Abstract
This paper makes the case for drastically reducing mathematics teaching in schools to the level of music teaching, and introducing specialized schools (i) to prepare future engineers and scientists, (ii) to prepare for all other professions who need mathematics and (iii) where all those children who are just interested in mathematics can go deeper into the subject.
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Notes
- 1.
Statistics which state that e.g. the numbers of UK A-level and AS-level Mathematics entries have increased (http://www.ima.org.uk/viewItem.cfm-cit_id=383105.html) don't change this assessment.
- 2.
An open question is the selection of children for such activities since they most often don’t know what their future profession is going to be. But this is one of the utopias in school—that it prepares for future professions. No, the choice goes mostly in the opposite direction: ‘I am good in X or Y so I would like to study or to work in a field related to this subject. Teachers do not stress the professions (except for those in professional schools). They stress the content. It is very seldom that somebody decides to go into a mathematics-related profession just because it is important or because there is a lot of money to earn—the run to departments/schools of engineering would otherwise be much bigger than it is now.
- 3.
It should also be possible to enter (and leave) such schools at any level of one’s education like one can enter a music school—this is only a matter of organization.
- 4.
I was made aware of a paper by Dolton and Vignoles (2002) the main claim of which is that individuals with advanced (UK-based A-Level) mathematics qualification earn between 7 and 10 % more than those without this qualification. This might be true but it is no argument against this proposal. Medical doctors also earn more than the average worker but we do not take this as an argument to introduce basic medicine in school. Medicine is an even better comparison than music—the utility is more than clear but there is no school subject related to medicine. For example it is completely unclear to me why there is no subject like preventive medicine in school. Maybe the pharmaceutical industry does not allow this to become a topic. If the salary argument holds then the specialized schools will be much sought after. This is a variant of the importance argument which has failed dramatically. And moreover, school children don’t care for statistical arguments; they care for role models, for specific arguments ad personam, which the Dolton/Vignoles article does not provide.
- 5.
But who knows? Maybe the majority will say: we do all mathematics needed with the help of technological tools who will “do the math” instead of us. This probably will not happen because of the idealists. But anyhow: any honest and open discussion about the necessary basic mathematical skills would be a real improvement.
- 6.
I am well aware that 1/3 does not have a finite decimal representation but I strongly doubt that the lack of knowledge about this fact has consequences in everyday life.
- 7.
The knowledge of the majority of undergraduate students about roots and logarithms seems very poor if not non-existent (and even worse: wrong, which is much more difficult to deal with) although these concepts are clearly a part of the school curriculum.
- 8.
Of course one can (as we say ‘without loss of generality’) replace music with painting or another art or ‘art and design’ as the subject is called in the UK.
- 9.
Maybe additionally a research mathematician or an engineer or a scientist could visit the class once a year, speak about what they are doing and present an interesting elementary proof or technique on which their more advanced techniques and proofs are based. As an alternative something already learned by the children might be discussed from a different perspective. This does not mean that the teacher could not prepare something interesting herself but it makes another impression to the children when they are confronted with a professional who might even tell something about his or her research. Concerning music, one can go to a concert, concerning fine arts, one can go to an exhibition but concerning mathematics? Yes, there is an increasing number of mathematics museums and exhibitions so a visit there could also serve as potential motivation.
- 10.
‘Mathematics is fun’ is a major theme in contemporary efforts to popularize mathematics but it fails because of the negative context in which school mathematics is embedded. …and, by the way, if we compare again with music, practicing is not always fun but unfortunately often a necessary prerequisite for future fun. I don’t know of any study among mathematicians giving information about what part of the occupation is just hard work and what part is fun. Like the occupation of a ballet dancer or of a basketball player—most of the time is occupied with hard training but the short moments of fun (the actual dance performance or the basketball game) are extremely rewarding and are enough motivation for the hard preparation. We don’t oblige every child to play basketball or to dance ballet. Why should we expect this hard work from everybody with respect to mathematics?
- 11.
One might also add the totally mechanical execution of algorithms but this is a more subtle issue which has to be discussed separately.
- 12.
This and the next emotional aspects are carelessly neglected. Hersh and John-Steiner (2011) mention in their book 'Loving and Hating Mathematics' a 2005 AP-AOL News poll (http://www.ipsos-na.com/news-polls/pressrelease.aspx?id=2756) of 1000 adults in the US according to which 37 % stated that they hated mathematics at school. Mathematics leads the list by a large margin. On the other hand site according to the same poll for 23 % mathematics is the favourite subject in school. A similar result comes also from a Gallup US-poll from 2013 (http://www.gallup.com/poll/164249/americans-grade-math-valuable-school-subject.aspx) where mathematics is considered by 34 % of the respondants as the most valuable school subject. If reliable this would mean that there will be enough candidates for the specialized schools as well as for the extracurricular courses.
- 13.
There has been research on this topic at least since the 1970s and it is interesting that one of the main proponents, Sheila Tobias, has published in 1978 a first book ‘Overcoming Math Anxiety’ (Tobias, 1978); then in 1994 there has been a revised edition of this book (Tobias, 1994) and recently in 2012 she has coauthored a book entitled ‘Banishing Math Anxiety’ (Tobias, 2012). The problem seems to be a persistent one….
- 14.
http://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/ I can’t resist giving the killer-counterargument which even somebody like Terence Tao cannot refute: ‘Of course it is easy for a genius to say that you don’t have to be a genius.’ This is a strange example of a self-reference (induced from outside) and deserves further attention—I suppose that it might show that the notion of genius is a social construct. By the way, the notion of ‘giftedness’ has a very strong religious touch: somebody possesses a ‘gift from above’. It is very strange that all the rational thinkers and all the analytic philosophers have not really written about this issue. Maybe they consider themselves as gifted….
- 15.
This sounds probably vague and not backed by empirical arguments. I myself have taught undergraduate mathematics to engineering students for more than 15 years and I have to say that I have to battle intensively against my upcoming cynicism in order to protect my own enthusiasm. The effort of not teaching to the test becomes harder and harder when one is permanently confronted with the one and only question: ‘Is this relevant for the exam?’. The proposal from this text does not address the issue of teaching to the test but, to be honest, testing students who don’t hate the subject and invest enough time in studying (which the new system would assure) has never been a problem but more a formal necessity.
- 16.
It becomes even more frightening when we consider the possibility that also some teachers and maybe even mathematics teachers have been among the majority of pupils described above. During my mathematics education studies I have been confronted with quite a number of such fellow students who most probably are school mathematics teachers now.
- 17.
Publicly! Privately they are well aware of the horrible situation and talks about this are always similar—angry laments. This does not help at all.
- 18.
http://www.oemg.ac.at/LS/Wuensche_Mathe_Unterricht.pdf …by the way, Tichy’s wishes (and I suppose other statements will be quite similar) can be easily fulfilled by the specialized schools proposed in this text.
- 19.
- 20.
There is also no fixed terminology. Aside from ‘bridging’ courses I know about ‘pre-college’ courses or programs (in English). In German there are at least three different notions. At my university they are called Warm-Up courses. This makes it even more difficult to collect information.
- 21.
http://www.khdm.de/veranstaltungen/vorkurstagung/ (in German).
- 22.
http://www.khdm.de/veranstaltungen/arbeitstagung/ (in German).
- 23.
- 24.
- 25.
And I include here all kind of brilliant texts such as W.T. Gowers’ ‘Mathematics: A very short introduction’ or his lecture entitled ‘The importance of mathematics’.
- 26.
I am aware of many names of suitable renown scientists but they are not presented as mass media heroes, as role models to be copied. The special situation of mathematicians’ depictions in mass media is the topic of Markus Pantsar’s text in this volume.
- 27.
One sees this very often with the opposite kind of exercise where a solution to an equation is given and the question is to check if this is really a solution. Very seldom do the students simply insert into the equation. In most cases, I have to report, they solve the equation from scratch and then compare their solutions with the given one. This normally takes much more time and is a source of more mistakes. Moreover their solution could look differently to the given one although they might be equivalent which again might be hard to check.
- 28.
Contrary to the quality of teaching the quality of learning is strangely neglected in the research on mathematics education. There are recent empirical studies by Hodds, Alcock and Inglis on self-explanation training in mathematics (Hodds et al., 2014).
The general question concerns the role of auto-didactics. There are almost no studies about auto-didactics as far as I know although this is the permanent activity in the research community after finishing regular studies. Scientists learn without teachers and I strongly suppose that the early decision (and training) to learn without a teacher is crucial. I would even dare to make a conjecture that all successful learning must necessarily include substantial parts of auto-didactics.
- 29.
- 30.
Innumeracy and the Fires of the Inquisition (Article in the newspaper “Izvestiya” 17 Jan 98). http://www.pdmi.ras.ru/~arnsem/Arnold/izvest.ps.gz.
- 31.
Notices of the AMS 37 (1990), 844–850.
- 32.
Notices of the AMS 57 (2010), Nr. 3.
- 33.
Auto-didactics!
- 34.
I strongly doubt this. If the specialized schools ought to be successful there must be lessons learned from the old failed system. The lessons must come from the few successes, from the allegedly ‘gifted’ pupils. It has to be figured out what the ‘gifted’ ones do and how. Our conception of mathematics has to change from something existing ‘outside’ to something that is in the human being or at least approachable by all humans. If we consider (as many suggest but the predominant analytic philosophy of mathematics does not and cannot take into account) mathematics as an art or as a foreign language or as a game (in the sports sense) then we should have a close look at how a skill for a game, an art, a foreign language is acquired. The consequence would be that the teachers will have to be renamed to trainers and act as such. The role of the trainer is not to do the training but to supervise the training and to watch the trainee to make him or her aware of mistakes that he or she cannot detect for various reasons. This will mean that we would need much more mathematics trainers than we have mathematics teachers today. The teaching part of the trainer will be to help the trainee to acquire autodidactic skills but a trainer has to care for fewer pupils than a teacher does nowadays so the total number of teachers/trainers might remain approximately the same. If this is done to the last consequence it will have a major influence on research as well. It will imply that communication in mathematics should not primarily be about results and proofs but about how they are reached, about individual training practices. It will be a revolution in the sense of Thurston (see his text ‘On Proof and Progress in Mathematics’, Bull. of the AMS, 30 (1994), 161–177) because the competition aspect of mathematics will have to be abandoned or at least significantly reduced. This issue is too broad to be dealt with in a footnote and I will stop here and leave it for another text.
- 35.
I was made aware by Brendan Larvor that if I take up the analogy to music I could consider (i) the social and ethnic narrowness of the players in elite orchestras or (ii) the psychological effects of hot-housing on children identified as talented or (iii) the well documented effect of identifying most schools as ‘ordinary’ i.e. mediocre. I have the feeling that these points bare a clear anti-elitarian touch and I am sure that this can be avoided if the system is designed well from the very beginning. The specialized schools have to offer enough places for every interested child. The division will happen later when it is seen who is willing to put in the necessary effort and time and who not. The elitarist argument does not hold for extracurricular courses in regular schools. The ideal non-elitarian situation will be when every child decides to go to the specialized school or to the extracurricular course—I strongly doubt that this will be the case.
- 36.
‘Der Standard’, Saturday/Sunday, February 26/27, 2005 and ‘Der Standard’, Wednesday, July 10, 2013.
- 37.
- 38.
- 39.
- 40.
By the way, auto-didactics does not need any contracts since the same person is teacher and pupil.
References
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Appendix
Appendix
I would like to make parts of the history of this text public. Initially it started with a request from Brendan Larvor to have a controversial and provocative contribution to the third Mathematical Cultures conference. I had had these ideas already for some years and had even a small article and parts of an interview on this in an Austrian newspaperFootnote 36 so I decided that this topic might suit the organizers’ wish. Below you can find the abstract which I submitted as well as the reviewer’s comments commented shortly by me. The talk from the conference can be viewed on YouTubeFootnote 37 and on the conference’s website.Footnote 38 The reactions to the talk were an additional motivation to write this article.
Abstract Submitted for the MC3 Conference
Is mathematics an issue of general education?
The link between mathematics and any other non-mathematical culture is biased by the fact that every representative of such a non-mathematical culture has been confronted with mathematics at school. The experience of a majority at school has been one of frustration, anxiety, incomprehensibility and this does not stimulate future contacts with mathematics.
The situation is known but from a professional mathematician’s viewpoint this issue is marginal. Only very few top-level mathematicians have dealt with this. There are e.g. texts by Arnold, Manin, Thurston or ThomFootnote 39 on the problematic situation in mathematics education, there is also a relatively recent blog-entry by T. Gowers. On a less ‘famous’ level there are articles like ‘A Mathematician’s Lament’ by P. Lockhart or ‘What is mathematics for?’ by U. Dudley. The problem is that these and other similar texts are individual statements, sometimes accompanied by individual actions but there is no bigger ‘movement’ driven by professional mathematicians with the aim to change the situation although in informal conversations almost everybody is unhappy (according to my experience). Professionals in mathematics education also seem unable to improve the situation (more or less since this profession exists). Maybe the assertion that mathematical competence is important for everybody is too idealistic and a more realistic approach is needed?
So why not question the general premise that mathematics should be an issue of general education? At least one could ask if it has to occupy such a large part of the school curriculum as it does now. In the talk I will present a suggestion to reduce the scope of mathematics in school to that of music. This must be linked to an increase of specialized schools out of which the demand for future engineers, scientists and mathematicians would be covered. This would dramatically change the public image of mathematics to the better and hence improve the basis for interchange between mathematical and non-mathematical cultures.
Comments by the reviewers with comments from the author
Review 1
Thought-provoking! There are many myths about how society will break down if general numeracy levels drop, and very little evidence that these scenarios are realistic. The education-general culture interface that the author discusses has a university-level equivalent: that of the content of mathematics programmes at the university and the actual mathematics that professionally trained mathematicians use. A lot of the debate is driven by myths like the one mentioned above and by the fact that the people involved in the debate have certain agendas. I like the fact that this talk links the themes of MC2 and MC3, and am looking forward to it.
It was not entirely clear to me from the abstract whether the “proposal” is (a) a serious proposal that the author wishes to make; (b) a proposal that serves a particular argumentative purpose; (c) a proposal that was proposed by someone else.
Comment: I doubt that by implementing the proposal the general level of numeracy will drop! There will be much more time available to secure basic numeracy when all the rest is omitted. And yes, the proposal is serious and by the author.
Review 2
This audacious paper questions the assumption, implicit in much hand-wringing about the need for increased mathematical literacy, that “mathematical competence is important for everybody.” What we mean by “mathematical competence” deserves more careful scrutiny, certainly, before we dismiss this assumption as too “idealistic.”
That compulsory mathematics education produces more frustration than understanding is a claim that also deserves further consideration (and more than anecdotal evidence). The suggestion is that it would be more “realistic” to expand applied mathematics in more courses students actually enjoy (e.g., increased instruction in music) and reduce required courses in abstract mathematics from which only a few benefit. But is this suggestion based on the assumption that mathematical capacity can be cultivated in nearly everybody provided the means by which it is delivered are engaging? If so, how is this shared mathematical capacity different from the (presumably) more narrowly possessed capacity to develop actual mathematical competence? These questions are worth further discussion and this talk will speak to many issues implicit in other talks that are more narrowly focused on particular historical case studies.
Comment: The reviewer speaks of ‘capacity’; I speak of ‘giftedness’ but we mean roughly the same. As mentioned in the text, I strongly doubt that there is something like ‘mathematical capacity’. On the other hand ‘mathematical competence’ can be trained which will increase ‘mathematical capacity’ in parallel in the sense of increasing the part of auto-didactics in the learning activity of the pupil who as a consequence could then be considered as more ‘gifted’ by those who have not undergone the proper training. By this I mean that ‘mathematical capacity’ can be trained. But this needs a lot of time consuming effort and it is absolutely unrealistic that one can expect from all pupils to undertake this effort. To paraphraze Lancelot Hogben: ‘Mathematics for the Million’—yes, but ‘Mathematics for the Billion’—no.
Review 3
This is an extremely ambitious proposal which sets itself the remarkable task of solving the problem of mathematics education. The author’s intriguing argument is that the amount of mathematics in the curriculum should be reduced (to the level of music which, in the UK at least, is almost zero). To cater for the need for mathematicians and engineers and the like, specialized schools would need to be set up.
Curiously, the latter part of the proposal is actually UK government policy (see e.g., http://www.kcl.ac.uk/mathsschool/), based on the Russian model of restricting mathematics to the elite.
My view is that the author’s proposal is a terrible idea for several reasons:
-
(i)
We know that a good knowledge of basic pre-university mathematics is associated with success in the labour market (even after you control for every imaginable confound: see doi: 10.1111/1468-0335.00273). Restricting the number of people who have access to this opportunity seems a backwards step.
-
(ii)
I suspect that in the future mathematics will become more essential for more academic disciplines (not just the science and engineering subjects mentioned in the proposal). For example, it is becoming increasingly impossible to be a social scientist without the ability to statistically model complex phenomena (http://www.nuffieldfoundation.org/why-q-step-necessary). Even apparently entirely non-mathematical subjects now have large quantitative components (e.g. corpus analysts in departments of literature). This trend is almost certain to continue, thus restricting the mathematics taught at school will severely restrict children’s post-study opportunities.
-
(iii)
The result of this would be to further increase and cement societal inequality between those who engage with an opportunity-enhancing mathematics education and those who don’t. I think this would be a bad thing.
-
(iiii)
My final complaint is that the proposal seems rather defeatist: “we don’t know how to teach mathematics well, so let’s give up”. Wouldn’t it be better to work out how to teach it well?
There’s no reason to believe that this is an intractable problem.
Anyway, the talk promises to be an interesting one, it should certainly be accepted.
Comment: (i) and (ii) One can go to a specialized school at any time if one wants a better job or intends a certain tertiary education. I have no problem if this is taken as one motivation to enter the specialized school. If there is an enormous rush on these schools the state can still increase their number. And there will be also the extracurricular courses at the regular schools. I would definitely prefer this approach to the mixed blessings of a well-meant torture as it is currently the case.
(iii) I think exactly the opposite. The specialized schools in order to deliver proper results have to be based solely on commitment and on the effort undertaken by the pupils and not on social background. If it is made clear in the compulsory phase that one needs hard work (and no gift and no special social background) in order to improve in mathematics, then the proposed system will give many more opportunities to reduce social inequalities.
(iiii) The problem is not how to teach it well. This is a marginal problem which is irresponsibly overrated. There are a lot of good proposals but they don’t work in reality (because there are two parties involved).
The real problem is how to learn mathematics well. To use Guy Brusseau’s notion of the ‘didactic contract’ Brousseau (1984), there is a lot of discussion about the teacher’s obligations but almost no discussion about the pupil’s contractual obligations. The basis of this proposal is that these obligations, if formulated realistically, cannot be fulfilled (again: realistically) because of the lack of willingness by the majority of pupils to commit themselves and to invest the time needed. If there was a serious intention to implement ‘didactic contracts’ it would probably lead to consequences similar to the current proposal if one wants really these contracts to be fulfilled.Footnote 40
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Simeonov, E. (2016). Is Mathematics an Issue of General Education?. In: Larvor, B. (eds) Mathematical Cultures. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-28582-5_24
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