Annelieke C Baller
email: a.c.baller@vu.nl

Martijn Van Ee

Maaike Hoogeboom

Leen Stougie

Complexity of inventory routing problems when routing is easy

Keywords: approximation, computational complexity, dynamic programming, inventory routing, periodic replenishment, pinwheel scheduling

In the inventory routing problem (IRP) inventory management and route optimization are combined. The traveling salesman problem (TSP) is a special case of the IRP, hence the IRP is NP-hard. We investigate how other aspects than routing influence the complexity of a variant of the IRP. We first study problem variants on a point and on the half-line. The problems differ in the number of vehicles, the number of days in the planning horizon and the service times of the customers. Our main result is a polynomial time dynamic programming algorithm for the variant on the half-line with uniform service times and a planning horizon of 2 days. Second, for nearly any problem in the class with nonfixed planning horizon, we show that the complexity is dictated by the complexity of the pinwheel scheduling problem, of which the complexity is a long-standing open research question. Third, NP-hardness is shown for problem variants with nonuniform servicing times. Finally, we prove strong NP-hardness of a Euclidean variant with uniform service times and an easily computable routing cost approximation, avoiding immediate NP-hardness via the TSP.

INTRODUCTION

This article studies the computational complexity of special cases of a variant of the inventory routing problem (IRP), in which a set of customers is supplied over a given time horizon by identical vehicles from a central depot. Each customer has a storage capacity, a fixed demand per day, a latest delivery day at the start of the planning horizon and a service time. The metrics that underlie the customer locations do not immediately imply intractability because of routing aspects. In particular, we consider the problem in which all customers are located in a single point, on a half-line and in the Euclidean plane, but the latter under a specific approximation of the tour length. On a half-line, the depot is located in the origin, that is, at one end of the half-line. The vehicles have a tour duration constraint which limits the number of time units per day (traveling plus service time). The objective is to minimize the total time spent by all vehicles over all days.

The motivation for this study stems from a business project in ATM replenishment in the Netherlands. ATMs need to be replenished regularly such that banknotes are sufficiently available to consumers. In practice, this is an involved problem in which service levels, safety regulations and uncertainty play a role. In this article, we study a stylized version of this problem.

In many vehicle routing problems (VRP) the vehicles are capacitated in terms of load, however, in the ATM replenishment problem, the time spent by a vehicle is often more binding [START_REF] Baller | The dynamic-demand joint replenishment problem with approximated transportation costs[END_REF]. Time is also more binding than load in, for example, online ordered package delivery and blood product distribution [START_REF] Hemmelmayr | Delivery strategies for blood products supplies[END_REF]. Therefore, we consider a tour duration constraint.

Almost any version of the IRP is NP-hard since it contains the well-known NP-hard traveling salesman problem (TSP) [START_REF] Karp | Reducibility among combinatorial problems[END_REF][START_REF] Papadimitriou | The Euclidean traveling salesman problem is NP-complete[END_REF] as a special case. We investigate the complexity of an IRP on metrics for which routing does not cause immediate NP-hardness through TSP. In the first part of the article, we consider the problem on a point, that is, the depot and the customers are all at the same location, and on the half-line. Furthermore, the problem variants studied differ in the number of vehicles available, the length of the planning horizon and the type of service times. In the second part of the article, we consider the respective problem in the Euclidean plane and choose a route length approximation function that avoids hardness through TSP. Still the problem is shown to be NP-hard. Some variants of the studied IRP are easily shown to be solvable in polynomial time and some are easily shown to be NP-hard (Section 3). In order to identify which aspects determine the computational complexity of the respective IRP variant, we search for borderline problem variants on a point and on the half-line, which are either maximally easy or minimally hard. A maximally easy problem variant is a variant that is polynomial time solvable, but if one feature is generalized it becomes hard or has an open complexity. Similarly, a minimally hard problem becomes easy or open if one feature is further restricted.

IRPs form a class of widely studied and still challenging problems in the Operations Research literature. An introduction to the IRP is given in the tutorials Bertazzi and Speranza [START_REF] Bertazzi | Inventory routing problems: An introduction[END_REF][START_REF] Bertazzi | Inventory routing problems with multiple customers[END_REF]. In their introduction to IRPs, Bertazzi et al. [START_REF] Bertazzi | The Vehicle Routing Problem: Latest Advances and New Challenges 43[END_REF] provide examples that give insight in the influence of holding costs, inventory capacities at the customers, and continuous consumption of goods at the customers. The authors state informally that limited storage capacity causes extra complexity in IRPs because of the implied time required between two deliveries. We will formalize this statement in this article.

Several literature surveys have been published since the 1990s [START_REF] Andersson | Industrial aspects and literature survey: Combined inventory management and routing[END_REF][START_REF] Baita | Dynamic routing-and-inventory problems: A review[END_REF][START_REF] Coelho | Thirty years of inventory routing[END_REF][START_REF] Cordeau | Vehicle routing[END_REF][START_REF] Federgruen | Analysis of vehicle routing and inventory-routing problems[END_REF][START_REF] Moin | Inventory routing problems: A logistical overview[END_REF][START_REF] Sarmiento | A review of integrated analysis of production-distribution systems[END_REF]. The two most recent surveys each have a different focus. Andersson et al. [START_REF] Andersson | Industrial aspects and literature survey: Combined inventory management and routing[END_REF] focus on industrial aspects of inventory routing and they propose a classification based on seven aspects that concern time, demand (deterministic or stochastic), routing, inventory, and fleet aspects. The survey is split into three parts based on the time horizon: instant, finite, and infinite time horizon. Coelho et al. [START_REF] Coelho | Thirty years of inventory routing[END_REF] state in their recent survey that there "does not really exist a standard version of the problem," since many variants with changing aspects are present in the literature. They propose a classification based on seven criteria including time horizon, routing, inventory and fleet aspects, and call all variants of the IRP that fit these criteria as "basic versions." These seven criteria do not include demand aspects, since the authors want to separate the structure of the problem from information availability (i.e., stochasticity of demand). The criterion added compared to Andersson et al. [START_REF] Andersson | Industrial aspects and literature survey: Combined inventory management and routing[END_REF] is the inventory policy used in the problem (maximum level or order-up-to level). Coelho et al. [START_REF] Coelho | Thirty years of inventory routing[END_REF] give an in-depth overview of models and solution methods for both "basic versions" of the IRP as well as extensions of this version. The solution methods are divided in exact methods and heuristic methods for the basic versions of the problem. Additionally, Desaulniers et al. [START_REF] Desaulniers | A branch-price-and-cut algorithm for the inventory-routing problem[END_REF] propose a structurally different problem formulation and an exact solution method for the IRP that gives promising results for the multiple-vehicle IRP. Alvarez et al. [START_REF] Alvarez | Iterated local search and simulated annealing algorithms for the inventory routing problem[END_REF] and Archetti et al. [START_REF] Archetti | A matheuristic for the multivehicle inventory routing problem[END_REF] present the most recent heuristic solution methods for the IRP.

The variant of the IRP considered in this article fits into the classification of Coelho et al. [START_REF] Coelho | Thirty years of inventory routing[END_REF] as follows. It has a finite time horizon and a one-to-many structure, which means that one vehicle can visit multiple customers in one route. Multiple homogeneous vehicles are available and each vehicle can serve multiple customers in one route. Inventory is replenished via an order-up-to-level policy, which means that inventory is filled up to capacity at each replenishment and all demand has to be satisfied, that is, back-orders or lost sales are not allowed. We assume that all demand information is available at the beginning of the planning horizon. Furthermore, suppose that in a given day all replenishments take place before demand occurs, which is a common assumption in IRP (see, e.g., Archetti et al. [START_REF] Archetti | Formulations for an inventory routing problem[END_REF]). Besides this classification, we take the service times of the customers as given and we do not consider inventory holding costs. Hence, our objective is to minimize the total traveling and service time. Moreover, instead of a vehicle capacity constraint in terms of units of goods, we consider a tour duration constraint which limits the time spent by a vehicle per day. In Section 4.4, we briefly discuss that our results with the tour duration constraint imply similar results for the case with a vehicle capacity constraint.

In this article, we study problem variants of the IRP in which customers are located on a point or on the half-line. A similar study was executed for the VRP by Archetti et al. [START_REF] Archetti | Complexity of the VRP and SDVRP[END_REF]. Specifically, the authors consider the VRP with unsplittable demand and a limited fleet on a line, a star, a tree, and a cycle. They show several hardness results using the relation with the weakly NP-hard partition problem. Here we derive similar hardness results by relating our problem variants to the strongly NP-hard bin packing problem (BPP).

Finally, we mention two papers that consider problems similar to the ones in this article. Das et al. [START_REF] Das | The train delivery problem-Vehicle routing meets bin packing[END_REF] study the train delivery problem with a single time period and multiple capacitated vehicles. The customers have weights and are located on the half-line. The goal is to assign customers to vehicles such that the vehicle capacity is not violated and the total distance traveled is minimized. The authors mention the NP-hardness of this problem, since it generalizes the BPP (cf., our Section 3.2.1). The main focus is on investigating approximation algorithms. Bosman et al. [START_REF] Bosman | Approximation algorithms for replenishment problems with fixed turnover times[END_REF] consider a replenishment problem on a tree and on general metrics over an arbitrary time horizon. Their main results also concern approximation algorithms, but some of their complexity results are similar to ours (cf., our Section 3.2.2).

The remainder of the article is organized as follows. In Section 2 the studied IRP variant is formally described, and the relevant problem variants are presented. We also present related problems and their complexity which will play a role in the complexity analysis of our problems. In Section 3, we present complexity results of some problem variants that are easily seen

α 1 = M α 1 = 1 α 1 = M α 1 = 1 α 4 = 1 α 4 = 2 α 4 = Z α 4 = 1 α 4 = 2 α 4 = Z
polynomially solvable (in P), or intractable. The main result on a maximally easy borderline problem, for which the complexity is not trivial, is presented in Section 4. We show that the problem can be solved in polynomial time using dynamic programming.

In Section 5, we present the hardness result for a variant of the problem in the Euclidean plane. Related problems and literature for that specific variant will be discussed in Section 5. Finally, Section 6 finishes the article with concluding remarks.

PROBLEM DESCRIPTION AND RELATED PROBLEMS

In the problems we study, we are given a metric space containing N customers and a depot r. Each customer has a service or replenishment time s i , a latest delivery day at the start of the planning horizon and a period p i , i = 1, …, N, which is the maximum number of days between two replenishments. The periods are defined directly by the customer's storage capacity and the fixed daily demand. At the depot, M identical vehicles are present that can each spend at most L time units per day on traveling plus service time. The vehicles need to return to the depot at the end of a day. We assume that travel time is equal to travel distance, that is, vehicles travel at unit speed. The length of the planning horizon is Z days. Customers can be replenished at most once per day, that is, split deliveries are not allowed. A solution to this problem consists of an assignment of customers to vehicles, and a route for each vehicle, for each day in the planning horizon. A solution is feasible if each vehicle spends at most L time units per day and the time between two consecutive replenishments of customer i is at most p i days. The objective is to minimize the total time spent by the vehicles.

Problem variants

We provide a concise description of the problem variants we study in this article, in much the same spirit as done for scheduling problems in Graham et al. [START_REF] Graham | Optimization and approximation in deterministic sequencing and scheduling: A survey[END_REF] and later for dial-a-ride problems in de Paepe et al. [START_REF] De Paepe | Computer-aided complexity classification of dial-a-ride problems[END_REF]. For all variants, a vehicle can spend at most L time units a day. Remaining features of the problems are stated in a 4-field notation 𝛼 1 |𝛼 2 |𝛼 3 |𝛼 4 . In this notation, 𝛼 1 denotes the number of identical vehicles which is equal to 1 or a given M > 1, hence 𝛼 1 ∈ {1,M}. The problem is studied on a point and on the half-line, which is indicated by 𝛼 2 , 𝛼 2 ∈ {half -line, point}. We use 𝛼 2 = point to indicate that all distances are zero, that is, both the depot and the customers are located in one point. Equivalently, we could say that the vehicles drive at infinite speed. We use 𝛼 2 = half -line to indicate that the customers are located on a half-line. On the half-line, we assume the customers are numbered in increasing distance to the depot; customer i is located at distance d i from the depot r and

d 1 ≤ • • • ≤ d N .
The type of service times are denoted by 𝛼 3 , 𝛼 3 ∈ {s, s i }, in which s indicates uniform service times and s i indicates that service times can differ per customer (arbitrary service times). The planning horizon is denoted by 𝛼 4 , which can be equal to 1, 2 or a given Z > 2 days, 𝛼 4 ∈ {1, 2, Z}.

Table 1 provides an overview of the complexity results obtained in Sections 3 and 4. Each entry of the table corresponds to a configuration of type of service times, whether the customers are located on a point or on the half-line, the number of vehicles and the length of the time horizon. All these problems are characterized as polynomially solvable, NP-hard, strongly NP-hard or "PSP-hard" which is defined in the next section. Table 1 contains references to the corresponding sections for all borderline problems and, additionally, for the problems in Section 3.1.

Most of the problems in Sections 3 and 4 consider a planning horizon of 2 days. For these problems, we define three types of customers: day 1-customers (D1-customers), day 2-customers (D2-customers) and period 1-customers (P1-customers). D1-customers and D2-customers need service latest on day 1 and 2, respectively. Note that D2-customers can also be served on day 1. P1-customers need service on both days 1 and 2. Similarly, define Dh-customers which need service latest on day h and Pm-customers need service every m days.

Related problems

To assess the complexity of some variants in the set of problems, we use hardness results from the BPP and the pinwheel scheduling problem (PSP).

Bin packing problem

Given are n items with weights w 1 , …, w n and bins with capacity B. Pack the items in a minimum number of bins such that each bin contains total item weight no more than B. BPP is strongly NP-hard [START_REF] Garey | Computers and Intractability: A Guide to NP-Completeness[END_REF]. The decision problem whether all items can be packed in two bins, with bins of capacity 1

2

∑ n j=1 w j , is known as the partition problem, and is a weakly NP-complete problem [START_REF] Karp | Reducibility among combinatorial problems[END_REF]. We use these hardness results in Section 3.2.1.

Pinwheel scheduling problem

Given are n tasks with integer periods p 1 , …, p n . Each time unit, one task can be scheduled. A schedule is feasible if the time between two consecutive moments at which task i is scheduled is at most p i time units. The goal is to find a feasible schedule.

A special feature of this problem is the following. If one would decide to schedule a task one time unit earlier, the next due date for the task is also shifted one time unit back. Hence, a decision for a given time unit influences directly the situation at a later moment in the schedule and can cause conflicts there.

The complexity of the PSP is a long-standing open question, which is mainly due to the compact input description. It was only shown to be in PSPACE by Holte et al. [START_REF] Holte | The pinwheel: A real-time scheduling problem[END_REF]. Recently, it was shown by Jacobs and Longo [START_REF] Jacobs | A New Perspective on the Windows Scheduling Problem[END_REF] that the PSP cannot be solved in pseudopolynomial time, unless there is a randomized algorithm for solving the well-known satisfiability problem in time n (log n log log n) . Since the latter is unlikely, the PSP is assumed to be intractable. Yet it is unclear if it is in NP or in co-NP. We define the class of PSP-hard problems as the problems that are at least as hard as the PSP. We use this hardness notion in Section 3.2.2.

PRELIMINARY RESULTS

In this section, we consider some problems for which its computational complexity is easily established. We first discuss two easy problems that are actually not borderline easy, but we think they will provide insight into the structure of optimal solutions of the problem M | half -line | s | 2, which is discussed in Section 4. We finish this section with presenting two classes of intractable problems.

Easy problems

The two easy problems are equivalent to special cases of problem M | half -line | s | 2. The first easy problem is the problem on a point, instead of on a half-line. In the second easy problem, the planning horizon is restricted to 1 day, instead of 2 days.

Problem M|point|s|2

There are M vehicles, the depot and all the customers are located in one point, the customers have uniform service times (s i = s ∀ i) and there is a planning horizon of 2 days. Let #P1, # D1, and #D2 be the number of P1-customers, D1-customers, and D2-customers, respectively. Since there is no travel time and all customers have equal service times, it is most efficient to serve each customer (with p i > 1) exactly once in the 2-day planning horizon. If the problem is feasible, this gives the optimal objective value. To check feasibility, we just need to check two things. First, whether M vehicles suffice to serve all P1-customers and D1-customers on day 1: #P1 + # D1 ≤ M⌊L/s⌋. And, if so, second, whether M vehicles suffice to satisfy all the customers' service requirements in the 2 days, taking into account that a D2-customer can be served on day 1: 2 # P1 + # D1 + # D2 ≤ 2M⌊L/s⌋. Thus, the running time of the algorithm is linear in N.

Problem M | half -line | s | 1

In this problem variant on the half-line, there are M vehicles, uniform service times (s i = s ∀ i) and a planning horizon of 1 day.

Recall that there are N customers which are numbered in order of increasing distance to the depot on the half-line. Define a "region" to be the interval in which a given set of customers is located on the half-line. Polynomial solvability follows from the following lemma, which is easily proved by a simple exchange argument, left to the reader.

Lemma 1.

In an optimal solution of the given problem, the regions of customer locations served by any two vehicles are disjoint.

The optimal solution is established as follows. Let a vehicle serve the farthest customer N and include, on the way to the depot, as many customers as possible with the highest indices. Let the next vehicle serve the farthest customer that is not served by the first vehicle and again, let this vehicle serve as many customers as possible. Continue until all M vehicles are used or all customers are served. This results in a solution in which each vehicle serves consecutive customers on the half-line. In case all vehicles are used and there are some customers left, there is no feasible solution. Otherwise, a feasible solution is found, which is clearly optimal by construction. Hence, if the order of the customers is given, this is a linear time algorithm.

Hard problems

The problem variants studied in this section are the problems having either arbitrary processing times s i or a planning horizon of Z days. Both problem variants are minimally hard, that is, restricting one of the characteristics makes a problem easy to solve.

Problem M|point|s i |1 and problem 1|point|s i |Z

Consider the class of problems with customers having arbitrary service times s i . For the problem on a point, any feasible solution has the same objective value, hence feasibility is the core problem. The problem variant that has 1 day and one vehicle (1|point|s i |1) is trivially in P (which also holds for 1 | half -line | s i | 1). However, if there are multiple vehicles and/or multiple days in the planning horizon, the problems are equivalent to BPPs (cf., Das et al. [START_REF] Das | The train delivery problem-Vehicle routing meets bin packing[END_REF]). For example, in the problem M|point|s i |1, the vehicles are equivalent to the bins in the BPP with bin capacity equal to duration limit L. Then the problem boils down to the feasibility question whether or not the number of available bins is sufficient to be able to assign each customer to a bin. Hence, this problem and more general variants are strongly NP-hard. The problem with one vehicle and a planning horizon of Z days (1|point|s i |Z) is by similar arguments also strongly NP-hard. The problem with one vehicle and a planning horizon of 2 days (1|point|s i |2) is a weakly NP-hard problem because of its equivalence to the partition problem. Concluding, any problem variant with arbitrary service times s i and multiple vehicles and/or days is an NP-hard problem. In Section 4, but also in the next subsection, hardness is avoided through bin packing by restricting to uniform service times.

Problem 1|point|s|Z

Consider the problem on a point with uniform service times (s i = s ∀ i), one vehicle, and an arbitrary long time horizon Z. A special case of this problem, in which a vehicle can replenish at most one customer per day, that is, L = s, is equivalent to the PSP. Hence, this problem variant is PSP-hard (cf., Bosman et al. [START_REF] Bosman | Approximation algorithms for replenishment problems with fixed turnover times[END_REF]). This does not imply that this variant is NP-hard, but that it is unlikely that it can be solved in polynomial time. Note that the recurrence of replenishments can occur because of the longer planning horizon. We avoid analyzing PSP-hard problems in Section 4 by restricting ourselves to a planning horizon of 2 days.

POLYNOMIAL TIME ALGORITHM FOR M | half-line | s | 2

In this section, we prove that problem variant M | half -line | s | 2 is solvable in polynomial time. In this variant on the half-line there are M vehicles, uniform service times (s i = s ∀ i) and a planning horizon of 2 days. All previously mentioned hardness results are avoided as follows. By restricting the problem to the half-line, we avoid hardness through TSP; by restricting to uniform service times, we avoid hardness through bin packing; by restricting to a planning horizon of 2 days, we avoid hardness through PSP.

Recall that P1-customers have to be served on both days, D1-customers have to be served on day 1, whereas D2-customers can be served on either of the 2 days. To minimize total travel and service time, it must be determined how many vehicles to use every day and which customers to serve with each vehicle, such that the tour duration limit (of L time units) is not exceeded. Obviously, in any optimal solution each customer that does not need service every day is served only once. Hence total service time is always equal in any relevant solution and we disregard it from the objective from here onwards.

A dynamic programming algorithm (DP) is designed to solve this problem to optimality. To simplify explanations, in Section 4.1 we first assume that there are no customers that need service on both days (no P1-customers). In Section 4.2 this assumption will be relaxed. In Section 4.3 the running time of the DP will be analyzed, an observation to speed up the DP is made and the complexity of extensions of the problem is discussed. Finally, Section 4.4 discusses the variant of the respective IRP variant with a vehicle capacity constraint.

Dynamic programming

Before discussing the DP in detail, we observe the following. Given the farthest customer that still needs to be assigned to a vehicle, there are only a limited number of options for the other customers that will be served by the same vehicle in an optimal Suppose customer A is the farthest customer that has not been assigned to a vehicle yet. Suppose that, because of the limitation on the time, at most three more customers can be served by the vehicle if customer A is the farthest served customer. For example, customers {A, B, C, D} can be served by one vehicle. Now, suppose customer A is assigned to a vehicle v that will serve customers on day 1. Then, by Lemma 1, it is not optimal to have vehicle v serving customer F but not serving customer B. A similar argument holds for serving D2-customers with vehicle v. If customer D is served by vehicle v, but C is not served by vehicle v, this means that another vehicle w has to drive up to customer C. By interchanging customers C and D a better solution is constructed.

Hence, in general, in an optimal solution, any vehicle serves a combination of consecutive D1-customers and consecutive D2-customers, that is, no vehicle skips a Dh-customer to serve another Dh-customer closer to the depot. This observation is used in the DP. The idea of the DP is to start at the customer farthest from the depot and work backwards to the depot. For every customer that needs service latest on day 2, a decision must be made whether this customer is served on day 1 or 2.

As a basis for the DP, Lemma 2 first generalizes Lemma 1 to prove the observation in the example above. Then, the DP is formulated.

Lemma 2. Starting from the farthest customer and moving toward the depot, no vehicle skips a Dh-customer to serve another Dh-customer closer to the depot, for h ∈ {1, 2}.

Proof.

First, consider the case of assigning customers that need service latest on day h to a vehicle v serving these customers on day h (case 1).

Case 1: the Dh-customers assigned to vehicle v will be consecutive on the half-line, otherwise the solution can be improved with the same interchange argument used in Lemma 1.

Second, consider the case of assigning customers that need service latest on any of the 2 days to a vehicle v on one of these days (such that the services are feasible). Define S(v) to be the set of customers served by vehicle v. Again, distinguish two cases: the customer in S(v) farthest from the depot is a D1-customer (case 2a) and a D2-customer (case 2b), respectively.

Case 2a: by case 1 it has been shown that D1-customers served by vehicle v are consecutive on the half-line. It remains to show that the D2-customers served by vehicle v are also consecutive on the half-line. Let A be the farthest D1-customer, B the farthest D2-customer and C the second farthest D2-customer from the depot such that d C < d B . Suppose there is an optimal assignment in which customer B is not assigned to vehicle v on day 1, but customers A and C are: (A,C ∈ S(v) and B ∉ S(v)). Then, for customer B it has to be decided on which day it is served and by which vehicle. If customer B is served on day 1 with vehicle w, this gives overlap between the delivery regions of vehicles v and w on day 1, which cannot be optimal by Lemma 1. Hence, customer B must be served on day 2 by vehicle u (A,C ∈ S(v) and B ∈ S(u)) and the cost of vehicles v and u is d A + d B . By interchanging customers B and C, the solution is still feasible, but the cost is d A + d C which is less than d A + d B since customer C is closer to the depot than customer B which contradicts the assumption of optimality.

Case 2b: by case 1 it is known that D2-customers served by vehicle v on day 2 are consecutive on the half-line. Hence, if no D1-customers are served by vehicle v the lemma has been shown. If any D1-customer is served by v, the same arguments used in case 2a prove the statement also for this case, and therefore complete the proof. ▪

It is easy to see that this lemma can be extended to h ≥ 3. We will use Lemma 2 to find an optimal solution for problem M | half -line | s | 2 in polynomial time. Again, assume that the customers on the half-line are numbered 1,2, …, N in increasing distance from the depot.

In the DP, in every state the customer farthest from the depot that has not been assigned to a vehicle yet is considered. The crucial observation is that given this farthest unassigned customer, n, and its latest service day h, the number of customers with latest service day 1 that is served by the same vehicle v as customer n defines the next state. Define C(𝓁) as the number of customers that can be served by a vehicle within its time limit L given that customer 𝓁 is the farthest customer served by the vehicle. Define k as the number of D1-customers that are served by a vehicle (including the farthest customer if it is also a D1-customer). The value of k ranges from 0 (if n is not a D1-customer) to C(n). Given a value of k, the optimal route for vehicle v is easy to find: select the customers farthest from the depot including exactly k D1-customers. If k ≥ 1 vehicle v has to ride on day 1. Note that servicing only D2-customers on day 1 is also an option. Therefore, if k = 0, it is yet to be decided in the DP on which of the 2 days vehicle v will ride, given that there are still vehicles left for both days. Concluding, the value of k, the availability of vehicles, and the decision on the delivery day, determine both the set of customers served by the same vehicle as customer n and the day on which these customers are served. Hence, to find the optimal solution it is sufficient to consider for each customer n all possible values of k and the decision whether to serve a set of only D2-customers on day 1 or 2.

Let the indices of the D1-customers, in increasing order, be denoted by 1 1 , 2 1 , …, I 1 , with I the total number of D1-customers. Similarly, D2-customers have indices 1 2 , 2 2 , …, J 2 , with J the total number of D2-customers. Hence, customer 1 (which is closest to the depot) is either 1 1 or 1 2 and similarly, N = I 1 or N = J 2 . Let n be the index of the customer farthest from the depot that still needs assignment to a vehicle, and let i 1 and j 2 be the indices of the D1-customer and D2-customer farthest from the depot that still need assignment. Hence, n = i 1 or n = j 2 . Let M be the total number of vehicles available per day and let m 1 and m 2 be the number of remaining available vehicles for day 1 and 2, respectively. Define the value 0 for the indices i 1 and j 2 for the case that no D1-customer and D2-customer, respectively, exists or none is still to be assigned and let x + = max{x, 0}.

A state of the DP is denoted by <i 1 , j 2 , m 1 , m 2 >. Let f (i 1 , j 2 , m 1 , m 2 ) be the minimal cost (time) of serving all customers in this state. The total minimal costs of an instance is given by f (I 1 , J 2 , M, M). We present two recursion formulas: for i 1 > j 2 , in which a D1-customer is the farthest unassigned customer:

f (i 1 , j 2 , m 1 , m 2 ) = min k=1,…,min{C(i 1 ),i} {2𝑑 i 1 + f ((i -k) 1 , ((j -C(i 1 ) + k) + ) 2 , m 1 -1, m 2 )}
and for the opposite case j 2 > i 1 :

f (i 1 , j 2 , m 1 , m 2 ) = min { min k=0,…,min{C(j 2 ),i} {2𝑑 j 2 + f ((i -k) 1 , ((j -C(j 2 ) + k) + ) 2 , m 1 -1, m 2 )}, 2𝑑 j 2 + f (i 1 , ((j -C(j 2 )) + ) 2 , m 1 , m 2 -1)
} .

The second recursion (implicitly) compares three situations: at least one D1-customer included, only D2-customers with service on day 1, only D2-customers with service on day 2. By the restriction of the choice of k in the recursion it is avoided that ik could become less than 0. But j -C(n) + k < 0 may occur (though never optimal in combination with ik > 0). Define the following starting conditions:

f (0, 0, m 1 , m 2 ) = 0 ∀m 1 , m 2 ≥ 0 f (i 1 , j 2 , x, m 2 ) = ∞ ∀i 1 > 0, m 2 ≥ 0, x ≤ 0 f (i 1 , j 2 , x, y) = ∞ ∀max{i 1 , j 2 } > 0, x, y ≤ 0.

Including customers with period 1

It remains to extend the proof to the general case in which customers with period 1 (service is required on both days) can be present. First, we argue that the same reasoning as in Lemma 2 still provides an optimal solution if there are customers with period 1. Second, the DP is adjusted to cover for these customers. First, interpret P1-customers as two customers, where the first is an additional D1-customer and the second customer must be served on day 2. This observation leads to three sets of customers. The first set of customers , with indices 1 1 , 2 1 , …, I 1 , has to be served on day 1. Note that set  contains both the original D1-customers and the converted D1-customers. The second set , with indices 1 2 , 2 2 , …, J 2 , contains the D2-customers. Finally, the third set of customers , with indices 1 3 , 2 3 , …, Λ 3 , must be served on day 2.

The need for having three sets is illustrated in Figure 2. Suppose that A 1 has already been assigned to a vehicle, so that A 2 is the farthest unassigned customer. Further, assume that C(B) = 3 and C(A 2 ) = 2. Now, it is optimal to serve A 2 and D 2 together on day 2 and to serve B, C and D 1 together on day 1. Observe that in the optimal solution there is a vehicle that does not serve consecutive customers of  ∪ . However, it does serve consecutive customers of every set as defined above, which holds for any optimal solution and is formalized in Lemma 3.

Lemma 3.

Starting from the farthest customer and moving toward the depot, no vehicle skips a customer in set  to serve another customer in set  closer to the depot, for  ∈ {, , }.

The proof of Lemma 3 is similar the proof of Lemma 2 and is omitted for reasons of conciseness. To describe the dynamic programming algorithm we define the following notation. Again, let n be the index of the customer farthest from the depot that still needs assignment to a vehicle, and let i 1 , j 2 and 𝜆 3 be the indices of the farthest customers in

Day 2 Day 1 A1 B D1 A2 C D2 FIGURE 2
Example in which elements of  are represented by circles, elements of  by squares, and elements of  by diamonds sets ,  or , respectively, that still needs assignment. Hence, n = max{i 1 , j 2 , 𝜆 3 }. Let M be the total number of vehicles available per day and let m 1 and m 2 be the number of remaining available vehicles for day 1 and 2, respectively. Redefine k to be the number of customers from set  that are served by a vehicle on day 1 and define 𝓁 to be the number of customers from set  that are served by a vehicle on day 2. Note that if n = j 2 , then depending on the day j 2 is served, either customers from  or customers from  can be served by the same vehicle that serves j 2 . Define the value 0 for the indices i 1 , j 2 and 𝜆 3 for the case that no customer exists in sets ,  or , respectively, equivalent to the definition in Section 4.1.

Denote the current state of the DP by <i 1 , j 2 , 𝜆 3 , m 1 , m 2 > and let f (i 1 , j 2 , 𝜆 3 , m 1 , m 2 ) be the minimal cost (time) of serving all customers in this state. The total minimal costs of an instance is given by f (I 1 , J 2 , Λ 3 , M, M). Three recursion formulas define the DP: the first one for max{i 1 , j 2 , 𝜆 3 } = i 1 , that is, a customer in  is the farthest unassigned customer:

f (i 1 , j 2 , 𝜆 3 , m 1 , m 2 ) = min k=1,…,min{C(i 1 ),i} {2𝑑 i 1 + f ((i -k) 1 , ((j -C(i 1 ) + k) + ) 2 , 𝜆 3 , m 1 -1, m 2 )}
the second one for max{i 1 , j 2 , 𝜆 3 } = j 2 :

f (i 1 , j 2 , 𝜆 3 , m 1 , m 2 ) = min { min k=0,…,min{C(j 2 ),i} {2𝑑 j 2 + f ((i -k) 1 , ((j -C(j 2 ) + k) + ) 2 , 𝜆 3 , m 1 -1, m 2 )}, min 𝓁=0,…,min{C(j 2 ),𝜆} {2𝑑 j 2 + f (i 1 , ((j -C(j 2 ) + 𝓁) + ) 2 , (𝜆 -𝓁) 3 , m 1 , m 2 -1)}
} and the third one for max{i 1 , j 2 , 𝜆 3 } = 𝜆 3 :

f (i 1 , j 2 , 𝜆 3 , m 1 , m 2 ) = min 𝓁=1,…,min{C(𝜆 3 ),𝜆} {2𝑑 𝜆 3 + f (i 1 , ((j -C(𝜆 3 ) + 𝓁) + ) 2 , (𝜆 -𝓁) 3 , m 1 , m 2 -1)}.
The following starting conditions hold:

f (0, 0, 0, m 1 , m 2 ) = 0 ∀m 1 , m 2 ≥ 0 f (i 1 , j 2 , 𝜆 3 , x, m 2 ) = ∞ ∀i 1 > 0, m 2 ≥ 0, x ≤ 0 f (i 1 , j 2 , 𝜆 3 , m 1 , x) = ∞ ∀𝜆 3 > 0, m 1 ≥ 0, x ≤ 0 f (i 1 , j 2 , 𝜆 3 , x, y) = ∞ ∀max{i 1 , j 2 , 𝜆 3 } > 0, x, y ≤ 0.

Running time and generalizations

The running time of the DP is polynomial. To see this, note that the number of states to be considered is (N 3 M 2 ) and each computation of the recursion takes (N) time. Moreover, we may assume without loss of generality that M ≤ N. Hence, our DP runs in (N 6 ) time.

Theorem 1. Problem M | half -line | s | 2 can be solved in (N 6 ) time.
A first possible generalization of problem M | half -line | s | 2 is extending the planning horizon to more than 2 days, for example 3 days. Then, immediately an essentially different ingredient is added to the problem: serving a D2-customer with period 2 on day 1 leads to the obligation of serving it again latest on day 3. This periodicity, asking for repetitive service, is the main issue in problems related to the PSP. As mentioned in Section 2.2, this is a badly-understood problem.

A second generalization concerns the underlying metric spaces. The DP can be adapted to yield a polynomial time algorithm for problem M|⋅|s|2 on a line or a cycle. Furthermore, the problem on a tree is NP-hard, even if the planning horizon is a single day and the tree is a star with the depot at the center. This follows again through equivalence to BPP, because the travel time to each customer can be different. Section 5 considers the IRP as defined in Section 2 in the Euclidean plane.

Vehicle capacity

Another aspect that can be incorporated in the studied IRP, is vehicle capacity in terms of load, that is, the maximum number of units demand that can be delivered by a vehicle in 1 day. To facilitate the exposition of the impact of vehicle capacity constraints, consider the case which discards service times.

In case the demand of each customer can be different, similar argumentation as in Section 3.2.1 for problems with arbitrary service times can be used to establish NP-hardness for all problems with more than one vehicle or more than 1 day. In case all demands are identical, the same argumentation can be followed as for equal service times (s i = s ∀ i) and tour duration limit L in all problem variants. Also the DP still holds for the equivalent problem by only adjusting the definition of the function C(𝓁) that defines the number of customers that can be served by a vehicle given that customer 𝓁 is served. Note that if a vehicle only has a capacity constraint instead of a tour duration constraint, the maximal number of customers that can be served C(𝓁) is the same for any customer 𝓁. By discarding service times and replacing them with uniform demands for the customers and a capacity constraint on the vehicle, solving the problem on a star is no longer hard since BPP in terms of service time is no longer an underlying hard problem.

INVENTORY ROUTING IN THE EUCLIDEAN PLANE

Consider the IRP as defined in Section 2 in the Euclidean plane with a single vehicle, uniform service times at the customers (s i = s ∀ i) and a time horizon of Z days, denoted by 1|plane|s|Z. Again, the travel time is equal to the total distance traveled. To avoid immediate NP-hardness from routing, we approximate the route length which provides an easy route length computation, instead of computing the exact optimal route length which requires solving TSPs [START_REF] Papadimitriou | The Euclidean traveling salesman problem is NP-complete[END_REF]. In spite of trivializing the routing cost computation, we show that the resulting problem is NP-hard.

This variant of the IRP is interesting to investigate theoretically, given the discussion on the PSP in Section 2.2, but also has a practical application [START_REF] Baller | The dynamic-demand joint replenishment problem with approximated transportation costs[END_REF]. For the tour length approximation, we use a result of Beardwood et al. [START_REF] Beardwood | The shortest path through many points[END_REF] who show that the tour length is asymptotically equal to 𝜙 √ A ⋅ N for large N, where 𝜙 is a constant and A is the surface of the area in which the N points can be placed uniformly at random. Chien [START_REF] Chien | Operational estimators for the length of a traveling salesman tour[END_REF] considers approximation functions with a similar functional form, but considers several areas for A which take the actual depot and customer locations into account instead of the area in which the customers can be located as in Beardwood et al. [START_REF] Beardwood | The shortest path through many points[END_REF]. As an approximation to the route length we use the same functional form as Beardwood et al. [START_REF] Beardwood | The shortest path through many points[END_REF] and Chien [START_REF] Chien | Operational estimators for the length of a traveling salesman tour[END_REF], and compute the area of the customers as the convex hull of the locations of the customers. The objective is to find a feasible solution, obeying customer periods and the tour duration constraint, that minimizes the total approximated route length. This section shows strong NP-hardness for the studied IRP with this tour length approximation as route length function.

Note that this IRP with the route length approximation is a generalization of the PSP in which the tasks are executed at different locations and more than one task can be scheduled per day. Hence, this section shows that we can prove NP-hardness of a generalization of the PSP in which the tasks are executed at different locations without using the hardness of TSP. Besides that, this IRP has features of the joint replenishment problem (JRP), which is an NP-hard problem [START_REF] Arkin | Computational complexity of uncapacitated multi-echelon production planning problems[END_REF]. The JRP is a multiperiod replenishment problem in which a fixed fee is incurred per customer replenishment and per period in which at least one customer is replenished. Since in the JRP a fixed fee is paid per served customer, the JRP is not a special case of the IRP with approximated route length because of the different cost structure.

A reduction from 3-partition [START_REF] Garey | Computers and Intractability: A Guide to NP-Completeness[END_REF] shows strong NP-hardness for the studied IRP in the plane with approximated route length. 3-Partition is defined as follows: given 3m integers a 1 , …, a 3m such that ∑ 3m i=1 a i = 𝑚𝐵, the question is whether there exists a partition in sets S 1 , …, S m such that |S j | = 3 and

∑ i∈S j a i = B for j = 1, …, m. The problem is hard even if B/4 < a i < B/2 for i = 1, …, 3m.
Theorem 2. IRP in the plane with approximated route length is strongly NP-hard.

Proof. Given an instance of 3-partition, create the following instance of IRP in the plane. Create customers with two different periods, period 1 and period m. Take (c 1 , …, c 3m ) as the 3m extreme points of a regular polygon with area P. One of these points is chosen as the depot. Let all other points contain a customer with period 1.

Second, choose another 3m locations outside the polygon, each making a triangle of area Q with two neighboring extreme points of the regular polygon, as depicted in Figure 3. Let each of these locations have a set of customers with period m. At the first such location a number of a 1 customers is located, corresponding to the integer value a 1 from the 3-partition instance, at the second location a 2 customers are located, and so on. Moreover, P and Q are chosen such that an angle 𝜃 in Figure 3 is at most 180 • . There is a single vehicle with tour duration limit 𝜙 √ (P + 3Q) ⋅ (B + 3m). Thus, there are ∑ 3m i=1 a i = 𝑚𝐵 Pm-customers spread over 3m locations. We will show that there is a 3-partition if and only if there is a feasible schedule in which no customer is out of stock and every day the approximated length of the tour is at most 𝜙 √ (P + 3Q) ⋅ (B + 3m). If there is a 3-partition, serve all customers at the location corresponding to integer a i for each a i in set S j on day j. Moreover, all 3m P1-customers are served every day. Clearly, this is a feasible solution for the IRP instance. It remains to show that the bound on the route length holds. The total area per day to be covered in the route length function is P for the 3m customers with period 1 and Q per set a i . Since there are exactly three such sets every day, the total area per day is P + 3Q. The number of customer services is 3m for the customers with period 1 plus ∑ k∈S j a k for the selected sets on day j which is exactly equal to B by the 3-partition. Hence, each day the approximated length of a tour is 𝜙 √ (P + 3Q) ⋅ (B + 3m). Reversely, if there is a feasible schedule for the planning horizon of m days for which the approximated route length on each day is at most 𝜙 √ (P + 3Q) ⋅ (B + 3m), a feasible 3-partition can be derived. First, since ∑ i a i = mB, B Pm-customers and 3m P1-customers should be served on average per day. Since a i < B/2, serving more than B + 3m customers can only be done by adding at least three times Q to the area of the convex hull. Hence, exactly B + 3m customers will be served every day. This means that the set of Pm-customers can be partitioned into subsets with exactly B customers each. Then, because B/4 < a i < B/2 for all i, on each day exactly three sets of customers a i are served. Hence, the schedule corresponds to a feasible 3-partition. ▪

CONCLUSION

The main positive result in this article is a polynomial time dynamic programming algorithm for the borderline problem variant M | half -line | s | 2. In this problem on the half-line the planning horizon is 2 days, there are M > 1 vehicles available to serve the customers with uniform service times s.

If we extend the planning horizon from 2 days to any number of days, the problems, even on a point, are at least as hard as the PSP, for which the complexity has not been determined. This is due to the fixed periods of the customers, which leads to a compact input description. The complexity of the considered IRP is open if the number of days is fixed but greater than 2, for example, if M | half -line | s | 3. These problem variants may very well be polynomially solvable, independent of the complexity of the PSP.

Less surprising is that allowing customers to have arbitrary service times introduces bin packing aspects into the problem, making the resulting IRP NP-hard, even when defined on a point. Our results show that not only the presence of a routing problem contributes to the complexity of the IRP, but also the service times and the periodicity of replenishments of the customers.

FFIGURE 1

 1 FIGURE[START_REF] Alvarez | Iterated local search and simulated annealing algorithms for the inventory routing problem[END_REF] Example for observation, the depot is represented by black squares, D1-customers are represented by circles, and D2-customers by squares

FIGURE 3

 3 FIGURE 3 Positions customers for m = 1

TABLE 1

 1 Overview complexity results, where problems are either in P (dots), NP-hard (diagonal lines), strongly NP-hard (vertical lines), or PSP-hard (horizontal lines) [Color table can be viewed at wileyonlinelibrary.com]

		Point, α 2 = point	Half-line, α 2 = half -line
	Unif. service times, α 3 = s		3.1.1	3.2.2	3.1.2	4
	Arb. service times, α 3 = s i	3.2.1	3.2.1	3.2.1	3.2.1

ACKNOWLEDGMENTS

A.C.B. and M.H. were funded by the Netherlands Organisation for Scientific Research (NWO), project number 407-13-050. M.v.E. was funded by NWO, project number 612.001.215. M.v.E. was employed by Vrije Universiteit Amsterdam when working on this article. Research by L.S. was partially supported by the NWO through the Gravitation Programme Networks (024.002.003). The authors want to thank the anonymous reviewers and the editors for their valuable comments.

This research was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek, 024.002.


Ruti Segal
email: rutisegal@gmail.com

Atara Shriki
email: atarashriki@gmail.com

Nitsa Movshovitz-Hadar

Boaz Silverman

Interweaving Mathematical-News-Snapshots as a facilitator for the development of Mathematical Knowledge for Teaching

Keywords: Specialized Content Knowledge, Knowledge of Content and Students, Horizon Content Knowledge, Mathematical News Snapshots (MNSs)

The study described in this paper accompanies a three-year R&D project that is currently in its final year. Participants are 25 high school mathematics teachers, of whom 14 have started their 2 nd year and 11 are currently in their 3 rd year of integrating Mathematical-News-Snapshots in their classes. This study examines the effect of integrating these snapshots in the ordinary high school curriculum on the development of several components of Mathematical Knowledge for Teaching. The findings indicate that teachers perceive the integration of Snapshots of contemporary mathematics as contributing to the development of their Specialized Content Knowledge, Knowledge of Content and Students, and Horizon Content Knowledge.

Theoretical background

Knowledge of mathematics teachers

Over the past few decades, the mathematics education research community has become interested in characterizing the required knowledge for mathematics teaching. [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF] recognized that the required teacher's knowledge is unique and suggested two main types of knowledge teachers required, not necessarily in mathematics: Subject Matter Knowledge (SMK) -a deep understanding of the subject matter, and Pedagogical Content Knowledge (PCK)a repertoire of representations, analogies, models, explanations, and demonstrations. Relying on Shulman's work, [START_REF] Ball | Toward a practice-based theory of mathematical knowledge for teaching[END_REF] attempted to distill the knowledge required from mathematics teachers. They defined the term "Mathematical Knowledge for Teaching" (MKT) as knowledge that crosses areas and levels of school mathematics, knowledge that supports connected ideas of the mathematics teacher, and emphasizes the ability to plan, evaluate, integrate and manage appropriate mathematical content for teaching. Following this work, [START_REF] Ball | Content knowledge for teaching: what makes it special[END_REF], presented six different components of mathematical knowledge for teaching. Three of them are particularly relevant to this study:  Specialized Content Knowledge (SCK). This component includes the mathematical knowledge and skills unique to teaching that is not typically needed for purposes other than teaching, such as searching for student errors, intelligent integration of examples during instruction;  Knowledge of Content and Students (KCS). This component combines knowing about students and knowing about mathematics, namely, knowing how students think, what may confuse them, what kind of mathematical task may be easy or difficult for them, and more. This kind of knowledge requires an interaction between specific mathematical understanding and familiarity with students and their mathematical thinking;

 Horizon Content Knowledge (HCK). "Horizon knowledge is an awareness of how mathematical topics are related over the span of mathematics included in the curriculum" (ibid p. 403). [START_REF] Ball | With an eye on the mathematical horizon: knowing mathematics for teaching to learners' mathematical futures[END_REF] suggested a more detailed definition of HCK: "knowledge as an awarenessmore as an experienced and appreciative tourist than as a tour guideof the large mathematical landscape in which the present experience and instruction is situated. It engages those aspects of the mathematics that, while perhaps not contained in the curriculum, are nonetheless useful to pupils' present learning, that illuminate and confer a comprehensible sense of the larger significance of what may be only partially revealed in the mathematics of the moment." (Ibid ,2009, p. 5). They present four components of HCK: A sense of the mathematical environment surrounding the current "location" in instruction; major disciplinary ideas and structures; key mathematical practices; core mathematical values and sensibilities. There is a strong connection between MKT components and mathematical quality of instruction [START_REF] Hill | Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study[END_REF][START_REF] Ma | Knowing and teaching elementary mathematics: Teacher's understanding of fundamental mathematics in China and the United States[END_REF]. Teachers with a strong MKT can present their students with rich mathematical links, representations and examples beyond what is given in most textbooks. These teachers are familiar with the curriculum and with ways to relate students to various subjects. In contrast, teachers with a poor MKT have a limited arsenal of examples to cater to diverse student needs.

The Mathematical News Snapshots project

Mathematics is a dynamic field that is developing constantly. The current number of publications in the mathematical reviews database is presently over three million, and over 80,000 additional reviewed papers are added to the database each year (American Mathematical Society, 2018). However, school mathematics curricula generally do not reflect this dynamic nature nor the constant struggle of mathematicians to discover new results. As a result, high school graduates often graduate while erroneously perceiving mathematics as a field in which all questions and answers are known in advance, and there is little space for further research [START_REF] Amit | Design and high-school implementation of Mathematical-News-Snapshots -An action research into 'Today's News is Tomorrow's History[END_REF]. In some countries, this situation has led to a decline in enrollment to mathematically intensive academic studies, and even developed into an economic, social, and national concern (Noyes & Adkins, 2017).

The Mathematical News Snapshots (MNSs) project, in which the current study is situated, was set up to provide a better perspective of the nature of contemporary mathematics by introducing highschool students to current frontiers in the rapidly growing area of mathematics [START_REF] Movshovitz-Hadar | Today's news are tomorrow's history -Interweaving mathematical news in teaching high-school math[END_REF]. The chosen method was to prepare high school teachers to interweave snapshots of mathematical news in their teaching on a regular basis. Each MNS is developed in the form of a PowerPoint presentation, with three main characteristics [START_REF] Amit | Design and high-school implementation of Mathematical-News-Snapshots -An action research into 'Today's News is Tomorrow's History[END_REF]): (i) A new mathematical result published in the professional literature in the past few decades; (ii) Exposition of about 25 minutes focusing on the new result, elaborating on its history, the main underlying ideas, and the people involved in the field; and (iii) Taking into account students' limited background, preferably linking it to some topics in the mandatory curriculum, without harming the progress in their teaching.

The current study focuses on one questionhow do high-school mathematics teachers perceive the contribution of integrating MNSs (in their mathematics lessons) to the development of their Mathematical Knowledge for Teaching?

This study is part of a larger research program that examines the experiences of teachers as well as of students participating in the MNS project.

Method

Setting: Mathematics teachers participating in the MNS project attend a dedicated three-year professional development program. This program involves both face-to-face meetings and online meetings in which teachers are getting acquainted with seven MNSs each school year (Grades 10-12) and hold discussions about the mathematical, historical, and innovative contents of the MNSs. Following the implementation of each MNS in their classes, teachers are required to document their experiences (e.g., their own reflections and their students' responses) into an online database via a dedicated proprietary software called "Ramzor" (lit. Traffic Light). These reports often entail information regarding students' comments and classroom atmosphere, as well as suggested accompanying activities, recommended videos, and questions regarding possible mistakes in the presentation. The project staff provides ongoing one-on-one support for each participating teacher.

To date, we have completed two and a half (out of three) years of our longitudinal study.

Sample:

The participants are 25 high school mathematics teachers who volunteered to participate in the MNS project: 14 teachers have integrated MNSs for one year ("beginning teachers" -Group A), and 11 teachers have integrated MNSs for two consecutive years with the same students ("continuing teachers" -Group B). (They are continuing this year, but the data s restricted to the 2 nd year).

Research instruments: Three data collection instruments are used in this study: Teacher questionnaires, teacher interviews, and field notes. Teacher questionnaires. At the beginning of the first year (prior to the students' first introduction to the MNSs) and at the end of each school year, each participating teacher is asked to complete a questionnaire. The questions involve three types of items: 5-level Likert-type scale items, multiplechoice items, and open-ended items. The intention was to trace the development of teachers' MKT following the integration of MNSs in their lessons. The questionnaires were constructed with, and validated by, a team of mathematics education experts. Participants' interviews. We conducted an informal semi-structured interview with each teacher at the end of each school year, in the form of a friendly conversation conducted online via video chat. The interviews dealt with the considerations that guided the teacher in integrating the MNSs. Field notes -Analysis included all (and only) comments made by teachers on the proprietary software and during conversations with the project staff.

Data analysis:

We utilized a qualitative research paradigm, implementing a process of open and axial coding, to identify the main categories and sub-categories [START_REF] Corbin | Basics of qualitative research: Techniques and procedures for developing grounded theory[END_REF]. At the first stage, the teachers' responses to the questionnaires were analysed, to generate preliminary categories. In the second stage, the data analysed from the observations and interviews in order to identify and map the initial categories. We identified a correlation between the reports received from the final evaluation questionnaire and the knowledge obtained from the interviews that supports the findings [START_REF] Miles | Qualitative data analysis: A sourcebook of new methods[END_REF]. In the third phase of the analysis, while reviewing the findings of the first and second stages, we focused on the data related to the development of key components of MKT.

Findings

The integration of the MNSs introduced teachers to diverse and innovative mathematics contents, in a systematic and structured manner. The teachers' experience in integrating the MNSs led to the development of various teachers' knowledge components, as they perceived it. Some of which are presented below.

Teachers' perceptions about Integrating MNSs as a lever for the development of their mathematical knowledge

In each end-of-year questionnaire the teachers were asked to relate to the contribution of the integration of the MNSs to their mathematical knowledge, both in the context of school curriculum and beyond the curriculum. Here are a few quotes (translated from Hebrew to English) from group A teachers' testimonials about the expansion of their mathematical knowledge: "The MNSs helped me understand, and present in class the distinction between proofs and refutation"; "I realized that there are proofs of impossibilities, such as the impossibility to trisect an angle"; "I learned about the existence of notions such as Mersenne number". As can be seen from Table1, both groups perceived the contribution of integrating the MNSs to their mathematical knowledge beyond the curriculum was greater than its contribution to knowledge related to the curriculum itself. Group A teachers maintained that: "I was exposed to mathematics theorems such as Fermat's last theorem and Kepler's theorem"; "I became familiar with interesting mathematical subjects that I did not know, like the connection between guarding an art gallery and mathematics". Similar to Group A, Group B teachers reported that: "My mathematical knowledge was expanded, through integrating MNSs for example I learned from the MNS about Efron cubes that the transitive rule does not always holds"; "Through integrating MNSs, I learned with my students a lot about contemporary mathematics "; "There are many mathematical contents I was not familiar with. For example, the frequency of the digit 1 as a leading digit". The findings from the interviews with the teachers reinforced the findings from the questionnaires. They indicated that all the interviewed teachers (Seven out of seven) perceived an expansion and deepening of their mathematical knowledge. "It was new to me that nowadays mathematicians employ technology not only for proving theorems but also for validating proofs". In addition, the teachers were exposed to the protracted work of proving mathematical theorem: "I did not know that to prove the Fermat Last Theorem it took 300 years"; "I am happy to learn new things in mathematics, as the MNSs power point presentation there are many references, to mathematical background materials and applets". The teachers in Group A reported that their recognition about the importance of exposing the students to the content of the MNSs motivated them to join the project to start with, and to carry on their participation in it. In addition, teachers in Group A, reported about their desire to change the students' image about mathematics as a dynamic, and constantly evolving field: "The exposure of the students is very significant to their image of mathematics"; "It's important for me that my students really recognized that mathematics is an ever-growing field". The teachers in Group B added: "It is important that students recognize the world of mathematics beyond matriculation exam". The teachers' awareness of the importance of exposing students to MNSs also related to an opportunity to deal with the fears of students from math: "One of my goals in integrating MNSs in my class is to try to remove fear of mathematics and try to develop a love for mathematics". In addition, integrating MNSs helped teachers emphasize the use and the connection of mathematics to daily life: "It was very important to me at every MNSs presentation to note the fact that mathematics connects to science, which has many uses and is constantly evolving". The teachers' interviews strengthened the findings from the questionnaire. Five out of seven interviewed teachers reported that they chose to participate in the project to expose themselves and their students to mathematics as a field of constant development: "Mathematics is perceived by students, as frozen and unchanging". Integrating MNSs enabled teachers to present mathematics in an orderly, interesting and exciting way: "the integration of MNSs gave me an opportunity as a teacher to present a story after story of things that can ignite students' imagination". With integrating the

Teachers' perceptions about integrating MNSs as an attribute for their recognition of the importance of exposing students to contemporary mathematical subjects beyond the curriculum

MNSs "I can open to my students horizons for what is happening in mathematics. I did not have the ability to connect things together without participating in this project"

Teachers' perceptions about integrating MNSs as a way to deepen teachers' knowledge about the relationships between mathematics, and its applications to science and society. The teachers in Group A, reported on important knowledge from integrating MNSs in their lessons, which links mathematics to its uses: "It is Important as a teacher to have the knowledge beyond the curriculum that connect mathematics to other fields such as: agriculture, the connection to taxes, music, world maps". Group B teachers also reported a rich expanse of examples illustrating the relationship between mathematics and science through integrating MNSs: "I was amazed to discover the connection between origami folds and the development of science and medicine". All the interviewed teachers reported about the contribution of the MNSs to their knowledge of the relationship between mathematics, sciences and society. The integration of the MNSs has given teachers the exposure, and understanding, the relationship between mathematics and scientific phenomena around them: "for example, soap bubbles were very interesting. Why the bubble is round, and it has to do with both science and mathematics, I knew that the Hungarian cube knew has math around it, but now I did deepen my knowledge about it. The MNSs enriched my knowledge, opened my horizons, and the integration of the MNSs in students instruction forced me to understand it more deeply". In addition, all the interviewed teachers reported the contribution of integrating MNSs, to their knowledge about the relationship between mathematics and computer science. An example of teachers report: "The fact that the computer becomes a significant player in the stages of proof is very impressive, such as the color of the map". The teachers reported a deepening of the relationship between mathematics and phenomena occurring in nature: "For example, in the Fibonacci series, I was very excited by the physical explanation of the series' shows in the flowering mechanism. MNSs help me to understand why, and the explanation took me a few steps forward".

Discussion

The present study focused on high-school mathematics teacher`s perceptions of the contribution of integrating MNSs to various component of their MKT during the first two years. The findings as appeared above n Tables 1,2, 3 indicate that teachers perceive the interweaving of MNSs in their mathematics lessons, as a catalyzer for the development of at least three components of their MKT. This goes hand in hand with the claim by [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF] and [START_REF] Ball | Content knowledge for teaching: what makes it special[END_REF] that this is the unique mathematical knowledge required for teaching mathematics. Teachers also expressed their perceptions that by integrating MNSs in their mathematics lessons, they developed their SCK as well as their KCS. By intelligent integration of MNSs during instruction teachers perceived that they deepened their knowledge about the connection between mathematics and other field, about different ways for proof theorems, hence they have expanded their SCK. By introducing various ways in which students' curiosity can be stimulated, and identified topics in which students were interested in the curriculum and beyond, teachers have expanded their KCS. Another component that the findings illuminate is the HCK as defined by [START_REF] Ball | Content knowledge for teaching: what makes it special[END_REF]. The findings indicate that teacher`s perceptions the significant contribution of integrating MNSs on their MKT component, related to HCK [START_REF] Ball | Content knowledge for teaching: what makes it special[END_REF]. By exposed to the MNSs, they strength and deepen their awareness of the dynamic, lively and ongoing mathematics development. The teachers found out about the connections between the development of mathematics and the development of science and society. According to HCK, defined by [START_REF] Ball | With an eye on the mathematical horizon: knowing mathematics for teaching to learners' mathematical futures[END_REF], by integrating MNSs in their mathematics lessons, the teachers reflected in the various HCK components: They developed a sense of the mathematical environment surrounding the current "location" in instruction, by identified links between topics taught in the mathematical curriculum to topics outside the curriculum. They were exposed to major disciplinary ideas and structures such as various possibilities of mathematical theorem, to the logical structure of mathematical content. Finally, the teachers stated they appreciated being exposed to core mathematical values such as, collaborative work and persistence, as expressed in the MNSs, with the description of mathematicians work with their colleagues in other fields.

In conclusion, this study will continue in examining more components as the third year of this study unfolds.

Table 1 : Teachers' grading of their mathematical knowledge development, according to groups and type of knowledge (in percentage)

 1 Table 1 presents the distribution of teachers' degree of agreement with the statement: To what extent did the integration of MNSs in your mathematics lessons contribute to the expansion of your mathematical knowledge?

	Degree of Agreement

  Table 2 presents the distribution of teachers' degree of agreement with the statements: To what extent integrating the MNSs contribute to your recognition of the importance of exposing students to contemporary mathematical subjects beyond curriculum?

			Degree of Agreement
	Teachers	Very Large	Large	Medium	Little	Not at all
	Group A (N=14)	21.43	50	21.43	7.14
	Group B (N=11)	18.18	63.64	18.18

Table 2 : Teachers' grading of their recognition about the importance of exposing students to contemporary mathematical subjects beyond curriculum (in percentage)

 2

Table 3 : Teachers' grading of their recognition of the relationships between mathematics and its applications to science and society

 3 Table3shows the distribution of teachers' degree of agreement with the statement: To what extent integrating the MNSs contribute to your knowledge about the relationships between mathematics and its applications to science and society?

		Degree of Agreement (in percentage)
	Teachers	Very Large	Large	Medium	Little	Not at all
	Group A (N=14)	14.29	35.71	42.86	7.14	0
	Group B (N=11)	18.18	36.37	27.27	18.18	0

Acknowledgment

This research was supported by the Israeli Ministry of Science and Technology.


Galit Shabtay
email: galit.shabtay@smkb.ac.il

Esther Levenson
email: levenso@tauex.tau.ac.il

Examining pedagogical and classroom discourse through the lens of figured worlds: The case of an elementary school teacher

Keywords: Figured worlds, explorative instruction, professional development

The problem solving approach to teaching and learning mathematics has for some time been the focus of professional development (PD) programs. This paper discusses a case study of an elementary school teacher who participated in such a program. We apply the lens of figured worlds when analyzing her pedagogical and classroom discourses to reveal which aspects of explorative figured worlds penetrate her discourse. Findings indicated that the teacher adopted some aspects of explorative figured worlds but resisted others. The lens of figured worlds can help us, as teacher educators, to plan continued PD.

Introduction

Although there is a consensus among researchers and teacher educators regarding the benefits of what may be called explorative instruction (i.e., inquiry-based, student-centered, and cognitively demanding instruction) [START_REF] Schoenfeld | What makes for powerful classrooms, and how can we support teachers in creating them? A story of research and practice productively intertwined[END_REF], studies have shown that teacher-centered, ritual instruction is still very prominent in many classes (e.g., [START_REF] Resnick | Talking to learn: The promise and challenge of dialogic teaching[END_REF]. Resistance to explorative instruction may even be found amongst teachers who have taken part in professional development specifically promoting this type of instruction [START_REF] Heyd-Metzuyanim | Conflicting frames: a case of misalignment between professional development efforts and a teacher's practice in a high school mathematics classroom[END_REF]. One way to explain this phenomena is by viewing it through the lens of 'figured worlds' [START_REF] Holland | Identity and agency in cultural worlds[END_REF]. Figured worlds are based on one's interpretation of situations and may be revealed through discourse.

This paper presents a case study of one teacher, Yifat (a pseudonym), and analyzes her pedagogical and classroom discourses after participating in a professional development (PD) program centered on problem solving and explorative instruction. Through the lens of figured worlds, we analyze her discourses and examine which aspects of explorative instruction she adopted and which were missing.

Theoretical background

This section begins by offering some background on the notion of figured worlds and how it relates to mathematics education. It continues by describing the problem solving approach to teaching mathematics. Finally, as this paper analyzes the teacher's pedagogical and classroom discourses, the section ends by presenting how discourse is understood in the commognitive approach.

A figured world is defined by Holland and her colleagues [START_REF] Holland | Identity and agency in cultural worlds[END_REF] as "a socially and culturally constructed realm of interpretation in which particular characters and actors are recognized, significance is assigned to certain acts, and particular outcomes are valued over others" (p. 52). People 'figure' out who they are through the activities of their world, in relation to the social types that populate these figured worlds, and in social relationships with the people who perform in these worlds. In addition, artefacts and signs are attributed meaning that might differ from how those outside of the figured world interpret them. To sum up, the people within the figured world take on roles that come along with responsibilities, and in addition, certain actions, outcomes, and objects are valued more than others.

Relating figured worlds to mathematics education, [START_REF] Boaler | Identity, agency, and knowing in mathematics worlds[END_REF] pointed out that the difference between ritual and explorative instruction may not just be a matter of certain teaching techniques or forms of classroom organization, but a matter of widely different figured worlds. They defined the problem-solving, group-work mathematics instruction found in some secondary schools as a "connected knowledge" figured world. In contrast, the more common teacher-centered instruction was conceptualized as a figured world based on "received knowledge". [START_REF] Ma | Learning to teach in the figured world of reform mathematics: Negotiating new models of identity[END_REF] showed that it was difficult for prospective teachers to move from the "traditional" figured world (in which they supposedly studied at school) to the "reform" figured world of mathematics instruction, supported and enacted in their training program.

Over the years, evidence has shown that the "connected knowledge" figured world is fruitful for mathematics learning [START_REF] Schoenfeld | What makes for powerful classrooms, and how can we support teachers in creating them? A story of research and practice productively intertwined[END_REF]. In particular, [START_REF] Hiebert | The effects of classroom mathematics teaching on students' learning[END_REF] highlighted two significant aspects of such teaching: Explicit Attention to Concepts (EAC), and Students' Opportunity to Struggle (SOS). These two aspects are part of the problem-solving approach to teaching mathematics [START_REF] Hino | Comparing multiple solutions in the structured problem solving: Deconstructing Japanese lessons from learner's perspective[END_REF]. This approach emphasizes meticulous planning of detailed mathematics lessons that include implementing rich problems in such a way that students are active participants who explore mathematics. The teacher's role is to create learning opportunities that support students' conceptual understanding through problem solving, as well as supporting student agency and authority [START_REF] Schoenfeld | What makes for powerful classrooms, and how can we support teachers in creating them? A story of research and practice productively intertwined[END_REF]. Relating to all of the above, Shabtay and Hed-Metzuyanim (2018) described the figured world of exploration as one where students are free to struggle, mathematics is at the center, and the teacher and students share responsibility for the learning process.

Learning, including learning mathematics, may be seen as the process of becoming a full participant in a certain discourse (Sfard, 2008). Such participation may be characterized as lying on an axis between ritual and explorative participation (Viirman, 2015). In Sfard's commognitive framework (Sfard, 2008), a basic assumption is that doing mathematics means engaging in mathematical discourse. Discourses, according to Sfard, are different types of communication "set apart by their objects, the kinds of mediators used, and the rules followed by participants, and thus defining different communities of communicating actors" (p.93). While these discourses are not stable or distinct, but rather overlapping and constantly developing, it still makes sense, within a given context, to see them as "distinct enough to justify talking about discourses [. . .] as reasonably welldelineated wholes" (p. 91). Regarding the classroom as a community, Tabach & Nachlieli, (2016) defined a pedagogue, as a person who assumes the role of the leading participant in the discourse. Thus, pedagogical discourse is a discourse about teaching and learning [START_REF] Shabtay | Teachers' discourse on students' conceptual understanding and struggle[END_REF]. Similar to any other discourse, it is made up of certain key words, narratives, and meta-rules that dictate what to teach students, how to teach them, and who can learn (or not learn). This view is based on Sfard's (2008) view of mathematizing as participating in a discourse about mathematical objects. The pedagogical (the how and for whom) is closely intertwined with the what [START_REF] Shabtay | Teachers' discourse on students' conceptual understanding and struggle[END_REF].

Through the lens of figured worlds, this study examines which explorative narratives may be found in a teacher's classroom discourse and in her pedagogical discourse, after participating in a PD program focused on the problem solving approach to teaching mathematics.

Methodology

Yifat is an elementary school teacher with a B.Ed. degree and a specialization in mathematics. At the time of the study, Yifat had 11 years of experience teaching mathematics to fourth and fifth graders (10-11 years old), and was the mathematics coordinator in her school. This case was chosen because it illustrates the gap between a teacher's classroom and pedagogical discourses. In addition, Yifat consented to be part of the study, and agreed that her lesson and interview be recorded.

Yifat participated in a PD program for mathematics teachers centered on designing and implementing lesson plans according to the problem-solving approach [START_REF] Hino | Comparing multiple solutions in the structured problem solving: Deconstructing Japanese lessons from learner's perspective[END_REF]. During the program, teachers engaged in cognitively demanding tasks, viewed videoed lessons demonstrating the problem solving approach, discussed the pedagogical gains and mathematical ideas that emerged from those lessons, and designed lesson plans modeled on the problem-solving approach. The PD program took place over a period of eight months, and included 10 sessions. In addition, each teacher met with the PD educator three times on an individual basis, receiving personalized instruction and advice for planning and implementing mathematics lessons using the problemsolving approach in their specific classes.

Relevant to the present study, the seventh meeting of the program dealt with the construction of prime and composite numbers. During that meeting, teachers, including Yifat, engaged with a cognitively demanding task that involved representing different prime number with different signs, and composite numbers with an appropriate combination of prime number signs. For example, a card representing the number 2 had a circle on it, the card for 3 had a triangle, and the card for six had both a circle and a triangle (see Figure 1 for the symbols which represented the numbers from one to ten). During the session, teachers also viewed a Japanese lesson, demonstrating how this activity may be implemented as part of a problem-solving teaching approach to learning these new concepts. At the end of this session, Yifat invited the teacher educator to come to her fourth grade class and observe her teaching this exact lesson. Yifat expressed her belief that she "finally understood the problem-solving approach," and that she felt students would enjoy the lesson and experience meaningful learning.

Data collection and analysis

Yifat was interviewed both before and after the observed lesson. The first interview dealt mainly with the goals of the lesson, the main mathematical ideas, and the planned pedagogical processes. The post-teaching interview revolved around achievement of goals, class-driven processes, and the mathematical ideas that emerged from students and the teacher during the discussions. Each interview was audio-taped and transcribed. The lesson was video-taped and transcribed.

Relating back to the theory of figured worlds described previously, a qualitative analysis was carried out on all transcriptions searching for four dimensions of figured worlds: characters, valued actions, outcomes, and artifacts. The characters, in this case the teacher and students, play roles that involve responsibilities. Thus, when analyzing characters, we analyze the roles Yifat plays and those she refers to during her interviews. Noting the value that Yifat places (or does not place) on these four dimensions, we describe Yifat's figured world and compare it to an explorative figured world.

Findings

In the transcripts below actions are underscored with one line, outcomes with a double line, and roles and responsibility are indicated with a broken line. Artifacts are in bold.

The first interview took place in the teacher's classroom, during the recess prior to the mathematics lesson. Yifat was anxious, and yet proud, to show the teacher educator what she had prepared. Interviewer What's new?

Yifat

Hi. (Yifat is preoccupied with a bag of flashcards).

Interviewer Do you need help?

Yif'at Look. I made these symbols. These (she places the flashcards on the table).

Interviewer Okay.

Yif'at You'll see. I will arrange it for them after they sort them like this. (She arranges the cards as in Figure 1. She has more cards that have not been arranged yet.) Then we will discuss the topic. That's okay, isn't it?

Interviewer What are you going to try to achieve today? How would you describe your objective?

Yifat Uh …That they know what a prime number is, and what a composite number is.

Interviewer Hmmm…and how did you plan the lesson?

Yif'at You'll see. I planned it just like the teacher in the clip. Don't worry (she is busy with the flashcards, putting them back into the bags)…I need the sticky stuff so that I can stick them on the board (gets up to get the glue).

From the lines above, we see Yifat's preoccupation with the flashcards, signifying their value in her eyes. Yifat also states her intent to emulate the teacher in the film. Emulation is an action that might indicate her tendency towards being a ritualistic learner, i.e., she watched a documented lesson during a professional development meeting, and would like to replicate the lesson. Thus far, it seems that Yifat has chosen an appropriate problem, and has brought along appropriate props (cards) that will help her achieve the goal of students "knowing" (Yifat's term) what prime and composite numbers are. Note that she does not relate to explorations and discussions as objectives, or valued outcomes.

The observed lesson was 47 minutes long. The teacher began by randomly sticking the cards on the board and telling the children to decide how the cards could be sorted. Yifat proceeded to spend 30 minutes listening to various suggestions of how to sort the cards, calling on different students to demonstrate their method on the board. Finally, she arranged the cards on the board in ascending numerical order as represented by the symbols, in accordance with her lesson plan (see Figure 1).

For the next 12 minutes, she again led a whole-class discussion, this time relating to the arrangement of the cards on the board. Yifat tried to steer the discussion towards explaining the arrangement of the cards, as depicted below.

Yifat

So wait a second. Let's make some sense out of this so we can understand what's going on. Michal said (referring to the circles) that 2 and 2 and 2 and other items were added. And Roni says … (referring to the triangles) it skips by 3.

In the above segment of classroom discourse, Yifat takes the lead, saying "Let's make things clear." Although she takes the lead in the discourse, her use of the word "let's" hints at her inclusion of the students in the process of making sense of the symbols. She also turns to different students and asks them if they agree, sharing with them the responsibility for learning. Yet, Yifat's questions are closed yes or no questions. She does not ask those students to expound on their thoughts. Finally, her valued outcome is that we will "understand what's going on." During the last five minutes of the lesson, Yifat wrote down the numbers from 1-10 under each of the symbols and introduced the notions of prime and composite numbers. It was only during those five minutes, that mathematics was at the center of the discourse.

In the interview following the lesson, Yifat talked about the differences between this particular lesson and her regular lessons:

"Look, this was a really different type of lesson. They haven't learned composite or prime numbers at all. So, it's the first level of acquisition. And also, it's…the first time that I…gave a lesson like this, when the whole lesson is…um…only the one problem. It's different from what we usually do. They always have a lot of practice, exercises… Here (pointing to the cards) … they are just symbols that you have to categorize…and understand."

The actions mentioned by Yifat are student actions. In the past, they practiced; in the present, they categorized. Most interesting are the outcomes she values. The outcome of her lesson plan is the "different" type of lesson she perceives as having enacted. The outcome for her students is that they will "acquire" the first level of understanding. The terminology Yifat uses gives us a mixed message regarding her figured world. On the one hand, she consciously attempts to break with her former lesson style. She chooses to implement a cognitively demanding task. Yet, she resorts back to words such as "acquisition," and "giving a lesson," that remind us of a traditional figured world.

Asked if she has achieved her goals for this lesson, the following exchange ensues: The actions Yifat values for herself (e.g., explaining, telling), go hand-in-hand with how she describes her "job", or role, as an "explainer". Although during the classroom discourse Yifat did ask students to share their ways of sorting the symbols, during the interview she ascribes a passive role for her students, claiming unequivocally that they would not be able to arrive at the mathematical conclusions on their own. This interview offers evidence that Yifat's figured world is mostly traditional.

Interviewer
Referring to the main characteristics of the problem-solving approach, Table 1 summarizes the similarities and differences between Yifat's figured world and the explorative figured world. Yifat does choose to present a cognitively demanding in her class. She perceives this lesson as being quite different from those she usually teaches. Additionally, Yifat led a discussion concerning the task she presented to her pupils. These are all part of the explorative figured world. However, Yifat did not offer enough of an opportunity for her students to struggle with the problem. Recall that after she randomly stuck the symbols on the board, she immediately began to call on students to share their ways of sorting, without giving them time to think about it. When she finally arranged the cards in order, she still had difficulty shedding her central role as the teacher who explains, and where the students are attentive. These are elements of a more traditional figured world. In other words, Yifat is struggling between two worlds.

Discussion

The aim of this study was to investigate what explorative narratives enter into a teacher's classroom and pedagogical discourses after attending PD focusing on a problem-solving approach to teaching. From the first, we get mixed messages. Yifat is very enthusiastic about the task, yet is preoccupied with the artifacts (the cards). Though not entirely insignificant, the cards are actually mere props, to be used in the ultimate aim of having students engage with the concept of prime and composite numbers. Yet, during the lesson, more time is spent discussing the artifacts than on the mathematical concepts they were supposed to elicit; not enough time is allowed for exploring the possible significance of the symbols and how they might be related to mathematics [START_REF] Hiebert | The effects of classroom mathematics teaching on students' learning[END_REF]. This is also reminiscent of [START_REF] Henningsen | Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning[END_REF], who argued that a teacher must not only select and appropriately set up worthwhile mathematical tasks, but must proactively and consistently support students' cognitive activity. In addition, although her classroom discourse shows her attempts at sharing authority, her second interview contradicts this view. These findings are in line with Sfard's approach (2008), that discourses are not stable, but rather overlapping and constantly developing.

While acknowledging that discourses may be overlapping, as teacher educators, we attempt to move teachers toward a more explorative discourse. [START_REF] Sfard | Ritual for ritual, exploration for exploration: Or, what learners are offered is what you get from them in return[END_REF] suggested that for teachers to be explorative in their teaching, they must have experienced explorative mathematics learning for themselves. Yifat had limited experience with this type of learning, most of which stemmed from the PD program. Indeed, Yifat explicitly states that her aim is to imitate the teacher in the filmed lesson. She does not say that her aim is to offer the students the same type of experience she has recently had in the program. From paying attention to her discourse, we might infer that additional time must be given for teachers to experience for themselves explorative learning.

By focusing on characters, actions, outcomes, and artifacts, we can see what parts of a figured world a teacher adopts, and which seem to be restricted. As teacher educators, we can utilize this lens to listen to teachers' discourses and identify aspects that perhaps need more attention. Yifat is ready to adopt appropriate tasks, and willing to listen to her students. At this point, we would say that more attention could be paid to authority, while at the same time placing the mathematics at the center. Insight into a teacher's figured world, can help teacher educators plan appropriate professional development.

Figure 1 :

 1 Figure 1: Cards with symbols representing the numbers from 1-10

  YifatLet's make things clear. You (meaning the class) said skip by twos, and Noa just said even numbers (referring to the cards with circles on them). Let's check everything. Imagine that I have more and more cards. And there's always a shape.

	Student A	(something unintelligible)
	Yifat	Here is another circle. And something. I am skipping on purpose. What do you
		say? Yoav, do you agree with what she said?
	Yoav	Yes.
	Yifat	Daniel, do you agree?
	Daniel	Yes.

There's a circle, and another shape, and a circle and something else, and a circle and something else. (Pause) And what's here?

with one symbol and there are cards with a number of symbols. …I

  'm sure that most of them understood. And whoever didn't, I will sit with them in a small group and explain it again. You know, it's only the beginning. Look. They can't arrive at it by themselves. My job is…I have to explain it to them and that's how they'll know. They would never get to it by themselves.

	Interviewer Um…what was your objective?
	Yifat	That they would know what a prime number is and what a composite number is.
		That they would know how to differentiate between the two.
	Interviewer Okay. How do you know that they have, in fact, made that distinction?
	Yifat Because, look. I told them. They also said that there are cards Interviewer Why?
	Yifat	Because it's too complex. Later, when things are clarified, I'll be able to let them
		come up with symbols for 13 or 17, and construct 12 or 18.

Okay. So, in your opinion…did you achieve your goal?

Yifat

Of course. Look, it's only the beginning, and [so] I had to explain…so…yes.

Table 1 : A comparison between Yifa's figured world and an explorative figured world

 1

	Explorative	figured	Yifat's	figured
	world		world