Chao Fang

Piao Dong

Yiping Fang
email: yiping.fang@centralesupelec.fr

Enrico Zio

Vulnerability Analysis of Critical Infrastructure under Disruptions: an Application to China Railway High-Speed

Keywords: vulnerability analysis, disruption, critical infrastructures, Monte Carlo simulation, spatially localized failures

Considerable attention has been paid to the vulnerability of critical infrastructures, because of the increasing occurrence of disruptive events, such as man-made or natural disasters. Even small disruptions could eventually affect the normal function of infrastructure systems. Enhancing the reliability of these systems and their robustness to disruptions is necessary and urgent. High-speed rail is a critical infrastructure that is subject to various disruptions, including component aging, malicious attacks, natural disasters, demand surges.

In this study, we analyze the topological centrality indicators of China Railway High-speed (CRH) network using network theory, and take real train flow information for assessing the importance of network components in terms of vulnerability to disruption. By Monte Carlo simulation, we analyze the risk of the CRH network under random attacks and spatially localized failures. The significance of taking pre-actions for protecting critical infrastructures by mitigating its vulnerability to disruptions is emphasized.

Introduction

Critical infrastructures are large dynamic systems made of interconnected components and sub-systems. They provide products and services, such as transportation, energy, communication, water supply, finance, and public health. Such critical infrastructures are vital to the normal functioning of society, and their failure can seriously affect public safety, economic development and social stability [START_REF] Murray | Critical infrastructure protection: The vulnerability conundrum[END_REF] . In recent years, critical infrastructures are facing increasing risks and security threats, e.g., operation failures, natural disaster, and terrorist attacks. Examples include the Deepwater Horizon oil spill, the accident at Fukushima nuclear power plant, the Haiti earthquake, and the Puerto Rico hurricane [START_REF] Park | Integrating risk and resilience approaches to catastrophe management in engineering systems[END_REF] . The vulnerability analysis of critical infrastructures has recently become of particular interest and many studies have been conducted on this subject. For example, some researchers [START_REF] Pursiainen | The Challenges for European Critical Infrastructure Protection[END_REF][START_REF] Yusta | Methodologies and applications for critical infrastructure protection: State-of-the-art[END_REF][START_REF] Marrone | Vulnerability modeling and analysis for critical infrastructure protection applications[END_REF][START_REF] Fang | Resilience-Based Component Importance Measures for Critical Infrastructure Network Systems[END_REF] discussed the protection of critical infrastructures and related challenges, of which Marrone et al. [START_REF] Marrone | Vulnerability modeling and analysis for critical infrastructure protection applications[END_REF] introduced a systemic view for the practical protection of critical infrastructures and demonstrated its effectiveness with an application to industrial settings. From the perspective of methodologies and applications, Zio [START_REF] Zio | Challenges in the vulnerability and risk analysis of critical infrastructures[END_REF] presented a vulnerability framework and indicated the importance of integrating different modeling approaches facing the dynamic complexities of critical infrastructure. Johansson et al. [START_REF] Johansson | Reliability and vulnerability analyses of critical infrastructures: Comparing two approaches in the context of power systems[END_REF] and Ouyang [START_REF] Ouyang | Review on modeling and simulation of interdependent critical infrastructure systems[END_REF] offered technical reviews of approaches for the reliability and vulnerability analysis of critical infrastructures.

Transport infrastructures, usually distributed on geographical extensions, are collections of many interacting man-made components assembled by design, for moving goods and people from one place to another. It can be termed critical infrastructure if its incapacity has a significant impact on health, safety, security, economics and social well-being [START_REF] Zio | Challenges in the vulnerability and risk analysis of critical infrastructures[END_REF] . The growing vulnerabilities of transportation systems force decision makers to prioritize their protection. In this respect, some studies [START_REF] Berdica | An introduction to road vulnerability: what has been done, is done and should be done[END_REF][START_REF] Jenelius | Importance and exposure in road network vulnerability analysis[END_REF][START_REF] Jenelius | Network structure and travel patterns: explaining the geographical disparities of road network vulnerability[END_REF] discussed the vulnerability analysis of road transportation systems with indicators used in the modeling that can be roughly divided into demand and supply. By using exhaustive search optimization and fuzzy logic, El-Rashidy and Grant-Muller [START_REF] El-Rashidy | An assessment method for highway network vulnerability[END_REF] presented a model that estimated the vulnerability changes of the highway network under different situations. Li and Cai [START_REF] Li | Empirical analysis of a scale-free railway network in China[END_REF] focused on the Chinese railway weighted network and found that the network has small-world and scale-free characteristics. Johansson and Hassel [START_REF] Johansson | An approach for modelling interdependent infrastructures in the context of vulnerability analysis[END_REF] modeled an electrified railway network, explored its five subsystems and analyzed the interdependencies among them for vulnerability assessment. Ouyang et al. [START_REF] Ouyang | Vulnerability analysis of complementary transportation systems with applications to railway and airline systems in China[END_REF] systematically analyzed the positive effect of interdependencies in complementary systems and proposed a network-based approach to model the vulnerability of transportation systems.

Bocchini et al. [START_REF] Bocchini | Resilience and sustainability of civil infrastructure: Toward a unified approach[END_REF] proposed a comprehensive method for assessing the vulnerability of a railway system under floods, and compared four vulnerability mitigation and maintenance strategies.

In general, a large number of the existing studies on vulnerability and reliability analysis of railway transportation systems have focused on rescheduling or reassignment of train services, following disruptions [START_REF] Chang | Measuring post-disaster transportation system performance: the 1995 Kobe earthquake in comparative perspective[END_REF][START_REF] Lee | Restoration of services in interdependent infrastructure systems: A network flows approach[END_REF][START_REF] Nurre | Restoring infrastructure systems: An integrated network design and scheduling (INDS) problem[END_REF][START_REF] Cavdaroglu | Integrating restoration and scheduling decisions for disrupted interdependent infrastructure systems[END_REF][START_REF] Zhu | Railway timetable rescheduling with flexible stopping and flexible short-turning during disruptions[END_REF] . In terms of economic efficiency and social function, preparedness to and mitigation of the disruption risks to decrease vulnerability is also considerably important. Furthermore, the performance metrics used in existing research are often topological, with only few exceptions considering railway flow [START_REF] Ii | Restoration of Services in Interdependent Infrastructure Systems: A Network Flows Approach[END_REF][START_REF] Jenelius | Road network vulnerability analysis of area-covering disruptions: A grid-based approach with case study[END_REF][START_REF] Wang | Vulnerability analysis of interdependent infrastructure systems: A methodological framework[END_REF][START_REF] Nicholson | Flow-based vulnerability measures for network component importance: Experimentation with preparedness planning[END_REF] . However, these studies usually refer to the number of train service lines affected as system performance indicator, which is insufficient because the importance of each service line is not the same, especially for passenger trains.

High-speed rail is an important part of a rail transportation system, the development of which has become a global trend since 1964 when the world's first high-speed railway, Japan's Shinkansen, opened to traffic. Regional integration, urban and regional agglomeration, and increasing labor productivity are key factors that push the emergence of high-speed rail worldwide [START_REF] Chen | A Sustainability Analysis on the Wuhan-Guangzhou High-Speed Railway in China[END_REF] . Given its characteristics of punctuality, comfort, energy efficiency and reduced pollution, high-speed rail has obtained wide support, leading to "the second railway age" [START_REF] Banister | High-Speed Rail in the EU27: Trends, Time, Accessibility and Principles[END_REF] .

With the importance and influence of high-speed rail worldwide, the ability to predict, withstand, and mitigate disruptions has raised considerable attention from managers and researchers. Given that high-speed rail provides services for mass passengers, the loss caused by a disruption is considerably more serious than that of freight trains. High-speed rail is also more sensitive to time, given the competition from civil aviation. The high-speed rail system is a crucial yet sensitive infrastructure of a nation's transportation systems and even a small disruption may lead to significant adverse effects not only on the economy but also on public safety.

Official data indicates that the railway mileage of China reached 124,000 km by 2017, of which high-speed rail operating mileage exceeded 22,000 km, accounting for more than 60% of the world's high-speed rail [START_REF]China's yearbook on transportation and communications[END_REF] . From a practical perspective, some researchers [START_REF] Banister | High-Speed Rail in the EU27: Trends, Time, Accessibility and Principles[END_REF][START_REF] Cao | Accessibility impacts of China's high-speed rail network[END_REF][START_REF] Jiao | Impacts on accessibility of China's present and future HSR network[END_REF][START_REF] Rongfang | The Role of State: High Speed Rail Development in China[END_REF] considered the characteristics and future development of the China Railway High-speed (officially named CRH). However, the analysis of CRH's vulnerability with quantitative metrics has yet to be conducted. In the present study, vulnerability is considered as decline in system performance. We analyze not only the system vulnerability of CRH using topological centrality indicators, but also consider the train flow information in operation to assess the importance of network components in terms of vulnerability to disruption. To indicate the difference in the importance of different trips, the mileage of train service line in operation is used in our flow-based vulnerability analysis. The application to the CRH network shows how these analyses can provide complementary information and knowledge of the vulnerability caused by disruptions and dynamic flows in railway systems. We use the Monte Carlo method and spatially localized failures (SLFs) [START_REF] Ouyang | Critical location identification and vulnerability analysis of interdependent infrastructure systems under spatially localized attacks[END_REF][START_REF] Ouyang | A Mathematical Framework to Optimize Resilience of Interdependent Critical Infrastructure Systems under Spatially Localized Attacks[END_REF] to simulate disruptions, like random attacks and natural disasters. The insights from this study can assist local and central government agencies in making decisions to enhance and protect CRH networks with respect to vulnerability mitigation.

The remainder of this paper is organized as follows. In Section 2, we introduce a network modeling of CRH, in which its topological characteristics are analyzed to gain a general structural understanding with regard to vulnerability. Section 3 presents a vulnerability analysis considering dynamic flows in the network. In Section 4, we simulate spatially localized failures using the Monte Carlo method and illustrate the vulnerability intensity chart of the CRH. Further managerial discussion is given in Section 5 and conclusions are drawn in Section 6. properties are accounted for in the links in the network. Finally, based on the CRH operation map [START_REF] Gaotie | High-speed rail network[END_REF] , we abstract it into a relatively simplified network diagram (see Figure 1), which

Notations

Topological analysis

Numerous studies have been conducted on the complexity characteristics and vulnerability of railway transport systems. These studies found that the vulnerability with regard to many types of disruptions in the system are closely related to the structural properties of the networks [START_REF] Bocchini | Resilience and sustainability of civil infrastructure: Toward a unified approach[END_REF][START_REF] Macchia | Maintenance management of railway infrastructures based on reliability analysis[END_REF][START_REF] Peeta | Pre-disaster investment decisions for strengthening a highway network[END_REF] . In this regard, we tailor some network indicators of betweenness centrality to characterize the topological properties of the CRH network with respect to vulnerability.

Betweenness includes node and link betweenness. It reflects the global connection importance of the components in the entire network. Here we consider the betweenness centrality based on shortest paths because in practice, the network design and route planning usually consider the shortest path as an important reference. The value of node betweenness is determined by the proportion of the number of paths that pass through this node in all the shortest paths. By defining 𝑛 𝑢𝑣 as the number of shortest paths that connect 𝑛 𝑢 and 𝑛 𝑣 , 𝑛 𝑢𝑣 (𝑖) as the number of shortest paths connecting 𝑛 𝑢 and 𝑛 𝑣 as well as passing through 𝑛 𝑖 , the betweenness of 𝑛 𝑖 is calculated by:

𝐵 𝑖 = ∑ 𝑛 𝑢𝑣 (𝑖) 𝑛 𝑢𝑣 𝑢,𝑣∈𝑁,𝑢≠𝑣 (1)
The calculation of a link betweenness is similar, where 𝑛 𝑢𝑣 (𝑖𝑗) represents the number of shortest paths connecting 𝑛 𝑢 and 𝑛 𝑣 , as well as passing through 𝑙 𝑖𝑗 :

𝐵 𝑖𝑗 = ∑ 𝑛 𝑢𝑣 (𝑖𝑗) 𝑛 𝑢𝑣 𝑢,𝑣∈𝑁,𝑢≠𝑣 (2)
Table 1 shows the ranking of the top-ten nodes and links based on betweenness centrality.

The first three columns indicate that Zhengzhou, Wuhan, and Changsha Stations are the top three nodes that possess the highest betweenness. These nodes have important topological positions and include many paths passing by. Figure 1 shows that these stations are central hubs in the network running north and south. For example, although the degree of Wuhan Station node is 3, it remains necessary for the connection of the north and south, due to its location in the central area of the network. Accordingly, the hub stations with high betweenness centrality are usually more sensitive to disruptions and once breakdown occurs, the entire network can be greatly affected. Table 1 shows also the top ten links with highest betweenness. The railway track from Wuhan to Changsha ranks premier. This is consistent with the results of node's betweenness, because these two nodes are in the central area of the network and each node has considerable betweenness value. These nodes and links are important in terms of topology and vulnerability, and hence mitigation and preparedness strategies should be considered in strengthening the components to resist potential disruptions.

Flow-based vulnerability analysis under disruptions

The impact caused by a disruption is mainly in the failure of services provided by the infrastructure, thereby causing inconvenience to people's lives and losses of finance. Different critical infrastructure systems have different physical flow characteristics of operation, such as electric flow in power systems [START_REF] Fang | Optimization of Cascade-Resilient Electrical Infrastructures and its Validation by Power Flow Modeling[END_REF][START_REF] Fang | Comparing Network-Centric and Power Flow Models for the Optimal Allocation of Link Capacities in a Cascade-Resilient Power Transmission Network[END_REF] , train services in railway transportation systems, and information flow in telecommunication systems. With regard to CRH, this study considers the flow of trains. Several train service lines are run back and forth to ensure people's travel, with duration of round-trips often less than 24 hours thanks to the high-speed. We have obtained the information of all train services from the website of China Railway Customer Service Center ("12306.cn"). We have selected all the high-speed trains that contain the prefix "G" for "GaoTie" (high-speed rail) and obtain 2,324 CRH train service lines in operation in a normal weekday. Given that the operating mileage of each train service line is not the same, its importance cannot be generalized when analyzing network vulnerability. Therefore, the train flow in this study takes into account the mileage of each train services, rather than simply the number of train service lines in operation.

By removing components in sequence from the network, we study the importance of each component separately. The vulnerability is defined here as the drop in system performance:

thus, the importance of a component is measured by the effect on the proportion of the mileages of train service affected by the component's failure (removal). Most often, the current duration of a disruption in CRH is no more than one day. Thus, we assume that the failure of the components can be recovered within 24 hours and network congestion is not taken into account here. Furthermore, some other assumptions simplify the calculation, such as the facility failures being binary (completely operational or completely failed).

From the topological analysis of Section 2.2, we gain an interpretation of the components' importance in CRH. To learn more about the vulnerability of the infrastructure network, we introduce performance metrics related to the flow of train services. Here we use the performance metric of the network connectivity efficiency. Connectivity efficiency is calculated by the average geodesic distance (𝑔 𝑖𝑗 ) between two nodes belonging to the network. The distance is geodesic, which we consider equal to the railway track length determined by the train services between two components. Thus, this metric can reflect the connectivity efficiency of the network from a realistic point of view. The equation to calculate connectivity efficiency is:

𝐸 = 1 𝑁(𝑁-1) ∑ 1 𝑔 𝑖𝑗 𝑖≠𝑗 . (3)
We indicate the connectivity efficiency of the network in the normal state as E. If a disruption on 𝑛 𝑖 occurs, which invalidates 𝑛 𝑖 , then 𝐸 𝑖 ′ should be defined by the network efficiency excluding 𝑛 𝑖 . Therefore, the efficiency-based vulnerability under the event is calculated as follows:

𝑉 𝑖 𝐸 = 𝐸 -𝐸 𝑖 ′ 𝐸 . (4)
On the basis of the values of 𝑉 𝑖 𝐸 , the ten most important railway stations of CRH are listed in Table 2. From the We also model the vulnerability regarding individual components from the perspective of affected operating mileage. If we assume a disruption on one node i, then the effect caused by the failure of this node should be quantified by the affected trains, as well as the affected cumulative travel mileages, denoted by 𝐶𝑇𝑀 𝑖 . TTM indicates the total travel mileage of all train services within network. Accordingly, the vulnerability of node i considering train flow can be expressed as follows:

𝑉 𝑖 𝐹 = 𝐶𝑇𝑀 𝑖 𝑇𝑇𝑀 . (5)
From the practical point of view, we assume that if the start/end station of a train service line is interrupted, then the entire line is affected. That is the common situation because the train cannot be prepared to depart normally or reach its destination for replenishment and maintenance for the return trip. Meanwhile, by field investigation and interviewing experts in CRH, we acknowledge that the interruption of intermediate stations along the line can possibly not affect the operation of the whole train service. That is because the train can choose not to stop at the disrupted station while moving on to the nearby dockable yard, or running across station in emergencies (especially for large stations having many linking routes). The train also has the option of returning in the network to ensure only a part of the original travel service.

Based on these assumptions, in this study, train services affected by the failures of stations can be divided into two types. When the start/end station 𝑛 𝑖 of the service line is disrupted, the impact caused by this failure of 𝑛 𝑖 should include the total travel mileages of these trains, as denoted by 𝑇𝑀 𝑖 𝑡 . The other situation is that the failure node is neither the starting point nor the terminal station of the train, where the impact is defined in our study by the travel mileages between this node and the adjacent stations of the affected trains, denoted by 𝐴𝑇𝑀 𝑖 𝑡 . Denote T as the total number of all trains, then we can calculate 𝐶𝑇𝑀 𝑖 by:

𝐶𝑇𝑀 𝑖 = ∑ 𝑇𝑀 𝑖 𝑡 𝑇 𝑡=1
(where 𝑛 𝑖 is start/end node for train 𝑡) + ∑ 𝐴𝑇𝑀 𝑖 𝑡 (otherwise ) 𝑇 𝑡=1 (6) For the case of link failure, as a link has absorbed a number of small yards not shown in the network structure, we assume that if the link between two adjacent stations is disrupted, then any train passing through it will be affected. Thus, the vulnerability of 𝑙 𝑖𝑗 should be quantified by the affected travel mileages for the proportion of the total travel mileages.

𝑉 𝑖𝑗 𝐹 = 𝐶𝑇𝑀 𝑖𝑗 𝑇𝑇𝑀 (7)
As illustrated in Table 3, the top-ten vulnerable stations are different from the previous topological rankings in Tables 1 and2. The first three columns of Table 3 indicate that Shanghai, Beijing, and Guangzhou Stations are the three most important stations. This can be easily explained by the fact that Beijing is the junction of the Beijing-Shanghai, Kazakhstan-Beijing, Beijing-Hong Kong, and the Jingkun channels. Meanwhile, Shanghai is the junction of Beijing-Shanghai channel, the coastal channel, and the channel along the Yangtze River.

Guangzhou has become a three main line intersection hub of the Kazakhstan-Beijing, Lan-Guang, and Guang-Kun channels. Once a disruption occurs in any of these three busiest stations, a greater effect will be incurred on the entire CRH system. The link failure analysis results reveal that the links connecting adjacent important nodes are also highly important in terms of vulnerability taking into account train flow, such as Shanghai--Nanjing track. In addition, some hub links with high topological betweenness centrality are also vulnerable, such as Wuhan-Changsha, Wuhan-Zhengzhou, that bear a lot of traffic flow in CRH.

Figure 2 is a visual representation of the results with the geographical location. We highlighted in the map the top-five nodes and top-ten links, whose failure can seriously affect the CRH operation flow. We can see that most top vulnerable nodes are connected by the top important links. Beijing, the capital, is ranked No.2 vulnerable station because of its numerous departure/terminal train services; however, the nodes and links connecting to Beijing are not ranked as high as expected because of its Siphon Effect: that is to say, the neighbor city stations around Beijing have been weakened in terms of their capacity of train flow in practice.

Simulation analysis of SLFs using Monte Carlo

As one of the most rapidly growing critical infrastructures in China, the CRH system plays an important role in ensuring economic development and social stability. However, at present, the system faces various types of disruption, including natural disasters, malicious attacks, and random disruptions. Such disruptions are generally highly uncertain events that are difficult to predict, estimate and model, thereby posing significant threats to the continuity of the system's operation. In Section 3, we conducted a train flow-based vulnerability analysis with respect to each individual component in the CRH network. Although the assumption of disruption on a specific component can simplify the calculations, it might not be sufficiently representative when considering certain types of disruptions that affect a whole area of the system. In these cases, the analysis of SLFs is necessary to fully understand the risk and vulnerability of the CRH network.

In reality, geological and hydrological disasters are two serious natural disasters that frequently disturb the CRH network. These two disasters are mainly considered here as disruption scenarios and the localized area is determined in terms of provinces. We consider 𝑥 𝑠 = {𝑥 1𝑠 , 𝑥 2𝑠 , … , 𝑥 𝑛𝑠 , 𝑥 (𝑛+1)𝑠 , … , 𝑥 (𝑛+𝑚)𝑠 }

More than one disruption can simultaneously occur in the CRH network during operation. Therefore, some scenarios can actually be superposed. Here, we consider a common disruption scenario that involves the superposition of attacks and natural disasters.

Components in the network face a series of initiating events, including various incidents. The specific calculation process is as shown in the flowchart of Figure 4:

Step 1: Define the number of iterations to be simulated; introduce a counter s and set

𝑠 = 1.
Step 2: Determine the provinces based on the 28 stations. From the China Statistical Yearbook from 2005 to 2015, determine the natural disaster probabilities for the 21 provinces and define its corresponding stations' natural disaster probability denoted by 𝑛𝑑𝑝 𝑖 .

Step 3: In the studied component set, randomly select 0%--10% of them and assume that as a consequence of the disruption they cannot function. Correspondingly, 𝑝 𝑖 𝑠 = 1.

Step 4: For each 𝑛𝑑𝑝 𝑖 , generate a uniformly distributed random number ε 𝑖 within [0, 1],

and compare 𝑛𝑑𝑝 𝑖 and ε 𝑖 : when ε 𝑖 is no larger than 𝑛𝑑𝑝 𝑖 , the province is considered to suffer a natural disaster and the stations inside the provinces fail, 𝑝 𝑖 𝑠 = 1; otherwise, 𝑝 𝑖 𝑠 = 0.

Step After the 10,000-scenarios simulation, we can obtain the mean value of 𝑝 𝑖 𝑠 for each node i, which gives the failure frequency of each station node. Taking risk as the product of probability and severity of adverse effects [START_REF] Aven | On some recent definitions and analysis frameworks for risk, vulnerability, and resilience[END_REF][START_REF] Aven | Risk assessment and risk management: Review of recent advances on their foundation[END_REF][START_REF] Aven | Quality of Risk Assessment: Definition and Verification[END_REF][START_REF] Flage | Concerns, Challenges, and Directions of Development for the Issue of Representing Uncertainty in Risk Assessment[END_REF] , we can calculate the risk of each component.

The risk consequence/severity is calculated by the sum of the affected train mileages as discussed in Section 3. The results are shown in Figure 5, in which the areas are filled with varying degrees of color: a darker shade implies greater vulnerability to the high-speed rail operations in that area. The scale value indicates the mileage sum of the CRH affected in that region, in units of 100 km. Given the larger disaster probability, the Hunan Province remains most vulnerable. Another area of equal degree of vulnerability is Guangdong Province, which is also one of the most prosperous areas. As shown in Table 3, within Guangdong Province, Guangzhou and Shenzhen Stations rank both high in terms of flow-based vulnerability.

Besides, the province itself suffers a relatively high frequency of natural disasters.

Discussions

The calculation results of the topological and flow-based vulnerability analyses shown some differences. For example, nodes that are critical for the flow-based vulnerability analysis may not be important in the topological analysis. We take Beijing for a comprehensive example. From Section 2.2, the betweenness of Beijing is only 0.07047, ranking 25th among the 28 stations, which can be explained by the city's geographic position.

However, in reality, Beijing is an important hub, because of the population, economy and also political positions. With the consideration of train mileage, the importance of this station increases greatly and ranks 2nd among the 28 stations. Given that many trains start or end at Beijing Station, if a disruption occurs, these service lines would be affected considerably.

With the consideration of train mileage, Shanghai is the most vulnerable station. However, similar to Beijing Station, this node does not stand out when analyzed from a topological or network efficiency perspective. The same difference can also be found in the importance comparison of links in terms of vulnerability. The most vulnerable link in terms of train flow is Shanghai-Nanjing. However, this link is not important in the topological analysis (only ranks 35th among all 42 links). This phenomenon can be explained by the different meaning of the three metrics: the betweenness centrality emphasizes the components' position and role as a hub in the network topology; the connectivity efficiency looks at the entire network for efficiently transmitting among the nodes and re-routing; the flow-based vulnerability analysis investigates the effect of each component's failure on the operating travel mileages.

With the consideration of spatially localized failures, and corresponding consequences, we analyze the vulnerability distribution on the CRH map, which considers the importance of each district from a comprehensive perspective. The areas with high vulnerability should be prepared and emergency responses are required to recover from the impact. The results of the analysis also provide insights for the investments of planning and maintenance of the CRH system. For example, in the southeast provinces, train stations must be designed to withstand the impact of typhoons; railroad tracks in flood-prone areas may need to be laid on higher terrains or viaducts; railroad tracks in the frozen areas need to be designed for frost protection.

Moreover, decision-makers can rely on the flow-based vulnerability analysis to plan contingency management, such as temporary scheduling and evacuation of passengers.

For the failure consequences shown in Section 3, nodes such as Shanghai and Beijing, indicate a high flow-based vulnerability because many train service lines start or end there. In the event of an interruption, these service lines are disrupted and, thus, seriously affect public safety, economic development and social stability. Managers often consider investment to reinforce the stations and increase train services in these important but already overburdened stations, while not realizing that it may result in high vulnerability. In terms of diversion, constructing two or more stations in prosperous and vulnerable cities is becoming common and convenient to reduce the risk of the entire CRH network. The failure probability of a train service line will disperse, that is, even when one of the stations breakdowns, the others can continue to operate as a substitute. Although many cities now have multiple stations, most of them are not adequate for the operation of high-speed trains. As shown in Figure 6, we select five nodes with multiple stations and calculate their train flow-based vulnerability separately.

The comparison indicates that the vulnerability of different stations in one city varies significantly. This is mainly due to the uneven distribution of high-speed train services for various reasons. For example, in Section 3 we found that Shanghai is an important site whose failure can seriously affect the operation of CRH. Calculating the vulnerability of each station in Shanghai separately, Shanghai Hongqiao Station accounts for a large proportion of more than 87% in terms of disturbed cumulative train mileage. Other than scattering the high-speed passenger train services and balancing the vulnerability, a solution of constructing a comprehensive hub is adopted in Shanghai, called Hongqiao Transportation Hub. Hongqiao integrated transport hub is an innovation in urban transport construction, which includes a variety of modes of transport such as aviation, high-speed railway, maglev, freeway and subway combined together. In this manner, passengers can be quickly evacuated to another system both for transit service, and for facing disruptions or emergencies. For example, travelers can decide to take airplanes when the high-speed rail services are unavailable in the same hub. Furthermore, the vulnerability of interdependent transportation networks will be more complicated and involve more factors.

Conclusion

In this study, we propose a comprehensive vulnerability analysis of CRH, which includes topological and train flow-based vulnerability analysis. Given the different importance of each train service line, the vulnerability defined here is the sum of affected train mileage, rather than the number of affected trains as done in traditional research. An SLF-induced vulnerability simulation is conducted using the Monte Carlo method. First-tier cities, such as Beijing, Shanghai, and Guangzhou, are vulnerable under disruptions due to their prosperous population and large number of high-speed passenger trains that depart from/arrive at the stations. Once they are interrupted, the service lines that include these stations will be affected considerably, thereby causing unexpected losses. Therefore, strengthening of these vital stations should be given attention. Moreover, dividing these start/end sites is feasible, such that the probability of failure in this line can decrease considerably. That is, if we construct two or more high-speed rail stations in these important cities, even if one of the stations breaks down, the others can continue to operate as a substitute. Another emerging solution of constructing transportation hubs integrating high-speed rail, aviation and other transportation systems is currently developed in various cities, after the success of Shanghai Hongqiao hub. From a managerial perspective regarding vulnerability, the traffic flow of nodes in the middle of service lines can be increased by enabling some trains to start from or end at these second-or third-tier cities. The current trend tends to increase the number of vehicles on stations that are already highly important and busy.

However, an optimal balance should be reached between the operational gains and vulnerability mitigation. As CRH has an obvious regional disaster characteristic according to the risk distribution map, specific defensive efforts and contingency response should be designed against disruptions.

In the future work, uncertainty propagation and control regarding vulnerability will be studied taking into account restoration and timetable rescheduling of trains under disruptions.

The proposed approach can also be extended to other critical transport infrastructures.

Furthermore, the cascading and complementary effects of interdependent transportation networks under disruptions will be considered.

  contains 28 nodes (stations) and 42 links (railway tracks). The text in the Figure represents the city name, such as BJ for Beijing and SH for Shanghai.

Figure 1 .

 1 Figure 1. Simplified network representation of CRH

  21 provinces including all the chosen stations in this study. Based on the data obtained from China Statistical Yearbook[START_REF]China's yearbook on transportation and communications[END_REF] , we consider the geological and hydrological disasters occurred in these provinces during the period from 2005 to 2015. We obtain the natural disaster probability 𝑛𝑑𝑝 ∈ [0, 1] based on the statistical mean value for each province per year. As shown in Figure3, a darker area implies more frequent geological and hydrological disasters occurred. The most serious area for disasters is Hunan Province (including Changsha and Hengyang Stations), which is subject to frequent hydrological and geological disasters. Statistics 29 indicate that 2,140 geological disasters occurred in Hunan Province in 2015, which caused direct economic losses as much as 878 million RMB. The areas suffering less serious natural disasters are those around Hunan Province, as shown in bright color. Some areas with lighter color are located in the plains or have small precipitation.

Figure 3 .

 3 Figure 3. Area distribution of geological and hydrological disasters in China

5 :

 5 Update iteration counter s. If 𝑠 < 10000, then 𝑠 = 𝑠 + 1. Return to Step 3; otherwise, the calculation is finished. Output the mean value of 𝑝 𝑖 𝑠 (each node's disruption probability).

Figure 4 .

 4 Figure 4. Flowchart of the Monte Carlo simulation under SLF

Figure 5 .

 5 Figure 5. Vulnerability of districts by simulation of SLFs in 100 km units

Figure 6 .

 6 Figure 6. Comparison of the flow-based vulnerability in separated stations Combined disruptions sometimes occur in the high-speed rail, which cause more than one component failed in the network simultaneously. Table 4 shows the calculation results under pair-disturbance scenarios. The first three vulnerable combinations in Situation 1 are Beijing with Shanghai, Guangzhou with Shanghai, and Wuhan with Shanghai. All these nodes are also most vulnerable separately, as shown in Table 3 and Figure 2. As Shanghai is the most flow-based vulnerable site, 9 out of top-ten pairs include Shanghai. The combinations in Situations 2 and 3 indicate consistent results, showing that components alone are also important when combined with others, such as in Shanghai Station and links of Wuhan--Changsha and Shanghai--Nanjing. It is noted that once these vulnerable components fail simultaneously, the entire network functioning may fall into chaos, and require significant resources and time to recover.

. Network modeling and topological analysis of CRH 2.1 Network modeling

	T:	total number of all train services, 𝑡 ∈ 𝑇
	TTM:	total travel mileage of all train services within the network
	𝐶𝑇𝑀 𝑖 :	cumulative travel mileage affected by failure of 𝑛 𝑖
	𝐶𝑇𝑀 𝑖𝑗 : cumulative travel mileage affected by failure of 𝑙 𝑖𝑗
	𝐴𝑇𝑀 𝑖 𝑡 : adjacent travel mileage of the train that passes 𝑛 𝑖
	𝑇𝑀 𝑖 𝑡 : travel mileage of the train that starts/ends from 𝑛 𝑖
	𝑇𝑀 𝑖𝑗 𝑡 : travel mileage of the train that includes 𝑙 𝑖𝑗
	2From the perspective of complex network theory, an infrastructure can be described as a
	network graph G (N, L), where N represents the set of components (nodes) of the system and
	L the set of connections (lines) between the components. Given the topology of the
	infrastructure system, represented by a graph-based model, we can analyze its structural
	properties with respect to suffered disruptions.
	On August 1, 2008, CRH began its debut operation. As of January 2017, the main
	skeleton of CRH's "four vertical and four horizontal" structure has been completed. This
	structure plays an important role in interregional passengers' transport in China. The
	infrastructure has been significantly developed and nowadays CRH is leading the
	development of high-speed railways worldwide. At present, CRH is operational in 296 cities,
	including 137 prefecture-level cities (municipalities, provincial cities and prefecture-level
	cities) and 159 county-level cities (in addition to counties outside the municipal districts). The
	CRH can be represented by a network G (N, L), where N is the set of stations and L is the set
	of rail links connecting the stations. In this study, we assume that multiple stations in the same
	city are combined into one site. For example, the city of Wuhan houses three railway stations,
	namely, Wuhan, Wuchang, and Hankou Stations, and we abstract them into one site called
	Wuhan Station. Although numerous small sites are not included in the nodes set, their
	N:	set of nodes (railway stations), 𝑛 𝑖 ∈ 𝑁
	L:	set of links (railway links connecting stations), l 𝑖 ∈ 𝐿
	G (N, L): railway network comprising a set of nodes N connected by a set of links L
	𝑔 𝑖𝑗 :	actual geodesic distance between 𝑛 𝑖 and 𝑛 𝑗
	𝑙 𝑖𝑗 :	link connecting two adjacent nodes i and j
	𝑛 𝑢𝑣 :	number of shortest paths that connect 𝑛 𝑢 and 𝑛 𝑣
	𝑛 𝑢𝑣 (𝑖): number of shortest paths connecting 𝑛 𝑢 and 𝑛 𝑣 , as well as passing through 𝑛 𝑖

TABLE 1 .

 1 TOP 10 IMPORTANT NODES AND LINKS IN TERMS OF BETWEENNESS CENTRALITY

	Ranking	Nodes	𝐁 𝒊	Links	𝐁 𝒊𝒋
	1	Zhengzhou (ZZ)	0.37681	Wuhan-Changsha(WH-CS)	0.28157
	2	Wuhan (WH)	0.36025	Wuhan-Zhengzhou(WH-ZZ)	0.26501
	3	Changsha(CS)	0.36025	Shijiazhuang-Zhengzhou(SJZ-ZZ)	0.25879
	4	Shijiazhuang (SJZ)	0.29193	Hangzhou-Nanjing(HZ-NJ)	0.14492
	5	Guangzhou(GZ)	0.21946	Tianjin-Jinan(TJ-JN)	0.12836
	6	Nanjing(NJ)	0.21532	Jinan-Nanjing(JN-NJ)	0.12629
	7	Hangzhou (HZ)	0.21118	Changsha-Guangzhou(CS-GZ)	0.11801

Table ,

 , Nanjing is the most important station in terms of connectivity efficiency. If a disruption occurs and shuts down Nanjing Station, then the connectivity efficiency of the network is reduced by 12.4%. The next important two stations after Nanjing are Tianjin (𝑉 TJ 𝐸 = 0.116) and Hangzhou (𝑉 𝐻𝑍 𝐸 = 0.110) stations. In CRH mapwe found that several nodes are connected to them and the geodesic distance is relatively small. For example, the geodesic distance between Nanjing and Hefei is only 175 km and that between Nanjing and Hangzhou is 276 km, whereas the geodesic distances between other nodes are often more than 300 km. The efficiency of the network is mainly determined by the geodesic distance. Moreover, if one of two close nodes fail, then the other one can only be reached through other remote nodes in the network, which may lead to a notable reduction in network efficiency. Therefore, these nodes are rather important in terms of network efficiency and should be given sufficient attention and protection.

TABLE 2 .

 2 TOP 10 IMPORTANT NODES IN TERMS OF NETWORK EFFICIENCY VULNERABILITY

	Ranking	Nodes	𝑽 𝒊 𝑬
	1	Nanjing(NJ)	0.12366
	2	Tianjin(TJ)	0.11568
	3	Hangzhou(HZ)	0.11028
	4	Shijiazhuang(SJZ)	0.10681
	5	Hefei(HF)	0.10463
	6	Guangzhou(GZ)	0.10429
	7	Changsha(CS)	0.09611
	8	Xuzhou(XZ)	0.09415
	9	Beijing(BJ)	0.09288
	10	Shangrao(SR)	0.08947

TABLE 3 .

 3 IMPORTANCE RANKING OF TOP 10 COMPONENTS IN TERMS OF VULNERABILITY CONSIDERING TRAIN FLOW

	Ranking	Nodes	𝑽 𝒊 𝑭	Links	𝑽 𝒊𝒋 𝑭
	1	Shanghai(SS)	0.28073	Shanghai-Nanjing(SS-NJ)	0.23488
	2	Beijing(BJ)	0.23519	Wuhan-Changsha(WH-CS)	0.18932
	3	Guangzhou(GZ)	0.13641	Jinan-Xuzhou(JN-XZ)	0.17250
	4	Nanjing (NJ)	0.13358	Wuhan-Zhengzhou(WH-ZZ)	0.16931
	5	Wuhan(WH)	0.13232	Changsha-Hengyang(CS-HY)	0.14632
	6	Changsha(CS)	0.11551	Shijiazhuang-Zhengzhou(SJZ-ZZ)	0.14437
	7	Xian(XA)	0.11196	Nanjing-Hangzhou(NJ-HZ)	0.13359
	8	Zhengzhou(ZZ)	0.09776	Guangzhou-Hengyang(GZ-HY)	0.11809
	9	Hangzhou(HZ)	0.07678	Xian-Zhengzhou(XA-ZZ)	0.11575
	10	Shenzhen(SZ)	0.07214	Tianjin-Jinan(TJ-JN)	0.11132

Figure 2. Simplified CRH network map with critical 5 nodes and 10 links highlighted

TABLE 4 .

 4 TRAIN FLOW-BASED VULNERABILITY OF CRH UNDER PAIR-DISTURBANCE SCENARIOS

	Ranki	Nodes	𝐕 𝑭	Nodes	𝐕 𝑭	Links	𝐕 𝑭
	ng	+ Nodes		+ Links		+ Links
	1	BJ + SH	0.46489	SH+ WH-CS	0.46768	SH-NJ + WH-CS	0.42184
	2	GZ + SH	0.41064	SH + WH-ZZ	0.45004	SH-NJ + WH-ZZ	0.40419
	3	WH + SH	0.40519	SH + CS-HY	0.42101	SH-NJ + CS-HY	0.38121
	4	CS + SH	0.37887	SH + SJZ-ZZ	0.41766	SH-NJ + SJZ-ZZ	0.37182
	5	XA + SH	0.36669	BJ + SH-NJ	0.41642	BJ-NJ + GZ-HY	0.35298
	6	BJ + GZ	0.36381	SH + BJ-HZ	0.40845	WH-CS + JN-XZ	0.34416
	7	ZZ + SH	0.36365	SH + GZ-HY	0.39511	SH-NJ + SJZ-BJ	0.34001
	8	HZ + SH	0.36128	SH + SJZ-BJ	0.38585	SH-NJ + NJ-HZ	0.33335
	9	NJ + SH	0.35725	BJ + WH-CS	0.38449	SH-NJ + SR-NC	0.33204
	10	SZ + SH	0.35287	SH + JN-XZ	0.37897	SH-NJ + JN-XZ	0.32651

The authors sincerely thank the editors and the anonymous reviewers for their invaluable comments and suggestions for improving this article. This work was supported in part by the NSFC under Grant 71731008, and by "Academic Team Building Plan for Young Scholars from Wuhan University".


Petra Scherer
email: petra.scherer@uni-due.de

Evelyn Kroesbergen
email: e.kroesbergen@pwo.ru.nl

Hana Moraová
email: hana.moraova@pedf.cuni.cz

Helena Roos
email: helena.roos@lnu.se

Introduction to the work of TWG25: Inclusive Mathematics Education -challenges for students with special needs

Keywords: Inclusive mathematics, special education, teacher education, special needs in mathematics education, differentiation, diversity

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction to the work of TWG25: Inclusive Mathematics Educationchallenges for students with special needs Introduction

For CERME11, a new TWG about inclusive mathematics education has been established, with its first meeting at CERME11. This group was created due to an arising need for a forum to discuss issues of inclusion in the light of special needs in mathematics (SEM) in educational research. This branch of research has been developed in relation to general educational research about special educational needs (SEN) over the last decades, and has come to include not only disability and psychological issues, but also social, cultural and educational issues [START_REF] Magne | Historical Aspects on Special Education in Mathematics[END_REF] Hence, the notion of SEM covers different epistemological fields as well as touching upon issues of normality [START_REF] Skovsmose | Inclusions, Meetings, and Landscapes[END_REF].

The scope and focus of TWG25 is hence SEM and inclusion, in the intersection of mathematics education research and special education research. Since this scope is broad, the TWG-papers compromised grades 1-12, teacher professionalization and teacher education programs, types of inclusive settings in mathematics, concepts and models for instruction and subject matter didactics, special educational needs and child characteristics, and content related decisions for inclusive mathematics education.

During CERME11 there were 17 participants in TWG 25 from 10 countriescolleagues from Europe, and also from Canada and Hong Kong. 10 papers and 2 posters were presented during the week, and the thematic schedule was oriented towards the topics of the submitted papers and posters (presenting authors below). The first session was spent on a discussion about the aims and objectives of this new TWG and an exchange of overarching issues of the situation of SEM in the different represented countries. In the following TWG-sessions, two or three papers were presented each time, under an overarching theme. The themes were: (1) Development of materials,tools,learning arrangements,settings,etc. (2) Research on classroom situations, outof school situations (3) Research on teacher education (pre-service and in-service).

The themes were discussed in the initial session in order to include all participants.

(1) Development of materials, tools, learning arrangements, settings, etc.

This theme includes discussions of how to develop tools to identify students in SEM, but also how to promote their learning, including the research aims: for example, seeking to know the place of mathematics education in dyscalculia research, how to reconcile approaches to reach a better understanding of the disorder and provide professional development activities for both, pre-service and in-service teachers, regarding the multicultural education and presenting all attributes of a designed learning software.

Introductory discussionoverarching issues of the situation of SEM

The world is changing very fast and so are the demands on teachers. School classrooms are growing increasingly heterogeneouswith respect to language, culture and abilities. This requires differentiation with regards to learning goals, instruction strategies and individual tasks, to accommodate the classroom to enhance every students learning [START_REF] Bishop | Diversity in Mathematics Education. Towards Inclusive Practices[END_REF].

Internationally, a trend is visible towards inclusion, a comprehensive school for all, where students with very different levels of abilities and different skills are educated together [START_REF] Bishop | Diversity in Mathematics Education. Towards Inclusive Practices[END_REF]. Some European countries have a very long tradition of inclusion, while in other countries it is relatively new, since students with more marked disabilities used to be educated in special schools. The change in the system generated a passionate discussion on the advantages and disadvantages that inclusive education brings, which reflects the negative attitudes that teachers might have towards inclusion (e.g. De Boer, Pijl, & Minnaert, 2011).

Inclusive education brings many challenges to schools. Often it is connected to a lot of administrative load in many countries. For example, some school systems and governments require detailed and individualized progress reports and education plans to include every student, and this has to be done in so-called multi-professional teams for which they might not be prepared (see [START_REF] Ritter | Multi-professional and monoprofessional collaboration and their association with teacher trainee's attitudes towards concepts of inclusive education[END_REF]. All this applies to mathematics education, too. A mathematics teacher is expected to cope with the demands within the inclusive classroom, to plan lessons where all students will receive appropriate support and challenge. At the same time, they often have to deal with more administration, and often with more rules and monitoring from the management or government. This could even lead to teachers feeling stressed and opposing the whole idea of inclusive education. Although large differences between countries exist, the problems are recognized to a smaller or larger extent in all countries. The work in TWG25 showed that there are ways that can help pre-service and in-service teachers and teacher educators to support these processes. If teachers are better trained for teaching in a heterogeneous classroom and have adequate tools, methods and techniques to make it work, they will be much more motivated and willing to support the system.

During almost all sessions in TWG25, one big question was circling, namely how do we understand inclusion in mathematics? Is it static? Is it depending on national and cultural settings? How do the teachers, the learners and/or the researchers describe and understand it? Do they understand it the same, or differently? Is the way, inclusion is understood affected by the perception of students in need, or with need? Is it dependent on the perception of the student? These questions are immersed and reflected (implicitly or explicitly) in the overarching themes that emerged from the discussions of TWG25, and described below.

Overarching issues of TWG25

Research in the field of inclusive mathematics covers a wide range. This can be seen in the directions taken in research covering disability, psycological, pedagogical and didactical issues. It can also be seen in how research covers aspects on either societal level or classroom level working with notions such as equity and diversity [START_REF] Roos | Inclusion in mathematics education: an ideology, a way of teaching, or both?[END_REF]; Kollosche, Marcone, Knigge, Godoy Penteado, & Skovsmose, 2019), and several trends in mathematics instruction were presented in the papers of TWG25. All papers focus on some aspects of mathematics teaching and learning in relation to SEM and deserve due attention. During the TWG-sessions based on the paper presentations, the following points were discussed, especially how they are linked with each other.

Inclusive system or special schools?

As stated in the introductory part, there are countries where special schools have a relatively long tradition (e.g. more than 100 years in the Netherlands) [START_REF] Evans | A Comparison of Inclusion practice in OECD countries[END_REF]. Their clear advantage is that the staff has a specialized knowledge to work with and help special needs children. The number of students in classes is relatively low and thus the teacher has the time to individualize work. These schools have been criticized in the past by the EU for their exclusive nature, and many countries extended or built up inclusive systems according to the realization of the UN conventions (see UN, 2006). Inclusive education means each child has the right to be educated in the local school. However, in some countries, there is still a debate whether this is for the benefit of the student.

Teacher education in relation to SEM

Depending on the tradition of inclusive education, SEM is taken into concideration to a greater or lesser degree in the mathematics classroom. In countries with a longer tradition the system has more experience and mathematics teachers are more experienced to take SEM into consideration in their teaching.

Without any doubt the education must adapt to the situation and much more attention has to be paid to special needs in mathematicstaking children with mental or physical disabilities, students in great access to mathematics or second language learners into account.

Change of teaching and learning strategies?

Does SEM imply using new teaching and learning strategies? Taking an inclusive standpoint in the classroom asks for differentiation and individualization of work and take the individual student as a point of departure as discussed in the TWG-paper by Roos. Then the teacher's role is changing to the role of a facilitator, to the role of the person who plans the lesson and activities and then monitors students' work and helps wherever is needed. One of the ways of allowing inclusive classes to learn is to focus on how tasks can be designed and used to differentiate between students of different levels, like the paper by Buró & Prediger shows. Teachers are not always aware what kind of adjustments are needed for the different levels amongst theirs students.

Another point of discussion is whether constructivist approaches could help as all students build their own knowledge in a way that is suitable for their needs. However, there is no clear empirical evidence that this is the best way to teach students in SEM as the research result might be contradictory (e.g. [START_REF] Chodura | Interventions for children with mathematical difficulties[END_REF][START_REF] Scherer | Productive or reproductive exerciseswhat is appropriate for low attainers[END_REF].

How can mathematics teachers be motivated to work with SEM?

The issue of motivating teachers to work with SEM is essential. Inclusive education is reality and teachers must be ready to face it. If they are well motivated, they are more likely to cope with the situation well. This, and how to work with teachers and different methods to use is discussed in the TWG-papers by Scherer, Fung & Wang and Korten et al.

One way to motivate teachers is to equip them with such tools that they will feel confident enough when facing a heterogeneous classroom with students with very diverse educational needs. Having diverse teaching methods and techniques that enable differentiation and motivation will help. Also having experience with managing a classroom where an assistant is present will help and will give the teacher the needed self-confidence. Not everyone is happy with having a "critical" eye in all lessons.

Another way to increase teachers' motivation is providing them enough opportunities to improve their knowledge and skills in the area. Pre-service teacher programs must pay enough attention to the issue, and high quality in-service teacher training programs should be available. If possible they should be designed and run by a number of specialists from different areas (special pedagogues, psychologists, but especially experienced teachers of a particular subject).

Once methodology and techniques of work with special needs students become more widespread and results will be tangible, motivation of teachers will become easier.

Who is in SEM?

The large heterogeneity of the students with special needs in mathematics was discussed in TWG25, ranging from low IQ to gifted, from specific learning disabilities like dyscalculia (discussed in the TWG-paper by Peteers & Ouvrier-Buffet) to behavioral and developmental disorders, including students with different cultural backgrounds (as discussed in the TWG-paper by Medová et al.). Also teachers' views on special needs as presented in the TWG-paper by Hamukwaya was discussed. The question arose why we use the term special, and what is perceived as 'normal' (see [START_REF] Skovsmose | Inclusions, Meetings, and Landscapes[END_REF]. Wouldn't it be better to talk about heterogeneity and take the individual differences as norm?

Conclusion and further directions of TWG25

Reflecting back on the arguments made in the papers of TWG25 and on the discussions, there are, as prior discussed, diverse issues concerning inclusion in mathematics education and challenges for students with special needs in mathematics. In the TWG-papers about development of materials, tools, learning arangements and settings, arguments are made for both developing sustainable ways of identifying dyscalculia in mathematics education (paper Peteers & Ouvrier-Buffet) and developing learning software for both, pre-service and in-service teachers (paper Medová et al.).

In the TWG-papers about research on classroom situations and out-of school situations the focus was on challenges and posiblities in the classroom from different perspectives. Strategies promoting mathematical activities in relation to students with language disorders were discussed: Too often teachers tend to reduce learning opportunities within mathematical tasks by focusing on their cognitive characteristics and behavior, although the reported research shows that if a teacher offers space for dialogue with students and questions them in order to trigger their mathematical activity it becomes a powerful driver of student activities in a classroom context where the teacher offers a space where all voices are legitimized (paper Bergeron & Perreault). Another issue in relation to how to work in classroom was how a classroom setting influences inclusion in mathematics, and the importance to consider how to plan the classroom to include everybody (paper Roos). In relation to work in the classroom, mathematical discourses of a teacher and a visually impaired student were investigated highlighting the interaction teacherstudent (paper Stylianidou & Nardi). Also a review highlighting the learning of hearing-impaired students and the challenges and possibilities of cognitive enrichment (paper Szücs) was discussed within this theme.

In the TWG-papers about research on teacher education the focus was on how teachers view inclusion and SEM, and how teacher education can work to help teachers work inclusively. The potential of substantial learning environments [START_REF] Wittmann | Developing mathematics education in a systemic process[END_REF] for inclusive mathematics in relation to student teachers' explorations with special needs students were reflected upon with the conclusion of the importance to include such type of explorations and reflections of inclusion in the teacher education (paper Scherer). In relation to this, it was discussed how in-service training can support teachers of different professions in the implementation of 'inclusive education' in the mathematics classroom (paper Korten et al.). Support to teachers was also discussed in relation to teaching mathematics to students with severe intellectual disability (paper Fung & Wang), with the conclusion that it is mathematics that must be in the centre of attention when discussing SEM, something that may sound too trivial to consider, yet often too easy to overlook. The development, testing, and refinement of didactical action plans involves studying various phenomena and their corresponding mathematical meanings, which can only be done from within mathematics Support to students by supporting teachers in relation to braille readers in reading and comprehending mathematical expressions and equations through specific instructions was problematised (paper van Leendert et al.).When looking into mathematics teachers in relation to inclusion and SEM the way teachers catagorise for differentiating in inclusive classrooms were highlighted, problematising low entrance (paper Buró & Prediger). Also teachers' perceptions in relation to learning difficulities in mathematics was problematised (paper Hamukwaya).

The major question, still left to be answered, is how we understand inclusion in mathematics, and how we can have a common ground in research despite the diversity of research directions and cultural and national differences. Still, it is interesting to see that even if there are very diverse issues concerning inclusion in mathematics education, there are similarities too. This can be seen in the themes of the TWG-papers. Hence, there seem to be opportunities to build a common ground regarding inclusive mathematics teaching and challenges for students with special needs. enrichment in inclusive mathematics classrooms.

Looking forward to CERME12 the community got a challenge to build further on the work at CERME11 and build a common ground to address issues within the scope of TWG25.

  What is the effect of improving the knowledge of braille display and Text-To-Speech synthesizer support of mathematics teachers on braille reader's achievement in mathematics? (paper Van Leendert) Petra Scherer: The potential of substantial learning environments for inclusive mathematicsstudent teachers' explorations with special needs students Laura Korten: An in-service training to support teachers of different professions in the implementation of 'inclusive education' in the mathematics classroom Chun-ip Fung & Dichen Wang: Teaching mathematics to students with intellectual disability: What support do teachers need?

	What challenges do mathematics teachers experience when teaching students with
	mathematics learning difficulties in an inclusive classroom and what potential measure do they
	use? (paper Hamukwaya)
	Florence Peteers: Diagnosis tools of dyscalculia -contribution of didactics of mathematics to
	numerical cognition
	and disability constructed in the discourses of teaching staff and pupils in mainstream
	mathematics classrooms? (paper Stylianidou & Nardi) How do SEM-students perceive their
	participation in terms of learning and teaching in an inclusive mathematics classroom to have optimal
	opportunities to learn? (paper Roos).
	Helena Roos: I just don't like math, or I think it is interesting, but difficult … Mathematics
	classroom setting influencing inclusion
	Laurie Bergeron & Audrey Perreault: Strategies that promote the mathematical activity of
	students with language disorders: an analysis of language interactions
	Kinga Szücs: Do hearing-impaired students learn mathematics in a different way than their
	hearing peers? Challenges and possibilities of cognitive enrichment in inclusive mathematics
	classrooms -a review
	Angeliki Stylianidou: Mathematical discourses of a teacher and a visually impaired student on
	number sequences: Divergence, convergence or both?
	(3) Research on teacher education (pre-service and in-service)
	This theme includes discussions of inclusive education and SEM in relation to teacher
	education and teachers' views. How are teacher education programmes related to inclusion
	and what are adequate strategies for teachers to develop inclusive mathematical settings.
	Examples of research questions within this theme are: How should didactical courses in
	teacher education be designed to address the topic 'inclusive mathematics'? (paper Scherer).

Janka Medová: Designing Mathematical Computer Games for Migrant Students (2) Research on classroom situations, out-of school situations This theme includes discussions of inclusion in relation to SEM students, mathematics classrooms and out-of school situations, where the focus lied in the challenges and possibilities in mathematics education. Examples of research questions within this theme are: How are inclusion Sarah Buró & Susanne Prediger: Low entrance or reaching the goals? Mathematics teachers' categories for differentiating with open-ended tasks in inclusive classrooms Annemiek van Leendert: Supporting braille readers in reading and comprehending mathematical expressions and equations Shemunyenge Taleiko Hamukwaya: K-12 Namibian Teachers' Views on Learning Difficulties in Mathematics: Some reflections on teachers' perceptions

  These different directions are discussed in the TWG-papers byvan Leendert et al., Hamukwaya, Stylianidou & Nardi, Roos, Szücs and Bergeron & Perreault. Also teacher education needs to prepare fresh graduates to know possible methods, techniques, and activities that will enable them to differentiate and individualize lessons. Teachers must have the theoretical background knowledge about different disabilities, in addition to practical examples of how to manage the classroom, what activities to use are essential. With all these special competences, the different forms of knowledge, subject specific knowledge included, is essential (cf.[START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF]. Teacher education should reflect the findings of current research on work with special needs children as this is a fast developing area. This is discussed in the TWG-papers byScherer, Korten et al. and Fung & Wang.


Sarah Buró
email: sarah.buro@tu-dortmund.de&prediger@math.uni-dortmund.de

Susanne Prediger

Low entrance or reaching the goals? Mathematics teachers' categories for differentiating with open-ended tasks in inclusive classrooms

Keywords: Inclusive education, natural differentiation, professional development, teacher knowledge

Open-ended inquiry tasks are considered a powerful approach for addressing the diversity of inclusive mathematics classrooms due to their potential for natural differentiation. However, this potential can only unfold when the teachers know how to work with the tasks. This article investigates teachers' personal categories for differentiating with an open-ended task, especially with respect to providing support for students with special needs. In a qualitative case study, a category-eliciting activity was conducted within a professional development session. Data gathering comprised 14 secondary mathematics teachers' and special needs teachers' video-taped group discussions and written answers, which were analyzed qualitatively. The results show that most teachers' ideas for support provided for the students with mathematical learning disabilities only addressed the low entrance, but not the core learning goals and the required basic conceptual knowledge.

Most German secondary schools only recently shifted to an inclusive system, so many secondary mathematics teachers and special needs teachers currently learn how to differentiate in inclusive mathematics classrooms. As effective PD programs need to take into account teachers' typical starting points, we investigate teachers' perspectives to deal with various differentiated teaching approaches, here specifically with respect to differentiating with open-ended inquiry tasks [START_REF] Scherer | Assistance of students with mathematical learning difficulties: How can research support practice?[END_REF].

The paper starts with presenting the teaching approach of open-ended tasks and its potential for natural differentiation [START_REF] Scherer | Natural differentiation in mathematicsthe NaDiMa project[END_REF]. The necessary teacher expertise for differentiating with open-ended tasks is conceptualized in the framework of [START_REF] Bromme | Der Lehrer als Experte[END_REF]. The qualitative case study based on this conceptual framework uses category-eliciting activities for pursuing the following research question in the empirical part of the paper:

Which categories and self-reported practices do teachers activate for differentiating in inclusive mathematics classrooms with an open-ended task? And how can this be supported by facilitation?

Background on classroom level: Open-ended inquiry tasks for natural differentiation

Inclusive mathematics classrooms call for differentiated instruction with joint whole-class experiences and specific support for students with special needs [START_REF] Lawrence-Brown | Differentiated instruction: Inclusive strategies for standards-based learning that benefit the whole class[END_REF][START_REF] Tomlinson | Differentiating instruction: A review of literature[END_REF]. One of the teaching approaches which have proven useful [START_REF] Scherer | Assistance of students with mathematical learning difficulties: How can research support practice?[END_REF], applies rich open-ended inquiry tasks with the potential of so-called natural differentiation:

Open-ended tasks with a low entrance and high ceiling provide the potential for natural differentiation if they allow for multiple representations, diverse solution pathways and different cognitive activities along the trajectory of discoveries [START_REF] Scherer | Natural differentiation in mathematicsthe NaDiMa project[END_REF].

One example for such an open-ended task for Grade 5 is printed in Figure 1. It aims at discovering the multiplicative structure of the volume of cuboids, and has a wide differentiating potential (see Figure 2). The task has a low entrance as all students succeed in finding at least one cuboid. The task's core goal is to discover that counting in rows and layers leads to a multiplicative structure of the volume. The high ceiling for students with strong mathematical potentials covers the combinatorial challenge to find all cuboids, usually by considering the multiplicative decomposition of 24 in three factors.  How can you be sure that you have found all? All possible cuboids: 1 1 24, 1 2 12, 1 3 8, 1 4 6, 2 3 4, 2 2 6. The degree to which rich and differentiating tasks are really productive for inclusive mathematics classrooms depends on the teacher's expertise for really ensuring responsiveness, not only in a reactive way (i.e. ad hoc repairing occurring obstacles), but also proactively in preparing specific support for students in need for it [START_REF] Tomlinson | Differentiating instruction: A review of literature[END_REF]. Especially, low expectations have often been problematized (Büscher, submitted) as research shows that students with mathematical learning disabilities can reach more than often expected if responsive, proactive support is provided [START_REF] Peltenburg | Mathematical potential of special education students[END_REF]. That is why the teachers must be prepared for these challenges. In order to base professional development offers on teachers' starting points, the study presented in this paper aims at capturing the teachers' current categories for differentiating with open-ended tasks.

Diverse ways of using material and structures

Personal categories are a crucial part of [START_REF] Bromme | Der Lehrer als Experte[END_REF] conceptualization of teacher expertise. He defines teacher expertise as the teacher's capability to cope with complex situations in subject matter classrooms, comprising (a) the teacher's practices by which they cope with situational demands (the so-called jobs), (b) the orientations, e.g. the (content-specific or content-independent) attitudes guiding the prioritization and interpretation of the jobs and (c) the categories which implicitly or explicitly guide their perception and practices (see Prediger, 2019 for a more general discussion of this conceptualization). Especially, Bromme suggests a powerful "heuristic to search for the 'natural' categories in expert knowledge" (Bromme, 1992, p. 88, translated by the authors) by analyzing the situational demands with respect to the relevant practices and their underlying implicit categories.

Based on the large literature on differentiating in inclusive mathematics classrooms [START_REF] Tomlinson | Differentiating instruction: A review of literature[END_REF][START_REF] Lawrence-Brown | Differentiated instruction: Inclusive strategies for standards-based learning that benefit the whole class[END_REF][START_REF] Scherer | Assistance of students with mathematical learning difficulties: How can research support practice?[END_REF], we identified three sub-jobs for the larger job of differentiating with open-ended tasks:

(1) analyzing tasks with respect to students' potentially diverse solution pathways and approaches

(2) identifying differentiating potential in a task and possible obstacles for reaching the core goals (3) providing support for students with specific needs for reaching the core goals.

Job 1 is a preliminary job for Job 2 and 3, which is necessary in order to unpack the mathematical structure and the possible steps on the trajectory of discovery (see Figure 2). The differentiating potential identified in Job 2 can refer to these steps and the multiple ways or representations, but also to possible obstacles, mainly those lying in required basis concepts. The later is crucial for proactively providing support in Job 3, especially for students with specific needs (in our case, students with mathematical learning disabilities, with or without the official status of having special needs).

Taking into account that effective differentiation is always knowledge-centered [START_REF] Tomlinson | Differentiating instruction: A review of literature[END_REF], the teachers should have a focus on the six steps of the trajectory of the discovery as well as on the required basic knowledge in understanding the underlying mathematical concepts (as Scherer et al., 2016 emphasize). In the concrete task, this concerns the concept definition of cuboid and, even more importantly, the mental model of multiplication as counting in groups which is required for the transition of counting separately (Step 1) to counting in rows or layers by multiplication (Step 2-4). Providing support for students with specific needs should not only guarantee the low entrance of the task, but also the possibility of reaching the core goal, here the multiplicative struc-ture of the volume. At the same time, the high ceiling (finding all cuboids with 24 cubes) provides challenges for the students with strong mathematical potentials.

Methods of the qualitative study for investigating teachers' starting points

Given that teachers' expertise is often implicit in their practices, the research question (Which categories and self-reported practices do teachers activate for differentiating in inclusive mathematics

classrooms with an open-ended task? And how can this be supported by facilitation?) was pursued in a qualitative study based on a category-eliciting activity.

Methods for data gathering by a category-eliciting activity on three jobs Sample. The sample consisted of 14 secondary math teachers who participated in their first session of a volunteer professional development series on inclusive mathematics classrooms. They had between 2 and 20 years of experiences in math teaching. 8 of them held a teacher degree as mathematics teachers (PCK + SNK -, i.e. with a formal qualification in pedagogical content knowledge in mathematics education, abbreviated PCK + , but no formal qualification in special needs knowledge on specific needs of students with learning disabilities, abbreviated SNK -), 1 of them was special needs teacher without a degree in mathematics education (PCK -SNK + ), and 5 of them were special needs teachers with a degree in mathematics education (PCK + SNK + ).

Category-eliciting activity. For eliciting teachers' personal categories and self-reported practices for differentiating with the open-ended task, the teachers' activity in Figure 3 was structured according to the jobs introduced in the next section. The teachers were asked to write down their ideas and discuss them in small groups during the PD session. The group discussions were video-taped and partly transcribed.

Methods for qualitative data analysis

The qualitative data analysis was based on deductive-inductive procedures [START_REF] Mayring | Qualitative content analysis: Theoretical background and procedures[END_REF], starting from the prospective analysis of differential potential and possible obstacles (see Figure 2), but open for teachers' further personal categories. The articulated personal categories were inductively subsumed under the most relevant categories (results listed in Table 2). The elicited categories along the three jobs were compared between the three subsamples PCK + SNK + , PCK + SNK -, PCK - SNK + .

The video data was analyzed qualitatively with respect to the inductively developed categories and their emergence in the discussions. The video data was of specific importance for determining not only what teachers miss, but also how the facilitator could activate inert knowledge after a while.

Students' task: Build many cuboids

Here, you have 24 wooden cubes. Which cuboids can you build with them? Document which one you have found. How many do you find?

Questions for teachers:

(1) Which solution pathways and approaches would you expect from your students for this task? (2) Which potential for differentiation does the task provide? (3) How would you support students with mathematical learning disabilities?

Main focus on low entrance in Step 1 instead of learning goals and basic concepts

Table 1 shows exemplary answers written by three teachers, together with the category assigned to the answers in the data analysis. Table 2 embeds these three cases into the complete group of 14 teachers (if only 13 teacher occur, one has not answered this part).

Table 1: Examples of teachers' written answers and the assigned categories

In Job 1 (Analyzing the task), Dieter and Hayat only focused on Step 1 of the trajectory of discovery (see Figure 2), only Melanie also considered further steps (even if condensed in one line without unpacking the necessary steps). Within the group of all teachers (see Table 2), Melanie was an exception, whereas many teachers (6 of 13 teachers) only focused on Step 1 of students' trajectory, but not how to reach the core goal of the task, discovering the multiplicative structure of the volume. Other categories teachers activated in Job 1 concerned social and affective factors, many of the teachers expressed low expectations like Dieter and two of them even expect that students get no possibility to build a cuboid.

In Job 2, it is interesting to see that Dieter (and with him three other teachers) could not identify the potential for natural differentiation and immediately started to provide specific support for weaker students. Those who identified potential for natural differentiation like Melanie and Hayat focused on the steps (7 of 13), the students' diverse ways of using material and structures (2 of 13), or on multiple representations for documentation (2 of 13). Teachers with PCK + SNK + could identify more potential for natural differentiation then the teachers with PCK + SNK -. The teacher with PCK - SNK + found none. The required basic concepts (cuboid, multiplying as counting in groups) were only addressed by one teacher in Job 2. With respect to Job 3, most teachers have some resources for planning support for students with mathematical disabilities, this is shown by the fact that only three teachers planned their support exclusively on surface levels of grouping strategies or taking larger wooden cubes.

However, as a consequence of the strong focus on Step 1 in Job 1, the planned support in Job 3 mainly focused on Step 1/2 again. That means, 2 of 5 teachers with PCK + SNK + and 4 of 8 with PCK + SNK -only provide support for guaranteeing the low entrance. By this focus, they miss the core goal of the task, which was discovering the multiplicative structure of the volume. As a consequence of the limited view on the necessary basic concepts, only three teachers with PCK + SNK + , provided support for overcoming limitations in the basic concepts, and none of PCK + SNK -did. Among the three who took care of basic concepts, two made sure that the definition of cuboid as shape with rectangular faces is accessible also for students with mathematical learning disabilities, and only one single teacher's support addressed the potentially missing mental model for multiplication as counting in groups.

ing the diverse aspects of potential for natural differentiation, she shifted the focus to the central learning goal, emphasizing the multiplicative structures of counting in rows and layers for determining the volume. She sent the teachers back into their group discussions with the following prompt which started an interesting process of eliciting further categories:

143 Facilitator What do we do know [means for the students with learning disabilities] that they can really…? So what kind of support can we give so that they can learn -not only what to calculate, but also understand why to use multiplication?

How to come to Step 2-5?

144 Melanie The levels while layering, there are the layers [shows the layers with her hands]

Step 2 …

Step 2+3a 149 Melanie And then coloring, well anyway, really draw the picture as three-dimensional and mark the layers and reflect, how many cubes are in one layer 150 Hayat I mean, when you have worked adequately, then they know this, this with lying the dot plates [she refers to rectangular arrays such as printed here]

Basic concept multiplication 151 Carmen I was just gonna say, the point system and with it, you can take the cuboids of one layer.

Step 3a

152 Hayat Yeah. 153 Carmen And then the point system behind, then they will get it, length times width.

Step The analysis of the transcript with respect to the implicitly or explicitly addressed categories from Table 2 shows that after the facilitator-initiated shift of attention from the low entrance to reaching the core goal, the group discussion at the observed table quickly turns to carefully thinking through all steps of the trajectory of discovering multiplicative structures (as marked next to the transcript). During this reflection, the teachers also identify the most important basic concept, the mental model of multiplication in the rectangular array (addressed by the teachers with different idiosyncratic termsthe dot plates, the point system, etc.but exactly that meaning).

It is only now that the teachers discover an important form of differentiating with the open-ended task: When planning how to provide support for students with mathematical disabilities, the basic concepts required for reaching the learning goals of the task must be identified and possibilities for integrated remediations of these basic concepts have to be searched. In addition, teachers can express higher expectations.

Discussion and outlook

Although the sample of 14 teachers is much too small to take the quantitative comparisons of teachers' different backgrounds as statistically representative, these first insights into an ongoing project can already point to teachers' resources and blind spots for differentiating with open-ended tasks:

 Open-ended tasks can only unfold their potential of natural differentiation if teachers can unpack this potential. In our sample, 9 teachers started to unpack it, which is a good starting point.

 Their focus is mainly on the start of the trajectory of discovery, and as a consequence, the provided support for students with mathematical learning disability is concentrated on the first steps, but not on the steps towards reaching the learning goal. As a consequence, also the support is not yet concentrated on the core goals.

 The observation that only three teachers provide support for overcoming obstacles posed by potentially missing basic conceptual knowledge is a very strong concern as students cannot reach the learning goals without getting access to the basic mathematical concepts.

 Within the PD session, the facilitator succeeded to shift the participants' focus from the low entrance to possible supports for reaching the core goals, so with adequate prompts, the teachers can activate categories in their PCK that help them. These include especially the basic concepts.

 Further questions: Which other blind spots for differentiation can be identified and how can they be processed?

Figure 2 :

 2 Figure 2: Differentiating potential of the open-ended cuboid task as empirically identified by Prediger (2009)

Figure

  Figure 1: Open-ended task for discovering the multiplicative structure of volumes (Prediger, 2009)

Figure 3 :

 3 Figure 3: Category-eliciting activity for three jobs

  I would remediate here, and then there are the layers, though. […] And then, if you color that, you are, easy peasy, in multiplication.

			4, High
			expectation
	154 Hayat	Exactly, I have first..#
	155 Carmen	#That is then, the remediation of the rectangle […]	basic concept
	157 Christiane Yes.
	158 Carmen	Area, isn't it […]
	160 Melanie	And then the rectangle.	Step 4
	161 Carmen	And then the multiplication […] You have the dot frames and the rectangle -	basic concept
	-165	that is what Step 4
	…
	169 Christiane You mean, decomposing it, actually, after it, decompose into the same forms.	Step 3b
	…
	173 Melanie	But then, of course, you are still in the hands-on, and you have to anyway, ehm.	Step 5
	174 Carmen	Or you come into the multiplication. […]	relevance of
	-177	IF you have remediated it thoroughly, before.	basic concept

Acknowledgment

The paper has grown within the research in the German National Center for Mathematics Teacher Education (DZLM, funded by the Deutsche Telekom Foundation). The research is organized in the PD research project MATILDA, funded by the German Ministry of Education (grant no. 01NV1704 to S. Prediger & J. Kuhl). We thank our project partners for the collaboration, especially Christian Büscher for the coordination.

Subsample

Shifting teachers' categories by prompts to further steps

In order to avoid a deficit-oriented inventorization of teachers' perspectives, it is crucial to consider how the teachers' activation of categories can be supported by facilitation in the PD session.

When the facilitator collected teachers' ideas of the aspects of natural differentiation and for supporting students with mathematical learning disabilities after the first group discussions in the PD session, she became aware of the exclusiveness of teachers' focus to the first steps of the trajectory. After acknowledging all teachers' efforts for guaranteeing low entrance for everybody and collect-