ck¢, a model to understand learners' understanding -- discussing the case of calculus

Cauchy, series of continuous functions1821

Nicolas Balacheff, CINESTAV , calculus meeting, September 2015

Cauchy, series of continuous functions1821 (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

Œuvres complètes p.372

Definition of continuitynote 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)

Definition of continuitynote 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)

Cauchy, series of continuous functions1821 (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.

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

The notion of a “infinitely small” is dynamic: an

infinitely small variable is a variable which has zero as a

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

“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?)Nicolas Balacheff, CINESTAV , calculus meeting, September 2015

Cauchy, from an interpretation to the modelling of a concept imageGibert 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)

From an interpretation to the modelling of a concept imageThree 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,

sense vs logicanal

cK¢, A MODEL TO UNDERSTAND LEARNERS’ UNDERSTANDING

discussing the case of Calculus

Nicolas BalacheffCNRS - Laboratoire d’Informatique de Grenoblenicolas.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

« ƒ is defined by f(x) = lnx + 10sinxIs the limit + in + ? »

with a graphic calculator 25% of errorswithout 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 ofmathematical 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 subjectsThe environment is reduced to those features that are relevant from an epistemic perspective:

the milieuthe learner’s antagonist system in the learning process

action

feedback

constraints

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 subjectsThe environment is reduced to those features that are relevant from an epistemic perspective:

the milieuthe learner’s antagonist system in the learning process

action

feedback

constraints

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

action

feedback

constraints

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

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 frameworkN. 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.

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

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]

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.

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]

RÉMI : So the polynomial is somewhere thereOLIVIER : 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

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 controlsWithout referent controls there are no means to assess the relevance and validity of action

cKȼ – an analysis frameworkN. Gaudin PhD.Criterion of choice

Curve conception

Analytical conception

Object conception

Plotting and computing

∑ – the curve and points (xi, yi) are visually closeR – draw curves and plot points

∑ – minimize 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 polynomialR – assess the expressions of fj(x)

∑ – decide on the regularity of the approximationR – 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 regularNicolas Balacheff, CINESTAV , calculus meeting, September 2015

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 and the or the points (xi, ) and the(xi, yi)

Global shape of the approximating curve and closeness of the and the or the points (xi, 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 registersFull use of Mapple as a tool for Calculus

Representation systems

Diagrams (plotting the functions)

Analytical and graphical

cKȼ – key role of controlsN. Gaudin PhD.

Formating data (discourse, actions, milieu, etc.

Referent controls - Shape of the curve of a 3rd degree polynomial- Closeness of and - Position of the curve with respect to the (xi, yi)

guide the resolution of the problemInstrumentation 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

cKȼ – key role of controlsB. 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?

Pedemonte 2002

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 034. Ludovic : yes this is true if we go on…37. Vincent : yes, but then the perimeter

… ?38. Ludovic: no, the perimeter is always the

same41. 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.

How does the perimeter vary at each stage? How does the area vary?”

Pedemonte 2002

31. Vincent : the area is always divided by 2…so, at the limit? The limit is a line, the segment from which we started … [A]

33. Vincent : yes, and then it is 034. Ludovic : yes this is true if we go on… [B]37. Vincent : yes, but then the perimeter? [C]38. Ludovic: no, the perimeter is always the

same [D] 41. Vincent: It falls on the segment… the

circles are so small. [A]42. Ludovic: Hmm… but it is always 2πr. [D] 43. Vincent: Yes, but when the area tends to

0 it will be almost equal… [A]44. Ludovic: No, I don’t think so. [D] 45. Vincent: If the area tends to 0, then the

perimeter also… [A] I don’t know… [E]

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

A B31. Vincent : the area is always divided by

2…so, at the limit? The limit is a line, the segment from which we started … [A]

33. Vincent : yes, and then it is 034. Ludovic : yes this is true if we go on… [B]37. Vincent : yes, but then the perimeter? [C]38. Ludovic: no, the perimeter is always the

same [D] 41. Vincent: It falls on the segment… the

circles are so small. [A]42. Ludovic: Hmm… but it is always 2πr. [D] 43. Vincent: Yes, but when the area tends to

0 it will be almost equal… [A]44. Ludovic: No, I don’t think so. [D] 45. Vincent: If the area tends to 0, then the

perimeter also… [A] I don’t know… [E]

Questioning controls to understand representations

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

formulaLudovic

Algebraicconception

Vincentsymbolic-arithmetic

conception

Questionning the sphere of practiceThe 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 know1. 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

Questionning the sphere of practiceWhat 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 ΣA study carried out in relation with Biehler’s prototypical categories.

The model cKȼ is used as a methological guide

Vilma Mesa

Questionning the sphere of practice 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 typesSymbolic 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,

ConclusionConception, knowing and concept

One more questionAre 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 pivotLet 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

ConclusionConception, 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

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

ConclusionA design frameworkcKȼ 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

reinforce

activate

destabilize

Learning as a journey in a graph of problems from an initial Ci to a targeted Ct (the content to be taught)

46Nicolas Balacheff, CINESTAV , calculus meeting, September 2015

ck¢, a model to understand learners' understanding -- discussing the case of calculus

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