs to be an environment where a CanDo attitude is developed by all pupils and ‘pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content.’ (National curriculum in England: Mathematics Programmes of Study). Challenge therefore comes through complex problem solving and skilful questioning to create an environment in which pupils are unafraid to grapple with the mathematics. Differentiation is therefore achieved though challenging depth of understanding rather than prematurely accelerating onto superficial understanding of new mathematical content.
A possible lesson design to embrace the National Curriculum aims and provide challenge in every part of the lesson is:
i) Can you ‘do it’? 
Focus on the ‘What it is?’ Simple, standard examples followed by nonstandard examples challenge procedural fluency. 
ii) Are you secure?

Focus on the ‘What it is not?’ Active argument tasks  eg True/False, Do you agree?  focus on misconceptions and mistakes to challenge conceptual understanding 
iii) Can you apply it and solve it? 
Solving familiar and unfamiliar problems including empty box/find the missing symbol, ‘Here’s the answer  what is the question?, Always/Sometimes/Never, etc challenge mathematical thinking 
A coherent scheme of work with clearly identifiable ‘Conceptual Themes/Big Ideas’ broken down into small key learning points is essential to support this lesson design. www.mathsnav.com provides a good example of such a scheme. Two lesson examples follow.
The acronym DEPTH is a useful reminder how to provide challenge through differentiating by depth rather than new content.
Do you agree? 
Explicit use of misconceptions and mistakes 
Probing Questions 
The missing digits/symbols 
Here is the answer what is the question? 
The explicit use of misconceptions and response to mistakes are at the heart of both the mindset and pedagogical challenges.
This diagram from Mindsetworks is used to focus the minds of both teachers and children when considering mistakes.
The teacher’s challenge is to create problems that will initiate stretch mistakes whilst the children’s responsibility is to eliminate the sloppy mistakes.
Struggling, making mistakes and having challenges are all essential to growing as a mathematician, developing sustainable understanding and building secure foundations for future learning. After all, FAIL is the First Attempt In Learning !