A note on designing and implementing richer mathematical tasks

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In this note I want to clarify for myself some aspects of designing and implementing richer mathematical tasks (i.e., tasks which involve more than just applying routine methods). For the purposes of discussion, closed tasks are taken to be those which have a clear goal and a unique answer; open tasks have a clear goal but more than one answer; investigative tasks are those which provide more scope for students to specify their own goals and research directions. These are to be distinguished from exercises which have one solution and a pre-specified solution method.

I have noticed numerous times in the past that many pupils suddenly become very engaged when I pose unusually challenging problems with unspecified and unclear (to them) methods of solution. These have typically been closed problems but they differed from the normal exercises pupils were used to. I have tried these problems on both low- and high-attaining groups and noticed a surprising degree of enthusiasm in both cases. For example, at the end of a lesson on substitution in algebra, I might put up a very complicated looking formula and ask the pupils what value of x would give the answer 4096 when substituted into it. After some initial expressions of puzzlement there will typically be a raised level of excitement and numerous suggestions will be made during animated discussions involving almost everyone in the class. Inevitably, someone will eventually find the correct answer by trial and error after some minutes. I have sometimes seen similar excitement among pupils faced with challenging problems in other mathematics teachers’ lessons.

In these situations, many pupils clearly became excited and intrigued by the more challenging tasks rather than being put off by them, including pupils who were usually difficult to motivate. I was sometimes amazed when solutions would be found much more quickly than I expected, or using interesting or unexpected approaches, or by unexpected pupils. What seemed to be happening was that I was suddenly enabling pupils to let their mathematical creativity and intuition run free over what was terra incognita for them. This contrasted with other parts of the lesson in which I had essentially taken out most of the fun by getting them to follow pre-specified rules in answering routine exercises. They seemed to feel liberated by being thrust into relatively unknown territory and clearly enjoyed it, even competing with each other to try to be the first work out an answer by any means possible. My realisation of this reminded me of a passage in a book by the physicist Leonard Susskind (2008, p.151) in which he ponders whether or not to include more demanding equations in his popular physics books. His publisher advised him not to, warning him that every additional equation would mean ten thousand fewer books sold. Susskind ignored the advice (and his books still became bestsellers), saying: “Frankly, that goes against my experience. People like to be challenged; they just don’t like to be bored.”

This issue is a relevant one because there seems to be an unnecessary divergence in practice between Ofsted’s desire to see pupils being stimulated with mathematically richer tasks in the classroom, and heads of departments’ (HoDs) desire to improve exam grades by ensuring pupils are well trained to answer short, closed, exam-type exercises. Often in practice the emphasis seems to be overwhelmingly on the latter, with one HoD describing his department to me as an ‘exam machine’. Teaching to the examination using routine closed exercises has been criticised by Ofsted (2009), and Ofsted instead praises schools (such as my old school – King Edwards VI Camp Hill School for Boys – see Ofsted, 2011) which provide pupils with more open and investigative tasks that require creative use of mathematics over extended periods. In this note I want to get a better understanding of the nature of different types of mathematical tasks, and the extent to which effective tasks can be designed which can simultaneously achieve both objectives.

Theories and views in previous literature

A large number of books, research articles and case studies have explored different aspects of enriching mathematics learning, and have documented the effects of different strategies on pupils. Hewitt (2002) introduced a useful distinction between arbitrary ideas in mathematics (those belonging to the realm of memorising things) and necessary ideas (those belonging to the realm of awareness). All students need to be informed of arbitrary ideas by someone else (e.g., that there are 360 degrees in a whole turn) but some students can then become ‘aware’ of necessary ideas by themselves (e.g., how many degrees there are in quarter-turns, half-turns, two-thirds turns, and so on). Hewitt argues that it is important for teachers to try to foster this ‘awareness’ rather than spoon-feeding necessary ideas as if they were arbitrary ones. This insight also has an important bearing on the design of mathematical tasks because it suggests that tasks involving more arbitrary ideas will involve more non-mathematical ‘memorising’ work by students. It would seem preferable in the limited amount of time available in maths lessons for pupils to be given tasks which allow them as much time as possible for exploring and finding out necessary ideas by themselves. It might be useful to classify mathematical tasks in terms of the extent to which they involve arbitrary vs necessary ideas, with those minimising the former and maximising the latter being preferred.

Another important reading for the purposes of understanding and classifying different types of mathematical tasks is Skovsmose’s (2002) Landscapes of investigation, which is the term he uses for learning environments which can support investigative work, such as project work. He contrasts this investigative paradigm with the traditional exercise paradigm of the typical maths lesson which involves closed problems to be solved by pre-specified methods. He also introduces a distinction in mathematical problems between references to pure mathematics, references to a semi-reality (a ‘virtual reality’ invented for the purposes of an exercise), and real-life references. He then defines six learning milieus in terms of the possible combinations of the two paradigms and three possible degrees of reference to ‘reality’, noting that the bulk of current mathematics education involves switching between pure maths and semi-real contexts within the exercise paradigm. Attempting to redress the balance are efforts like NRICH (http://nrich.maths.org) and Bowland maths (http://www.bowlandmaths.org.uk/), which now provide a wealth of investigative activities in pure maths, semi-real and real-life contexts. The matrix of learning milieus provides a useful classification scheme for mathematical tasks and also a useful analytical tool with which to assess the mix of activities in one’s teaching. Skovsmose advocates a type of mathematics education which moves between the six different milieus as appropriate, sometimes focusing on deep investigative work, sometimes on consolidation work using exercises. Although highly stimulating, Skovsmose’s ideas do not help much with the question of how to combine exam preparation with richer mathematical activities. Exam preparation would presumably fall under the exercise paradigm. Moreover, in trying to apply Skovsmose’s scheme to real-life situations, it could be argued that the exercise vs investigative distinction is too coarse, and that there are types of activity which lie between these two extremes. In particular, there are closed problems with unspecified solution methods which are not really routine exercises, and there are open problems with many possible answers which are nevertheless not quite landscapes of investigation. It might be useful to envisage a slightly more flexible version of Skovsmose’s classification scheme which includes exercises, closed problems, open problems and landscapes of investigation. When combined with Skovsmose’s three references to reality this gives twelve learning milieus.

Another useful classification scheme involves a three-dimensional cube (reproduced in Figure 1 below) as a model for integrating three dimensions of applications of mathematics: the context within which mathematics is to be used; the mathematical processes that are to be used; and the mathematical content (concepts, facts and techniques) that is to be used.

Fig1
Figure 1. Three dimensions of applying mathematics (Westwell and Ward-Penny, 2011, p. 26)

The model was introduced by Westwell and Ward-Penny (2011, p.26) to analyse how the three dimensions of any use or application of mathematics are either treated separately or embedded in various versions of the national curriculum. For example, they pointed out that a major change in the 2008 mathematics NC was a separation of the second dimension (processes) from the third dimension (mathematical content) in order to emphasise the need to help pupils develop a wider set of thinking skills as well as their functional mathematics abilities. However, the same model can be used for the purpose of understanding the nature of different types of mathematical tasks. Indeed, the context dimension is recognisable as the reference dimension in Skovsmose’s matrix of learning milieus. The other dimensions allow for a refinement of the classification of mathematical tasks according to the processes and mathematical content involved, so together with Hewitt’s and Skovsmose’s contributions there are now five criteria for classifying tasks.

The different ways in which tasks can be presented is also an important aspect of mathematical task design that affords yet another ‘degree of freedom’ in thinking about how to vary and enrich the learning experience of pupils. This is explored by Mason and Johnston-Wilder (2006) who provide a book-length treatment of issues surrounding the design and use of mathematical tasks. In Chapter 1 of their book they introduce the term dimensions-of-possible-variation to describe the scope for varying the numbers and other features in a mathematical task without altering the underlying structure. The authors say: “Altering the numbers in a task is the most obvious of several dimensions-of-possible-variation that transform a task from a single exercise into a class of problems or a ‘problem type’. Learners make progress when they become aware that what matters about a task is the method and the thinking involved, rather than the specific numbers. When they begin to think about a problem type, they are starting to think mathematically about tasks as well as within tasks.” (The term metacognition is often used with regard to the last point). However, what makes the concept of dimensions-of-possible-variation particularly useful is that it can be applied not only to tasks themselves, but also to ways of presenting tasks. This adds another dimension to the ways in which mathematical tasks can be varied to make them more interesting and challenging. To illustrate, in Chapter 3 of the book the authors explore various dimensions-of-possible-variation in the presentation of a classic task known as arithmogons. This is a versatile activity that consists of a network of connected circles and squares and only a single rule to remember: the number in each square must be the sum of the numbers in the circles on either side of it. Various arrangements can be used, as illustrated in Figure 2 below.

Fig2
Figure 2. Arithmogons (Mason and Johnston-Wilder, 2006, p. 44)

The authors discussed in detail a number of different ways of presenting this single task, including:

– doing and undoing calculations, as in Figure 2;
– starting from a non-school context to explain the activity (e.g., the authors suggested a story about an archaeological dig during which damaged clay tablets were discovered which had to be reconstructed using arithmogon-like techniques);
– distributed working and pooling resources;
– starting with a hard or complex version of an arithmogon, and encouraging pupils to make up simpler versions in order to ‘see how they work’ before tackling the more difficult one;
– asking pupils to say what is the same and what is different about two or three examples of arithmogons (a metacognition activity);
– starting in silence, i.e., the teacher completes a few examples of arithmogons in silence on the board, and then asks pupils to come up and offer similar examples;

These are all based on the same basic task, but the different ways of presenting it can provide different types of mathematical learning experiences for pupils. The concept of dimensions-of-possible-variation in the presentation of tasks is something to be borne in mind when assessing and classifying mathematical tasks, in addition to the five classification criteria identified earlier from Hewitt’s and Skovsmose’s insights and from the three-dimensional cube in Figure 1. Mason and Johnston-Wilder caution, however, that the degree of task variety that is provided in the classroom needs to be judged carefully. Too little variety in mathematical tasks can produce dependency (i.e., pupils find it difficult to deal with situations which are slightly different from the ‘norm’), but too much variety can be confusing and destabalizing. As Skovsmose (2002) indicates in his article, there needs to be an ongoing dialogue and negotiation in the classroom between teacher and pupils to decide what mix of activity types is most appropriate.

Yet another aspect of mathematical task design that needs to be considered, and that can be difficult to get right, is the appropriate level of difficulty of the tasks. Useful guidance about this is provided in the context of level differentiation by QIA (2007, p. 25). The article outlines four dimensions of mathematical tasks which determine their difficulty:

– complexity of the situation (e.g., number of variables, the variety and amount of data, the way in which the situation is presented, etc.);
– familiarity of the situation to the pupil (non-routine tasks are more demanding for pupils than routine activities they are familiar with);
– technical demand of the maths required to solve the problem (tasks which involve more sophisticated mathematics are more demanding for pupils than those which require only elementary mathematics);
– the extent to which the pupil is expected to tackle the problem independently (guidance from the teacher or from the structuring of a task into successive parts will make the task easier than if no such guidance is provided).

This a useful conceptual framework for mathematical task design, particularly for strategies involving modifying routine exercises to make them mathematically richer, while at the same time trying to avoid over-burdening pupils. For example, if a task is made more complex and non-routine and pupils are expected to work on it largely autonomously, it might be necessary to make the technical demands of the mathematics easier, to compensate for the other dimensions of difficulty. This framework thus provides a systematic way to adjust levels of difficulty, akin to turning four dials to ‘fine tune’ the level of difficulty appropriately. The level of difficulty provides a seventh criterion with which to classify different types of mathematical tasks.

A paper by Prestage and Perks (2007) is directly relevant to the question of how to combine exam preparation with more open and exploratory type work. It describes an approach for adapting and extending uninspiring tasks, such as short, closed exam-type questions, to make them mathematically richer. The approach involves:

(a) identifying the ‘givens’ in the task;
(b) changing, adding or removing a given;
(c) analysing the resulting maths including the choices for pupils and teachers;
(d) based on these explorations, choosing tasks which are appropriate for the classroom.

For example, Figure 3 below shows a typical question for work on algebraic substitution, with a range of suggestions for alternative tasks based on altering the ‘givens’. Prestage and Perks say: “Changing and removing givens reveals a wealth of different approaches to the mathematics and the potential for different but linked mathematics. The more you adapt and extend the more the different parts of the curriculum emerge…not as separate tasks but as continuous possibilities.” This approach seems promising as a technique for relatively quickly generating more interesting tasks for pupils as extensions of routine exam-type exercises, thus simultaneously satisfying the need both for exam preparation and for enriched mathematical learning.

Fig3
Figure 3. Generating alternative tasks (Prestage and Perks, 2007, p. 386)

Although not discussed by Prestage and Perks, one can even envisage asking pupils themselves to generate alternative mathematical tasks using an approach like the one in Figure 3, thus making the exploratory experience student-led rather than teacher-led. Something similar to this had in fact been suggested already by Watson and Mason (2000). They advocated asking students to generate their own examples of such things as linear equations, alternative notations, and even their own problems for assessment purposes. In a powerful concluding paragraph, Watson and Mason say: “We believe that all students come to class with immense powers to construe, and that it is vital that the teacher uses tasks which call upon students to use those powers. Otherwise students may become trained in dependency on the teacher or text to do the thinking, the generalising and the particularising. Having to generate examples as part of learning about concepts is one way to avoid such dependency.”

Another paper in this vein by Cai and Brook (2006) also seems particularly helpful in terms of generating richer tasks for pupils. As well as advocating enhancing mathematical learning by asking students to pose new problems for themselves and to make generalisations, the paper also recommends asking students to generate, analyse and compare alternative solution approaches to problems. The authors say: “Regardless of whether there is individual or group effort, it is important for teachers to guide students to reflect and compare various solutions because the comparison helps students recognise similarities and differences between solutions and enhances students’ understanding.” This strategy is particularly appealing as a way of converting routine exercises or exam questions into more exploratory activities because the exercises do not even have to be modified. Simply asking students to find alternative methods of solution and to compare and contrast them can transform a routine exercise into a much more interesting task that is somewhat like the kind of activity undertaken by mathematical researchers, who often seek alternative proofs for theorems. I have found this strategy particularly useful in advanced work with high-attaining students, as indicated in my previous blog posts.

The use of the term ‘richer’ when referring to mathematical tasks is being used broadly in this note to mean tasks which involve more than just routine applications of pre-specified methods. Whether they are closed, open or investigative, such richer tasks can be much more challenging and stimulating for pupils than routine drill exercises. However, attempts have been made in the literature to characterise the term rich task more narrowly, and more along the lines of what are being referred to as investigative tasks in the present note. A particularly useful article in this regard by Ahmed (1987, p. 20) suggested that the following should be regarded as the characteristic features of a ‘rich task’:

– it must be accessible to everyone at the start;
– it needs to allow further challenges and be extendible;
– it should invite children to make decisions;
– it should involve children in: speculating, hypothesis making and testing, proving or explaining, reflecting, interpreting;
– it should not restrict children from searching in other directions;
– it should promote discussion and communication;
– it should encourage originality/invention;
– it should encourage ‘what if?’ and ‘what if not?’ questions;
– it should have an element of surprise;
– it should be enjoyable.

This is a useful list of criteria with which to assess the extent to which particular mathematical tasks are ‘rich’. Similar lists of criteria for identifying rich tasks have been produced by others, for example Jenny Piggot, ex-director of NRICH (see McDonald and Watson, 2010, p. 4).

Other types of tasks, particularly closed tasks, should not be undervalued as vehicles for enriching mathematical learning, however. It can be argued that it is overly simplistic to regard open tasks and rich tasks as somehow ‘better’ than closed tasks. As indicated at the start of this note, closed tasks can also offer opportunities for richer mathematical experiences. There is some support for this view in the literature. For example, Foster (2011) argues in favour of using what he calls ‘closed but provocative’ questions to generate richer mathematical experiences in the classroom. These are closed questions which give rise to unexpected (or otherwise interesting) results, and thereby invite further investigation and questioning. He gives an example of such a closed question in the context of curves which happen to enclose a unit area, something which he says tends to surprise students and which then leads to further discussion and experimentation. The closed questions which I have experimented with myself were made more interesting by having unspecified and initially unclear solution methods, or by asking pupils to find alternative solution methods which were then compared and contrasted.

An interesting discussion in Foster (2013) on resisting pressures for reductionism in mathematics pedagogy helps to clarify a lot of the ideas discussed so far about enriching mathematics learning by putting them into the context of reductionist vs holistic teaching approaches. Reductionism is an approach to problem solving which involves breaking problems down into smaller and more manageable parts. Holism is an opposing idea that suggests some problems cannot be solved by looking at the constituent parts separately because their interactions are what are important, not the parts themselves. Foster argues that mathematics pedagogy itself is becoming increasingly reductionist, and that while applying a reductionist approach to mathematical problems is often productive, applying the same approach to teaching mathematics can be bad for students. He recommends a more holistic approach to teaching maths, involving “genuine and substantial mathematical activities, which bring into play general mathematical strategies such as abstracting, representing, symbolizing, generalizing, proving, and formulating new questions.” (Foster, 2013, p. 577). The use of richer closed, open and investigative tasks can be regarded in the context of Foster’s article as a more holistic approach to teaching maths, and the enthusiastic response of many students to the challenging tasks discussed in the introduction can be interpreted as a response to this more holistic invitation to use their ingenuity and mathematical skills. Foster’s ideas also very much echo views expressed by Barnes (2002, p. 96) on how best to provide pupils with ‘magical moments’ of mathematical insight and discovery. Barnes says: “We might surmise, however, that ‘magical’ moments would be less likely to occur in expository teaching, where work on problems is preceded by carefully structured preparatory explanations and guided practice. If instruction progresses by small, simple steps, and the teacher anticipates difficulties and provides immediate clarification, students will have less need to struggle and less occasion to make efforts of their own to achieve understanding and insight.”

A number of articles have discussed issues relating to task design in the context of degree level mathematics, and it is interesting to observe that similar concerns arise as in secondary schools regarding the inadequacy of commonly used mathematical task strategies. For example, Breen and O’Shea (2010, 2011) survey the literature on the types of tasks assigned to university mathematics students and find that the majority require only imitative reasoning (i.e., reasoning requiring only memorisation or the use of well-rehearsed procedures) as opposed to creative reasoning (i.e., thoughtful novel reasoning backed up by suitable arguments from appropriate mathematical foundations). They express a familiar concern that “students have knowledge but are not in a position to use it in unfamiliar situations”, and recommend that lecturers should “scatter throughout a course a considerable number of problems for students to solve without first seeing very similar worked examples.”

A number of articles bring to light some of the challenges that teachers can face when trying to implement richer mathematical activities in the classroom. These challenges can come from both the school system and from pupils themselves who have often been ‘trained’ for years to think of school maths as having to be done in a particular way and feel uncomfortable when attempts are made to expose them to richer mathematical experiences. In arguing for more holistic approaches to mathematics teaching, Foster (2013) attributes the rise of reductionist teaching approaches to a combination of assessment-driven and accountability-driven cultures in schools, which have the effect of de-professionalizing teachers and forcing them to ‘teach to the examination’ in bite-sized chunks. Foster says: “The senior teacher who ‘pops in’ might have time only for a 15-minute visit, in which they will make judgements that can have serious consequences for the teacher. So an understandable defensive strategy for the teacher against these intrusions is to break up the lesson into episodes of no longer than 15 minutes, during each of which some superficial public student assessment takes place, which no observer can fail to miss, and which highlights what students have achieved during this period.” These pressures often make it seem too risky to implement recommendations from mathematics education research involving richer and less fragmented classroom activities. Similar pressures are revealed in an article by Ward-Penny (2010), who was challenged by a colleague to justify his use of an ‘unorthodox’ classroom activity involving the construction of a geodesic dome using cocktail sticks and soft sweets. Ward-Penny says: “At first I tried to bluster my way to an answer — different types of triangles are on the syllabus, aren’t they? The emphasis on processes in the new edition of the national curriculum might also serve as a cover-all — after all, aren’t the students exploring and justifying why the geodesic dome cannot be constructed solely with flat, equilateral triangles? If that attempt at hand-waving failed to convince, I am sure I could justify the activity to any passing inspector by uttering the magic term ‘cross-curricular’. Upon reflection, though, I decided that there is another, and far better reason for using this activity: it is both mathematical, and fun.” In an interesting article reflecting on her experiences of teaching mathematics in an exam-driven all-girls school, Hagan (2005) tells of similar challenges, but also highlights how pupils themselves can make it difficult to implement richer and more interesting mathematical activities. She says: “I believe that these students wanted to be taught quick, ‘easy to follow’ rules and procedures because success to them is not about understanding what they are doing in mathematics, but about knowing how to do a particular type of question in a particular way so that the marker will put that magical ‘A’ (for Achieved) in the space on the right hand side of the examination paper. Any grand ideas that I had of making their mathematics a connected and relevant experience were greeted with a total lack of enthusiasm from the students. What I had to remember was that some of these girls had been indoctrinated in the culture of the school so by the time I became their mathematics teacher I had many years of examination focus and pressure to battle against.”

Conclusions

Based on the above literature I feel that seven dimensions of variation in mathematical tasks come across as particularly important:

– the balance of arbitary versus necessary ideas (Hewitt, 2002);
– the overall task paradigm: exercise, closed, open, or landscape of investigation (Skovsmose, 2002);
– the degree of reference to reality: pure mathematics, semi-reality, real-life (Skovsmose, 2002);
– mathematical content: number, algebra, geometry and measures, handling data (Westwell and Ward-Penny, 2011);
– mathematical processes: making decisions, communicating, reasoning (Westwell and Ward-Penny, 2011);
– mode of presentation (Mason and Johnston-Wilder, 2004);
– level of difficulty: complexity, familiarity of the situation, technical demands of the mathematics involved, degree of teacher support (QIA, 2007).

I have also gained a fuller appreciation of how useful closed tasks can be for generating rich mathematical learning experiences. Foster (2011) pointed out that there is a current tendency to regard closed tasks as somehow inferior to open and investigative ones in this regard. I agree with him that this is an overly simplistic attitude. I am particularly interested in the potential usefulness of closed questions because modifying routine closed exam-type exercises to make them mathematically richer and more interesting seems to be the best hope for simultaneously achieving the objectives of preparing students for exams and enriching their mathematical learning. HoDs, as well as parents, want pupils to be prepared for exams and this reality has to be accommodated somehow. Papers by Prestage and Perks (2007), Watson and Mason (2000) and Cai and Brook (2006) came across as particularly useful in terms of suggesting realistic avenues for extending the utility of closed tasks. In the past I have managed to use closed tasks to great effect to enrich my pupils’ learning experiences, particularly in the case of high-attaining students with whom I explored alternative solution approaches for advanced maths exam questions along the lines recommended by Cai and Brook (2006). The students thereby obtained greater familiarity with the exam questions they would be facing as well as richer mathematical experiences which will stand them in good stead in the future.

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