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THE CASE FOR CRUNCHY METHODS IN PRACTICAL

MATHEMATICS

Michael Wood

Department of Accounting and Management Science

Portsmouth Business School

Locksway Road, Milton

Southsea, Hants, PO4 8JF

England

email: michael.wood@port.ac.uk

Abstract

This paper focuses on the distinction between methods which are mathematically "clever", and those

which are simply crude, typically repetitive and computer intensive, approaches for "crunching" out

answers to problems. Examples of the latter include simulated probability distributions and

resampling methods in statistics, and iterative methods for solving equations or optimisation

problems. Most of these methods require software support, but this is easily provided by a PC. The

paper argues that the crunchier methods often have substantial advantages from the perspectives of

user-friendliness, reliability (in the sense that misuse is less likely), educational efficiency and realism.

This means that they offer very considerable potential for simplifying the mathematical syllabus

underlying many areas of applied mathematics such as management science and statistics: crunchier

methods can provide the same, or greater, technical power, flexibility and insight, while requiring

only a fraction of the mathematical conceptual background needed by their cleverer brethren.

Introduction

Imagine a girl with 155 pence to spend on chocolate bars costing 37 pence each. If she has a

thorough command of arithmetic she will simply divide 155 by 37, obtain 4.2 and realise that she

can buy four chocolate bars and be left with some change. If she is not sure about division but

understands about multiplication, she might proceed by guessing the answer and then checking by

multiplying:

37 x 3 = 111, and

37 x 4 = 148, but

37 x 5 = 185, so

she can buy four, but not five, bars. Alternatively she may add 37 to itself until she reaches a total

which is more than 155. It should then be obvious to her that the number of bars she can buy is one

less than the number of 37's she has added together:

37+37+37+37=148, but

37+37+37+37+37=185

so she can buy four bars. Finally, she could adopt a simpler strategy still by getting her 155 pence in

individual one penny coins, and then counting them out in piles of 37 to see how many complete

such piles she has. There are thus (at least) four approaches to this problem: division, guess-

multiply-check, repeated addition and counting.

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There is often a similar choice with more advanced problems. Algebraic equations may be

solved symbolically (sometimes), or by numerical, trial and error methods; definite integrals can be

evaluated symbolically (sometimes) or numerically; optimization problems can be tackled by the

methods of the calculus (sometimes) or by a search heuristic; probability distributions can be

investigated either by mathematical theory (sometimes) or by computer simulation; queues can be

modelled by means of probability theory (sometimes) or by simulation; and so on.

The common element in all these examples is the choice between using a crude repetitive

method to crunch out the answer, and using sophisticated (relatively speaking), or clever,

mathematics. This paper treats the crunchiness of a mathematical method as a general property,

and considers the advantages and disadvantages of greater or lesser crunchiness. The conclusion is

that crunchier methods have a number of substantial advantages over cleverer methods: these

include the fact that they are conceptually more straightforward and they tend to be of more general

applicability and require fewer restrictive assumptions (note the frequent occurrence of the qualifying

"sometimes" in the previous paragraph).

My perspective in this paper is not that of an educationalist. I am not interested in how to

ensure that the girl above, and her friends, understand division, but that they have available a useful

range of approaches to problems - which may or may not include the concept of division. Similarly,

managers wishing to use management science techniques and medical researchers using statistical

techniques are not concerned with learning mathematics for its own sake, but in the availability of a

practical and reliable cognitive and technological toolkit for approaching their problems. These

toolkits include artifacts of various kinds (calculators, computers, etc). Needless to say these

artifacts are a crucial, and changing, factor in the situation. At present, the most important of the

these artifacts are computer packages (e.g. spreadsheets, simulation packages) and, at a more

elementary level, calculators.

The arguments concerning crunchy methods have implications for the design of artifacts to

support cognitive processes, and for the mathematics curriculum - particularly for older students

who need to use mathematical techniques of various kinds for practical purposes. This paper argues

that crunchy methods have the potential to offer very substantial improvements in the user-

friendliness, power and reliability of the cognitive and technological toolkits available to these

students.

The use of mathematically based techniques for practical purposes by people who are not

experts in the techniques, or the mathematics underlying them, poses widely acknowledged

problems in areas such as management and engineering, and most of the many fields to which

statistics is applied (see, for example, Yilmaz, 1996; Mar Molinero, 1996; Stuart 1995; Romero et

al, 1995; Greenfield, 1993; Bailey and Weal, 1993; Altman and Bland, 1991). The symptoms of the

problems include dislike, often bordering on fear, of the techniques and the courses which teach

them, as well as the consequent under-use and mis-use of the techniques in practical contexts; the

remedies suggested usually centre on relating the subject more closely to real world applications.

One argument of this paper is that a move to crunchier methods provides a very powerful means of

tackling these problems.

Crunchy methods

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We must start by defining the term "crunchy". In general, "clever" methods employ a mathematical

theorem to deduce the corollaries of assumptions about a situation; crunchy methods use the

assumptions directly to work out the corollaries, sometimes with the aid of some common sense or

heuristic principles.

To be more precise, we can say that Method A is crunchier than Method B if Method A is

1More directly linked to the intuitive definition of the situation (more transparent); and

2Likely to involve more repetitive steps; and

3Dependent on fewer, or lower level, technical concepts and theorems, so can be followed

by users with a lower level of technical expertise (lower technical level)

than Method B.

The choice of the word "crunchy" stems from the phrase "number crunching" and is intended

to convey the idea of a crude "crunching" through of stages or possibilities. Several other phrases

have a similar meaning in specific contexts: these include brute force methods, iterative heuristics,

numerical methods, computer-intensive methods (based on simulation) in statistics (Noreen, 1989;

Simon, 1992) and trial and error methods. However none of these really has the right meaning

across the full range of situations I have in mind. The closest is probably "brute force", but this does

not seem appropriate to crude methods of avoiding division, and there is also no convenient

comparative form like "crunchier".

I will illustrate the notion of crunchiness in the context of the chocolate problem. The initial

assumptions are that the shopkeeper will want 37 pence for the first bar, and another 37 pence for

the second bar, and so on (discounts are not available). The cleverest of the four methods is the

division strategy. The guess-multiply-check strategy is crunchier than the division strategy because it

is more clearly linked to the idea that the price of four items is four times the price of one (point 1 of

the definition), is likely to involve some incorrect guesses so the guess-multiply-check sequence

needs repeating (point 2), and does not depend on an understanding of the concept of division -

(point 3). A similar argument demonstrates that the repeated addition strategy is crunchier still (by all

three criteria), and the counting strategy is the crunchiest of them all (again, by all three criteria).

Similar considerations apply to more advanced problems: for example the estimation of the

probability of a family of four children comprising two girls and two boys. This can easily be worked

out using the binomial probability distribution which gives an answer of 37.5% (making the usual

assumptions of independence and equal probabilities for boys and girls). Alternatively the same

answer can be obtained by computer simulation of, say, 10,000 simulated families. Figure 1 shows

the result of such a simulation: the proportion of two girl families in this simulated set of families is

37.1%.

Figure 1: Simulation of 10,000 families of four

0 girls XXXXXXX

1 girl XXXXXXXXXXXXXXXXXXXXXXXXXXX

2 girls XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

3 girls XXXXXXXXXXXXXXXXXXXXXXXXXX

4 girls XXXXXX

(X represents 93 families.)

Proportion of 2 girl families: 37.1%

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This latter approach is crunchier because:

1The simulation is defined directly by the definition of the situation: each simulated family has

four children and in each case there is a 50% chance of the child being a girl.

2It involves simulating the 10,000 families and so involves repeating the same process 10,000

times instead of using the "clever" binomial formula.

3It involves no more conceptual background than the assumption of statistical independence

between the sexes of different children in the same family, and the notion of a probability as

a long run proportion. Use of the binomial distribution, on the other hand, requires

knowledge of the addition and multiplication rules of probability and the idea of

"combinations" if the distribution is worked out from first principles. Alternatively, if the

binomial distribution is taken "on trust", the user needs to have mastered the concept of the

binomial distribution as a mathematical entity, the assumptions under which it is valid, and the

nature of the information necessary to use the binomial equations. The conceptual

background needed here is far richer and more extensive than for the simulation method.

Statistical process control (Shewhart) charts entail using probability theory to monitor an ongoing

industrial or business process. The conventional methods, based on clever formulae derived from

probability distributions such as the binomial, are difficult for the typically non-expert users to

understand and interpret meaningfully (Hoerl and Palm, 1992; Wood and Preece, 1992), and, in

addition, the probability models often fail to fit closely the patterns found in real processes. A

crunchy approach - resampling (Noreen, 1989, Simon, 1992) - has been suggested to make the

methods more user friendly (Wood et al, 1999), and to make them mirror reality more closely

(Bajgier, 1992; Seppala et al, 1995). Concepts such as the standard deviation, central limit theorem,

binomial distribution and the normal distribution are an essential part of understanding how the

conventional methods work, but are irrelevant to an appreciation of the crunchy method.

There are many other examples. One is the calculation of the economic order quantity in

inventory management (see, for example, Dennis and Dennis, 1991, 452-7). This is the order

quantity which minimises the total costs to a business of ordering and of carrying stock. There is a

simple formula, based on a number of assumptions, which can be used for this purpose (Dennis and

Dennis, 1991, p. 453) - this represents the clever method. Alternatively, formulae for the ordering

and carrying costs, and for the total cost, can be set up on a spreadsheet, and then different order

quantities can be "tried out" to see which makes the total cost as low as possible. (Most modern

spreadsheets have a "solver" or an "optimiser" which enables this process to be automated, although

the results are generally not infallible and users would be advised to check - for example - that the

suggested optimum is in fact reached from different starting points - see the appendix for an

illustration of this.) This approach is crunchier because it follows directly from a simple model of the

costs, because it involves a repetitive trial and error process (even if this is performed automatically

by the spreadsheet), and because users do not need to understand the formula for the economic

order quantity or its rationale (which involves the differential calculus).

A very similar argument applies to evaluating an integral numerically instead of symbolically,

solving an equation by trial and error rather than analytically, using bootstrapping and resampling

methods instead of probability theory for statistical inference (Noreen, 1989; Simon, 1992), and to

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many similar examples. The computer Deep Blue which has recently defeated the world chess

champion Gary Kasparov provides another illustration of the theme. The strength of the computer is

ability to calculate the consequences of more possible moves further ahead than can a human being.

Against this, skilled human players have a more extensive and flexible repertoire of clever, but often

intuitive, strategies. The crunchy approach adopted by Deep Blue has been shown to be more

effective than the cleverest human being.

These examples show that the use of crunchy methods is far from being restricted to

mathematical novices. In some cases, the "clever" method may not exist or may not have been

invented yet (e.g. some integrals and the roots of some polynomials cannot be expressed as

algebraic expressions of a convenient form), so a crunchier approach (e.g. numerical methods to find

an integral or the polynomial root) may be the only possibility. In these cases Method B in the above

definition would be a hypothetical method. Sometimes a crunchy method may be helpful as a quick

approach for exploring a situation even if a cleverer method is feasible - an example of this appears

in the appendix.

The notion of crunchiness thus has three dimensions: transparency, repetitiveness, and low

technical level. There is no logically conclusive reason why these three dimensions should always go

together, although the three examples above make, I think, a plausible case that they often do.

Sometimes the heuristics employed for a search procedure may be highly sophisticated - e.g. genetic

algorithms, the Simplex method for linear programming, the methods used by spreadsheet solvers

and optimizers - so the method may be crunchy in terms of criteria 1 and 2 above, but possibly not

on criterion 3. It is also important to note that the definition of each dimension is slightly vague: how,

for example, is the technical level to be defined? Similarly, if our guess-multiply-check shopper

strikes lucky with the initial guess, repetition may be unnecessary. Strictly we have defined a means

of comparing methods, and so a continuum of methods of varying crunchiness, and not absolute

definitions of crunchy and clever methods, but it is helpful to speak loosely of the crunchier methods

as being "crunchy".

My aim in this paper is to bring these ideas together under one umbrella, and explore the

implications for designing practical mathematical curricula and the artifacts necessary to support

them.

Properties of crunchier methods

I will start with three positive points, and then go on to one neutral and three negative ones.

1 Crunchy methods are conceptually simpler

The three criteria used to define the crunchiness of methods imply three important senses in which

crunchier methods are simpler: they are more transparent in the sense that they are more directly

linked to the definition of the problem, they involve repetition of similar steps, and they demand less

in the way of technical conceptual background from users. Each of these is likely to make crunchy

methods simpler to understand - in the deep or relational (Skemp, 1976) sense - than their clever

equivalents.

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2 Crunchy methods tend to be more reliable in the sense that inappropriate use - misuse,

misinterpretation, or failure to use when appropriate - is less likely. For this reason users

may regard them as more trustworthy.

This follows directly from the first point: if the methods are more transparent and depend on less

technical background, then users must be less likely to misuse them. (Like the first point, this is

"almost" a tautology. If these points were found not to be true of some supposedly crunchy method,

then, by definition, the method cannot be as transparent as supposed. The qualification implied by

the word "almost" is due to the three dimensional nature of crunchiness - e.g. it is possible that

problems due to the repetitive nature of a crunchy method might cancel out the advantages of

greater transparency.) This implies that, in practice, the applications of crunchier methods are likely

to be more rigorous (than cleverer methods) because the conditions on which they are based are

more transparent.

The difficulties experienced by beginners in subjects such as statistics (see Introduction

above) illustrate some of the problems due to the use of clever methods. Pfannkuch (1997), for

example, says that "it did not seem to occur to some students to use a significance test ... Underlying

this aspect seemed to be a lack of understanding of variation in relationship to significance testing."

Using a crunchy method such as resampling (Noreen, 1989; Simon, 1992), which involves

simulating the variation in question and requires none of the paraphernalia of mathematically defined

probability distributions, means that the rationale behind the method and its relationship to practical

situations are more transparent. This is not, of course, to claim that the problems of teaching

statistics will all be eliminated, but that some of the obstacles can be removed.

There is a counter-argument to this that crunchy methods may sometimes (but by no means

always) be less rigorous from a mathematical point of view - this is discussed under Property 6

below.

3 Crunchy methods tend to be more general and so more powerful and more able to model

complex phenomena

The essence of a clever method is that assumptions have to be made about the structure of the

situation in order to deduce a way of working out the answer. With crunchy methods, some of these

assumptions are often implicit in the fact that a step is repeated, but these tend to be less restrictive

and easier to override. For example, if the shopkeeper offers a discount for people buying more

than two bars of chocolate, this is easier to build into the crunchier methods. Similarly, discounts can

easily be built into the spreadsheet method for estimating the economic order quantity (see above),

but the standard formula is based on a number of assumptions, one of which is that there are no

discounts; the simulation approach to the family problem can easily (depending on the software) be

adapted to take account of the possibility of twins and triplets whereas the clever (binomial) formula

cannot; and the resampling approach to statistical process control charts avoids the (frequently

unrealistic) assumption that distributions are normal (Bajgier, 1992; Seppala et al 1995).

These examples illustrate the way that crunchy methods can avoid (to some extent) the

necessity to impose a simple structure on reality because this is necessary for the mathematics. There

is a growing feeling in some corners of academia (e.g. in mathematics - Stewart, 1990 especially pp

81-4 - and economics, physics, biology and other areas - Waldrop, 1994) that, contrary to the

optimism engendered by the Newtonian world view, God may not after all be a mathematician: the

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world is more complex and less structured than mathematicians would like it to be. The initial

impression that the world follows simple, linear mathematical patterns may be simply a function of

mathematicians ignoring anything which does not fit this assumption. To the extent to which this is

true, crunchy methods are, by definition, more suited to real problems in a complex world than are

clever ones. There is, for example, increasing use of heuristic methods for searching for solutions to

complex, "messy" problems in management science (Pidd, 1996, pp. 290-310).

Another aspect of the same point is that crunchy methods may be more general in the sense

that they may incorporate several conventional clever methods. The formulae for the binomial

distribution will only model this particular distribution; the crunchy method (as implemented on a

computer program) on the other hand (Figure 1) will also produce a (simulated) probability

distribution for the means of random samples from any empirically specified distribution, and could

easily be adapted to draw random samples without replacement - thus simulating the hypergeometric

distribution. The same software and essentially the same method can be used to derive bootstrap

confidence intervals for the mean (Gunter, 1991), and estimate action lines for mean and range, and

median and standard deviation, and many other control charts for quality control purposes (Wood et

al, 1999). Similarly, trial and error methods of solving equations work for any equation provided that

the user has a means of evaluating the two sides: clever methods on the other hand require that the

equation is of a particular type - e.g. there is one clever method for linear equations, another for

quadratic equations, another for trigonometric equations of a particular kind, and so on.

In addition to this, crunchy methods are often possible in situations where no clever method

has been invented, or even where it has been proven that none can exist (see above for examples).

4 Any method encourages the development of concepts which refer to the answers

produced. Crunchy methods, being different from clever methods, may lead to a different

set of concepts

Imagine someone who uses the counting strategy for problems like the chocolate one. If this method

becomes "interiorized to become a process [so that it is possible for] the individual to think about it

as a totality" (Cornu and Dubinsky, 1989), the individual in question is almost bound to develop,

implicitly or explicitly, a concept which refers to the answers produced by this counting strategy.

Someone tackling the chocolate problem with the operation of division and a calculator, on the other

hand, will have the concept of a "quotient" (or "answer produced by division") to describe the

answer produced by the method. The counting strategy does not produce quotients - the answer is

always an integer - and the concept developed is slightly different (and difficult to describe neatly).

Sometimes the clever and crunchy methods may lead to essentially the same concept for the

answer. The binomial simulation in Figure 1 leads to a proportion which is a binomial probability in

just the same sense as the answer produced by probability theory. The methods may be different but

the answers both refer to the same concept. On the other hand, if the same simulation software is

used to simulate other distributions (as described below), the concepts for the answers produced by

the crunchy method may be more general than those for the clever method.

These concepts are important. A very plausible theory of mathematical learning (Leron and

Dubinsky, 1995; Cornu and Dubinsky, 1989) asserts that the process of interiorizing an "action" or

a method so that it can be viewed as "a totality" or a "higher level" object is a vital part of

mathematical learning. In the language of cognitive psychology, viewing a method as a whole as a

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single "chunk" of information (a "maximal familiar substructure") is helpful for higher level thinking

about the method (or the results of the method) since human information processing can handle no

more than about seven chunks of information at a time (Simon, 1996). In the case of the

mathematical methods we have been discussing, a core aspect of the this higher level concept or

chunk of information is the nature and interpretation of the answer produced.

Sometimes concepts developed through the clever method may be more useful than those

developed by crunchy methods. The notion of a quotient which may be fractional, for example, is an

essential prerequisite for understanding the output of the crunchy binomial simulation. On other

occasions the greater generality of crunchy methods may lead to their underlying concepts being

more useful because of this greater generality.

One point to note here is that, for a variety of fairly obvious reasons, the vocabulary for

describing answers from clever methods is likely to be far richer and more established than the

corresponding vocabulary for crunchy methods: e.g. I could not think of a suitable term for the

answer to the chocolate problem from the counting strategy. It may be helpful to invent terms to

label some of these implicit concepts.

5 Crunchy methods tend to be computationally intensive or slow

This is a consequence of the repetitive nature of the methods. How serious a problem it is clearly

depends on the balance between the power of any computer support available and the number of

repetitions necessary.

6 Crunchy methods may only yield an approximate or tentative or unproven answer

Simulation methods will not yield an exact probability, and guess and check methods will yield an

exact answer in the chocolate problem (where the answer has to be an integer) but may not in

similar problems where the possible answers lie on a continuum. However, in both cases the method

can yield an answer to any desired degree of accuracy - which is all that is required in most practical

situations.

With optimisation problems, the typical crunchy search procedure may be fallible if, for

example, the search heuristic finds a local optimum rather than the required global optimum. A

cleverer method may find the global optimum, and provide a proof that it is in fact optimal. On the

other hand, the difficulty is often that there are no viable clever methods so there may be no

alternative to a crunchy heuristic approach (Pidd, 1996, chapter 10).

In practice, the potential advantages of the greater accuracy and rigour of some (not all)

clever methods compared to their crunchier alternatives, may be nullified if the clever methods are

misused or misinterpreted - see Property 2 above.

7 Crunchy methods do not provide a general answer and so may fail to provide insights

into the structure of the situation

For example, clever methods for solving polynomial equations (e.g. factorisation) demonstrate that

the maximum possible number of solutions is equal to the degree of the polynomial, and the formula

for the standard deviation of a binomial distribution indicates the relation between sample size and

the spread of the distribution. No such insights would be likely from the use of a search heuristic or

simulation method.

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The first three of these properties (the positive ones) are to some extent offset by the last

three. However it is important to stress the enormous practical advantages of the first three points:

the simplicity of crunchy methods, and their potential generality and ability to model complex

situations. To take two examples of the first point, simulation methods can be used as a substitute for

probability theory (see, for example, Simon, 1992) with the consequent simplification of large areas

of statistical theory, and the Solver built into the spreadsheet Excel will solve a very large variety of

equations and optimisation problems. The second point is equally important, and linked to the first

point in that the general nature of crunchy methods means that fewer such methods are necessary to

deal with a given range of problems with a subsequent simplification of the task of learners and

users. In addition crunchy methods will reach areas which clever methods cannot: this is illustrated

by the use of simulation and heuristic methods in areas as diverse as operational research,

meteorology, economics and so on. Clever methods can only work if the universe conforms to the

assumptions on which they are based; if it does not, crunching out the answers is the only option.

Computer support for crunchy methods

Clever and crunchy methods can both be implemented with or without machine support - as is

illustrated by the chocolate example. However, for more advanced problems, crunchier methods are

likely to be more dependent on computer support systems (CSSs) than are cleverer methods. It is

easy to use the binomial probability formula without the help of a computer. On the other hand,

while it is possible to simulate sufficient families of four to estimate the required probabilities without

a computer, it is not really a practical proposition. Many crunchy methods are only practically

feasible because of the ready availability of computers. Accordingly, it is helpful to consider the

computer systems in question. (I will assume that the artifacts for supporting crunchy and clever

methods are computer packages - which seems a reasonably assumption at the present state of

evolution of technology.)

A computer support system for a clever method fulfils some or all of three functions:

1it helps the user to develop or choose the appropriate method; and/or

2it implements the method; and/or

3it helps the user to interpret the results of the method.

For example the calculator used for the chocolate problem just supports function 2; on the other

hand many statistical packages give the user help with 1 and 3 too. However, at the present state of

the art, computers are far more helpful for 2 than they are for 1 and 3. The reasons for this are fairly

obvious: one major problem is often the lack of a common frame of reference for communication

between user and software system (Wood, 1989) - non-expert users of statistical packages, for

example, may lack a clear understanding of essential concepts like "significant", "interaction", "main

effect" and even terms like "variable" and "data".

In principle, a computer package to support a crunchy method could support the same three

functions. However, the enormous advantage of a crunchier method is that 1 and 3 (the functions

which are difficult to support) are much less problematic so support is less necessary - but this is

only likely to be true if the user knows what the CSS is doing. If some intuitive procedure is being

repeated many times, it is important that the user appreciates exactly what the repeated procedure

is. Then, given the facts that the rationale behind this follows directly from the intuitive understanding

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of the situation, and that the concepts involved are relatively simple, the whole method should be

clear so it should be obvious when it is useful and what the answers mean. In short, the problems

users typically face using a mathematical CSS are solved.

This means that a crucial feature of a crunchy computer support system is that the method

implemented should be transparent to the user. The obvious way to achieve this is to allow the user

the option of stepping through the method step by step. Then, when it is clear how it works, the

method can be accelerated and the details hidden from view.

How might this work in practice? A crunchy binomial support system allows users to

simulate the first few families individually, and see the results put on the histogram - as in Figure 2

below. Similarly the solver or optimiser on most spreadsheets allows users the option of seeing the

first few iterations.

Figure 2: First two simulated families of four

1st simulated family (girl=1, boy=0): 1 0 1 1

Number of girls in this family is 3

0 girls

1 girl

2 girls

3 girls X

4 girls

(X represents 1 family.)

Proportion of 2 girl families: 0%

2nd simulated family (girl=1, boy=0): 0 0 0 0

Number of girls in this family is 0

0 girls X

1 girl

2 girls

3 girls X

4 girls

(X represents 1 family.)

Proportion of 2 girl families: 0%

We may contrast this with a package for implementing a clever method. Here nothing would be

gained by stepping through the method a step at a time because the method is not directly linked to

intuitions about how the situation works, it is not repetitive and the conceptual background it links to

is relatively complex. A Help facility would be a possibility but the help is likely not to be very helpful

if users lack an extensive understanding of probability theory. In practice, users of a statistics

package such as SPSS tend to treat it as a black box and are often unaware of the methods used by

the package to obtain the answers. This is in strong contrast to the way a crunchy CSS can show

users the steps of the method and some intermediate results. This is another reason for the extra

transparency and reliability of crunchy methods when implemented with suitable software.

Strictly, the CSSs shown in Figures 1 and 2 are unlikely as practical CSSs because they

refer specifically to the family problem. (I have changed the output to clarify the argument of this

11

paper for readers not familiar with the subject area.) To be useful, a practical CSS needs to be able

to support the solution of a range of different problem types - not just problems about girls in

families of four, but about boys in families of any size, about defectives in samples taken for quality

control purpose and about scores in multiple choice tests - all of which can be modelled by the

binomial distribution.

In fact, Figures 1 and 2 (except words like "family" and "girls") are produced by a simple

program whose uses are far wider than the binomial distribution. The program, RESAMPLE, stores

a list of numbers (data) and then takes random "resamples" (with replacement) from this list and

analyses the mean, sum, standard deviation, median or any percentile of the resamples. The program

can be used to produce bootstrap confidence intervals (Kennedy and Schumacher, 1993; Gunter,

1991) for any of these statistics, estimate control chart limits (Wood et al, 1999), simulate various

probability distributions, and several other things as well. In effect the program is a crunchy method

for estimating the degree of variability between random samples drawn in similar circumstances. As

well as the advantages of transparency (see Figure 2), this crunchy method has the advantage that

the learner has only to master one software tool to cover a range of different (from a conventional

viewpoint) contexts.

Spreadsheets are another useful tool for implementing crunchy methods (see the Appendix

for an example; the family simulation in Figure 1 could also be implemented on a spreadsheet), and

there are obviously many other possibilities.

The practical detail of the software is clearly important. Software which is difficult to use

may hinder users; different interface styles (e.g. menus, commands) may have different strengths and

weaknesses; the words used to describe inputs, outputs and operations are obviously important;

there is a strong case for people using software with which they are familiar as much as possible.

There is, however, no space to pursue these issues here, except to point out their importance.

We turn now to some practical choices. We will consider three general situations. The first is

the situation where a CSS must be designed to support a given set of mathematical methods. The

second is where the CSS must be designed to support problem solving in a given area for a given

group of people with a given level of expertise - there is an extra element of flexibility in that different

methods may be considered. The third situation is like the second, except that the expertise of the

users is treated as part of the choice: what do people need to know to use the best possible

approach to a given type of problem? The first is a short term problem; the second is a medium term

problem in that it assumes that users can be weaned away from ingrained habits towards more

appropriate methods; and the third is the long term problem in that it assumes that the training and

education system can take account of the computer systems which are likely to be available.

Choice 1: support systems, given methods and user education

This the usual situation. The method is seen as given, and the CSS is designed to support this

method as well as possible given the expertise of the users. The difficult aspects are likely to be

phases 1 and 3 of the process (above): these are why the user's understanding of the method and its

interpretation is crucial.

Usually the method to be implemented is on the clever end of the continuum. There may, on

occasions be a possibility of utilising some of the user friendliness of the crunchier methods by

12

pretending to the user that the method is in fact crunchier than it is. We may call such methods

virtual crunchy methods.

As an example, consider the case of a computer systems for multiple regression such as

those provided by the tools in many spreadsheets. In fact, multiple regression, is based on some

clever mathematics and does not use an iterative method. This leaves the uneducated user (relatively

speaking) with the problem of understanding precisely what multiple regression is and when it is

appropriate to use it.

However it would be possible for a multiple regression CSS to start with a guess for the

coefficients of the least squares model, show the user how the squares are computed, and then

move on to another set of coefficients with a slightly lower sum of squares. In this way the user

would get the idea of the technique as a way of searching for a least squares solution, but then, when

the button is pressed to speed up the process and find the best answer, the standard algorithm

would be used. The idea of searching for a best answer can be used as a metaphor to explain to the

user what is happening. The justification for this is simply that the two methods are mathematically

equivalent.

(In fact, regression problems can easily be solved on a spreadsheet by using the Solver or

Optimiser to minimise the value of a function corresponding to the sum of squares. The real crunchy

method does work in this instance. Its disadvantage over the built in regression formulae is that it

takes time and some skill with spreadsheet formulae to set up; its advantages are its transparency

and the fact that the method can very easily be adapted to models other than the simple linear one

assumed by the built-in formulae.)

Choice 2: methods and support system, given user education

If we view the method as adjustable to suit the needs of the users and the situation, then a much

wider range of possibilities opens up. Clearly, given the advantages of crunchier methods, these

methods would often be the preferred ones for the computer to support, although there are counter

arguments as discussed above. Each case would need to be treated on its own merits, taking

account of the level of expertise of potential users.

So, for example, a suitable CSS for the very young child in the shop might use the crunchiest

approach - counting pennies; similarly for the family example, the simulation method (using whatever

software is convenient) would probably be more appropriate than the binomial distribution. On the

other hand for users with a higher level of expertise, the cleverer methods may be more appropriate.

There may be situations where there is a justifiable fear that the provision of computer

packages supporting crunchy methods may remove the incentive for users to master the more

advanced concepts required by the clever methods. This is the reason why I suspect that few would

support the use of computer-based counting software in primary schools. However, I do not think

this argument applies to many more advanced problems for the reasons discussed in the next

section. These conclusions may be complicated by convention and expectation. In practice problems

are often phrased in terms of the method or model and not the requirements of the situation;

academics may want the results of an analysis of variance, and production managers may want the

economic order quantity as defined by the standard formula (Dennis and Dennis, 1991, p. 453). In

each case custom or conventional advice decrees the appropriate piece of mathematics to be used,

13

and the situation is in effect, a choice of type 1 above. In the longer term, it may be possible to focus

on the actual requirements of the situation and so move towards the more powerful choice where the

method can be freely chosen.

We have mentioned above that one probable advantage of the crunchier methods is their

greater generality. This factor is likely to be extremely important for CSSs simply because a CSS is

of little use if users do not know of it, and in rough terms what it will do - and acquiring this expertise

inevitably takes time. General methods are likely to lead to a reduction in the time users need to

spend familiarising themselves with what is available, and also mean that the process of choosing an

appropriate approach to a given problem is likely to be easier simply because there are fewer

possibilities to choose from.

Choice 3: methods and support systems and user education: implications

for the curriculum

From a long term perspective we can ask what conceptual background education needs to foster

given the availability of CSSs implementing crunchy methods. Is the binomial probability distribution

a sensible part of the standard curriculum of elementary statistics given the availability of simulation

methods (which can be implemented on any suitable software including a spreadsheet)? Is division a

sensible part of the primary school mathematics curriculum despite the crunchier methods which can

be used?

My answer to the first of these questions would be no, and to the second yes - and I would

suspect that I am not alone in this judgment. There are three important differences between the two

situations: division is much more commonly used than the binomial distribution, people are likely to

want to use it when they have no CSS to hand, and it leads on to many other ideas which non-

mathematicians would expect to master, whereas the binomial distribution does lead on to further

ideas (e.g. the normal distribution) but exactly what and how would usually be considered to be of

interest to mathematical specialists alone.

In rough terms, if we view mathematical methods as forming a hierarchy in which the lower

levels are prerequisites for an understanding of the higher levels, the general conclusion is that there

is likely to be sense in using crunchy methods for the top level of the hierarchy since this does not

lead on to further ideas. This conclusion holds for both the novice and the expert mathematician.

It may also be reasonably to use crunchy methods in parts of the lower levels of this

hierarchy, if the concepts developed by the crunchy methods (see Property 4 above) are adequate

for the higher level developments. For example, the simulation of the binomial distribution leads to

the concept of a binomial probability distribution (but without the probability formulae). This gives

users a label and a concept which can be basis of further theory. The normal distribution can now be

viewed as a limiting case of this binomial distribution. The basic pattern of the normal distribution,

and the fact that this pattern is similar to many empirical distributions, is quite clear from the

simulations; the only thing missing is the formula.

The implication of this conclusion for education in mathematical methods is simply that many

clever methods may not be worth learning. Each clever method would have to be considered on its

merits, but my judgment is that this strategy of replacing clever methods (especially those at the

highest level of the cognitive hierarchy) by a few crunchy principles would lead to a very substantial

14

reduction in the technical content of mathematics education. However, the increased transparency of

the crunchy methods may lead to more powerful and realistic approaches to problem solving.

An objection

When I showed an earlier version of this paper to a colleague his reaction was indignation based on

the assumption that these ideas would, if taken seriously, "lower standards". Students would no

longer need to grapple with "proper" mathematics such as methods for solving polynomials and

mathematical probability distributions.

There were, I think, two points of difference between us. First we differed on the nature of

"proper" mathematics. From my perspective the ability to formulate models was far more important

than the ability to find a formula, and if some topics were discarded from the curriculum this was just

a part of the inevitable change that the progress of technology brings to a rational curriculum (see,

for example, Wood et al, 1997). In the words of Stewart (1990, p. 82):

"Formula? Who cares about formulas? Those are the surface of mathematics, not

the essence!"

The second difference between us was perhaps even more fundamental. My perspective is

that we want to help students develop as powerful and reliable a framework as possible with as

little pain and effort as possible. My colleague's implicit assumption was that we wanted to

develop as much understanding as possible of a given curriculum. For a given problem, from the first

perspective, it is generally sensible to encourage the adoption of the easiest method, but from the

second it may be more sensible to encourage learners to use the most difficult method since they

will then learn more. From my perspective the use of methods which are too difficult for learners to

grasp easily with the time and resources available is silly as it is likely to lead to doctors failing to

grasp the basics of statistics (Altman and Bland, 1991) and production managers failing to grasp

essential statistical principles of quality control. From my colleague's perspective the use of these

methods is necessary to "preserve standards" in academia, but not, unfortunately, in real life.

The fact that a method is easy does not mean that it is a bad method. Crunchy methods

should not be regarded as lacking in rigour; on the contrary, their transparency means that users can

see exactly what is going on and check the plausibility of any assumptions made.

Some related arguments

There are a number of other arguments about mathematics, education and computers, which I will

mention here very briefly to clarify their relationship with the argument of the present paper.

Computers and calculators can perform clever methods without help from users; therefore

users can concentrate on applications and interpretation, and do not need to concern

themselves with technical details.

This is regarded by some as a good thing, and by others as a bad thing. It is, however, quite

different from the argument of the present paper, which concerns crunchy methods which are

thoroughly understood.

Crunchy methods like simulation or trial and error methods are useful for helping

students develop the insights which will allow a deeper understanding of clever methods

15

This is doubtless true, but the argument of this paper is that it is often sensible to treat crunchy

methods as an end in their own right, not as the means to another end.

Computer programming is helpful for learning mathematics

Encouraging students to program mathematical activities and processes has been found to be an

effective tool for making them explicit and so helping students learn mathematics (Leron and

Dubinsky, 1995; Cornu and Dubinsky, 1989; Papert, 1993). This principle could be applied to

helping people understand the principles behind both crunchy and clever methods, although it may

apply more naturally to crunchy methods with their typically repetitive algorithms.

However, the argument in this paper is not about learning mathematics, but about which

mathematical methods are the most appropriate; about the content of the curriculum, not the

methods. The computer support systems discussed above are primarily for doing mathematics, not

for learning mathematics (although they may be helpful here too).

Conclusions

I have defined the crunchiness of a mathematical method in terms of:

*its transparency, and

*its repetitiveness, and

*the level of conceptual background required for its use.

Some methods may be crunchy according to some, but not all, of these criteria.

Crunchy methods are contrasted with clever ones. Crunchier methods are likely to be:

*conceptually simpler, and

*more general and so more powerful and more able to model complex phenomena.

This means that they are often preferable from the perspectives of user-friendliness, reliability and

realism. On the other hand crunchy methods:

*tend to be computationally intensive and slow, and

*may only yield an approximate or tentative answer on some occasions, and

*may in some situations be less useful for building further concepts and techniques.

Sometimes these disadvantages are important but often they are of little consequence.

It is fairly easy to devise methods which are crunchier than many conventional clever ones.

Examples include simulation, resampling and bootstrapping for problems in probability and statistics;

numerical methods for solving equations, optimisation and integration problems; simulation for

modelling queues; and so on.

Crunchy methods are ideally suited to computer packages for supporting mathematical

reasoning - particularly for non-expert users. To take advantage of the transparency of crunchy

methods it is important that users can step through a method slowly to see how it works. Crunchy

methods should then have the potential to enable novice users to develop useful methods for real

problems and interpret their results with genuine, reliable and consistent intelligence. This contrasts

strongly with the present situation where mathematical and statistical methods are frequently

misunderstood, misused and their results misinterpreted.

In practice, these opportunities have been taken up very little, although there is a limited

recognition of the advantages of resampling and bootstrap methods in statistics. Usually, computer-

16

supported mathematical reasoning follows the established methods. In part this is probably because

users' understanding of some of the methods is inadequate to enable them to distinguish the method

from the problem it solves; the problem is the method and possibilities for improvement are

unrecognised. The difficulties are exacerbated by the expectations of examiners and curricula.

The final section of the paper argues that the crunchier methods offer an opportunity to

simplify the mathematics curriculum very considerably. Many of the clever theorems of mathematics

are simply not necessary (for practical purposes: I am not referring to the study of pure mathematics)

if you have a computer to crunch out the answers. Much of the standard curriculum of mathematics

applied to statistics, management science and similar disciplines is unnecessary. In its place would go

a few crunchy methods implemented by suitable computer packages - e.g. one for simulating

univariate probability distributions, one for optimising numerical functions, and so on - and a more

thorough understanding of general principles and "lower level" concepts.

I will finish by raising a few questions prompted by these considerations. Is it possible to find

a crunchier alternative to any mathematical method? Does the idea just apply to mathematical

methods? Can we compile a short list of crunchy methods which subsume all commonly used clever

ones? As is often the case, I suspect that the answers to these questions depend on a careful

definition of the terms in which they are phrased.

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Appendix - Three methods for exploring a problem

I was recently writing a paper which puts forward a numerical model based on this equation:

c = (1-(1-p)n)v

c and p represent quantities which are determined by assumptions which can be intuitively linked to

the situation being modelled (market research) - typical values would be c = 0.8 and p = 0.1. The

purpose of the model is to estimate a suitable value of n - the size of a sample. Unfortunately v is

unknown and cannot, in principle, be determined.

If the range of values of n corresponding to all "reasonable" values of v was "reasonably

narrow", then I decided that the model would be still useful. Such a reasonable range of values for v

included all integers from 5 to 1000. What is the corresponding range of values of n? There are four

obvious methods for solving this.

The crunchiest approach to this question (Method 1) is to set up a spreadsheet and use trail

and error:

pvnc

0.1 5 50 0.97

18

The first three cells contain numerical values: the first two given by the constraints of the problem,

and the third (50 for n) is a first guess for an appropriate value for n. The fourth cell contains the

formula above to calculate the corresponding value of c. I then simply changed the vale of n until the

value of c was 0.8. This process took four trials and was quick and straightforward. The answer

was that n is 30. I then changed v to 1000 and repeated the process to find that n now has to be

80. (I decided this was a suitably narrow range.)

Method 2 was the same but using the spreadsheet (Excel) Solver to find the value of n

corresponding to a target value of 0.8 for c. This arrived at the same answers but took slightly longer

due to the time taken to set up the parameters of the Solver. When I tried again, starting from an

initial value for n of 1, the computer failed to find a solution - which is one of the problems with

heuristic methods.

Method 3 involves manipulating the equation and rewriting it as

n = log(1-c1/v)/log(1-p)

This formula can now be entered in the cell for n, and 0.8 entered in the cell for c which gives the

values of n directly without trial and error.

Method 4 would be to use the last equation to try to deduce, in general terms, how sensitive

n is to changes in v. In practice I could not see any easy way to do this, and as Method 3 had told

me all I needed to know, I abandoned Method 4.

The four methods are clearly in order of increasing cleverness and decreasing crunchiness.

Method 1 requires no ability to manipulate equations, and also seems the most trustworthy in that it

is direct and gives very little scope for errors. It is also a perfectly rigorous method and can give an

answer to any required degree of accuracy. It seems the obvious method for anyone who is not

good at manipulating equations.

As Method 1 was so quick Method 2 had few advantages in this situation. With more

variables, Method 2 may be preferable to Method 1 - but the Solver can never be regarded as fully

reliable.Method 3 was my preferred method. It enabled me to build up a table of values of n

corresponding to the different values of the other variables. On the other hand it is no more rigorous

than Method 1, and seemed less trustworthy in that I was not confident I had not made an error in

manipulating the equation.

The main drawback of method 4 was that I was not clever enough to do it.