Ümit V C ¸atalyürek

Kamer Kaya

Johannes Langguth
email: jlangguth@simula.no

Bora Uçar
email: bora.ucar@ens-lyon.fr

A Partitioning-based divisive clustering technique for maximizing the modularity

Keywords: 2010 Mathematics Subject Classification. Primary, 91C20; Secondary, 05C65, 05C70

We present a new graph clustering algorithm aimed at obtaining clusterings of high modularity. The algorithm pursues a divisive clustering approach and uses established graph partitioning algorithms and techniques to compute recursive bipartitions of the input as well as to refine clusters. Experimental evaluation shows that the modularity scores obtained compare favorably to many previous approaches. In the majority of test cases, the algorithm outperformed the best known alternatives. In particular, among 13 problem instances common in the literature, the proposed algorithm improves the best known modularity in 9 cases.

Introduction

Clustering graphs into disjoint vertex sets is a fundamental challenge in many areas of science [START_REF] Bern | Approximation algorithms for geometric problems, Approximation Algorithms for NP-Hard Problems[END_REF][START_REF] Everitt | Cluster Analysis[END_REF][START_REF] Jain | Algorithms for Clustering Data[END_REF][START_REF] Kannan | On clusterings: Good, bad and spectral[END_REF]. It has become a central tool in network analysis. With the recent rise in the availability of data on large scale real-world networks, the need for fast algorithms capable of clustering such instances accurately has increased significantly.

There is no generally accepted notion of what constitutes a good clustering, and in many cases the quality of a clustering is application specific. However, there are several widely accepted measurements for clustering quality called clustering indices. Among the most widespread clustering indices are expansion, conductance, and modularity. In the following, we will focus on modularity. See [START_REF] Kannan | On clusterings: Good, bad and spectral[END_REF] for a discussion of the former two indices.

Modularity was proposed in [START_REF] Newman | Finding and evaluating community structure in networks[END_REF] to analyze networks, and has recently grown in popularity as a clustering index [START_REF] Duch | Community detection in complex networks using extremal optimization[END_REF][START_REF] Fine | Spatially constrained networks and the evolution of modular control systems, From Animals to Animats 9[END_REF][START_REF] Fortunato | Resolution limit in community detection[END_REF][START_REF] Gaertler | Significance-driven graph clustering[END_REF][START_REF] Muff | Local modularity measure for network clusterizations[END_REF][START_REF] Ziv | Information-theoretic approach to network modularity[END_REF]. In addition, several heuristics based on greedy agglomeration [START_REF] Clauset | Finding community structure in very large networks[END_REF][START_REF]Fast algorithm for detecting community structure in networks[END_REF] and other approaches [START_REF]Modularity and community structure in networks[END_REF][START_REF] Reichardt | Statistical mechanics of community detection[END_REF] have been proposed for the problem. Although it was shown in [START_REF] Brandes | On finding graph clusterings with maximum modularity[END_REF] that these provide no approximation guarantee, for small real world instances the solutions produced by these heuristics are usually within a very small factor of the optimum.

In general there are two algorithmic approaches to community detection which are commonly known as agglomerative and divisive (see [START_REF] Newman | Detecting community structure in networks[END_REF] for a short survey of general techniques). Agglomerative approaches start with every vertex in a separate cluster and successively merge clusters until the clustering can no longer be improved by merging pairs of clusters. The divisive approaches on the other hand consider removing edges to detect the communities. They start with the entire graph as a cluster and successively split clusters until further splitting is no longer worthwhile. We follow the divisive approach by making extensive use of graph partitioning algorithms and techniques. A similar approach which reduces the clustering problem to a variant of the well-known MinCut problem is recently proposed [START_REF] Djidjev | Scalable and accurate graph clustering and community structure detection[END_REF].

Finding a clustering that maximizes a clustering index is often NP-hard. In particular, finding a clustering of the maximum modularity in a graph was shown to be NP-hard [START_REF] Brandes | On finding graph clusterings with maximum modularity[END_REF]. It remains NP-hard even if the number of clusters is limited to 2. In addition, APX-hardness was established recently [START_REF] Dasgupta | On the complexity of Newman's finding approach for biological and social networks[END_REF].

The remainder of this paper is organized as follows. We give some background in the next section. Section 3 contains the proposed divisive clustering method. The algorithm that we propose uses most of the standard ingredients of a graph (or hypergraph) partitioning tool: bisection, bisection refinement, and cluster refinement. We carefully put these together and explore the design space of a divisive clustering algorithm which makes use of those ingredients. We discuss, in the same section, a contrived example which shows that the divisive approaches can be shortsighted. We evaluate the proposed divisively clustering algorithm in Section 4 with different parameter settings to explore the design space. We compare the resulting modularity scores with the best known ones from the literature. Section 5 concludes the paper. Some more results from the challenge data set are provided in the Appendix A.

k , that is G(C k ) = (C k , C k × C k ∩ E, ω).
We define the weight of a vertex as the sum of the weight of its incident edges: ψ(v) = u∈V,{u,v}∈E ω(u, v), and we use ψ(C ) to denote the sum of the weights of all vertices in a cluster C . The sum of edge weights between two vertex sets U and T will be denoted by ω(U, T ), that is ω(U, T ) = {u,v}∈U ×T ∩E ω(u, v). The sum of the weights of all edges is denoted by ω(E), and the sum of the weights of the edges whose both endpoints are in the same cluster C is denoted as ω(C ). Furthermore, by cut(C) we denote the sum of the weights of all edges having vertices in two different clusters of C.

Coverage and Modularity.

We first define the coverage of a clustering, i.e., the fraction of edges that connect vertices in the same cluster:

(2.1) cov(C) = Ci∈C ω(C i ) ω(E) .
We can equivalently write that cov(C) = 1 -cut(C)/ω(E). Obviously, a good clustering should have high coverage. However, since the number of clusters is not fixed, coverage can trivially be maximized by a clustering that consists of a single cluster. It is therefore not a suitable clustering index. By adding a penalty term for larger clusters, we obtain the modularity score of a clustering:

(2.2) p(C) = cov(C) - Ci∈C ψ(C i ) 2 4 × ω(E) 2
The penalty term is such that the trivial clustering, i.e., C = {C 1 }, C 1 = V , has a modularity of 0. Like other clustering indices, modularity captures the inherent trade-off between increasing the number of clusters and keeping the size of the cuts between clusters small. Almost all clustering indices require algorithms to face such a trade-off.

Algorithms

We follow the divisive approach to devise an algorithm for obtaining a clustering with high modularity. The main motivation for choosing this approach is that for a clustering C with two clusters, the coverage is just 1 -cut(C)/ω(E) and the second term in (2.2) is minimized when clusters have equal weights. In other words, in splitting a graph into two clusters so as to maximize the modularity, heuristics for the NP-complete minimum bisection problem should be helpful (a more formal discussion is given by Brandes et al. [START_REF] Brandes | On finding graph clusterings with maximum modularity[END_REF]Section 4.1]). We can therefore harness the power and efficiency of the existing graph and hypergraph (bi-)partitioning routines such as MeTiS [START_REF]MeTiS: A software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings[END_REF], PaToH [START_REF]PaToH: A multilevel hypergraph partitioning tool[END_REF], and Scotch [START_REF] Pellegrini | SCOTCH 5.1 User's Guide[END_REF] in a divisive approach to clustering for modularity.

Algorithm 1 Divisive clustering approach using graph/hypergraph bisection heuristics Input: An edge weighted graph G = (V, E, ω) Output: K : the number of clusters; C = {C 1 , C 2 , . . . , C K }: the clusters;

p : the modularity score

1: K ← 1; p ← 0 2: C ← {C 1 = {v 1 , v 2 , . . . , v n }}
a single cluster 3: while there is an eligible cluster to consider do

4:

Let C k be an eligible cluster with the largest vertex weight 5:

C k1 , C k2 ←Bisect(C k , G) 6: if ω(C k 1 ,C k 2 ) ω(E) < ψ(C k ) 2 -ψ(C k 1 ) 2 -ψ(C k 2 ) 2 4×ω(E) 2 then 7: K ← K + 1 8: p ← p - ω(C k 1 ,C k 2 ) ω(E) + ψ(C k ) 2 -ψ(C k 1 ) 2 -ψ(C k 2 ) 2 4×ω(E) 2
update the modularity

9: C ← C \ {C k } ∪ {C k1 , C k2 } replace C k with two clusters 10: else 11:
Mark C k as ineligible for Bisect 12: K , C , p ←RefineClusters(G, K, C, p) 13: return K , C , p Algorithm 1 displays the proposed approach. The algorithm accepts a weighted graph G = (V, E, ω), and returns the number of clusters K and the clustering C = {C 1 , C 2 , . . . , C K }. It uses a bisection heuristic to compute a clustering of the given graph. Initially, all the vertices are in a single cluster. At every step, the heaviest cluster, say C k , is selected and split into two (by the subroutine Bisect),

if |C k | > 2.
If the bisection is acceptable, that is if the bisection improves the modularity (see the line 6), the cluster C k is replaced by the two clusters resulting from the bisection. If not, the cluster C k remains as is. The algorithm then proceeds to another step to pick the heaviest cluster. The clustering C found during the bisections is then refined in the subroutine RefineClusters that starts just after the bisections. The computational core of the algorithm is the Bisect routine. This routine accepts a graph and splits that graph into two clusters using existing tools that are used for the graph/hypergraph bisection problem. We have instrumented the code in such a way that one can use MeTiS, PaToH, or Scotch quite effectively at this point.

Unfortunately, there is no guarantee that it is sufficient to stop bisecting a cluster as soon as a split on it reduced the modularity score. As finding a bipartition of maximum modularity is NP-hard [START_REF] Brandes | On finding graph clusterings with maximum modularity[END_REF], it is possible that a Bisect step which reduces modularity, can be followed by a second Bisect step that increases it beyond its original value. As an example, consider the graph in Fig. 1 which shows a clustering, albeit a suboptimal one, that we will call C where C = {C 1 , C 2 }. This clustering has the following modularity score

p(C) = 5 10 - (3 + 4) 2 + (2 + 3 + 3 + 3 + 2) 2 4 × 10 2 = - 18 400 .
Since a trivial clustering {V } has modularity p({V }) = 0, we can easily see that the clustering C reduces the modularity to negative. Now, consider the clustering

C = {C 1 , C 21 , C 22 } which is obtained via a bipartition of C 2 as shown in Fig. 2.
Clustering C has the following modularity:

p(C ) = 4 10 - (3 + 4) 2 + (2 + 3 + 3) 2 + (3 + 2) 2 4 × 10 2 = 22 400 .
Thus, clustering C 2 has higher modularity than the initial trivial clustering {V }.

Of course, this effect is due to the suboptimal clustering C. However, since the bipartitioning algorithm provides no approximation guarantee, we cannot preclude this. Therefore, not bisecting a cluster anymore when a Bisect operation on it reduces the modularity score has its drawbacks.

The bisection heuristic.

Our bisection heuristic is of the form shown in Algorithm 2 whose behavior is determined by a set of four parameters: a, imb, b, and e. The first one, a, chooses which algorithm to use as a bisector. We have L, R ← apply Bisector a to G with imbalance tolerance imb 5:

for j = 1 to e do 6:

L, R ← RefineBisection(L, R, G(U )) 7: if ψ(U ) 2 -ψ(L) 2 -ψ(R) 2 4×ω(E) 2 -ω(L,R) ω(E) > mostIncrease then 8: mostIncrease ← ψ(U ) 2 -ψ(L) 2 -ψ(R) 2 4×ω(E) 2 -ω(L,R) ω(E) 9: L , R ← L, R
As shown in Algorithm 1, the bisection heuristic is called for a cluster C k of a clustering C with the modularity score p. Note that

C k contributes ω(C k ) ω(E) -ψ(C k ) 2 (4×ω(E) 2 )
to the modularity score. When we bisect

C k into C k1 and C k2 , the coverage of the clustering C ← C \ {C k } ∪ {C k1 , C k2 } becomes ω(C k1 , C k2
) less than the coverage of C, and the new clusters C k1 and C k2 contribute

ω(C k 1 )+ω(C k 2 ) ω(E) - ψ(C k 1 ) 2 +ψ(C k 2 ) 2 4×ω(E) 2
to the modularity score. The difference between the modularity scores is therefore the formula used at line 7 of Algorithm 2.

The vertex weights that are passed to the bisectors are simply the weights ψ(•) defined on the original graph. Balancing the sums of weights of the vertices in the two parts will likely reduce the squared part weights and will likely yield better modularity. This however, is not guaranteed, as the increase in the modularity score is also affected by the cut of the bisection. That is why trying a few imbalance parameters (controlled by imb), running the bisector multiple times (controlled by b) with the same imbalance parameter, and refining those bisections (controlled by e) is a reasonable approach.

The algorithm RefineBisection(L, R, G = (L∪R, E)) is a variant of Fiduccia-Mattheyses [START_REF] Fiduccia | A linear-time heuristic for improving network partitions[END_REF] (FM) heuristic. Given two clusters, FM computes a gain associated with moving a vertex from one cluster to the other one. The efficiency of the method is achieved by keeping these gains up-to-date after every move. In the standard application of this refinement heuristic (for the graph and hypergraph bipartitioning problems, see e.g., [START_REF] Aykanat | Hypergraph-partitioning-based decomposition for parallel sparse-matrix vector multiplication[END_REF][START_REF] Karypis | A fast and high quality multilevel scheme for partitioning irregular graphs[END_REF]), moving a vertex changes the gains associated with the adjacent vertices only. This is not true during the refinement process for improving the the modularity score. Consider a given weighted graph G = (V, E, ω) and a bipartition L, R of V . The contribution of the clusters L and R to the modularity score is

(3.1) ω(L) + ω(R) ω(E) - ψ(L) 2 + ψ(R) 2 4 × ω(E) 2 .
When we move a vertex v from L to R, the new modularity score becomes

(3.2) ω(L \ {v}) + ω(R ∪ {v}) ω(E) - ψ(L \ {v}) 2 + ψ(R ∪ {v}) 2 4 × ω(E) 2 . Subtracting (3.1) from (3.2) we obtain the gain of moving v from L to R (3.3) gain(v, L → R) = u∈R ω(v, u) -u∈L ω(v, u) ω(E) + 2 × ψ(v) ψ(L) + ψ(R) -ψ(v) 4 × ω(E) 2 .
As the gain of a move includes the cluster weights, a single move necessitates gain updates for all vertices. Thus, it is not very practical to choose the move with the highest gain in modularity at every step. We therefore designed the following alternative. We keep two priority queues, one for each side of the partition, where the key values of the moves are the reduction in the cut size (that is, the key values are the standard FM gains). Assuming uniform vertex weights, among the moves of the form L → R, the one with the maximum gain in the modularity will be the vertex move with the maximum gain in the cut size. This is due to the fact that the second term in (3.3) will be the same for all vertices in L. Similarly, among the moves of the form R → L, the one with the maximum gain in the cut size will be the one with the maximum gain in the modularity. Since vertex weights are not uniform, we need to be a little careful about choosing which vertex to move. Every time we look for a vertex move, we check the first move of both priority queues and compute the actual gain (3.3) and perform the better move tentatively. Realizing the maximum gain sequence of these tentative moves is done in the same way as in the standard FM heuristic. The bisectors (MeTiS, PaToH, or Scotch) are generally accepted to be of linear time complexity. The time complexity of the RefineBisection is, due to the use of priority queues and gain update operations, O(|E| log |V |) for a graph G = (V, E). Therefore, the running time of bisection step is O(|E| log |V |). However, we should note that depending on the parameter settings the constant hidden in the formula can be large.

Refining the clustering.

The last ingredient of the proposed clustering algorithm is RefineClusters(G, K, C, p). It aims to improve the clustering found during the bisections. Unlike the RefineBisection algorithm, this algorithm visits the vertices in random order. At each vertex v, the gain values associated with moving v from its current cluster to all others are computed. Among all those moves, the most beneficial one is performed if doing so increases the modularity score. If not, the vertex v remains in its own cluster. We repeat this process several times (we use m as a parameter to control the number of such passes). The time complexity of a pass is O(|V |K + |E|) for a K-way clustering of a graph with |V | vertices and |E| edges.

Experiments

We perform a series of experiments to measure the effect of the various parameters on the modularity scores of the solutions produced by the algorithm, and to evaluate overall performance of the approach.

To this end, we use a set of 29 popular test instances which have been used in the past to study the modularity scores achieved by clustering algorithms. The instances are from various resources [START_REF] Arenas | Network data sets[END_REF][START_REF] Barabasi | Emergence of scaling in random networks[END_REF][START_REF] Boldi | Ubicrawler: A scalable fully distributed web crawler[END_REF][START_REF] Boldi | Layered label propagation: A multiresolution coordinate-free ordering for compressing social networks[END_REF][START_REF] Boldi | The WebGraph framework I: Compression techniques[END_REF][START_REF] Geisberger | Better approximation of betweenness centrality[END_REF][START_REF]Network data[END_REF][START_REF] Staudt | A generator for dynamic clustered random graphs[END_REF][START_REF] Watts | Collective dynamics of 'small-world' networks[END_REF] and are available at http://www.cc.gatech.edu/dimacs10/.

We first test the algorithm using the standard parameter combination. It consists of m=5 refinement rounds at the end and a bipartition parameter of b=1. No refinements are performed during the algorithm (e=0). The results using PaToH, MeTiS, and Scotch partitioning are shown in Table 1 below.

As expected, the results for each instance are very close, with a maximum difference of less than 0.04. All partitioners provide good results, with PaToH delivering somewhat higher modularity scores. However, using MeTiS consistently yielded slightly inferior results. The same was true in preliminary versions of the experiments described below. Thus, MeTiS was not considered in the following experiments.

The slightly higher scores of PaToH can be explained by the fact that unlike SCOTCH, it uses randomization. Even though this is not intended by the algorithm design, when performing multiple partitioning runs during the Bisect routine, the randomized nature of the PaToH results has s light chance to find a superior solution, which is generally kept by the algorithm.

In the next experiment, we investigate the effect of the refinement algorithm RefineClusters on the final result. Table 2 shows the refined modularity scores using a maximum of m = 5 refinement steps at the end of Algorithm 1 for PaToH and Scotch partitioning, as opposed to the unrefined results (m = 0). On average, the effect of the clustering refinement step (RefineClusters) amounts to an improvement of about 0.01 for Scotch and 0.0042 for PaToH. Our preliminary experiments showed that increasing the number of refinement steps beyond m = 5 improves the end result only marginally in both cases. Although the improvement for Scotch is slightly larger, the difference is not sufficient to completely equalize the gap between the unrefined results for PaToH and Scotch. Since the computational cost of the refinement heuristic is low, we will continue to use it in the following experiments.

Furthermore, we investigate the influence of the number of repetitions of the Bisector step on the modularity score by increasing the parameter b from 1 to 5. Results are shown in Table 3 where we observe a slight positive effect for b = 5 as compared to b = 1. It is interesting to note that even though PaToH is randomized, selecting the best out of 5 bisections has almost no effect. This is partially because the RefineClusters operation finds the same improvements. Due to the refinement, the total effect can even be negative since a different clustering might preclude a particularly effective refinement. Overall, we conclude that b = 5 is worthwhile for Scotch, but not for PaToH. We also study the influence of applying the refinement process during the algorithm, as opposed to the standard refinement after termination of the main algorithm. This is done by setting the parameter e = 5, i.e., we use 5 passes of RefineBisection after each call of Bisector in Algorithm 2. Results that are displayed in Table 4 show that this approach does not help to improve modularity. Most likely, improvements that can be found in this manner can also be found by calling RefineClusters at the end of the algorithm. In addition, this technique is computationally expensive, and therefore it should not be used.

Finally, we compare modularity scores obtained by our algorithm with previous results found in literature. We compare the best score found by our algorithms with the best score found in the literature. These results are shown in Table 5. Compared to previous work, our algorithms perform quite well. For the small instances dolphins, karate, polbooks, and football the previous values are optimal, and the algorithms come quite close, deviating by only 0.00047 from the optimum values on average. The instance lesmis is a weighted graph and was treated as such here. Therefore the modularity score obtained is higher than the unweighted optimum computed in [START_REF] Cafieri | Locally optimal heuristic for modularity maximization of networks[END_REF]. It is included here for the sake of completeness, but it is not considered for the aggregated results. For larger instances, obtaining optimum values is computationally infeasible. Thus, the scores given here represent the best value found by other clustering algorithms. Our algorithm surpasses those in 9 out of 13 instances, and its average modularity score surpasses the best reported values by 0.01. Naturally, most clustering algorithms will be quite close in such a comparison, which renders the difference quite significant. Summing up, we conclude that the optimum configuration for our algorithm uses PaToH for partitioning with RefineClusters at m = 5. For the bipartition parameter, exceeding b = 1 is hardly worthwhile. In this configuration, RefineBisection should not be used, i.e., e = 0 should be selected.

Conclusion

We have presented a new algorithm for finding graph clusterings of high modularity. It follows a divisive approach by applying recursive bipartition to clusters. In addition, it makes use of a standard refinement heuristic. It can be implemented efficiently by making use of established partitioning software.

We experimentally established that the best modularity scores can be obtained by choosing the best out of multiple partitionings during the bipartitioning step and applying the refinement heuristic at the end of the algorithm. The modularity scores obtained in this manner surpass those of previously known clustering algorithms. A possible variant of the proposed algorithm that can be further studied would accept bipartitions of inferior modularity for a limited number of recursion steps, thereby alleviating the problem described in Section 3.

1 .

 1 Preliminaries. In the following, G = (V, E, ω) is a weighted undirected graph with ω : E → R + as the weight function. A clustering C = {C 1 , . . . , C k } is a partition of the vertex set V . Each C i is called a cluster. We use G(C k ) to denote the subgraph induced by the vertices in C

1 C 2 Figure 1 .

 121 Figure 1. Clustering C. All edges have weight 1. Vertex weights are given.

22 Figure 2 .Algorithm 2 3 :

 22223 Figure 2. Clustering C . All edges have weight 1. Vertex weights are given.

Table 3 .

 3 Comparison between bipartition parameter setting of b = 1 and b = 5. Using 5 steps improves the end result slightly. The best modularity for each tool with b = 1 and b = 5 is marked as bold.

Table 1 .

 1 Modularity scores obtained by the basic algorithm before the refinement. The difference between PaToH, MeTiS, and Scotch is visible. The best modularity for each row is marked as bold.

	Modularity score

Table 2 .

 2 Modularity scores and improvement after the application of the RefineClusters algorithm at m = 5. Improvements for Scotch partitioning are larger than those for PaToH. The improvements are given in the column "Improv.". The best modularity for each row is marked as bold.

			PaToH			Scotch
	Instance	Unrefined Refined Improv. Unrefined Refined Improv.
	adjnoun	0.2945 0.2977	0.0033	0.2946	0.2972	0.0026
	as-22july06	0.6683 0.6711	0.0028	0.6524	0.6578	0.0054
	astro-ph	0.7295 0.7340	0.0046	0.7183	0.7238	0.0055
	caidaRouterLevel	0.8641 0.8659	0.0019	0.8506	0.8540	0.0035
	celegans metabolic	0.4318 0.4436	0.0118	0.4343	0.4407	0.0064
	celegansneural	0.4855	0.4871	0.0016	0.4905 0.4939	0.0034
	chesapeake	0.2495	0.2595	0.0100	0.2624 0.2624	0.0000
	citationCiteseer	0.8160 0.8175	0.0015	0.8094	0.8119	0.0025
	cnr-2000	0.9116 0.9116	0.0000	0.8981	0.9026	0.0045
	coAuthorsCiteseer	0.8976 0.8982	0.0005	0.8826	0.8838	0.0012
	coAuthorsDBLP	0.8281 0.8294	0.0013	0.8115	0.8140	0.0025
	cond-mat-2003	0.8443 0.8456	0.0013	0.8329	0.8343	0.0013
	cond-mat-2005	0.7651 0.7674	0.0023	0.7507	0.7556	0.0049
	cond-mat	0.7293 0.7331	0.0038	0.7084	0.7170	0.0086
	dolphins	0.5155 0.5276	0.0121	0.5265	0.5265	0.0000
	email	0.5733 0.5776	0.0043	0.5629	0.5748	0.0120
	football	0.6009 0.6046	0.0037	0.6009 0.6046	0.0037
	G n pin pout	0.4565 0.4913	0.0347	0.3571	0.4740	0.1169
	hep-th	0.8494 0.8504	0.0010	0.8392	0.8409	0.0016
	jazz	0.4330	0.4450	0.0120	0.4289 0.4451	0.0162
	karate	0.4188 0.4198	0.0010	0.4188 0.4198	0.0010
	lesmis	0.5658 0.5658	0.0000	0.5540	0.5649	0.0108
	netscience	0.9593 0.9593	0.0000	0.9559	0.9559	0.0000
	PGPgiantcompo	0.8830 0.8831	0.0001	0.8726	0.8734	0.0008
	polblogs	0.4257 0.4257	0.0000	0.4247 0.4257	0.0010
	polbooks	0.5266 0.5269	0.0004	0.5242 0.5269	0.0027
	power	0.9394 0.9398	0.0003	0.9384	0.9386	0.0002
	preferentialAttachment	0.3013 0.3066	0.0053	0.2461	0.2815	0.0353
	smallworld	0.7838 0.7846	0.0008	0.7061	0.7451	0.0390
	Average	0.6465 0.6507	0.0042	0.6329	0.6430	0.0101

Table 4 .

 4 Modularity scores for refinement steps during the algorithm. After every bisection, up to 5 refinement steps are performed. The best modularity for each tool with e = 0 and e = 5 is marked as bold

			PaToH			Scotch
	Instance	e=0	e=5	Diff.	e=0	e=5	Diff.
	adjnoun	0.2977 0.3014 0.0037 0.2972 0.2941 -0.0031
	as-22july06	0.6711 0.6653 -0.0058	0.6578 0.6581 0.0003
	astro-ph	0.7340 0.7283 -0.0058 0.7238 0.7204 -0.0034
	caidaRouterLevel	0.8659 0.8627 -0.0033 0.8540 0.8483 -0.0058
	celegans metabolic	0.4436 0.4430 -0.0007	0.4407 0.4433 0.0026
	celegansneural	0.4871 0.4945 0.0074	0.4939 0.4944 0.0005
	chesapeake	0.2595 0.2658 0.0063	0.2624 0.2658 0.0034
	citationCiteseer	0.8175 0.8145 -0.0030 0.8119 0.8088 -0.0031
	cnr-2000	0.9116 0.9050 -0.0066 0.9026 0.9019 -0.0007
	coAuthorsCiteseer	0.8982 0.8971 -0.0011 0.8838 0.8829 -0.0009
	coAuthorsDBLP	0.8294 0.8276 -0.0018 0.8140 0.8106 -0.0033
	cond-mat	0.8456 0.8424 -0.0031 0.8343 0.8333 -0.0010
	cond-mat-2003	0.7674 0.7643 -0.0031 0.7556 0.7532 -0.0023
	cond-mat-2005	0.7331 0.7309 -0.0022 0.7170 0.7142 -0.0028
	dolphins	0.5276 0.5265 -0.0011 0.5265 0.5265 0.0000
	email	0.5776 0.5748 -0.0028 0.5748 0.5647 -0.0101
	football	0.6046 0.6032 -0.0013 0.6046 0.6032 -0.0013
	G n pin pout	0.4913 0.4921 0.0009	0.4740 0.4872 0.0132
	hep-th	0.8504 0.8472 -0.0031	0.8409 0.8412 0.0003
	jazz	0.4450 0.4451 0.0001 0.4451 0.4271 -0.0181
	karate	0.4198 0.4198 0.0000 0.4198 0.4198 0.0000
	lesmis	0.5658 0.5658 0.0000	0.5649 0.5652 0.0003
	netscience	0.9593 0.9551 -0.0042 0.9559 0.9558 -0.0001
	PGPgiantcompo	0.8831 0.8791 -0.0040 0.8734 0.8732 -0.0002
	polblogs	0.4257 0.4257 0.0000 0.4257 0.4257 0.0000
	polbooks	0.5269 0.5108 -0.0161 0.5269 0.5108 -0.0161
	power	0.9398 0.9373 -0.0024 0.9386 0.9346 -0.0040
	preferentialAttachment 0.3066 0.3058 -0.0008	0.2815 0.2952 0.0137
	smallworld	0.7846 0.7851 0.0005	0.7451 0.7857 0.0406
	Average	0.6507 0.6488 -0.0018 0.6430 0.6429 0.0001

Table 5 .

 5 Comparison between modularity score obtained by our algorithm and and scores reported in previous work. An asterisk indicates that this instances has been solved optimally. The best modularity for each row is marked as bold.

		Best found Best known
	Instance	Modularity Modularity Source Difference
	adjnoun	0.3014	0.3080	[26]	-0.0066
	caidaRouterLevel	0.8677	0.8440	[13]	0.0237
	celegans metabolic	0.4467	0.4350	[8]	0.0117
	celegansneural	0.4945	0.4010	[26]	0.0935
	citationCiteseer	0.8175	0.8037	[13]	0.0138
	cnr-2000	0.9119	0.9130	[13]	-0.0011
	coAuthorsDBLP	0.8306	0.8269	[13]	0.0037
	dolphins *	0.5276	0.5290	[8]	-0.0014
	email	0.5776	0.5738	[15]	0.0038
	football *	0.6046	0.6050	[8]	-0.0004
	jazz	0.4451	0.4452	[15]	-0.0001
	karate *	0.4198	0.4198	[8]	0.0000
	lesmis *	0.5658	0.5600	[8]	0.0058
	netscience	0.9593	0.9540	[26]	0.0053
	PGPgiantcompo	0.8834	0.8550	[8]	0.0284
	polblogs	0.4257	0.4260	[26]	-0.0003
	polbooks *	0.5269	0.5270	[8]	-0.0001
	power	0.9398	0.9390	[8]	0.0008
	Average	0.6414	0.6314		0.0100

Table 7 .

 7 The modularity scores of our algorithm ParMod in comparison to the two best ranked submissions CGGCi RG and VNS quality.

	Instance	ParMod CGGCi RG VNS quality
	333SP	0.9891	0.9887	0.9884
	as-22july06	0.6736	0.6783	0.6776
	astro-ph	0.7367	0.7438	0.7446
	audikw1	0.9173	0.9174	0.9180
	belgium.osm	0.9949	0.9949	0.9948
	cage15	0.8985	0.9032	0.3438
	caidaRouterLevel	0.8684	0.8720	0.8709
	celegans metabolic	0.4467	0.4521	0.4532
	citationCiteseer	0.8187	0.8239	0.8217
	coAuthorsCiteseer	0.8999	0.9053	0.9039
	cond-mat-2005	0.7380	0.7463	0.7451
	coPapersDBLP	0.8589	0.8668	0.8650
	email	0.5800	0.5819	0.5828
	eu-2005	0.9404	0.9416	0.9413
	G n pin pout	0.4936	0.5001	0.4993
	in-2004	0.9803	0.9806	0.9805
	kron g500-simple-logn16	0.0646	0.0637	0.0651
	kron g500-simple-logn20	0.0487	0.0504	0.0494
	ldoor	0.9694	0.9689	0.9691
	luxembourg.osm	0.9893	0.9895	0.9896
	memplus	0.6972	0.7005	0.6953
	PGPgiantcompo	0.8841	0.8866	0.8861
	polblogs	0.4257	0.4271	0.4271
	power	0.9401	0.9403	0.9409
	preferentialAttachment	0.3086	0.3023	0.3160
	rgg n 2 17 s0	0.9777	0.9781	0.9783
	smallworld	0.7872	0.7930	0.7930
	uk-2002	0.9767	0.9903	0.9901
	Average	0.7466	0.7496	0.7297

Table 8 .

 8 The execution time of ParMod for a few challenge instances. The times are the averages of 5 executions and given for reference purposes. The Ohio State University, Dept. of Biomedical Informatics and Dept. of Electrical & Computer Engineering, Email address: umit@bmi.osu.edu

	Instance	Time(sec) Instance	Time(sec)
	celegans metabolic	1.02 email	4.02
	power	11.18 polblogs	6.40
	PGPgiantcompo	27.44 as-22july06	58.73
	astro-ph	112.45 cond-mat-2005	232.82
	preferentialAttachement	524.85 smallworld	397.49

Acknowledgment

This work was supported in parts by the DOE grant DE-FC02-06ER2775 and by the NSF grants CNS-0643969, OCI-0904809, and OCI-0904802.

Appendix A. DIMACS Challenge results

After DIMACS Challenge was completed, we run the proposed divisive clustering algorithm on the challenge instances (we skipped a few of the largest graphs). In these runs, we did not try to explicitly optimize the other challenge metrics. Doing so would require some trivial changes to the proposed framework for each metric: the evaluation functions measuring the modularity, the functions that update the modularity score, and the refinement functions should now measure the desired metric(s). Without doing these changes, we measured the other clustering scores (of the partitions that took the modularity score as the criterion). The results are shown in Table 6. A comparison of the modularity scores obtained in the implementation challenge by the three best algorithms is shown in Table 7. Even though our ParMod algorithm provides the best score only for two instances, the differences in comparison to the scores of the best ranked algorithm CGGCi RG are very small. In fact they are smaller than the impact of several of the parameters studied in Section 4. The second ranked algorithm, VNS quality provides many top ranked results. However, due to an extreme outlier for the instance cage15, the average value is lower than that of the other algorithms. When disregarding the outlier, the average lies between those of ParMod and CGGCi RG. We also give the execution time of ParMod for some small challenge instances in Table 8.


Anette Bagger
email: anette.bagger@umu.se

Helena Vennberg
email: helena.vennberg@umu.se

Lisa Björklund Boistrup

The politics of early assessment in mathematics education

Keywords: Early assessment, preschool-class, development in mathematics, national assessment

One of the latest reforms in Sweden in order to increase equity and quality in education is making national assessment compulsory in preschool-class (age 6). The claimed political volition is all students' best possible mathematical development. In this paper, we examine the preparatory work, the assignment to the National Agency of Education, and the assessment material itself, searching for what meaning is inscribed regarding the student, mathematics and assessment. The results imply that the politics of the assessment might exaggerate a search for flaws and control instead of promoting all students learning and in addition contributing to the schoolification of preschoolclass.

immediate situation of assessment if students belonging to disadvantaged groups face obstacles in displaying knowledge. An example is when the teacher feels that securing multi-lingual students' equity could threaten the tests validity [START_REF] Bagger | Quality and equity in the era of national testing. The case of Sweden[END_REF]. These findings from earlier research indicate that it is urgent to continue to investigate how the assessment and the teacher are governed. In this paper, assessment in mathematics education is understood as an element of governing [START_REF] Foucault | The essential works of Foucault, 1954-1984[END_REF], where purpose and how it is carried out have impact on aspects related to quality and equity. [START_REF] Newton | Clarifying the purposes of educational assessment[END_REF] stresses that if several purposes of assessmentto evaluate education, to make grounds for decisions of recourses, and assessment of knowledgeare active simultaneously, conflicts of interests are built into the assessment, and a test will work poorly for all or mainly for some of the purposes. Further, how the teacher understands the student and the teaching content will have impact on the mathematical support [START_REF] Scherer | Assistance of students with mathematical learning difficulties: How can research support practice[END_REF]. Related to this, a study by [START_REF] Bagger | Quality and equity in the era of national testing. The case of Sweden[END_REF] indicates that the educational segregation in the teaching and learning of mathematics might occur earlier with earlier testing. Simultaneously, there are indications that if the preschool-class teachers get skilled in assessing mathematical knowledge, students at risk in preschool-class reach similar achievements as their peers in the national test in grade 3 [START_REF] Vennberg | Counting onlong term effects of an early intervention program[END_REF]. Our assumption is that assessment in the preschool-class constitutes a risk for but also an opportunity to levelling disadvantage.

The purpose of this study is to increase our knowledge of how policy documents govern the implementation of national assessment in preschool-class in mathematics. This is achieved through a systematic exploration of how some central aspects are discursively constructed: the student, mathematics, and assessment.

Theory and methods

In this paper, discourse is taken as representations of power, truth and knowledge that govern individuals and society. Truth and knowledge are further understood as social constructs and as such they evoke power relations (Foucalt, 1994). [START_REF] Popkewitz | The alchemy of the mathematics curriculum: Inscriptions and the fabrication of the student[END_REF] draws on Foucault's thinking and uses the concept of fabrication as a governing technology. Categories, for example central notions adopted in assessment practices, are conceptual constructs through which reality can be understood and is simultaneously fabricated. These fabrications communicate versions of truth, knowledge and power, and coincide with processes of exclusion and inclusion. Governing texts are regarded as inscription devices that attribute terms, possibilities, and characteristics [START_REF] Popkewitz | Numbers in grids of intelligibility: Making sense of how educational truth is told[END_REF]. In this case, the governing texts are the testing material, political preparatory work, and decisions, through which the student as a mathematician, the teaching content, and the assessment are understood as attributed with terms, possibilities and characteristics. In other words, the texts fabricate the student, the assessment and the subject, and also inscribe meaning into them. This might by extension affect how the student, subject and assessment are handled and understood in the practice and governing of national assessment in preschool-class. The governing may then concern what is considered as 'true', what counts as 'knowledge', how students' knowledge is valued, and how results should be used.

The texts selected to explore the fabrications of mathematics, the student and the assessment are preparatory work and political decisions regarding the assessment in preschool-class and the material for assessment (see Table 1). Initially, in the analytical process, statements concerning preschool-class were selected from the texts into a scheme in which each text was in its own column. After that, all utterances depicting the student, the mathematics, or the assessment were coded in different colours. The next step was to 'translate' such utterances into explanatory paraphrases. The fabrications were thereafter narratively construed with regards to how terms, possibilities, characteristics, and demarcations were attributed. An example of the analysis is displayed in Table 2.

Text

Responsible author and short description [START_REF] Skolverket | Devaluing knowledge: School mathematics in a context of segregation[END_REF] Comments to curriculum. Swedish National Agency of Education (SNAE). The purpose and content of mathematics in preschool-class. [START_REF] Skolverket | Devaluing knowledge: School mathematics in a context of segregation[END_REF] National support for assessment in mathematics in preschool-class Regeringen 2017a

Swedish Government (SG). The assignment to the SNAE to develop the national system for assessment including the material for assessment in preschool-class. [START_REF] Regeringen | Läsa, skriva, räknaen åtgärdsgaranti[END_REF] SG. A bill guarantying support in reading, writing and counting. Utredningen om nationella prov (2016). SG. A bill in which the need for assessment in preschool-class is put into the context of national assessment and the quality and equity of education. This is needed in order to identify students in need / at risk of not reaching goals three years later. "the assessment will help teachers to know what support to possibly put in" Teachers need help to know how to support.

"National material might increase the equity of the quality in the support given"

There is a lack of equity in the support nowanother aim with assessment. "… solutions should be evaluated by someone else than the teaching teacher"

The teacher should not evaluate the solutions of her/his own students.

Table 2: Example of the analytic procedure and construction of the narrative

Findings

A narrative was constructed by depicting how mathematics, the assessment and the student were fabricated. All texts are in Swedish and quotes have been translated by the authors.

The fabrication of mathematics

One aim of mathematics, as construed from the analysed texts, is to be a caretaker of curiosity and understanding: "The teaching shall take care of students curiosity and afford them the opportunity to develop their interest in mathematics and understanding for the possible use of mathematics in different situations" (Skolverket, 2016, p. 13). Also: "students shall be challenged and stimulated to use mathematical concepts and reasoning to communicate and solve problems in different ways through different representations and to explore and describe their surroundings" (p. 13). The areas of mathematics that preschool-class is supposed to pay attention to in order to reach the aim is: "Mathematical reasoning and different ways of solving problems" (p. 20) together with "Mathematical concepts and representations" (p. 21). Mathematical reasoning is defined as a "logical mathematical argument that is used for motivating an answer or a choice" (p. 20), and mathematical problems are defined as "situations or tasks in which the students do not know beforehand how to solve the problem" (p. 20). The mathematical contents that the students are supposed to be offered to meet are: the characteristics of natural numbers and how they can be used to designate cardinality and ordinality…. mathematical concepts and representations in order to explore and describe space, location, shape, direction, pattern, time and change. (Skolverket, 2016, p. 21) Mathematics is, through the above-mentioned statements, fabricated in the curriculum for preschool-class as a nurturing and caretaking assignment that will grant the student an opportunity to further develop an already existing curiosity, interest and understanding. The mathematical content is fabricated as a kind of founding building blocks needed in order to explore and investigate various problems and arguments. Problems and reasoning that could impose interest and curiosity are additionally fabricated as deriving from visuospatial elements of the student's surroundings. We interpret the intentions of mathematics education, as described in the curricula, as a way of supporting the students in experiencing themselves in the world, appreciating diversity in ideas and recognising their own competence to deal with problems.

In the title of the material for assessment itself, mathematics is labelled as mathematical thinking: "Find the mathematics. Evaluation material in mathematical thinking in preschool-class" (Skolverket, 2018, p. 1). In other words, it is fabricated as something that resides within the student. The material encourages the teacher to promote different arguments, reasoning, examples, and not to stop or be content with just one solution from students. This approach towards exploring is in line with the curriculum's intentions. The areas through which the mathematical thinking is supposed to be explored and evaluated are patterns, maps, counting, sorting, and volume. The focus on experiencing the world and relations in it, in both the curriculum and the material, fabricates mathematics as something concrete and visible that constitutes reality. We wish to connect this to Straehler-Pohls (2015) writings of distributive rules in relation to school mathematics;

The foundation of this distribution is a stratification of mundane and esoteric meanings. In this stratification, esoteric meanings inevitably take the dominant role, as it is these meanings that transcend the spatial and temporal materiality that bear the potentials to think yet unthinkable solutions that draw on (contextually) external frames of reference" (p. 3).

Mathematics as concrete and constituting is further interpreted as fabricating school mathematics in preschool-class as mundane, at the same time as the exchanged meaning making is supposed to take place in an esoteric form through an exchange of different understandings and reasoning. The promoted variation in reasoning and solutions connects to a fallibilist [START_REF] Ernest | What is mathematics, and why learn it[END_REF] view on mathematics in which knowledge is considered fallible and open for revision. In the assessment material, there is an attached observation scheme in order to register the student's knowledge. A text on this is: "date when the observation point is reached/finished" (Skolverket, 2018, p.1 in compilation form). This assumes knowledge to be possible to observe, reach and check at a certain date, fabricating mathematical knowledge as situated in an absolutist perspective [START_REF] Ernest | What is mathematics, and why learn it[END_REF] in which knowledge is secure, fixed and objective. This is opposed to the fallibilist view, represented in the previously described encouragement in the material and curriculum; to explore and negotiate solutions and arguments. It also fabricates the mathematical thinking, or knowledge, as something that could and should be reachedas if there were goals to achieve in preschool-class.

The fabrication of assessment

The title of the assessment material is: "Find the mathematics: Evaluation of mathematical thinking" [START_REF] Skolverket | Devaluing knowledge: School mathematics in a context of segregation[END_REF]. Assessment is thereby fabricated as needed in order to capture the inner world of the student, the thinking, or the lack thereof. The thinking is further fabricated as promoting or threatening a development of knowledge towards the goals in grade 3: "… is carried out during autumn in preschool-class so that the teacher can early identify students who show indications of not achieving the goals in learning that are set for grade 3" (p. 3). This message is also repeated in the preparatory work and the assignment to the Swedish National Agency of Education to develop this material. In addition, assessment is in these preparatory texts fabricated as something that will level inequalities in teaching and grant high quality and equal support earlier: "Nationally produced material might also increase the equity of the quality in the support given and contribute to a common terminology regarding assessment, progression and knowledge requirements which will lead to higher quality of judgements" (Regeringen, 2017a, p. 5). Teachers are fabricated as being responsible for detecting where in the development the student is and to adapt teaching so that the student can achieve the goals in third grade. This is seen in several parts of the texts and in the instructions to the material. For example: "to give teachers a clearer basis for assessment of the student's knowledge development and more information on what support to introduce" (p. 2). Also, the material will evaluate the teaching and help teachers to correct it through "affording preschool-class teachers and teachers support in seeing how opportunities and obstacles in the teaching impacts on students learning." (pp. 1-2). The proposition further fabricates assessment as insecure and invalid due to the teacher and relations involved: "solutions should be evaluated by someone else than the students' teacher, and such solutions should be deidentified" (p. 1). Another statement fabricating the teachers as the ones in need of support is in the assignment to the national agency of education: "the material will be completed with a supportive material for analysing evaluations" (p. 2). The assessment is then fabricated as a tool to control and support teachers in the assessment process.

The fabrication of the student

In the curriculum, the student is fabricated as already mathematically able and interested: "Teaching shall take students curiosity into consideration and give them the opportunity to develop their interest in mathematics and their understanding of how mathematics might be used in different situations" (Skolverket, 2016, p. 13). This harmonises with statements in the instructions to the material in which the use is presented as an opportunity in itself to learn mathematics: "By allowing students to meet activities that differ in character, they can develop trust in the ability to solve problems in different situations and contexts" (Skolverket, 2018, Instruction to material, p. 4). The student is thereby fabricated as a learner, rather than a test-taker. This stands in contrast to other statements in which the students are depicted as needed to display their knowledge. Such statements occur in the preparatory work and the material itself. The curiosity and interest that was fabricated as mathematical entities within the student and as something that the school should build on in the curriculum is instead fabricated in the material as a kind of mathematical knowledge that the student must display. "When the student shows curiosity and interest for the mathematical content, it reflects how the student participates in the activity" (p. 4). The participation is further fabricated as the notion that will inform the teacher about "where the student is in her/his development" (p. 5). In the preparatory work, the assignment, and the material, there is an overweight towards the student being fabricated as lacking or possibly missing knowledge, that those errors may grow and therefore need to be identified and corrected through adaptations of education.

Conclusions and discussion

A higher degree of goal achievement and increase of test-results have been central arguments in the evaluation of and revision of the national system for quality measuring [START_REF] Regeringen | Nationella provrättvisa, likvärdiga, digitala[END_REF]. This stands in contrast with the preschool-class's pedagogical assignment of being a bridge between preschool and school. National and compulsory assessment in preschool-class is a new phenomenon and it is crucial that it is implemented in a manner that is in line with this educational spirit of preschool-class. Otherwise, the assessment might not present an opportunity to nurture interest, curiosity and the identity of being a mathematical learner and explorer, but instead contribute to an approach in which it is the performed interest or competencies that counts. The preschool-class could risk being colonised by the prevalent neoliberal logics of governing that exists in compulsory school including schoolification of the assessment practice in mathematics.

A focus on mathematics as the observed and displayed activity, and the assessment of knowledge as making achievement visible, positions the 6-year-old preschool-class student as a test-taker and the teacher as the controller and corrector of knowledge. The mathematical thinking and development were fabricated in the data as having a certain order or being a destination. To identify where the student is positioned on this map or road, is claimed as a key for giving adequate support early. A question to raise is how the teacher is supposed to position the students' place in development through observing their displayed participation, as suggested (and we are reluctant to the idea of assigning students places in developmental 'paths', but that is not in focus in this text). It may be many factors that affect how and if a student participates, several of these have to do with the organisation of education and teaching. These are mentioned in the material with regard to the importance that students are allowed to express different ideas, which can also happen with other modes than spoken language. The stress laid on the freedom to communicate and try ideas indicates that it should be an explorative and playful situation. Also, knowledge can be displayed outside these situations or with other alternative and adaptive settings. These fabrications stand against the one of control and goal achievement. An adapted and creative mathematical teaching is supposed to be the possible outcome from the analysis of the material. We question how this can this happen, in a practice deriving from an assessment that is only assessing a few mathematical items. This is significant, not least since the risk of 'teaching to the test' is obvious. Another aspect is how the mathematical processes, selected and tested in the material, are promoted as the mathematics, which is limiting in itself. There is a risk that the material may contribute to the narrowing of implemented curricula, teaching to the test, and seeing students as their levels of achievement, as reported in previous research [START_REF] Wrigley | The testing regime of childhood: Up against the wall[END_REF]. Thereby, the material may unintentionally limit possibilities for development and knowledge in mathematics, which is opposed to the purpose of increasing equity and quality.

The described use of the assessment material is easy to interpret as being about a search for problems, rather than promoting education for all. The strive towards high quality and quality in equity is then becoming an issue of everyone doing the same thing, having the same focus and the same demarcation in regard to the support given and the mathematics taught. A final question is: Is this assessment material a material for teaching or a material for controlling, and if so, what is being controlled and thereby restricted, locked in, or included? What is excluded when it comes to kinds of students, kinds of mathematical knowledge and kinds of teachers' ways of teaching? Our hope is that the discursive fabrications, presented in this paper, of assessment, mathematics and who the student is, may contribute to revealing some of the underpinnings in the so-hard-to-grasp patterns or structures of disadvantage in mathematics education.

Table 1 : Overview of the five documents analysed in this paper Selected text Explanatory paraphrases Construed fabrication

 1

"To observe the students' mathematical abilities in different areas of importance for development of mathematical thinking"

The purpose of the assessment is to identify student's mathematical thinking Lack of mathematical thinking is fabricated as the supposed obstacle for goal achievement in grade 3. Assessment will level inequalities in teaching and grant high quality and equal support earlier. Another threat against equity is the teacher who draws wrong conclusions regarding support or the displayed knowledge. So, they also need support. "... early identify students who show indications of not achieving the goals in learning that are set for grade 3"


Laura Black
email: 1laura.black@manchester.ac.uk

Julian Williams
email: 2julian.williams@manchester.ac.uk

David Swanson
email: david.swanson@manchester.ac.uk

Sophina Choudry
email: sophina.choudry@manchester.ac.uk

Emilia Howker
email: emilia.howker@postgrad.manchester.ac.uk

'Critical uses' of knowledge and identity: Embedded mathematics as a site for/of class struggle in educational praxis

Keywords: Funds of knowledge, funds of identity, social class, mathematics

This paper considers what the praxis of 'Funds of Knowledge/Identity' (FoK/I) might offer to researchers and practitioners of mathematics education. Building on a critique of FOK/FOI as reflecting cultural capital (in Bourdieu's sense) we posit the notion of 'use value' in the knowledge and practices of oppressed communities: in knowing how to live poor, how to resist capital, and how to solidarise in social movements related to class, nationality, race, gender, sexuality etc. We focus on 'dark' FOK -defined as difficult or challenging experiences for learners and/or communities; we reconceptualise these as related to the objective relations of oppression under capitalism whereby 'dark funds' are the surfacing of class contradictions in our learners' experiences. The implications for critical mathematics education are led by these considerations: in the problems we choose to tackle, the partners we choose to work with, and the research methodologies we adopt.

Introduction

The Funds of Knowledge (FoK) (and more recently Funds of Identity -FoI) perspective has increasingly been utilized by those interested in addressing issues of inequality and oppression in education (including mathematics education, [START_REF] González | Bridging funds of distributed knowledge: Creating zones of practices in mathematics[END_REF]. Arguably, mathematics is a significant area of the curriculum for such work given the commonly accepted 'gatekeeper' role it plays in sorting and selecting who has access to high status careers/subjects and its function in excluding those from disadvantaged backgrounds [START_REF] Jorgensen | Structural exclusion through school mathematics: Using Bourdieu to understand mathematics as a social practice[END_REF]. In this paper, we present a narrative review of the literature on FoK/FoI in order to demonstrate how key critiques present important implications for how we conceptualise and how we 'do' critical mathematics education.

Funds of Knowledge/Funds of Identity in/for Mathematics Education

The 'Funds of Knowledge' (FoK) approach, originally developed by Moll and colleagues at the University of Arizona working with Latino immigrant communities and schools, has sought to emphasize the rich resources children have access to and experience in the home/community environment. In order to challenge the normative assumption that children from such communities are in some way 'deficient', it presents home activities (and the associated knowledge) as a rich resource for developing and teaching a school curriculum orientated towards social justice which are embedded in complex/rich social networks. The premise is that building a curriculum on these 'funds' will establish more meaningful connections between the school curriculum and learners lives beyond school. In one example, [START_REF] Moll | Funds of knowledge for teaching: Using a qualitative approach to connect homes and classrooms[END_REF] documented the development of a learning module around the FoK evident in the home of one child, Carlos, who engaged in buying and selling Mexican sweets to friends and relatives after visits to Mexico. This is presented as a hybrid curriculum since it initially involved bringing the home experience into the classroom (e.g., parents came into the classroom to make Mexican sweets with the children) but then incorporated formal curriculum knowledge (e.g., the ingredients of Mexican and American sweets and comparison of their nutritional value). According to [START_REF] Moll | Funds of knowledge for teaching: Using a qualitative approach to connect homes and classrooms[END_REF] the term 'funds' originated from the 'household funds' paradigm (in anthropology) and refers to the funds that households must juggle to make ends meetrent, social funds, ceremonial fundseach fund entailing a wider set of activities requiring a specific body of knowledge.

The essential cultural practices and bodies of knowledge and information that households use to survive, to get ahead or to thrive […] acquired primarily, but not exclusively, through work and participation in diverse labour markets. (p. 21) More significantly, [START_REF] Moll | Funds of knowledge for teaching: Using a qualitative approach to connect homes and classrooms[END_REF] were keen to point out that a FoK approach is not merely a culturally sensitive curriculum -or rather a 'static grab bag' of food, dances and celebrations which prove tokenistic (here they align with studies on ethnomathematics/culturally responsive pedagogies in mathematics education which also makes a similar critique). But rather this type of knowledge is strategic to households' or communities' functioning, development and well-being, and particularly may relate to the social, economic and productive activities of people in a local area.

More recently, Esteban-Guitart and [START_REF] Esteban-Guitart | Lived experience, funds of identity and education[END_REF] extended their approach to include 'Funds of Identity' (FoI) which refers to moments where a FoK is used to make some claim to be a certain kind of person: funds of knowledge become funds of identity when people actively internalize family and community resources to make meaning and to describe themselves. (p. 33) In this sense, they argue that education/teaching-learning should be about identity development as well as knowledge creation since the two are interwoven. Drawing on Vygotsky's notion of perezhivanie (defined as an emotional experience having developmental potential, [START_REF] Blunden | Translating perezhivanie into English[END_REF] they emphasise the subjective significance of the situation on the persons as they consume and use FoK. More so Esteban-Guitart and [START_REF] Esteban-Guitart | Lived experience, funds of identity and education[END_REF] define 'the term funds of identity … [as] a set of resources or box of tools and signs' (p. 37). The authors highlight five types of funds of identity: '(1) Geographical Funds of Identity…, (2) Practical Funds of Identity…, (3) Cultural Funds of Identity..., (4) Social Funds of Identity…, and (5) Institutional Funds of Identity' (p. 38).

FoK/FoI approaches have been utilised within mathematics education (Moll et al., 2009;Gonzalez et al., 2001) and clearly there is commonality with the ethnomathematics tradition (d'Ambrosio, 1985) and Critical Mathematics Education (CME) [START_REF] Gutstein | Reading and writing the world with mathematics: Towards a pedagogy for social justice[END_REF][START_REF] Skovsmose | Critical mathematics education[END_REF]; although CME is usually viewed as perhaps more explicitly political, involving critical interpretations of the world and political engagement (see, e.g., [START_REF] Skovsmose | What could critical mathematics education mean for different groups of students? For the Learning of Mathematics[END_REF]. The FoK approach is largely deemed critical because it rejects a deficit point of view on the learners (typically of immigrant or otherwise disadvantaged or marginal communities) vis a vis their 'capital'. Moll and colleagues (2009) argue, and their research findings establish this, that many children bring funds or resources that could be useful to teaching/learning, but that go unrecognized by the teachers/schools because of their ignorance of the culture involved. In this respect, this perspective also sits well with the Vygotskyan idea of the everyday/spontaneous practices of the learner as providing potential for 'making sense' of the formal, academic concepts of the school curriculum and its teachers.

Critiquing the 'Funds' [START_REF] Williams | Alienation in mathematics education: Critique and development of neo-Vygotskian perspectives[END_REF] has offered a critique of this work, drawing on Bourdieu, which illustrates how FoK/FoI has been used to refer to the surfacing of capital in the home/community as a means to reposition the learner with 'access' to the school curriculum. For instance, Moll et al. (2009) describe how the teachers-researchers involved in their projects discover one Mexican family's educational values and high aspirations for their children which were previously been unseen/unrecognised by the school. Here the identification of FoK may be to enable the redistribution of such 'capital' by making visible and legitimate home resources and by providing the machinery to 'scaffold' the movement of such capital from the home into the educational field as a means to access 'success' and capital growth. According to [START_REF] Williams | Alienation in mathematics education: Critique and development of neo-Vygotskian perspectives[END_REF], in such cases, identifying FoK does not necessarily challenge the process of capital exchange, but instead serves to strengthen its orthodoxy as 'natural' and open to all. As [START_REF] Zipin | Dark funds of knowledge, deep funds of pedagogy: Exploring boundaries between lifeworlds and schools[END_REF] notes, the very act of identifying 'everyday' or 'lifeworld' knowledge to facilitate access to more powerful or 'elite' cultural artefacts entails the loss of its use value (productivity) and transforms such knowledge into objects with exchange value to be mobilized for capital growth in the educational market (he defines this as an 'asset' perspective).

When schools operate as high-stakes competitive markets, the use values of diverse people's cultural assets are sadly diminished in relation to 'gold standards' of restricted exchange value. (p. 319) However, poor communities also have resources which may not be valued as 'capital' and may even function to resist capital exchange within the educational fieldfor example, parents who question how they are positioned by schools in terms of capital 'deficit' and express confusion as to why schools do not teach the 'life skills' they practice 'everyday' as a function of 'being poor' [START_REF] Howker | Investigate the role mathematics plays in the alienation of working class parents in the educational field[END_REF]. Both Bourdieu and Freirewere they alive todaymight argue for the need to disrupt the 'orthodoxy' which occurs when FoK/FoI approaches are used to merely surface capital in the home/community (thereby strengthening the process of capital exchange). For [START_REF] Bourdieu | An invitation to reflexive sociology[END_REF], it is the reflexive sociologist who, having made visible the processes of capital production/reproduction, might support practitioners/communities in identifying with a heterodoxa in a given field (i.e., an alternative frame which challenges orthodoxy and so threatens the 'doxa' in place). Similarly, [START_REF] Freire | Pedagogy of the oppressed (M. B. Ramos, Trans.)[END_REF] Pedagogy of the Oppressed suggests that emphasis should not be placed on resources 'latent' in the home/community practices which are useful to a formal curriculum of some sort but rather on how such resources can be brought to consciousness in order to 'name' (and challenge the limits of) the systems/relations/institutions which produce/reproduce oppression/inequality. Freire also outlines a process of conscientization where collective agency/solidarity and social movements are mobilized to fight struggles of injustice pertaining to class, race, etc. Through the pedagogy of the oppressed students can "perceive the reality of oppression, not as a closed world from which there is no exit, but as a limiting situation which they can transform" (p. 85). Here the agents are students/community members/teachers/activists rather than the social scientist (as per Bourdieu), although special status is given to the 'educator' whose role is to draw out the learners' view of reality and enable him/her to question it.

All of this suggests the need to situate FoK/FoI in relation to the objective relations of oppression, which under capitalism may be understood as relations of class position (dominated/dominating, working/leisured, disowned/owning class positions in Bourdieu). In identifying the types of knowledge embedded in homes/communities and the subjective experiences they engender (FoI), it is not sufficient to relate to 'domains' of activity (e.g., home/community etc.) in which they arise or even the social function that such activities serve (e.g., practical/institutional etc.). But rather we need to recognize how FoK/FoI function in relation to objective structures which (re)produce 'capital' (either through alignment or resistance) if we wish such an approach to challenge oppression/ alienation of those from disadvantaged communities.

Dark/Existential FoK/FoI

In relation to the above critique, [START_REF] Zipin | Dark funds of knowledge, deep funds of pedagogy: Exploring boundaries between lifeworlds and schools[END_REF] notes the difficulty of developing a curriculum based on the 'use-value' of FoK and anticipates the challenges and limitations presented by institutions whose function is to mechanize capital exchange [START_REF] Bourdieu | Forms of capital[END_REF]. In this sense, he critiques the tendency of researchers/teachers to gravitate to what he terms 'light' FoK or rather positive experiences and to omit what is seen as more difficult or challenging sides of living in disadvantaged or poor communities. As such, dark FoK are conceptualised as knowledge/resources/experiences which may be painful, challenging or difficult (e.g., bullying, poverty, incarceration, mental ill-health, exploitation, or other problems with social relations) and which [START_REF] Zipin | Dark funds of knowledge, deep funds of pedagogy: Exploring boundaries between lifeworlds and schools[END_REF] argues are significant if we wish to fully recognize how poverty and class position permeate the 'habitus' of children. More recently, [START_REF] Poole | Resituating Funds of Identity within contemporary interpretations of perezhivanie[END_REF] have made a similar argument in relation to existing accounts of FoI [START_REF] Esteban-Guitart | Lived experience, funds of identity and education[END_REF] critiquing the heavy focus on the learner's positive experiences, rather than capturing the whole range of human subjective experience. They suggest that difficult experiences or rather 'existential FoI' (identified through an 'experience as struggle' lens) are significant since they indicate a perezhivanie, that is emotional experiences which are/can be developmental in that they provoke the need to work through or reflect and subsequently overcome. Thus whereas [START_REF] Zipin | Dark funds of knowledge, deep funds of pedagogy: Exploring boundaries between lifeworlds and schools[END_REF] sees dark funds/difficult experiences as offering a potential FoK approach which can counter the logic of capital exchange in teaching and learning (since 'dark funds' are not merely a matter of identifying 'assets' for capital exchange), [START_REF] Poole | Resituating Funds of Identity within contemporary interpretations of perezhivanie[END_REF] suggest such potential be located at the subjective levelwith difficult or challenging experiences viewed as developmental in that they may bring about 'human growth': Existential funds of identity are defined as positive and negative experiences that students develop and appropriate in order to define themselves and to help them grow as human beings. Existential funds of identity are presented as an additional category of funds of identity that are designed to complement the typology consisting of geographical, social, cultural, institutional, and practical funds of identity as developed by Esteban-Guitart. (Poole & Huang, 2018, p. 126) In the above quote we can see how [START_REF] Poole | Resituating Funds of Identity within contemporary interpretations of perezhivanie[END_REF] equate existential FoI to other categories proposed by Esteban-Guitart and [START_REF] Esteban-Guitart | Lived experience, funds of identity and education[END_REF] as if they are merely an alternative form of resource/experience which individuals encounter. However, our own work on perezhivanie has begun to look at how emotional experiences (difficult/challenging or otherwise), although experienced subjectively, may also be related to objective structural conditions whereby contradictions arise. For instance, in [START_REF] Black | Identification with early primary school mathematics: A home-school activity theory conceptualisation and methodology[END_REF] we present the case of 6-year-old Nico who experiences his classroom mathematics as 'boring' and in contradiction with his home/community activities where he is able to 'see' some embedded mathematics 'in practice'. We have highlighted in this paper how this contradiction is experienced subjectively but also how it arises objectively in social structures, that is as a manifestation of the family habitus regarding school mathematics as 'not for them' which contradicts the embedded mathematics used by Nico, his family and his community in their everyday practice. Thus, we propose that dark/existential FoK need to be reconceptualised as manifestations of (and so refracted though subjective experiences of) objective relations of oppression under capitalism whereby 'dark funds' arise from the surfacing of class contradictions in learners' experiences. In general, the habitus of the oppressed is not well aligned with the cultural capital needed to resource the acquisition of power in the various fields and in the field of power generally. However, just as we must imagine a world beyond such arbitrary power structures, we must imagine the potential of such experiences to resource developments of new understandings and consciousness of ways to live differently. In this simple formula we may reposition Vygotsky's conception of perezhivanie in a critical class context, as the potential of experiences for development of newer and bolder class consciousness.

Implications for Critical Mathematics Education

Our critique of [START_REF] Poole | Resituating Funds of Identity within contemporary interpretations of perezhivanie[END_REF] conceptualisation therefore suggests that 'dark' experiences, if they are understood in the way we outline above, can give insight into how we might relate productively to them, and how they might be made to become developmental in a critical sense. This may entail finding 'uses' of mathematics in the home/community and in class struggles which can be mobilised to fight real issues of poverty or oppression in the way [START_REF] Freire | Pedagogy of the oppressed (M. B. Ramos, Trans.)[END_REF] envisaged. See, for example, [START_REF] Gutstein | Our issues, our people-math as our weapon": Critical mathematics in a Chicago neighborhood high school[END_REF] for a course designed with this explicit orientation. Examples where such an orientation integrates with actual struggles and campaigns are far rarer, however. [START_REF] Williams | Development: The dialectics of transgression and social transformation[END_REF] have more recently made the distinction between 'critical' and 'domesticated' forms of CH-AT in educational praxis: Our remarks in the previous paragraph suggest a critical perspective on perezhivanie to balance the more domesticated, personalized one attached to psychotherapy (i.e., that 'development' is a purely personal, therapeutic one rather than an emergence of class consciousness). The critical perspective may involve working with the experiential 'resources' that capital is not interested in, but that allows or encourages a different 'knowledge/know-how'. [START_REF] Zipin | Countering and exceeding "capital": A "funds of knowledge" approach to re-imagining community[END_REF] refer to such resources as 'knowledge that has use' as opposed to knowledge that 'has market-exchange value' thus: knowledge that has uses for social life in those settingsnot abstracted from living social use by a logic of accumulation of knowledge that, due to its scarcity, has 'market-exchange value' which can be parlayed by either individual 'entrepreneurs of the self' or networked collectives of 'social capitalists '. (p. 181) We should add to this 'knowledge that has use' in the struggles of the oppressed. Most obviously such 'resources' relate to how to 'live poor', but we also argue, following Freire, the need to recognize and legitimate practices and know how involved in sharing/mobilising solidarity with others who are oppressed and also how to engage in collective action, for example through social movements and community, union or party activity. One might also see how such resources can challenge the 'limits' Freire refers to in the education-industry, for example in asserting the need to overcome the disciplinary boundaries (in curriculum and pedagogy), spatial and temporal structures in educational institutions, and above all assessment and qualification structures [START_REF] Williams | Interdisciplinary mathematics education: A state of the art[END_REF].


Wajeeh Daher
email: wajeehdaher@gmail.com

Assessing students' perceptions of democratic practices in the mathematics classroom

Keywords: Democratic practices, mathematics classroom, questionnaire Methodology

Though there exists considerable literature exploring the connections between mathematics education and democratic society, much of this literature is theoretical about what could or should occur. This situation has led some researchers to call for the development of empirical research regarding democratic practices in the mathematics classroom. This paper attempts to advance quantitative empirical research in this area by presenting a questionnaire that examines students' perceptions of four democratic factors: freedom, engagement, equality and justice. The proposed questionnaire is factorable and explains 58.72% of the variance of a more general democracy score. The questionnaire was distributed to 398 ninth grade students, and the data collected was analyzed by computing a one sample t-test to assess participants' scores in relation to the four democratic factors. The results indicated scores are 'less than the good'.

Introduction

Democratic teaching practices may have a positive influence on students' learning outcomes in the mathematics classroom. This may be because successful democratic teaching and learning is conceived as situations where individuals are able to think for themselves, judge independently, and discriminate between good and bad information (Dewey as in [START_REF] Orrill | Mathematics, numeracy, and democracy[END_REF]. Therefore, it is desirable or necessary to enact democratic practices in the mathematics classroom. In addition, such democratic practices need to be assessed to ensure they are appropriately implemented, and to see if they influence specific variables of students' mathematics learning positively (e.g. emotions). One way to assess democratic teaching and learning practices and their relation with other variables is through quantitative methods, with a questionnaire as the main tool. The present research tries to do this and presents a questionnaire to assesses four democratic practices: freedom, equality, engagement and justice in the mathematics classroom. [START_REF] Vithal | Democracy and authority: A complementarity in mathematics education?[END_REF] argues that the literature exploring connections between mathematics education and democratic society is generally theoretical, which indicates the need for research that studies what happens actually in the mathematics classroom regarding this link. [START_REF] Aguilar | On the links between mathematics education and democracy: A literature review[END_REF] agree with Vithal and point to the need for empirical studies to test and expand such theoretical ideas. The present research offers one step towards providing such empirical research, in that it offers a questionnaire to assess students' perceptions of democratic practices relating to four democracy components: Freedom, engagement, equality and justice. It does so in the context of the Palestinian ninth grade mathematics classroom.

Research goals and rationale

Literature review

Democracy in general is described as consisting of distinct components. [START_REF] Kesici | Teachers' opinions about building a democratic classroom[END_REF] found that teachers, who adopted democratic practices, talked about democratic classrooms where fair behaviors are demonstrated towards students, students' range of personal freedom is enlarged, and students are provided with equality of opportunity. Thus, Keisci (2008) presents three categories of democracy: fairness, freedom and equality, where fairness can be related to justice and freedom and equality can be related to equity which includes the consideration of individual needs. Regarding equality, [START_REF] Kesici | Teachers' opinions about building a democratic classroom[END_REF] emphasizes that equality does not mean that teachers should treat all students the same way, but they should give equal opportunity to all students, so to meet their needs.

Democracy in the mathematics classroom [START_REF] Aguilar | On the links between mathematics education and democracy: A literature review[END_REF] identified three links between mathematics education and democracy. Firstly, mathematics education can provide students with the skills to analyze critically real life phenomena. Second, the mathematics classroom can encourage values and attitudes essential to building and sustaining democratic societies. Third, mathematics education can function as a social filter that restricts students' opportunities for development and civic participation.

The possible links between mathematics education and democracy have attracted the attention of mathematics education researchers for at least three decades, where different aspects of teaching and learning practices have been studied in the mathematics classroom. Some of these aspects are: authority [START_REF] Amit | Authority and authority relations in mathematics education: A view from an 8th grade classroom[END_REF], students' voice [START_REF] Daher | Student voice in the mobile phone environment[END_REF]Kaur, Anthony, Ohtani, & Clarke, 2013), the right to equal access to mathematical ideas [START_REF] Allen | Mathematics as thinking: A response to 'Democracy and school math[END_REF][START_REF] Ellis | Preparing teachers for democratic mathematics education[END_REF], promoting equality [START_REF] Croom | Mathematics for all students: Access, excellence, and equity[END_REF], promoting democracy [START_REF] Allen | Mathematics as thinking: A response to 'Democracy and school math[END_REF][START_REF] Ellis | Preparing teachers for democratic mathematics education[END_REF], diversity of curriculum and classroom [START_REF] Ellis | Preparing teachers for democratic mathematics education[END_REF], revisiting old ideas in new ways [START_REF] Ellis | Preparing teachers for democratic mathematics education[END_REF], dialogue [START_REF] Ball | The role of mathematics instruction in building a socially just and diverse democracy[END_REF][START_REF] Skovsmose | Linking mathematics education and democracy: Citizenship, mathematical archaeology, mathemacy and deliberative interaction[END_REF], proving [START_REF] Almeida | Are there viable connections between mathematics, mathematical proof and democracy[END_REF][START_REF] Skovsmose | Linking mathematics education and democracy: Citizenship, mathematical archaeology, mathemacy and deliberative interaction[END_REF], and engaging in ethnomathematics [START_REF] Ball | The role of mathematics instruction in building a socially just and diverse democracy[END_REF][START_REF] Skovsmose | Linking mathematics education and democracy: Citizenship, mathematical archaeology, mathemacy and deliberative interaction[END_REF]. This literature has indicated the complexity of the construct of 'democratic practice' in teaching and learning mathematics, which indicates the need for an assessment tool to better understand this construct.

Assessing democratic practices in the classroom

Educational researchers have offered various questionnaires to assess democratic practices in the classroom [START_REF] Daher | Democratic practices in a constructivist science classroom[END_REF][START_REF] Kubow | Fostering democracy in middle school classrooms: Insights from a democratic institute in Hungary[END_REF]. For instance, [START_REF] Ahmad | Empirical evidence on the relationship between democratic classroom and social skills development of students[END_REF] designed a questionnaire to assess the relationship between democratic classroom practices and students' social skills development. In addition, the tools used to collect data related to democracy in the mathematics classroom have included open ended questions [START_REF] Daher | Student teachers' perceptions of democracy in the mathematics classroom: Freedom, equality and dialogue[END_REF] and questionnaires with Likert items [START_REF] Bulut | The relationship between mathematics teachers' democratic behavior and students' mathematic success[END_REF]. The questionnaire suggested by [START_REF] Bulut | The relationship between mathematics teachers' democratic behavior and students' mathematic success[END_REF] consists of a single scale to assess democratic practice. In the present research we suggest a questionnaire that offers multiple factors relating to democratic teaching and learning practices described in the literature. The mathematics teacher ignores some students' discussions.

3.39 1.46 -.37 The mathematics teacher does not give the same time to all students to answer questions.

1.40 -.01

The mathematics teacher does not give the same time to all students during discussions.

1.31 -.07

The mathematics teacher does not give different solutions to accommodate differences between individual students.

1.53 .11

The mathematics teacher ignores some students' solutions.

3.41 1.41 -.35 The mathematics teacher encourages us to give new mathematical ideas.

2.58 1.29 .48 The mathematics teacher encourages us to justify our mathematical ideas.

2.48 1.29 .45 The mathematics teacher encourages us to give different answers to a mathematical question.

1.46 .52

The mathematics teacher encourages us to engage in mathematical discussions.

1.26 .73

The mathematics teacher encourages us to ask questions.

2.25 1.24 .62 The mathematics teacher does not clarify the reasons for giving marks on the mathematics exam.

3.34 1.20 -.14 I cannot have explanations in the mathematics classroom on the issues that I have not understood.

3.69 1.25 -.61

Analysis method of the level of democratic behavior scores in the Palestinian ninth grade

To calculate democracy mean scores for the students participating in the research, we computed first grouped frequency distributions [START_REF] Stockburger | Introductory Statistics: Concepts, Models, and Applications[END_REF]. Then we compared the resulting democracy mean scores with a 'good democracy score' and a 'normal democracy score'. We computed the normal democracy score by dividing 4 by 5 (4 items divided by 5 possible scores where 1 is the lowest score of any item, and 5 is the highest). This resulted in0.8 which was used to calculate intervals on a scale presented in Figure 1. We considered point (1.8) to be a 'low democracy score', (2.6) to be a 'normal democracy score' and (3.4) to be a 'good democracy score' on this scale. Thus, the scores between 1 and 1.8 were considered 'very low democracy scores', the scores between 1.8 and 2.6 'low democracy scores', the scores between 2.6 and 3.4 'normal democracy scores', the scores between 3.4 and 4.2 'good democracy scores', and the scores between 4.2 and 5 'very good democracy scores'. To compare the students' mean scores with the 'normal democracy score' and the 'good democracy score' we used a one-sample t-test and compared with the appropriate democracy value.

Results

Initially, the factorability of the 18 democracy items was examined. Several well recognized criteria for the factorability of a correlation were used. Firstly, it was observed that all the items correlated at least .3 with at least one other item, suggesting reasonable factorability. Secondly, the Kaiser-Meyer-Olkin measure of sampling adequacy was .89, above the commonly recommended value of .6, and Bartlett's test of sphericity was significant (χ²(153) = 2562.90, p < .001). The diagonals of the anti-image correlation matrix were also all over .5. Finally, the communalities were all above .4, further confirming that each item shared some common variance with other items. Given these overall indicators, factor analysis was deemed to be suitable with all 18 items. Table 3 presents factor loadings based on rotated principal components analysis with Oblimin rotation for 18 items from the above scale (N = 218).

Principal component analysis was used because the primary purpose was to identify and compute composite scores for the factors underlying the scale. Initial eigenvalues indicated that the first four factors explained 33.56%, 9.96%, 7.81% and 7.39% of the variance respectively. The fifth factor had an eigenvalue 5.02, while the other factors (from the sixth to the eighteenth) explained less than 4% of the variance.

Our adoption of the four-factor model depended on the results of the principal factor analysis, including the total-variance-explained table and the scree plot (see Figure 2 below). There was little difference between the four-factor Varimax and Oblimin solutions, thus both solutions were examined in subsequent analyses before deciding to use a Varimax rotation for the final solution.

According to this solution, the four factors explained 58.724% of the variance in the democracy scores.

Component

Item

F En Eq J We can express our opinion regarding teacher's presentation.

.78 We can express our opinion regarding the teacher's solution methods.

.75 We can express ourselves freely in the mathematics lesson.

.74 We can give different solution methods for a mathematical problem.

.74 We can even give a faulty solution method for a mathematical problem.

.47 The mathematics teacher encourages us to justify our mathematical ideas.

.75 The mathematics teacher encourages us to engage in mathematical discussions.

.73

The mathematics teacher encourages us to ask questions.

.72 The mathematics teacher encourages us to give new mathematical ideas.

.71 The mathematics teacher encourages us to give different answers to a mathematical question.

.68

The mathematics teacher does not give the same time to all students to answer questions.

.77

The mathematics teacher does not give the same time to all students during discussions.

.77

The mathematics teacher does not give different solutions to accommodate differences between individual students.

-.45

.63

The mathematics teacher ignores some students' discussion.

.60 The mathematics teacher ignores some students' solutions.

.54 The mathematics teacher does not clarify the reasons for giving marks on the mathematics exam.

.77 I cannot have explanations in the mathematics classroom on the issues that I have not understood.

.72

The teacher gives additional time for solving.

-.49

Validity and reliability analysis

Validity and reliability analyses were performed for the four factors. To ensure validity, the firstversion of the questionnaire was presented to experts in mathematics education or in the social aspect of learning to analyze it and thus verify its validity for data collection. After this analysis, the necessary corrections were made to the scale in accordance with their comments, which gave the present 18-item 5-point Likert type rating scale. To ensure reliability, Cronbach's Alpha was computed for each of the four democratic factors. This computation gave .81 for Freedom, .82 for Engagement, .78 for Equality, and .54 for Justice. These reliability results indicate good reliability for the constructs: freedom, engagement, and equality [START_REF] Field | Discovering statistics using SPSS[END_REF] because these reliabilities are around .80. However, the reliability of justice is not sufficient, and we suggest the need to add additional items to the questionnaire in order to improve the reliability of this construct [START_REF] Field | Discovering statistics using SPSS[END_REF]. The Cronbach Alpha computation for the whole questionnaire is .88 which indicates high reliability.

Level of democratic practices in Palestinian middle schools

To assess the level of democratic practices in the Palestinian mathematics classrooms, we computed means and standard deviations for the four factors: freedom, engagement, equality, justice s. We also conducted a one-sample t test to assess the statistical significance of the variation of each factor with a low, normal and good level of democratic practice (see

Discussion and conclusions

Empirical research regarding democratic practices in the mathematics classroom is still in its infancy. One way to develop such empirical research is to develop quantitative tools, such as questionnaires, to assess the level of democratic practice in mathematics classroom. The present research offers a questionnaire that assesses four democratic factors: freedom, engagement, equality and justice. These four constructs/factors explain 58.72% of the variance in the overall democracy score across our sample. Previous research, using questionnaires to assess democratic practice has suggested one scale consisting of only one factor that explains 47.701% of the variance of an overall democratic score. I argue that a scale with four factors addresses the concerns regarding scales with only one or two factors, which may not provide an accurate representation of the construct [START_REF] Pett | Making sense of factor analysis[END_REF]. Also, further democracy constructs, such as autonomy and decision making, could also be assessed in future research.

The questionnaire suggested in the present research may help researchers study the level of democratic practice in mathematics classrooms. Here, I used the suggested questionnaire to examine the democratic practices in Palestinian ninth grade mathematics classrooms. The results indicate that the level of democratic practice in this context is less than good. There may be various reasons that explain this level of democratic practice, such as authoritarian styles of teaching by teachers [START_REF] Ramahi | Education in Palestine: Current challenges and emancipatory alternatives[END_REF]. These authoritarian styles cannot foster critical or independent thinking, which may impact negatively the future functioning of students. In addition, the 'less than good' level of democratic practice may be a result of students' and teachers' limited awareness of the student's right to a democratic space in the mathematics classroom.

The present research indicates the need to improve the level of democratic practice in the Palestinian mathematics classroom. This could be done by different means. First, by holding workshops for mathematics teachers with the goal to increase their awareness of democracy factors and discuss with them ways to increase democracy in their classrooms. These workshops need to be based on pedagogic approaches that enhance student agency and voice, foster creative and critical thinking, and enable students to collaborate, share their mathematical ideas, and negotiate their decisions regarding their mathematical learning [START_REF] Daher | Student voice in the mobile phone environment[END_REF][START_REF] Ramahi | Education in Palestine: Current challenges and emancipatory alternatives[END_REF]. Moreover, these workshops need to be supported by the ministry of education as it calls for inclusive education in the schools (Ministry of Education & Higher Education, 2015). Second, special emphasis needs to be put on enhancing students' awareness of their democratic rights. These workshops to advance democratic practices in the Palestinian context could benefit from similar attempts around the world (see, e.g., [START_REF] Varnham | Democracy in schools: Encouraging responsibility and citizenship through student participation in school decision making[END_REF].

Figure 1 :

 1 Figure 1: Intervals related to the democracy scores of any item

Table 1 : The distribution of the sample according to gender and level in math

 1 The descriptive statistics of the questionnaire items are given in Table 2. Item M SD Skew We can express ourselves freely in the mathematics lesson. 2.84 1.35 .19 We can express our opinion regarding the teacher's presentation. 2.67 1.30 .32 We can express our opinion regarding the teacher's solution methods. 2.70 1.29 .32 We can give different solution methods for a mathematical problem. 2.55 1.47 .281 We can give a faulty solution method for a mathematical problem. 2.46 1.30 .77 The mathematics teacher does not give additional time for solving problems. 3.62 1.34 -.66

				Level in math
		Poor	acceptable	good	very good	Excellent	Total
	Gender male	41	34	33	22	10	140
	female	59	65	76	52	6	258
	Total	100	99	109	74	16	398

Table 2 : Means, standard deviations and skewness of the questionnaire items (N=398)

 2

Table 3 : Factor loadings based on Oblimin-rotated principal components analysis (N=398), F=Freedom, En=Engagement, Eq=Equality, J=Justice Figure 2: Scree Plot of the items factorization

 3

Table 4

 4

	).
	T value

Table 4 : Level of democratic practices in the Palestinian mathematics classrooms (N=398), ** p<.000

 4