sustainable-enquiry.comsustainable-enquiry.com/wp-content/uploads/2018/01/maths-pt-… · Web viewCalculation plays a small part in ‘everyday’ or ‘professional’ use of mathematics.

Learning Mathematics and practising P4C: Towards complementary social practices

Part 3: A mathematical curriculum to complement P4C practice. DRAFT

Doing mathematics

Calculation plays a small part in ‘everyday’ or ‘professional’ use of mathematics. Most calculation can easily be automated. That is not to say that understanding of what ‘multiplication’, ‘division’ etc. are is not important, these are among concepts central to mathematical fluency. However, understanding of these concepts is not supported by undertaking multiple calculations.

As an example of this point consider the fact that at one stage in the history of mathematics curricular it was seen as essential to teach people how to find the square root of a number using a manual method. This algorithm is rather like long division only more complicated. It would now be difficult to find anyone who could tell you how to perform such a calculation manually. Further down the line people found a square root using a slide rule, then log tables. Now, of course, it is a matter of pressing the appropriate calculator button. Understanding of what a square root is not supported by any of the methods cited above. What does help with such understanding? Drawing and measuring is one possibility (although unlikely to be very accurate). What does work well is using a calculator to perform multiple trial and error calculations that inch ever closer to an answer. In fact there is much value in this approach in helping the learner to find out about degrees of accuracy and to see how square and square root form inverse functions.

The mathematics that people use is mainly about problem solving, reasoning, questioning, using intuition and only partly about calculating. Learning mathematics is more about enquiry than calculation. Problem solving using mathematics plays a limited part in the school curriculum. The emphasis in school is largely upon calculation, probably because competency in this is easy to assess and probably because of some outmoded thinking about the value of arithmetic.

Conrad Wolfram describes the four stages to problem solving Pose a question in the world Turn into a mathematical model Perform a calculation (most of school maths is this aspect) Interpret the answer back in a real situation

Again this is not to deny the value of all calculation. Mental calculation, in particular, is of value within the vital area of developing ‘number sense’, of estimation, of assessing error and accuracy. Certain ‘big ideas’ are central to mathematics. We do not necessarily have to cover the whole body of mathematics because by delving into one key concept in depth the connections reach out to others. As Davis puts it ‘by tugging at one corner of mathematics the whole thing explodes in your face’.

1

The importance of Spatial Reasoning

In previous sections it was argued that spatial reasoning is teachable and a very important aspect of mathematical development generally. Davis et. al. 2015 found that combining spatial reasoning with a contextual narrative was particularly effective in promoting understanding. It is important, they found, to connect with opportunities to ‘embody’ the mathematics, to use movement, position and orientation with young children linked to a story of events to allow them to ‘be in’ the frame of activity. LOGO programming provided opportunities to develop spatial and symbolic reasoning together and is one example of a very successful attempt to make mathematics learning more exploratory. (see Seymour Papert) Some key spatial reasoning moves are:

Construction (linked to narrative to be more effective)VisualisationOrientationDecomposing/recomposingComparing

Early years practice often incorporates spatial reasoning. Our maths curriculum tends to undervalue the use of enactive and iconic representations for older students. This is quite possibly a mistake. For example, understanding algebraic expressions can progress with visual activities such as that developed by Geoff Giles:

It is argued that Spatial Reasoning and Number Sense are two important skills which need to be developed and nurtured. It is clear that there need to be opportunities for skill-building as well as longer mathematical enquiry. There is debate about how this should best be achieved in a mathematics curriculum. An idea I want to promote, and this is not original by any means, is that skills development should proceed in short and frequent sessions which

2

often take the form of interactive, whole class work led by the teacher while concept development and problem solving is part of a longer-term plan. I draw on two parallel cases to advance this idea. First, that of language learning, which typically uses short daily vocabulary and structure-building sessions along with longer conversation pieces. The second comparison, which I think has even more appeal in this case, is that of learning to play a musical instrument. I describe this further in the next section.

A musical metaphor

Learning to play a musical instrument and becoming a musician could be exemplified by three levels of activity.

Developing skills by working on scales and exercises. Playing tunes to build on these skills and to provide interest and enjoyment in their

own right. Taking part in performance, either as an individual player or in groups.

Scales and exercises are part of daily practice and develop a ‘muscular memory’ and an ear for particular sound patterns. Tunes develop musicality focused on musical forms and ideas, they are enjoyable in their own right. They demand attention to loudness, phrasing etc in keeping with the atmosphere of the piece. Performance (individual or with others, in large arenas or small, private settings) is something of an aim for all the work put in on exercises and tunes gives purpose to the development of fluency.

A developing musician will engage regularly at all three levels. Each level supports and gives meaning to the other two. There is no sense that performance comes after a long period of tune playing which is preceded by extensive scale and exercise practice. The other point to note is that exercises don’t need to be ‘boring’. They can be adventurous and ‘tune-like’. Furthermore, tunes merge into performance, and in fact all three interrelate (note for example the Bach Cello suites which he wrote as ‘practice exercises’ and which are highly prized as performance pieces.

The reason I think this metaphor might be useful is because of the focus it can bring to our mathematical and philosophical activity. There is a tendency in mathematics lessons to emphasise skill building whilst relegating ‘using and applying’ to the role of enrichment. Further, investigations are often presented as structured exercises affording little or no responsibility to the student for progress towards a satisfying resolution. If focused investigations represent the ‘tunes’ of mathematical development then ‘performances’ are almost entirely lacking. Seldom are mathematical processes combined with other skills across the curriculum to address a substantial questions or problems the solutions which are presented in some format.

3

If this is a useful metaphor in mathematics education then it might also be of use in P4C practice. Consider what ‘tuneful’ P4C might look like:

An asideUsing the metaphor in P4C

Scales and exercises:Exemplifying 4Cs thinking and the Good thinking guide

Tunes:Discussion plans Concept stretchersExploring historical arguments

Performance:Enquiries and follow-up activities.

Using the metaphor in mathematics

So what might ‘tuneful’ mathematics look like? I make some suggestions below:

Scales and exercises: Internalising the important building blocks of mathematics

Understanding and having a feel for number.Number bonds and combinationsSpatial reasoning activities………

Some examples of regular exercises to develop number sense

As part of general teaching, drawing the class together undertake activities such as:

Good mistake –where did that come from?If we know this what else do we know?Give me, draw me, tell me…and another, and anotherOdd one out, why? Why another?The answer is, what was the question?Zoning inGive me a silly answer and say why it cant beAlways, sometimes, never true? distinctionsGive me a POG (peculiar, obvious, general) answer6+4=7+3,  does 6-4 =7-3 ? Questions??

4

(Talk Maths 2006 Camden Training and Media Service,)

Another such skill building activity is the twentyfour game: Using the four digits shown and answer 24. This is played a bit like snap. Players sit in a circle. A card is revealed. The first person to come up with a solution can claim the card (and state their solution)

The ATM, BEAM and SMILE resources cited in the reference section (part 4) have extensive skill-building activities. Such quick-fire activity was also a feature of the National Numeracy Strategy in England.

Fluency with Number facts.

Mike Askew argues that certain number facts need to be internalised in order to allow the problem-solving mind to operate effectively. He identifies factual fluency and procedural fluency in his book Transforming Primary Mathematics.

Factual Fluency

Adding or subtracting a single digit to any numberAdding a multiple of 10 or 100 to any numberCounting on or back in twos, tens, or fives from any starting numberRecalling rapidly the number facts up to 10 X 10 and then, if need be, the 11 and 12 times tablesMultiplying any number by 2 or 10

Procedural fluency

Knowing what to add to a number to make it up to a multiple of 10 or 100Halving any numberMultiplying any number by 5 ( x ten then halve)Knowing the division facts associated with the multiplication facts

Many of the exercises in this section are aimed at establishing such fluency in an enjoyable way

5

Short, regular activities are the answer. Marvin Minsky suggests one such strategy. Create flash cards with all the multiplication facts to 10 x 10 with the answers on the back. Get pairs of pupils to test each other, when a fact is correctly answered put it to one side so that only the ‘difficult’ ones are being re-tested.

Tunes: Focusing on the ‘big ideas’ of mathematics through questions and/or Concept Studies. (see the end of this paper for some further ‘big ideas’ from Mike Askew)

Some of these big ideas are:

PatternSpace and shape (line, perimeter, area, volume, tessellation)Movement (Location, orientation)Change (growth, operation, inverse)Probability and ChanceQuantity (Number, and number systems)Measurement (estimation, error)Part/whole (proportion, ratio)Equivalence, equalsFunctions (Symbolism, equations, inequalities, variable, limit, constant)Tending to a limitRecursionRepresenting and interpreting dataMathematics for social justice

The example in part 2 of the concept study on the ‘circle’ is an example of what I am calling a ‘tune’. It involves focus on a particular big idea in the first instance making links to other mathematical (and every-day ideas) and utilizing building-block skills. Such activity is likely to involve a mix of individual, small-group and whole class work. Underpinning this will be the ability to engage in dialogue.

The CAME material (cognitive acceleration through maths education) could also be regarded as tunes. Key mathematical ideas are explored through semi-structured starting points leading to more open-ended work, but always with the expectation that certain ideas will be highlighted. There is little room in these activities for accommodating new interests.

Marion Bird’s work with young children comes at the more ‘improvisational’ end of the tune category. She establishes a ‘mathematical situation’ such as paper folding and then encourages pupils to devise questions, hence very much in the vein of P4C stimulus/question/enquiry model.

A stick of CubesIn Mathematics for Young children. Routledge 1991 Marion Bird relates the following activity with four and five year olds. She gave them the shape below made from unifix.

6

6

Working in pairs she invited them to re-arrange the shape. Most wanted to reduce the number of cubes by 1. There was discussion about which one to take off and how to rearrange the others. Some wanted to change the number at the top. Some were encouraged to explore ‘families’ of such towers in light of a particular pattern which they devised.

Growth: A further example of a ‘tune’.

Start with a story about the King of Siam and then lead into some discussion with some questions

The King of Siam wanted to reward a loyal subject for a job well done. The King asked:‘What can I give you as a reward?’‘I don’t want much, your highness, just a little rice,’ his subject replied.‘A bag, two bags, name your price.’ said the KingThe King’s subject pointed: ‘Do you see that chessboard over there? Just let me have a grain of rice on the first square, then double on the second, double again on the third, and so on, keep doubling until we get to the last square. Then if we can just gather up all the rice calculated for each square, I’ll take that.’‘I would’ve thought you’d be better to have the two bags, my boy.’ The King answered, looking slightly puzzled.

What do you think? Would you rather have the bags of rice or the total of the rice on the chessboard?

What would have happened if the subject had asked for just two more grains on each square?

(You could model the loyal subject’s suggestion by using a calculator in the following way:Key in: 1 X 2 = = =) 8X8 chessboard or 1+2====

Following on from this students can be invited to raise questions about how things grow.A number of prompts can be used if questions dry up. For example, modelling the growth of a forest fire using squared paper and/or making links with growth and the Fibonacci sequence. Possibly the easiest entry to this sequence is by using a diagram of rabbit reproduction.

7

Noting the growth pattern 1,1,2,3,5,8,13 and the fact that this pattern occurs when there is binary division (ie cell growth).

Types of growth and decay are of fundamental importance to the human species and the environment.

There are questions to raise about why exponential growth doesn’t totally overwhelm us (predator prey relations and forms of controlling growth)Is growth a good thing? Is it inevitable?What is the same/different about additive and multiplicative growthWhat can/should be done about growth in different areas.Who decides etc.Questions about cause and effect, how different decisions lead to different outcomes.How we can tell a story by looking at starting ideas and outcomes, then infer what happened in between. (see Let’s Think Through Maths p50 year 5/6 ‘Something Happens’, for example)

Two more focused stimuli give examples of how question-building could be stimulated as part of work on a big mathematical idea. From

http://www.inquirymaths.co.uk

As part of work on place value and structure of number:

As part of work on operations and inverse relations

8

For further ‘Tune-type’ activities see also the ATM ‘Big Ideas’ book and ‘Practicing Maths’CAME ‘Thinking Maths’ and ‘Let’s Think Through Maths’See also NRICH maths website: http://nrich.maths.org/frontpage

Some of the above ideas start look more like performances.

Performance: Using and applying mathematics along with other disciplines and skills to make some progress with a problem or question.

The performance category emphasises the fact that mathematical thinking is an essential ingredient of problem solving in the ‘real world’. This is both engaging and empowering. It takes mathematics out of the purely instrumental realm. Engagement with real and important problems lends importance to mathematics as a discipline. The point made by Jo Boaler and others is that becoming mathematically literate is an entitlement and an empowerment. All of us are capable of thinking and reasoning mathematically. Having such capacity developed enables people to take part in critical decisions and to ask important questions. Providing opportunities to engage with such central problems and ideas helps, in turn, to motivate learning in mathematics. My expectation is that pupils would activate a community of enquiry when appropriate within the project work. Hopefully being the instigators of such an enquiry when considered appropriate. Teachers might have some ideas about where enquiry-appropriate issues were likely to arise,

Example stimuli briefs for performance:

Developing a brief for setting up a Mars colony (perhaps use the film ‘The Martian’ as a stimulus?)Addressing the problem of waste, energy conservation etcSaving a species from extinctionUnderstanding predator/prey relationships and the effect on populationsReorganising traffic systemsSafe routes to schoolHealthy eatingOrganising a stall at a school feteMathematics and Social Justice

The above simply give a flavour of what a school might embark upon. There would be a need to work across subject areas, indeed this would be the strength of such an approach. Students would need to have input into project title and design. Possible project work could

9

arise out of P4C enquiries. For example, using the book ‘If the World Were a Village’ and focus on food distribution might lead students to think about where food comes from and what some of the underlying issues around food production and transportation are. Opportunities to ‘drop into’ a P4C enquiry could arise as a natural consequence of questions that arise. For example fair trade is likely to emerge as a consequence of thinking about how food is traded. The question could arise: ‘Is Fairtrade Fair?’

Comparative graphs showing the proportion of return made by different groups is shown below from Pete Wright ‘Teaching Mathematics for Social justice’.

Share of Chocolate Revenue in the 1980s

Share of chocolate revenue in 2012

The visual comparison is, of course compelling but proportions of revenue is not the whole issue. What is fair is not simply a matter of amount, there are issues about investment, expertise, need. Many such issues require discussion, definition, argument

10

A project such as ‘organize a stall for a school fete’ can lead to reflective dialogue and discussion. Instead of treating such an activity as just a bit of fun, why not take the chance to have pupils consider ‘how much profit is reasonable, appropriate, worth-while etc?’ This can lead to discussion about the functioning of society and about world and life priorities.

Possible sources of ideas:

Bowland Maths?MIMA‘Maths all Week’. Published by BEAM‘Maths for Social Justice.’ World newsThe InternationalistThe Imaginative Education Research Group http://ierg.ca

Founded by Kieran Egan at Simon Frazer University in Canada

I am not suggesting strict divisions between the three categories or holding on to a strict analogy between the musical and the mathematical. My framework is merely an heuristic device to loosen the more didactic ties in mathematics classes. I think the musical example shows the importance of the variety of educational experience. Davis makes the point that, in the West, we have tended to divorce mathematical competency from our physical selves. He puts the blame on Rene Descartes and the mind/body distinction which he believes has pervaded Western Science for 400 years. The call for more spatial reasoning and embodied mathematics is part of a move to put the learner into mathematical enquiry mode. There are arguments and evidence to support the effectiveness of this approach, but also, gains can be seen in terms of sheer engagement.

The following from Davis is example of the contrast between what could be called a ‘calculation mastery’ approach which emphasises factual recall and algorithmic competence and a second lesson format which is richer in developing conceptual understanding. The second approach involves ‘engaging with the world through posing questions, identifying patterns, expressing observations, varying the questions, contriving explanations, defending interpretations…’

A calculation mastery lesson might require pupils to perform a set of written calculation exercises using the number bonds for 15. The second approach might start with a question: ‘In how many ways can you add two numbers together to get 15?’ This question led pupils to explore a range of different combinations including negative numbers and in asking searching questions about how many such calculations were possible. In fact in the second lesson pupils ended up doing five times more ‘sums’ than those in the first.

Big Ideas in Mathematics and making a start on collecting starting point for development of concept and connections. (from Askew 2016)

The number system

11

Big ideasNumbers have a unique position on the number linePlacing numbers on a number line connects discrete and continuous quantitiesThink about place value in two ways additively or multiplicativelyTwo routes to fractions, measurement and sharingConnections Counting to calculationPlace value in multiplicative terms leads to decimal fractionsFractions, decimals, measuring and division are all connectedSee broken tape activity (how can we use reason to get information when it is not all in front of us?)

Additive reasoning

Big ideas additive and multiplicative thinkingEg rice on the chessboardAddition in any order subtraction different?

See inquiry maths activityAdditive usually situations about more or less

Connections roots of addition and subtraction, change, part-part-whole and compare

eg something changes input outputpart whole using bar diagramscomparing size biggest, smallest is this important?

a multitude of real world problemsfractals and iteration at different scales

Multiplicative reasoning Multiplication in any order but not division

Ration and proportionMother frog and baby frog jums 3 to 1

Arrays inverse

Geometric reasoning

Big ideas2D and 3 D shapesposition and movement

robot activity, spatial awareness seeing from another position

connections:what changes and what stays the same (invariance)similarity and congruencegeometric properties are independent of measurementimagining and visualising

12

classification - sortingvisualisation, construction, reasoningwhat nets will make a cube?Max box, max area etc

Algebraic reasoning

Geoff Giles activitiesHinged squares axa and bxb

MeasuresMeasurement makes continuous quantities countableUsing the same unit (informal and formal – comparison)Estimation, benchmarks to show if reasonable

Smaller units lead to greater accuracyScales lead to multiplicative reasoningPerimeter, area, volumeGiant’s beltTile arrangementsFractal perimetersRipped and tornRecipes, ratioStrong as an ant ant weighs .004g and can lift .2g

Data handling and statisticsBig ideas - data handling cycle, focus on a question is key!

Different types of testSampling and validityIdentify and refine a question or issueDecide what data to collect and how – honesty in collectionCollect dataPresent itInterpret it in the light of the question – honesty in interpretationMeaning of words etc

Same data can be presented in different waysPresenting data for different effects (is fair trade fair?)

SortingFair gameUncertaintyChanceRisk

Word length CAME

13

Recycling etcAnalysing arguments

References:

Askew, M. 2012 ‘Transforming Primary Mathematics: understanding classroom tasks, tools and talk.’ Abingdon: Routledge.

Askew, M. 2016 A practical Guide to Transforming Primary Mathematics: activities and tasks that really work.’ Abingdon: Routledge.

Bird, M. 1991 Mathematics for Young children. London: Routledge.

Boaler, J. 2016 Mathematical Mindsets. San Francisco, CA: Josey-Bass

Davis B 2011 Complexity and maths education Brent DavisPlenary address to Western Conference on Science Education 07/07/2011http://ir.lib.uwo.ca/wcse/WCSEEleven/Thur_July_7/16/Dr. Brent Davis, Werklund research professor at the University of Calgary

Davis, B. and Renert, M. 2014 The Math Teachers Know: Profound Understanding of Emergent mathematics. Abingdon: Routledge

Davis, B. and the Spatial Reasoning Group 2015 Spatial Reasoning in the Early Years. New York: Routledge

Mowat, E. and Davis, B. 2010 ‘Interpreting Embodied Mathematics Using Network Theory: Implications for Mathematics Education’. Complicity: An International Journal of Complexity and Education. Volume 7 (2010), Number 1  • pp. 1-31 •www.complexityandeducation.ca

Papert, S. 1980 Mindstorms: Children, Computers and Powerful Ideas. Brighton: Harvester Press.

Rod Cunningham January 2018

14