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Smith and JonesMr Smith and Mr Jones are two maths teachers, who meet up one day. Mr Smith lives in a house with a number between 13 and 1300. He informs Mr Jones of this fact, and challenges Mr Jones to work out the number by asking closed questions.

Mr Jones asks if the number is bigger than 500. Mr Smith answers, but he lies.

Mr Jones asks if the number is a perfect square. Mr Smith answers, but he lies.

Mr Jones asks if the number is a perfect cube. Mr Smith answers and (feeling a little guilty) tells the truth for once.

Mr Jones says he knows that the number is one of two possibilities, and if Mr Smith just tells him whether the second digit is 1, then he'll know the answer. Mr Smith tells him and Mr Jones says what he thinks the number is. He is, of course, wrong.

What is the number of Mr Smith's house? www.nrich.maths.org – April 2004

What is mathematics enrichment anyway?

Jennifer PiggottJuly 2005

www.nrich.maths.org

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Outline

Proposals

Consequences

Time to reflect

Key words“powerful tools” transferable skills 8(6)“comprehension”broadening, breadth and depthactivities - rich tasks that yield lots of questions 1.2.12seeing patterns 8(5)very involved thinkingCodifyingGeneralisingGeneralising does not need algebra but leads to algebra (1.1)AnalogyExploring/explorationCreativeMediationExtensionProblem solvingMathematical thinkingFun/enjoyable. Although the notion of fun was concerning to

some as this may imply a lack of rigour or the importance of immediate gratification and this was not the view considered by the team

Conjecturing ProvingStructureMediationSymbolsGroup workConnection

Talk/discussionCommunitySpeculationMotivationGeneralising and communicatingLettersCodifyingNotation

Phrases and connectionsCommunicating though diagrams, breaking down problems aformulae are not enrichmentexplanation=enrichmentthinking about and around a problem – implies some structure

to the thinking and knowing what to look for.Logic Digging deeper–looking underneath-exploringEnrichment can also be harder because you are using harder

maths, harder problem solving skillsAwareness of structures, going inside-to break things down-to

make connections 8(2)Tying things togetherCombining areas of maths-integrating knowledge (multiply

complex situations)Developing from pupils thinking they know everything - e.g.

evens and odds and make 37Not recipe mathsSomething from which maths might be derivednon standard problemsLearning from mistakes

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Some proposals Depth

Breadth

Balance

Relevance

Acceleration

Extension

Extra to normal classroom practice

Provision for the most able

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Depth

The measurement from the top down, from the surface inwards, or from the front to the back;

difficulty, abstruseness;

comprehensively, thoroughly or profoundly;

intensity of emotion.

Extracts from the Oxford English Dictionary

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Breadth

The distance from side to side of a thing;

extent, distance, room;

freedom from prejudice or intolerance

Extracts from the Oxford English Dictionary

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Balance

An amount left over;

harmony of design and proportion;

offset or compare one with another;

establish equal or appropriate proportions;

choose a moderate course or compromise;

zodiacal sign.Extracts from the Oxford English Dictionary

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Possibilities so far:

Harmony of design and proportion (balance)

Extent (breadth)

Freedom from prejudice and intolerance (breadth)

Thorough and comprehensive (depth)

Emotional involvement (depth)

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Four-points

There are four points on a flat surface:

How many ways can you arrange those four points so that the distance between any two of then can be only one of two lengths:

Example:

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Relevance

Bearing on or having reference to the matter in hand.

Real world:

Actually existing or occurring

The Oxford English Dictionary

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Acceleration - Extension

Acceleration is the intentional exposure of pupils to more advanced standard curriculum subject matter with the specific aim of examination on that material in advance of chronological age.

Extension is the exposure of pupils to content not normally found in standard curriculum and which might be considered appropriate to that chronological age or older:

the opportunity to learn new mathematical content or techniques

application of an area of mathematics to different contexts not normally covered within the curriculum;

the study of mathematics as a cultural, social or historical phenomenon .

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And finally…

Extra to normal classroom practice:

Trips

Activities

Clubs

Aspiration raising

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Aspiration

Long term gains for pupils in terms of their attitudes to and understanding of what it is to be mathematical by

improving pupil attitudes,

developing an appreciation of mathematics as a discipline.

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Consequences – a view of enrichment

Content

Teaching

Aspiration raising

Audience

What I have described involves a level of engagement with the subject on a personal and social as well as an intellectual level, which in turn has implications for:

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Content Engaging contexts

Extend knowledge

Challenging knowledge and conceptions

Makes connections

Offers opportunity for a developing interest

Involves problem solving, problem posing and mathematical thinking.

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Teaching

Encourages pupils to be mathematical by building on appropriate content and

uses effective mediation;

engages with the mathematics as a community communicating;

encourages independent, critical thinkers;

values the individual and different approaches but also encourages critical evaluation of efficient methods;

makes use of metacognition and misconceptions.

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Problem solving

Understanding the problem

Devising a plan

Carrying out the plan

Looking back

Polya 1957

CAPE model

Planning and executionConsidering novel approaches and/or solutions,Planning the solution/mental or diagrammatic model,Execution of solution,

ComprehensionMaking sense of the problem/retelling/creating a mental image,Applying a model to the problem,

Analysis and synthesisApplying facts and skills, including those listed in mathematical thinking (below),Identifying possible mathematical knowledge and skills gaps that may need addressing,Conjecturing and hypothesising

EvaluationReflection and review of the solution,Are there more questions to answer?Self assessment about ones own learning and mathematical tools employed,Communicating results,

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Using Subgoals

1. Place the numbers 1-9 in a 3x3 magic square

2. How many zeros appear at the end of 100! ?

3. Find the sum of all the mulitples of 4 or11 in the integers from 1 to 1000

4. Consider the groupings (1), (2,3), (4,5,6), (7,8,9,10), …What is the sum of the digits in the kth grouping?

5. How many rectangles can be drawn on a 17 x 31 magic grid?

Shoenfeld 1985

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Mathematical Thinking

Mathematical strategies that are employed in solving the problems Type I examples

Generalising (as identifying patterns –general or common patterns – formula – looking for an essential shape or form)

Being systematic;

Mathematical analogy

Type II examples Introducing variables;

Specialising, looking for a particular case (specific action that comes out of the problem – doing a particular thing to help to simplify, e.g. paper folding)

Solving simpler related problems;

Working backwards.

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Purposes of problem solving

For:- problem solving seen as mathematical activity in its own right, often with problems designed to extend or connect mathematical concepts and undertaken explicitly for the purpose of being mathematical;

About:- involving the overt teaching of problem solving skills, teaching about how to problem solving;

Through:- teaching mathematical concepts through problem posing.

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For - Pentagonal

Can you prove that the sum of the distances of any point inside a square from its sides is always equal (half the perimeter)?

Can you prove it to be true for a rectangle or a regular hexagon?Does the hexagon need to be regular?

Can you show the same is the case for a regular pentagon? Does the pentagon need to be regular?

www.nrich.maths.org – June 2005

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About - Isometrically

How many unique symmetrical shapes can you make by shading four small triangles?

www.nrich.maths.org – Oct 2003

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Through – Subtended Angles

Choose two points on the circumference of the circle. Call them A and B.

Join these points to the centre, C. What is the angle at C?

Join A and B to a point on the circumference. Call that point D. What is the angle at D?

If the angle at D is acute, what do you notice about the angles at C and D?If the angle at D is obtuse, what is its relationship with the reflex angle at C?

What happens if you choose a different point D?What happens if you choose a different pair of points for A and B?

Would the same thing happen if you started with any two points on the circumference of any circle?Can you prove it?

www.nrich.maths.org – July 2005

Related to the initial impact of the problem or context: uses succinct clear unambiguous language, draw the solver in and offers intriguing contexts such that solving them

feels worthwhile, gives opportunities for initial success but have scope to extend and

challenge (low thresh hold high ceiling problems).Related to the experience for the solver: encourages solvers to think for themselves and to apply what they know

in imaginative ways, gives “the solver” a sense of slight unease at first Related to the problem: allows for different methods which offer opportunities to identify elegant

or efficient solutions, opens up patterns in mathematics and leads to generalisations, reveals underlying principles and can lead to unexpected results, requires a solution that calls for a good understanding of process and/or

concept draws together different mathematical concepts or branches of

mathematics.

Descriptions of a good problem situation:

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The trick

“What is the missing term in: 6, 11, 12, … , 110?”

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Enrichment

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Big wheel

100miles

100 mph

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Squirty

Using a ruler and compass only it is possible to fit a square into any triangle so that one side of the square rests on one side of the triangle and the other two vertices of the square touch the other two sides of the triangle:

 

www.nrich.maths.org – May 2004