Frederick Benaben

Matthieu Lauras

Audrey Fertier

Nicolas Salatgé

INTEGRATING MODEL-DRIVEN ENGINEERING AS THE NEXT CHALLENGE FOR ARTIFICIAL INTELLIGENCE -APPLICATION TO RISK AND CRISIS MANAGEMENT

Artificial Intelligence (AI) is currently on top of the hype regarding simultaneously research publications and industrial development. However, the current status of AI makes it quite far and different from the current understanding of Human intelligence. One suggestion that is made in this article is that Model-Driven approaches could be considered as an interesting avenue to complement classical visions of AI and to provide some missing features. Specifically, the use of Model-Driven Engineering tools (such as metamodel and model transformation) could benefit to the domain of AI by introducing a way to extend the apprehension of unknown situations. To support that proposal, an illustrative example is provided regarding the domain of risk and crisis management. 1

INTRODUCTION

Artificial Intelligence (AI) is at the center of most of the main developments in both research in computer science and industrial innovation in software systems. This is an amazing trend of nowadays and out of any cynical "buzz word" consideration, it is obvious that both the research in computer science and the software system industry are being transformed by this emergence. However, one can argue that it has been going very fast (especially due to availability of dormant past results in the domain of neural networks and machine learning) and that what we call AI should actually be consider as quite far from the simple notion of intelligence.

In Legg andHutter (2007a, 2007b), a lot of definitions of intelligence are studied, including some extracted from AI domain. As a conclusion, the authors tries to provide a synthesis which includes the following statements regarding stable features of definitions:

 "Intelligence is seen as a property of an individual interacting with an external environment, problem or situation" which includes an idea of perception.  "intelligence is related to the ability to succeed" which implies the existence of objectives.  "there is a strong emphasis on learning, adaptation and experience" which requires knowledge.

In addition, in this article, the following definitions will be considered for intelligence as well:

(1) The ability to learn or understand or to deal with new or trying situations (merriam-webster.com) (2) the ability to apply knowledge to manipulate one's environment or to think abstractly as measured by objective criteria (merriam-webster.com)

(3) the ability to perceive or infer information, and to retain it as knowledge to be applied towards adaptive behaviors within an environment (wikipedia.org) (4) the ability to acquire and apply knowledge and skills (oxforddictionaries.com) (5) the ability to learn, understand, and make judgments or have opinions that are based on reason (dictionary.cambridge.org) (6) the ability to use memory, knowledge, experience, understanding, reasoning, imagination and judgment in order to solve problems and adapt to new situations (AllWords Dictionary).

From these definitions, four main characteristics can be considered as the root of intelligence:

 Perception: directly connected to the first statement from Legg and Hutter, and corroborated by the use of words such as "learn", "understand", "perceive or infer information", this basically means that intelligence requires to be able to gather data about the context of the faced situation.  Interpretation: this idea is not provided by the analysis of Legg and Hutter but it is a consequence of the previous idea of "perception" (and probably included in their first statement). In this article, the gathering of signals from the situation and the understanding of these signals are to different points: perception for the first one and interpretation for the second one.  Exploitation: similar to the second statement of Legg and Hutter, this idea implies that there are objectives to reach by using the perceived and interpreted signals. The use of words like "apply", "make judgment or have opinions", "solve problems" confirm this statement. This would often be a decision to take, with the support of the situation understanding.  Knowledge: echoing the third statement of Legg and Hunter, another principal idea regards the capitalization of experience. This is also embedded in the notion of decision tacking. The previous definitions show that trend with the use of terms like "apply knowledge", "retain it as knowledge", "based on reason", "memory, knowledge, experience".

Based on these considerations, this article considers the following definition of "intelligence":

Definition 1 Intelligence can be seen as the ability to use one's perception senses to construct an understanding of any given (and potentially unknown) situation and to confront it to an existing stored knowledge to enact optimally, according to objectives. Figure 1 shows the idea of climbing abstraction levels from gathering signals about the situation (data level) to understanding that situation with regards to an existing knowledge (information level), in order to take decisions based on the understanding of the situation and the stored knowledge. Basically, the knowledge is used at every level and could be enriched by all levels. This vision of intelligence is quite close to the results presented in [START_REF] Rowley | The Wisdom Hierarchy: Representations of the DIKW Hierarchy[END_REF] as the wisdom hierarchy or the DIKW hierarchy (where D stands for Data, I for Information, K for Knowledge and W for Wisdom).

The following of that article discusses in section 2 how model-driven engineering can be used to complete traditional AI tools to provide a more relevant intelligence, while section 3 illustrates this framework within a use-case of risk and crisis management.

TOWARDS A MORE INTELLIGENT AI USING MODEL-DRIVEN ENGINEERING

Classical Artificial Intelligence approaches

Artificial Intelligence is mainly based on Data Science. According to [START_REF] Jagadish | Big Data and its Technical Challenges[END_REF], Data Sciences are mainly based on Data Management and Data Analytics. On the one hand Data Management includes Acquisition, Content Extraction and Data Integration and Representation. On the other hand, Data Analytics concerns Analysis and Human-Based Interpretation. This human-based interpretation is not to be confused with the interpretation mentioned as the transition to the second level (Information) of the intelligence framework presented in section 1. With regards to the framework presented in the previous section and illustrated through Figure 1, it is easy to see that the Data Management step is dedicated to perform the perception feature. The Data Analytics is dedicated to provide the user with a content that can be interpreted by him to take decision: the Analysis sub-step focuses on detecting patterns and correlations (for statistical purpose) to feed the Decision level for the Human-Based Interpretation.

Figure 2 illustrates Data Science on the Intelligence framework introduced in Figure 1: Basically, the most important finding from that analysis is that Data Science and more broadly general approaches of AI actually do short-cut the Information level. These approaches focus on providing the decision level with material that would actually support the making of a decision by establishing statistical reports on the data. This definitely allows such systems to deal with Volume and Velocity of data.

Model-Driven Engineering

Model-Driven Engineering (MDE), as described in (Da Silva 2015) mainly relays on two basic activities:

Modeling and Model transformation. It is important to notice at this point that MDE is absolutely not dedicated to software engineering (though it is used a lot in that domain), and as stated in [START_REF] Dietz | The Enterprise Engineering Series[END_REF]) can be used in Industrial Engineering contexts. Modeling can be defined as:

Definition 2 Modeling is the activity of building a representation of a known or unknown entity according to shared concepts and vocabulary.

In an operational context, modeling is dedicated to the following objectives (which may obviously be combined, depending on the context):

 Modeling to manage the behavior (simulation): exploiting models to anticipate on the behavior of a subject (e.g. weather models, heat diffusion equations, etc.).  Modeling to build appropriately (conception): defining the static and dynamic structure of a system to support its construction (e.g. furniture plan, algorithm, etc.).  Modeling to explain or understand (communication): describing a system to make it understandable (e.g. marketing presentation of a product, meeting schema, etc.).

Besides, one can notice that the three previous usages of models can also be considered with regards to the formal and rigorist degree of the models: simulation requires very formal models, communication usually exploits more informal models, while conception is in between and can use hybrid models.

Figure 3 illustrates schematically but very accurately the activity of modeling (independently from the domain, the subject, the objectives or the type of the concerned modeling): This vision of modeling is based on the following steps:

1. Boundary of the subject: definition of the exact perimeter of the system to be modeled. 2. Choice of the modeling point of view: definition of the why, for whom and for what purpose of the modeling activity. 3. Choice of the projection plan: selection of the modeling language that fits the point of view (basically that is relevant to the goal of the model and understandable by the future users). 4. Projection of the subject on the projection plan: this is the model building step. 5. Choice of the granularity level: definition of the appropriate precision level of the model.

One important remark concerns the question of consistency between point of views. Actually, it is very common to have to realize several models of the same system (plan of a building VS energetic behavior VS budget and costs). It is so very important to understand, especially in "conception" that any choice on one point of view has some impacts on other points of view: esthetic and aerodynamic considerations regarding the design of a car must take into account the plans of the engine, otherwise there is a major of risk of the engine not fitting under the hood… It is important to notice that step 4 requires a metamodel. There are lot of definition of what is a metamodel. (OMG 2017) describe it as "a model that defines the language used to define a model", it is so the model of a modeling language. More pragmatically, a metamodel can be seen as "the description of all concepts of a language, their semantics and the syntactic related to the use of these concepts" (Chapron 2006). Based on this and on [START_REF] Bezivin | On the Unification of Power of Models[END_REF], we will keep the following definition:

Definition 3 A metamodel is the description of concepts and relationships between these concepts that allows to defines models where: the models conforms to the metamodel and the metamodel is represented by the models (NB: a metamodel is usually the high-level structure of an ontology). The concept of metamodel is particularly pregnant in the domain of model transformation. Transformation rules are actually defined between metamodels. On a very schematical point of view model transformation is the tool used to travel the model-driven engineering cycle. For instance the cascading Model-Driven Architecture provides transitions between Computer Independent Model (CIM, mostly representing requirements and expectations of a software) and Platform Independent Model (PIM, mainly figuring the theoretical system responding to the CIM), and then between PIM and Platform Specific Model (PSM, representing the concrete vision of the implementation of the PIM on a chosen platform). Each of these transitions is supposed to be assumed by model transformations, describing transformations rules between source metamodels and target metamodels.

The objective of a model transformation is to exploit the knowledge embedded in a source model to create (systematically and thus, potentially automatically a target model. The source model is built according to a source metamodel (which defines modeling rules for the source domain). The target model, similarly, must conform to a target metamodel (which defines the target domain). A key principle is that the two metamodels, source and target, should not be totally disjointed in order to share a number of concepts (as shown in Figure 5). Because of this common conceptual space between the source space and the target space, a set of rules can be established. These rules must describe how the concepts of the target metamodel (the ones located in the common space) can be obtained. from the concepts of the source metamodel (also located in this common space). These rules will constitute the transformation rules (or mapping rules). Considering the existence of these concepts shared between metamodels, the source model contains modeling elements resulting from these shared concepts (the shared part of the source model) and others resulting from the specific concepts (the specific part of the target model ). From this source model we will then be able to identify a capitalized knowledge (corresponding to the material resulting from the specific part) and an extracted knowledge (resulting from the shared part). The knowledge extracted will concretely constitute the raw material for the transformation of the model strictly speaking whereas the capitalized knowledge will rather be stored for later use. The mapping rules can then be applied to the extracted knowledge to transform the elements of the source model resulting from the shared concepts into elements of the target model also referring to these shared concepts. The result of this operation is the obtaining of a transformed knowledge. Additional knowledge must then be imported in order to fill the gaps concerning the specific concepts of the target metamodel (since only the shared concepts have been covered by the application of the mapping rules). The transformed and additional knowledges are combined to create the shared part and the specific part of the target model. Figure 5 illustrates schematically the mechanism of model transformation: It should of course be noted that these different "knowledges" may be more or less consistent depending on the relevance and quality of the transformation of models. If the capitalized knowledge and the additional knowledge are not significant (compared to the knowledge extracted and transformed), one can suppose that the considered transformation of models is relatively convincing (in the sense of the potential of automation). On the other hand, if the extracted knowledge and the transformed knowledge are weak compared to the capitalized and additional knowledge, it is very likely that the domains considered are too distant and that the transformation turns out to be very manual (i.e. not programmable by generic rules). The first of the two "asymptotic" (i.e. at the limit) situations associated with these considerations corresponds to a transformation of models between two conceptually disjoint domains (called pure exogenous), which would in fact be a creation of a target model on the basis of the only additional knowledge (the transformed knowledge being empty because of the disjointed nature of the two spaces of departure and arrival). It would simply be a "from scratch" achievement of the target model. The second "asymptotic" situation corresponds to a transformation between two conceptually confused domains (socalled endogenous), which would concretely result in a creation of the target model on the basis of transformed knowledge alone (the additional knowledge being empty due to the confused nature of the two spaces of departure and arrival). It would ultimately be a transformation of a first model into a second model rigorously equivalent, but expressed in a different language (for instance the transformation of a process model expressed as an Petri net into a BPMN model).

Moreover, a transformation is called "vertical" if it involves different levels of abstraction within the model-driven engineering cycle (typically, the transformation of PIM to PSM in an MDA cycle is a vertical transformation). Similarly, a transformation is said to be "horizontal" if it remains at the same level of abstraction within the model-driven engineering cycle (the transformation of a GRAI digraph into a UML activity diagram presented in [START_REF] Grangel | Transformation of Decisional Models into UML: Application to GRAI Grids[END_REF]) is a horizontal transformation). Thus, by freely taking inspiration from [START_REF] Combemale | L'Ingeniérie Dirigée par les Modèles (IDM) -état de l'art[END_REF], Figure 6 describes a summary table aiming at linking the orientation of model transformations (horizontal or vertical) with the origin of model transformations (endogenous or exogenous) with regards to their frequency of occurrence: Finally, regarding MDE, the question about the location of MDE on the intelligence framework presented in Figure 1 should be considered to echo the proposal of Figure 2. With regards to that intelligence framework, it can be stated that the Modeling step (based on metamodel) is dedicated to perform the interpretation feature. The Model Transformation step is dedicated to use the obtained model to generate new content that would be more useful for the user to take decision. This is the exploitation feature Figure 7 illustrates the location of MDE on the introduced Intelligence framework: Basically, the most important finding from that analysis is that Model-Driven Engineering actually do not consider the data level (and the perception feature) out of human consideration. This is basically the human modeler that must make the link between signals he or she can gather from the observed situation and concepts of the metamodel to be instantiated. Then, the obtained model(s) can be exploited automatically through model transformation for decision support. This definitely allows such systems to deal with Variety and Value of data. It is also important to notice that the obtained model(s) can also be used in various types of other exploitation types (usually dedicated to decision support but not only) like visualization or simulation. Visualization is not necessarily based on generated models and can be obtained directly from data. However, simulation absolutely requires formal models to be performed.

Complementarity and limits of both approaches

Based on both conclusions of subsections 2.1 and 2.2, and if we consider that the question of Veracity of data is out of scope due to the fact that we only consider data that is already clean and trustable (the topic is not about data source discovery or data filtering and cleaning), then, it is easy (and potentially too simple) to state that data science could take care of Volume and Velocity, will model-driven engineering could tackle Variety and Value. In any case, both approaches struggle with specific conditions as soon as we extend the range and the amount of data to manage:

 Data science can not deal with unknown type or content: just like a shark, able to detect a drop of blood in millions of litters of sea water, but which would be lost with a Rubik's Cube®. This is due to the fact that the data associated to the Rubik's Cube ® is out of the scope of its perception and it can only apply a default behavior (like attack, ignore or escape).  Model-Driven Engineering can not manage a large amount of data: just like a baby, able to start to play with a Rubik's Cube® because of its shape, color, touch, etc. but would be unable to detect a drop of lemonade in his bath among all other tastes and odors of the water.

These considerations have been largely discussed in [START_REF] Benaben | Big-Data and the Question of Horizontal and Vertical Intelligence: A Discussion on Disaster Management[END_REF]) and are represented on Figure 8 in a general perspective mapped on the framework of Figure 1:

IMPLEMENTATION OF THE ARTIFICIAL INTELLIGENCE FRAMEWORK ON A RISK AND CRISIS MANAGEMENT USE-CASE

Rapid description of the Loire Flooding Use-Case

The Loire flooding is one of the three most feared disasters on the French Territory (with the Seine flooding and an earthquake on the French Riviera). The Loire river is characterized by significant variations in flows (DREAL 2016) and its significant rates of containment and urbanization. In 1856, such a flood had flooded 110,000 hectares where there are today more than 300,000 people, 13,600 companies and 72,000 jobs (DREAL 2016). To enable the simulation of such a scenario, we have collaborated with the Loire-Cher-Indre and Maine-Loire Services of Forecast of Floods (In French SPC). Two models of propagations have been developed: a five-year flood and a 100-year flood on the Middle Loire, between the Bec d'Allier and the Bec de Maine. In this example, we focus on a 100-year flood event that corresponds to the highest known flood event for a given region [START_REF] Medd | Les inondations[END_REF].

A hydro-meteorological scenario, reported in [START_REF] Dolidon | Synoptique Scenario GéNéPi. Deliverable of the French funded Project GéNéPi[END_REF], realistically models the water inflow of the six tributaries present in this part of Loire in the case of a 100-year flood. Thus, three daily flow and water level (coast) surveys were developed for the seven reference hydrometric stations and the 11 expected flood days. Each survey contains forecast flow and coast over the next 56 hours and three probability levels: high, median and low.

Metamodel for crisis management

As described in [START_REF] Lauras | Towards a Better Management of Complex Emergencies through Crisis Management Meta-modelling[END_REF], the COSIMMA (Collaborative Situation MetaModel of Albi) defined for crisis management is structured according to two layers: (i) the core layer which actually describes concepts and relations of any collaborative situation, and (ii) the specific layer containing concepts that inherit from the core concepts and that describe more precisely concepts of the domain. The core is agnostic while the layer is dedicated to crisis management domain (in some other research works, other layers have been defined about supply chain and health care.

The core layer of the metamodel has been organized in five complementary parts:

 Context system (light grey) including components and characteristics of the considered environment, and also opportunities or threaten specific to these environment characteristics.  Objective system (medium grey) containing characteristics of collaborative network (common objective and facts that the collaboration has to manage).  Partner system (strong grey) expresses the different resources and know-how of the partners. This includes notably capabilities, patterns, instructions, resources (information, material, people, etc.), flows (linking capabilities) and mediator able to orchestrate the different business processes.  Behavior system (dark grey) describes the concrete operations, which are deployed to concretize the collaboration. This includes business processes / activities.  Performance system (white) assesses the overall performance of the collaboration by comparing dedicated Key Performance Indicators (KPIs) to the measures on the field.

The crisis management metamodel is structured as presented in Figure 10. The core metamodel at the center and four packages (Context, Partners, Objectives and Behavior) contain concepts dedicated to crisis situation inheriting from the core concepts. In the following, only concepts dedicated to interpretation of data (Context, Partners, objectives packages) are described as they are currently implemented in the knowledge base:

Context package:

 Good: any human-made elements (e.g. building, road, etc.).  People: any group of person (e.g. students of a school, employees of a plant, etc.).  Natural site: any natural element of the environment (e.g. lake, forest, etc.).  Civilian society: any social actors (e.g. media, intellectual society, etc.).  Territory: any administrative area (e.g. county, island, etc.).  Danger: any specific dangerous characteristic of the environment.  Intrinsic risk: any permanent risk due to identified danger (e.g. earthquake, riot, etc.).

Partners package:

 Actor: any stakeholder involved in crisis management (e.g. firemen, EMS, etc.).  Resource: any resource used by actors (e.g. truck, decontamination tent).  Service: any capability of actors (e.g. evacuate victims, treat injured people, etc.).  Actor service: any service specifically provided by actors.  Mediation service: any service provided by Mediation IS.

Objectives package:

 Emerging risk: any risk specifically emerging due to the crisis itself.  Effect: any direct consequence of the crisis itself (e.g. 10 injured people, fire, etc.).  Mission: any objective directly responding to identified risk or effect of the crisis.  Event: any event occurring during crisis management that must be considered as triggering.  Gravity factor: any characteristic of the current situation that may change the gravity of the crisis.  Complexity factor: any characteristic of the situation that may change the type of the crisis.

Illustration with the RIOSUITE software tools

The following describes the way the RIOSUITE suite of tools is used to deal with crisis management, implementing the Artificial Intelligence framework presented above and using the metamodel presented in the previous section 3.2. The presented screenshots do show the way RIOSUITE climbs the abstraction levels of the Artificial Intelligence Framework: (i) Perception, i.e. data gathering, (ii) Interpretation, i.e. from data to information, and (iii) Exploitation, i.e. from information to decision.

Perception -data gathering

The perception aims at collecting data related to the faced crisis situation. In the case of the Loire flooding, considering that there is actually no real crisis, the Loire flooding is simulated through the use of simulation sensors, sending "fake" data extracted from the Services of Forecast of Floods and replicating exactly the data from the scenario they defined. In Figure 11, we can see a screenshot describing on the left part the map of the area with the location of "fake" sensors and on the right part the curves of water level and water flow that the sensors are actually sending. The data sent is respecting the WSN (web-service notification) standard.

Figure 11: The Perception level in RIOSUITE.

Interpretation -from data to information

The interpretation is dedicated to automatically build the situation model at the information level, i.e. to instantiate concepts of the metamodel related to context, partners and objectives.

For the context, the mechanism is simply based on extracting information from OpenStreetMap® based on a pre-filtered selection of amenities. For the partners, this is mainly by interviews of the stakeholders that the actual capabilities (and their relationships) can be identified. Regarding the objectives, the used mechanisms are based on formal rules that have been described in [START_REF] Fertier | Interprétation Automatique de Données Hétérogènes Pour la Modélisation de Situations Collaboratives: Application à la Gestion de Crise[END_REF]). These rules are dedicated to detect patterns and to instantiate dangers mainly. Then the RIOSUITE system identify (thanks to the context model) the stakes that are in the area of the danger and generates risks according to the susceptibility of these stakes to that danger. Figure 12 illustrates the objective interpretation. On the upper left part of the picture are the risks that have been interpreted from the presence of buildings in the area of the danger of flooding identified thanks to the triggering of some rules due to the values of some sensors. On the bottom left part are these risks on the map. On the right are the curves representing the sent data through the "fake" sensors.

Figure 12: The Interpretation level in RIOSUITE.

On a practical point of view, the complete obtain models is similar to the next Figure 13 where on the left part are the three models (from up to bottom: objectives, context, partners) and on the right part is the graph database representing the embedded knowledge base.

Figure 13: The obtained models from the interpretation level in RIOSUITE.

Exploitation -from information to decision

The exploitation is dedicated to use the obtained models (at the information level) to provide decision support to the crisis managers. Basically, as presented on Figure 14, there are three main features at this level. First, the RIOSUITE platform is used to deduced a collaborative process model (see bottom-left part of Figure 14). This is based on a model transformation from the models of context, partners and objectives to a model of behavior. This model transformation has been completely described in [START_REF] Mu | Collaborative Process Cartography Deduction based on Collaborative Ontology and Model Transformation[END_REF]. Second, there is an orchestration engine able to run the obtained collaborative business process among the partners as personalized to do lists (see the upper-left part of Figure 14). Third, there is an agility management tool in charge of the detection of any disruption with regards to the expected situation by continuously comparing the expected situation and the field situation (see the right part of Figure 14). The main benefits illustrated through the previous example basically concern the inclusion of models between data gathering and decision support. Thus the proposed approach is able to deal with potentially unknown situations (by casting raw data into concepts) and to use a formal material (the models) to perform complex reasoning and decision support.

CONCLUSION

This article tried to show how the evolution of Artificial Intelligence could benefit on the integration of Model-Driven Engineering principles. The mechanisms provided by the use of metamodel and models during a decision making process based on artificial intelligence could definitely bridge the gap between the current vision of AI on the one hand, mainly based on data science, and that the conclusions of this article trend to see like "Artificial Reflex" or "Artificial Instinct" more than an actual Artificial Intelligence, and, on the other hand, the expectations of an AI able to deal with more complex situation. The illustration presented in the third section, based on a risk and crisis management scenario have shown how much this hybrid vision, mixing data science and model-driven engineering could be worth. Finally, on a broader perspective, this article can lead to the following bet: just like big-data brought artificial intelligence back on the foreground, the next achievement of artificial intelligence, particularly in the domain of complex decision making, could bring model-driven engineering back on the stage.

Figure 1 :

 1 Figure 1: Big Picture of the definition of Intelligence suggested and used in this article.

Figure 2 :

 2 Figure 2: Location of Data Science in the Intelligence Framework.

Figure 3 :

 3 Figure 3: A generic representation of the modeling activity.

Figure 4

 4 Figure 4 illustrates the classical abstraction layers of modeling/metamodeling:

Figure 5 :

 5 Figure 5: Model transformation basic principle.

Figure 6 :

 6 Figure 6: Frequency of the different model transformations.

Figure 7 :

 7 Figure 7: Location of Data Science in the Intelligence Framework.

Figure 8 :

 8 Figure 8: Limits of both Data Science and Model-Driven Engineering in the Intelligence Framework. The line of attack is, first, to use Data Science to perform the modeling step by interpreting the flow of incoming data thanks to Data Analysis tools. Then it is expected to instantiate concepts of a metamodel to create situation model(s). Second, the goal is to use Model-Driven Engineering to exploit the generated

Figure 10 :

 10 Figure 10: The Collaborative Situation MetaModel of Albi (COSIMMA).

Figure 14 :

 14 Figure 14: The exploitation level in RIOSUITE.


Carl Winsløw
email: winslow@ind.ku.dk

Professional and academic bases of university mathematics teaching for the 21st century: the anthropological approach to practice-based research

Keywords: university mathematics teaching, ATD, practice-based research

Based on an anthropological approach to the notions of profession and métier within an institution, we show how the label "mathematics" could relate to both notions in the setting of universities. We also propose a finer characterization of segments of the métier. Finally, we revisit some examples of how our own research have addressed different segments so identified. We finally return to the question of how different forms of scholarship interact with the métier and its segments, and formulate a proposal for professionalizing the métier in view of current and future needs.

INTRODUCTION

Mathematics as a university discipline has hugely expanded in the 20 th century, both at the level of research and at the level of education. This development is inseparable from the interaction of mathematics with other -and equally evolving -disciplines, many of which have not only drawn on, but also contributed to the advances of mathematics both as a field of research and as a matter to be taught. It is neither possible nor helpful to try to draw sharp lines between mathematics and other disciplines. This is so not only for research but also in higher education contexts like Engineering and Natural Sciences.

The growth of mathematics from an institutional point of view is visible in the existence and expansion of mathematics departments in virtually any university type institution. One informal, frequently implicit, characterization of a "university mathematician" is certainly an individual who works at such a department. In mathematics departments, we find a great variety of scholarly activities, often with labels such as "pure mathematics", "applied mathematics", "statistics" and so on, each with further subdivisions and overlaps; and sometimes also "mathematics education", "history of mathematics", "data science" and more. Another informal characterization of a "university mathematician" could be university faculty member with an advanced degree labeled "mathematics", but in practice, this is quite similar as labels often result from the name of departments where they are obtained. In fact, people whose teaching or research are centered on mathematical contents may have other affiliations. We can think, for instance, of a specialist in mathematical education who teaches mathematics methods courses in an education department, or of a statistics researcher employed at a Medical school to teach statistics to future physicians.

From the point of view of university mathematics education, the notion of a university mathematics teacher is more relevant than the departmental categorization. Here, labels and descriptions of teaching units can be used to clarify more precisely what is being taught by a given university teacher. We shall pursue this delimitation in the next sections. We also return to the important co-existence and interaction of teaching and research in universities in the case of university mathematics teachers.

THE NOTIONS OF PROFESSION AND OF MÉTIER IN THE FRAMEWORK OF THE ATD

The English language offers many terms to designate a position within an institution that is characterized by a responsibility to carry out certain types of tasks. We shall define use and two of them, profession and métier, which both have their roots in Latin (via the French language). Our definitions may not correspond to dictionaries that explain common usage. However, they are, at least to some extent, consistent with distinctions made by [START_REF] Chevallard | Quel avenir pour la formation des professeurs ? Éléments pour la transition didactique[END_REF] and [START_REF] Stigler | The teaching gap[END_REF] to discuss the general status of teachers in Western and Asian societies, respectively.

By a métier, we mean simply mean the set of positions within an institution defined by a family of types of tasks, as explained above. Members of the métier are individuals holding one of these positions. Naturally, to stay as member of a métier, one will normally need minimal capacity to carry out the involved types of task; there may be no further requirements.

A profession is a special case of a métier, in which the praxis, and knowledge on the praxis, is made explicit and shared among its members. The resulting discourse is a logos Λ describing and justifying the use of certain methods or techniques, using a more or less specialized professional discourse. The combined praxis and logos P = (Π,Λ) is regulated and developed by the members of the profession (possibly by other agents too). Admission to the profession is strongly linked to sharing P at some level, obtained through formal training carried out by members of the profession (again, possibly by other agents as well).

In terms of the anthropological theory of the didactic (see e.g. Chevallard, 2019 for details), P is a collection of praxeologies, including both praxis blocks Π i (types of tasks, techniques) and logos blocks Λ j (discourse about techniques, and theory to support and justify the discourse). If p belongs to the métier defined by Π = (Π i ), we can assume some minimal relation R I (p, Π) of p within the institution I at which the métier is exercised. Meanwhile for p to belong to a profession we have further requirements on R I (p, P ), including the role of p in establishing R I (p', P ) for the position p' of newcomer to the profession.

Examples of professions, which are well established in most developed societies, include: lawyers, doctors and engineers. Scientific researchers within specific fields also constitute professions -certainly, disciplines of modern science have extensive and explicit logos blocks, and scientists are trained by other scientists of the same discipline and within the same institution. The case of teachers in, say, primary school is less clear -the development of explicit knowledge about teaching, as well as the training that gives access to it, may only to a very small extent be carried out by primary school teachers.

Considering now the case of university mathematics teachers, many of them do belong to professions of research (in a broad range of specialties, as we have seen), although not necessarily, and not necessarily the same. However, one can hardly call the set of university mathematics teachers a profession, as there is little shared and explicit knowledge about how to carry out the types of didactical tasks that characterize this métier, and even less training and development of such shared knowledge that is carried out by the members of the métier. Of course, individual members of the métier have shared their views and principles about the métier (e.g. [START_REF] Halmos | I want to be a mathematician[END_REF][START_REF] Krantz | How to teach mathematics[END_REF], and in some countries, conferences and committees on university mathematics teaching are being organized with and by members of the métier (e.g. Burn, [START_REF] Appleby | Teaching undergraduate mathematics[END_REF]Saxe & Braddy, 2025). Such works are usually written to push new agendas which are not widely shared in the profession, and may remain unknown to most members. On the other hand, the interest and willingness to share knowledge on teaching is growing among university mathematics teachers in many institutions and countries. In 1988, a mathematician noted that when a mathematician speaks about teaching, colleagues smile tolerantly to one another in the same way family members do when grandpa dribbles his soup down his shirt (Clemens, cited in Krantz, 2015, p. xi). This is certainly not universally true anymore. Still, university mathematics teaching is hardly a profession in the sense defined above. In the rest of this paper, we try to characterize the métier, and then consider if and how it might become a profession.

THE UNIVERSITY MATHEMATICS TEACHER MÉTIER

Mathematical practice and knowledge can itself be modelled in terms of praxeologies ω, and within university institutions U, didactical tasks are all about establishing certain relationships R U (σ, ω) for individual in a student position σ within U. If the university mathematics teacher position is called τ, a minimal requirement for R U (τ,ω) is certainly that R U (τ,ω) ⊇ R U (σ,ω). But more is required: σ must have practical knowledge about the didactical praxis of establishing R U (σ, ω), and if this praxis is called Π(ω), there are thus additional requirements for R U (τ,Π(ω)) which may be assessed by the extent the praxis of τ actually succeeds to establish R U (σ, ω). Indeed, university institutions usually have rather explicit and established ways to assess the latter kind of relation, and the observed student performance is frequently also used to assess R U (τ,Π(ω)). Newcomers to the métier will often have to develop R U (τ,Π(ω)) more or less through building up personal experience with Π(ω) and possibly drawing on their own experience as students. However, as the student position σ may be different from positions they have themselves held -for instance, in the case of a background as graduate of pure mathematics, who is faced with the praxis of teaching applied mathematics to populations of some other discipline. In this case, both R U (τ,Π(ω)) and R U (τ,ω) have to be developed based on R U (τ,ω') where ω' consists of mathematical praxeologies somewhat similar to ω. While all of this appears at first sight a bit theoretical, it all was eminently concrete to someone who, like the author of this paper, started out in the métier with a background as researcher in pure mathematics, faced from day one with the task of teaching mathematics and mathematical biology to future biologists. While the mathematical elements were all very familiar, the full praxeologies to be taught, as well as relevant didactical techniques to do so, were largely to be acquired by the teacher.

One way to characterize the métier is thus in terms of the didactical tasks, closely related to the relationship R U (σ,ω) to be established, and in particular in terms of the student positions σ and the praxeologies ω concerned. Even when teaching students in positions that the teacher has actually occupied, R U (τ,ω) is evidently of central importance to R U (τ,Π(ω)). In the frequent absence of external support (specific training) to establish R U (τ,Π(ω)), and therefore also of shared logos blocks Λ(ω), we see the clear traits of a métier in the establishment and function of the position τ.

Considering that the set of mathematical praxeologies ω σ to be taught to students in position σ depend largely on σ, a first rough "topology" of the métier can thus be given in terms of the student populations: a teacher is in position τ σ if she must engage in Π(ω σ ). In brief, the métier can be subdivided according to positions τ σ for which R U (τ σ ,Π(ω σ )) must then satisfy some minimal requirements, more or less assessed trough R U (σ,ω σ ). Still, formal training aiming to support entrance into the position τ σ is often generic (see Winsløw, Biehler, Jaworski, Rønning & Wavro, to appear), corresponding to the assumption that R U (τ σ ,Π(ω σ )) is not only independent of σ, but also that Π(ω) is independent of ω. The techniques from Π(ω) assumed to be independent of ω are basically pedagogical and concern, for instance, how to prepare and conduct interactive lectures on a generic praxeology ω, relate to a generic student independently of her actual position σ, and so on. While any training on this generic practice Π will then also involve some form of logos block Λ, the relation R U (τ σ ,(Π,Λ)) may indeed fail to establish R U (τ σ ,Π(ω σ )), even when combined with the relationship R U (τ σ ,ω') that τ σ may hold to praxeologies ω' that are somewhat similar to ω σ , or even include ω σ .

The access to the métier is, nevertheless, to a large extent based on developing R U (τ σ ,ω') through the mathematical training of τ σ , which (in the case of researchers) may be assumed to largely guarantee that R U (τ σ ,ω σ ) can be established satisfactorily by any person in position τ σ , irrespectively of the student position σ concerned. This could seem justified at least in case where τ σ has previously developed R U (σ,ω σ ) successfully. Even in this case, the establishment of R U (τ σ ,Π(ω σ )) remains, and the assumption that some R U (τ σ ,(Π,Λ)) will suffice to complement R U (σ,ω σ ) is hardly without raising concerns -especially in the frequent case where existing didactical practices Π(ω σ ) appear, for students in position σ at large, in need of improvement in some sense. Again, taking measures of R U (σ,ω σ ) as measures of the quality of Π(ω σ ) may at best help to realize such a need, while a logos block Λ(ω σ ) is needed to identify possible shortcomings of Π(ω σ ), to device innovations of Π(ω σ ), and to conduct systematic experiments of these innovations -and finally, to share the results in a way that conveys the position τ σ with professional traits.

Considering again the variety of student populations that university mathematics teaching may address, and the idea that these somehow provide a structure on the métier as a whole, we will draw on an idea initially presented in the thesis of [START_REF] Kim | Les besoins mathématiques des non-mathématiciens : quel destin institutionnel et social ? Études d'écologie et d'économie didactiques des connaissances mathématiques[END_REF] and further developed by [START_REF] Chevallard | Introducing the anthropological theory of the didactic: an attempt at a principled approach[END_REF]. They consider that a human population P at large can be subdivided as P 1 ∪P 2 ∪P 3 . Here, P 1 consist of people who engage in production of new mathematical knowledge based on a postgraduate degree in some mathematically intense discipline (like physics, pure or applied mathematics, statistics etc.). Population P 2 consist of people who do not engage in production of new mathematics, but whose work is nevertheless crucially based on a postsecondary study of mathematical disciplines (like most secondary level mathematics teachers, but also most engineers, business specialists etc.). Finally, P 3 is "the rest" and certainly the largest portion of P. While there may be some grey zones left from these somewhat informal definitions, we can nevertheless identify corresponding positions σ i (i = 1,2,3) of students at university, who are in some sense preparing for adult life in population P i . These are not, in general disjoint, at least in early stages of university studies: especially σ 1 and σ 2 may be required to build the same relations to basic mathematical praxeologies from, say, linear algebra, and thus be taught together, depending on the institution U; and even future members of P 3 may at some point face such requirements. At later stages, specializations may be more common. Nevertheless, university mathematics teachers may be roughly assumed to occupy positions τ i (i = 1,2,3) corresponding to the students they face, with both overlaps (as mentioned) and with further specializations (e.g. τ 2,t when the didactical tasks is specialized for the case of future secondary teachers t, τ 1,µ when the public are future researchers µ in a field of pure mathematics, etc.).

We note here that it is common for individuals at U to occupy certain positions, such as τ 1,µ and µ, simultaneously. Indeed, the simultaneous occupation of positions as scholar or researcher, and as teacher, is both currently and traditionally considered a hallmark of university institutions -associated with the more radical Humboldtian ideal of Einheit von Lehre und Forschung (unity of teaching and scientific research, cf. [START_REF] Madsen | Relations between teaching and research in physical geography and mathematics at research intensive universities[END_REF]. Before delving further into research based on the subdivisions of university teaching métier that were introduced above, we shall dwell on research into how such double occupancy of teaching and research positions may influence and shape the métier, and in particular how the first position may lend and draw professional traits from the latter.

RESEARCH ON THE TEACHING-RESEARCH NEXUS

The position µ as researcher of mathematical sciences (including, for instance, statistics and various fields of applied mathematics), clearly is a profession which, from the19 th century, has become firmly established at university institutions all over the world. Mathematical knowledge and expertise was of course of great societal important even before, but was often developed with and under the auspices of other disciplines such as astronomy, geodesy, mechanics and (from the 18 th century) engineering. However, far into the 20 th century most university mathematicians were primarily teachers. According to Tucker (2013, p. 699), who focuses on American universities: While in the early 1950's most faculty at doctoral institutions still saw undergraduate teaching as their primary mission, by 1970 that mission had changed with research becoming the primary focus of these faculty.

This reflects a more general development in many (not only American) university institutions over the 20 th century [START_REF] Cuban | How scholars trumped teachers: Constancy and change in university curriculum, teaching, and research, 1890-1990[END_REF]: the increasing priority of research over teaching both in the tasks characterizing university faculty positions, and in the selection of individuals to fill those positions. In particular, occupying position τ can be a mere corollary of occupying position µ. Halmos (1985, p. 400) talks about in the following quote: I spent most of a lifetime trying to be a mathematician -and what did I learn? What does it take to be one? I think I know the answer: you have to be born right, you must continually strive to become perfect, you must love mathematics more than anything else, you must work at it hard and without stop, and you must never give up.

The position µ is clearly what

The role of commitment and personal ability is evident also in similar descriptions of the research profession by other mathematicians. At first sight, it would look like being a mathematician is merely a personal affair. At the same time, the professional character of the research métier is evident for mathematics for the same reason as in other research areas (system of training and regulating access to position µ are both internal to the métier, as is the system for developing new explicit knowledge relative to the métier).

Halmos, who was a famous textbook author and an eminent lecturer, feels guilty after a day where he taught well but did not do research (ibid., p. 322): Despite my great emotional involvement in work, I just hate to start doing it; it's a battle and a wrench every time. Isn't there something I can (must?) do first? Shouldn't I sharpen my pencils, perhaps? (…) Yes, yes. I may not have proved any new theorems today, but at least I explained the law of sines pretty well, and I have earned my keep.

Here, "work" is evidently research, and "sharpening pencils" becomes a metaphor for distractions from µ-tasks -including teaching ("the law of sines"). Certainly, filling τ well does not make up for failures to accomplish the tasks of µ.

Despite this competition for time, it is also a classical idea that teaching and research can somehow resource and inspire each other -in the higher education literature, one speaks of a teaching-research nexus [START_REF] Neumann | Perceptions of the teaching-research nexus: a framework for analysis[END_REF], reflecting that the link between the two is by no means simple or one-way. Of course, we often think about "research based teaching" as teaching that somehow draws on research, but to the extent knowledge is produced by research, this is somewhat trivial. It becomes less trivial if we think of the individual occupant of µ and τ -so that somehow the concrete research activity of the individual influences the same person's teaching. In between, one could conceive of how the teaching of a larger or smaller institution (department or university) is affected by its research activity. Finally, there might be influences from teaching on research at both individual and institutional level, such as including students in research activities.

For the case of mathematics, not many studies exist of the teaching-research nexus. [START_REF] Madsen | Relations between teaching and research in physical geography and mathematics at research intensive universities[END_REF] modeled it in terms of praxeologies of research and teaching, P R and P T for short. Each of these centrally include mathematical praxeologies (used or developed during research and teaching) but not be limited to them -for instance, research praxeologies could be broadly conceived to include practices of publishing, funding, communication, etc. In the study, five researchers in pure mathematics were interviewed on teaching practice, their research practice and connections between them (in that order). Not surprisingly, the links between P R and P T turn out to be rather "indirect", in the sense that the mathematical praxeologies involved in the two are normally quite distant. As one of the informants says (ibid., p.

756):

How long did it take me before I had an impression of what is going on in the research area that interests me? Well it took 5 years, after I had graduated.... You can't tell a bachelor student what it is about can you?

In other words, the mathematical praxis and logos in undergraduate teaching is very far from that involved in the research of the informant.

At the same time, four out of five informants insist with considerable energy and many examples that considerable indirect links exist at the level of types of task. For instance, this involves the didactical task of constructing challenging assignments for students, and also the similarity between the activities aimed at for students (when working on the assignments) and the research task. While solving teaching tasks, one may discover points that later, maybe by association or further development, can be used in research. Many other links are mentioned to illustrate the experience expressed by one informant: "I feel that I can get things forth and back between the two parts." We can say that the teaching-research nexus is largely implicit and indirect, and it concerns mainly the level of practices which are not the same but somehow similar, while it does reach the level of logos.

The fact that the experience of students in university mathematics is not always very close to mathematical research was emphasized as a motivation for Burton's (2004, p. 27) interview study with 70 researchers of mathematics:

It was my hope that a gap, between how mathematicians themselves came to know and how they promoted learning in others, if confirmed in the study, would help to explain student disaffection with the discipline, their difficulties in learning it, and the gender bias in those who take it up as a career.

The study itself considers mainly the creative part of research practices, the "enquiry", through which the individual researchers are supposedly learning (new) mathematics. This activity is found to be quite different from what university students are asked to do, for instance because it involves intuition, room for different "thinking styles" and so on. The study does not investigate the experience of university students but relies, for this part, on the author's personal experience. A main value of this study is the detailed analysis of a large and exciting survey of how P R is described by people in position µ. Direct observation of these praxeologies is obviously difficult to arrange, which explains the use of interview methods in both of the above-mentioned studies.

Similar methods were used in [START_REF] Misfeldt | Mathematical Writing[END_REF] study of the writing practices of researchers in pure and applied mathematics, also in view of informing didactical practices at university to enhance students' learning in this area. However, in this study, interviews were carried out in the presence of material traces of current research of the informants, in the form of pieces of writing. In particular, three distinct modes of writing were found: exploratory writing (often involving more diagrams and the like, than linear text), "first drafts" (attempts to write out proofs etc. in a linear fashion, often by hand) and "article text" (produced with TeX or similar software).

From all three studies, we can see that possible links between students' and researchers' praxeologies are quite indirect, and maybe even missing (Burton's "gap"). The motivation of all three studies is that when university teachers are also researchers one could hope that those teachers could somehow produce a didactic practice that could make the link closer. In fact, much of recent didactical engineering research on university mathematics is more or less explicitly based on this idea: draw on teacherresearchers' experience with research to create a didactical practice that somehow enable students to do mathematics in ways that are similar to mathematical research. It could also be that students are to engage in mathematical activities close to what is found in another professional field or discipline that they will subsequently encounter. This brings us back to the different teacher positions (τ 1,µ , etc.) in the university mathematics teaching métier, and research on controlled experiments with corresponding didactical practices.

THE TEACHING MÉTIER BY STUDENT POPULATIONS

We now return to take a closer look at experimental research involving cases of τ i (i = 1,2) and the didactical praxeologies undertaken by teachers in these positions. We note in passing that in some university type institutions, such as liberal arts colleges in the US, there is a strong tradition for teaching mathematics to "all" students, including those whose study and professional aims do not strictly need postsecondary mathematics. Such teaching often focuses on historic and otherwise humanistic aspects of the discipline [START_REF] Fried | History of Mathematics, Mathematics Education, and the Liberal Arts[END_REF]. At other universities (including, to my knowledge, most European universities) many students have no mathematics courses, and positions σ i (and hence τ i ) may not exist at all.

Research-like situations in undergraduate analysis

A recent survey of undergraduate pure mathematics programmes (Bosch et al., to appear) confirms that their structure offers many similarities, and all programmes analyzed include a calculus-analysis sequence of modules as a central component. The calculus vs. analysis distinction is in general somewhat blurred and contingent upon local conditions. However, a quite common interpretation can be made precise in terms of praxeologies [START_REF] Winsløw | Transformer la théorie en tâches : la transition du concret à l'abstrait en analyse réelle[END_REF]Gyöngyösi, Solovej & Winsløw, 2011;[START_REF] Winsløw | Mathematical analysis in high school: a fundamental dilemma[END_REF][START_REF] Kondratieva | Klein's Plan B in the early teaching of Analysis: Two theoretical cases of exploring mathematical links[END_REF]. Calculus praxeologies ω involve types of tasks concerning functions that are given in closed form, including functions of one or more real variables, real and complex valued sequences, and so on; differential and integral calculus are among the central sectors. In calculus courses, the student relationship R U (σ,ω) aimed at (and especially, assessed) is often focused on students' mastery of techniques corresponding to a well-defined set of type of tasks. Thus, students will learn techniques to evaluate certain integrals, find extremum points of functions in certain situations, etc.; the logos block is on the other hand more informal, compared to analysis courses. These, on the other hand, focus on theory, and first analysis courses may merely complement calculus courses with formal theory, and thus in a sense extend R U (σ,ω); we call this type of extension a type I transition. In more advanced Analysis courses, students face type tasks that are formulated in terms of previous logos blocks (e.g. investigate whether this or that function is a complete metric on 𝑅𝑅 n ), thus working with entirely new and generally more "abstract" praxeologies. We call this kind of passage to new praxeologies, in which tasks concern logos from previously developed praxeologies, a type II transition.

To support type I transitions, it is evidently crucial to create links between familiar praxis from Calculus, and the new theoretical superstructure. One strategy to do so, pursued by Gyöngyösi et al. (2011), is to design student assignments which involve new theoretical material which are explored based on computer supported experiments with objects from Calculus. An example was to "explore the convergence properties of the sequence of functions given by f n (x) = 1/[1+exp(n(2-x))]". Based on plots in software like Maple, the students quickly see the pointwise convergence of (f n ) to a non-continuous function, and infer that the convergence is not uniform. They can also, based on Maple calculations, verify that the limit function has the same integral over any interval as the limit of the integrals of f n , and hence that uniform convergence is only a sufficient but not a necessary condition for the interchanging limit and integral. Experiments with such designs in a first-year analysis course showed that a "middle group" students effectively improved their results when such assignments were included, while it made no difference for high-and low-performing students.

The transition of type II could be attacked with similar ideas for task design, but naturally going further than the mere illustration and application of new theory. [START_REF] Grønbaek | Thematic projects: a format to further and assess advanced student work in undergraduate mathematics[END_REF] experimented so-called thematic projects which are relatively long assignments that proceed from more closed tasks (of the type found at the end of chapters in text books) towards open questions that require students to device and prove a theoretical result, which is naturally supported by the first parts. Instead of an oral exam based on students' presentation of material directly extracted from a textbook, the students should now present one of their thematic projects. Experimentations over several years with thematic projects in a real analysis course demonstrated a significant increase in students' work, satisfaction and results, as measured by the standard exam; however, the increased work for the course teachers (both to create new assignments and to provide adequate supervision to students) made the format less viable outside of a funded project.

Very similar experiments were made later in the context of a less advanced real analysis course [START_REF] Gravesen | Task design for students' work with basic theory in analysis: the cases of multidimensional differentiability and curve integrals[END_REF]. In a project funded by a University of Copenhagen grant to further the connections between research and undergraduate teaching, we defined a number of research-like activities, and constructed a collection of exercises that would engage students in some of these (for each exercise). Of course, mathematical research is not limited to "prove that" activities, while these dominate end-of-chapter exercises in many post-calculus textbooks. Among the activities explicitly focused on in this design, students were to use special cases to investigate an abstract hypothesis or question, to formulate a hypothesis for a given question, to formalize relations between two or more results, to produce or validate ε-δ type definitions, etc.

Another idea for more advanced courses, developed by Kondratieva et al. ( 2018) but yet to be tested at larger scale, is to link calculus praxis with proofs of major theoretical results in analysis. As an example, a student assignment was developed in which the so-called Basel problem (convergence and value of Σ(-1) n /n 2 ) is solved by calculus techniques, and then the same sequence of techniques is used to give an elementary proof of Dirichlet's theorem on Fourier series. The construction of assignments that relate different domains in mathematics, or (as here) basic and more advanced courses in the same domain, is proposed as a strategy for task design research linked to Klein's idea of "Plan B" (cf. Klein, 2016, p. 83).

Perhaps we can formulate two overall conclusions emerging from these and many other experiments with task design that aims to create "research like" situations for undergraduate students:

1 The revised didactic practice Π(ω) can certainly be realized and as a result, more ambitious aims for R U (σ,ω) are in fact realized;

2 It is much less straightforward to establish R U (τ,Π(ω)) for the position τ as such (rather than for an individual teacher in privileged circumstances), like when Π(ω) requires time-consuming design (e.g. of new assignments).

We close this section by briefly examining a more famous and generic parallel to (1), which offers in some sense also a counterexample to (2): the so-called Moore method (described, for instance, by Halmos, 1985, pp. 255 ff.) to teaching theoretical mathematics. Moore was a professor of mathematics at the University of Texas from 1920 to 1969, and a legendary teacher and doctoral supervisor. Over the course of the past century, his methods of teaching expanded and developed several variations in several North American universities. According to the "Moore method" article on Wikipedia (as it looked in December 2020), dozens of professors and departments use some version of it even today.

The core of Moore's method is to let students (re)construct proofs of given theorems, with no use of books or other sources, but referring only to a handout with Definitions and Theorems carefully prepared by the teacher. The method apparently works for any specific mathematical praxeology or domain, except for the clear focus on formal proof (which is anyway common in almost any post-calculus course in pure mathematics).

In that sense the method is a set of pedagogical techniques to teach proof, while the didactical practice Π(ω) comes with the concrete handout for a given set of praxeologies ω. It would be very interesting to investigate the institutional and historic conditions that enabled the success of this approach. It is a rather certain hypothesis that one important condition has been the existence of a well-developed "logos" on the didactical techniques, disseminated in several books written for and by members of the métier. The method has not only been transmitted but also further developed by some of these members (see for instance [START_REF] Chalice | How to Teach a Class by the Modified Moore Method[END_REF]). It appears from some of these writings that not only descriptions of the pedagogical techniques, but also examples of handouts for concrete praxeologies ω, have been disseminated widely. It remains that the method also shares the challenge of design by the teacher, to the extent Π(ω) has to be constructed for a given unit of teaching, in view of concrete student populations and specific praxeologies ω.

According to the literature referred to, the Moore method is found to offer an excellent experience for students in position σ 1 , as considered in this section. However, at the undergraduate level, such students usually mix with students in position σ 2 , for whom training to prove theorems may not be as important. Apparently, the method works best with advanced courses and hardworking students, who are more or less clearly in position σ 1 . It is still remarkable as a case of sustained, explicit development of shared didactical practice by the métier itself, which moreover connects clearly to an important aspect of mathematical research, the construction of formal proofs. It is a fair hypothesis that this implicit or even explicit link between P R and P T contributed to the success of Moore's way to organize P T .

We now turn to studies of the métier of teaching mathematics to students in position σ 2, focusing on two variants of this type that are both more common than σ 1 in most university institutions.

Klein's second discontinuity: the case of teacher students

We first consider the case of σ 2,t or, for short, σ t : university students who prepare to become mathematics teachers outside of the university, generally at primary or secondary level. In some universities, the position σ t is found in specialized programs apart from σ 1 . In other universities -including many universities in Europe, particularly when it comes to future secondary mathematics teachers -the two positions are indiscriminate at least in the first years of study. In other words, the same requirements are made for R U (σ t ,ω) and R U (σ 1 ,ω), for a good deal of the mathematical praxeologies ω that are taught at U. In this situation, which appears common in many universities even today, Klein (2016, p. 1) identified a major problem as early as 1908:

The young university student finds himself, at the outset, confronted with problems, which do not remember, in any particular, the things with which he had been concerned at school. Naturally he forgets all these things quickly and thoroughly. When, after finishing his course of study, he becomes a teacher, he suddenly finds himself expected to teach the traditional elementary mathematics according to school practice; and, since he will be scarcely able, unaided, to discern any connection between this task and his university mathematics, he will soon fell in with the time honoured way of teaching, and his university studies remain only a more or less pleasant memory which has no influence upon his teaching.

Klein identifies these two phases of "forgetting previous mathematics" as the first and second discontinuity. The first is a general problem of transition from school to university that has been the subject of much (if not most!) research on university mathematics education, given the struggles in which many students σ find themselves in (irrespectively of future orientation). The second discontinuity, from university to school, is specific to σ t , or rather to the passage

R U (σ t ,ω) → R S (t,o)
where S is naturally the school institution and t the position as teacher and where o designates school praxeologies. The change of praxeologies correspond to the fact that what Klein calls "traditional elementary mathematics according to school practice" is at best somehow related to mathematical praxeologies ω met at U, "his university mathematics". In fact, Klein also emphasizes that the position t requires not only a relation to o but also the "task" to "teach" it "according to school practice"; a more accurate representation of this passage is thus

R U (σ t ,ω) → R S (t, Π(o)).
If we represent the full story of Klein's unfortunate character, who starts out in the position as school student s, we then get

R S (s,o) → R U (σ t ,ω) → R S (t, Π(o))
under the assumption that o, "elementary mathematics according to school practice" does not change too much while our friend is at university (which may, in fact, be somewhat incorrect in times of curriculum change).

The point of Klein's book is that universities need to take more responsibility when it comes to enrich R U (σ t ,ω) with explicit links between ω and o. In fact, most of the text consists of revisiting elements of o -especially within the domains of arithmetic, analysis and geometry -from the "higher standpoint" (as the title says) of ω. The text, indeed, resulted from Klein's own lectures to future teachers during the preceding decades, following his inauguration as professor at the University of Erlangen in 1872.

We could represent this effort as an attempt to "smoothen" the second discontinuity by adding a relation to be developed (cf. Barquero and Winsløw, in preparation):

R S (s,o) → R U (σ t ,ω) → R U (σ t , ω∪o) → R S (t, Π(o))
where the subject matter of the "Klein course" is naturally not supposed to be a disjoint union ω∪o, but to emphasize links and overlaps.

The emergence of Didactics of Mathematics (or mathematics education research, in Anglophone countries) as a scientific discipline, both results and departs from this project, particularly from the sixties onwards. On the one hand, Klein type courses were established at many universities (in Germany, often specialized in domains, labeled Didactics of Analysis and so on; in USA as so-called "capstone courses" which are also offered at the end of several other professional university degrees). Still, the last passage R U (σ t , ω∪o) → R S (t, Π(o)), may remain somewhat discontinuous, given that Π(o) is more than o. In many countries, official systems of "induction" into the teaching métier are offered (see e.g. [START_REF] Britton | Comprehensive teacher induction[END_REF] to take care of the passage to the praxis Π(o), with more or (often) less attention to the specificity of o. This, in fact, means, that yet another relationship is added to smoothen the second discontinuity, between R U (σ t , ω∪o) and R S (t, Π(o)). This may involve both university course units, given by specialists of Didactics of Mathematics or Pedagogy, who may introduce more or less subject specific elements of logos Λ(o) related to elements of practice Π(o) in school. One could then pose the complete model of mathematics teacher education that exists today, with local variations (such as leaving out entire relations aimed at):

R S (t,o) → R U (σ t ,ω) → R U (σ t , ω∪o) → R U/S (σ t /t, (Λ(o),Π(o))) → R S (t, Π(o)).
More can be said about this last extension, and especially of the frequent absence of logos in the last relationship (see [START_REF] Miyakawa | Paradidactic infrastructure for sharing and documenting mathematics teacher knowledge: a case study of "practice research" in Japan[END_REF]. However, from the point of university mathematics education, which is assumed in this paper, the second passage R U (σ t ,ω) → R U (σ t , ω∪o) is of special interest, as it concerns university teaching of mathematics. It is still important to bear in mind that this passage is very often followed by training more directly related to Π(o).

In this vein, let us first recall the considerable body of research which, beginning with [START_REF] Begle | Teacher knowledge and student achievement in algebra[END_REF] first demonstrations that R U (σ t ,ω) → R S (t, Π(o)) does not succeed better (in terms of performance of the students of t) simply because ω (measured as numbers of advanced courses taken) was larger. Later studies refined his results and nuanced the view both from mere volume to a closer look at contents. Without going into details that are better explained elsewhere, the following recommendation seems still to be of current, consensual value, at least in the United States:

Prospective high school teachers of mathematics should be required to complete the equivalent of an undergraduate major in mathematics that include three courses with a primary focus on high school mathematics from an advanced viewpoint (CBMS, 2012, p. 18).

We are thus faced essentially with the proposal of Klein, when it comes to the university responsibility to prepare R S (t, Π(o)), in the case of upper secondary school S: establish R U (σ t , ω∪o) with a "primary focus" on o, but linking it to the "advanced standpoint" of ω. It is an important challenge for the university mathematics métier to identify what R U (σ t , ω∪o) could best function as stepping stone towards R S (t, Π(o)), and to implement didactical practices that can establish such R U (σ t , ω∪o). The complexity of this task is evident, and probably more acute that in the time of Klein, where very small minorities reached the position s.

To solve this task evidently requires a teacher relation R U (τ, Π(ω∪o)) which is not immediately derived from R U (τ,ω), although it also involves this relation. But in addition to that, to design Π(ω∪o), requires a relation R U (τ, R S (t, Π(o))), where the complexity is even more evident. Such expertise, on the other hand, is in principle held by the faction of the university mathematics métier who engage in empirical research not only on mathematics teaching in secondary school, but also on secondary mathematics teacher knowledge. This field, of course, is currently under development, and is only slowly getting specialized enough to capture specific praxeologies o. At any rate, we can summarize this theoretical discussion by agreeing that devising and adjusting courses (three, perhaps?) is an excellent opportunity to combine and mix expertise from both teaching and research in mathematics including but also beyond the classical domains of mathematics. We return to this in Section 6.

As an example of evidence from newer "Klein type" courses, [START_REF] Winsløw | Klein's double discontinuity revisited: Contemporary challenges for Universities preparing teacher to teach Calculus[END_REF] conducted an analysis of student challenges in such a course at the University of Copenhagen, based on praxeological analysis along the lines outlined above. One of the striking observations was that even R U (σ t ,ω) may have to be developed in such a course. Within the same context, Barquero et al. (in preparation) will delve further into specific challenges when it comes to students' perception on and challenges with praxeologies ω∪o related to the real number system.

The education of future school mathematics teachers may be of special interest to scholars in university mathematics education, as they are often also teacher educators.

However, when it comes to the métier of university mathematics teaching -our object of research -there are other target métiers, if not professions, which are equally important, if not more so. We now turn to a major example: future engineers.

Authentic Problems of Engineering in first year mathematics

As for teacher education, the role of mathematics in engineering education has been the subject of numerous policy papers. Naturally engineering programs include different specialties and academic levels, whose mathematical needs vary significantly. An engineering student σ e encounters mathematical praxis and logos in many different settings of the university study, but "mathematics courses" (given by members of the university mathematics métier) appears mostly in the first year or two of undergraduate studies. Whatever mathematical praxeologies ω that σ e studies then, the aim for R U (σ e ,ω) is to prepare and facilitate the establishment of relationships of type R U (σ e ,ε) where ε is some praxeology from engineering courses at large, in which mathematical practices or logos related to those of ω appear.

A main problem for university mathematics education in this context is that the transition (or knowledge transfer) from R U (σ e ,ω) to R U (σ e ,ε) is not automatic, even when ω and ε are actually bridged by the expert (or teacher) of both. For instance, in the context of a signal theory course, Hochmuth, Biehler and Schreiber (2015) investigated specific ruptures between the mathematical model of "Dirac impulse" treated (and calculated with) in this course, and the technology associated to functions, limits, and distributions in mathematics courses. The techniques required to solve associated problems in the signal theory course, which involve operating with functions that assume the value ∞ at isolated points, "do not fit with higher mathematics discourses (technologies)". The students have somehow to learn that they should neglect specific aspects from those discourses" (ibid., p. 696).

Another, related problem, concern the specialized métier τ e of teaching mathematics to students in position σ e . The classical solution is that the mathematical praxeologies ω to be taught are simply some subset of what is taught to σ 1 (including extensive work on formal logos with proofs etc.). While this model still exists in some countries, it seems to disappear in many places due to problems with students' motivation, attrition and transfer [START_REF] Pohjolainen | Modern Mathematics Education for Engineering Curricula in Europe[END_REF]. Hernandes-Gomes & González-Martín (2016, 2020) found that teachers' relationship R U (τ e ,ω) with basic calculus topics depended significantly on their scholarly background (which included pure mathematics, mathematics education, mechanical engineering and electrical engineering), and on the teachers' corresponding experiences as undergraduate students. For instance, only the university mathematics teachers with an engineering background had precise ideas about how specific mathematical techniques appear (or do not appear) in the engineering program. In the context of supervising capstone projects, professional experience from engineering institutions outside of the university is also of considerable importance, even when it comes to the ways in which teachers assist students with mathematical techniques. These case studies mainly suggest that different institutional backgrounds offer somewhat different qualities to the position of τ e .

To improve students' motivation to develop, apart from exam requirements, and also to prepare the transfer of type R U (σ e ,ω) → R U (σ e ,ε), it is an interesting strategy to integrate some concrete tasks from ε in the mathematics course, which can be used to show the relevance to engineering of techniques and logos from ω. In his doctoral thesis, [START_REF] Wolf | Anwendungsorientierte Aufgaben für Mathematikveranstaltungen der Ingenieurstudiengänge: Konzeptgeleitete Entwicklung und Erprobung am Beispiel des Maschinenbaustudiengangs im ersten Studienjahr[END_REF] carried out an ambitious project on designing and experimenting application-oriented exercises in a first-year mathematics course for students of mechanical engineering. The applications were "authentic" in the sense that problems and data were taken from professional contexts of machine construction. The authenticity was ensured by collaboration with university teachers of engineering.

Schmidt and Winsløw (to appear) investigate a similar, but more longitudinal and entirely spontaneous collaboration pattern, focused on designing authentic problems of engineering assignments for a first-year mathematics course with more than 1100 students every year. They describe the explicit principles that have developed, through practice but also from leadership in the position τ e to facilitate collaboration with scholars of engineering fields, who often produce a first draft of the assignment, which is subsequently revised and implemented by (mostly) mathematics faculty in position τ e . The assignments appear in a first-year course on calculus, linear algebra and differential equations, and the main challenges for students are thus to be of a mathematical nature, combining several praxeologies taught in the course. Still, the mathematical model is built up from an authentic problem of engineering. Here, "authentic" in that it comes from recent publication in scholarly engineering. To organize the systematic collaboration with institutions of scholarly engineering, informal didactical logos was developed from the position τ e , adding a professional trait to this specific form of the university mathematics métier.

A PROPOSAL FOR THE FUTURE

In the preceding two sections, we have analyzed instances of the variety of positions and knowledge bases that the university mathematics métier is currently based on. Both scholarly knowledge, coming from engaging in research praxeologies, and knowledge built from didactical praxis, contribute to this knowledge basis. In some cases, didactical practice is supplied with more or less strongly developed logos, which often takes on relatively generic forms. Experiments initiated by scholars specializing in university mathematics education research appear to be mostly punctual, while we have also identified instances of more sustained and explicitly framed efforts to develop the métier from positions τ 1 and τ 2 . As is the case for mathematics teaching in other institutions, the development of professional -shared, explicit and practice-specific knowledge -remains quite limited and local. Professional journals focusing on university mathematics teaching do exist in some countries like the USA, but even then, there seems to be a considerable distance, in terms of logos and readership, between these and scholarly publications in the field of university mathematics education research (such as the present volume).

Indeed, the university mathematics teaching métier remains, to a large extent, a secondary occupation of various professions of scholarly research in the mathematical sciences, interesting new forms of collaborations emerge especially in the position of type τ 2 . Still the formal preparation for occupying such positions seems to be mainly pedagogical, as a complement to the more substantial training for a scholarly profession. Reactions to external reform requirements often take the form of more or less minimal reconstruction of external didactical transpositions at least when it comes to undergraduate programs in pure mathematics (Bosch at al., to appear). In the parts of the métier catering to students in positions σ 2 , more significant developments appear, while in all cases, massive challenges with attrition and failure remain evident and perhaps even growing in many universities.

As a result, the impact problem for research in university mathematics education is pointed out in several recent syntheses (e.g. Winsløw, Gueudet, Hochmuth and Nardi, 2018, p. 71). The current institutional model separates, largely, such research from the university mathematics teaching métier. To seek impact of an external scholarly field on a métier of teaching implies two risks that are very well known from the teaching métiers at primary and secondary levels in many countries. The first is to continue to fail. The second is to succeed, at least to some extent, but to have merely "robbed teachers of the opportunity to participate in the development of new knowledge about teaching" (Stigler et al., 1999, p. 174).

To avoid these risks, a new nexus between teaching and research seems necessary. Of course, various blends of scholars and teachers have appeared spontaneously under current institutional conditions, as has now and then also appeared in some of the efforts outlined here. However, the vast majority of the professional training of PhDs in mathematical sciences remains totally disjoint from the preparation of PhDs in Didactics of Mathematics, including those focusing on university mathematics education. Why would the latter not include some level of further mathematical education and experience with research in some mathematical domain? And why not include elements of education and research in the didactical domain? Should future teachers of university mathematics not be prepared to engage in (rather than, hypothetically, be mute consumers of) didactical research on university mathematics? What are the institutional and intellectual conditions under which it would be realistic to establish mixed doctoral programs (of various compositions) that could prepare for shared and fruitful professional development of the métier in all its different forms?