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MSOR Connections Vol 10 No 3 Autumn Term 2010
Chris Sangwin
The truth, nothing but the truth...
but not the whole truth!
Chris Sangwin
School of Mathematics
University of Birmingham
c.j.sangwin@bham.ac.uk
I enjoy solving problems. Indeed, it forms a central part of my professional business
and of my personal identity. Hence, I spend a significant proportion of my time
getting ‘stuck’ or simply being confused. In the past, my technology was a pencil and
paper, together with algebra, calculus and where necessary tables of integrals and
special functions. Currently, when I am stuck I usually reach for more contemporary
technology such as the dynamic environment of GeoGebra, or a computer algebra
system (CAS). This is what I mean when I talk about ‘technology’. Communications
and collaboration, data capture, online journal access, internet search and storage
technology are key components of my practice but I don’t comment on these here.
“You don’t need to understand how the car works to drive it.”
Indeed not. But you drive the car on a very carefully prepared surface! Certainly, the
difficulties of using the technology should be reduced through good design. However,
the ground on which you drive your car is relatively simple, and all the hazards, e.g. low
bridges, are clearly sign-posted. When you travel to parts of the world where standards
are different you find a lot more accidents.
Poor design leads to answers from a CAS such as
x k+1
∫x kdx = . (1)
k+1
Arguably this is better written as
{
x k+1
∫x kdx = k + 1 + c, k ≠ –1 .
1n(x) + c, k = –1
where c is an arbitrary constant. This sort of conditional answer tends to make further
calculation messy. Ideally we want a single formula upon which we can potentially
carry on our computation. Hence, we might prefer a result such as
∫x kdx = lim x –1 + c.
l+1
l→k l + 1
This is correct, [8], but is likely to be very confusing to all but the most sophisticated
user. An alternative is to ask the user if k = – 1, but this stalls automatic calculation.
Furthermore, if k is itself an arbitrary constant generated automatically and internally
from some previous step the question ‘is k = – 1 ?’ may be utterly confusing. So (1) is
certainly the truth, but is not the whole truth! Surely, objections such as these can be
resolved through good design, and an interesting catalogue of similar issues is given
by [10]. For now they remain real, and a source of potential errors for users.
I fully support the call for an increased emphasis on modelling, rather than on fluency
of computational technique. I constantly have to remind myself when teaching
29
MSOR Connections Vol 10 No 3 Autumn Term 2010
that the subject is all about problem solving, not about {2xy – ay – bx = 0, x2 – y2 = 0, –2y2 + 2by + a2 –1 = 0,
only tidy, well presented knowledge and the practice of 2y2 – 2by + b2 – 1 = 0, –2y2 + by + ax = 0}
tutorial examples or exercises. However, very few problems
which is equivalent to the original system. From this we
can be solved exactly. Perhaps these constitute the ‘well
isolate the equation in only x and y, which gives us our
prepared surface’ on which our students are to drive their
expected solution
mathematics as they gain confidence and competence. Real
problems are messy. Making choices in modelling requires x2 = y2, i.e. y = ±x.
a deep knowledge of the sorts of models which are likely
Does this calculation constitute an answer? Again, this is
to produce fruitful results. I wrote about this recently, with
the truth, but not the whole truth, since the correct answer
some examples, [9]. The example I suggested in a recent
to the constrained real problem is a segment. While the
meeting is the ‘square trammel’.
segment is part of the line, not all the points on the line
Constrain the opposite corners of the unit square to be correspond to legitimate positions of the real square. So
on the x and y axes respectively. What are the loci of the what is going on? What is the algorithm calculating? Is,
other two corners? for example, an underlying assumption that all variables
are complex, giving us too many solutions? Are we really
An experiment with GeoGebra strongly suggests that the confident that the system has given us all solutions or only
loci are segments on the lines y = ±x centred at the origin, some of them? The mathematician’s concerns over existence
2
of length 2√¯. Does this constitute an answer? and uniqueness of solutions is precisely to ensure complete
and correct results are obtained.
This is certainly the kind of problem which is simple to
state and once a reasonable experiment is done the
correct answer readily suggests itself. The CAS calculation
is routine. Of course, we have behaved in a sophisticated
way. However, I would venture to suggest that similar
problems ought to be something which are accessible to
later years of school mathematics if we reduce the emphasis
on computation as an educational goal. While there may be
other ways to solve this problem1, to derive this answer in
an algebraic way we have needed to do the following.
• Use the Pythagorean theorem to model this situation.
• Understand that ‘solve’ means eliminate a, b, and expect
to be able to achieve this.
• Technical knowledge of how to use the CAS.
• An ability to interpret the output to isolate the solution.
If we throw out calculation, and include more modelling,
Fig 1 – van Schooten’s locus problem then presumably these are exactly the kinds of critical
judgments we would expect students to become adept at
Let us model this problem algebraically by assuming the rather than performing heroic calculations by hand? It all,
unit square APBQ has corners with coordinates A = (a, 0), of course, depends on what you want, and what you think
B = (0, b) and P = (x, y). The constraints are constitutes the nature of the subject. This won’t be the same
a2 + b2 = 2, for all groups at all moments, but it does raise the question:
what is the point of the activity?
(x – a)2 + y2 = 1,
1. Getting an answer.
x2 + (y = b)2 = 1.
2. Understanding and explaining why the answer is true.
The goal is to find (x, y). Since we have three equations in
four variables we expect to be able to eliminate a and b and 3. Understanding the limitations of the answer or method.
write a single equation which relates x and y. This is what When the purpose of the activity includes more than just
we mean here by solve. Perhaps you might like to try this getting the answer, then I feel very let down by current CAS
by hand, or at least to verify the potential solution y = x. technology. Too much is done for me, and I have few, if any,
One way to ask a CAS to do this is to calculate the reduced
Gröbner basis [1] of this system of equations with respect to
1
The constrained square is a special case of the locus problem of Franciscus
an appropriate order. This results in the system: van Schooten (1615--1660), see [7] for more details of the Gröbner basis
approach, [3, 47] in which a geometric proof is presented, and also [11].
30 The truth, nothing but the truth... but not the whole truth! – Chris Sangwin
MSOR Connections Vol 10 No 3 Autumn Term 2010
ways of expanding every function in progressive levels to knowledge is serial in nature. For example, consider the
see what is going on inside. The sorts of activities I often following sequence of topics:
need to undertake include:
1. calculation with integers;
• running steps of algorithms;
2. algebra of polynomials;
• accessing local variables within algorithms; and,
3. algebra with other functions, including trig formulae;
• altering and running parts of algorithms.
4. differentiation;
Having written some software I know that such an
5. symbolic integration (as the inverse of differentiation);
interface will be significantly more difficult to write than
the algorithms which underlie them. Of course, if I could 6. integration by parts;
have written such an interface myself, I probably would 7. Fourier transforms; and,
have done so already! I may simply be asking for something
impossible. Nevertheless, current CAS are still not giving 8. the frequency domain representation of a function and
me easy access to the whole truth. There is a more the power spectrum.
serious objection to the proposal to open up the CAS: the Without facility in one level progress is impossible. Of
algorithms actually used are not those you learn as tutorial course facility is a long way short of mastery, and indeed it
examples! Again, we are hidden from the whole truth. One is argued that the way to gain facility is to move ahead and
stark example is the algorithm used in symbolic integration, start to use the ‘level’ below. For example, you get a lot of
[6]. Even where some CAS do show “steps” in working, practice with integer calculation when you perform algebra
these packages are additional to the algorithms underlying with polynomials: [4]. So progress is not a linear journey and
the actual CAS. The number of people who really need is not simply a matter of assembling a portfolio of isolated
to understand all workings of this algorithm is minuscule mechanical skills in a rote manner. Perhaps we can jump
compared to those who will need to obtain a symbolic to the last stage more quickly by typing a problem into a
anti-derivative of a particular expression. So should we just CAS, without facility with all the intermediate steps, but I
throw this out, and let the machine do the work anyway? sincerely doubt it.
Arguments about the use of technology when teaching What might an examination question look like under these
mathematics are at least 375 years old. circumstances? It is clear that access to computational tools
“That the true way of Art is not by Instruments, but by in examinations would enable a much broader range of
Demonstration: and that it is a preposterous course of applied mathematics questions to be tackled, including
vulgar Teachers, to begin with Instruments and not with the statistics with real data. But what about pure mathematics?
Sciences, and so instead of Artists to make their Schollers I am not in a position to suggest examination questions for
only doers of tricks, and as it were Juglers: to the despite school, mostly because I am not sufficiently familiar with
of Art, losse of presious time, and betraying of willing and what is possible with school level students. However, for
industrious wits unto ignorance and idlenesse.” [5] first year university examinations the following is perhaps
one potential example.
On the other hand...
“All are not of like disposition, neither all (as was sayd Find
before) propose the same end, some resolve to wade, ∫sin(t) cos(t)dt
others to put a finger in onley, or wet a hand: now thus to
in three different ways. In each case express the answer
tye them to an obscure and Theoricall forme of teaching, is
as powers of sin(t), not as multiple angles.
to crop their hope, even in the very bud.” [2]
I don’t believe we need to be ‘good at integration’ for any However, we have just decided to throw out integration
practical purposes. Just as we threw away our slide rules, we techniques, so another question might look like the following:
can throw out much of the nineteenth century techniques
of integrating by hand. I already have: I use tables and a CAS Let p(x) = x3 – 6x2 + 10x – 3.
when integrating in anger but we must keep the underlying 1. Find the equation of the tangent line to p(x) at x = 2.
important techniques in mind. These include integration
2. Find the remainder of p(x) on division by (x – 2)2.
by substitution and integration by parts. I would expect
my students to be able to complete reasonable examples 3. Expand p(x 2). Let q(x) be the truncation of this
by hand in the traditional way. Not because the answer by removing all terms with powers greater than 1.
is important anymore, but because the understanding Expand q(x 2).
they generate is key, in its own right and in subsequent
4. What do you notice about the answers to (1), (2) and
mathematics. Mathematics is unusual in the extent to which
(3)? Why is this true?
The truth, nothing but the truth... but not the whole truth! – Chris Sangwin 31
MSOR Connections Vol 10 No 3 Autumn Term 2010
At least here the computation is certainly not the point of the
question. Indeed, hand calculation would take quite a long
time and would be error prone. Thus some students would
have nothing to ‘notice’ in (4), and would simply be confused.
Furthermore, if a CAS is available the conjecture could be
tested with other functions to perform a mathematical
experiment. Of course, the real pure mathematics work
comes in (4). A supplementary question ‘for what other
functions is this true?’ would be a significant mathematical
challenge, with an interesting historical precedent.
I think the issue of examination questions in an
environment where computation is not the focus is difficult
and of the upmost importance. The challenge, of course, is
striking the most productive balance between the extreme
and artificial positions of CAS = good vs CAS = bad.
References
1. Adams, W. W. and Loustaunau, P.(1994) An Introduction
to Grobner Bases, volume 3 of Graduate Studies in
Mathematics. American Mathematical Society.
2. Bryden, D. B. (1993) A patchery and confusion of
disjointed stuffe: Richard Delamain’s grammelogia of
1631/3. Transactions of the Cambridge Bibliographical
Society, 6:158--166.
3. Dorrie, H. (1965). 100 Great Problems of Elementary
Mathematics: their history and solution. Dover.
4. Hewitt, D. (1996) Mathematical fluency: the nature of
practice and the role of subordination. For the learning
of mathematics, 16(2):28--35.
5. Oughtred, W. (1634) The circles of proportion and the
horizontal instrument. Oxford.
6. Risch, R. H.(1969) The problem of integration in finite
terms. Transactions of the American Mathematical
Society, 139:167–--189, May 1969.
7. Sangwin, C. J. (2009) The wonky trammel of
Archimedes. Teaching Mathematics and its Applications,
28(1):48--52, March 2009. DOI: 10.1093/teamat/hrn019.
8. Sangwin, C. J. (2010) Intriguing integrals. Plus Maths
online magazine, 54, 2010. Available via: http://plus.
maths.org.uk/issue54 [Accessed 20 October 2010].
9. Sangwin, C. J. (2011) Modelling the journey from
elementary word problems to mathematical research.
Notices of the American Mathematical Society [to be
published in2011].
10. Stoutemyer, D. R. (1991) Crimes and misdemeanors in
the computer algebra trade. Notices of the American
Mathematical Society, 38(7):778--785, September 1991.
11. Sullivan, J. M. and Wetzel, J. E. (2010) An ancient
ellipse locus. The American Mathematical Monthly,
February 2010.
32 The truth, nothing but the truth... but not the whole truth! – Chris Sangwin
