1. The Road Less Travelled
Jennifer Piggott

2. TWO roads diverged in a yellow wood,
And sorry I could not travel both
And be one traveller, long I stood
And looked down one as far as I could
To where it bent in the undergrowth;
Then took the other, as just as fair,
And having perhaps the better claim,
Because it was grassy and wanted wear;
Though as for that the passing there
Had worn them really about the same,
And both that morning equally lay
In leaves no step had trodden black.
Oh, I kept the first for another day!
Yet knowing how way leads on to way,
I doubted if I should ever come back.
        
I shall be telling this with a sigh
Somewhere ages and ages hence:
Two roads diverged in a wood, and I—
I took the one less travelled by,
And that has made all the difference.
By Robert Frost

3. Mathematics and Personal learning and Thinking Skills
Independent Enquirers
Creative Thinkers
Team Workers
Self-managers
Effective participators
Reflective learners

4. NC - Key Concepts Competence Creativity
Applications and implications of mathematics
Critical understanding
Competence
Applying suitable mathematics accurately within the classroom and beyond.
Communicating mathematics effectively.
Selecting appropriate mathematical tools and methods, including ICT.
Creativity
Combining understanding, experiences, imagination and reasoning
to construct new knowledge.
Using existing mathematical knowledge to create solutions to unfamiliar problems.
Posing questions and developing convincing arguments.
Applications and implications of mathematics
Knowing that mathematics is a rigorous, coherent discipline.
Understanding that mathematics is used as a tool in a wide range of contexts.
Recognising the rich historical and cultural roots of mathematics.
Engaging in mathematics as an interesting and worthwhile activity.
Critical understanding
Knowing that mathematics is essentially abstract and can be used to
model, interpret or represent situations.
Recognising the limitations and scope of a model or representation.

5. NC - Key Processes 2.1 Representing 2.2 Analysing
Use mathematical reasoning
Use appropriate mathematical procedures
2.3 Interpreting and evaluating
2.4 Communicating and reflecting
2.1 Representing
identify the mathematical aspects of a situation or problem, choose between representations, simplify the situation or problem in order to represent it, athematically, using appropriate variables, symbols, diagrams and models
select mathematical information, methods and tools to use.
2.2 Analysing
Use mathematical reasoning
make connections within mathematics, use knowledge of related problems, visualise and work with dynamic images
identify and classify patterns, make and begin to justify conjectures and generalisations, considering special cases and counter-examples
explore the effects of varying values and look for invariance and covariance, take account of feedback and learn from mistakes, work logically towards results and solutions, recognising the impact of constraints and assumptions, appreciate that there are a number of different techniques that can be used to analyse a situation, reason inductively and deduce.
Use appropriate mathematical procedures
make accurate mathematical diagrams, graphs and constructions on paper and on screen, calculate accurately, selecting mental methods or calculating devices as appropriate, manipulate numbers, algebraic expressions and equations and apply routine algorithms use accurate notation, including correct syntax when using ICT, record methods, solutions and conclusions, estimate, approximate and check working.
2.3 Interpreting and evaluating
form convincing arguments based on findings and make general statements, consider the assumptions made and the appropriateness and accuracy of results and conclusions, be aware of the strength of empirical evidence and appreciate the difference between evidence and proof
look at data to find patterns and exceptions, relate findings to the original context, identifying whether they support or refute conjectures, engage with someone else’s mathematical reasoning in the context of a problem or particular situation, consider the effectiveness of alternative strategies.
2.4 Communicating and reflecting
communicate findings effectively, engage in mathematical discussion of results, consider the elegance and efficiency of alternative solutions
look for equivalence in relation to both the different approaches to the problem and different problems with similar structures, make connections between the current situation and outcomes, and situations and outcomes they have already encountered.

6. Problem solving Understanding the Problem Devising the Plan
What is known?
What is given?
What do I need to find?
Devising the Plan
Have I seen something like this before?
What mathematical approaches might be helpful?
Carrying out the Plan
Check each step
Estimate and assess – does this look right?
Look Back
Check the result
Could I use this idea somewhere else?
Communicate
Share your findings and consider what others have done
Could I have got there another way?
Would I do something like this differently next time?

7. NRICH Website Mapping documents Teachers’ menu
Content and process mapping documents
Teachers’ menu
Problems with teachers’ notes
Student solutions
Case studies
Teachers using NRICH, for example:
CPD

8. Embedding PLTS
Identify questions to answer and problems to solver, planning and carrying out research
Generating ideas and constructing mathematical models, exploring and varying values
Discussing and writing up findings
Reflecting and communicating using appropriate representations
Organising time and resources and prioritising
Comparing other people’s methods and approaches with their own
Planning and preparing ways forward, breaking problems down

14. Some problems Seven squares Stringy quads Hundred square
Stringy quads
Hundred square