EUREQua Andreea Cosnita
email: cosnita@univ-paris1.fr

Jean-Philippe Tropeano
email: tropeano@univ-paris1.fr

Negotiating remedies : revealing the merger efficiency gains

Keywords: merger control, structural merger remedies, asymmetric information JEL: D82, L41 Résumé contrôle des fusions, remèdes structurels, information asymétrique

This paper aims to contribute to the normative economic analysis of merger control by taking into account the possible efficiency gains for the design of structural merger remedies. We show that a larger asset transfer should be requested from a less efficient merged firm than from a more efficient one, which conforms with the recommendations of competition policy practitioners. However, since cost savings are private information of merging firms, the Competition Authority will require them to reveal their efficiency gains, so as to tailor the optimal remedy. We propose a revelation mechanism combining the use of divestitures with the regulation of their sale price. We discuss the opportunity of such a merger policy tool, and argue that in practice it may be used to signal the efficiency gains of notified mergers.

Introduction

Faced with a potentially anticompetitive merger, the Competition Authority (CA) has basically the choice among three alternatives: rejecting it, accepting it if the efficiency gains are overwhelming, or accepting it provided that corrective remedies are adopted. A recent study of the European Commission DG Comp 1 counted no less than 190 concentrations cleared with commitments from a total of 2469 mergers since the introduction of the European Merger Regulation in 1990. It might look like a small number, but a closer look at the Commission's statistics reveals that remedies mostly apply to important mergers [START_REF] Bensaid | Competition, Cooperation and Mergers: Economic and Policy Issues[END_REF] . Moreover, about 80% of those commitments address horizontal concerns. Divestitures, i.e. structural remedies, are typically employed for horizontal mergers, and account for more than 60% of all remedies. Structural remedies are thought to be easier to apply than behavioral ones, since they change the allocation of property rights within the industry, and therefore need no monitoring once implemented. Yet, their application and effects have often been subject to questioning, because "the fashioning of merger remedies is not materially governed by case law[...]" and therefore "is subject to standards that are not well-defined or consistent" (see Blumenthal (2001)).

To begin with, this paper aims to contribute to the economic analysis of structural merger remedies by taking into account the possible efficiency gains for the design of divestitures. Our model formalizes the intuition of a link between the amount of efficiency gains that the merging partners can achieve and the amount of assets they will have to divest for the merger to be accepted. Such an idea is in line with the current consensus among competition policy practitioners, according to which the required divestiture should neither exceed the net competitive harm caused by the merger, nor prove insufficient to correct it (that is, both overfixing and underfixing should be avoided [START_REF] Besanko | Contested Mergers and Equilibrium Antitrust Policy[END_REF] ). Consequently, the required divestiture should obey the proportionality principle, meaning a larger transfer being requested from a less efficient merged firm than from a more efficient one. The important consequence is that in order to tailor the optimal remedy, the CA needs to learn the efficiency gains generated by the merger, which are private information for the merger partners.

Thus, the second objective of the paper is to shed light on the design of optimal structural divestitures when merging firms are better informed than the CA with respect to the synergy level of the merger. The U. S. Merger Guidelines acknowledge that "mergers have the potential to generate significant efficiencies", but warn at the same time that "efficiencies are difficult to verify and quantify, in part because much of the information relating to efficiencies is uniquely in the possession of the merging firms" [START_REF] Blumenthal | Reconciling the Debate over Merger Remedies: A Discussant's Proposed Decision Rule[END_REF] .

Competition authorities do try to extract this private information. A possible "approach in screening mergers would be to implement a revelation mechanism through the institution of merger license fees to be paid to the government", as Röller et al. (2000) suggested. An alternative powerful way to extract private information on efficiency gains is quoted by Brodley (1996), who reminds that unfounded efficiency claims may be deterred by bonding procedures. He provides the suggestive American example of the Pennsylvania versus Providence Health Sys., Inc. case [START_REF] Brodley | Proof of Efficiencies in Mergers and Joint Ventures[END_REF] , where the consent decree negotiated between the Pennsylvania Attorney General and the undertakings provided that if argued efficiencies do not translate into net cost savings directly passed on to consumers five years later, the merging parties engaged to pay to the Treasury the shortfall from the claimed efficiency gains. This solution is nevertheless subject to the specific critique against behavioral remedies, concerning the ex-post monitoring cost.

We do not formalize here the implications of such a "put-up-or-shut-up" clause. In turn, we model a competition authority requiring the merging firms to reveal efficiency gains so as to tailor the optimal structural remedy. We propose a revelation mechanism combining the use of divestitures with the regulation of their sale prices. We supply as a matter of fact a possible answer to the following question: "What role, if any, should a competition authority play in the pricing of assets to be divested?" [START_REF] Compte | Capacity Constraints, Mergers and Collusion[END_REF] .

For the time being, Competition Authorities do not tamper with the sale price of the divested assets. However, in this paper we intend to check the relevance of a regulated sale price as a possible screening device, even if this item rather belongs to a regulator's instruments. As a matter of fact, merger control is typically a mixed area of antitrust and regulation, where the Competition Authority is supposed to directly impact on the market structure by rejecting anticompetitive mergers (see Motta et al. (2002)). Besides, since divestitures alone may not suffice to screen merger proposals, it is natural to look for complementary screening devices that might prove effective. Note that as far as merger control is concerned, the scope of instruments employed has constantly increased. For instance, the very structural remedies themselves represent a rather recent instrument introduced in the 70s by the FTC [START_REF]Horizontal Merger Guidelines[END_REF] . In addition, whenever the CA appoints a trustee to implement the divestitures it requires, merger control comes close to an indirect determination of the price of divested assets: according to the European Merger Regulation8 , the trustee may apply the fire-sale clause and organize the sale of the divested assets without a minimum price, subject only to the Commission's approval. After all, though in a different context, the CA would rather interfere with the price setting process, although maybe not quite ready to do it: indeed, in the battle to force Microsoft to open up the market for media-playing software, the European Commission may not yet be likely to order a lower price for Windows without Media Player, even if it does ponder it [START_REF] Dixit | The Role of Investment in Entry-Deterrence[END_REF] . Concerning merger control, our screening mechanism incidentally lends itself to an equivalent and appealing signaling-like interpretation: notwithstanding that competition laws do not currently allow antitrust agencies to explicitly fix the price of divestitures, the latter might nevertheless reveal efficiency gains. More precisely, a more efficient merged entity will accept a lower price for any given level of divestiture than a less efficient one, and by doing so it will signal itself as such.

Our model builds on a simple framework, which nevertheless allows a consistent formal treatment of both efficiency gains and structural remedies. We consider a Cournot competition game with homogenous good, constant marginal costs and capacity constraints in a three-firm framework. Following a two-firm exogenous merger, the merged entity may enjoy cost savings. The latter stem from the efficiency gains brought about by the merger. The CA requires in turn asset transfers to the outsider, so as to fulfill its objective. Divestitures alter the distribution of capital assets between firms and thereby the capacity constraints and merger profitability for firms in the industry. We look for the optimal transfer to implement according to the objective of the CA: maximizing total welfare under the constraint that the Consumers' Surplus does not fall. ((In our framework with capacity constraints, the same transfer will keep the market price constant. We show that the first best divestiture will be proportional to the level of cost savings, meaning that less efficient mergers will need to divest more. But as the cost savings are private information for merging partners (also called insiders), the latter are likely to cheat when declaring the amount of efficiency gains generated by the merger, so as to avoid higher asset transfers. Thus the divestiture alone will not be effective as a screening device.

We argue here that a non linear tariff for the divested assets can be employed as a screening device. More specifically, we consider a two-type model, where the insiders are either highly efficient (low-cost) or poorly efficient (high-cost). We show that the first best divestiture contract that the CA proposes to the merged entity consists of a menu of two tariffs. The first tariff consists of a lower divestiture for a lower average sale price than the second one. The efficient merged entity values its capacity more than the inefficient one does. As a result, the low-cost insiders are more reluctant to divest a large quantity of assets than the high-cost insiders, even though the corresponding monetary transfer is high. By the same token, since assets have lower value for the inefficient merged firm, the latter will find it profitable to divest a large quantity of assets for a higher average price.

In other words, the CA clears the merger provided the merged entity pays a "licence to merge", although not to the CA, we should stress, since the monetary transfer actually takes place between the insiders and the buyer of divested assets. The optimal licence schedule proposed by the CA takes two forms. The firms pay either by divesting a large quantity of assets, or by divesting a low quantity of assets at a depreciated price. Hence the licence is either a monetary payment whenever few assets are divested at a low price, or an asset payment whenever a large quantity of assets is divested. Facing this two possible types of licences to merge, the more efficient merged entity is induced to choose the least distorting form of payment, the monetary one. Basically, we provide an answer to Rey's (2000) informal suggestion that "The firms could be asked to 'pay' [...] for any negative external effect of the merger, so as to ensure that only socially desirable mergers are proposed". The effectiveness of the sale price as a screening device (which is what basically our model shows) reveals that it can be successfully be employed by merging firms to signal their efficiency gains. Notwithstanding the reluctance of competition authorities to openly interfere with the pricing of the divested assets, close attention should be paid to the asset prices as they result from the divestiture process, since they may very well signal the merger synergies.

As early as 1993, Yao and Dahdouh clearly stated that "In merger review, the selective provision of information creates problems for government antitrust officials because much relevant information is held privately by merging parties" (Yao and Dahdouh (1993), p.24). The asymmetric information problem for competition policy has been already tackled by Faulli-Oller and Corchon (1999), who study the implementation of socially optimal mergers with standard tools in dominant strategy implementation, but without allowing for merger remedies.

To our knowledge, few papers deal with structural merger remedies. Our model takes on Rey (2000) and Gonzalez (2003), who both account for the possibility of divestitures, besides their corrective role, to be used for screening. As compared with Gonzalez (2003), whose incentive mechanism relies on the choice of the market where the divestiture will apply if accepted, we restrict the use of divestiture to the same market on which the competitive harm is witnessed, and propose a second revelation instrument, the sale price. Medvedev (2004) proposed the first formalization of the intuition that the amount of asset transfer necessary to remedy the competitive harm depends on the amount of efficiency gains. We show in turn that this result can be obtained within a quite general framework with constant marginal costs, without assuming a particular relationship (substitutability in his case) between the capital-based cost-savings and the synergies following the merger.

This paper also belongs to a strand of literature analyzing the effects of capital transfers between firms. For instance, Farrell and Shapiro (1990,a) raise precisely the question of how capital transfers between Cournot oligopolists affect total industry profit or welfare. Compte et al. (2002) and Vasconcelos (2005,a) also look into the effects of asset transfers, but from a collusive industry point of view, whereas Vasconcelos (2005,b) models the CA's incentives to apply divestitures within an overfixing yet symmetric information merger control framework. We completely overlook the possibility for collusion, but focus only on the unilateral effects of the concentration.

We present first the market equilibria both before and after merger, taking simultaneously into account efficiency gains as well as asset transfers. We then go on to present the game between industry firms and the CA. For the design of optimal remedies, we begin by the symmetric information benchmark, then deal with the asymmetric information framework. Finally, we comment our results and conclude on their relevance. Technical proofs are grouped in the Appendix.

The model

We present first the pre-merger equilibrium as a benchmark and then the post-merger framework.

Pre-merger market equilibrium

We consider as starting point a homogenous good, three-firm perfectly symmetric industry. Demand is linear: P (Q) = 1-Q, where Q is total output. Firms maximize individual profit. We place ourselves in a situation where firms face capacity constraints as in Dixit (1980). Explicitly, we assume that a two-stage capacity-quantity game took place before merger, where firms acquire first capacity at a positive unit cost c k and then compete à la Cournot with a constant production marginal cost equal to c. The Subgame Perfect Equilibrium of such a game is given by capacity and output equal to 1-c-c k

4

. We take this to be the operational capacity of a firm before merger and denote it by k. Consequently, pre-merger equilibrium yields Q = 3k and P (3k) = 1 -3k. We denote Π the pre-merger individual profit, which writes Π = k(1 -3kc). We assume that the unit cost for the acquisition of additional capacity is prohibitive in the short-run.

Post-merger market framework

Since we only deal with exogenous market concentration, and that initially the three firms are identical, merger is assumed to take place between any two of them. We shall index the merged entity, i.e. the insiders, by M, and the outsider by o respectively.

Following the merger, M's capacity constraint changes, since now it holds the double of the pre-merger capacity. As far as the outsider is concerned, its capacity is unchanged, as well as its marginal cost. The insiders in turn may benefit from merger-specific cost savings 10 . More precisely, the marginal cost of the merged firm, denoted c M , satisfies the following: c M = cα, where α ∈ [0, c] measures the amount of cost savings. Our framework is general enough to lend itself to different interpretations of this parameter. For instance, α stands for the synergies that arise from the merger, i.e. substantial efficiency gains that would not have been obtained without it. More generally, α measures the positive effect of an essential complementarity between the merger partners that allows them to lower their common marginal cost [START_REF]on the control of concentrations between undertakings[END_REF] . Following Röller et al. (2000), these synergies can typically be due to complementarity between technological or administrative capabilities of firms. For example, firms may own complementary patents, which if employed together further improve the production process.

In the post-merger framework, firms play a standard Cournot game -we rule out in our model any possibility for post-merger collusion. Firms obey the following capacity constraints [START_REF]Merger Remedies Study[END_REF] : q M ≤ 2k and q o ≤ k. Taking into account the cost savings for the merged entity, their Best Reply functions write as follows:

BR M (q o ) = min ¡ 1+α-c-q o 2 , 2k ¢ and BR o (q M ) = min ³ 1-c-q M 2 , k ´. It is straight-
forward to show that post-merger equilibrium price is less or equal to P (3k) only if α ≥ 5k -1 + c. In other words, a merger increases market price whenever its cost savings fail to exceed this minimum threshold.

To sum up, mergers have an ambiguous impact on the economy. Indeed, if the efficiency gains are sufficiently high, the merged entity is induced to use all its capacity and thus the price is unchanged with respect to its pre-merger level. In turn, with lower efficiency gains, the insiders hold slack capacity and the price is higher than its pre-merger level. In such a case, divestitures might be used by the CA to modify the post-merger market equilibrium so as to prevent any drop in Consumers' Surplus. These divestitures depend on the efficiency gains, and their design is studied in the next section.

Remedies as a screening device

Objective of merger control and terms of divestiture

As far as the CA's objective goes, we wish to closely follow the current trend in merger control in as much as we assume total welfare maximization under the constraint of no drop in Consumers' Surplus. Competition Authorities actually rather accept or reject mergers on the basis of the likelihood of a price increase. To put it short, a merger is approved if consumers will not be hurt. However, due to our modeling choice of capacity constraints, keeping the price constant is the best that can be done, so maximizing Consumers' Surplus is equivalent to requiring the above constraint to bind [START_REF] Farrell | Negotiation and Merger Remedies: Some Problems[END_REF] .

We use a two-type model, with the synergy parameter α ∈ {α, α}, where α stands for high efficiency gains, and α for low efficiency gains. Explicitly, α = cc M and α = cc M , where c M and c M are the marginal costs of the low-cost merged firm and of the high-cost one respectively. Objective probabilities are ρ and 1ρ respectively.

In our setting, divestitures will consist in transfers of assets to the outsider, whom we consider here to be the only possible buyer. We make this hypothesis for the sake of simplicity, but the framework lends itself well to the introduction of a new entrant on the market [START_REF] Farrell | Asset Ownership and Market Structure in Oligopoly[END_REF] . Moreover, by not allowing entry on the market, we want to remind that besides capacity, a certain know-how and experience of the market are necessary to guarantee actual competition on behalf of the buyer of divested assets.

We determine first the optimal divestiture when information on the merger type is symmetric, and then go on to study the role of asymmetric information for the design of the remedy.

Optimal divestitures with symmetric information

Given the screening stand we take in our model, the game we consider between the firms and the CA is the following:

In the first stage, the merging firms learn their efficiency gains level α and submit a merger proposal to the CA. When information is symmetric the parameter α is also observed by the outsider and by the CA. (Later on we detail the information structure of the game with asymmetric information).

In the second stage, the CA evaluates the consequences of the merger taking into account its own merger control objective. It proposes a divestiture contract accordingly, if such a contract exists; if not, it rejects the merger.

In the third stage, the insiders accept or reject the divestiture. If they accept, assets will be transferred to the outsider on a take-it-or-leave-it basis.

In the fourth stage, the outsider decides whether to take over or not the divested assets.

In the fifth stage, conditional on the divestiture contract being accepted, the Cournot market equilibrium is determined taking into account the amount of asset transfer required by the CA. If any of the parties rejects the contract, the merger project falls through.

This last assumption is quite in line with the current unfolding of a divestiture negotiation process. Indeed, whenever a divestiture injunction is being settled upon, it has previously gained approval of all involved parties: the divesting firms, the buyer, and the CA. The failure of such a three-party negotiation ends either in the appointment of a trustee to carry out the divestiture, or in the merger itself falling through.

At the last stage of the game, firms play a standard Cournot game. Note that before the divestiture requested by the CA, firms' capacities amount to 2k for M and k for o. Once the transfer of ∆ is made from M to o, firms obey the following capacity constraints: q M ≤ 2k -∆ and q o ≤ k + ∆. Taking into account both the cost savings and the required divestiture, the Best Reply functions of the Cournot game write as follows:

BR M (q o ) = min ¡ 1+α-c-q o 2 , 2k -∆ ¢ and BR o (q M ) = min ³ 1-c-q M 2 , k + ∆
´.

Profits are denoted Π M (∆; α) and Π o (∆; α) respectively, and depend on the cost savings parameter and the amount of divestiture.

Actually, for a given divestiture 0 ≤ ∆ ≤ (1c) -4k, i.e. not too high so as to have the outsider operate at full capacity, profits write

Π M (∆; α) = ½ ¡ 1+α-c-k-∆ 2 ¢ 2 , if α ≤ 5k -∆ -(1 -c) (1 -3k -c + α) (2k -∆), if α > 5k -∆ -(1 -c) and Π o (∆; α) = ½ ¡ 1-α-c-k-∆ 2 ¢ (k + ∆) , if α ≤ 5k -∆ -(1 -c) (1 -3k -c) (k + ∆), if α > 5k -∆ -(1 -c)
Clearly, when the efficiency gains are high enough ( i.e. α > 5k -∆ -(1c)), the merged firm is also led to employ all of its capacity.

However, if the divestiture exceeds the threshold above mentioned, regardless of the level of merger synergies, the outsider will never operate to full capacity. Thus, for ∆ > (1c) -4k,

Π M (∆; α) = µ 1 + 2α -c -2k + ∆ 2 ¶ (2k -∆) and Π o (∆; α) = µ 1 -c -2k + ∆ 2 ¶ 2
At stage four, as far as the outsider is concerned, the decision to accept to take over the divested assets depends on his willingness to pay for them, equal to Π o (∆; α) -Π. We denote by P the price of divestitures proposed by the merged entity and observed by the CA. The outsider accepts the divested assets iff Π o (∆; α) -P ≥ Π.

At the stage before (stage three), the insiders make a take-it-or-leave-it offer to the outsider. Thus if the assets are transferred to the outsider, the insiders set a price equal to the outsider maximum willingness to pay:

P = Π o (∆; α) -Π.
Moreover, the merged entity agrees to divest iff Π M (∆; α) + P ≥ 2Π.

At the second stage, since the CA observes the type of the merger submitted for approval, it makes its decision based on the following programme:

max ∆≥0 W (∆; α) (S) s.t.    CS(∆; α) = CS 0 Π M (∆; α) + P ≥ 2Π Π o (∆; α) -Π = P
where CS 0 stands for the Consumers' Surplus level before merger. Since the Consumers' Surplus constraint is binding, we will denote by ∆ F B (α) the solution of this programme, which is the lowest positive asset transfer that ensures production at full capacity on behalf of both firms [START_REF] Farrell | Horizontal Mergers: An Equilibrium Analysis[END_REF] . We remind that the market price increases whenever firms hold slack capacity. Thus, successful remedies in our context are necessarily those that make firms produce up to their full capacity. However, given that total capacity in the industry is fixed, firms can never produce more than they did before merger. Consequently, in our framework, maximizing consumers' surplus after merger and keeping it constant are equivalent. Finally, the positivity constraint of this programme simply states that whenever there is no asset transfer able to keep constant the price, the merger is rejected.

We characterize the First Best asset transfers as follows:

Lemma 1. (i) For k ∈ ³ 2(1-c) 9 , 2-c 9 ´, there exists a threshold b α = 9k -2(1 -c), b α ∈ (0, c), such that: • for any α ≥ b α, the merger is accepted with divestiture ∆ F B (α), where ∆ F B (α) = max(0, 5k -α -(1 -c)). Also, ∆ F B (α) < ∆ F B (α).
• for any α < b α, the merger is rejected.

(ii) For k ≤ 2(1-c) 9 , b α ≤ 0, therefore all mergers are accepted with

∆ F B (α) (iii) For k ≥ 2-c
9 , b α ≥ c and therefore all mergers are rejected. Whenever the divestitures ∆ F B (α) and ∆ F B (α) exist, the assets are transferred to the outsider at prices equal to P F B = (P (3k)c) • ∆ F B (α) and P

F B = (P (3k) -c) • ∆ F B (α) respectively.
See proof in the Appendix.

The threshold b α corresponds to the shut-down limit. In the case where this threshold is zero, for all mergers there exists a positive transfer such that the CA's objective is fulfilled. In the last case (case (iii)) the shutdown range covers the whole interval, so there are no transfers that can make both firms operate to full capacity. Henceforth we shall only consider k ∈

³ 2(1-c) 9 , 2-c
9 ´, so as to deal with an interesting case [START_REF] Fauli-Oller | To Merge or Not To Merge: That Is the Question[END_REF] .

This lemma shows that our mechanism replicates the outcome of CAs' behavior to the extent that merger control decisions obey threshold criteria. Here, whenever the notified merger does not generate enough cost savings, the CA rejects it.

Moreover, if firms anticipate the CA's decision making process, only efficiently enough mergers will be proposed, and therefore all submitted mergers shall be accepted. This is a self selection effect. In turn, if α < b α, no merger is submitted. Henceforth we shall consider only the case where α > b α. Finally, the required transfer is higher for the less efficient merger. The reason is quite simple: in this model with fixed total capacity, for the price to be kept constant, firms need to produce to their full capacity. But the more efficiency gains it generates, the more capacity will employ the merged entity. Therefore, more efficient insiders hold less slack capacity, so the price increase will be lower for a more efficient merger, and the corresponding divestiture as well. This conforms with the proportionality principle advocated by competition policy practitioners, and justifies the fact that the average sale price of divested assets is actually constant, since

P F B ∆ F B (α) = P F B ∆ F B (α)
. Nevertheless, such a proportionality principle is subject to an implementation problem if efficiency gains captured here by the parameter α are not observed by the CA.

Optimal divestitures with asymmetric information: a regulated sale price mechanism

The game with asymmetric information is basically the same as before, taking into account the changes due to the inobservability of the merger type:

In the first stage, the merging firms learn their efficiency level and submit a merger proposal to the CA. The parameter α is now private information of the merging firms, but the latter may or may not report it truthfully.

In the second stage, the CA evaluates the consequences of the merger taking into account its own merger control objective. It proposes a divestiture contract accordingly, if such a contract exists; if not, it rejects the merger. The divestiture contract will contain the amount os assets to be divested ∆ and the corresponding sale price P .

In the third stage, the insiders accept or reject the divestiture. If they accept, assets will be transferred to the outsider at the price P determined by the CA.

In the fourth stage, the outsider, having observed the insiders' choice to accept or not the divestiture contract, decides whether to take over or not the divested assets.

In the fifth stage, conditional on the divestiture contract being accepted, the Cournot market equilibrium is determined taking into account the amount of asset transfer required by the CA. The merger is abandoned whenever one of the parties rejects the contract.

Note that the CA and the outsider have a common prior on the merger's types (ρ and 1ρ). At stage four, the outsider observes the menu of contracts proposed by the CA as well as the contract chosen by the merged entity and thus revises its prior beliefs.

Following the revelation principle we restrict to truthful direct revelation mechanisms, and look for a separating perfect Bayesian equilibrium of the game.

When information is asymmetric and the only instrument employed is the asset transfer, the CA is no longer able to make the insiders reveal truthfully their efficiency level, i.e. their type, since they always choose the lowest level of divestitures that is proposed. Indeed, should the CA propose the former levels of divestiture (∆ F B (α) and ∆ F B (α)), associated with the previous First Best prices, i.e. equal to the outsider's willingness to pay, we can show that the high-cost merged firm has incentives to choose the low asset transfer destined to the lowcost merged entity. Explicitly, the following holds:

Π M (∆ F B (α) ; α) + Π o (∆ F B (α) ; α) -Π | {z } =P F B < Π M (∆ F B (α); α) + Π o (∆ F B (α); α) -Π | {z } =P F B
See proof in the Appendix.

In other words, given the first best levels of divestitures, mimicking is more profitable than truth-telling for the less efficient type. Indeed, when optimal divestitures are sold for a price equal to the outsider's willingness to pay (P F B or P F B ), the less efficient merged entity prefers the lower asset transfer, which allows it to hold spare capacity and thus increase its profit by means of a price raise. As a result, in order to induce truthful revelation, a second instrument is needed.

We model here a CA regulating the monetary transfer between the insiders and the outsiders that accompanies the asset takeover. In case this instrument should trigger the critique that the CA behaves as a sheer market regulator, we claim that it is worth analyzing it for three main reasons. Firstly, it allows us to test its theoretical relevance for the merger control. Secondly, because the frontier between pure regulation and merger control has already been blurred by the very use of structural divestitures, which are basically meant to modify the market structure (see Motta et al. (2004)). After all, "introducing the possibility of remedies ... puts the merger control office in a position close to that of an industry specific regulator" - Rey (2003, p.130). Thirdly, regulating asset prices already concerns other competition-related fields than pure sector regulation, such as intellectual property rights, where the price of the licence would be set by the Patent Office [START_REF] Gonzalez | Divestitures and the Screening of Efficiency Gains in Merger Control[END_REF] .

The incentive contract fixed by the CA will thus contain a given sale price P for an amount of divested assets ∆. Two Incentive Constraints (IC) are added to the programme of the CA to induce revelation of information. Hence, the programme of the CA writes:

max { (∆,P ),(∆,P ) } ∆≥0,∆≥0 ρW (∆; α) + (1 -ρ) W ¡ ∆; α ¢ (AS) s.t.                    CS (∆; α) = CS ¡ ∆; α ¢ = CS 0 Π M (∆; α) + P ≥ Π M ¡ ∆; α ¢ + P Π M ¡ ∆; α ¢ + P ≥ Π M (∆; α) + P Π M (∆; α) + P ≥ 2Π Π M ¡ ∆; α ¢ + P ≥ 2Π Π o (∆; α) -P ≥ Π Π o ¡ ∆; α ¢ -P ≥ Π
where the contracts (∆; P ) and ¡ ∆; P ¢ are destined for types α and α respectively.

Note that unlike a standard screening programme, there is no direct transfer between the agent and the principal. Instead, the CA fixes the monetary transfer between the agent and a third party, the outsider, and uses it as an incentive device. The lump-sum transfer between firms does not affect total industry profit, and is different from the suggestion of Röller et al. (2000) to introduce an explicit licence to merge, to the extent that in our model the monetary transfer does not benefit directly the CA. In this way we actually avoid the direct implication of the CA in the merger process as an explicit regulator, although the revelation mechanism is indeed based on the taxation principle.

Note equally that since the contract we look for induces information revelation, i.e. separation of types, in equilibrium the priors of the outsider necessarily coincide with its revised beliefs. As in the symmetric information configuration, CA employs a supplementary choice variable to solve the screening problem, namely the shut-down policy (i.e. refuse the merger of the less efficient type), hence the positivity constraints on the asset transfers. Finally, the equality constraints on Consumers' Surplus as well as the participation constraints correspond actually to a three-party negotiation. Indeed, the additional instrument we propose to screen mergers, namely the sale price of the divested assets, is merely a lumptransfer between the outsider and the insiders, hence it stands for a particular distribution of total industry profit. To achieve screening, the contracts proposed by the CA need to ensure the industry firms' participation. Thus, the above programme is actually designed to make the industry parties involved agree on the incentive-compatible sharing of their total profit. Before presenting the optimal contract we give the following lemma:

Lemma 2. For any ∆ < ∆, Π M (∆; α) -Π M ¡ ∆; α ¢ > Π M (∆; α) -Π M ¡ ∆; α ¢ .
See proof in the Appendix. This inequality stands for a standard single crossing condition. It states that the efficient merged firm benefits more from a low asset divestiture than the inefficient firm. In other words, the efficient merged firm attaches more value to capacity than the inefficient one. The intuition goes as follows: following merger and the ensuing increase in capacity, the efficient insiders are always able to produce the same quantity as the inefficient ones, and can thus guarantee themselves the same revenue. Yet, thanks to the synergy cost reduction, their profit increase is actually higher, hence their willingness to receive w.r.t. the monetary payment of a given divestiture is lower. As in any standard Principal Agent model, this lemma makes room for screening. We can therefore derive the optimal contract summarized in the following proposition: The optimal contract has two main characteristics. First, there is no distortion of asset divestitures and the merger clearance decision is unchanged as compared with the case of symmetric information. Second, the price of the low divestiture is distorted downwards, whereas the price of the high divestiture is not distorted at all (meaning it still equals the outsider's willingness to pay), so that the average divestiture price increases now with the level of divestiture. To sum up, we obtain no distortion at all in terms of asset transfers, and distortion 'at the top' for the monetary transfers.

The intuition behind the design of these contracts proceeds in two steps.

First of all, we have previously emphasized that the CA must distort asset prices and possibly divestiture levels so as to obtain separation of types. Otherwise, both firms are induced to choose the low level of divestiture. Note however that since the asset price is a lump-sum transfer between firms without impact on the CA's objective, the CA can distort prices at no cost, so as to induce the inefficient firm to choose the high divestiture.

Second, it remains to show that price distortion alone is sufficient to lead both firms to choose the optimal levels of divestiture disclosed in Lemma 1. For that purpose the CA must lower the price of the low level of divestitures so as to deter the inefficient firm from choosing such a contract. According to Lemma 2, a given divestiture is more distorting for the efficient merged firm than for the inefficient one, and thus we have seen that the willingness to receive for the First Best ∆ is lower for the efficient entity. Hence we can find a price for this low level of asset divestiture that induces the inefficient entity to give it up, and this price will consequently need to be lower than the outsider's willingness to pay. To sum up, it is enough to keep the price of the high First Best divestiture equal to the outsider's willingness to pay, and in turn to set for the low First Best divestiture a price inferior to the outsider's willingness to pay.

Moreover, in one particular configuration, the asset price can be negative, so that the CA would require the efficient firm to subsidize the outsider. Specifically, if the efficient firm has very substantial cost savings, the optimal level of divestiture required to the efficient firm is so low, that in order to prevent the inefficient merged firm from choosing it, the corresponding distortion on the sale price will make it negative.

Finally, the CA might have to reject the less efficient merger due to the absence of optimal divestiture, i.e. when α < b α. But, with asymmetric information and not distorted prices, both types of insiders will submit their merger. Again, to extract information, the CA will distort downwards the price for the low divestiture, so as to incite the submission of the highly efficient merger only. Point (ii) of our Proposition actually gives the value of P that violates the participation constraint of α, while still ensuring that of α. For this price, only the efficient merger will be submitted.

A low price of divested assets is often interpreted as a signal of failure of the divestiture process. The European Commission's Merger Remedies Study, October 2005 (p.103), argues that "remedies were less effective in at least three divestiture cases where the purchaser had acquired the divested business for free, or at a negative price" [START_REF] Lagerlöf | On the Desirability of An Efficiency Defense in Merger Control[END_REF] . We claim however that a more efficient merger can signal itself as such by accepting a low average asset price combined with low quantity of assets divested, whereas an inefficient merged firm reveals itself as such by divesting a large quantity of assets for a high average price. In the present model, the inefficient entity owns more slack capacity and values assets less than the efficient one. This is the reason why the inefficient insiders accept to divest a larger quantity of assets, while the efficient ones prefer a low price, provided that the quantity of divested assets remains low. Hence, one way to interpret our result is that a trustee appointed by the CA might tell an efficient merger proposal from an inefficient one using such a non linear tariff for asset divestitures.

Note that this mechanism bears no risk of inefficient or distorting lobbying activities on behalf of parties involved. Indeed, our programme is designed to make all three parties agree on an incentive-compatible distribution of industry profit. Thus, the more efficient merged entity will transfer less assets at a depreciated price, and the profit sharing is favourable to the outsider, who pays less than his maximum willingness to pay. In other words, our mechanism has the initial informational conflict between the CA and the insiders solved as soon as the diverging interests of the insiders and the outsider are reconciled. Thus, by making use of a three-party incentive-compatible negotiation, the CA can make sure that no useless lobbying activities occur.

Within the debate on the frontier between competition policy and regulation, our proposition is meant to draw attention to the best instruments that should be used to address the anticompetitive effects of merger. We argue that while price distortion is considered as highly interventionist and prohibited in an orthodox view of competition policy, in a merger control context the most distorting tool is more likely to be the asset transfer, rather than a lump-sum monetary transfer between industry firms. Indeed, whereas the monetary transfer does not affect market behavior of firms, the level of assets divested has a direct impact on firms' production decisions. As a result, in order to induce firms to reveal efficiency gains, the use of monetary transfer appears less interventionist than the transfer of physical assets.

Our proposition suggests that firms be asked to pay a kind of licence to merge. Indeed, the divestiture implies a cost imposed to the merged entity which will give up some of its assets. The way it will pay to be allowed to merge depends on its level of efficiency. Basically, the inefficient merged firm pays by giving up assets, whereas a very efficient one pays by means of a monetary transfer to the outsider.

Conclusion

This paper contributes to the economic analysis of merger remedies. We propose a revelation mechanism allowing the design of optimal merger divestitures when information is asymmetric between firms and the CA with respect to the synergy generated by the merger. Our revelation mechanism replicates the typical behavior of a CA, namely decision making based on thresholds of announced efficiencies. In our framework, shut-down of the least efficient type is possible. Basically, mergers will only be accepted if they generate enough synergies, and this is what the merger control practically aims at.

Our results show that the sale price of divested assets is a powerful screening device. Despite the information asymmetry, types are perfectly screened, and only divestiture sale prices get distorted. Complete distortion, i.e. also affecting divestiture levels, would be possible in our framework if several modifications were performed, such as introducing a no longer prohibitive cost of capacity acquisition, or imposing a positivity constraint on asset sale prices.

We acknowledge of course the modelling of a CA actively modifying the market structure, but then any structural merger remedy is precisely meant to do this. Taking into account the reluctance of competition authorities to actually employ this instrument for the screening of merger projects, we can nevertheless insist on the ability of the price of divested assets to signal on behalf of merging partners the efficiency potential of their merger.

for o: BR o (q M = 2k -∆) = 1-c-(2k-∆) 2 ≥ k + ∆ ⇔ ∆ ≤ (1 -c) -4k = ∆ 2
It is straightforward to check that for ∆ < ∆ 1 , the outsider produces up to its full capacity, but the merged firm holds slack capacity, whereas for ∆ > ∆ 2 the reverse is true. Note that ∆ 1 > 0 as long as α ≤ 5k -(1c), a necessary condition being k ≥ 1-c 5 so as to have positive cost savings. Proof of Lemma 1. As seen before, successful remedies belong to

[∆ 1 , ∆ 2 ], with ∆ 1 = 5k -α -(1 -c).
The First Best level of divestiture is actually ∆ 1 , since we take it to be the lowest positive asset transfer that satisfies the objective:

∆ F B (α) = 5k -α -(1 -c).
The existence of ∆ F B (α) is ensured as long as the interval [∆ 1 , ∆ 2 ] exists. Define b α as the threshold value of cost savings for which

∆ 1 = ∆ 2 : b α = 9k - 2(1 -c). This threshold is positive provided that k ≥ 2(1-c)
9 . Whenever α < b α, we have ∆ 1 > ∆ 2 therefore the CA rejects all mergers, since there is no transfer ∆ for which firms produce both to full capacity. In turn, for α

≥ b α, ∆ F B (α) = 5k -α -(1 -c) and since α > α we obtain directly ∆ F B (α) < ∆ F B (α).
Since assets are divested to the outsider at its willingness to pay, prices are given by: P

F B = Π o (∆ F B (α); α)-Π = (P (3k)-c)∆ F B (α) and P F B = Π o (∆ F B (α); α)- Π = (P (3k) -c)∆ F B (α).
Proof -incentive to mimick. Proof of Proposition 1.

Π M (∆ F B (α); α) + Π o (∆ F B (α); α) -Π | {z } =P F B ≤ Π M (∆ F B (α); α) + Π o (∆ F B (α); α) -Π | {z } =P F B ⇔ [P (3k) -c + α] • ¡ 2k -∆ F B (α) ¢ + [P (3k) -c] • ¡ k + ∆ F B (α) ¢ < £ P (k + ∆ F B (α) + BR M α (k + ∆ F B (α))) -c + α ¤ • BR M α (k + ∆ F B (α)) -[P (3k) -c] • ¡ k + ∆ F B (α) ¢ ⇔ [P (3k) -c] • ¡ ∆ F B (α) -∆ F B (α) ¢ < £ P (k + ∆ F B (α) + BR M α (k + ∆ F B (α))) -c + α ¤ • BR M α (k + ∆ F B (α)) -[P (3k) -c + α] • ¡ 2k -∆ F B (α) ¢ ⇔ (1 -3k -c) • (α -α) < µ 1 -3k -c + α + α 2 ¶ 2 -(1 -3k -c + α) 2
• No shut-down For P = Π o ¡ ∆; α ¢ -Π and P = Π o (∆; α) -Π, both firms prefer the contract (∆, P ), since the following inequality holds for the inefficient type:

Π M ¡ ∆; α ¢ + Π o ¡ ∆; α ¢ > Π M (∆; α) + Π o (∆; α)
In turn, if P = 2Π -Π M (∆; α) , then Π M (∆; α) + P < Π M ¡ ∆; α ¢ + P so both firms prefer the contract (∆, P ). By continuity of P , there exist P and

P = Π o ¡ ∆; α ¢ -Π with Π o (∆; α) -Π ≥ P > 2Π -Π M (∆; α)
such that Π M (∆; α) -Π M ¡ ∆; α ¢ = P -P This latter condition ensures that P -P > Π M (∆; α)-Π M ¡ ∆; α ¢ , i.e. separation of types, thanks to the single-crossing condition (see Lemma 2).

We show next there exists e α such that for α > e α, P < 0 First of all, note that for ∆ = 0, P < 0 necessarily, because: Last but not least, we can show that a sufficient condition for this threshold e α to be < c is to have k < 1 5 , which is compatible with our condition for positive cost savings, namely k ≥ 1-c 5

• Shut-down of less efficient merger When α < b α, the optimal response from the CA is to reject the α, simply because it is the only way to keep the price constant -see Lemma 1, which shows that when there is no transfer ∆ that can keep the price constant, the CA rejects the merger. To prevent therefore the submission of the α merger, it is enough to set P = 2Π -Π M (∆; α), which violates the participation constraint of the α type.

Proposition 1 .

 1 Denote (∆, P ) , ¡ ∆, P ¢ the divestiture contracts proposed with asymmetric information when they exist. The solution of the programme (AS) is such that: (i) No shut-down When α > b α, then ∆ = ∆ F B (α) and ∆ = ∆ F B (α). Prices are P = Π o (∆; α)-Π and P = P -Π M (∆; α) + Π M ¡ ∆; α ¢ . Moreover, there exists a threshold e α such that for α > e α, P < 0. (ii) Shut-down of less efficient type When α < b α, type α merger is rejected by offering a single contract: ∆ = ∆ F B (α) and P = 2Π -Π M (∆; α) See proof in the Appendix.

⇔ 1 4 2 , 0 • 0 0 • 0 ∂ 2 Π•∂ 2 P

 42000022 (αα) • (α + 3α) < 0 which is true since 0 ≤ α < α Proof of Lemma 2. A necessary condition for the two incentives constraints to hold isΠ M (∆; α) -Π M ¡ ∆; α ¢ > Π M (∆; α) -Π M ¡ ∆; α ¢ .The profit of the merged firm writes generally as follows:Π M (∆; α) = £ P (k + ∆ + BR M (k + ∆)c + α ¤ • BR M (k + ∆)whereBR M (k + ∆) = min ³ 1+α-c-(k+∆) 2k -∆ Ẃe show next that ∂Π M (∆;α) ∂∆ < 0 and ∂ 2 Π M (∆;α) ∂∆∂α < Show ∂Π M (∆;α) + ∆ + BR M (k + ∆)c + α ¤ +BR M (k + ∆) • ∂P (k + ∆ + BR M (k + ∆)) ∂∆Note that ∂BR M ∂∆ < 0 always, and that ∂P (k+∆+BR M (k+∆))∂∆ ≤ 0. Actually, if BR M (k + ∆) = 2k -∆ then P (k + ∆ + BR M (k + ∆)) = P (3k) therefore ∂P (k+∆+BR M (k+∆)) ∂∆ = 0, whereas if BR M (k + ∆) = 1+α-c-(k+∆) 2 then P (k + ∆ + BR M (k + ∆)) = 1+c-α-k-∆ 2 and thus ∂P (k+∆+BR M (k+∆)) ∂∆ < To sum up, ∂Π M (∆;α) ∂∆ < Show ∂ 2 Π M (∆;α) ∂∆∂α < + ∆ + BR M (k + ∆)c + α + ∆ + BR M (k + ∆)c + α ¤ ∂P (k + ∆ + BR M (k + ∆)) (k + ∆ + BR M (k + ∆)) ∂∆∂α | {z } ≤0 Conclusion: since α > α, the cross derivative ∂ 2 Π M (∆;α) ∆; α) -Π M (∆; α) ¢ < 0, ∀∆ ⇔ for any ∆ < ∆, Π M (∆; α) -Π M ¡ ∆; α ¢ < Π M (∆; α) -Π M ¡ ∆; α ¢ , q.e.d.

P

  = P -¡ Π M (∆; α) -Π M ¡ ∆; α ¢¢ = Π o ¡ ∆; α ¢ -Π -¡ Π M (∆; α) -Π M ¡ ∆; α ¢¢ = (P (3k)c) • ¡ k + ∆ ¢ -(P (3k)c) • k -¡ (P (3k)c + α) • (2k -∆) -(P (3k)c + α) • ¡ 2k -∆ ¢¢ = (P (3k)c) • ∆ -(P (3k)c + α) ¡ ∆ -∆ ¢ ⇒ for ∆ = 0, this yields (P (3k)c) • ∆ -(P (3k)c + α) ∆ < 0 Moreover, ∆ = 0 ⇔ 5kα -(1c) = 0 ⇔ α = 5k -(1c)To sum up, for α = 5k -(1c) (which by the way is > b α), P < 0 But, for α = α, P = P , thus P > 0 Therefore, by continuity and monotonicity[START_REF] Llobet | Rewarding Sequential Innovators: Prizes, Patents and Buyouts[END_REF] of P , there exists a e α > b α such that for α > e α, P < 0[START_REF] Llobet | Rewarding Sequential Innovators: Prizes, Patents and Buyouts[END_REF] It is straightforward to show that P is decreasing with α

See the Horizontal Merger Guidelines, available at http://www.usdoj.gov/atr/public/horiz_book/4.html

See O'Connor, FTC Hearings, "Efficiencies: Should Current Antitrust Policy Be Changed?" November 7,1995, availbale at http://www.abanet.org/antitrust/committees/stateantitrust/efficiencies.pdf

[START_REF] Compte | Capacity Constraints, Mergers and Collusion[END_REF] See OECD report on Merger Remedies, DAF/COMP(2004)21, December 2004, p.33.

See the Hart-Scott-Rodino Act.

See the Commission notice on remedies acceptable under merger regulation, OJEC 2.3.2001.

The Wall Street Journal Europe,May 26, 2005.

Our treatment of cost savings involves the marginal cost, so as to consistently follow the current CAs' treatment of allowable merger efficiency gains.

For an explicit modeling of cost savings through the use of complementary assets by merger partners see[START_REF] Bensaid | Competition, Cooperation and Mergers: Economic and Policy Issues[END_REF]

We consider here that in the short run firms cannot increase their production capacity. This result remains true as long as the unit cost of capacity acquisition c k is high enough, which is precisely our assumption here.

See also theoretical contributions supporting the choice of a pure Consumers' Surplus standard instead of the Total Welfare one, such as Besanko and Spulber (1993),[START_REF] Neven | Consumer Surplus vs. Welfare Standard in a Political Economy Model of Merger Control[END_REF], and[START_REF] Lyons | Could politicians be more right than economists? A theory of merger standards[END_REF].

[START_REF] Farrell | Asset Ownership and Market Structure in Oligopoly[END_REF] This would nevertheless go beyond the primary purpose of our model, since it would require the specification of the entrants' marginal cost and a detailed case discussion to determine then the optimal asset buyer from the CA's pont of view.

According to the competition policy theory and practice, merger remedies are to be the slightest modification possible that is able to restore market competition.

If the industry capacity is large enough, a duopoly is never induced to produce at full capacity even if the cost of M is zero, whereas if the capacity is low enough, a duopoly always produces at full capacity, even without efficiency gains.

Such a "buy out" mechanism is formally analyzed inLlobet et al. (2001).

See also[START_REF] Farrell | Negotiation and Merger Remedies: Some Problems[END_REF], or the Antitrust Division Policy Guide to Merger Remedies, by the U.S. Department of Justice, October 2004.

Appendix

Post-merger Cournot equilibrium. Successful Remedies:

These are the transfers for which both firms produce up to their post-merger capacity: for M, 2k -∆, and for o, k + ∆. Checking that the Best Reply function yield in equilibrium precisely the post-merger capacities allows us to compute the limits of the relevant range for the divestiture:

for M: BR M (q o = k+∆) = 1+α-c-(k+∆)


domain: math shs

text:
Maryvonne Priolet
email: maryvonne.priolet@univ-reims.fr

Eric Mounier
email: eric.mounier@u-pec.fr

Mathematics textbooks and teaching activity

Keywords: Mathematics textbooks, primary school, didactic, resources

This article focuses on the relations between the activity of the teachers and the contents of textbooks and the teacher's manuals. Through the observation of lessons, we analyse and discuss how the teachers follow the recommendations written by the authors of teacher's manuals. We describe the adjustments made by these teachers, comparing

Introduction

The present study constitutes a part of a larger research project investigating the place of mathematical textbooks in the French publishing market and the role teachers assign them in the daily practice [START_REF] Mounier | Les manuels scolaires de mathématiques à l'école primaire -De l'analyse descriptive de l'offre éditoriale à son utilisation en classe élémentaire[END_REF]. In France, teachers may decide for themselves and in each of their classes whether they want to resort to textbooks or not, and which resources they wish to use, on the condition of respecting the national curricula. The resources and their uses have been already the object of plentiful scientific literature [START_REF] Pepin | Re-sourcing teachers' work and interactions: A collective perspective on resources, their use and transformation[END_REF][START_REF] Fan | Textbook research in mathematics education: development status and directions[END_REF][START_REF] Matić | Teacher and textbook: Reflection on the SDT-model[END_REF][START_REF] Lenoir | Le manuel scolaire et l'intervention éducative: Regards critiques sur ses apports et ses limites[END_REF]. This paper examines how two teachers interact with mathematics textbooks in teaching the same topic; it focuses on the use of number lines. The choice of this theme seems to us relevant, with regard to the works of Hamdan and Gunderson (2017, p. 587) that show how "the number line plays a causal role in children's fraction magnitude understanding, and is more beneficial than the widely used area model".

Theoretical framework and research question

The teaching activity

The teacher has "to prepare the course", "to handle the class" and "to teach the class" (Amigues, 2003, p. 11). His activity results from a compromise between his objectives, his own purposes, his constraints, and the resources of his work environment (Goigoux, 2007, p. 47). So, it exceeds the context of the classroom. Then, the teaching activity has to be considered outside and inside the classroom, but, in this article, we mainly reserve the expression "teaching activity" for the activity of teaching in the presence of pupils.

Resources, textbook and teacher's manual

Within the framework of his teaching activity, the teacher interacts with a set of resources. To define the concept of resource, Adler (2010, p. 25) refers to two meanings of a word: a "reserve" from which the teacher can draw, and the action "to be nourishing again". We define the resource as "product of the human activity, developed to join a finalized activity" [START_REF] Rabardel | Les hommes et les technologies; approche cognitive des instruments contemporains[END_REF], quoted in Gueudet & Trouche, 2010, p. 58). As produced by an author, the textbook and the teacher's manual are resources for the teacher. The resource "Textbook" is intended for the pupils in the class and it is in connection with the curricula. The "teacher's manual" is the documentation annexed to the textbook, intended for the teacher and who allows "to understand better the transactions of the teachers with the curricular resources in mathematics" (Remillard, 2010, p. 201).

Teacher-resources relation

The teachers are differently positioned with regard to the use of the resources according to "modes of commitment" (Remillard, 2010, p. 214) being able to be shaped by particular expectations, convictions, habits or past experiences. This positioning may have an important effect when the use of the resource leads to the adoption of a didactic device structuring the session. The didactic devices in a primary class come down to three devices (Rey, 2001, pp. 31-35). In the first one named "explanation-application", some part of knowledge, for example the definition of a mathematical object, is presented to the pupil. Practical exercises follow the presentation of this knowledge. In the second named "observation-explanation-application", in the first instance the pupil is asked to observe an object, for example a geometrical figure then to generalise from this observation. Practical exercises are then proposed. In the third named "problem-explanation-application", in the first instance the pupil starts with the active manipulation of material or conceptual resources is brought in to the apply to problem-solving. This phase is followed by the shaping of the knowledge and then by the series of exercises. Some teachers sometimes make some adjustments. If the teacher's degree of expertise and the level of training are not sufficient, these adjustments can lead to "problems of coherence between objects of teachings, processes and activities" (Arditi, 2011, p. 361). Besides, from a generative document, expert teachers can proceed to relevant adjustments whereas, for lack of self-important training, the novice teachers are sometimes going to bring modifications going against the intentions of the authors of textbooks, the specialists of didactics [START_REF] Margolinas | Usage des manuels dans le travail de l'enseignant : l'enseignement des mathématiques à l'école primaire [Use of textbooks in the work of the teacher: the teaching of mathematics at the elementary school[END_REF][START_REF] Priolet | The use of mathematics textbooks in the training of elementary school teachers in France[END_REF].

Research questions

Considering the above research, we question the relations between the activity of the teachers and the contents of mathematics textbooks and of teacher's manual. Do teacher-users of a medium operate different types of adjustments during the activity of teaching in the presence of their pupils, and do they follow the model led by the teacher's manual?

Methodology

In order to answer our research questions, we provide a qualitative approach based on observation of practices used by teachers or semi-structured interviews with them.

Participants

This case-study involves two female teachers, Teacher B and Teacher A. Both of them teach at the 4 th level of elementary school (9-10 year-old pupils), in two schools located in two small towns in the centre of France. Teacher B has been teaching for 15 years and Teacher A for 10 years. None of them has studied higher education in mathematics. They both teach all school subjects.

Both teachers belong to a sample of 10 teachers of the 4 th level of elementary school who declared using mathematics textbooks and being volunteers to participate in our research. We had chosen this level regarding the introduction of fractions and decimal numbers. The ten teachers had agreed to be observed, by one of the two researchers, in their class, during a lesson of mathematics concerning the numbers, then to be interviewed during a semi-directive interview. For this case-study, we chose Teacher B and Teacher A among these 10 teachers for two reasons. Firstly, Teacher B and Teacher A have the same textbook1 in their class. Secondly, when we observed them teaching in their classroom, both presented a lesson on the theme of fractions and decimal numbers.

Method

In the classroom of Teacher B, the observation lasted 57 minutes and the interview 36 minutes. In the classroom of Teacher A, the observation lasted 33 minutes and the interview 44 minutes. We did not film the learning sessions, but some photographs have been taken, related to the use of the textbook or other artefacts. An observation table has been assigned in two parts :

 The observation of the classroom with identification of the different moments of the learning session (total duration, duration of each phase), duration of the phases of use of the texbook by the pupils, identify the moments while the teacher uses her teacher's textbook.  The database about pupil and teacher documentation. Following this observation of sessions, a second data collection was made through semi-structured interviews. An interview guide was set up on these subjects: preparation of the observed session, place taken by the manual during the session, manual's choice, general use of the teaching and pupil's guide and during the session, and finally, teacher training. The interviews often relied on the photographs that we had taken during the session concerning the use of the manual by the teacher or by the pupils. They can be linked in a methodological way with the self-confrontation method [START_REF] Theureau | Les entretiens d'autoconfrontation et de remise en situation par les traces matérielles et le programme de recherche «cours d'action» [Self-Confrontation Interviews, Re-Situating with Material Traces, and the 'Course of Action' Research Program[END_REF].

For the "fractions and decimal numbers" topic, the classroom manual has eight sessions numbered from five to twelve in its summary. The selected lessons for the analysis are lessons 8 (Teacher B) and 9 (Teacher A), because both of these lessons refer to the use of number line.

Both interviews were transcribed. All the data collected through observation and interviews have been analysed [START_REF] Bardin | L'analyse de contenu[END_REF] in order to extract those concerning the presence and frequency of use of the textbook and the resources used by the teacher for the conception of his teaching. The times of effective use of the textbook by the pupil have been converted in percentages of the total duration of the lesson.

Results

For each teaching activity, we present below a lesson in which we can spot the relation that each teacher has with the textbook and with the teacher's manual in her teaching activity.

Teacher B

Teacher B herself chose the textbook given to each of the pupils of her classroom. She reports using it frequently in class, mainly for the exercises. The Teacher's manual is present in the classroom. Teacher B reports that she doesn't use it because she has been disappointed by its general contents. She organizes the distribution of the lessons of the year herself.

Today, she proposes the following situation: she shows on the board a big number line she has prepared herself (Figure 1). In the first part of the lesson, she explains to the pupils how she made this number line: "the unit is here (u), so here between 0 and 1 there are 10 parts". She tells them that point A is equivalent to fourtenths of one. Then she asks pupils to write on their board the fractional numbers to which the points placed on the number line are associated. In the second part of the lesson, pupils open their textbook to do the 4 th exercise (Figure 2). For this lesson related to session 8 of the manual (À portée de maths CM1) page 42-43 and named "decimal fractions", the teacher's manual first planned a research path with an individual preparatory report (Figure 3) to the activity "Let's look together" in the pupil's manual (Figure 3). In Teacher A's classroom, each pupil has got a textbook. This textbook has been chosen by one of the colleagues predecessors of Teacher A in that school. She declares to have adopted this textbook which was already present in the class before she came. Teacher's manual is present in the classroom. Teacher A reports: "Mathematics is absolutely not my field. I refer a lot to the teacher's manual but after this I try to appropriate it". She says that she follows the annual distribution of the lessons in the manual. She also uses the manual's exercises. For this lesson related to session 9 of the manual (À portée de maths CM1) page 44-45 named "Decimal fractions", the teacher's manual first planned a research path with the number line drawn at the board (Figure 4) to prepare the activity "Let's look together" in the pupil's manual (Figure 4). Teacher A reports referring to the teacher's manual to build the "Let's look together". While the teacher's manual suggests as support for each exercise a number line increased without digital marks, this teacher writes a number line increased in tenth marks-units from 0 to 3, on the blackboard. Then she asks the pupils to indicate which fraction corresponds to such a graduation (yellow arrow).

On the board, she writes two answers and (Figure 5) proposed by two pupils. She asks them to explain their process. Then Teacher A asks all the pupils to open their textbook to individual work on exercise number 2. In this second part of the lesson, and especially with this exercise number 2, the pupils have to use a number line (Figure 6).

Analysis and discussion

We use Rey's model (2001) to analyse the didactic set up plan in each of these two classes. We compare it with the model underpinned by the instructions provided by the teacher's manual's research path. Although the authors declare in the preface (page 3, pupil's textbook) that the "teacher is a professional that chooses and assumes his pedagogy" and in the preamble (page 3, teacher's guide) that "the guide is conceived in order to give the teacher the freedom of his own ways", the instructions which are supplied in the scenario of the teacher's manual about the research path of both consider lessons, seem to lead an approach of the "observation, explanation, application" type. Our observations show that Teacher B operates the didactic device "explanation-application", whereas Teacher A tends to use the "observation-explanation-application" device. Teacher A modifies the starting situation support by converting the teacher's manual (number line increased without digital marks on a number line increased in tenth marks-units from 0 to 3). This modification of the support does not favour the devolution [START_REF] Brousseau | Théorie des situations didactiques [Theory of Didactical Situations in Mathematics[END_REF] of this problem to the pupils. It has transformed, by reducing it, the difficulty of the task planned by the authors' textbook: to question on the density of decimal numbers which constitutes an epistemological obstacle to the pupils' understanding. Although the authors of the textbook declared that teachers keep their pedagogic freedom, in both of the observed situations, both teachers do not commit the pupils in an approach of type "problemexplanation-application".

Conclusion

In order to analyse the relations between the teaching activity and the contents of mathematics textbooks and of teacher's manual, we have referred to the didactic model of [START_REF] Rey | Manuels scolaires et dispositifs didactiques [Textbooks and didactic devices[END_REF]. Our purpose was to detect the adjustments operated by two 4 th level of elementary school teachers who use the same mathematics textbook. We observe that both do not follow all the recommendations of the authors of the teacher's manual. For example, the teachers redefine the task planned by the authors of the textbook, then changing the planned didactic device, from a model of "observationexplanation-application" led by the teacher's manual, into a model of "explanation-application" [START_REF] Rey | Manuels scolaires et dispositifs didactiques [Textbooks and didactic devices[END_REF]. This change may reduce, in a way, the pupils of the understanding of the density of the order of decimal numbers. Thus, our analysis reveals a problem of coherence, already pointed by [START_REF] Arditi | Variabilité des pratiques effectives des professeurs des écoles utilisant un même manuel écrit par des didacticiens [Variability of the effective practices of primary school teachers using the same textbook written by professors in mathematics didactics[END_REF] between the adjustments operated by the teacher-user and the authors-designers of the textbook. This echoes the question of the validation of the collected knowledge [START_REF] Bruillard | Le passage du papier au numérique: le cas du manuel scolaire [Transition from paper to digital technology: the case of the textbook[END_REF], in particular in the context of the development of the recourse to the digital resources.

In conclusion, we notice that the logic of the teacher and the logic of the textbook cannot be the same. We observe that this gap can interfere with the aimed knowledge, from which we conclude in the necessity for the teacher to exercise an epistemological and didactic vigilance on pupils' understanding. It seems to us essential, following the example of [START_REF] Charles-Pézard | Installer la paix scolaire, exercer une vigilance didactique [To create school peace, to exercise didactical vigilance[END_REF], to include this issue in the training of the teachers.

Figure 1 :

 1 Figure 1: number line showed on the board (Teacher B)

Figure 2 :

 2 Figure 2: Exercise number 4, page 43 of the pupil's textbook

Figure

  Figure 3: teacher's manual page 37

Figure 4 :

 4 Figure 4: Teacher's manual p. 39 (Beginning of the research path)

Figure 5 :

 5 Figure 5: Photography of the board during the lesson (Teacher A)

Figure 6 :

 6 Figure 6: Exercise n°2 page 44 of the pupil's textbook

Proceedings of CERME10

Whereas in our study(Mounier & Priolet,

2015), there are at least 23 different textbooks in France for this level of teaching.Thematic Working Group 19Proceedings of CERME10


text:
Miguel Ribeiro

Edelmira Badillo
email: edelmira.badillo@uab.cat

Glória Sánchez-Matamoros
email: gsanchezmatamoros@us.es

Miguel Montes
email: miguelmontesnavarro@gmail.com

Genaro De Gamboa
email: genaro.degamboa@uab.cat

Intertwining noticing and knowledge in video analysis of self practice: The case of

Keywords: Mathematic teachers' knowledge, measurement, video analysis

This paper discusses a primary prospective teacher analysis of practice from a video episode she selected from her own practiceaimed at exploring the use of non-standard length measurement units. The analysis focuses on the revealed knowledge (MKT) while in practice, on the reflection when justifying the choice of the episode, and on her analysis of such episode. The results reveal aspects of PT knowledge associated with anticipating students' difficulties, but also her difficulties in interpreting and give meaning to student use of non-standard measurement units in a non-standard way. From the analysis, the need for an improvement in the use of video-based tasks in teachers' education is discussed, as well as the importance and impact of analyzing and discussing the analysis made by PT in and for educators' professional development.

Introduction

Mathematics teachers' knowledge and professional development has acquired an important relevance for research, enhancing its complexity. Particularly, the research focusing on mathematics prospective teachers' (PTs) education brings to the fore the role and importance of the relationships between teachers' mathematical knowledge and their knowledge of the content and students (e.g., [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF]. Following such focuses, new trends (conceptualization and implementation of interventions) for accessing, understanding, and developing teachers' and PTs knowledge are being developed (e.g., [START_REF] Ribeiro | Prospective teachers' knowledge in/for giving sense to students' productions[END_REF][START_REF] Santagata | Professional development processes that promote teacher change: the case of a video-based program focused on leveraging students' mathematical errors[END_REF][START_REF] Sherin | Selecting Video Clips to Promote Mathematics Teachers' Discussion of Student Thinking[END_REF].

More recently, analyzing video episodes has been both a focus of attention and a source for teacher education (e.g., [START_REF] Llinares | Prospective primary mathematics teachers' learning from on-line discussions in a virtual video-based environment[END_REF][START_REF] Van Es | Learning to notice: scaffolding new teachers' interpretations of classroom interactions[END_REF]. In that sense, the reflection and discussion upon one's own practice through video analysis is perceived as a powerful path for the development of a teacher's knowledge and awareness-focusing essentially on the mathematically critical features, both for teachers and students (e.g., mathematical content, competencies, interactions). One such critical aspect for students concerns measurement (e.g., [START_REF] Sarama | Evaluating of a learning trajectory for length in the early years[END_REF], particularly with regard to length. Considering the core role of teacher knowledge in student learning a focus on such knowledge is essential to better understand students' difficulties.

This paper aims at contributing to a broader and deeper understanding of the hows and whys sustaining the intertwining of teachers' knowledge (in the sense of Mathematical Knowledge for Teaching-MKT; [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF] and professional competency of noticing (in Mason's 2002 sense). Therefore, the research question in this study is what kind of knowledge is mobilized by the teacher when analyzing students' interactions with the mathematical content in a videotaped episode of measurement. For doing so, a video clip of a primary PT practice on length measurement is discussed, as well as her analysis of such video. The results reveal powerful trends concerning both the video analysis process as well as PT knowledge concerning length measurement.

Theoretical framework

As in other mathematical topics (e.g., adding or dividing fractions), the understanding of the mathematical whys of the measurement process is not straightforward, the understanding of such process being much more complex than the process itself. [START_REF] Piaget | Judgment and reasoning in the child[END_REF] mentions that acquiring the notion of magnitude requires going through different stages, from the use of words to express the magnitude (correctly) until one has the knowledge about the measurement of such magnitude. Going through such stages is not a straightforward path, and developing a broader and deeper understanding of the concept image and definition involved (in the sense of [START_REF] Tall | Concept image and concept definition[END_REF]) is a core aspect of such development.

Two aspects in measuring a length are crucial: the dimension and the distance. The dimension is connected with the use of physical resources, and the distance concerns the space between two points/objects. Although both notions are perceived in an intertwined and inseparable way, due to its nature, one can't approach them in a single identical manner with students [START_REF] Clements | Measurement in pre-K to grade 2 mathematics[END_REF]. Such a measuring process requires the choice of the unit to use and perceive the quantity of units (or unit parts) are needed (how many) to go from one point to the other. Ultimately, one would need to combine both processes in order to get a more approximated value for the considered magnitude. In that sense, measuring is linked with two core ideas: the inverse relationship between the size of the unit used/number of units needed and the need for using the same unit in the same process of measuring (e.g., [START_REF] Clements | Measurement in pre-K to grade 2 mathematics[END_REF], applying it using a certain algorithm. Measuring the length thus requires knowing the standard measure(s) used, as well as the differences from using different measurement units (e.g., hands, foot, fingers) and the possibility of using these non-standard measurement units in a non-standard way.

Teachers' MKT on measurement is essential for developing students' knowledge and awareness of the topic. Among the MKT subdomains, and due to the aim and context of the work reported here, we consider for discussion the Common and Specialized Content Knowledge (CCK and SCK) and the Knowledge of Content and Students (KCS) subdomains. For doing so, examples linked with length measurement are used. The Common Content Knowledge (CCK) is associated with the mathematical knowledge required by teachers, including being able to use the instruments to correctly perform a certain measurement, knowing that no empty space must be left, as well as that there shouldn't also exist any overlays (instrumental knowledge). In doing so, knowing what instruments to use (in the sense of standard instruments/measurement) for measuring different entities and the differences between the entities they measure is involved. It corresponds to knowledge of how to perform, assuming a user's perspective (knowing how to measure). Complementary to this CCK, following the MKT conceptualization, teachers are required to be in possession of mathematical knowledge specifically linked with the tasks of teaching [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF]. Such Specialized Content Knowledge (SCK) includes knowing the mathematical whys justifying the different measuring processes (considering all the stages mentioned by [START_REF] Piaget | Judgment and reasoning in the child[END_REF]. In addition to requiring knowledge of the different ways of measuring different entities, from one side, teachers' knowledge should also include knowing the whys of using such different forms of measuring (and the associated units).

Complementarily, it should include knowing possible different units for measuring the same entity and the ways of doing so (e.g., length), as well as the whys associated with the use of (non)standard units. In that sense, such knowledge does not include only knowing the whys associated with the procedures, but also includes the concepts (both image and definition in Tall's 1988 sense) and the whys associated with such concepts (e.g., the inverse relationship between the size of the unit and the number of iterations needed). Thus, it corresponds to a core aspect of the knowledge that allows teachers to give meaning and interpret students' solutions and comments (part of the interpretative knowledge, e.g., [START_REF] Ribeiro | Prospective teachers' knowledge in/for giving sense to students' productions[END_REF] while in practice.

Both CCK and SCK (and HCK, although it is not discussed here) give support for teachers' developing their practice-conceptualizing and implementing the tasks, hearing students' comments, and interpreting them in order to decide the path to follow at each moment. Those decisions are also informed by the knowledge teachers have of their students' difficulties or what they consider easier.

Intertwining knowledge of the content and knowledge of the students' learning processes concerns the KCS. It includes knowing that one of students' difficulties concern the measuring process (e.g., the need for using units with different natures), which is related to the complexity of understanding the measuring process (e.g., Clement & Stephan, 2004). In such subdomain of knowledge, one can also include the knowledge allowing teachers to anticipate students difficulties in differentiating the measurement instrument (non-standard unit, e.g., the hand) and the measuring unit, or on ways of using non-standard measuring units (e.g., using the finger length or width).

Recent research has shown the need for designing instruments/resources for teacher education allowing them (and the researcher) to characterize the knowledge in action when analyzing their own practice (e.g., [START_REF] Kersting | Teachers' Analyses of Classroom Video Predict Student Learning of Mathematics: Further Explorations of a Novel Measure of Teacher Knowledge[END_REF]. One such resource is video analysis (of classroom episodes), which allows a focus on how and which knowledge (prospective) teachers bring to front when interpreting, analyzing, and reflecting upon the recorded interactions (e.g., [START_REF] Kersting | Teachers' Analyses of Classroom Video Predict Student Learning of Mathematics: Further Explorations of a Novel Measure of Teacher Knowledge[END_REF][START_REF] Van Es | Learning to notice: scaffolding new teachers' interpretations of classroom interactions[END_REF]. When focusing on selecting and characterizing a video episode, [START_REF] Sherin | Selecting Video Clips to Promote Mathematics Teachers' Discussion of Student Thinking[END_REF] consider three dimensions: (i) window, (ii) depth and (iii) clarity. In the particular case we address here, focusing on length, windows dimension is related to evidences of students' different levels of comprehension of the measurement of length; depth dimension is related to evidences of interactions in which students participate in the decision making process about choosing the measuring unit, the instruments and the measurement procedure and clarity is related to evidence of student's arguments that transparently show their comprehension of measurement of length.

Reflecting upon what seems to be happening, and discussing grounded on the analysis elaborated, is perceived as a pathway for developing teachers' knowledge and professional noticing [START_REF] Sherin | Selecting Video Clips to Promote Mathematics Teachers' Discussion of Student Thinking[END_REF]. Such professional noticing includes identifying what is important in a teaching situation, using what one knows about the context to reason about a situation, and making connections between specific events and broader principles of teaching and learning [START_REF] Van Es | Learning to notice: scaffolding new teachers' interpretations of classroom interactions[END_REF]. Such noticing is thus linked with the ability for examining practice pinpointing the significant aspects in order to be better informed at the time of making their pedagogical decisions.

The context of the study

Aiming at identifying and deepening the understanding of the content of teacher knowledge and how it intertwines with teacher actions and beliefs several case studies have been developedin different contexts and involving different school levels, mathematical topics, and competencies, contributing also to conceptualizing tasks of different natures focusing on developing teachers' knowledge (e.g., [START_REF] Ribeiro | Prospective teachers' knowledge in/for giving sense to students' productions[END_REF].

Method

We first present the participant of this study, and afterward, the specific context and analysis stages and process. Here we focus on the knowledge and awareness about the topic of length measurement of a primary PT (Carla), who was part of a bigger research project in which her professional development through video analysis and reflection was the aim. Carla was in the last year of the teachers' training program at University Autònoma of Barcelona, and she was teaching the field practice to grade 2. Previously, she had some training concerning classroom analysis, identifying and interpreting relevant events in a video from a novice teacher. As part of the field practice, PTs have to record the 10 classes they teach (one hour each) and choose one episode to analyze and reflect uponwhich has to be transcribed. The selection of the episode needs to be justified, mentioning the whys associated with its mathematical richness and referring explicitly to the mathematical goal pursued. For selecting and analyzing such an episode, Sherin et al.'s (2009) criteria (windows, depth and clarity) should be used (which have been previously explored with the PTs). Therefore, the analysis should focus on the mathematical content approached, student-teacher interaction, and students' understanding.

For selecting the episodes, PTs' were advised to look for situations they perceived as involving a "high level" of students mathematical knowledge and discussions/argumentations. Carla's choice was an episode aimed at introducing "the measurement of the length and width of objects using nonstandard units" (e.g., the length of the classroom using the foot; the length of a glue package using the finger). An example of the proposed tasks is: using an unconventional length measure (finger), determine the length of the glue package. Although Carla chose a 15 minutes long video associated with the goal "review the content," she only considered it important to analyze and reflect upon the last six minutes -sustaining such choice on her perceived richness, in terms of Sherin et al.'s criteria (2009). In the video, one can perceive how the students assign the corresponding number (amount of units) to the length or width of the objects as a result of comparing such amount with the unitary measurement they consider as a reference. In the discussion, the inverse relationship between the length to be measured and the unit used is also explored. Complementary to the video and Carla's analysis, Carla (as all the other PT) were also asked to justify (in a written form) their actions in the video they selected.

The analyses were made in different stages and each of them followed the same structure. First, each researcher focused on the divergent aspects, then there was a joint discussion focusing on them. Such joint discussions contributed to a refinement of the analysis and increased the researchers' own mathematical awareness and "interpretative knowledge." Here we expand the notion attributed to this expression by [START_REF] Ribeiro | Prospective teachers' knowledge in/for giving sense to students' productions[END_REF] by including the researcher's ability to read, hear, and understand the interactions and knowledge in action. The first stage focused on analyzing Carla's revealed knowledge in the part of the video she considered important to analyze and reflect upon-the last six minutes. In the second stage, the focus of attention was Carla's analysis of her own practice using the same criteria PTs use for doing the analysis (Sherin et al.'s 2009 criteria). The third stage focused on the mathematical aspects associated with what happened in the first part of the video-which Carla chose not to include in her analysis-and the possible whys concerning her knowledge and awareness leading to such choice.

Analysis and discussion

When analyzing Carla's practice (the 15' video) with a focus on the CCK, SCK, and KCS, different aspects of her revealed knowledge are identified that sustain her awareness and professional competency of noticing. She reveals knowing the importance of measuring with different nonstandard units (CCK), leading to the inverse relationship between the number of units and the size of such units (CCK):

Carla: I gave an example, showing to the students that is not the same measuring with bigger o smaller hands, so I put my hand in the sheet they had to measure and asked a student to put his hand next to mine, to compare the different measures of both hands, telling them: "Do you see that his hand is smaller? Then my measure will be smaller than his".

Also, the students' difficulties in measuring without leaving empty spaces between units is anticipated (KCS) when interacting with students:

Carla: Of course, another thing is how we put our hands. If some of you put them like this (partially opened) and some of you put them as Isaac did (completely opened), Isaac will get less . . . . But it doesn't mean that it is wrong; it simply indicates that we have different hands and we have measured differently.

However, we can see a potential conflict identifying the unit of measurement, arising at least four different possible unit combinations: kin's and teacher's hand, both in a close or open position. In spite of Carla's ability to identify the previous situation as an interesting topic to be discussed in the class (noticing), it is noteworthy that there is a lack of awareness of the necessity of exhaustiveness when re-covering the measurable object [START_REF] Clements | Measurement in pre-K to grade 2 mathematics[END_REF].

While teaching, Carla indistinctly uses dimension and distance-the two aspects of length [START_REF] Dickson | El aprendizaje de las matemáticas[END_REF]-revealing aspects of the content of teachers' knowledge that need to be a specific focus of attention in training. Such knowledge would sustain the conceptualization and implementation of tasks aimed explicitly at exploring both concepts and their complexity. Not overcoming such difficulties, and thereby enriching teachers' SCK on measurement, would contribute to a low level of professional competence of noticing, particularly concerning the students' difficulties in distinguishing perimeter and area (and volume)-KCS.

Focusing on her analysis of the students' reasoning (only the last six minutes of the video), the fact that she can differentiate various aspects of understanding from different students reveals Carla's (advanced) level of professional competency of noticing according to Sherin et al.'s (2009) criteria. Such can be linked with her KCS as well as her interpretative knowledge [START_REF] Ribeiro | Prospective teachers' knowledge in/for giving sense to students' productions[END_REF]. In that sense, she takes into consideration (CCK and SCK) the dimension, the absence of empty spaces between the use of the measurement units, the use of anthropomorphic units, and the inverse relationship between the size of the unit and the number of iterationson the written justification Carla wrote: Hugo and Daniel notice that the different results of the measurement depends on the size of the hands; Miguel, on the other hand, concludes that the result depends on the ways the hands are placed).

When justifying the choice of the episode to analyze, Carla's reflection reveals an awareness of the relationship between the SCK on the content, her decisions, and her ability to anticipate the students' difficulties and understanding of the topic (KCS).

Carla: I chose this particular part of the class because it reveals the moment when the pupils become aware that when measuring with non-standard instruments and units, they get different results…Through the students' reflection and reasoning, they get that the size of the hands matter when measuring the length (thus getting different results) and that not only the size of the hand matters, but also the ways one uses it. I also have to note that after the mentioned reasoning emerged from students, I registered the different values on the blackboard, and I could then use it to explain to the whole class. During this period, I also use different teaching strategies, including the registration on the board (with the number and the unit used), giving time for students to present their results, and promoting reflection concerning the measurement technique used, supported by visual examples, as they could have difficulties in the content concerning both the measuring process and finding the relationship between length and units used.

However, in the remainder time of the episode Carla didn't chose to analyze (the first nine minutes), something curious and mathematically important happened. Analyzing it can shed some light on teacher decisions and associated specialized knowledge as well as concerning the need for a change in Sherin et al.'s (2009) criteria for selecting and analyzing he video episodes.

During the first part of the episode, one student (Miguel) provides a completely different answer to the posed questions (e.g., length of the class in steps, when the rest of the class provided answers around 30, his answer was 14). Carla interpreted it as a misunderstanding of the measurement process, showing him in each case "how to do it correctly." But the consistency of Miguel's answers reveals a high level of understanding (concept image and definition, [START_REF] Tall | Concept image and concept definition[END_REF] of measurement [START_REF] Clements | Measurement in pre-K to grade 2 mathematics[END_REF], using non-standard units. When Carla discussed the question "How many fingers are needed to measure the glue package?" Miguel again disagrees with the other students and with Carla's validation of a length of eight fingers. With the glue package on the top of the table, on the vertical, the following dialogue occurs:

Teacher: No? How many fingers did you get?

Miguel (putting the finger vertically along the glue package): One! Although the goal of the class was "to use non-standard length measures," such measures have been used and explored by Carla only in a standard way (the one with which she was used totaking into consideration the cultural aspects). Such exclusive use, linked with the content of SCK, makes it difficult to anticipate and understand (KCS) Miguel's answers--use of non-standard measurement units in a non-standard way. The fact that when reflecting upon her practice and analyzing the video she did not point to this aspect as problematic sustains the need for a change in the way the video analysis task has been conceptualized. And, in an intertwined way, makes us wonder about Sherin et al.'s (2009) criteria for professional competency, as when focusing only on her own analysis (of the last six minutes), Carla could have been considered to have an advanced level of such competence; when looking at the part of the video she did not analyze, a different conclusion could be drawn. In that sense, there is also evidence of the interdependent nature of such competence and teachers' knowledge.

Carla seems to not be aware of the inexplicit use of a one-dimensional measurement unit on a threedimensional object, leading thus to some contingency moments. To overcome such moments, she opted (grounded in her own revealed knowledge of measurement) to tell the student "how to do it". Although aspects of professional awareness and competencies of noticing are present both in her practice and in her analysis of the video, her knowledge shaped such aspect, leading to a partial view of the students' understanding of the measurement length and units. This supports the need for a complementary discussion and reflection upon the video analyses PTs make in order to focus also on the mathematically critical aspects they are not aware of, and which are a barrier to completely achieving the advanced level of professional competency of noticing expected when using Sherin et al.'s (2009) criteria.

Final comments

From the work focusing on the video analysis, three aspects can be enhanced: teachers' knowledge and professional noticing abilities; reflection and awareness capabilities; video-based task design and its potentialities. Although of different natures, these aspects have in common the educator's responsibility in changing the training process. Through this case study, one can better understand aspects of the content of the different subdomains of MKT, both revealed in practice as well as expressed when reflecting upon such practice when analyzing the video. Other aspects concern the impact of the interactions and the prevalence of non-standard length measurements in standard ways, and that revisiting the practice didn't become a prompt for awareness and noticing development, obviously linked with previous experiences and MKT. This calls our attention to the need for an exterior element to pinpoint these critical features, leading them to become a starting point for developing such knowledge and awareness.

Although the video analysis performed by Carla with the provided instrument did not accomplish completely the defined aim, our analysis of her practice and of her own analysis of the video was a prompt for inquiring our own practice as educators in terms of interpretative knowledge [START_REF] Ribeiro | Prospective teachers' knowledge in/for giving sense to students' productions[END_REF] and of awareness of the educators professional noticing abilities. One future research path concerns thus the develop of a complementary instrument for video analysis that would allow trainees to dig deeper into their awareness of the mathematical whys sustaining what happens and to focus also on the educators specialized knowledge, awareness and noticing abilities.

Proceedings of CERME10

Acknowledgements: This research has been carried out as part of research projects EDU2015-65378-P, EDU2014-54526-R and EDU2013-44047P, funded by the Spanish Ministry of Economy and Competitiveness and the SGR-2014-GIPEAM. It has also been partially supported by the Portuguese Foundation for Science and Technology (FCT), project code (UID/SOC/04020/2013).