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Some criteria for understanding and teaching Mathematics from a Christian perspective
Type of Material
Article.
Target group
Secondary school (high school) Mathematics teachers or lecturers in Mathematics.
Author and country of origin / contact
Andrew Palfreyman, England.
HYPERLINK "mailto:andrew@namyerflap.fsnet.co.uk" andrew@namyerflap.fsnet.co.uk
Original language
English.
Summary
The aim is to give some criteria for looking at mathematics. There are three main sections: philosophical issues about the subject, practical issues across various topics, and practical issues that relate to particular topics. This is very much workinprogress; any improvements gratefully received!
Reference to reality tests
The practical content is drawn from my personal reflections in teaching Mathematics over the last 15 years. The philosophical content is related to a thesis that I have just completed in Philosophy for which I was awarded an MPhil.
Some Criteria for Understanding and Teaching Mathematics from a Christian Perspective
I have been interested in the overlap between Christian faith and mathematics now since I was at university and have just completed a thesis addressing issues in the philosophy of mathematics from a theistic perspective. Although this thesis is in philosophy rather than the philosophy of education, some of the principles are applicable here, and so I have endeavoured to combine them with some more practical thoughts. There will be three main sections. Firstly I will deal with some issues of a philosophical nature. Then I will move on to some issues that are ones that relate to the subject across various topics. Lastly I will consider ones that are of relevance to particular areas within the subject. To a large extent this is a working document, and I would like to amend it with suggestions of other elements that I have so far overlooked, and also would like to expand upon the sections in which I sense that more can be said than I have so far formulated.
Issues relating to the subject as a whole
Discovered or created?
One of the key divisions within the philosophy of mathematics at present relates to the nature of mathematics itself. Stewart Shapiro notes that there are two different positions on the contemporary scene in the philosophy of mathematics. One group understands mathematics in a realist sense, and will be happy to speak about discovering mathematical truths; existence in mathematics is not thought to be unlike other cases of existence. The members of the other group value mathematics, but are sceptical of talk of the existence of mathematical entities, and try to reformulate the subject without having to commit themselves to the existence of abstract objects (1).
The problem can be elaborated like this. If mathematical truths are discovered, they must exist. Now, we know what it means to discover a coin down the back of the sofa or that woodpeckers visit the local park. A coin and a woodpecker are both objects which can be perceived. On this analogy, the truth that 7+5=12 would have to be perceived in some way. Given that such abstract truths do not physically exist within the universe, some philosophers talk of abstract entities existing outside of space and time, although clearly others will be hostile to talking of anything outside of space and time!
Such talk of mathematical entities may satisfy our desire to retain language of discovery but we then have the problem of how we gain access to such objects. Even if we were to suggest that a form of mathematical perception exists, analogous to sense perception, or philosopher of religion William Alstons mystical perception (2), there is then the problem of how such perception fits with our experience of doing mathematics. Perception seems to be rather more passive than understanding how mathematical statements are correct. Mathematics, in fact, seems quite the opposite. In involves a lot of time, effort and practice on the part of the person trying to advance their knowledge, either of the subject as a whole or regarding a particular problem within it. Even if mathematics were thought to function like a really complicated map, so that it takes a lot of time and thought to interact with the map, we are still left with the question of what the map corresponds to. We can see the difference between a map and the streets and parks they represent, and given enough time and skill could construct our own map from the reality of the area being studied. However, how would we start in trying to make a correct mathematical map? Our test of mathematical correctness involves understanding, but lacks a perception of the alleged mathematical realm to check whether such understanding is accurate or not! Mathematics seems more downtoearth and this worldly than according to a position that wishes to say that the mathematics that we are studying is about abstract objects outside of space and time.
Nevertheless, an antirealist alternative to this platonistic understanding of mathematics does not fair any better. If mathematics is not a perception of something out there, some may question in what sense it can be said to be true at all. Is it just a meaningless shifting of symbols that is agreed by communal consensus? From a Christian perspective, we want to defend that mathematics is true and not some form of relativistic symbolism, however well respected; a pragmatic acceptance that whatever mathematical truth is, we had better get on with it is not good enough. In addition, there is what is termed the indispensability argument, that idea that mathematics is indispensable to science. Science attempts to understand the way the world is. How could mathematics be of aid in this respect if it were merely a human creation, or a meaningless shifting of symbols? Nevertheless, if the platonistic position is true, and we have a problem with accessing truths that are hidden away in some divine realm, how could they be of relevance to a physical world down here anyway?
This is where a belief in God is helpful. We are not faced with a choice between inaccessible eternal truths that seem unrelated to our existence in our world, and truths that seem to be powerless to explain either why they should be accepted as true at all or why they are relevant to the world about us. Instead we can accept that we are made in the image of God, and that as such, he has given us the ability of understand the quantitative aspects of the world about us. We do discover mathematical truths, and so our position is therefore a realist one, but this is not via some mysterious mathematical perception, but through thinking about the world around us. If the same God has designed the world and our human mind, it is not surprising if one is relevant to the other! As Christians we can take such a position to be the start of an incarnational understanding of mathematics  transcendent truth has been revealed and made concrete within our own space and time, and this is how we gain access to it, rather than ascending up into heavenly realms.
Intuition and proof
Given our rejection of a platonistic mathematical perception, it is important to emphasize what this does not mean. This does not mean that mathematics is to be understood in a narrowly logical and mechanical way, where students learn skills and techniques to gain control over mathematical questions. Michael Polanyi writes:
No teacher will be satisfied with imparting a chain of formulae connected by formal operations as constituting a mathematical proof, and no student of mathematics should be satisfied with memorizing such sequences. To look at a mathematical proof by merely verifying each consecutive step  says Poincar  is like watching a game of chess, noting only that each step obeys the rules of chess. The least that is required is a grasp of the logical sequence as a purposeful procedure: what Poincar describes as the something which constitutes the unity of the demonstration. It is this something  perhaps in the form of an outline embodying the main steps of the proof  for which the student will grope, if baffled by a sequence of operations which convey no sense to him, and it is again this outline, embodying the general principle or general structure of the mathematical proof, which will be remembered when the details of the proof are forgotten (3).
Mathematics is not a collection of rules to be learnt  although clearly it does involve learning rules and formulae and arguably memorization is underplayed in a society that places so much emphasis upon the ease with which technology can be a laboursaving device. It is rather an art to be mastered, and this will involve insight as well as logic in a more stagebystage sense. A person giving advice on how to juggle may say things like You need to catch that ball higher up or Try to catch the ball without looking at it. Such advice is important but it is only worthwhile when the learning internalizes such advice via practice. In mathematics, we might consider two minuses make a plus to be a useful rule, but the student needs to learn that this does not mean that 52=7. Life cannot be reduced to seven rules for success and happiness or whatever the latest selfhelp book claims. Similarly maths cannot be simplified to formulae to solve equations without a prior stress that understanding is more important. Having a tentative sense of how to solve a question is better than following stepbystep a solution without any overall sense of why it is being done. In downplaying perception in the rather mystical platonistic sense, in no way do we want to deny the active role of intuitive glimpses of truth that occur as we seek further understanding. Mathematics can therefore be thought to work by a form of fides quaerens intellectum, or faith seeking understanding. The alternative would be to think that reality could be mechanized  a reality that unfortunately some people in society seem to think is possible with their emphasis upon quantitative data without a reason for why such structures are being used in the first place.
Pure mathematics
Mathematics is often separated out into those parts which are judged to be pure and applied. The idea seems to be that mathematics in itself is abstract and meaningless and needs to be applied to particular problems in everyday life. Now, to some extent there is an element of truth in such a division. A question dealing with conditions for continuity in analysis is clearly more abstract than one wishing to model to traffic flow in central New York during rush hour. However, there are difficulties with a strict division between pure and applied mathematics. Morris Kline writes about the nineteenth century attitude regarding the foundations of mathematics, saying that mathematicians could not leave the subject on pragmatic lines: Their prestige was at stake. How could they otherwise distinguish their nobleminded activity from that of earthgrubbing engineers and artisans? (4). When I was at university I occasionally heard jokes about a mathematician, a physicist and an engineer. The physicist was generally safe from criticism; either the engineer was at the end of the joke for not being exact enough, or the mathematician was for being too pedantic. The assumption seemed to be that mathematicians and engineers use the same techniques but inhabit different worlds. The mathematician has her eyes upon higher things  beauty and proof  whilst the engineer scuttled around merely trying to make things work.
We have here another form of a division between the transcendent and the created order that we found in the section on whether mathematics is discovered or created. However, if we have an incarnational perspective upon mathematics, there should not be a rigid division between usefulness and abstraction. Now admittedly different mathematicians may have a preference for a particular form of mathematics, which may have more or less immediate uses. However, a Christian should not have an aversion to usefulness, nor on the other hand, place the subject in servitude to the workplace or the economy. A Christian approach should not emphasize purity and such narrow usefulness. It should stress the importance of love: the appreciation that springs from understanding the beauty of the reality that is being studied, and a love for the people who are being served by the application of the subject.
It can also be noted that in practice it is hard to decide what is pure and what is applied. When I was at school doing my A Levels, I took for granted that calculus was a more theoretical subject. However, at university its manifestation as partial differential equations seemed far more practical in comparison with the logic, set theory and group theory on offer. So is calculus pure because it is used in hydrodynamics, or is it applied because topology can be applied to it? On reflection even logic should not be thought to be a pure area of study because it should be related to our rational thought processes, and help us to understanding how we should make decisions.
Some issues of a practical nature across various topics
Whilst these more philosophical perspectives may help to form a perspective upon the subject, they will not usually impact the subject directly in terms of what to object to with a syllabus that a Christian teacher is requested to teach. From my experience, it is unlikely that textbooks nowadays will put forward a view of mathematics that is unnecessarily abstract, where pupils are asked to solve algebraic equations for their own sake without any sense that they might be of vital relevance in other domains. In fact the textbooks that I have used do a pretty good job in terms of gaining a balance of applications  after all, from a pedagogical point of view it is good practice to use, say, pie charts for financial, scientific, environmental, social and leisurebased based questions to keep pupils alert to the variety of ways that mathematics is useful.
However, there still remains the issue of the degree of balance even though a textbook will have a variety of questions; it can be asked whether the balance of questions should be shifted in a particular direction. If a main textbook is being used (which seems pretty unavoidable unless pupils are given a large amount of photocopied sheets, which seems neither a coordinated resource for pupils to refer to, nor a good use of the environment from a Christian perspective), the Christian teacher can pick questions not just according to the mathematical techniques that are needed, but by which questions positively reinforce particular values. Consequently, a third question on gaining a bargain in a sale might be a rather poorer choice, other factors being equal, than a question that, say, deals with the percentage change of aid given to the developing world.
There are other issues other than that of balance that can communicate values. Cartoons in textbooks can do this. Often these can provide lighthearted contexts that help to break up the seriousness that the subject generally communicates. (A big problem with mathematics is that, whilst pupils may not understand questions that well in other subjects, they can at least usually attempt an answer. If they cannot understand a mathematical question, they often cannot even make a start, leading to anxiety and eventually fear of the subject.) However, sometimes a cartoon might be less appropriate and not worth drawing attention to. Another factor is the range of names that we use in examples. As Christians we are meant to welcome the stranger and show love to those who are from a different background to us. Consequently, it is good practice to ensure that, say, teachers in English schools do not always use merely AngloSaxon names, but also names from African and Asian backgrounds, especially in areas that have a high degree of ethnic diversity.
Of course, if this is done too much this could be seen as too much political correctness, but nonetheless Christians should ensure that all people in our communities feel welcomed, and that we encourage a worldview that is outwardlooking rather a form of national or Christian tribalism. For instance, whilst Celtic art may have Christian roots that are worth commenting upon, clearly Islamic art is also a great cultural product and geometrical resource. However, there are other issues that may cause us to challenge the views of society by the people used. If society were being reflected accurately, questions would probably have to include a reasonable proportion of cohabiting couples. Nevertheless, on this issue, I believe that as Christians strongly believe in the institution of marriage, questions should presuppose that people who live together are families, married or friends, rather than subconsciously accepting sexual partnerships that are not accepted in Scripture. I have not seen examples in textbooks about people living together yet (or samesex couples) but I can guess that it will be only a matter of time before this could become an issue, given the direction in which the West is heading.
Issues relating to particular topics
We have already mentioned in passing questions that relate to money. There does appear to be a balance here that needs addressing. A good proportion of Christs teaching was upon our use of money, and so it is not surprising if the balance in secular textbooks is not on the same wavelength. I rarely see questions about giving, for instance. When I do, it is generally something like a grandmother has died and has left money to her grandchildren in a particular ratio according to their ages, and the students are asked to find the amounts gained. The question thus is not ultimately about giving at all, but reinforcing consumerist values about getting as much as you can. Consequently, whilst students clearly need to be able to work out a percentage decrease when it applies to items in a shop, it is also worth giving examples in terms of people giving to charity as well, or perhaps even about tithing to introduce the simple percentage of 10%. (I believe that the Church of England has recommended that members give 5% to the church, which is another useful percentage.) There is also a common assumption in textbooks that the item that is the best value is the one that has the lowest unit cost. Again whilst not underplaying the importance of being able to calculate unit costs, densities, exchange rates, and so on, it is worth pointing out when a lower unit cost is not a better value. If a person lives on there own, for instance, it may be pointless buying more than they will be able to consume  the buy three for the price of two slogan does not question whether a person actually needs two to start with! More significantly, there are wider social factors: should we not rather be buying produce that is fairlytraded, food that is healthier, and consider whether the product has been tested on animals or not? And then there are perhaps issues to do with supporting local community shops rather than large supermarkets, or buying some product that is produced locally rather than being flown in from abroad merely to satisfy the consumerist demand for food products when they are not in season.
Another issue that impacts a Christian worldview is that of probability and handling data. From the theoretical point of view, the question needs to be borne in mind as to what is actually certain or impossible. I do not like the common assumption in textbooks that I will die is accorded a probability of 1. Clearly from a Christian point of view, Jesus could return before this occurs, and incidentally Muslims also believe that Jesus will come again to judge the world, so this is not an issue of us merely pushing our own religion! Another issue is that of the issue of gambling. I have shown at least one class that the chance of winning the National Lottery in the UK is 1 in 13,983,816. This would mean that even if a person were to buy a ticket twice a week, she would still on average have to wait over 130,000 years before gaining the jackpot, which raises issues regarding the responsible use of Godgiven resources! I also believe that more could be done on the central concept of the nature of chance itself and how it fits in with our belief that God has a purpose for the world. Any ideas?
A couple of questions have also stuck in my mind that assumed that people should use mathematics to further their own advantage. One dealt with some statistics which were then commented upon by a union leader and the management of a firm, and the reader was asked to say which argument they would use if they were either character! Now, I have no problem with pupils needing to understand how statistics can become biased in a fallen world, but it is not our job to encourage them to engage in such partiality  we should be encouraging them to be as truthful as possible! Another question asked which supermarket layout was better from the perspective of the customer and from the perspective of the shopowner. The assumption behind this question was that a supermarket design that forced the shopper to take a detour around the shop, rather than providing a quicker route, would encourage the customer to see more items and thus buy more products, and so would be better for the owner!
A last issue that I wish to mention is very much a tentative one. How should we understand infinite decimals? Clearly , once it is thought of in terms of base 10, must be approximated by 3.141592654, if ten decimal places are required, and so must have the digit 9 in the fifth decimal place. However, does equal an infinite expansion? At a simpler level, one approaches the limit of 1/3 by considering the sequence 0.3, 0.33, 0.333, 0.3333, ..., but does this legitimate saying that 1/3 is equal to 0.333... ? I am in two minds on this issue. On the one hand I am a realist and do not believe that mathematics is a mere human creation; it reflects the fact that God is a mathematician and that he has created a world that is consistent with our mathematical discoveries. In addition, as a Christian I have no problem with the infinite existing, as those of an empiricist persuasion might do. However, my question is whether as human beings we have the ability to complete an infinite sum, as, taken literally, this implies that we are capable of an infinite action  not an obviously human capability! At university, sequences were taught in terms of limits, so that for whatever small accuracy you were interested in, your sequence would get closer than this. However, this is different from saying that the limit is the result of an infinite calculation, rather that the sequence can be seen to approach it (but never arrive). Would it not be more accurate to say that 1/3 is actually inexpressible in base 10, albeit tended to by the sequence 0.3, 0.33, 0.333, ... , rather than equating 1/3 to a recurring decimal? Or am I being too pedantic? Should people instead be happy to use such language about recurring decimals as a useful image without taking it too literally? (Nevertheless, there does appear to be an the assumption that such infinite expressions actually exist in Cantors diagonal argument, which provides the basis for showing that the real numbers, unlike the integers and the rational numbers, are uncountable. Do any implications for this argument exist if the reality of such infinite expressions is questioned?)
I hope that this essay provides a basis for further reflection by mathematics teachers and I would be interested in other areas of overlap between our faith and mathematics!
References
(1) Stewart Shapiro, 2000, Thinking about Mathematics. Oxford and New York: O.U.P., pp. 20102.
(2) William Alston, 1991, Perceiving God: The Epistemology of Religious Experience. Ithaca and London: Cornell University Press.
(3) Michael Polanyi, 1962, Personal Knowledge: Towards a PostCritical Philosophy. London and New York: Routledge, pp. 11819.
(4) Morris Kline, 1982, Mathematics: The Loss of Certainty. New York: O.U.P., p.173.
(5) e.g. D. Smith and B. Carvill, 2000, The Gift of the Stranger: Faith, hospitality and foreign language teaching. Grand Rapids: Eerdmans.
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