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Healy
L., Hoyles
C. (1998) Research Report |
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Proof is at the heart of mathematical thinking, and deductive reasoning, which underpins the process of proving, exemplifies the distinction between mathematics and the empirical sciences. Developing the ability to recognise and construct chains of logical argument based upon agreed rules and procedures is fundamentally important for those aiming for careers that rely upon mathematical literacy. The process of building a valid proof is clearly a complex one: it involves sorting out what is given -- the mathematical properties that are already known or can be assumed -- from what is to be deduced, and then organising the transformations necessary to infer the second set of properties from the first into a coherent and complete sequence. Research in mathematics has consistently highlighted students' difficulties in engaging with formally-presented, analytical arguments and understanding how these differ from empirical evidence. The current National Curriculum for mathematics prescribes an approach to proving, maybe as a response to these student difficulties, in which the introduction of formal proofs is reserved for 'exceptional performance', and thus delayed until after students have progressed through early stages of reasoning empirically and explaining their conjectures. Most of the requirements to explain and justify take place within investigations driven by numerical data, as part of the attainment target, Using and Applying Mathematics. The project, Justifying and Proving in School Mathematics (1) , started in November 1995 with the aim to examine the impact of the National Curriculum on high-attaining Year 10 students' views of and competencies in mathematical proof. In particular, it set out to: - describe the characteristics of mathematical justification and proof recognised by high-attaining Year 10 students; Two questionnaires were designed, piloted and refined, a student proof questionnaire and a school questionnaire. The proof questionnaire comprised a question to ascertain a student's views on the role of proof, followed by items in two domains of mathematics -- arithmetic/algebra and geometry -- presented in open and multiple-choice formats. In the former format, students were asked to construct one familiar and one unfamiliar proof in each domain. In the latter format, students were required to choose from a range of arguments in support of or refuting a conjecture in accordance with two criteria: which argument would be nearest to their own approach if asked to prove the given statement, and which did they believe would receive the best mark. The school questionnaire was designed to obtain data about a school and about the mathematics teacher of the class selected to complete the proof questionnaire. These teachers also completed all the multiple-choice questions in the proof questionnaire, to obtain their choices of argument and to identify the proof they thought their students would believe would receive the best mark. After piloting with 182 students, the survey was administered to 2,459 Year 10 students (14 or 15 years-old) from 94 classes in 90 schools. All the students were in top mathematics sets or chosen as high-attaining by the mathematics departments. Key Stage 3 test scores (2) of the students who completed the questionnaire were provided by the schools and these ranged from Level 5 upwards with an average of 6.56. The schools were spread across England and Wales and included mixed and single sex schools in different locations (urban, rural, suburban), operating diverse forms of selection procedures on entry. The following student outputs were analysed: scores on the four constructed proofs and the forms of argument used; scores for assessing the correctness and generality of the arguments presented in the multiple-choice questions (one score for algebra and one for geometry); choices in the multiple-choice questions; and views of the role of proof. Descriptive statistics were used to describe patterns in student response, followed by multilevel modelling using data from the school questionnaire to identify factors associated with performance and how these varied between schools. In presenting these findings, we have chosen to interpret some associations causally, while recognising that these interpretations must be treated with caution. 1. High-attaining Year 10 students show a consistent pattern of poor performance in constructing proofs.Overall, the performance of the students in the constructed proof questions was very disappointing. The average score for a constructed proof was less than 1.5 (half of the maximum score) and for the unfamiliar questions, well below 1. It is important to stress that a proof was scored merely on the basis of the correctness of the argument, and not on its presentation. Many students were unable even to begin to construct a proof (between 14% and 62% scored 0) and, if they did make a start, between 28% and 56% could only indicate relevant information unconnected by logical argument, thus scoring only 1. The percentage of students showing evidence of deductive reasoning varied according to the mathematical content of the question, with rather more, 40%, in the familiar algebra proof and very few in the harder questions in both domains (about 10% of students). Empirical verification was the most popular form of argumentation used by students in their attempts to construct proofs, and in problems where empirical examples were not easily generated, the majority of students were unable to engage in the process of proving. Among those students who did not rely exclusively upon empirical evidence, arguments written in narrative form were more common than formal presentations, and in general were associated with a higher incidence of deductive reasoning. However, in the case of the unfamiliar geometry proof, it was amongst the tiny group of students, (about 10%), who attempted to construct a formal proof, that the highest proportion of deductive arguments was found (54%). 2. Students' performance is considerably better in algebra than in geometry in both constructing and evaluating proofs.Although there was little difference between domains in the (very small) percentage of students able to produce a completely correct proof, the picture of proof construction was rather more positive in algebra than in geometry, in that students seemed better able in algebra to identify the relevant mathematics and begin to construct logical arguments. When trying to prove the familiar conjecture -- that the sum of two odd number is always even -- 40% of students used some deductive reasoning, whereas only 24% used any deduction when proving the equally familiar statement -- that the sum of the angles of a quadrilateral add up to 360o. Where the content was less familiar, a larger proportion (56%) of students was able to isolate relevant pieces of knowledge in algebra as compared to only 28% in geometry, where 62% did not know where to begin. Students were also considerably better in algebra than in geometry at assessing whether an argument was correct and whether it held for all or only some cases within its domain of validity. In fact, not one student amongst the sample was able to assess correctly the ten geometry arguments, while 20 managed to do this in algebra. 3. Most students appreciate the generality of a valid proof.Despite difficulties in evaluating particular arguments, the majority of students were aware that, once a statement had been proved, no further work was necessary to check if it applied to a particular subset of cases within its domain of validity. In contrast to all other questions, more students answered this question correctly in geometry (84%) than in algebra (62%). 4. Students are better at choosing a valid mathematical argument than constructing one, although their choices are influenced by factors other than correctness, such as whether they believe the argument to be general and explanatory and whether it is written in a formal way.Significantly more students were able to select a correct proof than to write one. However, they were influenced by their view of the generality of an argument and how far they judged it to be convincing: they were more likely to choose arguments that they believed to be general and which they found helpful in clarifying and explaining the mathematics in question. Students were also likely to make rather different selections depending on the two criteria for choice: the argument which most closely resembled the approach they would adopt or the one they believed would receive the best mark. For best mark, formal presentation was chosen frequently and empirical argument infrequently -- even when the latter would have provided a perfectly adequate refutation. Such empirical counter-examples were much more common when students were selecting the refutation closest to their own approach. In algebra, students were less likely to choose an empirical argument than to construct one, and in all three multiple-choice questions, arguments presented in a prose-style were the most popular choice for own approach, with symbolic-algebraic forms the least popular. In geometry, patterns of choice for own approach were less clear-cut, although formal arguments were selected more frequently than in algebra. 5. General mathematical attainment has a consistent influence on students' views of proof and their competencies in proving, although it is never the only significant variable associated with performance.Of all the factors associated with student responses, students' general mathematics attainment as measured by Key Stage 3 test score exerted the most consistent effect. In both domains, students with high, as compared to lower, Key Stage 3 scores constructed better proofs, were less likely to rely upon empirical evidence in their constructions and selections, and were better at evaluating arguments in terms of correctness and generality. However, Key Stage 3 test score was never the only significant variable associated with student performance in proving, and other student factors, along with a range of particular characteristics of school and curriculum, were also found to be influential. 6.Students' views of proof and its purposes account for differences in their responses.Students' views of proof and its purposes were associated with performance in a variety of ways: students with little or no sense of proof (over one quarter of the sample) were more likely to choose empirical arguments; those who recognised the generality of proof and its role in establishing the truth of a statement (over half of the sample) were better at constructing proofs and evaluating particular arguments; and in algebra, students who believed that a proof should be explanatory (over one third of the sample) were less likely than others to try to construct formal proofs and more likely to present arguments in a narrative form. 7. In algebra, girls and boys perform significantly differently, with girls constructing better proofs than boys and choosing different forms of argument.In algebra, girls and boys performed significantly differently, with girls scoring higher in their constructed proofs. Girls also showed preferences for different forms of argument than boys, both in their own proofs and the arguments they chose, although there was no obvious pattern to these differences. Boys also appeared more susceptible to school influences, with scores on the familiar algebra proof varying according to the school attended, whereas girls' performances were similar across all schools. 8.Teacher characteristics are not associated with students' competencies in proving.There was no variation in student response according to teacher variables, such as qualifications, sex and teaching experience, although it must be noted that almost all of the teachers in the sample were well-qualified mathematically. Neither did the teachers' responses to the survey in terms of their own choice of approach or their predictions of their students' choices for best mark appear to influence student response. 9. A range of school and curriculum factors are associated with performance.School and curriculum factors influenced students' competencies in proving, although no one factor had an effect across all questions, even in the same mathematical domain. However, some general trends are identifiable.
One factor consistently influential in student proof constructions and evaluations was the percentage of students expected to sit the higher-tier GCSE (3) paper: those from classes with a large percentage of students expected to sit this paper were likely to be better at constructing proofs and evaluating arguments in terms of their correctness and generality than similar students from classes where more would be entered for the middle-tier paper.
Curriculum factors were significant variables in student performance: in particular, the hours of mathematics teaching each week, the textbook used and examination syllabus followed affected the responses students made, especially in the multiple-choice questions. For example, more mathematics teaching reduced the likelihood of students choosing an empirical argument. These influences were all more apparent in algebra than in geometry.
In response to some questions, specific emphases on teaching proof appeared to improve students' performances: students in classes expected to write formal geometry proofs were more likely to be able to do so, and those from classes where proof was taught as a separate topic were better than others at evaluating arguments in algebra. 10. After taking into account all the factors found to influence student performance, there remains unexplained variation in the responses of students attending particular schools.Although there was more unexplained variation in performance within schools than between schools, school variation was found, and outlier schools identified whose students performed significantly better or worse than predicted on more than half of the scores. School variation was more evident in geometry than in algebra, with schools in the former case differentially affecting students' choices for their own approach, the forms of argument students used in constructed proofs, student preferences for formal arguments and student ability to assess the correctness and generality of an argument. 11.Summary and conclusions.The major finding of the project is that most high-attaining Year 10 students after following the National Curriculum for 6 years are unable to distinguish and describe mathematical properties relevant to a proof and use deductive reasoning in their arguments. Most are inclined to rely upon empirical verification. However, students perform more successfully when it comes to choosing rather than constructing correct proofs. The majority also recognise that a valid proof is general and accord high status to formally-presented arguments, even while valuing arguments that convince and explain. The research indicates that the ability to construct, assess or choose a valid proof is not simply a matter of general mathematical attainment. Clearly this has an influence, but at least some of the poor performance in proof of our highest-attaining students may simply be explained by their lack of familiarity with the process of proving. Far too many students have little idea of this process and no sense of proof, which, our findings suggest, can hinder their ability to construct and correctly evaluate proofs. The study was unable to identify teacher characteristics associated with different student responses, although school and curriculum factors did prove to be influential. Student performance in geometry was consistently poor and is a major cause for concern. This again, we suggest, is a matter of curriculum emphasis. The high-attaining students in our survey had little familiarity with geometrical structures and relationships, even of the simplest kind, and were certainly unused to explaining geometrical phenomena. It could also be argued that the fact that school effects were more apparent in geometry than in algebra was a result of their relative emphases in the curriculum &endash; as so little is prescribed in geometry, more leeway is available for some teachers to make quite a difference if their situation makes it possible and they so decide. In contrast to the absence of any curriculum requirement to engage in geometrical argument, students, under our existing guidelines, gain plenty of experience in constructing empirical verifications and refutations. They are accustomed to number/algebra investigations, where results have to be presented and explained, but where the focus of explanation appears to be less on the mathematical properties and relationships which underpin constructions than on the output data. The research suggests however, that many students do come to value general and explanatory arguments through these investigative activities, but this fertile ground is not exploited to introduce mathematical proof and face students with the challenge of setting out a mathematical argument in a coherent and logical manner. Particular curriculum influences on student responses were apparent in the survey, although generally their effects varied from question to question, suggesting that familiarity with mathematical content rather than general competencies in proving was the dominant influence. Nonetheless, the study does identify influential factors which suggest that more challenge and more attention to proving could enhance performance: students in classes with a larger proportion taking the higher- rather than the middle-tier GCSE paper, or where proof is explicitly addressed and the writing of formal proofs encouraged, do better than their counterparts in other classes. Taken together the results of our study suggest that, in the forthcoming review of the National Curriculum for mathematics, attention should be paid to the coverage of geometry and more generally to the approach to proof. We suggest that more explicit efforts should be made to engage students with proof while discussing with them the idea of proof at a meta-level, in terms of its meaning, generality and purposes. This would involve finding ways of balancing the need to produce a coherent and logical argument with the need to provide one that explains, communicates and convinces. This implies that alongside the curriculum emphases on measurement, calculation and the production of specific (usually numerical) results, more consideration should be given to appreciating mathematical structures and properties, the vocabulary to describe them, and simple inferences that can be made from them. Our evidence suggests that students could well respond positively to the challenge of attempting more rigorous and formal proofs alongside informal argumentation, and that developing approaches where this might be accomplished in the context of geometry as well as of algebra, would be a useful way forward.
(1) Funded by ESRC,
Project Number R000236178 [back]
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