Posted on Nov 28, 2004 in Issue no.1, November 2004, Volume 04

*By Maria de Hoyos Guajardo, Ph.D. Candidate, M.Sc., B.Eng.*

The theory that is presented below aims to conceptualise how a group of

undergraduate students tackle non-routine mathematical problems during a

problem-solving course. The aim of the course is to allow students to experience

mathematics as a creative process and to reflect on their own experience.

During the course, students are required to produce a written ‘rubric’ of their

work, i.e., to document their thoughts as they occur as well as their emotions

during the process. These ‘rubrics’ were used as the main source of data.

Students’ problem-solving processes can be explained as a three-stage process

that has been called ‘solutioning’. This process is presented in the six sections

below. The first three refer to a common area of concern that can be called

‘generating knowledge’. In this way, generating knowledge also includes issues

related to ‘key ideas’ and ‘gaining understanding’. The third and the fourth

sections refer to ‘generating’ and ‘validating a solution’, respectively. Finally, once

solutions are generated and validated, students usually try to improve them

further before presenting them as final results. Thus, the last section deals with

‘improving a solution’. Although not all students go through all of the stages, it

may be said that ‘solutioning’ considers students’ main concerns as they tackle

non-routine mathematical problems.

An important activity in students’ problem-solving process is to generate

knowledge about the situation; i.e., to generate relevant data and information

and to gain understanding. This is usually conducted at the start of the process,

particularly if students know little or nothing about the situation. For this reason,

generating knowledge and understanding seems a good place to start the

discussion on students’ problem solving processes. However, it must be made

clear that the need to generate knowledge will continue to emerge throughout

the process and that students respond to this need in ways that will be

discussed in this section.

A common strategy that students use as they try to generate information and

understanding is to reduce the complexity of the situation that they are dealing

with. By reducing complexity, students “start at the beginning” and focus on

intentionally simplified or even trivial versions of the situation. Students’ aim

behind reducing complexity is to start gathering the information and

understanding that will allow them to eventually move on to more sophisticated

cases. Reducing complexity may help students gain access to complex

situations by reducing them to simpler, more manageable ones.

*Numbers which can be expressed as a single prime to a power may be*

* a good place to start…*(Oscar, Liouville, p. 2)

*Right, let’s think about this. Start simple and work my way up*…(Hillary,

Steps, p. 1)

Students generate information and gain understanding about the situation in

many ways. Thus, it is hypothesised that the only limit for students as they try to

generate useful information and understanding might be the one imposed by

their own creativity and mathematical abilities. The following is a brief list of the

types of activities that students conduct for this purpose. The list is not extensive

and other activities may be included from further research:

• A common way in which students generate information and understanding is

by ‘specialising’, i.e., by looking at particular aspects of the situation. When

students specialise, they focus on isolated aspects of the situation and thus on

simplified versions of the problem. For this reason, it may be said that

specialising is intrinsically about reducing complexity. Most students specialise

at one point or another in their processes and the choice seems to be made in

a ‘natural’ way (“My instinct to this problem is to start from the easiest case.”)

However, during the course, students were specifically introduced to Mason’s

(1982) idea of specialising. This fact may account for the students’ tendency to

specialise and to label their activity in that way.

*I will start by specialising and using squares, since they seem more*

* straightforward, and then progress to rectangles. *(Hannah, Cartesian

Chase, p. 2)

• In order to start making sense of the situation, students sometimes ‘import’

ideas or information from sources other than the problem and the situation that

it presents. These ideas may be relevant to the problem and in the sense that

they may help students to better understand the situation and deal with it.

Recalling past knowledge or experience are common ways of importing

information.

*I know a similar problem. Diagonals of a Rectangle, which seems to be*

* related and I think I can use my solution.* (Emilio, Visible Points, p. 1)

*Fault line – brings to mind brick walls. In a brick wall you couldn’t have*

* such a line because the wall would be too weak. Conjecture that brick*

* laying pattern may prove the answer. I will carry on specialising and will*

* come back to this conjecture later. *(Kirk, Faulty Rectangles, pp. 1–2)

Students may also import information from other sources such as their notes (or

any bibliographical reference, from that matter). Sharing ideas with fellow

classmates may also be a way of gaining information and/or understanding.

Importing requires borrowed ideas to be evaluated in terms of their relevance

and applicability to the present situation. Importing can provide useful

information but also presents the risk of considering irrelevant ideas that may

have to be abandoned at a later time.

• Another way of generating knowledge is by taking a ‘hands-on’ approach and

carrying out the basic operations that are relevant to the situation. For

instance, in ‘Faulty Rectangles’ students physically constructed rectangles with

pieces of domino and observed what combinations could lead to fault-free

rectangles. Another example can be given in relation to ‘Ins and Outs’, where

students conducted hands-on investigations by folding pieces of paper and

observed the sequences of folds that were generated. Hands-on investigations

provide students with first hand experience of the situation and may lead to

gaining important knowledge and understanding.

*Shall I try playing it? Use a chessboard and a pawn. *(Jules, Cartesian

Chase, p. 1)

• A way of generating useful information and possibly understanding is by

organising the data that is available. This may involve arranging available

information in a convenient way so that further information becomes more

evident and easier to spot. Tables of values are a common example of

organising the data, but any other method for visualising the situation can also

be of help.

*I will use a table to search for some patterns:*(Keith, Sums of Diagonals, p. 2)

[please see PDF version for all diagrams]

*If I try to draw a diagram of the possible outcomes this may help give me a*

* better idea of what is happening and may lead to further development*. (Lila,

Steps, p. 1)

• An important way of finding more about the situation is by carefully analysing

the information that is available or that has been made available. In some

cases, information and understanding may emerge easily by looking at the

data. In other cases, however, students have to make conscientious efforts in

order to generate knowledge. By insistently considering (or reconsidering)

available information and trying to understand it, it may be possible to derive

further information and understanding from it. This may involve reviewing the

data and making deliberate efforts at drawing out observations and ideas.

*Ok, let’s look at our previous example.*

*N=4*

*Stage 1: 1, 2, 4 [Divisors of N]*

*Stage 2: (1), (1, 2), (1,2,4) [Divisors of divisors of N]*

*Is there any significance about the numbers at stage 1?* (Jared,

Liouville, p. 6)

*Can’t see anything from 3 folds. Only – I guess that the sequence that*

* happened in the previous fold would happen in the current fold again,*

* so 4 folds should start with in in out in in out out, and something else. I*

* want to guess more detail about the 4 folds because I want to prove my*

* prediction is correct. But this is what I can see now.* (Patrick, Ins and

Outs, p. 1)

It is not uncommon for students to combine these activities by either conducting

them at the same time or by sharing information from one activity to another. For

instance, students may take a hands-on approach as they gather information for

a table of values. Another example is when students conduct a close analysis of

information that has been generated after a period of specialising. As said, there

is no imposed limit to what students can do in order to generate information and

understanding.

The need to generate knowledge will continue to emerge throughout the

process. New information and understanding may be required at any stage, from

situations in which students are looking for new ideas to situations where they

are trying to take an idea further. In other words, students may incur in the

activities discussed above at any time during their process.

Finally, students make reference to the information they observe in the form of

written or verbal observations. Trying to gain knowledge about the situation

leads students not only to noticing but also to ‘making a note’ on those new

pieces of information that may be relevant in terms of generating a solution. The

next subsection looks at the observations that students make as a result of

dealing with the data.

The information and understanding that students generate may become

manifest in the form of observations. Observations are facts or ideas about the

situation that students may find interesting or relevant, and that they choose to

point out in a written or verbal way. In some cases, these observations may lead

directly to an initial solution.

*AHA! The pattern behind the centre is just a pattern of the previous one,*

* while those behind is just the opposite way around […]*

*Therefore, if we repeat this, we would be able to generate a sequence*

* after 10 folds.* (Karina, Ins and Outs, pp. 1–2)

In other cases, however, observations may involve information that may or may

not be used at a later time.

This is to say that not all observations will be useful in the same way. Some may

inform students about ways to generate a solution (like in the example above)

and some may provide less central (though not necessarily unimportant)

information. In some cases, important observations are easily identified as such.

In other cases, it may take the student time and effort to be able to tell whether

a certain piece of information is relevant or not.

*Slope=(4-1)/(4-1)=1.*

*AHA! The gradient of slope 1 is 1. I can use the same method and*

* apply it to slope 2.*

*Slope=(9-1)/(5-1)=2.*

*Aha! I got it!* (Patrick, Sums of Diagonals, p. 2)

*Obviously, I can only pull out the numbers 1 and 2 and the difference*

* between these is 1.*

*Hmm… could this always be the case (wild guess)? Or is it too early to*

* tell.* (Aminta, Hat Numbers, p. 1)

When students come across an observation, sometimes they adopt what can be

called a ‘pragmatic’ approach. Adopting a pragmatic approach involves focusing

not only on the observation itself but also on how it can be used for generating a

solution. When students adopt a pragmatic approach towards making

observations they ask themselves questions like “How can this [idea, fact, etc.]

be used?” A pragmatic approach can help students decide more efficiently

whether an idea is useful and how.

The examples below (as well as Patrick’s example above) illustrate cases where

students considered observations in a pragmatic way. As the second example

below suggests, a pragmatic approach may help students discriminate

unimportant ideas and thus may help in making their process more efficient.

Thinking in terms of how ideas can be used seems to lead to starting to

generate a solution sooner than if observations are made without considering

their usefulness or applicability.

*The answers for 2 and 5 give the answers for 10. Does this work for*

* other numbers?* (Julia, Liouville, p. 5)

*Points (i, j), where i, j are positive.*

*Defined to be BELOW (m, n) where m, n are positive when ≤m and ≤n.*

*∴ (i, j) is below itself – not particularly important.* (Dylan, Visible

Points, p. 1)

Having discussed how students generate knowledge about the situation and

how this knowledge becomes manifest, this section will look at ‘key ideas’ as

knowledge that is crucial to solving the problem and that students employ

directly to generate a solution. The first subsection discusses ‘looking for

patterns’ as ways of looking for key ideas by investigating the situation in a

particular way. The second sub-section discusses ‘key searching’ as a way of

looking for key ideas in a more direct way.

As said in the previous section, some of the observations that students make

during problem solving lead directly to generating a solution. Since these

observations usually refer to crucial aspects of the situation they can be called

key ideas. Students usually base their solutions on a key plan or idea that

provide hints as to how a solution can be obtained. In order to deal with

‘Diagonals of a Rectangle’, for instance, students used the fact that there is a

relationship between the highest common factor of the rectangle’s dimensions

and the number of rectangles crossed. This fact was the key idea on which most

(if not all) students who provided a solution for this problem based their

processes.

Key ideas sometimes emerge as sudden realisations of important aspects of the

situation. These ideas may appear as important breakthroughs (as the student

below suggests) and give students the feeling of having discovered how to

generate a solution.

*AHA! This is a huge breakthrough! Anything that happens before the*

* row marked (*) is not important. As long as we can guarantee that our*

* opponent moves to (*), we have won, since we can then move to a*

* definite win position.* (Leonard, Cartesian Chase, p. 5)

In other cases, key ideas emerge as less of a surprise. In these cases, key

ideas may come gradually as knowledge and understanding increase.

In either case, it seems that being able to arrive at a key idea requires a good

deal of understanding of the situation. When students are able to see a key

idea, they are also able to see its significance, its importance in relation to the

situation and how it can be of use. In relation to this, Raman (2003) observed

that the key ideas that more experienced solvers use to provide a mathematical

proof “give a sense of understanding and conviction” and show “why a particular

claim is true” (p. 5). In more general terms, Barnes (2000) suggested that when

students and more experienced mathematicians are able to see a key idea the

following takes place:

*…there is a claim to a sudden realisation of new knowledge or*

* understanding. Usually this knowledge is ‘seen’ with great clarity, or*

* experienced with a high degree of confidence or certainty*. (Barnes,

2000, p. 34)

Key ideas can be seen as the product of gathering sufficient relevant knowledge

and understanding to be able to start generating a solution. The following subsections

look at ways in which students generate and search for key ideas.

Looking for patterns can be considered as a way of learning about the situation

that can lead to finding key ideas. When students look for patterns, they are

usually looking for particular features of the situation can lead them to start

generating a solution. Students look for patterns hoping that, when they find

one, they will be able to transform it into a formula or to make a general

statement about the situation.

*I shall look for patterns which might lead me to a formula of some kind.*

(Lila, Sums of Diagonals, p. 1)

Looking for patterns can be a useful activity that generates relevant information.

For instance, noticing a pattern in the way the creases were formed in the ‘Ins

and Outs’ problem allowed students to tell how the creases for the 10th fold

would look like. Furthermore, as students look for patterns, they may also gain

understanding and learn about the situation. Thus, in many cases, looking for

patterns can be a fruitful activity.

However, looking for patterns can also become a ‘blinding’ activity that prevents

students from gaining the necessary information and understanding. When

students focus mainly on looking for patterns and neglect trying to see other

aspects of the situation, the possibility of gaining useful information seems to

decrease. In the example mentioned above, most students were able to see

how creases were formed and thus were able to tell how the 10th fold would

look like. However, very few students were able to provide a general (nonrecursive)

formula for this sequence. Students that were able to provide a

general formula did so not by looking for patterns but by gaining a deeper

understanding of how the sequence of ‘ins’ and ‘outs’ was generated. In

contrast, students that focused mainly on looking for patterns (as illustrated

below) were able to provide a recursive formula but failed to provide a general

one.

*I can’t see a pattern or anything jumping at me.*

But by counting the number of ‘ins’ and ‘outs’ in any number of folds I can see

that each one seems to be an odd number.

E.g.,

*Just comparing the difference between the number of ‘ins’ and ‘outs’*

* seems to show that they are powers of 2*. (Rita, Ins and Outs, p. 2)

Thus, it may be said that looking for patterns can provide some very useful

information. In order to provide a more satisfactory solution, however, further

information and understanding need to be generated as well. Focusing on trying

to find particular information about the situation can lead to a dead end as it

prevents students from genuinely learning about the situation. ‘Key searching’,

as will be discussed in the next sub-section, is a way of looking for key ideas

that is related to this aspect of looking for patterns.

As mentioned above, key ideas allow students to start generating a solution.

Finding a key idea is certainly related to successful problem solving, and

students seem to be aware of this. For this reason, students may look for key

ideas by looking for patterns. Another way of looking for key ideas is by ‘key

searching’. Key searching means looking for key ideas in a direct way by trying

discover special features about the problem or by trying to find “what is so

special” about the situation.

*I’m looking to see if the number left in the hat has some special*

* quality…*

*Still stuck! Maybe I should go back and try the odd numbers. After all,*

* as this may be the missing clue to the solution…*(Aminta, Hat Numbers,

pp. 2–4)

As students try to gain knowledge and understanding of the situation, it is very

likely that they will eventually come across key ideas. Paradoxically, however,

key ideas are less likely to emerge if students focus on actively seeking them.

The reason for this may be that searching for key ideas may divert students’

attention from trying to learn about the situation. During key searching, students

seem to be so concerned about trying to find some “special” clue or quality that

they may neglect other important information. In the case of the Liouville

problem, for instance, some students spent most of their process trying to figure

out what was so special about sequences of numbers that if added and then

squared give the same value as when they are cubed and then added. In these

extreme cases, students were unable to make any significant progress and were

not able to identify any of the key ideas that allowed other students to generate

a satisfactory solution.

When students search for key ideas, they may ignore important information that,

if not a solution in itself, can be used towards that end. Furthermore, in some

cases, students that search for key ideas seem to ponder on the problem rather

than on trying to gain a broader understanding of the situation.

In general, not all students incur in key searching and those who do may

eventually abandon this activity and try to generate information and

understanding. However, the implications of key searching make this activity an

important one to consider. There is no evidence to suggest that key-searching is

related to mathematical background. What can be suggested is that keysearching

may be related to the features of the problems involved. This

hypothesis is supported by the fact that more students key-searched in the

‘Liouville’ problem than in any other. There is not sufficient evidence to state take

this hypothesis further. This issue can only be suggested for further research.

The above sections deal with the way students generate knowledge during their

problem-solving processes. This knowledge constitutes the information and

understanding that will allow them to deal with the problem and eventually to

achieve a solution. This section deals more closely with the issue of gaining

understanding. This issue plays an important role in being able to generate a

solution and most students will seek to gain understanding about the situation.

However, as it is discussed below, students may also ignore or avoid trying to

gain understanding and concentrate on manipulating data.

A good place to start a discussion on the characteristics of gaining

understanding during problem solving is by considering the following quote from

Thurston:

*On a more everyday level, it is common for people first starting to*

* grapple with computers to make large-scale computations of things they*

* might have done on a smaller scale by hand. They might print out a*

* table of the first 10,000 primes, only to find that their printout isn’t*

* something they really wanted after all. They discover by this kind of*

* experience that what they really want is usually not some collection of*

* answers – what they want is understanding.* (Thurston, 1995, p. 29;

emphasis in the original)

Although Thurston’s assertion was made in reference to professional

mathematicians, it may be said that it applies to many students as well.

Gaining understanding is an important aspect of the problem solving process.

Most students try to gain understanding of the situation to be able to start

generating a solution. As a student put it, it is easier to generate a solution by

“understanding the underlying principles” of the situation. In general, it seems

that having a better understanding of the situation empowers students and

allows them to generate a solution and take it further.

*I can’t believe how I missed how every entry in the grid is the product of*

* its coordinates…*

*This means that given any coordinates we can work out what the entry*

* is.* (Nadia, Sums of Diagonals, p. 4b)

An important way of gaining understanding is by reasoning in terms of how the

data is created, or how it stems from the situation. Although not all students try

to gain understanding in this way, and those who do may not do so all the time,

it may be said that thinking in terms of how information is created is a common

practice. Thinking in terms of how the sequences of ‘ins’ and ‘outs’ were created,

for instance, provided students with useful understanding of the situation. In

most cases, this allowed them to generate an initial solution for the ‘Ins and

Outs’ problem. The following quotes illustrate the type of reasoning that was

conducted in an attempt to gain understanding in relation to this problem.

*What I’m going to do is take the five folds sequence and identify which*

* creases come from which fold.* (Lydia, Ins and Outs, p. 7)

*Maybe I should start to think about things on a more subtle level. What*

* actually happens every time I add a crease of paper? I’ll try to get this*

* into a diagram. (*Leonard, Ins and Outs, p. 4)

When students try to think in terms of how the data is created, they usually gain

a kind of understanding that allows them to make informed decisions on what to

do next. In other words, they achieve what Skemp (1976) called ‘relational

understanding’. This type of understanding allows students to know “both what

to do and why” (p. 20) and for this reason it is usually an important asset during

problem solving. The understanding achieved by the students in the following

examples is relational in the sense that it provides information that can be useful

for understanding the situation and deciding what to do next. Furthermore, their

understanding seems to have been generated by reasoning in terms of how

what they observe stems from the observed situation:

*Let’s try to think logically about specifically when a diagonal would pass*

* through a corner.*

*AHA! I think the diagonal will pass through a corner when n and m have*

* a common factor greater than 1. This makes a lot of sense because it*

* implies that the rectangle can be split up into smaller rectangles with*

* the same diagonal, and therefore the diagonal would pass through the*

* corners.* (Hannah, Diagonals of a Rectangle, pp. 3–4)

Finally, considering the benefits of trying to think in terms of how the data is

created may look as if all students worked naturally in this way. However, this is

not the case. Students with stronger mathematical backgrounds are usually

keen on reasoning in terms of how the data stems from the situation. On the

other hand, students for whom mathematics is not a main subject seem more

prone to look for patterns without considering the situation that gives rise to the

data. The reasons for this behaviour are difficult to trace. It can be speculated

that thinking in terms of how data relates to the situation requires students to

combine thinking about the situation while, at the same time, trying to identify

useful patterns. Thus, some students may unconsciously avoid such an

increased complexity and choose to focus on only one task at the time. In such

situation, they may prefer to work on the simpler one which will be, presumably,

trying to spot patterns. This, however, is a tentative explanation; a more

grounded explanation certainly requires further research.

The previous sections looked at how students generate knowledge about the

situation. It was discussed how students make key ideas available and what

courses of action may hinder their emergence. Some ways in which students

gain understanding about the situation were also considered. In spite of its

importance, it may be said that generating knowledge is not the final aim of

problem solving but a means of making necessary resources available. The aim

of problem solving is to generate a solution and students will start attempting to

do this as soon as sufficient knowledge has been gathered. Two ways in which

students may try to generate a solution is by reasoning deductively and

inductively. Reasoning in terms of how data is generated from the situation can

also play an important role in generating a solution.

In order to generate a solution, students may rely on deductive reasoning. In

other words, they may follow logical implications from one idea to another until a

conclusion is reached. Reasoning deductively seems to be held in high regard

by most students since, whenever possible, they will try to arrive at a solution in

this way. In the Liouville problem, for instance, most students’ first attempt at

generating a solution involved providing some version of the following deductive

argument.

*A prime number n has divisors 1 and n only, by definition.*

*1 has one divisor (1)*

*n has two divisors (1, n)*

*The sum of the number of divisors or divisors is therefore 1+2=3 and*

* squared this is 9.*

*The sum of cubes of the number of divisors or divisors is 13+23=9.*

*So the two numbers are equal for prime numbers.* (Julia, Liouville, p. 2)

Also, as one student put it:

*I generally try to use deduction. Deduction is ‘more valid’ in mathematics*

* although I often use inductive arguments*. (Leonard, informal interview)

When students reason deductively, they sometimes base their arguments on a

relevant piece of mathematical knowledge. This piece of knowledge may consist

of mathematical concept or a procedure. In other words, students may build a

deductive argument by applying a concept or a definition in an ingenious way or

by making use of a familiar mathematical procedure. In the example above, the

student based her deduction on the mathematical definition of ‘prime number’.

The way she made use of this definition allowed her to generate a logical chain

of reasoning and to achieve an initial solution. As for applying a mathematical

procedure, the Arithmagons problem provides a good example. In most

solutions to the ‘Arithmagons’ problem it was common for students to base their

arguments on procedures for solving systems of linear equations. Although

making use of procedures may be more straightforward than deciding how to

apply a concept, in the sense of constructing logical chains of reasoning, the

former can also be considered a deductive argument.

Whenever there is the possibility of generating a deductive argument from the

knowledge and information available, students will usually follow this route.

When this is not the case, one option is to continue trying to generate

information and understanding until it is possible to generate a deductive

argument. Another option is to start trying to generate a solution by induction.

Reasoning inductively involves making tentative conjectures or generalisations

out of the information that is available. Making deductions involves deriving

ideas that are a logical consequence of the information available. In contrast,

when students reason inductively, they not only consider the information that is

available (and the logical implications of this information) but also draw upon

other less factual sources such as previous (possibly informal) knowledge and

experience. This knowledge and experience may arrive in the form of insight or

intuition, or in the form of ‘intuitive guesses’, as Fischbein and Grossman (1997)

put it. It is the combination of empirical data with other sources of knowledge

what usually makes inductive reasoning a fascinating process.

*All the results are in a range 48–63…*

*Notice that the last two results are equal.*

*Conjecture 1: the percentage of visible points converges to a number.*

*Conjecture 2: the convergent number x=48.7%*. (Aminta, Visible Points,

p. 4)

Generating ideas inductively may lead to inaccuracies or even to incorrect

solutions. This is not to say that deductive reasoning is foolproof. What this

suggests is that, due to the nature of inductive reasoning, students sometimes

have to accept, and deal with, the fact that they are working with imperfect

results. However, this is usually not a serious problem since ideas can be reexamined

and modifications can be made. Moreover, checking whether a

tentative solution is correct and makes sense allows students to improve their

solution and increases their knowledge and understanding of the situation. This,

together with the fact that an initial solution – i.e., a starting point – is already

available, seems to outweigh the possible drawbacks of generating a solution in

an inductive way.

As said, most students will try to work deductively if at all possible and if not

they may choose to work inductively. However, inductive and deductive

reasoning are not mutually exclusive as this generalisation may suggest. In fact,

it may be said that students combine both approaches and that they

complement each other. For instance, after reasoning inductively and generating

some feasible conjectures, students may recur to deductive reasoning to show

that these are always true.

Besides reasoning inductively and deductively, students may generate a solution

as a result of reasoning in terms of how data is created. The previous section

discussed how thinking in terms of how data is created may provide students

with information as to what to do next and why. Since this information is easily

translated into a solution, reasoning in terms of how data is created can be

considered as another way of generating a solution that is different to both

induction and deduction. Simon (1996) observed a similar situation. He

suggested that students may invent or infer situations to explain how data is

created and that this may allow them to generate a solution. The following

example illustrates the case of inventing a situation to explain how data is

created and how the understanding that it provides can be used to generate a

solution.

Ms. Goodhue: *Mary, could you make an isosceles triangle by specifying*

* two angles and the included side?*

Mary pauses and then punches in equal angles.

Ms. Goodhue: *Can you tell me what you did?*

Mary: *Well, I know that if two people walked from the ends from this*

* side at equal angles towards each other, when they meet, they would*

* have walked the same distance.*

Author [Martin Simon]: *What would happen if the person on the left*

* walked at a smaller angle to this side?*

Mary: (Without hesitation) *Then that person would walk further [than the*

* person on the right] before they meet…* (From Simon, 1996, p. 199)

Thinking in terms of how data is created can be seen as a way of gaining deep

understanding of the situation that helps generating a solution. Solutions

achieved in this way tend to be more ‘transparent’ than solutions arrived at by

deduction or induction. When students reason in terms of how data is created, it

may become evident how a solution should look like and why.

It was mentioned before that tentative solutions that are generated inductively or

in any other way are usually a good place to start generating a more

comprehensive solution. However, there does seem to be an exception to this

case. In some cases, students’ apparently inductive reasoning can be better

explained as ‘guessing’. When students guess a solution, their reasoning is

unclear and it is usually difficult to tell where ideas come from.Yet, from the

comments that students make, it usually becomes evident that they may be

testing their luck and proposing ideas without going through conscientious

reasoning about the situation.

*Try completely new approach. Convert sequence into a straight number*

* using binary representation (might get lucky).* (Sebastian, Ins and

Outs, p. 5)

*We can see by looking at the diagram that there are three points that*

* would not be visible. Could I work this out algebraically so that it applies*

* to any size grid square?*

*Maybe it could be (i–j)/j, that would be (9–3)/3=6/3=2. That doesn’t*

* work!*

*Maybe (i–j)/i would be better: (9-3)/2=3.Would this work for other (i, j)?*

* […]*

*There only seems to be two points which means that my formula is not*

* correct.* (Gina, Visible Points, pp. 3–4)

Ideas that are arrived at by guessing are usually ungrounded, i.e., they are more

the product of inventiveness than of carefully analysing the data. Although the

relation between guessing and ungrounded ideas is somewhat evident,

guessing a solution is not the only way in which students may generate this type

of ideas. Trying to invent a situation to explain how data is created may also lead

to generating ungrounded ideas, particularly when used without considering

sufficient empirical data. In other words, in an attempt to provide an account of

how data is generated or of how the “system in question works”, students may

fall into ‘making up’ an explanation that is more the product of their ingenuity

than of what they know about the situation.

Ungrounded ideas tend to be inconsistent and thus can lead to problems and

frustration. This was the case of a student that provided an interesting

explanation as to why it is not possible to build a fault-free rectangle (see the

‘Faulty Bricks’ problem). Since fault-free rectangles can be built, and since the

explanation was the result of the student’s creativity, she found it hard to

elaborate the argument further. In general, although ungrounded ideas can be

problematic, a positive aspect is that the frustration that they cause may

become, in some cases, a good place for starting to learn about the situation.

Summarising, students may generate an initial solution by reasoning

deductively, inductively or in terms of how data is generated. Although students

may have a ‘predilection’ for deductive reasoning, it seems that this predilection

is based more on their beliefs about mathematics (deductive reasoning being

‘more valid’) than on the results that they obtain from reasoning in this way.

Inductive reasoning may allow students to generate initial solutions that can

later be improved. Thinking in terms of how data is generated is a good way of

generating ‘transparent’ solutions. Although the last two types of reasoning may

not be the students’ first choices, they can be efficient ways of generating

results.

Once a solution is generated, it may be validated and/or improved. The next two

sections look at ‘validating’ and ‘improving’ results, respectively.

During their problem solving processes, students look for ways of validating the

ideas that they are generating. To do this, they may try to validate their results in

terms of whether they are correct and make sense. In other words, students try

to verify that their results are correct and seek to explain why this is the case.

When students validate their results in this way their main concern is being on

the ‘right track’ and having a clear understanding of the situation. Thus, the

arguments that they produce can be considered as personal ‘proofs’ aimed at

convincing themselves, their peers and possibly even a sceptical reader trying

to follow their process (i.e., convincing oneself, a friend and an enemy, in

Mason’s (1982) terms).

Once students have achieved a satisfactory solution, they sometimes seek to

provide a formal mathematical proof of their work. However as the quote below

suggests, providing a formal argument seems to have a different purpose than

making sure that a solution is correct and makes sense.

This certainly seems to hold for all m, n [where m and n are natural

numbers], but whether or not I can prove it is a different matter.

(Leonard, Diagonals of a Rectangle, p. 19)

It seems that trying to provide a formal mathematical argument that proves that

a solution is true is more a way of improving a solution than of making it

convincing for themselves and for others. For this reason, providing a formal

proof will be discussed in the next section below (‘Improving Results’).

Students may validate their results by verifying that their ideas are correct and

make sense. In order to verify that results are correct, students may review their

reasoning and look for any errors or inconsistencies. For instance, they may

check that suitable procedures were chosen and that they were properly

conducted. Besides verifying their procedures, students may check to see

whether their generalisations work in particular cases. If the results obtained

from particular cases are as expected or match with previous data, then they

can be accepted. Verifying that results are correct allows students to move on,

whereas noticing any inconsistencies will require them to go back and try to

correct them.

*Now I want to check it again that my result is right before I go any*

* further from here. Therefore I count the number of grid squared that are*

* touched by the diagonal again from the grid squares that I have already*

* drawn. And it’s correct.* (Anibal, Sums of Diagonals, p. 4)

*I will now see if it works for the numbers I have so far*. (Jasmine, Sums

of Diagonals, p.6)

*Check: Does this match the examples I have tried so far?* (Julia,

Liouville, ca. p. 10)

Students may verify that the ideas being generated make sense by looking for

explanations as to why they must be true. Explaining why an idea is true

reassures students that the solution that they are generating is congruent with

their knowledge and with what they know so far about the situation.

Furthermore, when students try to make sure that their generated ideas make

sense they may resort to thinking in terms of how data is created.

Understanding of how the situation ‘works’ and how the data is derived from the

situation provides students with ideas that can be used to explain why a solution

must be true.

*Why does this work? Aha! Looking at any diagonal, moving down one*

* adds 1 to the first element, 2 to the second, etc. And then finally one*

* more element equal to the new ‘x’*. (Marcus, Sums of Diagonals, p. 3)

Trying to verify that results are correct and making sure that they make sense

are related activities that are usually combined. In many cases, after checking

that their results are correct, students may proceed to explain why this is the

case. The following quote illustrates this situation.

*This looks like the number of creases is 2a-1.*

*Check for a=6.*

* From previous formula creases = 31+32=63=26-1.*

*I can see this would be true because each time I am doing n+(n+1) to*

* get the next term which is equal to 2n+1, so each time I am doubling*

* the previous number (which is less than 2n as 1 is one less than21=2)*

* which would give me 2n=2 and then adding one so I get 2n-1.*

(Jasmine, Ins and Outs, p. 4)

This is not to say, however, that verifying that generated ideas are correct

implies that students will proceed making sure that they make sense. After all,

not all students are able to conclude their process by saying:

*My calculations do work and make sense, and I think the answer is*

* reasonable*. (Hannah, Faulty Rectangles, p. 11b)

In some cases, students may not be interested in explaining why ideas are true

so long as they seem correct. In other cases, students may be able to verify that

their results are correct but may find it difficult to provide an explanation as to

why this is the case.

*It does seem to be the case that the Liouville results are always*

* identical, regardless of the chosen starting number. Sadly, I have no*

* theories as to why this occurs.* (Conrad, Liouville, p. 5; emphasis

added)

Continuously trying to verify that ideas are correct and make sense ensures that

inconsistencies are brought to the fore and provides an opportunity to amend

them. In fact, it seems that verifying that ideas are correct and make sense, and

making the necessary modifications, plays an important role in successful

solutioning. Inglis and Simpson (in press) suggest that it is error-correcting –

rather than error-free processes – that may account for the fact that

mathematicians perform better than non-mathematicians in logic tasks.

Furthermore, in a study of collaborative problem solving in combinatorics,

Eizenberg (2003) found that it was not peer collaboration that was directly

related to successful problem solving but that successful problem solving is

closely related to ‘control behaviours’, i.e., to constantly monitoring whether

ideas are correct and making the necessary modifications. In the author’s

words:

*Our study provides evidence that success in problem solving in*

* combinatorics is not a direct outcome of collaborative problem solving. It*

* is mostly a result of enhanced control behavior*. (Eizenberg, 2003, p.

399)

In spite of the benefits of validating results, students do not always stop to verify

that their results are correct and make sense. As said, validating solutions in the

ways discussed here may help to reassure students that they are on the ‘right

track’ in terms of the ideas that they are generating. This, in turn, will allow them

to continue with their solving process or, in other words, to ‘move on’. In some

cases, however, being able to move on can be more important than whether

results are correct and make sense. In such cases, students may simply avoid

trying to validate their results or will do it in superficial ways. For instance, they

may check that results are true in one or two known cases. In this way, even if

results are inaccurate, this will not necessarily prevent them from continuing to

work towards a concluding solution.

*Number of rectangles formed is 3n+(n–4). E.g., when 5 dominoes are*

* used 15+1=16.*

*That seems to work! I will test the formula out when more dominoes are*

* used.[Continues to work with 3×1 rectangles]* (Gina, Faulty Rectangles,

p. 3)

Being able to validate a result may provide students with an acceptable solution.

However, unless the student had already been working on improving this

solution, it is very likely that it will not be final but one that needs to be

improved. The next section looks at ways in which students may seek to improve

a solution once it has been achieved.

This section looks at what can be considered as the last stage of the solutioning

process. Once a solution is achieved, students usually acknowledge the need to

improve their results. This is particularly true when students feel that their

answer is correct but not ready to be presented as it is. If time and mathematical

knowledge allow, they may try to improve their results by providing a formal

mathematical proof or by extending their results to other domains. Alternatively,

they may try to express their solution in more concise ways.

*OK – I’m happy that’s worked out in that case. I’m definite there is a*

* more elegant explanation which might be worth looking for. Argument*

* sounds a little awkward to me at the moment – could do with being*

* more persuasive.*

*Right. Review here – there’s a few different ways to go…*

* Have shown for odd x even, if I could show for even x even I’d be done!*

(Rafael, Faulty Rectangles, p. 12)

*I wonder if I could improve this further by rewriting my formula as a*

* closed expression, i.e., an equation in x and n with no summation signs.*

(Hillary, Sums of Diagonals, 15)

Improving a solution can be a straightforward task that involves making simple

modifications or additions. However, this is not always the case and the work

that students need to conduct to improve a solution can vary from being

straightforward to very laborious and time-consuming. In most cases, improving

a solution will involve dealing with situations that are more complex than when

an initial solution was generated. Having to deal with progressively more

complex situations can make it difficult – or even impossible – for some students

to improve their solutions further. The probability of this being the case seems to

be higher when students lack the necessary mathematical background to deal

with more sophisticated mathematical ideas. Lack of time or energy can also

prevent students from improving their solutions. Under these circumstances,

some students will decide to stop their process and will present their solution as

it is.

*Reached a dead end at the moment so I am unable to progress any*

* further. If I had been able to solve this problem properly I could have*

* also extended it to look at the rest of the items on my brainstorm.*

(Lydia, Cartesian Chase, p.13)

Students who are able to improve their solutions recognise that it is almost

always possible to take them even further. However, they only have to continue

improving their solution until a seemingly acceptable solution is found. Such a

solution is one that is clearly (and if possible, formally) stated and that accounts

for a variety of cases.

One way in which students may seek to improve their results is by attempting to

produce a formal mathematical proof of their work. Once a satisfactory solution

or initial solution is generated, students may try to improve it by providing a

more rigorous argument. Providing a formal mathematical argument is a way of

putting an already satisfactory solution in such a way that it can be presented as

a final product to others. In other words, providing a formal mathematical proof

involves elaborating a deductive argument that not only satisfies the student’s

understanding but also satisfies certain mathematical requisites.

Producing a formal mathematical proof is something that some students do as

part of their processes. For instance, in ‘Sums of Diagonals’ various students

proved their general formulas by mathematical induction. However, in general, it

may be said that providing a rigorous mathematical proof is usually considered

a secondary aim. For some students, the fact that the results are reliable should

be evident from the way they were generated and validated.

*I believe I have the correct answer, although I have no concrete proof. I*

* believe that, as a possible extension, it would be possible to get an*

* answer involving trigonometry…This would be a concrete ‘proof’ of the*

* answer but it isn’t very easy to show. Other extensions [could be]…*

(Roberto, Diagonals of a Rectangle, p. 5)

*My formulas are very general and because of the way they were*

* obtained they don’t really need any formal proof or justification, as these*

* are evident in the method.* (Nadia, Sums of Diagonals, p. 7b)

In general, students seem more concerned about producing arguments that are

convincing, both for themselves and for a sceptical reader than of providing a

formal mathematical proof. Moreover, when it comes to improving their solution,

they seem to be more concerned about extending their results, as will be

discussed next.

Once students generate a solution, it is not uncommon for them to try to

improve it by extending it. Students extend their solutions by showing that they

account for all possible cases or by making their results valid for a wider

domain.

When students generate a solution, they sometimes notice that the ideas or the

methods that they used can be applied to other situations as well. In other

words, they notice that some of their ideas can be transferred and thus be made

useful for solving, or dealing with, other cases – i.e., for extending.

*Aha! If I can do this for a number with two divisors that are prime, I*

* could probably do it for a number with exactly 3, 4, … or more nontrivial*

* divisors, all which are prime.* (Jason, Liouville, p. 3)

*Can I use the same process as earlier to generate more even x even*

* fault free rectangles?* (Camille, Faulty Rectangles, pp. 2b–3)

Although transferring means that previously developed ideas will be used in

other situations, this is not necessarily a simple task. Transferring may require

students to make some changes to the ideas or procedures to be transferred to

make them suitable for the new situation. These changes can be relatively

simple, such as when students decide to introduce a new, more efficient

notation.

*The largest secret number ‘a’ was found by adding the two largest side*

* numbers and subtracting the remaining side numbers…I think [this] rule*

* is most likely to work with arithmagons with >3 sides.*

*As I am seeing a general rule for arithmagons with n sides, I will need*

* to alter my notation for improved clarity. Instead of x, y and z for the side*

* numbers I will use s1, s2, s3, …, sn…* (Jules, Arithmagons, p.5)

In other cases, adapting previously used methods or ideas can be complicated

or even impractical.

*My proof that there was a path came from visualising, again, what the*

* path should be, since anything other than the circle seemed unlikely,*

* and bearing in mind the complete symmetry of the circle. Unfortunately,*

* this reliance on the symmetry of the circle meant I couldn’t extend the*

* theory to irregular circles very easily.* (Albert, Jogger’s Dog,

Commentary)

In some cases, adapting can be a considerably complicated activity. In situations

like this, students will find that looking for new ways of generating a solution may

be a better option. In a way, finding new ways of solutioning may suggest that

students will need to start the solving process all over again. However, this is

not the case. The knowledge and understanding that students have gained

about the situation are very likely to make this ‘new’ process a more efficient

one. Of course, this will be the case only if students persist in extending their

solutions. They may well decide to stop their process at this stage.

APPENDIX 1 – THE PROBLEMS (IN ALPHABETICAL ORDER)

[please see PDF version for appedix problems]

**Maria de Hoyos Guajardo, Ph.D. Candidate, M.Sc., B.Eng.**

Research Fellow

Institute for Employment Research

Warwick University

Coventry UK

CV4 7AL

Phone: +44 (0) 779 6614243

E-mail: maria.de-hoyos@warwick.ac.uk

Barnes, M. (2000). “Magical” Moments in Mathematics: Insights into the Process of

Coming to Know. For the Learning of Mathematics, 20(1), 33-43.

Eizenberg, M. M., & Zaslavsky, O. (2003). Cooperative Problem Solving in Combinatorics:

The Inter-Relations between Control Processes and Successful Solutions. Journal of

Mathematical Behavior, 22(4), 389-403.

Fischbein, E., & Grossman, A. (1997). Schemata and Intuitions in Combinatorial

Reasoning. Educational Studies in Mathematics, 34(1), 27-47.

Inglis, M., & Simpson, A. (in press). Mathematicians and the Selection Task. Proceedings

of the 28th International Group for the Psychology of Mathematics Education, Bergen.

Mason, J. H., Burton, L., & Stacey, K. (1982). Thinking Mathematically. Avon: Addison-

Wesley.

Raman, M. (2003). Key Ideas: What Are They and How Can They Help Us Understand

How People View Proof? Educational Studies in Mathematics, 52(3), 319-325.

Simon, M. A. (1996). Beyond Inductive and Deductive Reasoning: The Search for a

Sense of Knowing. Educational Studies in Mathematics, 30, 197-210.

Skemp, R. R. (1976). Relational Understanding and Instrumental Understanding.

Mathematics Teaching, 77, 20-26.

Thurston, W. P. (1995). On Proof and Progress in Mathematics. For the Learning of

Mathematics, 15(1), 29-37.