Understanding mathematics

UNDERSTANDING MATHEMATICS
Keith Porteous
University of Hull
k.porteous(at)hull.ac.uk

ABSTRACT
The word “understanding” is widely used in discussion about learning and doing mathematics.  It can mean many things.  This paper will concentrate on a meaning which it will be claimed is central to the whole issue of mathematics education, concerning understanding as a sort of mental state, or a set of behavioural capacities which constitute a mental state.  Consideration will be given to various attempts that have been made to provide a positive description of what such understanding is, and it will be argued that these are lacking in helpful advice and guidance for teachers of the subject.  Instead, an alternative approach to the concept will be proposed, which will more-or-less equate understanding with knowing, and it will be argued that this helps to clarify the concept of understanding and also makes it easier for practitioners to interpret the requirement that they teach for understanding.

 

The central concept

To develop a clear focus it is, perhaps, helpful to start by eliminating a collection of meanings of “understanding” which are not directly relevant.  Common parlance recognises all of the following, none of which will be of  immediate concern to this discussion.

• “Understanding”, as equating to “being given to believe”.  Example: I understand that the train on platform 7 is the 1355 to Crewe.  Such formulations are often precursors to questions of the form “Am I right?”.  They indicate a tentative belief with a clear denial of actual knowing.
• “Understanding”, as a tacit agreement.  Example: we have come to an understanding about the role of mediators in this dispute.  This use does exemplify an aspect of understanding which is important to the later discussion – that it entails some sort of mutuality in some people’s thinking about a matter – but it deals only with the coinciding of ideas of the two parties concerned, and as such is not helpful.
• “Understanding”, as empathy.  Example: I understand why you did that, or I understand how you feel about this.  Such uses have the purpose of indicating to someone else your appreciation of their motivations, emotions, or such like.
• “Understanding”, as comprehending.  Example: I understand what “Je ne suis pas stupide” means.  This is all about an individual making sense of some sort of attempted communication, or some sort of input.  “Understand” is a verb, and this use of “understanding” is concerned with something someone does, at a particular time.  It describes a successful performance of some sort: that of receiving some sort of message, or other input, and making sense of it appropriately.  We hope that our mathematics students will do this, too, when they are dealing with mathematical input, but this meaning of understanding is not central to this discussion.

 

What will be tackled is the notion of understanding which is implicit in statements like the following:

• She has a good understanding of place value.
• This answer shows a good understanding of the different measures of the average of a set of numbers.
• He clearly does not understand the significance of a logical proof.
• Her understanding of Newtonian mechanics has improved greatly this term.

Such statements point to understanding as a noun.  Even when the statement contains “understand” as a verb, as the third does, it is understanding as a state of affairs that is being considered; “he clearly does not understand” can be replaced by “he clearly does not have an understanding of”.  (What looks superficially like a statement about an event that may or may not have occurred is surely more about the quality of a person’s mental capacity.)  Understanding is an aspect of someone’s mental life; it is something like a mental state; or, probably more usefully, it refers to someone’s capacity to behave in appropriate ways which indicate the possession of understanding.

Some further explanation of what this state of understanding is will be attempted later.  It is instructive, first, to examine the manifestations of the concept in some high-profile documents and the writings of other authors.

 

Understanding in official curriculum guidance

The English national curriculum for mathematics frames the specifications for the teaching of pupils in terms of “knowledge, skills and understanding in the programme of study (which) identify the main aspects of mathematics in which pupils make progress” (DfEE/QCA, 1999, page 6).  The phrase “knowledge, skills and understanding” is taken directly from the Education Act of 1996, section 353a, and it heads the specified programmes of study for each of the key stages: the knowledge, skills and understanding described form the greater part of each section, and constitute “what has to be taught in the subject during the key stage” (ibid, page 12).  The National Strategy guidance that follows the national curriculum specifications, and expands on what is expected, echoes this formulation.  An impressive collection of examples is provided, the purpose being “to illustrate the level of difficulty of each teaching objective in the teaching programmes through a selection of what pupils should know, understand and be able to do by the end of the school year” (DfEE, 2001, page 4).  The “know, understand and can do” formulation is now very familiar to teachers in England.  It seems that if we can deliver the curriculum under these three headings satisfactorily, then we have done our job, and the pupils will have learned what they should have learned.

We may be tempted to ponder, though, on what the three terms mean.  As an initial reflection it may be easy to see what is meant by knowing something (although some analysis of this will be helpful, a little later), and also by being able to do something.  It is rather more puzzling to determine what may be required of understanding.  For a pupil whose knowledge of a topic is sound, and who can do the things required when working with the topic, what could it be that is required if they are also to understand it?

A little digging into the detail of the National Strategy document furnishes examples such as the following.  Year 9 pupils should

“understand upper and lower bounds.  For example:

• For discrete data such as – the population p of Sweden to the nearest million is 15 million – know that the least population could be 14 500 000 and the greatest population could be 15 499 999; understand that this can be written as
• For continuous data such as the measurements of distance – the distance d km from Exeter to Plymouth is 62 km to the nearest km – know that the shortest possible distance is 61.5 km and the longest possible distance is 62.5 km, which can be written as ” (DfEE, 2001, page 47)

What is interesting here is that an attempt is being made to explain what understanding is (in relation to upper and lower bounds), and that it turns out to be knowledge.  It is the word “know” in the two examples that is crucial.  Even the re-appearance of the word “understanding” in the first bullet point carries little weight: it could as readily have been written as “know that this can be written as ….”.

It is not at all clear that anything very definite is intended by the inclusion of the need for understanding, rather than “mere” knowledge and skill.

This sort of formulation was not born, however, with the English National Curriculum.  In “Mathematics from 5 to 16”, in an HMI series of “Curriculum Matters” papers, we find the following:

“It is damaging to pupils’ mathematical development if they are rushed into the use of notation before the underlying concepts are sufficiently developed and understood.”  (p10)

What are the concepts like for the learner if they are sufficiently developed but not understood?

“Counter-examples can be used…. to give fresh insights and understanding.”  (p 23)

What are those pupils losing who only gain fresh insights?

“Nothing should be included for which there is not a sound, clearly defined purpose, understood and appreciated by the pupils.” (p 28)

If one appreciates the sound, clearly defined purpose, what extra is one supposed to gain from understanding it?

It is as if the concept of understanding in thrown in regularly to make sure that everything one might want from learners is covered.  It is a catch-all.

 

Understanding as a positive characteristic

It is not the purpose of this paper, though, to argue that the concept of understanding is not very relevant to the learning and teaching of mathematics.  Those who take mathematics education seriously are surely unanimous in promoting learning with understanding.  We just need to get clearer about what it is.

Much of what is written about understanding in the learning of the subject takes for granted that there is already a shared idea of what it is.  Many writers will emphasise the need for understanding, and stress its importance in many ways (see, for example, French, 2002), but will not give a very full or convincing account of what it is they are talking about.  What they have to say is very valuable, and their drive for understanding is to be applauded; what is needed, though, is a clearer view of what it really is that they are driving for, in order that teachers can see more clearly how to do it.

Some writers, on the other hand, have tried to give a positive characterisation of understanding, and it is worthwhile giving some consideration to some of these.

Skemp (1976) made an invaluable contribution to thinking about mathematics learning when he drew the distinction between relational and instrumental understanding, the former being “knowing both what to do and why” (Skemp (1976) page 20), the latter being the application of rules without knowing why.  The points he made have as much validity today as when he made them: rather than dwell on the matter, though, let it simply be noted that he focuses on the key aspect, as in the quote above, by means of the word “knowing”.

The Cockcroft Report, Mathematics Counts (Cockcroft, 1982), was a landmark publication for mathematics teachers in the UK.  It was seen as drawing together the best thinking about teaching the subject, and pointing to the best way forward for ongoing developments.  It had relatively little to say about understanding, as such, however, which is perhaps a disappointment given that its main thrust was to promote learning with understanding.  Skemp’s relational-instrumental analysis is criticised, understanding is acknowledged to be not a black-and-white, all-or-nothing, matter; and we find this, as the nearest the report gets to an explication of understanding:

“…understanding in mathematics implies an ability to recognise and make use of a mathematical concept in a variety of settings, including some which are not immediately familiar.”  (Cockcroft, 1982, p 68)

Presumably the use of “implies”, rather than the simpler “is”, indicates the author’s unwillingness to commit to anything which might be taken as a complete description of what understanding is.  One is left, still, with a strong impression that understanding is seen as a pervasive and powerful characteristic of a good mathematician, but one which remains vague.  We still do not know what it is.

Some authors have attempted a positive description of understanding. Haylock (1982), for example, argues that “a simple but useful model for discussing understanding in mathematics is that to understand something means to make (cognitive) connections” (p 54).  The connections are to be among the following: concrete situations, pictures, symbols, and mathematical language.  As stated, it is the performative meaning of “understand” that is being described, but it is easy to imagine that the same criterion could be applied to a state of understanding: to understand something means to have made connections.  This making of connections is most appealing, and many writers have emphasised it in various ways.  One immediate conclusion from this approach is that, because the connections can be limited or extensive, understanding can be limited or extensive, too: it is not all-or-nothing, and indeed it is rash ever to claim full understanding.  May there not be further connections yet to be made?

Perhaps one of the most sustained and detailed attempts to develop a positivist account of understanding is that made by Ormell.  He, also, promotes the centrality of connections: “at a rough level of description one might say that understanding X consists in knowing how X relates to, or connects with, other aspects of the world” (Ormell, 1974, p 13).  He proposes a test for understanding: can the student “answer a limited, but continuous, connected range of ‘if…then’ questions about the situation”? (ibid, p 14).  This appealing idea gives us an operational criterion which reflects the key notion  of understanding as a set of cognitive connections.  It is not without weaknesses, however.  The ability to answer “if…then” questions could readily be seen as dependent upon the cognitive abilities of the student as well as the cognitive connections they have made.  Their intelligence (to put it bluntly) will affect their success with such questions, as much as their understanding.  Also, the hypothetical nature of these questions gives us cause for concern when assessing the understanding of young people, in particular, because (if we give any credence to Piaget’s model of development) the pre-hypothetical child will necessarily find them difficult, or impossible.  These considerations seem to suggest that understanding is to be a characteristic only of older and brighter children and adults.  We may well be reluctant to lose the possibility of talking meaningfully about the mathematical understanding of young children.

Sierpinska (1994) has produced a major work which aims to throw light on the nature of mathematical understanding.  She draws the distinction between “mental experiences, which we might call ‘acts of understanding’” and “’an understanding’ which is a potential to experience an act of understanding when necessary” (p 2).  It is the latter which is closest to the subject of this paper, but her view of “understandings”, that they “seem more to belong to the sphere of knowing: they are the ‘resources’ for knowing” (p 2), expresses a more complicated relationship between understanding and knowing than will be promoted here.  Indeed the bulk of her work deals with the acts of understanding, and the process of understanding which she conceives of as a longer term aggregation of acts of understanding.

 

“Understanding” as a trouser-word

Notwithstanding the examples sketched in the preceding section, there seems to be a reluctance, among the majority of writers who use the term, to offer a comprehensive account of what understanding is.  It is also clear, though, that the concept is ubiquitous in discussion about learning mathematics.  This seems to suggest that the general approach works on the supposition that we all know what understanding is, really, and that we do not need to explain it.  It will now be argued that this is almost correct: we all know what not-understanding is.  In this way “understanding” can be seen as a trouser-word (in the sense of J L Austin, 1962), any cases of non-understanding can be corrected by imparting knowledge, and thus if there is a positive characterisation of what understanding is, it is this: understanding is knowledge.

First, what does Austin mean by a trouser-word?  His example, which is so clearly and appealingly described, is worth quoting in full.

“…’real’ is what we may call a trouser-word.  It is usually thought, and I dare say rightly thought, that what one might call the affirmative use of a term is basic – that, to understand ‘x’, we need to know what it is to be x, or to be an x, and that knowing this apprises us of what it is not to be x, not to be an x.  But with ‘real’…. it is the negative use that wears the trousers.  That is, a definite sense attaches to the assertion that something is real, a real such-and-such, only in the light of a specific way in which it might be, or might have been, not real.  ‘A real duck’ differs from the simple ‘a duck’ only in that it is used to exclude various ways of being not a real duck – but a dummy, a toy, a picture, a decoy, etc.; and moreover I don’t know just how to take the assertion that it’s a real duck unless I know just what, on that particular occasion, the speaker has it in mind to exclude.  This, of course, is why the attempt to find a characteristic common to all things that are or could be called ‘real’ is doomed to failure; the function of ‘real’ is not to contribute positively to the characterisation of anything, but to exclude possible ways of being not real – and these ways are both numerous for particular kinds of things, and liable to be quite different for things of different kinds.”  (Austin, 1962, p 70)

This is a highly persuasive account of the use of the word “real”.  It is a concept that may not be applicable to many words – Waks (1968) has argued that “fact” is a trouser word, and in the determinism versus free will debate the word “free” has been convincingly portrayed as a trouser-word – but to conceive of “understanding” as a trouser-word gives a useful perspective on the project of teaching for understanding.

The consideration in the “official curriculum guidance” section of this paper, of the use of “understanding”, highlighted the way in which the word is used in an irretrievably vague way.  It is a catch-all, seemingly intended to make sure that what is being described covers all the appropriate knowledge, behaviours and other characteristics that are relevant.  It could almost be deliberate, the way the word is added to an otherwise perfectly acceptable description, to make open-ended the inclusion of all possible valued aspects that are of concern.  Thus the only way to see clearly what the appending of the need for understanding demands, is to contemplate particular ways in which the person concerned may not understand.

This approach is not unique to official curriculum guidance, sadly.  Teachers also adopt it, probably unconsciously, although the phenomenon manifests itself differently.  They are always ready to point out that a learner does not really understand.  A pupil who appreciates a purpose, grasps the motivation behind an action, and is fully conversant with the reasoning relating to it, but who does not know that such-and-such is the case (some minor peripheral detail) could well be said not to understand.  Or at least their understanding is limited, or at fault: they do not fully understand.  A concept which is fully developed, and which has been successfully used in a variety of circumstances, but which fails to be employed in a particular instance, betrays a lack of understanding.  When the person concerned learns the things they had been supposed not to know (that such-and-such was the case, or that the concept is applicable in this instance) then this is acclaimed as a gain in understanding.  Let it be noted, here, that what, specifically, is gained is an item of knowledge.

Consider a specific example – what it is to be able to perform with understanding the four operations of arithmetic.  The inclusion of the term “understanding” clearly indicates that performance on its own is not enough.  What extra is demanded if understanding is required?  A list could begin with: knowledge of the applicability of operations in appropriate problems; the ability to spot short cut methods when appropriate; appreciation that the multiplication of a number by a whole number is simply repeated addition of the number; awareness that addition and subtraction, and multiplication and division, are pairs of inverse operations; and so on.  Where does this list end?  In any educational setting the practical question of what is included is settled by reference to a variety of parameters, not least being the age and attainment level of the learner.  But the list could, in theory, be continued indefinitely.  Graduate mathematicians would be expected to know about the field structure of real numbers.  It would be rash to claim that any finite list was complete.

What would the negative use of the word “understanding” tell us in this example?  To say “he does not perform the four operation of arithmetic with understanding” would require for its justification some definite evidence.  A twelve-year-old could calculate the difference between nine 7s and eight 7s by doing two multiplications and a subtraction (instead of spotting that it is clearly just 7).  An A-level student may not be aware that p/q is undefined for q=0.  It is the negative use which relates to specifics; the positive use is vague and all-embracing.  Austin can be paraphrased:

A definite sense attaches to the assertion that someone understands, only in the light of a specific way in which they might not have understood.

The trouser-word parallel between “real” and “understanding” is not, however, exact.  “Real” can be contrasted with “dummy”, “toy”, “picture”, “decoy”, and so forth, and these terms are not closely related to one another.  “Understanding” cannot boast such a wealth of mutually-excluding terms.  To claim to understand is to claim a lot of things, but these things are all of a type.  In other words, there are many ways in which evidence of lack of understanding can be found, but varied as these may be they are not different in kind.  Failure to realise that 9x7 – 8x7 is simply 7, and failure to appreciate that p/q is undefined for q=0, are not qualitatively different failures.

But if all the instances in which evidence of lack of understanding is present have a common factor, a similarity running across the whole range, is this a feature which can be identified and analysed?  In particular, can the unifying thread in failures of understanding be positively described, in contrast with understanding, which cannot?

 

Knowledge

The claim being made here is that cases of not-understanding can be fully characterised as cases in which certain specific knowledge is lacking.  To make this claim as strong as it needs to be, it is worth expanding on what it means to know something.

The standard formulation for this varies a little among those who have considered the issue, but these variations are not crucial to the arguments developing here  (See, for example, Woozley 1949, Ayer 1956).  Let us take Ayer’s formulation: that the necessary and sufficient conditions for a person A to know X are,

• X should be true;
• A is sure about X;
• A has adequate evidence about X.

Thus knowledge is fully described as justified true belief.  It is a formulation which has attracted a massive amount of attention, in particular that generated by the criticisms of it made by Gettier (1963).  For the present purposes, however, the many refinements and reservations that have been made will be set aside: their impact is small on our project of trying to see what teaching for understanding in mathematics might mean.  Let us simply take these criteria as a good enough working model to guide our thinking about teaching and learning mathematics, and see what they entail for our conception of understanding.

Consider an example.  Suppose that Jack and Jill have been learning about the Theorem of Pythagoras.  Jack knows that in a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides.  Jill understands the theorem.  What does Jill have that Jack lacks?

First, it has to be noted that Jack has quite a lot.  He genuinely believes the statement about right-angled triangles, which implies that he knows what the theorem states, and he has good reason to believe it, which means for our purposes that he appreciates a proof of the theorem.  This is a fairly high level of mastery: many teachers, for many students, would be happy with this outcome of their teaching.  The contrast between knowing and understanding that has been made by some writers now appears a good deal more difficult to justify.  Ormell, for instance, gives this example:

A child can give evidence of knowing the meaning of the words of Mark Anthony’s post-assassination speech by memorising the notes in his text-book.  But whether he really understands it is another matter.
(Ormell 1979, p35)

If the child really knows the meaning of the speech, he has got quite a lot.  In a later publication he makes the case more directly:

It hardly needs an elaborate argument to show that understanding and knowledge are different things.                             (Ormell 1984, p97)

He adds:

…there are many things which we know, but do not understand, in areas like medicine, astro-physics, particle physics, geology, the evolution of species.                                                 (Ormell 1984, p97)

The intention here seems to be to point out that we may know that such-and-such is the case, but not know why it is the case, or not know how this fact links to other facts which should have some bearing on it.  If this interpretation is correct, then it suffices to point out that what Ormell sees as lacking, by virtue of our merely knowing something, comprises further knowledge: his “understanding” covers “knowing why”, or “knowing how this relates to other facts”.  The extras that he expects from understanding can quite adequately be describes in terms of knowing.

Getting back to Jack and Jill, what has Jill got that Jack has not got?  Maybe Jill knows how to apply the theorem in a range of situations, or knows of the significance of the theorem in the controversies in ancient Greek philosophy, or she knows several different proofs, or she knows that it also tells us about the areas of other shapes that can be erected on the sides of a right-angled triangle, or…..  All of these extras can be comfortably described in terms of knowledge.  If, as teachers, we are concerned that Jack should also come to know these things that Jill has already mastered, then describing them in these terms, as knowledge, as justified true belief, is just the right way to set about thinking of how to teach him.

So, to know something is to be quite accomplished, and to contrast understanding with mere knowledge is essentially to fail to recognise this.  If we wish understanding to encompass more, then we can specify the extras perfectly well in terms of further knowledge to be gained.  If  this description of the relationship between knowledge and understanding is accepted, then the project of teaching for understanding becomes a good deal easier to grasp.

Implications for teaching and teachers

This brings the argument nicely to the question of what this all entails for teachers of mathematics.  If understanding is to be seen as essentially comprising knowledge, does this in any way have an impact on classroom practitioners?

First, we should beware the response which is all to easy for teachers - that of seeing a failure in one of our students to tackle some task satisfactorily as evidence of some sort of insurmountable problem.  Little Johnny cannot do something which he has been taught, and the immediate conclusion is that he simply does not understand, this carrying the full weight of the vagueness, and open-endedness, of the concept of understanding, and engendering a state of fatalistic despair.  He just does not understand, and that is that.  But if understanding is seen as an aggregation of knowledge, and knowing is believing, with good reason, that something is true, then cannot Johnny’s problem be identified, with proper focus given to establishing the knowledge that is lacking, applying all our teaching skills in helping him to acquire it?  Understanding is built bit by bit, so let the teaching focus be on each bit in turn as it becomes the key bit to learn.

Second, we should acknowledge that a learner’s intelligence does have an impact on their ability to tackle problems: the quality of their understanding or knowledge is not the only determining factor for success.  If a student fails to make links between this and that, it may be due more to their limited cognitive abilities than to any lack of knowledge.  If Jane can successfully work out such things as 36% of $450, and knows what such things mean and can justify her working, but then fails when asked to calculate 17 ½ % of £200, this may be a sign of her weakness with solving problems, and thinking out new problems, assimilating them to existing structures.  What she probably needs is some further explanation or other experience which helps her to make the connections required.  It does not follow that her understanding itself is flawed.  This issue is similar to the first.  Understanding is grown bit by bit.

Third, we should not feel overwhelmed by what seem to be unmanageable demands from policy-makers and curriculum writers.  When the official government guidance asks that pupils acquire the knowledge, skills and understanding for such-and-such a purpose; when the curriculum specifications want pupils to know about a topic and to be able to do such-and-such with understanding; and when pupils are to have mathematical concepts that are fully developed and understood; we should just remember that the concept of understanding is appealed to only because the writer wants to capture everything that they find impossible to describe in more specific terms.  They are all fearful that teaching should be aimed at rote-learning, which of course it will not be, as this cannot generate knowledge.  It does not even generate belief.  It is not appropriate to worry about these apparently unbounded demands: all they want is that students should learn properly, to acquire knowledge, to come to believe, with good reason, some things that are true.  It is much simpler to think of it that way.

 

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