The research aims to identify trends to guide the selection of extension activities that produce significant benefit whilst minimising negative impact on the motivation of less able members of the class. In the first part of the study, two extension activities were carried out. For each activity, exercises were marked to give quantitative data on the students’ progress and questionnaires were completed by the students both before and after the activities to gauge the impact on the group’s attitudes to learning. The second part of the study sought to gain a greater understanding of the motivation of the students undertaking extension tasks by giving free-choice between three activities, with the students completing a questionnaire afterwards to give their perception of the activities and the impact of them on their wider enjoyment of mathematics and confidence in the subject. It is concluded that the perception of both extension activities and mathematics as a whole appears to be based upon the students’ wider impressions, rather than easily influenced by individual sessions.
Exceptional ability students are necessarily taught with less able peers, even within generally high-ability groups. The choice of extension activities must be made carefully if the potential of the most able is to be realised without creating a negative experience for the less able.
In a class with a significant fraction of exceptionally able students (over 25% receiving Mathematical Olympiad medals in the current year), there is a clear need to develop beyond the typical Year 9 scheme of work if these students are to maintain their interest in mathematics and fulfil their potential. However, not all students in the set fall within this category and experience informs that some extension activities and methods might not be to everyone’s benefit. The aim is to foster the perception that extension is interesting, although may be challenging as an activity that is so difficult as to be demoralising is not at an appropriate level.
As the school has an additional intake at Year 9 from a number of local preps, we must be aware that the students have a variety of previous learning experiences. The recent conversion of the 11-16 section of the school to co-education has also resulted in 15% girls within the class, somewhat below that of the wider year-group. Whilst it would be difficult to consider this small group separately with any statistical certainty, we should remain sensitive to any apparent disparities between the reactions of this group and those of the class as a whole.
There has been extensive debate concerning the best approach for students showing a high-ability in mathematics, and the concept of ability itself remains controversial. We are not seeking to enter into that latter argument here, but instead accept that, at any given point, some members of the group will engage more quickly with tasks and more easily progress to higher levels of learning. If we take Bloom’s taxonomy as a structure for these levels (Krathwohl, 2002), then previous CamSTAR research by Smith describes well an experience with which many teachers will be familiar when seeking to encourage pupils to move beyond merely applying rules to greater analysis and creativity (Smith, 2012).
The “acceleration versus extension” debate surrounding the best approach to encouraging development in mathematics has also proved divisive, although a useful review of the possible virtues and pitfalls associated with both approaches was published by White et al. under the Local Government Association educational research programme (White et al, 2003). Recent publication of a comparison of the mathematics skills of school children in England and East Asia (Jerrim and Choi, 2013), does also suggest that a certain amount of what we consider to be acceleration might constitute a “normal” pace elsewhere. Hence a balance between moderate acceleration within the national context, and a programme of extension activities, could provide a stronger approach than either acceleration or extension in isolation. Although this research considers only classroom-based extension, an interesting complementary study of out-of-school programmes for this age- group has recently been made by Feng (2010).
The research question
In seeking to answer our main research question,
“What characterises a good extension activity for a high-ability year 9 mathematics set?”, we need to expand further on the meaning of “good” within this context. For the purposes of this research, the following aspects have been considered:
- What is an appropriate level of extension activity?
- How is the group’s confidence affected by extension?
- Which methods of delivery maximise positive impact?
In the first section of the research, two different extension activities were chosen in topics where the group had no significant prior knowledge but was able to progress from a basis of established skills and techniques through a number of levels of increasing complexity. The learning outcomes were assessed quantitatively in order that the students’ progress could be compared with a benchmark data set of the results from recent year-wide tests. For both activities, qualitative survey data was also used to monitor pupil perception of the extension tasks and their impact on attitudes to learning, with questionnaires being completed before and after the activities in order that changes could be mapped accurately.
The first activity was teacher-led, aiming to develop the students’ skills in sketching graphs of quadratic functions, starting with the application of familiar techniques to find intercepts and developing the method of completing-the-square to locate the turning points before finally making a sketch of the quadratics. Exercises to practice completing the square and sketching curves were then completed individually in class, with a further set for homework. The homework exercises were taken in and marked to provide quantitative data about performance in this activity.
In the second activity, the pupils worked through a guided booklet studying the gradients of curves and their tangents, leading to some basic ideas connected with the derivative of a function. The activity was presented as an investigation, and collaborative work with peers was encouraged. Two teachers also circulated throughout the activity to provide assistance and further discussion of the topic. The investigation booklet concluded with a page of exercises based on the material introduced through the earlier narrative. The marks from these exercises provide the quantitative assessment of the students’ progress in this activity.
A second section of research then sought to explore the students’ motivation within extension work and the effects of granting autonomy. The students were offered free-choice of three extension activities, which were carried out in the next double lesson:
- Learning about permutations and combinations, taught in a small class group;
- A computer-based lesson on sequences and series using myMaths lessons and tasks;
- Investigating complex numbers and fractals using a worksheet and spread sheet from the Royal
Institution Maths Masterclasses.
The three topics had been chosen to be of approximately equal demand, but with different methods of delivery. No quantitative assessment was made of the pupils’ progress with the activities but, in the following lesson, all were asked to complete a questionnaire to look at the motivation of their choice and reaction to the activity.
The first activity, curve sketching, appears to have had little effect on the attitude of the class towards mathematics, or their perception of whole-class extension activities. The questionnaire that the students completed asked them to state on a scale of 1-4 how happy they felt in advance of, and during, maths lessons; the results are shown in Figure 1a for the surveys conducted before and after the activity. Figure 1b shows the variation in perception of whole-class extension work, also on a 4 point scale, ranging from “very easy” to “very hard”. The scores in the marked exercise are shown in Figure 1c, plotted against the students’ results from recent year-wide progress tests. Here little correlation is noted, with a large number of very high marks in the extension exercise, but with a not-insignificant few achieving a relatively poor result.
Figure 1a. Showing the class’ attitude towards maths lessons before and after the first intervention, with happiness with mathematics marked on a four-point scale from “unhappy” to “very happy”.
Figure 1b. Showing the class’ perception of extension work before and after the first intervention, with the difficulty of the work marked on a four-point scale from “very easy” to “very hard”.
Figure 1c. Showing the marks from the assessed exercise of the first intervention plotted against the students’ recent scores in year-wide tests.
The second activity produced a similarly small shift in the attitude of the students towards mathematics and their perception of extension work. The differences seen in Figure 2a are due to a small number of absences on the day of the survey rather than a significant alteration in the overall shape of the distribution of the responses, with a similar pattern also seen in Figure 2b.
Figure 2a. Showing the class’ attitude towards maths lessons before and after the second intervention, with happiness with mathematics marked on a four-point scale from “unhappy” to “very happy”.
Figure 2b. Showing the class’ perception of extension work before and after the second intervention, with the difficulty of the work marked on a four-point scale from “very easy” to “very hard”.
Result of Year 9 Tests (%)
Figure 2c. Showing the marks from the marked exercise at the end of the second intervention activity plotted against the students’ recent scores in year-wide tests.
Figure 2c is rather more revealing than might be expected from the students’ relatively unchanged perception of extension work. The marked exercise formed the final section of the booklet that the students had been using to guide them through their investigations. The students were told that this section of the work would be collected and checked, but that the marks would not “count” as feedback to them would be in terms of comments rather than a score. Some students, who had made less progress during the activity, were hence content to hand in fairly incomplete work and we see a significant scatter of marks as a result.
As the second section of the research projects involved looking at the students’ motivations when faced with a free-choice of activity, it is first interesting to note the numbers selecting each option:
Table 1. Uptake of the three free-choice activities.
The students were asked to select the most important factor in their choice of activity from a list of five options. The responses are shown in Table 2, separated by activity, with the numbering following from that given in Table 1.
|Most important factor in choice of activity||Activity 1||Activity 2||Activity 3|
|Perceived level of challenge||3||1||2|
|Medium of delivery||0||2||1|
|Topic of the activity||0||3||7|
|Opportunity to work with others||0||0||1|
|Choices of peers||0||5||0|
Table 2. Showing the factor deemed by the pupils to be the most important in their choice of activity, subdivided to show variation by activity chosen.
Finally the students were asked to say whether the activity they selected has affected their confidence in mathematics. The responses can be summarised:
|Confidence Increased||Confidence Unaffected||Confidence Decreased|
Table 3. Summary of responses to being asked to state how the activity has affected the individual’s confidence in mathematics.
Returning to the first activity, and Figure 1, it appears that on the whole the uptake of the material was good, with most pupils finding it accessible. There is no significant change in attitude caused by the activity, although a small shift is seen towards finding extension work easily approachable. Anecdotal evidence suggested that that not a few of the low marks in the quantitative section were caused by students having disengaged from putting their usual effort into the work due to close proximity to the Christmas holidays and the fact that the work was acknowledged to be extension. It is hence concluded that the topic was at an appropriate level but that alterations to its delivery might have helped to keep the whole class fully engaged. In part, this outcome led to the use of an “investigation” style for the second activity, and the development of the second section of the project to consider the students’ motivation in more detail.
The second activity, as an investigation, allowed the pupils much more scope to proceed at the pace they desired. Some highly-motivated and able pupils completed the work within the allocated double and were set further extension tasks to consider by the members of staff circulating around the group. Notably, some other pupils found the investigation very challenging and asked for a reasonable amount of assistance from the staff: these pupils might have found greater benefit had the activity been more teacher-led as they struggled to progress to the higher levels of learning encouraged by the investigation format of the activity. A third group made very slow progress, spending a disproportionate amount of time on the early stages of the investigation, and so spent little time attempting the marked questions. It was this group that therefore produced some of the lowest scores, appearing to lack the motivation to work through an multi-stage task requiring a variety of skills for a reasonably extended period.
The survey of attitudes does not necessarily reflect the diversity of responses seen as there was once again little alteration in the students’ reported perceptions and attitudes after the extension class. The level of this activity was higher than the initial one but the students were not given the impression that they were being tested on the material (work was returned with comments but not numerical marks). This removed the potential demotivating factor of performing badly, but simultaneously also acted to lower the motivation of some of the students to complete the activity.
From the first two activities, it is clear that student motivation has a very direct impact on their progress, although not necessarily on their perception of the task. The second section of the project was therefore developed to explore further their choices and reasoning. By giving the option of three activities of equal difficulty but different methods of delivery, it is hoped to gain an understanding of the role choice plays in motivating the pupils as well as seeing the effects of the choices being made by their peers.
Although no assessment was made of the students’ performance in the tasks, it was noted that most of the weaker students, who typically struggle the most to maintain focus, chose the computer-based activity; the RI Masterclass activity appealed to the majority of the rest of the class, including all the girls, who seemed to very much enjoy working together as a group. It was somewhat surprising that just three students chose the small-group taught class, but may be worth noting that this group included the two top-scorers from this year’s Olympiad. It would be interesting to explore further whether this preference for teacher-led activities was repeated amongst those with high marks on other occasions, even if the perceived level of challenge was their nominal motivation. The majority of the students were content with their choice of activity although a number of those choosing the Masterclass noted that they found the topic contained some difficult concepts.
These observations should also be placed in the context of the reasons given by the students as having been most significant in their choices. The choices of peers played a major role in influencing the group choosing the computer-based activity and the data suggests that one or two boys, keen to use the computer, took a large group of friends with them.
The Masterclass group show a strong motivation by the topic of the activity which hints at a basic interest in aspects of the subject that can dominate over other factors. Perhaps it is this same interest that caused the perception and attitude data to be little influenced by the first two interventions and betrays a valuable enthusiasm within a significant section of the class that deserves to be nurtured, regardless of whether the individuals concerned are currently the very top performers. It is to be hoped that allowing the free selection of activities, granting the students some autonomy in their learning, did help to promote this goal. Interestingly, over a quarter of the class felt that their chosen activity has benefited them with improved confidence in mathematics which suggests that individual extension activities produce small changes in perception, which are not easily detected on a coarse four-point scale, but have a cumulative effect on attitude to learning. It would be interesting to develop this research further by following these small changes over a longer period of time as further extension work is undertaken.
Summary and development
In summary, the students’ perceptions of mathematics seem remarkably robust to the experience of individual extension activities, instead reflecting the totality of their experiences of extension to date. Inevitably perceptions and attitudes are influenced by many external factors and the students’ wider background so the time scale of this research is perhaps rather too short to see any significant impact from the extension programme as a whole. It might be more useful to look at the on-going progress of the set, for instance in terms of the number going on to A-Level Further Mathematics and beyond, although disentangling the role of Year 9 extension in this would be very challenging!
In terms of the specific activities, the first, curve-sketching, was well-suited to that stage of the year and could easily be tweaked to improve the engagement of some members of the class. To some extent, we might need to take a longer term view of the success of the second activity, introducing the derivative, as any benefits of this groundwork may only become visible in assessment outcomes when calculus is returned to formally.
The work to understand the students’ motivations confirmed a good number of ideas that on-going observation of the group had suggested and some of the peer pressures that exist within it. Questionnaires were used rather than a focus group in order to keep the quantity of data manageable, but perhaps the quality of the feedback suffered as a consequence. Interviews with a focus group would certainly have allowed for additional perspective to be gained in more complex areas such as how the motivation and perspective of the girls varies from that of the boys and would be a useful way to develop this research further, along with the other possible areas for development suggested previously.
Feng, W. Y. (2010) ‘Students’ Experience of Mathematics Enrichment’, Proceedings of the British Congress for Mathematics Education, vol 30, pp. 81-88.
Krathwohl, D. (2002) ‘A Revision of Bloom’s Taxonomy: An Overview’, Theory into Practice, vol. 41, no.4, pp. 212-218.
Smith, S. R. (2012) Developing the problem solving and thinking skills of gifted and talented pupils in mathematics by use of Blooms Taxonomy, http://www.camstar.org.uk/wp- content/uploads/2012/12/Problem-solving-in-Maths-Rob-Smith.docx
White, K., Fletcher-Campell, F., Ridley, K. (2003) What Works for Gifted and Talented Pupils: A Review of Recent Research (LGA Research Report 51) Slough: NFER.
Jerrim and Choi, (2013) The Mathematics Skills of School Children: How Does England Compare to the High-performing East Asian Jurisdictions?, http://repec.ioe.ac.uk/REPEc/pdf/qsswp1303.pdf