This paper draws attention to some major problems in mathematics education. These problems are partly based on a misunderstanding of the nature of higher level mathematics. Illustrations and anecdotes are drawn from over thirty years teaching experience in seven tertiary institutions. Solutions lie in awareness of important distinctions: Community awareness for major improvements; Individual awareness for more local improvements.
In this paper, several major problems for mathematical education are considered. Some of these and their solutions may possibly be paradigmatic for problems in other areas of education.
Many of the problems, however, will not appear so obviously in other subjects, since the extreme features of mathematics occur in some other subjects either not at all or in very mild form. The first step is to point out some of these extreme features.
At more elementary levels, deductive reasoning and problem solving have to be applied to lesser extents than is necessary later. Indeed, the notation of mathematics also has to be developed and the idea that it is universal across languages and cultures will not apply initially. It applies more and more as the specialised language and notation of mathematics is developed.Thus evidence that the algorithms of elementary arithmetic differ from culture to culture has no bearing on the universality of higher mathematics. It adds to the picture. Similar remarks apply to evidence that some languages may more readily encourage the abstract concepts of mathematics than others. An interesting example of this has been reported by Bishop concerning a group in Liberia known as the Kpelle [Bishop, G. C. , (1967)]. As a summary from such evidence we may say we recognise
Cultural diversity in elementary mathematics and cultural convergence in higher mathematics
Failure to recognise the fundamental differences in focus between what is quite appropriate for elementary mathematics and what is vital for genuine higher mathematics is at the heart of many of the problems in mathematical education.
The distinction between these two kinds has for a long time been recognised in mathematical circles. A common way to refer to it is to reserve the term "mathematics" for the higher kind and refer to the other as "arithmetic". The great mathematician Gauss in the preface to his monumental work "Disquisitiones Arithmeticae" makes the distinction very clearly. Gauss was dealing with the properties of whole numbers mathematically, an area now called Number Theory. Ironically, whole numbers might seem to be only relevant in very elementary arithmetic so Gauss pointed out that there is a distinction. Gauss observes that "what is commonly called arithmetic hardly extends beyond the art of enumerating and calculating". He concludes that " it seems proper to call this subject Elementary Arithmetic and to distinguish it from Higher Arithmetic." Gauss noted that in his "present volume" he considers "only Higher Arithmetic" [Gauss, C.F.(1870)].
In a class of First Year Surveying students, the one who seemed to be the most able wished to find the radius of convergence of a power series. The student knew that he needed to take the limit of the ratio of successive terms but as the terms themselves were expressed in the form of algebraic fractions, he needed to divide by fractions. This he was quite unable to do. One gets used to widespread poor background and wholesale forgetting by many students, but in such an otherwise very good student it was a surprise. After some inquiry, I found that many primary schools did not teach the manipulation of vulgar fractions on the grounds that they were unnecessary. Students were taught to handle fractions only as decimals and that especially by use of calculators.
The particular area of fractions is not the full problem but rather a symptom of unfortunate attitudes towards mathematics. Even the terminology can be misleading.
The term "literacy " has long been used to refer to a person's reading and writing skills in language. In more recent years the term " numeracy" has been used to denote mathematical skills. Sadly, people are counted as "numerate" if they can operate with numbers in a mechanical way doing what is usually known as arithmetic. Numeracy is easily and quite naturally interpreted as dealing with numbers themselves rather than the patterns associated with them. It is not number development alone that is valuable for mathematics. An elementary additional valuable area of mathematics is geometry. This is usually introduced to students early on in their education. Very elementary geometry need not involve numbers. It is for these reasons that I wish to distinguish basic mathematical education from numeracy and to refer to the basic skills as ' mathematical literacy '.
It true that strict mathematical proofs, if unmotivated, can be quite difficult for learner mathematicians to follow. Overcoming this is the task of teachers. The logical presentation with all the details covered may well sensibly come after a psychological presentation. This may give highlights, perhaps the big picture and crucial sticking points which need to be overcome in a complete proof. Interestingly, many mathematicians inexperienced at teaching find great difficulty at times in explaining to students things that are obvious to themselves. Having proved the result by a logical argument, to them, there is nothing else available for explanation. A related matter is the experience of students who later on perceive how obvious the answer to their question is. Some such students may be embarrassed by their foolishness. Teachers, however, have an opportunity to point to this as a small part of a wider phenomenon whereby much more of mathematics can become "obvious" though originally seeming to be a bizarre conglomeration of unrelated parts.
Several years ago as part of a summative assessment a short test was administered to first year calculus students studying in the evening. In order to encourage the students to make an effort, it was planned to be relatively easy but not trivial. The necessary work for the test was completed the previous week and a practice test paper given out to students. Since the classes were held only once every week , model answers to the practice test were explained not long before the test itself. Students asked questions about the practice test and when ready we started the test itself. Neither notes nor books were permitted in the test, but I had deliberately left some of the model answer to the practice test visible on the board. All of the test questions held some resemblance to something on the practice paper and so for one question at least they had a reminder of a solution easily seen. I certainly noticed some looking at it. The test was marked and the results given to students the next week. The following week a student who was repeating the subject and had been feeling that failure once more was inevitable excitedly told me how having scored 13 out of 20 on the test had given her hope to pass the subject. She stressed that she was aware it was not a wonderful result but enough to restore her confidence. That week she was so motivated, that on returning home after the class she studied the subject. Classes finished about 10pm and evening students often have already put in a day's work before classes commence.
Experience seems to make it irrefutable that confidence plays an enormously important role in mathematics. The simple process of solving mathematical problems illustrates very well one way this works. People who are convinced they cannot solve problems are likely to seek help without making much effort themselves toward solutions. The opposite applies to people who are confident. These latter will pursue strategies and persistently work at the problem, and, in the extreme cases continue in spite of minor setbacks thinking the solution is just around the corner. Such people frequently do solve the problem and this experience reinforces their confidence. Those who lack confidence mostly will not solve the problem themselves and their experience also reinforces their attitude- this time a conviction about their lack of ability to solve problems.
Since problem solving is at the heart of mathematics, one could expect a major effect on the distribution of apparent ability in mathematics. Perhaps there is some evidence of this at least anecdotally. It is a very widespread experience of mathematics teachers that tests with a reasonable number of independent items scored approximately equally give rise more readily to a distribution of raw scores that are bimodal than to a unimodal "bell shaped" distribution. This is just what one would expect if some basic potential ability were distributed unimodally and if success increased actual ability while failure diminished it. The early actual abilities would follow the potential abilities but the more able would mostly experience success while the less able would not. In the middle region it might initially be just chance, but, as time progressed, most would be clearly either confident solvers or convinced incapables.
Confidence is a key to success in mathematics. To count as mathematics here, the requirements expected of students need to include applying general results to particular instances, as well as solving some problems differing in some small way from the routine examples given. For such mathematics , confidence in one's own ability is valuable. Confidence that one can solve a problem leads to persistent efforts if necessary before it is solved. Once it is solved, the satisfaction derived from that success reinforces both the willingness to again persist and indeed the confidence felt. Conversely , lack of confidence leads to a very limited effort to solve the problem without assistance. Continuing effort is then readily identified with wasted effort. Again there is reinforcement of the attitude with every apparent failure.
In both these cases there is a cycle- a valuable cycle in the one case and a quite undesirable cycle in the other case. Anecdotal evidence of the role of confidence comes from my own experience. Firstly in my own case as a solver of mathematical problems, my confidence that I can solve them ensures my persistence to do so if they ever turn out to be more difficult than at first sight. My teaching experience also produces similar messages. The reactions of numerous first year students aspiring to honours in mathematics at the University of Queensland during the years from 1967 to 1983 illustrate well the effect of confidence. At this time period such students were given a test after quite a few weeks. The test was purely formative but based on the results, students were encouraged either to continue or to change from the subject to the ordinary version of introductory mathematics. By far the greatest majority of students found the test very challenging and inevitably some students were quite ready to drop their aspirations. It was strikingly noticeable at the time that there were students still confident in spite of their poor results and furthermore there was a gender bias. It seemed that many young men doing poorly retained confidence in greater proportion than the women with poor results. Indeed it was not uncommon to find a young woman with a mediocre result feeling that the subject was beyond her while there was a young man with a quite poor result bristling with confidence. It is tempting to associate known and purported differences mathematically between men and women to differences of confidence . The confidence success cycles would be more easily established in one group and the lack of confidence failure cycle in the other.
There can of course be over-confidence effects, but these are more easily corrected than the opposite case. For whatever reason, physiological, sociological or other, the men seemed to be more mathematically confident than the women even for some cases where the results pointed the other way. McLeod (1992) discusses confidence and makes reference to Mura (1987) on the matter of gender and confidence. The area concerning confidence is dealt with in a number of systematic studies. The negative side called mathematics anxiety has been considered in a number of studies. Quite an array of of literature resources for this area is provided by Berebitsky, R. D. (1985) An extreme form of lack of confidence though not necessarily a question of anxiety is the idea of learned helplessness. McLeod (1992) refers to Diener and Dweck (1978) who "described a pattern of behaviour called learned helplessness where students attributed failure to lack of ability." Furthermore McLeod finds from Dweck (1986) that these " students tended to demonstrate a low level of persistence and to avoid challenges wherever possible".
It is clear that confidence related problems can emerge in later years because of failure in previous courses or subjects. However that does not exclude first year students since subjects can be by semesters and results from semester1 can then influence attitudes to second semester subjects. Indeed, failure prior to first year is quite a common thing, since any mathematics taught has prior learning involved.
Time and topics driving the tempo.
The commercial atmosphere settling down around some Australian Universities makes certain kinds of educational improvements difficult to introduce. A good illustration of this occurs at QUT with Engineering students. Firstly, the requirements of professional bodies result in substantial topics being included in mathematics courses for engineers. However, competition for students ensured that when the University of Queensland dropped Mathematics C as an Engineering Entry requirement, QUT did so also. Internal credit allocations within QUT between Faculties ensured that no addition formal class time could be set aside for students of Mathematics to help them cope with difficult topics for which many of them are ill prepared.
For confidence building, a slow initial pace may be needed. This is not reasonably available as students afraid of failure may well complain of their fears of not covering the course. An alternative face saving approach is often resorted to whereby most topics are covered in a highly superficial way. Selected topics are then concentrated on. These are the topics on which the major assessment is later on based. By using very generous marking schemes and also by using a simple order-preserving tranformation of the scores, for example adding 10% to all raw scores, results may be made to seem respectable.
One engineering student formally complained that the lecturer in the subject asked a question in a final examination that had not been covered in the course. It emerged that the lecturer had covered the course completely and had done various examples of questions including a complete past paper that reflected the questions which students could expect in their final paper. The student's complaint hinged on the fact that in the worked paper, the shape used for a rigid body question was not identical to the shape the final paper asked about. Over a number of years, it was known that many students would attempt to use "template" solutions from examples where the shape was the same but different information was given and so different calculations were needed.
In effect with this situation, certain work is prescribed by topics for a subject unit which is misleading. This is because normally one could interpret a pass in the subject to mean that students had gained a mathematical understanding of these topics. However, a true description of the student's learning differs from that prescription.
There is a prescription description gap.
Students and even staff themselves might believe the propaganda about understanding topics when they only know a few recipes or, tragically, knew once but now have forgotten them. One colleague whose service course for business students was required to include Lagrange Multipliers use to tell students that they ought not to think that will really know about the topic even if they know all she was teaching them. Should they ever need to use them, she would say, they ought to immediately consult a mathematician. Some years ago the business faculty cancelled a service mathematics subject on the grounds that the topics were covered in school mathematics. This explanation denied that the tertiary subject treated the topics at a level significantly deeper than did the school subject. Another example comes from the University of Papua and New Guinea during 1988 where all first year mathematics students did a foundation course. This essentially covered the topics of final year school mathematics but going further. The full group assembled for lectures but were in much smaller groups for expanding and illustrating the material in a classroom teaching situation. Set assignments were discussed and indications of how to proceed given. In some cases, students with a good background from school assumed they knew the work and would skip some classes before assignments whereas those with lesser backgrounds attended faithfully. The fact of an extension of the school work was not apparent to these more informed students who did worst in assignments than one might expect.
One would naturally think that the name of a topic should be the same for all cases. It is not surprising to educators that understanding can come at various depths. However, for people uninformed in mathematics it is easy to see mathematics as very constrained and so a very well defined area. Since opinion plays such a small role in standard mathematics what often comes to be regarded in other subjects as higher level mental activities are not part of the mathematics. In fact mathematicians can talk among themselves of "elegant" proofs, "amazing results" and areas of mathematics being "beautiful". These are personal aspects of mathematical experience that have a role to play in inspiring students and good teaching can make appropriate use of it. However, the main sense of deeper learning is in seeing results as part of a big picture. Results are often just a part of more general results and facts in one area imply facts in other areas.
Many of the problems that arise can be summarised as resulting in a butterfly mentality for mathematics. That is, the teaching moves rapidly from one isolated topic to another without doing anything in depth. These examples illustrate how failure to recognise the essential nature of mathematics and tertiary mathematics in particular as a logically tight system leads to teaching and learning problems.
Four test papers are given. One final paper testing selections from all the work and three others all testing work to date but increasing in difficulty and contribution to assessment. For each of the first three tests a mock test is given with solutions. For the first test the mock test is a very close indicator and students are informed of this. Later tests can vary from the mock test more than the first case. For all tests there is plenty of time. They are all deliberately so administered.
The inevitable success in the first test leads to confidence and motivation to go on. Final results for the subject are predefined by scores from the tests and are not determined by competition among the students. Naturally, presentation and teaching generally has to be good also and I make myself very available to students for consultation.
For everyone, whether teaching mathematics or not, promoting awareness of the vital distinctions between numeracy and mathematical literacy and those associated with cultural convergence for higher mathematics could make a difference.
Bishop, G. C. (1967). The new mathematics and an old culture. Mathematics Education Library Vol.6, Kluwer Acamedic Publishers: Dordrech/Boston/London.
Diener, C. J. and Dweck, C. S. (1978). An analysis of learned helplessness, continuous changes in performance, strategy, and achievement motivation cognitions following failure. Journal of Personality and Social Psychology, 36, 451-462.
Dweck, C. S. (1986). Motivational processes affecting learning. American Psychologist, 41, 1040-1048.
Gauss, C. F. (1870). Disquisitiones Arithmeticae. Springer-Verlag.
McLeod, D. B. (1992). Research on affect in Mathematics Education: A reconceptualization. From Grouws, A. D. [details missing]
Martens, E. (1994). Tertiary teaching and cultural diversity. South Australia: Centre for Multicultural Studies, Flinders University.
Mura, B. (1987). Sex-related differences in expectations of success in undergraduate mathematics. Journal for Research in Mathematics Education, 18, 15-24.
Author: Dr R.N. Buttsworth, School of Mathematics, Gardens Point Campus, Queensland University of Technology Telephone 3864 5241 Fax 3864 2310 Email b.buttsworth@qut.edu.au Please cite as: Buttsworth, R.N. (2001). Problems and solutions in mathematical education - a spectrum extremity. In L. Richardson and J. Lidstone (Eds), Flexible Learning for a Flexible Society, 99-105. Proceedings of ASET-HERDSA 2000 Conference, Toowoomba, Qld, 2-5 July 2000. ASET and HERDSA. http://www.aset.org.au/confs/aset-herdsa2000/procs/buttsworth.html |