DIFFERENTIATION INMATHEMATICS CLASSROOMSMike Ollerton discusses the wider implications of differentiation in themathematics classroom

L earning is messy. It does not occur inneat linear steps; it is a complex process.Sometimes we miss seeing the most obvious

idea, whilst at other times we make intuitive leapsof understanding, often when we least expect it.My favourite example was waking in the middleof the night when studying an Open Universitymathematics degree course and finding I hadworked out the solution to a problem involving themost complex piece of mathematics I had everengaged with then, or since.

In A-level mathematics the concept of difterentiation*i'/5;< is basically about rate of change based uponinfinitesimally small changes to a function; to theslope of a curve. This is a useful analogy to considerwith regard to difterentiated learning; similarlystudents learn at difterent rates according to all kindsof small, often imperceptible changes. This happensboth because, and irrespective of, what teachers do.This is not an attempt to write the teacher out of theequation, far from it; the quality of stimuli teachers'öfter students impacts upon the quality of thinkingand the depth of sense-making difterent studentsachieve.

Difterentiated learning happens; full stop. It'simpossible not to achieve differentiation in theclassroom, no matter what you as a teacher decideto do. Brown (1995, p33). Because difterentiationoccurs irrespective of what we do this means evenwhen asking a closed question such as:"What is 23 plus 197" Although this has a singleanswer, students will think about the answer atdifterent processing speeds and have difterentdegrees of conftdence to öfter their answer. I am notfor a moment advocating the value of asking such aclosed question to a whole class, merely using thisas an example of the inescapability of difterentiation.Of course we can also turn this question around andask more open versions such as

i) Discuss with a partner how many different ways

you can think of finding the answer to 23 + 19, or

ii) "If the answer is 42 what might be the question?"

Indeed when I asked this question to a class of7 to 8 year old learners I was delighted one ofthem answered: "The answer to life, the universeand everything", going on to explain that his Dadwas reading the Hitch Hikers Guide to the Galaxyto him. Returning to the question, there are of

course difterent answers and these, in turn, createopportunities for difterent students to providedifterent answers oftering difterent degrees ofcomplexity.

Furthermore, difterentiation does not happen atsome spurious notion of three difterent levels; ithappens at as many difterent levels of cognitionand depth of sense-making as there are studentsin a class. Difterentiation, therefore, is probably themost complex and important issue for teachers toengage with. It is omnipresent. How we embracedifterentiated learning in our planning-for-teaching,and in our interactions with students are crucialconsiderations.

Because in any class, as suggested above, therewill be as many difterent levels of engagement,confidence, speeds and depths of understandingas there will be students in the class. A fundamentallyimportant issue, therefore, is how to plan for theinevitable difterentiation that exists?

Planning for the inevitability of differentiatedoutcomes

The following quotation from a seminal publicationMathematics from 5 to 16 is unequivocal aboutthe importance of planning: Differentiation ofcontent, if well planned, facilitates progressionfor all pupiis. DES (1985, p26). The quality of thetasks we provide/ofter/set up in terms of their initialaccessibility and potential extendibility cannotbe understated. Seeking to öfter tasks which areintended to provoke active engagement, becausethey are accessible and sufticiently interesting forstudents to engage with in the first instanceis imperative. Planning the difterent depths a taskmight be taken to, whilst recognising that some ofthe more powerful tasks are those which studentsextend themselves, is a further key criterion.As an example I öfter the Sums and Products task,which I first met in an Association of Teachers ofMathematics (ATM) publication. Points of Departure,(book 3, idea 23) which works as follows:

Choose a number and ask each student to write fouror five partitions of that number.

Several students could be asked to provide oneof their answers and these could be written on theboard/screen. For example, using the number 10students might öfter:

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DIFFERENTIATION IN MATHEMATICS CLASSROOMS

5 + 5

2 + 3 + 5

4 + 5 + 1

6 + 4

1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1

This would clearly be accessible to the vast majorityof students more than seven years old.

The next part of the task is to turn each additionsign into a multiplication sign, and for learners tocalculate the resulting products gained, i.e.

5 X 5 = 25

2 x 3 x 5 =30

4 X 5 X 1 = 20

6 X 4 = 24

1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 = 1

The intention now is for students to see the differentrange of products possible. One aspect of teacherquestioning might be to ask students to producethe partitions that gave rise to the answers providedby asking other students. For example, if someoneoffers an answer of 32 the teacher could askstudents to consider what the calculation was. Thevalue of this is to cause students to work in reverse.

The main point of this task is for students to explorewhich partition produces the maximum product -which will be 36, where the calculation will be3 X 3 X 4, or 3 X 2 X 3 X 2. As this result is achievedthe task develops by asking students to choose theirown starting number and repeat the above process,again looking how to maximise the product.

There will be opportunities for the teacher to discusswith students the use of exponents to write some ofthe calculations in mathematical shorthand.

Thus 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 can bewritten as 1̂ °,

likewise 2 x 2 x 2 x 2 x 2 as 2^.

Posing the problem were students look for a way ofbeing able to predict how to create the maximumproducts for any number will be a substantialundertaking, especially as 'the answer'Wes withinthe domain of modular arithmetic and exponents.

Another key quotation from HMI mathematics from5 to16 engages with the intrinsic value of extensiontasks as a way of supporting differentiated learning:Mathematical content needs to be differentiated tomatch the abilities of the pupils, but according tothe principle quoted from the Cockcroft report, thisis achieved at each stage by extensions rather thandeletions. DES (1985, p 26). Thus, with regard to thetask above, different extension tasks can be worked

on, for example:

i) What happens if the rule for partitioning intonon-integer values is allowed.What will the maximum product be under thiscircumstance?For example 2.5 x 2.5 x 2.5 x 2.5 = (2.5)" and(2.5)" = 39.0625; which is clearly bigger than theprevious product using integer values, but is thisthe largest possible answer.

ii) Another avenue to explore might be to considerwhat happens if we only use values using just onepartition for example,5 + 5, 3 + 7, 9 + 1 etc.Here we achieve products of 25, 21, and 9.However, students could be asked to considerwhat the graphs look like when the followingco-ordinate pairs are plotted: (5, 5), (3, 7), and(9, 1 ) - clearly these would sit on the linex + y = 10.

iii) A more complex graph would be to considerjust one of the partitions against the product forexample, (5, 25), (3, 21), (7, 21), (9, 9), and(1, 9). This will produce a negative quadraticgraph which has the equation y = A-(10 - X ) .

There are clearly different extension tasks whichdifferent students can develop to different degreesof complexity; thus possibilities for differentiationabound.

This task is problem-solving by nature. Anotherdescription which has caught hold in the last fewyears is that of a rich mathematical task. This notionhowever was first defined in Better Mathematics(1987, p20) at an inset course by a group ofteachers as:

• musí be accessible to everyone at the start;

• needs to allow for further challenges and beextendable;

• should invite children to make decisions;

• should involve children in speculating, hypothesismaking and testing, proving or explaining,reflecting, interpreting;

• should not restrict pupils from searching in otherdirections;

• should promote discussion and communication

• should encourage originality / invention;

• should encourage 'what if and 'what if not'questions;

• should have an element of surprise;

• should be enjoyable.

Each of these elements describes quality teaching,which, in turn, automatically embraces differentiatedlearning. In the same publication, p15, there is

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DIFFERENTIATION IN MATHEMATICS CLASSROOMS

another interesting, entertaining yet pedagogicallysound list which compares 'Junk Food and 'Junkmathematics']

Differentiation and whole class discussion

The essential issue of difterentiation with regardto whole class discussion is about the depthof understanding any student takes from thediscussion.

Ouestioning strategies which do not require atraditional 'hands-up' approach can be powerful interms of keeping everyone on their toes, and forcreating the expectation/culture that everyone hassomething to öfter. The 'think, pair, share' strategywhich, I believe, is grounded in effective practicehas been used in some primary schools for verymany years and is beginning to be utilised insecondary and in tertiary education. There are otherapproaches such as the use of a random namegenerator, the 'card' trick, the 'two dice' idea. Thecard trick works as follows:

The teacher takes a pack of cards and shows adifferent card to each student as they enter theroom, possibly asking them to write their card in theback of their books or their mini-whiteboards. Theteacher öfters students a stimulus, such as a picture,a graph, a random collection of numbers, a shapeetc then turns over the first card and that student isexpected to say something about the stimulus.

I have seen this 'card' approach used mosteftectively in a Year 8 class; the two dice approachworked superbly in another class where the teacherhad organised the classroom into six work 'stations';students knew their table number and the numberon the first die identified the table number. Eachstudent then had their own number for their tableand this was identified by the number on the seconddie; so 5, 2 was table 5 student 2. Thus the roll ofthe die determined who was going to speak next,but this gave the teacher a great opportunity; havingidentified who is to speak then next to decide whatkind of question to ask. Perhaps stronger studentsmight be asked to justify the previous answer,perhaps another student could be asked to 'giveanother example' of what a previous student hadoffered. I saw both of these strategies used to greateffect in a Year 8 and a Year 7 class, and I felt bothapproaches strongly supported learning.

Whilst both approaches undoubtedly worked wellthis does not mean these, or any other approacheshave to be rigidly adhered to. For example, if duringthe period of time when students are engaging witha task the teacher has interesting exchanges withsome of the students, then an added approach couldbe for the teacher to invite the student(s) to explainto everyone else what they had told their teacher.Again we are working with difterentiated outcomes.

Differentiation and the 'Mantle of theexpert'/'flipped' classroom

I have noticed recently on Twitter the concept of'Mantle of the expert' has begun to appear, though itis a phrase constructed many years ago by DorothyHeathcote (1926 - 2011 ) who was a Drama inEducation teacher. Dorothy worked for many yearswith all kinds of organisations for example, BritishGas, and taught on PGCE and Masters Levelcourses at the University of Newcastle-upon-Tyne. Ihad the good fortune to meet her in 1983. Anyway,"Mantle of the expert' as it suggests is aboutconferring on learners a mantle of expertise. Thismeant that learners engaged in research, discussionand sharing in order to gain the knowledge forwhich they had been designated experts of. Overthe past couple of years, to my knowledge, theconstruct of a 'flipped classroom' has emerged ineducational parlance. This is where learners aregiven responsibility to teach something to the rest oftheir class, having previously been tasked to do soby the teacher. This means the student(s) have toprepare something in advance of 'the flipped lesson';a strategy firmly grounded in 'Mantle of the expert'.In turn, the flipped classroom is just one strategywhich sits comfortably within the wider construct ofenquiry-based learning (EBL).

The 'flipped classroom' has taken on an increasedsignificance over the past two or three years andis in part associated with the increasing amount ofinformation available on the internet, which includessites such as the Khan academy:www.khanacademy.orgOne intention is to develop students' personalqualities, such as independence, and responsibility,for their own learning.

Recently, in September and October 2013, I wasinvited to work with a couple of classes on the topicof percentages. I had no idea what students alreadyknew, or who the least and most confident studentswere, or in fact anything else other than they were12 to13 years old. I decided, therefore, to employ myown version of a flipped classroom, and to do this Igave students the information sheets which appearin Appendix 1. The intention was for students tochoose: with whom they wished to work in a groupof 3, and which problem to work on. I explained thatproblem 1 was the easiest, and that problem 6 wasthe most challenging.

Differentiated learning occurred both in terms of thetask students chose to work on and the outcomesachieved. At one point in the first lesson I noticedtwo of the groups had chosen problem 6, and Imade a professional judgement that both groupswould have benefited from my input with regard todeveloping the task. To achieve this I asked all sixstudents to join me in a space in the classroom and

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asked them if they could think of a way of shorteningtheir calculation of continuously multiplying by 1.1, inorder to achieve each 10% increase. Ouite quicklyone student asked if they could use exponents.I asked the student to explain to the others what hemeant, and it was clear that the groups were able toreturn to their tasks with this enhanced knowledge.The whole intervention took less than two minutes.In Appendix 2 are some further tasks which Iindicated would be tasks for later lessons. Task c) ofAppendix 2 is in fact a subsequent addition, whichI did not offer, but would do if I were to use the ideaagain in the future.

Note: All appendices to this article can be foundonline at www.atm.org.uk/mt240 or by clicking on thered QR code at the end of the article.

Quality planning, quality teaching

Returning to the issue of quality, teachers whoplan for the inevitable differentiation recognisethe impact the stimuli they offer has upon studentlearning. What follows might read like a 'formula' forsuccessful teaching, however, I base my thinkingupon firstly my learning as a consequence of when ateacher has enabled me to shine and, secondly, myteaching when I believe I have enabled students toshine.

The conditions upon which I believe quality teachingis based are:

• Planning which begins with a question such as:"How can I cause my students to get activelyinvolved with and engaged in...?"

• Stimuli, which are initially accessible and easilyextendable by students or the teacher.

• Teaching strategies based upon more open-endedquestions and stimuli where students, perhapsworking in pairs, are given a couple of minutes todiscuss their response to such questions.

• Problem-solving approaches which enablestudents to deepen their thinking.

• Projects which students can develop over timeand see for themselves the progress they make.

• Planning which causes students to go rightback to their basics in order to remind them ofwhat they need to know to move forward. Ofcourse 'basics' is a changeable feast; basics forEYFS students will be about counting, numberrecognition, and shape recognition. For Year 12students some basics will be their knowledgeof sequences, functions and graphs, which willenable them to engage with, and make sense of,calculus.

At the beginning of this piece I used the analogyof mathematical differentiation as a metaphor forlearning at different rates of change. I cannot missthis opportunity to complete the piece by using theidea of integration within calculus, as a metaphor forinclusion. Mathematical integration is about the areaunder a curve; educational inclusion, meanwhile,is about including everyone in a class in learningwhatever concept is under scrutiny. The followingquotation from Bruner (1972, p122) comes to mind:

With respect to making accessible the deepstructure of any given discipline, I think the ruleholds that any subject can be taught to any childat any age in some form that is both honest andpowerful. It is a premise that rests on the fact thatmore complex abstract ideas can in fact be renderedin an intuitive, operational form that comes withinreach of any learner to aid him towards the moreabstract idea yet to be mastered.

Bruner throws down a challenge which I believe werise to by seeking to include everyone.Inclusion is fundamentally about how we engagein our planning-for-teaching and our moment-by-moment interactions inclassrooms, constantlytaking the key issue ofdifferentiationinto account.

Mike Ollerton is a Freelance MathematicsEducation Consultant

References

Ahmed, A, (1987), Better Mathematics: a curriculumdevelopment study, London, HMSO.

ATM (1989) Points of departure, Derby, Associationof Teachers of Mathematics.

Brown, L. (1995) What would it be like not todifferentiate?, Mathematics Teaching 152, Derby,Association of Teachers of Mathematics.

Bruner, J. S. (1972) The relevance of education,Trowbridge: Redwood Press Ltd.

DES (1985) Mathematics from 5 to 16, HMI SeriesCurriculum Matters 3 London, HMSO.

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