The document discusses various perspectives on understanding in mathematics. It focuses on understanding geometric transformations as an example. Understanding a concept involves structural, operational, semiotic, descriptive, functional, and epistemological aspects. The structural aspect involves understanding formal definitions and statements. The operational aspect involves using concepts appropriately. The semiotic aspect involves recognizing different representations of concepts. The descriptive aspect involves understanding properties and relationships. The functional aspect involves applying concepts to solve problems. The epistemological aspect involves making conjectures and having a critical perspective. Understanding requires considering all these interrelated aspects to have a rich comprehension of mathematical concepts.

1. 1
REFLECTIONS ON THE ISSUE OF UNDERSTANDING
IN MATHEMATICS
1
Driss LAMRABET, Rabat
Keywords: Understanding / Comprehension – Learning – Problem solving -
Didactics – Mathematics education- Geometric transformations.
I INTRODUCTION
Many educaional psychologists and didacticians (such as Bloom, Skemp, Bergeron and
Herscovics) have been interested in the problems of understanding in mathematics.
Without pretending to fully deal with the issue, we will simply present a few
considerations that might clarify it. From an example, we will highlight a number of
aspects which, we believe, contribute to the understanding of a concept, a property, an
algorithm; in short, common objects in mathematics. The objective of the following is
to identify, by taking geometric transformations (GT) as examples, the aspects whose
consideration allows to identify and delimit a mathematical concept ( or property, or
algorithm,...) and to make it easier to understand, as well as to highlight, in connection
with geometric transformations, the importance of the notion of change of perspective
in mathematics as an aid for problem solving.
Please note that the aspects thus identified are not the result of a study based on some
theoretical framework or on rigorous experimentation; they are the result of only
personal reflections.
II SOME RESEARCH ON UNDERSTANDING MATHEMATICS
2.1 Understanding considered from the point of view of the required intellectual
activity: Bloom's taxonomy
Understanding (comprehension) is the second category in Bloom's taxonomy , which
includes six: knowledge, comprehension, application, analysis, synthesis, and
evaluation
Comprehension is divided into:
Transposition: Example: expressing a relationship in ordinary language through
symbols.
Interpretation: Example: from its graph, extract information about the properties of a
function.
Extrapolation: Example: provide the general term of a sequence given its first three
terms.
1
Mostly based on an (unpublished) lecture given (in French) at the Maghreb symposium on the
didactics of science, Algiers 2002.

2. 2
2.2 Understanding from the Perspective of the Learner's Opportunities and Needs:
Skemp's Model
- Instrumental understanding (the how): knowing how to use a concept, a theorem or
an algorithm without being necessarily able to justify why it leads to the desired
result.
Examples: use correctly the increasing/decreasing test for a differentible function on
an intervall without necessarily knowing the facts underlying this test or their
justification; use statistical tests without knowing their mathematical basis.
- Relational understanding (the why): being able to establish the link between the
result obtained by applying a theorem, an algorithm, ... and the underlying
mathematical concepts and properties. Justify, conjecture,...
2.3 Understanding viewed through the way mathematical concepts are presented:
the expanded constructivist model of Bergeron and Herscovis (1988)
Two interrelated tiers:
Understanding of the logico-physical concepts
Based on three different levels of understanding of the preliminary physical concepts:
1- Intuitive understanding (visual perception,...)
2- Logico-physical procedural understanding
3- Logico-physical abstraction (logico-physical invariant, reversibility, generalization,...)
Understanding of the emerging mathematical concepts
Concerns reflections on actions and transformations affecting mathematical objects: 3
non-hierarchical components:
1- Logico-mathematical procedural understanding
2- Logico-mathematical abstraction
3- Formalization (axiomatization, formal proofs, formal definitions,
mathematical symbolization,...)
For our part, understanding will be considered according to a number of aspects found
in usual mathematical objects.
III ANALYSIS OF WHAT IT MEANS TO UNDERSTAND A GEOMETRIC
TRANSFORMATION
Understanding a geometric transformation (or having a "good" understanding of it)
covers many meanings. In particular, this may mean:
- Be able to reproduce, recognize and identify the definition of this transformation.
- Identify this transformation among other mathematical entities.
- Know the various constituents of the definition.

3. 3
- Recognize the various forms in which this transformation can occur (natural
language, matrix; mathematical symbols; analytic form; geometric configuration...).
- Recognize this transformation as a particular case of a more general case; or as the
generalization of a particular case.
- Construct the image of a given figure.
- Mention the properties of the transformation in question.
- Recognize, in a given configuration, that one figure is the image of another.
- Exploit this transformation and its various properties to solve a given mathematical
problem.
- …
Identify the GT
as a particular
case
of another
Recognize
various forms
of the GT
Identify,
reproduce the
definition
Identify the GT
among
other objects
Identity the
components
of the definition
Draw the image
of a figure
under the GT
Enumerate the
properties
of the GT
Recognize a
figure as image
of another
under the GT
Utilize the GT
and its
properties in
Problem solving
...
Identify a GT as
a general case
of another
GEOMETRIC
TRANSFORMATIONS
...
As this example shows, if "a good definition" is useful, if not necessary, in the teaching
of a mathematical object, it alone may not be sufficient to delimit all that makes the
richness and fertility, but also the complexity, of that object2
. This is mainly due to its
ability to be (re)presented under accoutrements that sometimes make it almost
unrecognizable, to maintain more or less apparent links with other objects, and to
intervene in various mathematical and extra-mathematical situations. To account for this
ability, we propose that six aspects be considered for understanding in mathematics;
these are: structural, operational, semiotic, descriptive, functional and epistemological
aspects. Let's take a closer look at each of these aspects.
2
To lighten the text, this term will refer primarily to: (i) a notion, (ii) a statement that is a definition, a
theorem, a property, an algorithm/rule, a demonstration, a formula, a problem text and all that is
assimilated to it, iii) and when the context permits: a graph, a geometric figure, a drawing, a digram
and everything similar to it. In addition, the terms "concept" and "notion" will be considered
synonymous.

4. 4
IV ASPECTS OF UNDERSTANDING
As already mentioned, this, of course, is merely an attempt to circumscribe these
aspects without any pretensions of completeness.
4.1 Structural aspect
It refers to the formal statement of a definition, a theorem, a property, an algorithm, ...
Understanding this aspect includes being able to:
- Reproduce, identify, rephrase/paraphrase the statement in question;
- Identify the elements that form the statement (terms, symbols, ...) ;
- Identify, reproduce, reformulate the relationships that compose the statement (what
is given or assumed to be known, what is established or has just been defined, the
arguments and their arrangement in the case of a demonstration, what is requested in
the case of a problem,...).
It also refers to digrams, graphs, tables,... identification of the components, of their
articulation.
Created initially as a tool, a mathematical notion is subsequently decontextualized by
necessity of communication and generalization, and it settles in the body of
mathematical knowledge: it becomes an object [Douady, 1986]. As a tool, it can be used
to solve problems other than the one for which it was originally forged. Thus, the
concept of a group, originally developed to study the roots of algebraic equations,
proved extremely fruitful as a tool in all areas of mathematics. Its formal definition no
longer has any connection with the primitive problem which gave rise to it
[decontextualization, see Brousseau, 1984?]: it has acquired the status of a mathematical
object.
4.2 Operational aspect
It concerns the mobilisation of an object (definition, theorem, rule...) with a view to:
-Provide an example that meets the criteria for the object (such as a descriptive
definition. For instance: give the example of an odd function).
-Use a statement to construct the object described (as in the case of a constructive
definition3
).
For example: i) calculate the derivative of a function at a point using the definition;ii)
execute an algorithm (such as: perform a "standard" geometric construction -e.g. an
3
There are generally two main types of definitions: i) Analytical (or descriptive) definitions that define a
mathematical object by stating its carachteristic properties. Examples: definitions: of a bijection, of
continuity,... (ii) Genetic (or constructive) definitions that define the mathematical object by providing a
means of obtaining it or a rule of its formation - i.e. a sort of a pattern governing it -.
Examples: definition of derivative number
h
afhaf
af h
)()(lim)(' 0
−+
= →
; a matrix of elements in a
ring.

5. 5
angle bisector- construct the image of a geometric figure under a translation, calculate
the roots of a quadratic equation according to conventional formulas).
- Establish whether or not an object meets the statement (e.g., demonstrate
derivability, or non-continuity of function at a point using a definition).
It should be noted in passing that prior understanding of the structural aspect is
necessary.
4.3 Semiotic aspect
Mathematics uses natural language, but also symbols, graphs, geometric figures,
diagrams, ... A mathematical object is likely to exist under either of these forms, and
we will say to simplify that they represent its semiotic aspect.
Understanding this aspect, about a given object, includes being able to:
- Reproduce, identify the various forms in which this object is likely to be
(re)presented. In particular, identify this object in various real situations.
Examples: recognize symmetry in a natural or artificial object; a fraction in a visual
model.
- Move from one mode of expression to another.
Thus, a geometric transformation can be described in various forms:
- Geometric (configuration). Example: The figure below shows that A'B'C is the
symmetric of ABC's with respect to X.
- Matricial:
Example: ( )aa
aa
cossin
sincos − is the rotation R(O,a) with centre O and angle a.
- Analytic (coordinated).
Example: (x, y) → (-x, -y) is a point reflection in the origin.
- Numerical (with complex numbers).

6. 6
For example, f(z) -z-a (a and z complex) is a translation; z'=z is a point reflection in
the origin.
- Verbal (natural language). Example: "M' is called the image of M under a point
reflection in O if O is the middle of segment [MM']."
Understanding an object, therefore, means being able to identify it in whatever form it
occurs, so that it can be effectively exploited when the time comes. And more
specifically being able to move from one mode of expression to another. This is
reminiscent of R. Douady's notion of change of framework (changement de cadre). In
some cases, however, it is not necessary to change the framework, but only the point
of view, by interpreting the concept or property in question in a different way. We
will call this activity a change of perspective..
Let's illustrate this with a very simple example. The entry DCBA

= expresses the
equality of two vectors. But it may be interesting to know it also expresses that the
ABDC quadrangle is a parallelogram, or that D is the image of C in under the vector
translation BA

. Similarly, it may be useful to consider an equality such GEFE

2= as
expressing that F is the image of G by the homothety of centre E and ratio 2, or that G
is the middle of EF. Similarly, a change in perspective makes it easy to calculate
1lim1 −→ x
Logx
x
through viewing this ratio as the derivative of the function Log x at x= 1.
4.4 Descriptive aspect
The understanding of this aspect means in particular to be able to:
- Formulate, identify, reformulate the properties of an object.
- Identify, state links of this object with other mathematical or extra-mathematical
objects.
-Quote specific cases of this object
- Quote generalizations about the object.
- Justify a procedure, an algorithm, … Elaborate mathematical proofs .
Once a mathematical object created, we discover concerning it a certain number
of properties, links with other concepts, interpretations in terms of other notions,
examples illustrating it, ... The latter play an important role in students' conceptions of
mathematical objects. For Bauersfeld (1985), the best learning is learning through
examples. He further argues that "In terms of memory there are no general or abstract -
i.e. context-free - concepts, strategies, or procedures. The person can think (produce)
relative generality in a given situation. But the products are not retrievable from memory
in the same generality or abstractness." Examples must, however, be sufficiently varied
not to obscure the generality of a notion. See in this regard the geometric conception of
the notion of continuity of a numerical function in secondary school students, resulting

7. 7
from the latter remaining attached to particular examples (curves) which were used to
introduce this notion (El Bouazzaoui, 1988, pp.349- 351)
4.5 Functional aspect
It concerns the recourse to an object to solve a given problem.
Understanding this aspect is mainly, in the face of a problem:
- Be aware and decide that the problem can be solved by appealing to the object in
question.
- Use this object and its properties effectively to come up with the solution
- Decide that the solution obtained is satisfactory.
The success of this undertaking requires an understanding of the four aspects above,
as well as the epistemological aspect that will be presented below.
There may be a need to examine how the recourse to the object in question is used to
solve the problem. Either the statement of the problem explicitly indicates the use of
this purpose, or (a far more interesting case) it is left to the student's initiative.
This aspect is overlooked in the case of GT. Indeed, as El Hasnaouy's (1996) work on
symmetries has shown, geometric transformations, once their chapters completed,
rarely return in the following chapters. Even in basic problems, the solution is looked
for through more "classic" means (including congruence of triangles), while the use
of GT can be far more elegant and economical. Without an explicit invitation, our
students do not spontaneously use GT to solve a problem.
The functional aspect is, however, the raison d'être of any mathematical notion,
because it is as a tool to solve given problems that a concept arises, before being
institutionalized, decontextualized and acquiring the status of a mathematical object.
This aspect differs in particular from the operating aspect in which respect for the
algorithmic procedure normally leads to the result.

8. 8
4.6 Epistemological aspect
Understanding the previous aspects cannot be satisfactory without taking into account
the importance of certain activities and attitudes in mathematics.
In particular, we include in this aspect the ability to:
- Make conjectures about an object.
If it is necessary to know how to successfully mobilise an object, it is also important
to be able to say why it works in this way, to speculate on how it works if a particular
element were to be modified or omitted.
- Have a critical attitude towards mathematical objects.
A satisfactory understanding of a mathematical object implies stepping back from that
object, being aware of its reach and its limitations, that is, the problems it solves, and
those for which it seems less effective. This is part of metacognition.
- Be aware of the major steps in the evolution of certain mathematical concepts.
It may be interesting to consider the history of a notion, a property,... the problems for
which it was originally forged, the potential epistemological obstacles inherent in it.
The education of this aspect in the student is likely to develop in him a vision of
mathematics as a dynamic, living and evolutionary science, and a critical attitude
towards this discipline and its objects.
It is not enough, of course, to separately consider each aspect, which would lead to a
crumbling of the object. It is necessary to ensure articulation between these different
aspects, in particular by effectively exploiting changes of frames (numerical,
algebraic, graphic, symbolic, geometric -configurations-, ...) and highlighting the
dialectic tool-object of the concept. In particular, we noted, for example, that
understanding the structural aspect is a necessary (but not sufficient) prerequisite.
While it is difficult to establish a total order between these various aspects, it seems
that some links are possible, as outlined in the following diagram:

9. 9
Note:
The first four aspects can be considered to be related to an object's past and present:
how was it constructed from other objects? What properties does it enjoy? How does
it occur?
The last two aspects are more about its future: What does this object look like? What
problems can it help solve? How will it behave if you operate on this or that
modification? ...
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10. 10
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