The Road Less Travelled Jennifer Piggott jsp38@cam.ac.uk

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

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

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.

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.

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?

NRICH Website Mapping documents Teachers’ menu http://nrich.maths.org/public/viewer.php?obj_id=5665 Content and process mapping documents Teachers’ menu http://nrich.maths.org/public/viewer.php?obj_id=6318 Problems with teachers’ notes Student solutions Case studies Teachers using NRICH, for example: http://nrich.maths.org/public/viewer.php?obj_id=6388 CPD http://nrich.maths.org/public/viewer.php?obj_id=2695

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

Some problems Seven squares Stringy quads Hundred square http://nrich.maths.org/public/viewer.php?obj_id=2290 Stringy quads http://nrich.maths.org/public/viewer.php?obj_id=2913 Hundred square http://nrich.maths.org/public/viewer.php?obj_id=2397