domain: math shs

text:
Annamarie Conner
email: aconner@uga.edu

An application of Habermas' rationality to the teacher's actions: Analysis of argumentation in two classrooms

Keywords: Argumentation, proof, geometry, teaching

In this paper, I argue that Habermas' components of epistemic, teleologic, and communicative rationality provide insight into the differences in teachers' support for collective argumentation. I examine the teacher's supportive actions in two different classrooms. In their interactions with students, the teachers emphasize different components of rationality. I suggest that teachers may act in ways to support students' development of components of rationality by asking different kinds of questions and raise the question of whether it is useful to consider the components separately.

Introduction

It is generally accepted that argumentation and proof are crucial to the study of mathematics. Argumentation has been shown to be particularly important to the learning of mathematics through social interaction. Numerous examples in the mathematics education literature have unpacked aspects of arguments in elementary and secondary classrooms (e.g., [START_REF] Krummheuer | The ethnography of argumentation[END_REF][START_REF] Pedemonte | How can the relationship between argumentation and proof be analysed[END_REF]; these cases have focused on the learning of mathematics through participation in argumentation, the similarity of argumentation to the structure of proof, the analysis of proof as argument, and the role of the teacher within argumentation. Recent research has examined "successful" argumentation within classroom discussions [START_REF] Boero | Argumentation and proof: Discussing a "successful" classroom discussion[END_REF], argumentation that does not meet expectations [START_REF] Cramer | Argumentation below expectation: A double-threefold Habermas explanation[END_REF], and different aspects of rationality with respect to argumentation [START_REF] Boero | Habermas' construct of rational behavior in mathematics education: New advances and research questions[END_REF]. This paper explores the differences in collective argumentation that can be observed in classrooms. It addresses a temptation to characterize the argumentation in one classroom as productive and that in the other as problematic and suggests an explanation for the teacher's actions in each case can be found in Habermas' (1998) constructs of rationality as described by [START_REF] Boero | Habermas' theory of rationality as a comprehensive frame for conjecturing and proving in school[END_REF].

Background

In this paper, we explore the teacher's role in argumentation through the combined lenses of our interpretation of Toulmin's (1958[START_REF] Toulmin | The uses of argument[END_REF] description of arguments in multiple fields, our framework for teacher support of collective argumentation [START_REF] Conner | Teacher support for collective argumentation: A framework for examining how teachers support students' engagement in mathematical activities[END_REF], and [START_REF] Boero | Habermas' theory of rationality as a comprehensive frame for conjecturing and proving in school[END_REF] description of Habermas' (1998) components of rationality.

Habermas' (1998) components of rationality have been applied to argumentation in several ways. [START_REF] Boero | Habermas' theory of rationality as a comprehensive frame for conjecturing and proving in school[END_REF] analyzed a seventh grade student's argument (and the reactions of teachers to the argument) using three interrelated components introduced by Habermas: epistemic, teleologic, and communicative rationality. Boero gave the following explanation of these components.

Epistemic rationality is related to the fact that we know something only when we know why the statements about it are true or false…the crucial requirement is that the person has elaborated an evaluation of propositions as true and is able to use them in a purposeful way and to account for their validity. The teleologic rationality is related to the intentional character of the activity, and to the awareness in choosing suitable tools to perform the activity…The communicative rationality is related to communication practices in a community whose members can establish communication amongst them…rational means that the subject has the intention of reaching the interlocutor in order that he/she can share the content of communication, with an adequate and conscious choice of tools to make it possible. (p. 189-190) Boero concluded that the student acted in a rational way, using all three components of rationality, while the teachers' behavior did not meet these criteria for rationality.

Recently, multiple researchers have taken up Habermas' components of rationality to examine a range of issues with argumentation (see [START_REF] Boero | Habermas' construct of rational behavior in mathematics education: New advances and research questions[END_REF]. Within [START_REF] Boero | Habermas' construct of rational behavior in mathematics education: New advances and research questions[END_REF] research forum report, Douek introduced the construct of rational questioning, suggesting that teachers can ask students questions in order to "organize the mathematical discussion according to the three components of rationality" (p. 1-210). The teacher plays an essential role in organizing and supporting argumentation in classrooms. In this, Habermas' construct of communicative rationality is key, but the teacher can also influence the teleologic and epistemic rationality of the classroom community.

We follow [START_REF] Krummheuer | The ethnography of argumentation[END_REF] in adapting Toulmin's (1958[START_REF] Toulmin | The uses of argument[END_REF] description of argumentation to collective argumentation in mathematics classrooms. We define collective argumentation broadly as any instance in which students or teachers make a mathematical claim and support it with evidence. Our adaptation of Toulmin's diagrams (see Figure 1) includes the use of color (line style) to denote the contributor(s) of components of an argument and the addition of contributions and actions of the teacher that prompt or respond to parts of arguments (teacher support). Our framework for teacher support of collective argumentation includes three main kinds of supportive actions: direct contributions of argument components, questions, and other supportive actions such as gestures or diagrams [START_REF] Conner | Teacher support for collective argumentation: A framework for examining how teachers support students' engagement in mathematical activities[END_REF]. We defined a teacher's support for collective argumentation as any teacher move that prompted or responded to an argument component. We used Toulmin's (1958[START_REF] Toulmin | The uses of argument[END_REF] model to classify the direct contributions of argument components, and we used an inductive approach to develop categories of questions and other meaningful supportive actions the teacher used. More details about the development of the framework are available in [START_REF] Conner | Teacher support for collective argumentation: A framework for examining how teachers support students' engagement in mathematical activities[END_REF].

Methods

The analyses in this paper are based on data collected from a project that investigated the beliefs and argumentation practices of a cohort of secondary prospective teachers in the southeastern United States. In particular, the data for this paper include video recordings, field notes, and other artifacts from two days selected from a unit of instruction in each of two student teachers' (Ms. Bell and Ms. Carr, all names are pseudonyms) classrooms. These days were selected as representative of the variety of instructional moves observed in each teacher's instruction. We diagrammed every episode of argumentation from these days of instruction and categorized every supportive action of the teacher that we identified. We used Habermas' (1998) components of rationality to examine the actions of the teacher within our framework categories.

Episodes from two classrooms

The episodes presented and diagrammed below capture essential qualities of the instruction in each teacher's classroom. For each teacher, we present an excerpt of an episode of argumentation, our interpretation as captured by a partial diagram, and a summary of the teacher's support for argumentation in the class. We then examine the teacher's (and students') actions using Habermas' (1998) components of rationality and argue that the teacher's actions with respect to argumentation reflect her teleologic rationality. That is, we examine the teacher's supportive actions as tools to infer her goals for students' learning and contributions to class and her classroom norms.

Ms. Carr's Class

This episode occurred when Ms. Carr and her students were at the beginning of a unit on congruence. The students had not yet learned any of the triangle congruence theorems. Thus they were proving figures congruent by their definition of congruence, which required all corresponding sides and all corresponding angles to be congruent. Ms. Carr posed the problem in Figure 2 Figure 3 shows the diagram of this excerpt of an argument. Notice that three parts of the argument were contributed by students with a significant amount of support from the teacher. Ms. Carr asked a question that prompted each of the argument components, pointed at or wrote something on the board for each of them, and restated or affirmed each part as well. A little more than five minutes later, Ms. Carr and her students had compiled all of the information about the figure into congruence statements. They ended the proof construction by verifying that they had three pairs of congruent segments and three pairs of congruent angles, warranting the claim that the triangles were congruent with the definition of congruent triangles. In the diagram for this excerpt of argument (Figure 4) we see that the teacher contributed the final claim, the teacher and students jointly contributed the data, and a student contributed the warrant. Ms. Carr prompted both the data and warrant, and she supported each of these components with actions such as repeating, pointing, and writing on the board.

Ms. Carr supported her students in making arguments by contributing many argument components, including approximately one-half of the warrants in her class. In addition, she prompted most argument component by asking questions (primarily factual answer and elaboration questions, [START_REF] Conner | Teacher support for collective argumentation: A framework for examining how teachers support students' engagement in mathematical activities[END_REF], and she provided additional support for these argument components using other several kinds of supportive actions (including focusing, evaluating, informing, and repeating actions, [START_REF] Conner | Teacher support for collective argumentation: A framework for examining how teachers support students' engagement in mathematical activities[END_REF]. The importance of Ms. Carr's choices in supporting her students' arguments becomes clear as we reference Habermas' (1998) components of rationality. Ms. Carr asked several questions (line 541, lines 547-548) that requested a factual answer and then asked for elaboration by asking the students to justify that answer in line 550. In this interchange we see an assumption by Ms. Carr of her students' epistemic rationality. She invited them to participate in the argument and indicated by her questions that they should have reasons for their statements. This may be an instance of Douek's rational questioning, as described in [START_REF] Boero | Habermas' construct of rational behavior in mathematics education: New advances and research questions[END_REF], although Douek's rational questioning seems to presuppose all three aspects of rationality. Ms. Carr seemed to focus on epistemic rationality for her students, while Ms. Carr's own statements and actions indicate a focus on communicative rationality for herself. She repeated or restated (and often wrote the statement on the board) all of the student-contributed components of the arguments. If we consider the teleological rationality of Ms. Carr's actions, they appear to be very goal-directed. Her goal was student understanding of concepts and procedures. In search of that goal, her goal or focus for students was on epistemic rationality. She intended to make sure that they knew the reasons for the statements that were made. Across the class periods, this was evidenced by her many questions prompting argument components as well as her pervasive prompting and providing of warrants for arguments.

Ms. Bell's Class

In Ms. Bell's class, the excerpt exemplifying her instruction involved a task in which students had measured the interior angles of several polygons. Students were asked to find a formula for the sum of the interior angles of an n-sided polygon. So same thing; he got it a different way. When he got to the 540--so he got this 540 out when he plugging in 3 for his s, and he got 540. We wanted to get 180 when we plug in a 3. So he said, 'how am I going to get from 540 to 180?' So he found the difference between them and subtracted from this product. Do y'all see that? 1482 Martin:

It works with all of them too.

Figure 5 shows the diagram of this argument. In this argument, Martin (the student) contributed all of the components except one warrant. Ms. Bell prompted three of the components with questions, and she supported five of the components by restating or rephrasing Martin's contributions. In general, Ms. Bell asked questions of multiple kinds to prompt argument components, and she contributed some components of arguments, but only about one-eighth of the contributed warrants. Students in Ms. Bell's class seemed to contribute more autonomously to arguments, as evidenced by components that were neither prompted by nor responded to by Ms. Bell. We see evidence of Ms. Bell's teleologic rationality [START_REF] Habermas | On the pragmatics of communication[END_REF] in her actions and questions in support of her goal of engaging students in doing mathematics. Ms. Bell modeled actions related to all three components of rationality, and she seemed to encourage all three components of rationality in her students. Ms. Bell's actions show a strong emphasis on developing her students' communicative rationality, not only in her communication with her students (see line 1475 in which she restates the student's argument), but in her encouragement of her student to communicate his ideas more clearly (e.g., line 1458) and in the student's instinctive actions and statements (e.g., lines 1444 and following in which he explained the entries in the chart he drew on the board), which illustrate norms established in this class. Several times after the student gave a claim and warrant, Ms. Bell seemed to slow down the presentation to make sure it was clear to others, enhancing their understanding of communicative rationality. But instead of giving all the information herself, she asked the student to do so (line 1467). She seemed to be balancing engaging in acts of communicative rationality herself and encouraging her student to do so. In addition, Ms. Bell assumed epistemic rationality in her student and encouraged him to express it (line 1458). The beginning and end of the episode evidence a classroom norm regarding goal-directed behavior and use of appropriate tools (lines 1444-1455; line 1482). The student indicated by his final statement that he had intentionally completed his goal of finding, expressing, and justifying the formula for the sum of the interior angles of a polygon, showing the teacher's encouragement of behavior exhibiting teleologic rationality. This episode illustrates a teacher's use of rational questioning, bringing the students' voices into the discussion and encouraging their implementation and understanding of all three components of rationality (Douek in [START_REF] Boero | Habermas' construct of rational behavior in mathematics education: New advances and research questions[END_REF].

Discussion

The argumentation we observed in the two classes was very different. One classroom was characterized by a focus on students' epistemic rationality and the teacher's communicative rationality. The argumentation in this class seemed to be both somewhat shallow and more formal and proof-like. The other classroom was characterized by a more balanced focus on students' epistemic, communicative, and even teleologic rationality, and we saw the argumentation in this class as somewhat informal but characterized by student autonomy. The second classroom also illustrated some intentionality and awareness of components of rationality (although not with those words) in the interactions, as Douek suggested was necessary [START_REF] Boero | Habermas' construct of rational behavior in mathematics education: New advances and research questions[END_REF]. The teachers also used different kinds of tasks in their classrooms. The choice of tasks in each classroom may also be related to the teachers' intentions with respect to the components of rationality; more research is necessary to examine this question.

Differences were observed in the kinds of questions each teacher asked. Ms. Bell asked a wide range of questions, while Ms. Carr asked primarily factual answer and elaboration questions. Perhaps the kinds of questions teachers ask may indicate their focus on a particular component of rationality.

It is an open question as to the significance of these components of rationality in a mathematics class, but if we want to encourage students to view mathematics as rational and to act in rational ways when engaging in the study of mathematics, then it seems that it would be helpful for teachers to act in ways that encourage all components of rationality at appropriate points (as Douek suggested, to engage in rational questioning, [START_REF] Boero | Habermas' construct of rational behavior in mathematics education: New advances and research questions[END_REF]. As [START_REF] Boero | Habermas' theory of rationality as a comprehensive frame for conjecturing and proving in school[END_REF] suggested, teachers can model the components of rationality for their students at appropriate times. Perhaps introducing these components of rationality to teachers could provoke a wider focus. Examining the kinds of questions teachers ask in conjunction with their argumentation shows promise for revealing which components of rationality are privileged in their classes. And these components of rationality provide a useful explanatory mechanism for the differences in support for argumentation observed in classrooms. Future research will have to examine how important it is for a teacher to engender all components of rationality and whether it is possible or productive to address each component separately.

Figure 1 :

 1 Figure 1: Adaptation of Toulmin's (1958/2003) Diagram for an Argument

  to her class; the students and she worked together to mark relevant parts of the figure, and when we enter the discussion, they had modified the figure as shown. (They had extended segments BC, AB, and CD, and they marked angle ABE and angle DCE as angles of interest.)

Figure 2 :

 2 Figure 2: Initial Problem in Ms. Carr's Class and Modification 541 Ms. Carr: Okay. So, what I have marked up here in green, we said are what? What is their special relationship? 543 Alice: They are alternate interior angles. 544 Ms. Carr: They are alternate interior. Ok. So, let's write that down. ABE, let's call it, and angle, what is it? Angle DCE…[writes ABE DCE on board, leaving space between the two angles] [unrelated conversation/interruption] Now, I left some space in there. What symbol needs to go, what do we know about these? 549 Students: {congruent} {congruence} 550 Ms. Carr: Awesome. They are congruent. Why do we know that? 551 Cameron:Because they are alternate interior angles.

Figure 3 :Figure 4 :

 34 Figure 3: Diagram of First Excerpt of Argument in Ms. Carr's Class

Figure 5 :

 5 Figure 5: Diagram of Argument from Ms. Bell's Class

  The brief snippet of class we examine occurred when a student was presenting his group's work at the end of class. Prior to this excerpt, a student representing a different group presented a solution. Martin, the student in this episode, asked to present his solution because his group found the solution in a different way from the first student.

	1471 Martin:	[writes 180 x 3 = 540] 540. But then I just subtracted 180 from 540 and it
		equals 360. Yeah. So, subtract 360. [Writes f(s) = 180s -360]
	1475 Ms. Bell:
	1444 Martin:	All right. I had the chart. This is the sides of the figure. That would be the
		sum of the interior angles.
	…	[Martin talks as he constructs a chart containing numbers of sides and
		corresponding sums of interior angles for polygons with three to eight sides]
	1456 Martin:	And then it changes by 180 degrees each time.
	1458 Ms. Bell:	So Martin, the fact that it changes by the same number each time, when you're
		going up by one side, tells you what?
	1460 Martin:	That it has--that that's the slope.
	1461 Ms. Bell:	That's the slope. Which means it's? Karin, you said it earlier. What does that
		mean when it's?
	1463 Martin:	Linear.
	1464 Ms. Bell:	Linear. It's linear, right? If it changes the same amount each time, when you're
		going up by 1, it's going to be a linear function.
	1466 Martin:	So I did f(s) = 180s
	1467 Ms. Bell:	What is that 180?
	1468 Martin:	It's the slope. But that doesn't work out right, because 180 times 3 is like
	1470 Ms. Bell:	540

Proceedings of CERME10


Vincent Calvez
email: vincent.calvez@math.cnrs.fr

King-Yeung Lam

Uniqueness of the viscosity solution of a constrained Hamilton-Jacobi equation

In quantitative genetics, viscosity solutions of Hamilton-Jacobi equations appear naturally in the asymptotic limit of selection-mutation models when the population variance vanishes. They have to be solved together with an unknown function I(t) that arises as the counterpart of a non-negativity constraint on the solution at each time. Although the uniqueness of viscosity solutions is known for many variants of Hamilton-Jacobi equations, the uniqueness for this particular type of constrained problem was not resolved, except in a few particular cases. Here, we provide a general answer to the uniqueness problem, based on three main assumptions: convexity of the Hamiltonian function H(I, x, p) with respect to p, monotonicity of H with respect to I, and BV regularity of I(t).

Introduction

This note is intended to address uniqueness of the viscosity solution of the following Hamilton-Jacobi equation, under a non-negativity constraint:

         ∂ t u(t, x) + H(I(t), x, d x u(t, x)) = 0 , t ∈ (0, T ) , x ∈ R d , min x∈R d u(t, x) = 0 , t ∈ (0, T ), u(0, x) = g(x) , x ∈ R d , (1.1)
where

H : R × R d × R d → R is of class C 2
, and the initial data g ∈ W 1,∞ loc (R d ) satisfies min g = 0. This problem arises naturally in the analysis of quantitative genetics model in the asymptotic regime of small variance [START_REF] Diekmann | The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach[END_REF][START_REF] Perthame | Transport equations in biology[END_REF][START_REF] Barles | Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics[END_REF][START_REF] Perthame | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF][START_REF] Barles | Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result[END_REF][START_REF] Lorz | Dirac mass dynamics in a multidimensional nonlocal parabolic equation[END_REF].

The main difficulty beyond the classical issue of weak solutions in the viscosity sense stems from the role played by the scalar quantity I(t) which is subject to no equation, but is attached to the non-negativity constraint min u(t, •) = 0. Moreover, its regularity in context is usually low, typically of bounded variation [START_REF] Perthame | Transport equations in biology[END_REF][START_REF] Perthame | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF][START_REF] Barles | Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result[END_REF]. Interestingly, we shall see that BV seems to be the natural regularity ensuring uniqueness of the constrained problem. In order to give a sense to problem (1.1) in such a setting, we use the theory of viscosity solutions for equations with a measurable dependence in time which was first studied by H. Ishii [START_REF] Ishii | Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets[END_REF] and then by P.L. Lions and B. Perthame [START_REF] Lions | Remarks on Hamilton-Jacobi equations with measurable time-dependent Hamiltonians[END_REF].

Motivation and previous works

A special case of constrained Hamilton-Jacobi equation (1.1) arises in the asymptotic limit of the following quantitative genetics model proposed in [START_REF] Barles | Concentrations and constrained Hamilton-Jacobi equations arising in adaptive dynamics[END_REF][START_REF] Perthame | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF] (see also [START_REF] Perthame | Transport equations in biology[END_REF][START_REF] Barles | Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result[END_REF][START_REF] Lorz | Dirac mass dynamics in a multidimensional nonlocal parabolic equation[END_REF]):

∂ t n = n R(I (t), x) + 2 ∆n , t > 0 , x ∈ R d , I (t) = R d ψ(x)n (t, x) dx, (1.2)
where n (t, x) is the density of a population structured by a d-dimensional phenotypical trait x. The reproduction rate R of a given individual depends both on its trait x, and the environmental impact of the population I . Individuals may burden differently, and that burden is weighted by the function ψ which is bounded below and above by positive constants. The key point is that I is a scalar quantity, so that individuals compete for a single resource. As is natural for biological populations, the density-dependent feedback is negative, meaning that R(I, x) is decreasing with respect to I.

The Hopf-Cole transformation u = -log n transforms (1.2) into the following equation:

∂ t u + R(I (t), x) + |d x u | 2 = ∆u , which yields formally (1.1) in the vanishing viscosity limit → 0, with H(I, x, p) = R(I, x) + |p| 2 :

         ∂ t u + R(I(t), x) + |d x u| 2 = 0 , t ∈ (0, T ) , x ∈ R d , min x∈R d u(t, x) = 0 , t ∈ (0, T ), u(0, x) = g(x) , x ∈ R d . (1.3)
In fact, locally uniform convergence to a viscosity solution was established under suitable assumptions on R and the initial data, but along subsequences n → 0 [START_REF] Perthame | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF][START_REF] Barles | Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result[END_REF][START_REF] Lorz | Dirac mass dynamics in a multidimensional nonlocal parabolic equation[END_REF]. Therein, compactness of {I } usually follows from a uniform BV estimate. The constraint min u(t, •) = 0 can then be derived from natural properties of the integral I = ψ exp(-u / ) dx being uniformly positive and bounded in , as a consequence of the negative feedback of I on growth. However, the convergence of u n → u does not bring any direct information about the limit function I in the limit n → 0, except that the constraint min u(t, •) = 0 must be satisfied at any time.

The uniqueness of the limiting problem (1.3), if available, enables to obtain the convergence of the whole family of solutions {u } as → 0. It interesting in the mathematical as well as biological perspectives, since the limiting problem (1.3) determines much of the Darwinian evolutionary dynamics of the population model (1.2). When is small, the population n concentrates at the point(s) where the limit function u reaches its minimum value 0.

The uniqueness was first treated in [START_REF] Perthame | Dirac concentrations in Lotka-Volterra parabolic PDEs[END_REF], for the particular case when R(I, x) is separable in the following sense:

R(I, x) = B(x) -D(x)Q(I), or R(I, x) = B(x)Q(I) -D(x),
with positive functions B, D, and a monotonic function Q such that R is decreasing with respect to I.

Later on, the uniqueness for (1.3) was treated in [START_REF] Mirrahimi | A class of Hamilton-Jacobi equations with constraint: uniqueness and constructive approach[END_REF] under convexity assumptions on R(I, x) and the initial condition g, essentially: R decreasing with respect to I, and concave with respect to x, plus g convex. It was proved that convexity is propagated forward in time so that the solution u(t, x) to (1.3) is a solution in the classical C 1 sense. Hence, it has always a unique minimum point x(t), which is a smooth function of t. As a result, I(t) is necessarily smooth in that setting, since it can be determined by the implicit relation R(I(t), x(t)) = 0. Biologically interpreted, their results describes very well the Darwinian dynamics of a monomorphic population as it evolves smoothly towards a (global) evolutionary attractor. Recently, the preprint [START_REF] Kim | On the uniqueness for one-dimensional constrained Hamilton-Jacobi equations[END_REF] tackled the uniqueness of (1.3) when the trait space is one-dimensional, with a mixture of separable and non-separable growth rate R(I, x), without convexity assumptions. However, the uniqueness result is also restricted to the case of continuous functions I(t), which is not guaranteed in the absence of convexity.

Assumptions and main result

In this paper, we establish uniqueness of solutions to problem (1.1) under mild conditions. In distinction with previous works, we assume neither (i) separability of the Hamiltonian H(I, x, p) in any of its variables; nor (ii) convexity in the trait variable x ∈ R d . In particular, we can handle solutions (u, I), allowing possibly:

-x → u(t, x) to possess multiple minimum points {x i (t)}; and -the Lagrange multiplier I(t) to be discontinuous. Both of them are natural and attractive features of the solutions of population genetics models.

We restrict to C 2 Hamiltonian functions H(I, x, p) which are convex and superlinear with respect to the third variable p:

(H1): (∀I, x, p) d 2 p,p H(I, x, p) > 0 , (∀I, x) lim |p|→+∞ H(I, x, p) |p| = +∞ .
Our uniqueness result strongly relies on the following monotonicity assumption:

(H2): (∀I, x, p) ∂H ∂I (I, x, p) < 0 .
As we are dealing with convex Hamiltonians, it is approriate to reformulate the problem using suitable representation formulas: For a given function I(t), we define the variational solution V (t, x) of (1.1) as follows:

V (t, x) = inf γ∈AC(0,t) γ(t)=x t 0 L(I(s), γ(s), γ(s)) ds + g(γ(0)) , (1.4) for (t, x) ∈ (0, T ) × R d . Here L : R × R d × R d → R is the Lagrangian function, i.e.
the Legendre transform (or convex conjugate) of H defined as:

L(I, x, v) = sup p∈R d {p • v -H(I, x, p)} . (1.5)
It is such that d p H and d v L are reciprocal functions. In this formulation, the problem (1.1) becomes the determination of I(t) so that the value function V (t, x) satisfies the constraint: min V (t, •) = 0. In the formulation (1.4), the role played by the scalar quantity I(t) is perhaps more apparent: it should somehow be adjusted in an infinitesimal and incremental way to satisfy the following constraint at each time, among all γ ∈ AC(0, t) irrespective of the endpoint of γ:

(∀t) inf γ∈AC(0,t) t 0 L(I(s), γ(s), γ(s)) ds + g(γ(0)) = 0 . (1.6)
Our methodology relies on the Lagrangian formulation of the constraint (1.6). Due to the above variational reformulation, it is more appropriate to write the assumptions on the Lagrangian function: Assumptions (H1)-(H2) can be recast as:

(L1): (∀I, x, v) d 2 v,v L(I, x, v) > 0 . (L2): (∀I, x, v) ∂L ∂I (I, x, v) > 0 .
We need two supplementary conditions, to be satisfied locally in I ∈ (-J, J) for any constant J > 0:

(L3): There exist a constant C Θ , and a super-linear function Θ : R + → R + so that

(∀I, x, v) ∈ (-J, J)×R d ×R d L(I, x, v) ≥ Θ(|v|)-C Θ , lim r→+∞ Θ(r) r = +∞ .
(L4): For each K > 0, there exists positive constants α K , β K such that

(∀I, x, v) ∈ (-J, J) × B(0, K) × R d |d x L(I, x, v)| ≤ α K + β K L(I, x, v) .
Finally, we assume the initial data g to be locally Lipschitz continuous, nonnegative and coercive:

(G): g ∈ W 1,∞ loc (R d ) , min x∈R d g(x) = 0 , and lim |x|→∞ g(x) = +∞.
Remark 1. The super-linearity in (L3) holds true pointwise in (I, x) by the very definition of the Legendre transform (1.5). The main point here is the uniformity with respect to x.

Theorem 1. Under the assumptions (L1)-(L4), suppose that I 1 and I 2 are two non-negative BV functions associated with two variational solutions V 1 and V 2 of (1.4) with the same initial data g satisfying (G). Then, I 1 and I 2 coincide almost everywhere, and so do V 1 and V 2 .

To make the connection with viscosity solutions, we state the following auxiliary result: Theorem 2. Under the assumptions (L1)-(L4), given a function I(t) of bounded variation, the variational solution V (t, x) given by (1.4) is the unique locally Lipschitz viscosity solution of (1.1) over [0, T ) × R d .

The reason for separating these two results is to emphasize the use of the variational formulation in our proof. It would be of considerable interest to by-pass the variational formulation and derive uniqueness from PDE arguments only. Uniqueness of unbounded solutions generally requires stringent conditions on the growth of the solution and the Hamiltonian [START_REF] Barles | An introduction to the theory of viscosity solutions for first-order Hamilton-Jacobi equations and applications[END_REF][START_REF] Crandall | Remarks on the existence and uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations[END_REF], but here this issue is mediated by the fact that the Hamiltonian function is convex, and the solution is nonnegative by definition. We could not find a reference containing precisely Theorem 2, but [START_REF] Maso | Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities[END_REF] is close, and we adapt their proof to our context in the Appendix.

Examples

First, we apply our result to the special case presented in Section 1.1.

Corollary 3. Consider the problem (1.3) with the initial data g as in (G). Assume that R ∈ C 2 (R + × R d ) satisfies (∀I, x) ∂R ∂I (I, x) < 0 and (∀I) sup x∈R d R(I, x) < +∞ , (1.7)
then the solution pair (u, I) to the constrained Hamilton-Jacobi equation (1.3) is unique, in the class of locally Lipschitz viscosity solutions u, and BV functions I.

The second condition in (1.7) is natural from the biological viewpoint as the net growth rate is presumably bounded from above. Since the three other conditions (L1), (L2) and (L4) are straightforward, it is sufficient to verify (L3). Indeed, the Lagrangian L is given by 1 4 |v| 2 -R(I, x). Moreover, since the range of the BV function I(t) lies in some bounded interval [0, J], it is sufficient to restrict R to [0, J] × R d to address uniqueness over a bounded time interval (0, T ). Since it is immediate from (1.7) that R is uniformly bounded over [0, J] × R d , (L3) follows.

Our result also includes relevant examples that were not covered by the previous contributions, particularly non-separable Hamiltonian functions H(I, x, p). For instance, consider the following quantitative genetics model:

∂ t n = 1 K x -x B(I (t), x )n (t, x ) dx -n D(I (t), x),
where I (t) is the same as in (1.2), and K is a probability distribution function that encodes the mutational effects after reproduction: if the parent has trait x , and gives birth at rate B(I , x ), the trait x of the offspring is distributed following

K (x -x ) = 1 K x-x
. Assume that K is symmetric, and has finite exponential moments, and denote by K its Laplace transform:

K(p) = K(z) exp(p•z) dz.
Then, the limiting problem as → 0 is (1.1) with the following Hamiltonian function [START_REF] Barles | Concentration in Lotka-Volterra parabolic or integral equations: a general convergence result[END_REF]:

H(I, x, p) = B(I, x)K(p) -D(I, x) . (1.8)
L(I, x, v) = B(I, x)L v B(I, x) + D(I, x) ,
where L(v) is the Legendre transform of K(p). Let I 1 , I 2 be the two functions that are involved in the uniqueness test. Let [0, J] be a bounded interval in which both I 1 and I 2 take values. Condition (L3) is clearly verified with Θ = 1 + L, because L is decreasing with respect to the value B, sup [0,J]×R d B < +∞, and L satifies (L3) automatically (see Remark 1). The justification of (L4) requires more work. We begin with the following inequality:

d v L(v) • v ≤ 2 (1 + L(v)) . (1.9)
To derive it, consider the following pointwise inequality: for all X ∈ R, cosh X ≤ 1 + 1 2 X sinh X, which in turn implies the following one by symmetry of K:

K(p) = K(z) cosh(p • z) dz ≤ 1 + 1 2 K(z)(p • z) sinh(p • z) dz = 1 + 1 2 K(z)(p • z) exp(p • z) dz .

By applying this estimate to

p = d v L(v) ⇔ v = d p K(p) in the dual Legendre transformation L ↔ K, we deduce, by the fact K(p) + L(v) ≥ p • v, that d v L(v) • v -L(v) ≤ K(p) ≤ 1 + 1 2 d v L(v) • v .
This yields the simple estimate announced in (1.9). The technical condition (L4) is reformulated in this context as follows, after division by B > 0:

d x (log B)L v B -d x (log B) d v L v B • v B + d x D B α K 1 B + β K L v B + D B .
It is indeed guaranteed for a suitable choices of α K , β K . The main arguments besides (1.9)

Regularity of the minimizing curves

The purpose of this section is to establish BV regularity of the derivative γ of any minimizing curve γ in (1.4). Such regularity is crucial in our argument of uniqueness. First, we establish the following consequence of the convexity and the super-linearity of the Lagrangian: Lemma 5. Assume (L1) and (L3), then

(∀I, x) lim |v|→+∞ d v L(I, x, v) • v |v| = +∞ ,
and the limit is uniform over (I, x) lying in compact subsets of R d+1 .

Proof. Fix K > 0 and let (I, x) ∈ [-K, K] × B(0, K). Let M > 0 be given, and choose r 0 = r 0 (K) > 0 large enough so that for all r ≥ r 0 ,

Θ(r) -C Θ -L(I, x, 0) ≥ M r .
Then we have, for r ≥ r 0 and e ∈ S d-1 , L(I, x, re) -L(I, x, 0) ≥ M r .

By convexity of L in v, we have, for all e ∈ S d-1 and r ≥ r 0 ,

d v L(I, x, re) • e = d dr L(I, x, re) ≥ L(I, x, re) -L(I, x, 0) r ≥ M .
Therefore, for each M there exists r 0 such that

d v L(I, x, v) • v ≥ M |v|, provided that |v| ≥ r 0 and (I, x) ∈ [-K, K] × B(0, K).
We will now establish the BV estimate of γt,x for (t, x) ∈ (0, T ) × R d . For the remainder of this section, we fix T > 0 and J so that sup (0,T ) |I| ≤ J. For ease of notation, dependence of various constants on T and J will be omitted. Lemma 6. For each K, there exists a constant C K such that, uniformly for (t, x) ∈ (0, T )×B(0, K), any minimizing curve γ t,x ∈ AC(0, t) associated with (1.4) satisfies

[ γt,x ] BV (0,t) ≤ C K t + [I] BV (0,t) .
(2.1)

Proof. The proof is divided into three steps, wherein classical arguments are recalled for the sake of completeness. For the sake of notation, we drop the superscript of γ t,x , assuming that the pair (t, x) ∈ (0, T ) × B(0, K) is fixed throughout the proof.

Step #1: L ∞ bound on γ. We deduce immediately the following bound from (L3):

t 0 Θ( γ(s)) ds -Ct + g(γ(0)) ≤ t 0 L(I(s), γ(s), γ(s)) ds + g(γ(0)) ≤ t 0 L(I(s), x, 0) ds + g(x) , (2.2)
from which we deduce a non-optimal W 1,1 estimate

t 0 | γ(s)| ds ≤ Ct + t 0 L(I(s), x, 0) ds + g(x) -g(γ(0)) .
by using the super-linearity of Θ in a crude way, namely, L ≥ |v| -C. Furthermore, we deduce from g ≥ 0 that γ belongs to L 1 (0, t), so that

γ(s) L ∞ (0,t) ≤ C K provided (t, x) ∈ (0, T ) × B(0, K).
Let A = A(g, K) be the (local) Lipschitz bound on g in the ball with radius C K . By updating the constant C, we can assume that L(I, x, v) ≥ (A + 1)|v| -C. Back to (2.2), we deduce that

(A + 1) t 0 | γ(s)| ds ≤ Ct + t 0 L(I(s), x, 0) ds + A t 0 | γ(s)| ds ,
We obtain as a consequence the following updated estimate:

1 t t 0 | γ(s)| ds ≤ C + 1 t t 0 L(I(s), x, 0) ds ≤ C , (2.3)
where the bound C is uniform for (t, x) ∈ (0, T ) × B(0, K), and I(s) taking values in [-J, J].

Step #2: L ∞ bound on γ. We deduce from (2.3) that there exists a subset S t,x ⊂ (0, t) of positive measure, such that for all ŝ ∈ S t,x , | γ(ŝ)| ≤ 2C . Since γ is a minimizing curve, it satisfies the following Euler-Lagrange condition in the distributional sense:

- d ds (d v L(I(s), γ(s), γ(s))) + d x L(I(s), γ(s), γ(s)) = 0 . (2.4)
Let Leb(f ) denote the set of Lebesgue points of the function f . Let s 1 and s 2 belong to Leb(I) ∩ Leb( γ). Let {ρ n } be a family of mollifiers. We can test (2.4) against s 0 (ρ n (s -s 2 ) -ρ n (s -s 1 ) ds . After integration by parts, we find that:

t 0 d v L(I(s), γ(s), γ(s)) (ρ n (s -s 2 ) -ρ n (s -s 1 )) ds ≤ t 0 |d x L(I(s), γ(s), γ(s))| ds ,
d v L(I(s), γ(s), γ(s))ρ n (s -s i ) ds = d v L(I(s i ), γ(s i ), γ(s i )) .
(2.6)

Then, we can specialize s 1 ∈ Leb(I) ∩ Leb( γ) ∩ S t,x because the latter has positive measure. We deduce from (2.5)-(2.6), and the definition of S t,x that

|d v L(I(s 2 ), γ(s 2 ), γ(s 2 ))| ≤ C + t 0 d x L(I(s ), γ(s ), γ(s )) ds ,
for all s 2 ∈ Leb(I) ∩ Leb( γ). Multiplying by the unit vector γ(s 2 ) | γ(s 2 )| , and using (L4), we find:

d v L(I(s 2 ), γ(s 2 ), γ(s 2 )) • γ(s 2 ) | γ(s 2 )| ≤ C + t 0 d x L(I(s ), γ(s ), γ(s )) ds ≤ C + t 0 α K + β K L(I(s ), γ(s ), γ(s )) ds ,
where K = γ L ∞ (0,t) . We deduce from the uniform bound of γ L ∞ (0,t) , and the minimizing property of γ that the right-hand-side is uniformly bounded for (t, x, I) ∈ (0, T ) × B(0, K) × [-J, J]. Hence,

d v L(I(s), γ(s), γ(s)) • γ(s) | γ(s)| ≤ C a.e. s ∈ (0, t),
and the boundedness of γ is a consequence of Lemma 5.

Step #3: BV bound on γ. Back to (2.4), we see that

p(s) = d v L(I(s), γ(s), γ(s))
is Lipschitz continuous as I, γ, γ ∈ L ∞ , and L is C 2 . By the Fenchel-Legendre duality, we have γ(s) = d p H(I(s), γ(s), p(s)).

Therefore, γ is BV .

From the chain rule involving BV functions (see [START_REF] Ambrosio | A general chain rule for distributional derivatives[END_REF] and references therein, as well as [2, Theorem 3.96]), we deduce as a by-product that

[ γ] BV (0,t) ≤ [d p H] Lip([-J,J]×B(0, γ ∞)×B(0, γ ∞)) [p] BV (0,t) + [γ] BV (0,t) + [I] BV (0,t) .
Both p and γ are Lipschitz continuous, so there exists a constant C such that max([p] BV (0,t) , [γ] BV (0,t) ) ≤ Ct. Hence, we have obtained (2.1).

Uniqueness of the variational solution (proof of Theorem 1)

Let I 1 , I 2 ∈ BV (0, T ). There exists a constant J such that I 1 and I 2 takes value in [-J, J]. Recall the definition of the variational solutions V 1 , V 2 :

         V 1 (t, x) = inf γ 1 ∈AC(0,t):γ 1 (t)=x t 0 L(I 1 (s), γ 1 (s), γ1 (s)) ds + g(γ 1 (0)) , V 2 (t, x) = inf γ 2 ∈AC(0,t):γ 2 (t)=x t 0 L(I 2 (s), γ 2 (s), γ2 (s))ds + g(γ 2 (0)) .
Lemma 7. There exists C > 0 such that for j = 1, 2,

V j (t, x) ≥ min |x| 2 , min |x |≥|x|/2 g(x ) -Ct .
Proof. By definition of the variational solution and (L3), we have

V j (t, x) = t 0
L I j (s), γ t,x j (s), γt,x j (s) ds + g γ t,x j (0)

≥ t 0 | γt,x j (s)| -C ds + g γ t,x j (0) ≥ x -γ t,x j (0) -Ct + g γ t,x j (0) ≥ min |x|/2, min |x |≥|x|/2 g(x ) -Ct
where we used the fact that γ t,x (t) = x and that |x| ≤ |x -γ t,x (0)| + |γ t,x (0)|, so that: either |x -

γ t,x (0)| ≥ |x|/2 or |γ t,x (0)| ≥ |x|/2.
By Lemma 7 and (G), we deduce that x → V j (t, x) attains minimum in some bounded set, say B(0, K), uniformly for t ∈ (0, T ). Let x t 1 (resp. x t 2 ) be some minimum point for V 1 (t, •) (resp. V 2 (t, •)) -this might not be unique -and let γ t 1 (s) (resp. γ t 2 (s)) be an optimal trajectory ending up at x t 1 (resp. x t 2 ). We deduce from Lemma 6 that γt j lies in BV , uniformly with respect to t ∈ (0, T ): max

j=1,2 γt j BV (0,t) ≤ C . (3.1)
The optimality of γ t 1 and γ t 2 , together with the constraints min V 1 (t, •) = 0 and min V 2 (t, •) = 0 implies the following set of inequalities:

0 = V 1 (t, x t 1 ) = t 0 L(I 1 (s), γ t 1 (s), γt 1 (s)) ds + g(γ t 1 (0)) ≤ t 0 L(I 1 (s), γ t 2 (s), γt 2 (s)) ds + g(γ t 2 (0)) = t 0 L(I 2 (s), γ t 2 (s), γt 2 (s))ds + g(γ t 2 (0)) + t 0 φ 1 (t, s) (I 1 (s) -I 2 (s)) ds , = - t 0 φ 1 (t, s) (I 2 (s) -I 1 (s)) ds .
i.e.

t 0 φ 1 (t, s) (I 2 (s) -I 1 (s)) ds ≤ 0 , (3.2)
where the positive weight φ 1 is given by

φ 1 (t, s) = 1 0 ∂L ∂I (1 -θ)I 1 (s) + θI 2 (s), γ t 2 (s), γt 2 (s) dθ . (3.3)
Similarly, by exchanging the roles of the two solutions, we obtain

t 0 φ 2 (t, s) (I 2 (s) -I 1 (s)) ds ≥ 0 , (3.4)
where the positive weight φ 2 is given by

φ 2 (t, s) = 1 0 ∂L ∂I (1 -θ)I 2 (s) + θI 1 (s), γ t 1 (s), γt 1 (s) dθ . (3.5)
By Assumption (L2) and the uniform boundedness of (I j (s), γ t j (s), γt j (s)) (by (3.1)), there exists λ > 0 such that:

(∀t, s) min j=1,2 φ j (t, s) ≥ λ . (3.6)
Functions of bounded variations have left-and right-limits everywhere. Here, we focus on the value of the right-limit at the origin. This is expressed in the following statement.

Lemma 8. Let φ 1 , φ 2 be defined as in (3.3) and (3.5). Then

lim t→0+ [φ 1 (t, •)] BV (0,t) + [φ 2 (t, •)] BV (0,t) = 0.
Proof. Our first observation is that BV regularity of {I j } j=1,2 implies the following smallness estimate: lim

t→0+ [I 1 ] BV (0,t) + [I 2 ] BV (0,t) = 0. (3.7)
The important point here is that the left point of the interval is fixed to 0. The same conclusion would not be true if the interval (0, t) would be replaced with (-t, t) due to possible jump discontinuity at the origin. To prove (3.7), let us decompose I 1 , say, into a difference of non-decreasing functions

I 1 = I + 1 -I - 1 . Then, [I 1 ] BV (0,t) ≤ I + 1 t 0+ + I - 1 t 0+ -→ t→0+ 0 , (3.8)
simply because I + 1 and I - 1 have right limits at the origin. By (2.1), we get that this vanishing limit can be extended to γt j as well:

γt j BV (0,t) ≤ C t + [I j ] BV (0,t) -→ t→0+ 0. (3.9)
Consequently, we are able to estimate [φ 2 (t, •)] BV (0,t) as follows. To keep the idea concise, we will compute the derivative ∂φ 2 ∂s of the BV function φ 2 (t, •) in the sense of a finite measure on (0, t), so that [φ 2 (t, •)] BV (0,t) = (0,t) | ∂φ 2 ∂s (t, s)| ds. We shall adopt this convention for the remainder of the paper.

∂φ 2 ∂s (t, s) ≤ 1 0 d 2 I,I L(Γ(s)) (1 -θ) İ2 (s) + θ İ1 (s) dθ + 1 0 d 2 I,x L(Γ(s)) γt 1 (s) dθ + 1 0 d 2 I,v L(Γ(s))γ t 1 (s) dθ ≤ C | İ1 (s)| + | İ2 (s)| + | γt 1 (s)| + |γ t 1 (s)| ,
where we have used the shortcut notation Γ(s) = ((1 -θ)I 2 (s) + θI 1 (s), γ t 1 (s), γt 1 (s)). We may integrate the latter over the open interval (0, t) to obtain

[φ 2 (t, •)] BV (0,t) ≤ C [I 1 ] BV (0,t) + [I 2 ] BV (0,t) + t + γt 1 BV (0,t) ≤ C [I 1 ] BV (0,t) + [I 2 ] BV (0,t) + t
where we used (3.9). By (3.8), we deduce that lim t→0+ [φ 2 (t, •)] BV (0,t) = 0. The proof for φ 1 is analogous.

We are now in position to prove Theorem 1.

Proof of Theorem 1. Let µ = I 2 -I 1 and suppose to the contrary that µ = 0 on a set of positive measure in (0, T ).

We claim that we may assume, without loss of generality, that µ = 0 in a set of positive measure in (0, t), for each t ∈ (0, T ). To see this claim, let t 0 := sup{t ≥ 0 : µ(s) = 0 a.e. in (0, t)}.

If t 0 = 0, we are done. If t 0 > 0, then the variational solutions V 1 (t, x) ≡ V 2 (t, x) are identical for (t, x) ∈ [0, t 0 ] × R d . By Lemma 7, x → V j (t 0 , x)
is coercive, and the other conditions in (G) are clearly satisfied. Thus we may re-label the initial time to be t 0 . In any case, it suffices to derive a contradiction assuming µ = 0 in a set of positive measure in (0, t), for each t ∈ (0, T ). Using t 0 φ 2 (t, s)µ(s) ds ≥ 0 (by (3.4)), we may integrate by parts to obtain

φ 2 (t, t-) t 0 µ(τ ) dτ = (0,t) ∂φ 2 ∂s (t, s) s 0 µ(τ ) dτ ds + t 0 φ 2 (t, s)µ(s) ds ≥ (0,t) ∂φ 2 ∂s (t, s) s 0 µ(τ ) dτ ds
Taking the negative part, we deduce the following partial estimate,

φ 2 (t, t-) t 0 µ(s) ds - ≤ sup s∈(0,t) s 0 µ(τ ) dτ (0,t) ∂φ 2 ∂s (t, s) ds (3.10)
Similarly we deduce from (3.2) that

φ 1 (t, t-) t 0 µ(τ ) dτ = t 0 ∂φ 1 ∂s (t, s) s 0 µ(τ ) dτ ds + t 0 φ 1 (t, s)µ(s) ds ≤ t 0 ∂φ 1 ∂s (t, s) s 0 µ(τ ) dτ ds
Taking the positive part, we deduce the following complementary estimate, Then, taking supremum in (3.12) for 0 < t < t 1 , we have

φ 1 (t, t-) t 0 µ(s) ds + ≤ sup s∈(0,t) s 0 µ(τ ) dτ t 0 ∂φ 1 ∂s (t,
λ sup t∈(0,t 1 ) t 0 µ(s) ds ≤ λ 2 sup s∈(0,t 1 ) s 0 µ(τ ) dτ .
This implies t 0 µ(s) ds = 0 for all t ∈ [0, t 1 ]. Hence, µ(t) = 0 almost everywhere on (0, t 1 ). This is in contradiction with the assumption that µ = 0 on a set of positive measure in (0, t 1 ), and we conclude that I 2 -I 1 = µ = 0 a.e. Finally V 1 ≡ V 2 by virtue of the variational formulation.

The Pessimization Principle: I(t) is non-decreasing

The pessimization principle [START_REF] Diekmann | A beginners guide to adaptive dynamics[END_REF] is a concept in adaptive dynamics, which says that if the environmental feedback is encoded by a scalar quantity I(t) ∈ R at any time, mutations and natural selection inevitably lead to deterioration/Verelendung. In the setting of this paper, it can be formulated by claiming that the population burden I(t) is a non-decreasing function.

In this section, we give an additional assumption that guarantees this claim.

Theorem 9. Under the assumptions (L1) -(L4), let (u, I) be the unique solution pair (u, I) to (1.1). Assume, in addition, that

(L5): (∀I, x, v) L(I, x, v) ≥ L(I, x, 0).
Then I is non-decreasing with respect to time.

Note that (L5) is equivalent to d v L(I, x, 0) = 0 by convexity, and thus to d p H(I, x, 0) = 0 by duality. It is clearly verified for the two examples in Section 1.3 by symmetry of the Hamiltonian function with respect to p = 0. Corollary 10. Under the assumption (1.7), let (u, I) be the unique solution pair to (1.3), with the initial data g as in (G). Then I is non-decreasing with respect to time.

Corollary 11. Under the assumptions of Corollary 4, let (u, I) be the unique solution pair to the problem (1.1) with the Hamiltonian (1.8), and the initial data g as in (G). Then I is non-decreasing with respect to time.

Proof of Theorem 9. We start by choosing the right-continuous representative of I without loss of generality. For each t > 0, let x t be a minimum point of x → u(t, x) as before, and let γ t be an associated minimizing curve ending up at x t .

Step #1: L(I(t-), x t , 0) ≤ 0 for all t. It follows from the non-negativity constraint and the dynamic programming principle that 0 ≤ u(s, γ t (s)) = s 0 L(I(s ), γ t (s ), γt (s )) ds + g(γ t (0)) for 0 < s < t, and the equality holds when s = t. Hence, we deduce that L(I(t-), γ t (t-), γt (t-)) ≤ 0. Since γ t (t-) = γ t (t) = x t , we may use (L5) to deduce that L(I(t-), x t , 0) ≤ L(I(t-), γ t (t-), γt (t-)) ≤ 0.

Step #2: I(t-) ≤ I(t+) for all t. Fix t > 0, let x t and γ t be as above. We define

γ 1 : [0, t + 1] → R d by γ 1 (s) = γ t (s) for 0 ≤ s ≤ t, x t for s > t.
Then γ 1 ∈ AC[0, t + 1] and for all 0 < h < 1, L(I(s ), x t 0 , 0) -L(I(t 0 -), x t 0 , 0) ds .

0 ≤ u(t + h, γ 1 (t + h)) ≤ t+h 0 L(I(s ), γ 1 (s ), γ1 (s )) ds + g(γ 1 (0)). Since 0 = u(t,
However, this is in contradiction with (4), in view of the fact that L is strictly increasing in I (L2).

A Variational and viscosity solutions coincide (proof of Theorem 2)

Given I ∈ BV (0, T ), let V (t, x) denote the corresponding variational solution of (1.4), and let u denote a locally Lipschitz viscosity solution of (1.1). The purpose of this section is to show that u ≡ V .

As the Hamiltonian is convex with respect to p, sub-solutions in the almost everywhere sense, and viscosity sub-solutions in particular, lie automatically below the variational solution [START_REF] Barles | An introduction to the theory of viscosity solutions for first-order Hamilton-Jacobi equations and applications[END_REF][START_REF] Fathi | Weak KAM theorem in Lagrangian dynamics[END_REF]. We include a proof here for the sake of completeness.

Proposition 12. Assume that u is locally Lipschitz, u(0, x) ≤ g(x) for all x, and that the following inequality holds for almost every (t, x) ∈ (0, T ) × R d , ∂ t u(t, x) + H(I(t), x, d x u(t, x)) ≤ 0 a.e.

(A.1)

Then, u ≤ V .
where d + u(t, x)(µ, v) is the one-sided directional differentiation in the direction (µ, v):

d + u(t, x)(µ, v) = lim sup s→0+ u(t, x) -u(t -sµ, x -sv) s .
We can interpret (A.6) as follows: there exists an element which is common to the partial epigraph of v → L(I(t), x, v):

E t,x = Epi v (L(I(t), x, v)) = (v, ) ∈ R d × R : r ≥ L(I(t), x, v) ,
and to the hypograph of v → d + u(t, x)(1, v):

Hypo v (d + u(t, x)(1, v)) = (v, ) ∈ R d × R : r ≤ d + u(t, x)(1, v) .
For technical reason, we consider the full hypograph of d + u(t, x), taken with respect to variables (µ, v) ∈ R × R d :

H t,x = Hypo (µ,v) (d + u(t, x)(µ, v)) = Hypo (µ,v) lim sup s→0+ u(t, x) -u(t -sµ, x -sv) s . (A.7) In contrast with Hypo v (d + u(t, x)(1, v)), H t,
x is a cone because the quantity in (A.7) is positively homogeneous with respect to (µ, v). In fact, it coincides with the definition of a contingent cone, up to a change of sign. If S ⊂ R N is a non-empty subset, and z ⊂ R N , recall that the contingent cone of S at z, denoted by T S (z), is defined as follows [3, Definition 3.2.1]:

w ∈ T S (z) ⇐⇒ lim inf s→0 + dist(z + sw, S) s = 0 .
Then, we claim the following equivalence:

H t,x = -T Epi u (t, x, u(t, x)) . (A.8)
For the convenience of readers, the equivalence (A.8) is illustrated in Figure 1 for a scalar function u. Now we show (A.8). Indeed, (µ, v, ) belongs to -T Epi u (t, x, u(t, x)) if and only if there exist subsequences s n → 0+ and (t n , x n , u n ) such that:

     t -s n µ = t n + o(s n ) x -s n v = x n + o(s n ) u(t, x) -s n = u n + o(s n ) , and
u n ≥ u(t n , x n ) .
The latter inequality is inherited from the choice S = Epi u. Reorganizing the terms, and using the Lipschitz continuity of u, we obtain:

u(t, x) -s n ≥ u(t -s n µ, x -s n v) + o(s n ) , u(t, x) -u(t -s n µ, x -s n v) s ≥ + o(1) .
The latter is precisely (A.7). i.e. (µ, v, ) ∈ H t,x and this proves (A.8). Summarizing, we are seeking an element (1, v) which is common to H t,x and to {1} × E t,x . The latter is a convex set, but the former is not necessarily convex.

H z 0 H z 1 H z 2 u(z)
Figure 1 -Illustration of various shapes of cones as they may appear for a scalar function u (in opposition to the text where the domain of u is genuinely multidimensional). The set H is represented in shaded green, whereas the corresponding contingent cones T = -H are depicted in shaded orange.

Therefore, we are led to consider its convex closure co(H t,x ) in order to use the separation theorem. Next, we shall use the viability theory to remove the convex closure, exactly as in [START_REF] Maso | Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities[END_REF].

Step #3: Finding the backward velocity: the separation theorem. We wish to avoid separation of the two convex sets co(H t,x ) and {1} × E t,x . We argue by contradiction. If the two sets are separated, then there exists a linear form q • + p, • such that (i) co(H t,x ) lies below the hyper-plane {(µ, v, ) : = qµ + p, v }, and (ii) {1} × E t,x lies strictly above it [START_REF] Rockafellar | Convex analysis[END_REF]. We deduce from the latter condition (ii) that q + p, v ≤ L(I(t), x, v) -δ for all v ∈ R d and some δ > 0. This can be recast as q + H(I(t), x, p) ≤ -δ from the definition of the Legendre transform. On the other hand, we deduce from condition (i) that lim sup s→0 + u(t, x) -u(t -sµ, x -sv) s = d + u(t, x)(µ, v) ≤ qµ + p, v , for all (µ, v) ∈ R d+1 . Consequently, (q, p) belongs to the subdifferential of u at (t, x). By applying the usual criterion of viscosity super-solutions (for continuous Hamiltonian functions), we find that q + H(I(t), x, p) ≥ 0. This is a contradiction. Thus, the two convex sets are not separated, i.e.

(∀t, x) co(H t,x ) ∩ ({1} × E t,x ) = ∅ . (A.9)

Step #4: Finding the backward velocity: the viability theorem. Note that (A.9) is equivalent to (∀t, x) co(-T Epi u (t, x, u(t, x))) ∩ ({1} × E t,x ) = ∅ . (A.10) holds for all (t, x, U ) ∈ Epi u, and not only for points (t, x, u(t, x)) on the graph of u. This is immediate, as T {Epi u} (t, x, U ) = R d+2 for U > u(t, x). Finally, all the assumptions of the viability theorem are met. As a consequence, we can remove the convex closure in (A.10), and thus in (A.9), so as to obtain:

(∀t, x) H t,x ∩ ({1} × E t,x ) = ∅ .

In particular, for each (t, x) there exists a vector v(t, x) such that (A.6) holds true.

Step #6: Building the backward trajectory up to the initial time. Now that we are able to make a small step backward at each (t, x), let > 0 be given, and start from (t 0 , x 0 ). There exists (s 0 , v 0 ) such that u(t 0 , x 0 ) ≥ s 0 L(t, x, v 0 ) + u(t 0 -s 0 , x 0 -s 0 v 0 ) -s 0 .

By choosing s 0 small enough, we can even replace the right-hand-side by: u(t 0 , x 0 ) ≥ s 0 0 L(t -s, x 0 -sv 0 , v 0 ) ds + u(t 0 -s 0 , x 0 -s 0 v 0 ) -2 s 0 .

(A.14)

In particular, we have u(t 0 , x 0 ) ≥ inf γ t 0 t 0 -s 0 L(s , γ(s ), γ(s )) ds + u(t 0 -s 0 , γ(t 0 -s 0 )) -2 s 0 , where the infimum is taken over all γ ∈ AC[0, t] such that γ(t 0 ) = x 0 . As a result, the set Σ = τ ∈ (0, t 0 ) : u(t 0 , x 0 ) ≥ inf γ t 0 τ L(s , γ(s ), γ(s )) ds + u(τ, γ(τ )) -2 (t 0 -τ ) .

is non-empty, and τ * := inf Σ ∈ [0, t 0 -s 0 ] is well-defined. We wish to prove that τ * = 0. Suppose, for contradiction, that τ * > 0. Then, the estimates obtained in Lemma 6 allows to extract a converging sequence {γ n } such that γ n is defined on the time span (τ n , t), with τ n τ * , and { γn } is uniformly BV [ 1] . Hence, we can pass to the limit γn → γ a.e. by Helly's Selection Theorem, and then use Bounded Convergence Theorem to prove that u(t 0 , x 0 ) ≥ inf t 0 τ * L(s , γ(s ), γ(s )) ds + u(τ * , γ(τ * )) -2 (t 0 -τ * ) .

By applying again the single step backward as in (A.14) at (τ * , γ(τ * )), we can push our lower estimate to an earlier time τ * * ∈ (0, τ * ), and obtain thoroughly a contradiction. Thus, τ * = 0 and u(t 0 , x 0 ) ≥ inf γ t 0 0 L(s , γ(s ), γ(s )) ds + g(γ(0)) -2 t 0 = V (t 0 , x 0 ) -2 t 0 .

By letting → 0, we have established that u ≥ V in the case when t → I(t) is Lipschitz continuous.

Corollary 4 .

 4 Consider the problem (1.1) with the Hamiltonian (1.8), and the initial data g as in (G). Assume that K is the Laplace transform of a symmetric p.d.f. with finite exponential moments, and that B, D ∈ C 2 (R + × R d ) are nonnegative functions which satisfy the following conditions: B > 0 everywhere, and(∀I) B(I, •) ∈ L ∞ (R d ),accompanied with the following monotonicity conditions: (∀I, x) ∂B ∂I (I, x) < 0, and ∂D ∂I (I, x) > 0 , then the solution pair (u, I) to the constrained Hamilton-Jacobi equation (1.1) is unique, in the class of locally Lipschitz viscosity solutions u, and BV functions I. Proof. There are a few items to check in order to apply Theorem 1. Firstly, the Hamiltonian function H (1.8) clearly verifies (H1) and (H2), hence (L1) and (L2) follows. Secondly, the Lagrangian function L associated with the Hamiltonian H (1.8) is

( 2 . 5 )

 25 where we have used that | s 0 (ρ n (s -s 2 ) -ρ n (s -s 1 ) ds | ≤ 1 for n large enough. Using the definition of the Lebesgue points, we find that lim n→+∞ t 0

  This work was initiated as the first author was visiting Ohio State University. VC has funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 639638). KYL is partially supported by the National Science Foundation under grant DMS-1411476.

are: both |d x (log B)| and |d x D/B| are locally uniformly bounded from above, 1/B is locally uniformly bounded from below, and D is non-negative.

Acknowledgment.

  Dividing by h, and letting h → 0+, we obtain L(I(t+), x t , 0) ≥ 0. Comparing with L(I(t-), x t , 0) ≤ 0 (by Step #1), we deduce from the monotonicity of L in I (L2) that I(t-) ≤ I(t+) for all t > 0.Step #3: Conclusion. Suppose to the contrary that I(t 2 ) < I(t 1 ) for some t 1 < t 2 . Since I is right-continuous, there exists t 3 > t 2 such that I(t) < I(t 1 ) for all t ∈ [t 2 , t 3 ). Let t 0 = sup{t ∈ [t 1 , t 3 ) : I(t 1 ) ≤ I(t)}. Then t 0 ≤ t 2 < t 3 , and I(t) < I(t 1 ) ≤ I(t 0 -) for t ∈ (t 0 , t 3 ).

	γ 1 (t)) =	t 0 L(I(s ), γ 1 (s ), γ1 (s )) ds + g(γ 1 (0)), we have
	t+h		t+h
	0 ≤	L(I(s ), γ 1 (s ), γ1 (s )) ds ≤	L(I(s ), x t , 0) ds .	(4.1)
	t		t
	Now, using Step #1 and (4.1) from Step #2, we have
			t 3
	0 ≤
			t 0

The family { γn} is even uniformly Lipschitz as I is assumed to be Lipschitz here, see the proof of Lemma 6.

Proof. The proof is adapted from [START_REF] Fathi | Weak KAM theorem in Lagrangian dynamics[END_REF]Section 4.2]. A more direct proof can be found in [4, Section 9] but the latter assumes time continuity for H, which does not hold in the present case. A first observation is that (A.1) makes perfect sense as u is differentiable almost everywhere by Rademacher's theorem. We shall establish that u(t 2 , γ(t 2 )) -u(t 1 , (γ(t 1 )) ≤ for all curves γ ∈ W 1,∞ . Thus, the result will follow immediately by taking the infimum with respect to γ, and invoking regularity of minimizing curves, as in Lemma 6.

To prove (A.2), we proceed by a density argument. The case of a linear curve γ = x + (s -t 1 )v is handled as follows: firstly, we deduce from (A.1) that

Secondly, by Fubini's theorem one can find a sequence x n → x such that (A.3) holds almost everywhere in the line {(s, x n + (s -t 1 )v)} for each n. Therefore, we can apply the chain rule to u(s, x n + (s -t 1 )v), so as to obtain:

We deduce that (A.2) holds true for all linear curves by integrating (A.4) from t 1 to t 2 and taking the limit n → +∞. Consequently, (A.2) holds true for any piecewise linear curve. The conclusion follows by a density argument of piecewise linear curves in the set of curves having bounded measurable derivatives.

It remains to show that viscosity super-solutions lie above the variational solution. The criterion for super-solution for time-measurable Hamiltonians that we adopt is the following one. (See [START_REF] Ishii | Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets[END_REF][START_REF] Lions | Remarks on Hamilton-Jacobi equations with measurable time-dependent Hamiltonians[END_REF] for various other equivalent definitions.) Definition 13 (Viscosity super-solution). Let φ ∈ C 1 (R d ) be such that the minima of u(t, •) -φ are reached in a ball of radius R for all t ∈ [0, T ]. Let M(t) be the set of minimum points of u(t, •) -φ, and m(t) = min u(t, •) -φ. Then, it is required that the following inequality holds true in the distributional sense:

Proposition 14. Assume that u is a locally Lipschitz viscosity super-solution, in the sense of Definition 13, and that u(0, x) ≥ g(x) for all x. Then, u ≥ V .

Proof. We follow the lines of [START_REF] Maso | Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities[END_REF] which is essentially based on convex analysis. We adapt their proof in our context for the sake of completeness. We will first prove the proposition in the special case of I ∈ W 1,∞ (0, T ). This assumption will be relaxed to I ∈ BV (0, T ) at the end of the proof.

Step #1: Finding the backward velocity: setting of the problem. The key is to find, for each (t, x), a particular direction v(t, x), such that the following inequality holds true:

We wish to use the viability theorem [3, p. 85] (see also [START_REF] Maso | Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities[END_REF]Theorem 2.3]):

Theorem 15 (Viability). Suppose that G : R N R N is an upper semi-continuous set-valued map with compact convex values. Then for each closed set S ⊂ R N , the following statements are equivalent:

Further compactness estimate is required in order to apply Theorem 15. We claim that we can restrict (A.9) to a compact set:

(the choice of r K is possible due to the superlinear growth of Θ), and m, M are respectively m = min L, M = max L where both minimum and maximum are taken over the set J × {x} × B(0, R |x| ).

To this end, consider the following two options: either the dual cone (H t,x ) -is empty or non-empty. In the first case, it implies co(H t,x ) = R d , so that any element of E t,x is appropriate. In this case we have

In the second case, (H t,x ) -is non-empty. Hence, there exists a linear form q • + p, • such that co(H t,x ) lies below the linear set {qµ + p, v } as in Step #3. Therefore, every common point (1, v, ) ∈ co(H t,x ) ∩ ({1} × E t,x ) (and there is at least one such point) must satisfy L(I(t), x, v) ≤ ≤ q + p, v .

By the facts that (i) L grows uniformly super-linearly (by (L3)), and (ii) (q, p) is bounded as it belongs to the subdifferential of the locally Lipschitz function u, i.e. max{|q|, |p|} ≤ [u] Lip(J×B(0,|x|)) , we deduce

By the choice of r K in (A.11), we must have |v| < r K with K = |x|.

i.e. co(

By (A.12) and (A.13), and our choice of R |x| := max{1, r |x| }, we find that

To conclude, it remains to remove the additional continuity assumption on I(t). Let I ∈ BV (0, T ). First of all, we approximate I(t) from below by a sequence of Lipschitz functions I k (t) I(t) converging pointwise [ 2] : It follows from (H2) and (A.5) that u is also a super-solution associated with I k (t). Hence we have u ≥ V k in (0, T ) × R d , (A. [START_REF] Lorz | Dirac mass dynamics in a multidimensional nonlocal parabolic equation[END_REF] where V k is the variational solution associated with I k .

On the other hand, the compactness estimates on minimizing curves obtained in Lemma 6 combined with Lebesgue's dominated convergence theorem guarantees that V k

V . Thus, we may let k → ∞ in (A.15) to deduce u ≥ V . This completes the proof.


text:
Paolo Boero Disfor
email: boero@dima.unige.it

Cognitive unity of theorems, theories and related rationalities

Keywords: Theorems, conjecturing and proving, cognitive unity, proving as rational behavior

The construct of cognitive unity of theorems was introduced twenty years ago to identify suitable conditions for students' "smooth" approach to proving. In this paper the Habermas' construct of rationality, adapted to mathematics education in previous research, is used to identify some factors in the activation of cognitive unity of theorems. In particular, I consider the dependence of cognitive unity on the specific rationality (e.g. analytic geometry rationality, or synthetic geometry rationality) according to which a conjecturing and proving problem is dealt with. The analysis of some examples will provide evidence for it, together with hints for further research.

Introduction

"Cognitive unity of theorems" (CUTHE) is a construct introduced in Garuti, [START_REF] Garuti | Challenging the traditional school approach to theorems: A hypothesis about cognitive unity of theorems[END_REF] to account for a phenomenon detected in a grade 8 (13-years-old students) classroom engaged in a conjecturing and proving activity, concerning a theorem of space geometry contextualized and verbally expressed in terms of Sun rays (instead of straight lines) and Sun shadows (instead of shapes projected on a plane according to parallel projection rules). The conjecturing task (see [START_REF] Garuti | Challenging the traditional school approach to theorems: A hypothesis about cognitive unity of theorems[END_REF] may be shortly reported this way: "Is it possible that the Sun shadows of two non-parallel sticks are parallel on the ground? If yes, under which conditions?". After comparison and standard re-phrasing ("if… then…") of their conjectures, students were asked to validate their statements by "general reasoning". We observed that, while trying to validate their conjectures, several students resumed some pieces of personal reasoning (e.g. ways of looking at the Sun rays and the Sun shadows) developed during the production of the conjecture and the search for reasons for its validity, and arranged them in a deductive chain of statements. The ways of looking at the Sun rays and the space relationships had been different for different students; those ways corresponded to the different ways of proving the theorem by them. After having found other theorems (in geometry, and in elementary arithmetic) for which students behaved in a similar way, we defined "cognitive unity of theorem" (CUTHE) what happens for some theorems when: during the production of the conjecture, the student progressively works out his/her statement through an intensive argumentative activity functionally intermingled with the justification of the plausibility of his/her choices. During the subsequent statement-proving stage, the student links up with this process in a coherent way, organizing some of the previously produced arguments according to a logical chain (Garuti, Boero, & Lemut, 1998, p. 345).

The CUTHE construct was also extended to the case of the relationships between the exploratory phase of proving a theorem, and the subsequent construction of a proof for that theorem [START_REF] Garuti | Cognitive unity of theorems and difficulties of proof[END_REF]: indeed, the exploratory phase of proving shares some common aspects with conjecturing (as re-construction of the meaning, and appropriation, of a statement; and identification of elements for its validity). The construct of cognitive unity resulted in various research developments. [START_REF] Pedemonte | How can the relationship between argumentation and proof be analysed[END_REF][START_REF] Pedemonte | Argumentation and algebraic proof[END_REF] performed studies in which (given a theorem for which CUTHE is accessible to students) the mechanism of arranging arguments produced in the exploratory phase Thematic Working Group 01 does not result in a proof for some students, due to their difficulty of re-arranging inductive or abductive arguments into deductive arguments. These difficulties are not likely to emerge in algebraic conjecturing and proving [START_REF] Pedemonte | Argumentation and algebraic proof[END_REF], while they frequently emerge in the case of plane geometry [START_REF] Pedemonte | How can the relationship between argumentation and proof be analysed[END_REF]. Leung and Lopez Real (2003) investigated CUTHE in the case of computer-based learning environments, which change the nature of students' exploration and make CUTHE difficult to activate, finding out ways of activating it in the new situation. [START_REF] Fujita | Students' geometrical constructions and proving activities: a case of cognitive unity[END_REF] studied conditions under which CUTHE may be activated in the field of synthetic geometry: they "analyze the circumstances when students unite, or not, their conjecture production and proof construction"; the potential of geometrical constructions for the activation of CUTHE was explored. The quoted studies suggest the opportunity of investigating the conditions for the activation of CUTHE for a given theorem. [START_REF] Garuti | Cognitive unity of theorems and difficulties of proof[END_REF] started a discussion on it, taking into account both the student (her skills, her knowledge and expertise in a given field of mathematics) and the field of mathematics in which a given statement is dealt with. Douek (1998) analyzed the individual variety of exploration strategies and their effects on conjecturing and proving; at present (personal communication) she is further deepening the idea of subject-relativity of CUTHE, together with the relationships between the quality of student's exploration (including its semiotic features) and the construction of the proof. In this paper I will try to identify some aspects of the relativity of CUTHE referred to a given system of discursive practices that concern the truth of statements, the ways of producing and validating them, and the ways of communicating with others -i.e. a "rationality", according to [START_REF] Habermas | On the pragmatics of communication[END_REF].

Theoretical assumptions

Mathematical theory

It is possible to define a mathematical theory (shortly, a theory) by its characteristic components: primitive notions, and definitions related to them; postulates; inference rules to get true statements from the postulates and other statements proved as true. 'Characteristic components' depend on the historical period and, in a given historical period, on epistemological assumptions that may be different, according to different fields of mathematics. The case of Euclidean geometry before and after Hilbert's Grundlagen der geometrie is a well-known paradigmatic example of historical change in the ways of considering the requirements of a mathematical theory. In this paper, we will consider the following theories: Synthetic geometry (in particular, Euclidean geometry); Analytic geometry (including the algebraic treatment of conic sections); Elementary, verbal-semantic number theory (evidence for truth and inference rules rely on properties of the concept of number and its representations); Elementary, algebraic-formal number theory (evidence for truth of statements comes from the interpretation of an algebraic expression derived, through suitable syntactic transformations, from the algebraic expression which represents the problem situation). [START_REF] Mariotti | Introduction to proof: the mediation of a dynamic software environment[END_REF] defines a theorem as a statement and its proof with reference to a theory (and related inference rules). The definition results in the possibility of considering different theorems with the same statement (in particular, when different proofs referring to different theories are available). The definition encompasses theorems related to various kinds of theories and related inference rules: Euclid's as well as Hilbert's geometry; analytic geometry; graph theory, with its reference to visual objects; 19th-century probability theory as well as Kolmogorov's axiomatic theory, etc.; and the different ways of considering proof since the Greeks, including verbal-semantic proofs (like in Euclid) and modern algebraic-formal proofs of arithmetic statements.

Theorem

CUTHE and Habermas' rationality

In this paper we are interested in CUTHE, one possible aspect of the conjecturing and proving process, in order to identify for which theorems (in Mariotti's sense) it may be easily activated, thus we need a comprehensive frame to deal with the process of proving and its relationships with the product (proof) to be built up in a given theory. Habermas (1998, pp. 310-317) deals with the complexity of discursive practices according to three interrelated components, concerning: knowledge at play, and the answer to "why is it true" questions in a given cultural context (epistemic rationality); action and its goals, and strategies to achieve them, to be evaluated (teleological rationality); communication and related, intentional choices in a given social context on a given subject (communicative rationality). In [START_REF] Boero | Habermas' construct of rational behavior in mathematics education[END_REF] a detailed elaboration of the reasons for adapting Habermas' construct to mathematics education is presented, with references to how it has been used in different studies. In the case of proof and proving, according to Mariotti's definition of theorem, the adaptation of the Habermas' construct concerns:

 criteria for validity of inferences and truth of statements within a theory, and their dependence on historical periods, mathematical domains, and institutions and cultures. Inferences may rely on visual evidence, or conceptual meaning, or syntactic transformations, etc.;  problem solving strategies that may be adopted to reach the goal of proving, along with their effectiveness; strategies may use analogies, abduction, and so on. Strategies and exploration are not constrained within the border of the reference theory;  the choice and use of appropriate communication means for proof in a given context, together with the relationships among them, taking into account the goal of the proving processa proof, conforming to requirements specified for the first and the third components. The expression 'rationality frame' will be used to put into evidence the system of epistemic constraints, strategies and forms of communication, which works as reference for proving and proof in a given theory.

Examples

Moving to the school, the role of the following examples is to provide evidence for the hypothesis that CUTHE depends, for the same statement, on the specific rationality frame in which a conjecturing and proving problem is dealt with by the student; and also to provide elements for further investigation. The examples will include some excerpts from students' think aloud solving processes. Italic is for written texts. (…) is for omitted sentences. … is for a pause in oral speech.

Example 1

The same conjecturing and proving problem was proposed in grades VIII and IX: "Consider all the products of three consecutive natural numbers. What is their GCD? Prove that it is their GCD". S-A is a grade VIII (13-years-old) student not yet familiar with the use of letters to prove:

Student S-A: 1•2•3=6 2•3•4=24 3•4•5= 60 10•11•12=1320;
it is evident that 6 is the GCD of the first three products, because it is the greatest divisor of the first product and a divisor of the other products. Is it a divisor of 1320? … Yes, 1320 is an even number divisible by 3 because the sum of its digits is a multiple of 3. Then 6 might be the divisor of all the other products too. But why? Probably, by looking at these four products, all the products are even… But why? OK, one factor is always even! Even numbers go two by two, thus among three numbers one number … one number at least is even, and they may be two, like in the case of 2•3•4. Look at, three is there! And a multiple of three is in the last product! Why? In the case of 2, multiples go two by two … In the case of 3, numbers go three by three. That is the reason! Now I try to write down the general reasoning:

The greatest common divisor is 6 because every product is divisible by 6 because every three consecutive numbers contain one even number (multiple of 2) and one multiple of 3, because multiples of 2 go two by two, and multiples of 3 go three by three (The teacher writes the following question: Why greatest?) (after a while) Because the first product is divisible by 6, and no greater divisor is there.

S-A resumes the examples, which conjecturing was based on, to identify general reasons for the truth of the conjecture. The intention of proving is related to the emerging conjecture, through "But why?" self-posed questions of epistemic relevance. A narrow intertwining between epistemic, teleological and communicative components of rationality allows the student to move continuously from exploration to the production of the conjecture, to proof construction by exploiting relevant elements got during the exploration, and then to proof writing. We may consider S-A's solution as an example of CUTHE in the rationality frame of verbal-semantic elementary theory of numbers. S-B is a grade IX student who tries to solve the problem after some classroom work (about 10 hours) on the use of letters to prove in an algebraic way. Note that he would be free (according to the didactic contract) to choose another way of solving the problem, as other schoolmates do: n+1)=n(n 2 -1).… I see nothing! I am not able to prove it! S-B tries to solve the conjecturing and proving problem in the rationality frame of elementary algebraic theory of numbers; the difficulty to produce a conjecture in that frame is overcome by moving to the rationality frame of verbal-semantic theory, where afterwards he will also try to dispel a doubt on the truth of the conjecture by considering a further, more elaborated example. Differently from S-A, no effort is addressed to find general numerical regularities that might be exploited to build up a verbal-semantic proof. In terms of rational behavior, this is an example of lack of connection between two different strategies (teleological aspect): to produce the conjecture and afterwards to provide some empirical evidence for it; and to produce a general reasoning for proving. As a consequence, CUTHE does not work in the rationality frame where it could have been activated (verbal-semantic theory). The same happened with the other students who tried to build up an algebraic -formal proof. Note that an algebraic -formal validation of the statement may be performed either in combinatorics, or in modular arithmetic. Some schoolmates get the conjecture in the rationality frame of verbal-semantic theory of numbers, then they consider the products (n+1)(n+2)( n+3) or (n-1)n( n+1) and realize that in these products one number is divisible by three and at least one number is divisible by two; thus proving still relies on semantic considerations related to the number line and the positions of multiples of 2 and 3 in it, like in the case of S-A. The algebraic expression of the product is only a device to favor the transition to a general reasoning. CUTHE works thanks to the intention of finding general regularities and a proof in the same rationality frame of verbal-semantic theory, where the conjecture had been produced.

Student S-B: (n+1)(n+2)(n+3)=(n2+2n+n+2)(n+3)=(n 2 +3n+2)(n+3)

Example 2

A conjecturing and proving problem was proposed by the same teacher in grade XI, in three parallel classes, as an individual task: "Among the triangles with a given side and the same perimeter, find the triangle with the greatest area". Those classes were familiar with conjecturing and proving in number theory (both in a verbal-semantic way and in an algebraic way), and in Euclidean geometry.

The first class at that moment was familiar only with proving in plane Euclidean geometry; according to the conjecturing style of Euclidean geometry, some students (one third of that class) got the conjecture (the solution of the problem is the isosceles triangle) by considering that, after drawing some triangles, an isosceles triangle looks as the "widest" one (students say: "the fattest") among the drawn triangles (but three students got the conjecture of a right-angled triangle with the same considerations); a few students got the conjecture through a "limit & symmetry" consideration related to the fact that, when the triangle becomes strongly asymmetric, the surface within it becomes very "small", if we want to keep the same perimeter. During the discussion on the produced conjectures, after disproving (through measures) the conjecture concerning the right angled triangle, some students proposed to consider another triangle with the same height of the isosceles triangle (thus with the same area), and to try to prove that its perimeter is longer than in the case of the isosceles triangle. But a rigorous proof is not easy to build up, and in fact no student built it up, in spite of a long time spent for it in the classroom, by working in small groups (and then at home as well!); a relatively easy proof needs an auxiliary construction and the use of related theorems. The exploration to get the conjecture only suggests a first step of a proving process, and does not provide the ingredients to build up the proof: CUTHE does not work.

The second class had already met conic sections in synthetic geometry (they knew that an ellipse is the locus of points whose sum of distances from two given points is constant, and its basic properties concerning symmetry, axes, etc.). In this class, the conjecture was produced in a similar way as in the first class; but one fourth of students, thanks to the drawings of some triangles with approximately the same perimeter, arrived also to make a link with the ellipse in synthetic geometry. Students shared what had been discovered; then (by working in small groups) four groups out of six were able to solve the proving problem by considering the properties of an ellipse in synthetic geometry. The exploration provided students with a visual link with the ellipse in synthetic geometry, thus bridging conjecturing with provingeven if proving did not rely on the considerations ("fatness" of triangles) that has generated the conjecture (and thus CUTHE did not work). Here is an excerpt from S-C's think aloud process:

Student S-C: (…) Now I have a reasonable conjecture. How to prove it? (student C draws three more triangles, with the same side in common with the four previously drawn triangles, and approximately the same perimeter). It is even more evident that the isosceles triangle has the largest area. But it seems to me that all those triangles have something in common. Their free edges are …Yes! I understand: the same perimeter means that the free edges are on an ellipse. Thus I may try to see if I succeed to build the proof by using the ellipse. (…)

The We may observe how (as it usually happens in analytic geometry) conjecturing and proving are dealt with at the same time, thus CUTHE works. Exploration is driven by the goal to be attained through algebra, thus the initial figures are not exploited to get a conjecture. The first trial is abandoned after interpretation of the algebraic result, the second one develops and brings to the conclusion. Epistemic control works on formalization, choice of syntactic transformations, and interpretation of results (see Morselli & Boero, 2011, pp. 455-456).

Conclusion and discussion

The aim of this paper was to elaborate the idea of cognitive unity of theorems (CUTHE) by relating it to the rationality frame available to (or chosen by) students to solve a conjecturing and proving problem. Through the examples (particularly Example 1, S-A and also Example 2, S-D) we have seen how the same statement may be produced in a particular rationality frame and then proved in the same rationality frame by exploiting some elements produced during the conjecturing phase, in a continuous process where the intention to achieve the conjecture and ascertain why it is true drives the attention of the student to relevant aspects of the problem situation, useful to build up the proof. While the same statement of Example 1 resists S-B's effort of proving it in another rationality frame. The same for the statement of Example 2 in the frame of Euclidean geometry.

This paper brings some elements of novelty in the field of research, which deals with the relationships between the exploratory phase of conjecturing and proving (or of proving a given statement), and the phase of proof construction. Through the use of the rationality construct, the hypothesis of dependence of activation of CUTHE on the theory chosen as reference for conjecturing and proving, already briefly presented in [START_REF] Garuti | Cognitive unity of theorems and difficulties of proof[END_REF], is further elaborated, with a counterpart in some examples from classroom activities. The rationality perspective provides a lens to compare (and distinguish between) different rationalities in mathematics, with different opportunities to validate the same statement by activating CUTHE. The chosen examples (particularly in the case of S-B if compared with S-A and with some S-B's schoolmates) also suggest to move to a deeper consideration of the relationships between the student's intention (i.e. the teleological component of her rational behavior) and the production of those elements, which might be arranged in a deductive chain in order to get a proof. Another, possible research development (related to Douek's present work) concerns a connection with what is called "semantic proof production" in Weber (2005, p. 356-357): in his reported example the student produces a visual-graphic representation of the sequence (an)=(1,0,1,0,1, …) and a horizontal band, which 'shows' that the sequence is not convergent to a limit; that "informal representation" suggests and guides "the formal inferences that (she) would draw". CUTHE does not work: elements produced during the exploration are not resumed as steps of the construction of the proof in the rationality frame of formal Calculus. But those elements allow to bridge the exploration of the proving situation with the construction of a proof in terms of the teleological component of rationality, with some analogy with the case of S-C (in Example 2); both cases suggest to widen the idea of CUTHE by including that kind of productive relationships between exploration and proof construction.

  third class had already constructed, under the teacher's guide, the equations of a circumference and a parabola by translating into algebraic equations the characteristic conditions of those geometric loci. They had not yet met the equation of an ellipse, or the notion of an ellipse in synthetic geometry. The teacher suggested to use algebra to solve the problem. Student S-D is a representative of those students (about one third of the class) who succeeded in finding the conjecture and proving it. S-D draws three triangles with (approximately) the same perimeter: Student S-D: I must maximize an expression for the area of the triangle, when x changes: The maximum is when x=0. Perhaps this is the solution! But I have not considered the condition a+c=K. And what I found is … it is obvious: x=0 means the rectangle triangle. Obvious: in that case the side of length a is vertical, namely, maximum height of the triangle. But that side has always the same length. But in this problem a is related to c. I should find how to take the condition a+c=K into account. Perhaps it should be good to compare two expressions for the height of the triangle, perhaps… in order to get the area depending only on b and K. Good! Given K and b, the area depends only on x (algebraic calculations follow) Now it works: I see the equation of a parabola; … if x=b/2 I get the the vertex of the parabola, it means the maximum… the maximum of the area. (S-D draws an isosceles triangle) OK, it looks fine: the isosceles triangle looks as the widest one!

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