cK¢, a model to understand learners' understanding -- discussing the case of Calculus

Cauchy, series of continuous functions
1821
1Nicolas Balacheff, CINESTAV , calculus meeting, September 2015

1821 (ref.Arsac 2013)
x is not explicit in the writing
 the notion of function can be
both practically close to the
modern one and conceptually
reflect the dominant
understanding of the time
un and x are two variables, but
x is the independent variable
on which depends un
Nicolas Balacheff, CINESTAV , calculus meeting, September 2015

3
1821 (ref.Arsac 2013)
Œuvres complètes p.372
Definition of continuity
note  the notation f(x) is known
- in the neighborhood of a point
- related to a “vision” of continuity
of a curve
- a kinematic expression of limit
- the domain of definition is not
defined
(discontinuity corresponds to points where the
function is not defined)

4
1821 (ref.Arsac 2013) Definition of continuity
note  the notation f(x) is known
- in the neighborhood of a point
- related to a “vision” of continuity
of a curve
- a kinematic expression of limit
- the domain of definition is not
defined
(discontinuity corresponds to points where the
function is not defined)

1821 (ref.Arsac 2013)
Monotonous evolution, the notion
of limit is controlled by a kind of
kinematic “concept image”
(inherited from Neper and Newton
and common at that time)
 Arsac notices that Cauchy did
not pretend that this is a
mathematical proof, as used to
do elsewhere in the course, but a
remark.

1853 (ref.Arsac 2013)
6
Cauchy recognized that
there are “exceptions”,
mainly those of the Fourier
series, and revised the
remark (or the proof?)
These exceptions were
pointed by Abel and Seidel.
The exception Cauchy
mentioned
sin(𝑛𝑥)
𝑛
The notion of a “infinitely small”
is dynamic: an infinitely small
variable is a variable which has
zero as a limit

1853 (ref.Arsac 2013)
7
the variable x remains implicit in
the expression [again embedded
in the terms of the series]
x
- the order of the text is not
congruent to the logical order it
expresses
- n depends on ε and not on x
∀ ε ∃ N ∀ x
This is a non-modern expression
of the Cauchy criterion of Uniform
convergence
∀ ε ∃ N ∀ x ∀ n>N ∀ n’>n
|sn-s n’ |< ε

1853 (ref.Arsac 2013)
8
“always”, following Arsac,
should be interpreted as
“∀ x”
The expression is still in terms
of variables, one independent
and one dependent, and their
co-variation underpinned by a
kinematic concept image.
The style of the text makes it still closer to a
remark than to a mathematical proof in the
modern way. The rigor is there, as a willing,
but this willing encounter obstacles: the
algebraic formalism of Calculus is yet not
available and the kinematic concept image
still dominant in the mathematical
community of that time.
(NB: but isn’t rigor always a willing?)

Cauchy, from an interpretation to the
modelling of a concept image
Gibert Arsac interpretation of Cauchy’s understanding is based on a
critical and rigorous analysis of the text taking into account the
situation of Calculus in the first half of the XIX° century:
1. The notion of variable dominates the notion of function
(dependent variable) with a kinematic vision of convergence
which impact the concepts of limit and continuity
2. Inequality (<, >) is rarely used and the algebraic notation of
absolute value is absent
3. The notion of continuity is still under construction, being defined
on an interval and not a point, tightly linked to a vision of the
graphical continuity of a curve.
4. Quantifiers are not in use (one have to wait for the XX° century)
making difficult to identify the dependences introduced by their
order in a statement, and the negation of a statement which
involves them (e.g. discontinuity as a negation of continuity)
9
analysisbasedonArsac2013

From an interpretation to the
modelling of a concept image
Three dimension of analysis drives the interpretation and
may allow to model the thinking underpinning the case of
Cauchy’ concept of uniform convergence:
- the nature of the problem addressed (convergence of
series of continuous functions)
- the available tools to solve this problem which include
those to manipulate rational numbers, variables,
function, limit, continuity
- the semiotic systems including natural language,
algebraic representation as available at that time,
representation of curves
- the controls like the Leibniz law of continuity, the
repertoire of known functions,
10
sense vs logic
analysisbasedonArsac2013

cK¢, A MODEL TO UNDERSTAND
LEARNERS’ UNDERSTANDING
discussing the case of Calculus
Nicolas Balacheff
CNRS - Laboratoire d’Informatique de Grenoble
nicolas.balacheff@imag.fr

Understanding learners’ understanding
“Asking a student to understand something means asking a
teacher to assess whether the student has understood it.
But what does mathematical understanding look like?”
(Common core state standard initiative retrieved 11/10/13)
With the objective of contributing to a response , let’s start from the following two
theoretical postulates:
From a didactical perspective teaching design consists of
producing a game specific to the target knowledge among
different subsystems: the educational system, the student system,
the milieu, etc.
(Brousseau 1986)
From a developmental perspective, a concept is altogether: a set of
situations, a set of operational invariants, and a set of linguistic
and symbolic representations.
(Vergnaud 1980)

Note about the vocabulary (1)
Misconceptions, naïve theories, beliefs have been largely documented in an attempt
to make sense of learners’ errors and contradictions
13
« ƒ is defined by f(x) = lnx + 10sinx
Is the limit +  in + ? »
with a graphic calculator 25% of errors
without a graphic calculator 5% of errors
(Guin & Trouche 2001)
Decisions are situated
Distributed in space and time, decisions which are never brought face to face
in practice are practically compatible even if they are logically contradictory
(paraphrasing Bourdieu)
Contradictions and errors appear when learners are involved in situations
foreign to their actual practice but in which they have to produce a response

Note about the vocabulary (2)
“many times a child’s response is labeled erroneous too quickly
and […] if one were to imagine how the child was making sense of
the situation, then one would find the errors to be reasoned and
supportable” (Confrey 1990 p.29).
Learners have conceptions which are adapted and efficient in
different situations they are familiar with. They are not naïve or
misconceived, nor mere beliefs. They are situated and
operational in adequate circumstances.
They have the properties of a piece of knowledge.
 Knowledge is a difficult English word which can refer to implicit or
explicit mental constructs, it can express the familiarity of someone
with something or be authoritative with a theoretical status.
Instead of “knowledge” I will use “knowing” as a noun, leaving “knowledge” (saber) for those
“knowings” (conocimiento) which have a social and institutional status.

Behaviors and understanding
 what does mathematical understanding look like?”
understanding cannot be reduced to behaviors, whereas it
cannot be characterized without linking it to behaviors
This is a classical feature in psychology
A behavior is
- a product of mental acts (ways of understanding)
(Harel 1998)
- a component in an activity (it is intentional)
- a response to a situation (it is situated)
it has explicit (what) and implicit (why) dimensions
- a construct not a given

Behaviors and understanding
 what does mathematical understanding look like?”
understanding cannot be reduced to behaviors, whereas it
cannot be characterized without linking it to behaviors
This is a classical feature in psychology
A behavior is
- a product of mental acts (ways of understanding)
(Harel 1998)
- a component in an activity (it is intentional)
- a response to a situation (it is situated)
problems as revealers of
mathematical understanding

the “learner/milieu system”
A learner is first a person with her
emotions, social commitments, imagination,
personal history, cognitive characteristics.
He or she lives in a complex
environment which has physical, social
and symbolic characteristics.
However, for the sake of the modelling objective
and with in mind the practical limitations it will
entail…
Learners are considered here as the
epistemic subjects
The environment is reduced to those features
that are relevant from an epistemic perspective:
the milieu
the learner’s antagonist system in the learning
process
17
action
feedback
constraints
S M

the “learner/milieu system”
A learner is first a person with her
emotions, social commitments, imagination,
personal history, cognitive characteristics.
He or she lives in a complex
environment which has physical, social
and symbolic characteristics.
However, for the sake of the modeling objective
and with in mind the practical limitations is will
entail…
Learners are considered here as the
epistemic subjects
The environment is reduced to those features
that are relevant from an epistemic perspective:
the milieu
the learner’s antagonist system in the learning
process
18
action
feedback
constraints
S M
A conception is the state of
dynamical equilibrium of an
action/feedback loop between
a learner and a milieu under
proscriptive constraints of
viability

Conception (2) a characterization
a “conception” is
characterized by a
quadruplet (P, R, L, Σ)
where:
 P is a set of problems
sphere of practice
 R is a set of operators
 L is a representation system
 Σ is a control structure
19
action
feedback
constraints
S M
the quadruplet is not more related to S than to M: the representation system allows the formulation and use of
operators by the active sender (the learner) as well as the reactive receiver (the milieu); the control structure allows
assessing action, as well as selecting a feedback.

Representations and the challenge of
translation / interpretation
20
Egyptian computation of 10 times 1/5
for 4055/4093 one will get the shortest and unique
additive decomposition:
[1/2 + 1/3 + 1/7 + 1/69 + 1/30650 + 1/10098761225]
Unfortunately, Egyptians could not write the last term.
What is denoted by the signs are parts of
the whole, hence integers but integers
which could not be added as integers are.
Scribes used tables to establish the
correspondence between two numbers to
be multiplied and the result.

cKȼ – an analysis framework
N. Gaudin PhD.
The following yi provide values with possible errors (+/-10
%). These values come from a 3rd degree polynomial which
coefficients are unknown, evaluated at a series of points xi.
Five approximations (f1 … f5) are proposed.
You have to choose the one with approximate the best this
polynomial:
 on the interval [0;20]
 on [0 ; +∞ [
Explain why you choose or not each of this approximations.

N. Gaudin PhD.
f1(x) = 1.2310 + 0.0752 x + 1.789 × 10-
3 x2
f2(x) = 1.2429 + 0.06706 x +
2.833×10-3 x2 – 3.48 ×10-5 x3
f3(x) = 1.2712 + 0.0308 x + 0.0115 x2 –
7.1626 ×10-4 x3 + 1.704 ×10-5 x4
f5(x) = 8,817×10-5x3 - 0.00160x2 +
0.10977x + 1.2200 with f5(0) = 1,22 ;
f5(6) = 1,84 ; f5(13) = 2,57 et f5
(20)=3,48
22
f4 defined by: (1) it passes through each point (xi, yi); (2) on
each interval [xi ; yi], it is a polynomial of a degree equal or
less than 3; (3) it is twice differentiable and its second
derivative is continuous; (4) its algebraic representation is the
following on each interval [xi ; yi]): [3rd degree polynomials]
Maple

N. Gaudin PhD.
Gather data about the subject/milieu
interactions and the discourse
Create atoms composed of:
 an action which is performed
 a statement about an action
 a statement about a fact
Atoms are classified depending on their role
(operator, control) and gathered when they
correspond to the same action or
judgement.

N. Gaudin PhD.
24
RÉMI : So the polynomial is somewhere there [A26]
OLIVIER : Yeah. The best approximation could be
outside [A27 a]. So we have not made so much
progress [A27 b].
RÉMI : It depends how we define the best. It depends
if you consider that a point out of there is a bad thing
or if you consider it on average… if it is the set of point
which ok… [A28] You see what I mean? So we try to
draw all the polynomial, you see? We draw all
OLIVIER : all in a raw? [A29]
RÉMI : Not sure that it will be easy to see anything, but we can try,
and use the colors.
OLIVIER : You will remember that the yellow is the first? Can you
write it? Then green… blue , we have to chose the colors… red. May be
we avert yellow. Try « teal », it’s the best color which exists [A30]

N. Gaudin PhD.
RÉMI : So the polynomial is somewhere there
OLIVIER : Yeah. The best approximation
could be outside [A]. So we have not made
so much progress [B].
RÉMI : It depends how we define the best. It
depends if you consider that a point out of
there is a bad thing or if you consider it on
average… if it is the set of point which, ok…
[C] You see what I mean? So we try to
draw all the polynomial, you see? We draw
all
OLIVIER : all in a raw? [D]
RÉMI : Not sure that it will be easy to see
anything, but we can try, and use the
colors.
OLIVIER : You will remember that the yellow
is the first? Can you write it? Then green…
blue , we have to chose the colors… red. May
be we avert yellow. Try « teal », it’s the best
color which exists [E]
 A assessment of a fact
 B judgment
 C assessment of the judgement
 D decision on an action
 E assessment of an action
Several statement may be gathered
within one atom
One statement may split into several
atoms

N. Gaudin PhD.
 Two types of controls:
 Referent control  to identify objects in order
to characterize them by their properties
 Instrumentation control  to establish a
relation between referent controls and
operators to be used
Tight dependence between operators and controls
Without referent controls there are no means to assess
the relevance and validity of action

N. Gaudin PhD.
27
Criterion of choice Curve conception Analytical
conception
Object conception
Plotting and
computing
∑ – the curve and
points (xi, yi) are
visually close
R – draw curves
and plot points
∑ – minimize
[𝑓𝑖 𝑥𝑗 − 𝑦𝑗]²
R – make an
evaluation of
∑ – minimize the
difference (f-P)
R – make an
evaluation of
Regularity ∑ – continuity,
less than 2
variations
R – draw the
curves, plot points
∑ – f(x) is a 3rd
degree polynomial
R – assess the
expressions of fj(x)
∑ – decide on the
regularity of the
approximation
R – assess the
irregularity of fj
Uncertainty f1, f2, f3 are
equivalent
approximations
f2 is the best
approximation
no best
approximation
without a purpose,
but f1 and f2 are
the most regular

N. Gaudin PhD.
28
Curve conception Analytical
conception
Object conception
Referent
controls
Global shape of the
approximating curve
Visual closeness of the
approximating curve to the
(xi, yi)
Closeness of the 𝑓𝑖 𝑥𝑗
and the 𝑦𝑗 or the
points (xi, 𝑓𝑖 𝑥𝑗 ) and
the(xi, yi)
Global shape of the
approximating curve
and closeness of the
𝑓𝑖 𝑥𝑗 and the 𝑦𝑗 or
the points (xi, 𝑓𝑖 𝑥𝑗
and the(xi, yi)
Instrumentation
controls
Related to the use of
Mapple to plot the
functions
Selecting the formula
Related to the use of
Mapple for the
calculations
Integration of the
algebraic and
graphical registers
Full use of Mapple as
a tool for Calculus
Representation
systems
Diagrams (plotting the
functions)
Analytical and
graphical
Analytical and
graphical

cKȼ – key role of controls
N. Gaudin PhD.
 Formating data (discourse, actions, milieu, etc.
Referent controls
- Shape of the curve of a 3rd degree polynomial
- Closeness of 𝑓𝑖 𝑥𝑗 and 𝑦𝑗
- Position of the curve with respect to the (xi, yi)
guide the resolution of the problem
Instrumentation controls
- Distance between the approximating curve and points (xi, yi)
- Criterion of best approximation
guide the choice of operators adequately to the referent controls
 controls are more often than not implicit

B. Pedemonte PhD.
Construct a circle with AB as a
diameter. Split AB in two equal
parts, AC and CB. Then
construct the two circles of
diameter AC and CB… and so on.
How does the perimeter
vary at each stage?
How does the area vary?
30
A B
C
Pedemonte 2002

B. Pedemonte PhD.
31. Vincent : the area is always divided by
2…so, at the limit? The limit is a
line, the segment from which we
started …
32. Ludovic : but the area is divided by two each
time
33. Vincent : yes, and then it is 0
34. Ludovic : yes this is true if we go on…
37. Vincent : yes, but then the perimeter … ?
38. Ludovic: no, the perimeter is always the
same
41. Vincent: It falls on the segment… the circles
are so small.
42. Ludovic: Hmm… but it is always 2πr.
43. Vincent: Yes, but when the area tends to 0 it
will be almost equal…
44. Ludovic: No, I don’t think so.
45. Vincent: If the area tends to 0, then the
perimeter also… I don’t know…
46. Ludovic: I will finish writing the proof.
31
A B
Construct a circle with AB as a diameter. Split AB in two
equal parts, AC and CB. Then construct the two circles of
How does the perimeter vary at each stage?
How does the area vary?”
Pedemonte 2002

B. Pedemonte PhD.
started … [A]
time
34. Ludovic : yes this is true if we go on… [B]
37. Vincent : yes, but then the perimeter? [C]
same [D]
are so small. [A]
42. Ludovic: Hmm… but it is always 2πr. [D]
will be almost equal… [A]
44. Ludovic: No, I don’t think so. [D]
perimeter also… [A] I don’t know…
[E]
32
A B
Construct a circle with AB as a diameter. Split AB in two
equal parts, AC and CB. Then construct the two circles of
How does the perimeter vary at each stage?
How does the area vary?”
Pedemonte 2002

cKȼ – Structuring data
 The cKȼ modelling framework
- provides means to elicit key features of learners’ conceptions
- but does not account for their structure.
 Argumentation in relation to a conception
- drives the process (e.g. referent controls)
- provides "reasons“ (either epistemic, logical or referent)
- but does not necessarily back validity from a mathematical
perspective (e.g. taking into account teacher expectations).
Use of the Toulmin’s schema because of the role of control
(warrant, backing) in shaping a conception and driving
problem-solving

Bridging cKȼ and the Toulmin’s schema
 the rebuttal could take
the form of an external
feedback (e.g. feedback
from a peer)
 the warrant could come
from the conception or
not (e.g. an element of the
control structure or a hint
provided by the teacher)
 controls could be part of
the warrant (e.g.
instrumentation control)
or backing (e.g. referent
control)

Bridging cKȼ and the Toulmin Schema
35
A B
started … [A]
time
34. Ludovic : yes this is true if we go on… [B]
37. Vincent : yes, but then the perimeter? [C]
same [D]
are so small. [A]
42. Ludovic: Hmm… but it is always 2πr. [D]
will be almost equal… [A]
44. Ludovic: No, I don’t think so. [D]
perimeter also… [A] I don’t know…
[E]

Questioning controls to understand
representations
36
The symbolic
representation works
as a boundary object
adapting the different
meanings but being
robust enough to work
as a tool for both
students.
The differences lie in
the control grounding
their activity.
Algebraic frame
area /perimeter
formula
Ludovic
Algebraic
conception
Vincent
symbolic-arithmetic
conception

Questionning the sphere of practice
The origin of conceptions is in their
mobilization in teaching-learning situations
and problem-solving activities.
For most students functions as
mathematical objects are met in the
classroom (what does not mean that the
concept is not relevant in other contexts but
rarely used or necessary)
Then, it is important to know
1. What conceptions are induced by
textbooks?
2. Are the patterns of conception
different in different countries?
3. Is there a relation between the
conceptions induced by textbooks and
students performance
Vilma Mesa

What relevance and which use of functions in
problems?
determination of the sphere of practice P
What learners need to solve these problems?
determination de R
Which representations are required?
détermination de L
How learners can know that their solution is
correct?
détermination de Σ
A study carried out in relation with Biehler’s prototypical
categories.
The model cKȼ is used as a methological guide
Vilma Mesa

 Establishing a coding procedure, testing
against bias (consensus of coders),
 Sample of textbooks from 48 pays
 2304 énoncés
 P - 10 categories
 R - 39 items
 L - 9 items (graphical, numerical, verbal)
 Σ - 9 items
Conception : Symbolic rule, Set of Ordered Pairs, Social data
(controlled by context), Physical phenomena (modelling control),
controlling image (multiple representations)
Vilma Mesa

cK¢, a guide for textbooks analysis
“What should the problems look
like so that important aspects of
function are at stake?”
“What combinations of
operations, representations,
and controls should be available
to the students, so that they can
effectively put those aspects
into action?”
“The scarcity of controls
available to the students is
probably one of the most
pressing problems to address.”
Dominating types
Symbolic rule 20 %
Ordered pair 14 %
Social data 7 %
Physical phenomena 4 %
Controlling image 3 %
Vilma Mesa
“ the TIMSS items, as a set, do not
share the same characteristics as
those depicted by the tasks in the
textbooks”

Conclusion
The rich case of functions
 multiple problématiques
 as such within mathematics (as an object or a tool),
as a modelling instrument (e.g. physics, economy,
etc.)
 multiple representations
 algebraic, numerical, graphical, geometrical
 dialectic of the graphical and the symbolic
 multiple operators and classes of controls
 algebraic, logical, numerical, geometrical
 a large complex of related conceptions
 real numbers, functions, variables, continuity, limit,

Conclusion
Conception, knowing and concept
One more question
Are the conception we diagnose and the one we “hold” referring to the same
“object”?
Difficult in mathematics where the only tangible things we manipulate are
representations, but Vergnaud’s postulate (1981) offers a solution:
problems are sources and criteria of knowing
Representation a pivot
Let C, C’ and Ca be three conceptions such that it exists functions of representation ƒ:
L→La and ƒ’: L’→La
[C and C’ have the same object with respect to Ca if for all p from P it exists p’ from
P’ such that ƒ(p)=ƒ’(p’), and reciprocally]
Conceptions have the same object if their spheres of practice can
be matched from the point of view of a more general conception
 which in our case is the conception of the researcher/teacher

Conclusion
Conception, knowing and concept
 “To have the same object with respect to a conception Ca” sets an
equivalence relation among conceptions.
 Let’s now claim the existence of a conception Cµ more general than
any other conception to which it can be compared
(pragmatically this is the role of a piece of a mathematical theory as a reference)
A “concept” is the set of all conceptions having the
same object with respect to Cµ.
This definition is aligned with the idea that a mathematical concept is not reduced
to the text of its formal definition, but is the product of its history and of practices
in different communities, esp. the mathematical one.
A “knowing” is any set of conceptions.
In other words: a conception is the instantiation of a knowing by a situation (it
characterizes the subject/milieu in a situation) or a conception is the instantiation
of a concept by a pair (subject/situation).

Conclusion
44
cK¢ proposes a modeling framework to
provide
 an analytical tool
 to “represent” mathematical understanding
 to address the complexity of accounting for
learners ways of understanding
 a unifying formalism
 to inform the design of learning material and
learning situations, including technology
enhanced learning environments

Conclusion
A design framework
cKȼ is a tool to drive the design of a
learning situation
For a given content to be taught
identify
- the most relevant class of
problems and situations
- the tools/operators accessible to
the students & those made
available by the milieu
- the semiotic means available to
the student & at the interface
with the milieu
- the controls available to the
learner in order to take decisions
and to make judgement & the
kind of feedback the milieu may
provide
45
C
C
C
C
C
C
P
P P
P
reinforce
activate
link
destabilize
Learning as a journey in a graph
of problems from an initial Ci to a
targeted Ct (the content to be
taught)

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