Haringey Session 1 Supporting fluency and reasoning through rich tasks

HaringeySession 1Supporting fluency and reasoning through rich tasks 8 October 2014 Lynne McClure Director, NRICH project

National Curriculum 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.

National Curriculum Reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language

Reach 100 Choose four different digits from 1−9 and put one in each box. For example: This gives four two-digit numbers: 52,19, 51, 29 In this case their sum is 151. Can you find four different digits that give four two-digit numbers which add to a total of 100?

What is the mathematical knowledge that is needed? • Who would this be for? • What is the ‘value added’ for higher attaining children/struggling children.

Strike It Out 6 + 4 = 10 10 take away 9 makes 1 1 add 17 is 18 18…… Competitive aim – stop your partner from going Collaborative aim – cross off as many as possible

What is the mathematical knowledge that is needed to play? • Who would this game be for? • What is the ‘value added’ for able children/struggling children of playing the game? • How could you adapt this game to use it in your classroom?

How do these rich tasks contribute to fluency? reasoning?

Efficiency An efficient strategy is one that the student can carry out easily, keeping track of sub-problems and making use of intermediate results to solve the problem.

Efficiency Accuracy depends on careful recording, the knowledge of basic number combinations and other important number relationships, and checking results.

Efficiency Accuracy Flexibility requires the knowledge of more than one approach and being able to choose appropriately between them (Russell, 2000 http://investigations.terc.edu/library/bookpapers/comp_fluency.cfm)

Procedural & conceptual fluency Automaticity Automaticity with recall

Fluency

Using the same rules is it possible to cross all the numbers off? How do you know?

Two types of reasoning Inductive reasoning • Can be incorrect • Can’t be used to ‘prove’ Deductive reasoning • Follows rules of logic • Can be used to prove

In a problem: • Reasoning is necessary when: • The route through the problem is not clear • There are some conflicts in what you are given or know • There are some things you don’t know • Theres no structure to what you’re given • There are different possible solutions • All of which require mental work….

Reasoning is… • A critical skill to knowing and doing maths • Enabling – it allows children to make use of all the other mathematical skills – it’s the glue that helps maths to make sense.

Structuring children’s reasoning • Questioning: closed v open • Listening • Acknowledging • Improving • Modelling KS1: good 'because' statements, short chains • KS2: logic, convincing

Session 2 Problem solving

National Curriculum Can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions

Historically • learning the content v problem solving • theory versus practice, reason versus experience, acquiring knowledge versus applying knowledge. • problems seen as vehicles for practicing applications ie computation procedures are acquired first and then applied • problem-based learning

Dominoes • Dominoes – have a play…. • Have you got a full set? • How do you know? • Can you arrange them in some way to convince yourself/others that you have/ haven’t got full set?

What number knowledge/skills did you use? • What mathematical processes did you use? • What ‘soft skills’ did you use?

Amy has a box containing ordinary domino pieces but she does not think it is a complete set. She has 24 dominoes in her box and there are 125 spots on them altogether. Which of her domino pieces are missing?

What number knowledge/skills did you use? • What mathematical processes did you use? • What ‘soft skills’ did you use?

Order of events • Free play –Montessori ‘work’ • Closed activity: structure of the apparatus • Task which uses that knowledge • Multistep • With or without apparatus • Ruthven’s • Exploration • Codification • Consolidation

Dominoes v houses Sort – have you got them all? How do you know? Tasks using that knowledge Guess the dominoes/ houses

Rich tasks…. • combine fluency, problem solving and mathematical reasoning • are accessible • promote success through supporting thinking at different levels of challenge (low threshold - high ceiling tasks) • encourage collaboration and discussion • use intriguing contexts or intriguing maths

allow for: • learners to pose their own problems, • different methods and different responses • identification of elegant or efficient solutions, • creativity and imaginative application of knowledge. • have the potential for revealing patterns or lead to generalisations or unexpected results, • have the potential to reveal underlying principles or make connections between areas of mathematics (adapted from Jenny Piggott, NRICH)

Tasks • Non-routine • Accessible • Challenging • Curriculum linked • Rich tasks/LTHC tasks Implications for your teaching?

Valuing mathematical thinking • Process as well as end product • Talk as well as recording • Questioning as well as answering • …………

Purposeful activity Give the pupils something to do, not something to learn; and if the doing is of such a nature as to demand thinking; learning naturally results.John Dewey

Session 4 Games are more than fillers

3 4 2 1 5 3 5 1 2 Dotty 6 Green wins!

What is the mathematical knowledge that is needed to play? • Who would this game be for? • What is the value added of playing the game? • Could you adapt it to use it in your classroom? • Contribute to F, R, PS?

Board Block

Four Go

Nice and nasty

“If I ran a school, I’d give all the average grades to the ones who gave me all the right answers, for being good parrots. I’d give the top grades to those who made lots of mistakes and told me about them and then told me what they had learned from them.” Buckminster Fuller, Inventor

What were these children’s views of maths? • Would you get the same answers?

Session 3 Maths Working Group

Purpose of study Mathematics is a creative and highly inter-connected discipline that has been developed over centuries, providing the solution to some of history’s most intriguing problems. It is essential to everyday life, critical to science, technology and engineering, and necessary for financial literacy and most forms of employment. A high-quality mathematics education therefore provides a foundation for understanding the world, the ability to reason mathematically, an appreciation of the beauty and power of mathematics, and a sense of enjoyment and curiosity about the subject.

interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas. • make rich connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems • apply their mathematical knowledge to science and other subjects.

The new National Curriculum What’s important to teachers?

Haringey Session 1 Supporting fluency and reasoning through rich tasks