Axel Tonini
email: axel.tonini@ec.europa.eu.

Roel Jongeneel

Regression and Qualitative Choice Models < C3 -Econometric Methods: Multiple/Simultaneous Equation Models < C -Mathematical and Quantitative Methods, P20 -General < P2 -Socialist Systems and Transitional Economies < P -Economic Systems, Q10 -General < Q1 -Agriculture <

Keywords: Dairy, Farm size, Poland, Markov chain, Generalized cross entropy dairy, farm size, Poland, Markov chain, generalized cross entropy

This paper sets out to analyse the evolution of the dairy farm structure of Poland during the post-socialist period. After focusing on how the farm structure has changed over time, an instrumental variable generalized cross entropy estimator is used to develop and estimate a Markov model in order to explore how farm structure will probably develop in the coming decade. The estimator exploits both sample data and prior information, including general and plausible information on farm mobility and structural adjustments based on independent literature. Next, several statistical indicators are computed for farm mobility and for which farms are likely to survive.

Finally, milk projections are made and related to policy scenarios. The projections show that the number of dairy farms will continue to decline, but the number of medium and large farms will increase. In the coming decade, subsistence dairy farms are expected to leave the sector slowly. Milk projections show that under the status quo, milk quotas will be binding and overrun, whereas under the 'soft landing' scenario they appear to be only binding after 2010.

INTRODUCTION

In this paper we set out to analyse the evolution of the dairy farm structure of Poland during the post-socialist period. This analysis is of interest to policy makers, in providing insight into how the farm structure is likely to evolve; a particularly relevant issue is what will happen to the subsistence and semi-subsistence farms in the restructuring process. The analysis is also of interest to the upstream and downstream industries that have to decide on investments in dairy processing capacity, milk collection schemes, and providing farm input supplies.

We have four objectives: to examine how the farm structure has changed over time and what path it is likely to follow in the coming decade by making several projections; to test whether the evolution of farm size is explained by non-stationary effects; to compute several statistical indicators of farm mobility and of which farms are likely to survive; and finally, to make milk projections for the coming decade, based on the projected number of dairy farms and to compare them with two possible policy scenarios: 1) status quo milk quota and 2) a gradual phasing out of the milk quota.

We use a Markov probability model [START_REF] Lee | Estimating the Parameters of the Markov Probability Model from Aggregate Time Series Data[END_REF] of farm size distribution which is able to analyse movements of individuals between different states when only aggregate data on finite size categories are available for a given time period. A generalized cross entropy (GCE) estimator is used (see [START_REF] Golan | Maximum Entropy Econometrics: Robust Estimation with Limited Data[END_REF][START_REF] Mittelhammer | Econometric Foundations[END_REF]. Entropy estimators are particularly suitable when dealing with limited data, as is often the case for empirical applications on Central Eastern European Countries (CEECs). Our paper further extends the approaches of [START_REF] Golan | Estimation of non-stationary social accounting matrix coefficients with supply-side information[END_REF], [START_REF] Courchane | Estimation and evaluation of loan discrimination: An informational approach[END_REF], Karantininis (2002) and [START_REF] Jongeneel | Dairy farm size distribution in East and West: Evolution and sensitivity to structural and policy variables, paper Agri-Food System[END_REF] by allowing for a heteroscedastic version of the set of Markov equations and for seemingly unrelated regressions (SUR) estimation. Assuming a common and constant variance matrix across the different Markov states, as done, for example, in Karantininis (2002) and [START_REF] Jongeneel | Dairy farm size distribution in East and West: Evolution and sensitivity to structural and policy variables, paper Agri-Food System[END_REF], could easily create bias on the estimated Markov transition probabilities affecting related indicators as well as projections.

The remainder of this paper is organized as follows. Section 2 describes the farm structure of Poland, with a focus on dairy farming. Section 3 specifies the Markov chain entropy formalism. Section 4 discusses the sample data as well as prior information. Section 5 discusses results. Section 6 presents the associated milk projections and relates these two policy scenarios. In Section 7, the conclusions are presented.

FARM STRUCTURE IN POLAND, WITH A FOCUS ON DAIRY FARMING

Poland is one of the most important dairy producers in the European Union (EU). In 2006 it accounted for about 8 per cent of the total EU-27 cow milk production, being the fourth EU milk producer after Germany, France and United Kingdom. In the last five years, dairy cow numbers have declined by 9.4 per cent and milk yields have improved by 15.1 per cent (FAOSTAT, 2006). Since the demise of the socialist regime, the Polish dairy sector has presented a highly fragmented dairy farm structure, with a large number of small private family farms, just as in other sectors of agriculture. In 1987, about 67 per cent of the dairy farms had only 1-2 cows and these accounted for 41 per cent of the national herd. The number of private dairy about 80 per cent of the national milk production was being produced from farms with 10 cows or less (Sznajder, 2002, pp. 242-244).

In Poland, dairy producers after the transition reform can be classified into three main categories: farmers with 1-2 cows, producing milk mostly for the farm household (i.e. subsistence dairy farms); farmers with more than 3-4 cows, who produce milk for sale in local markets and for their own needs (i.e. semi-subsistence dairy farms); and farmers with more than 10 cows, who produce almost exclusively for the dairy industry (Sznajder, 2002, p. 248). In 1996, about one quarter of Polish milk was produced by almost 1 million individual farms keeping 1 to 3 cows, while half was produced by farms with 3 to 9 cows (European Commission, 1998, p. 36).

This underscores the great fragmentation of Polish milk production even after transition. In 2005 there were about 700 000 dairy farms: a decline of about 51 per cent as compared with the number of farms in 1995. Also in 2005, about 65 per cent of the farms with dairy cows were subsistence farms with 1-2 cows (Figure 1) and about 53 per cent of the dairy cow stock was concentrated in farms with 1-9 cows.

The Polish Ministry of Agriculture expects the number of total farms to fall by 76 per cent between 1996 and 2010 (AgraEurope, 2000, pp. 18-19). At first sight, Figure 1 suggests that the evolution of Polish dairy farms has proceeded without being affected by the EU milk quota system which was announced in 2004 and effectively introduced in 2006. In addition, it appears that the size class with 3 to 9 cows is the 'switch' class: farms with smaller herds (i.e. 1-2 cows) show a tendency to decline, whereas for farms with larger herds (i.e. more than 10 cows) the opposite holds. This suggests that some of the dairy farms in the size class with 3 to 9 cows will go out of business, or scale down, or scale up.

CHAIN

The Markov chain approach is very suitable when the only data available are count data in the form of observable proportions or aggregates rather than data at the level of micro units. Movements from state to state are represented by a stochastic process and are typically modelled by estimating the so-called Markov transition probabilities. It is often the case that the proportions/count data are only available for the total aggregate and not for the net shifts, so that the number of unknowns in terms of transition probabilities to be estimated might exceed the number of available data points (i.e. ill-posed problem). In addition, the proportions/count data may be potentially correlated (i.e. ill-conditioned problem). In this context, the maximum entropy (ME) algorithm developed in [START_REF] Golan | Maximum Entropy Econometrics: Robust Estimation with Limited Data[END_REF], [START_REF] Fomby | Advances in Econometrics, Applying Maximum Entropy to Econometric Problems[END_REF] and [START_REF] Mittelhammer | Econometric Foundations[END_REF] is a suitable candidate for extracting the maximal signal from an initial 'out-of-focus' problem. [START_REF] Fraser | An application of maximum entropy estimation: The demand for meat in the United Kingdom[END_REF] used maximum entropy estimators to estimate the demand for meat in the United Kingdom under severe multicollinearity problems. He showed that maximum entropy estimators relying on minimal underlying distributional assumptions perform well where traditional econometric approaches are unsatisfactory.

Our paper is based on a GCE formalism which is founded on the directed divergence or minimal discriminability principles of [START_REF] Kullback | Information Theory and Statistics[END_REF][START_REF] Good | Maximum entropy for hypothesis formulation, especially for multidimensional contingency tables[END_REF]. GCE is suitable when some 'educated' guesstimates based on previous data, experiments or economic theory are available (i.e. prior estimates). As discussed by [START_REF] Golan | Information and entropy econometrics X Editor's view[END_REF], GCE is an information theory distance measure of the information contained in the posterior estimates as compared to the information contained in the prior estimates. Out of all the feasible solutions, GCE selects the one that minimizes the divergence between the data and the priors, the final solution being the closest to the data and priors. Considering the dynamic farm growth process in a Markov problem, it seems likely that farm growth can be explained by non-stationary effects.

Several economic variables are then expected to affect the unknown transition probabilities 1 . Applying the formulation as developed in [START_REF] Golan | Estimation of non-stationary social accounting matrix coefficients with supply-side information[END_REF] and [START_REF] Courchane | Estimation and evaluation of loan discrimination: An informational approach[END_REF] 2 , it is possible to assess the impact of key variables on the Markov transition probabilities, therewith potentially improving the explanatory power of the model. In formalizing the problem, the non-stationary GCE Markov problem can be formulated as follows:

( ) ( ) ( ) + = t k h tkh tkh tkh l k lk lk lk tkh tkh lk lk u w w q p p u w q p I / l n / ln , , , min (1)
subject to the following constraints: For example, a literature review suggests that out of all possible covariates the following appear likely to affect the transition probabilities of dairy farms: technological shift, milk price, feed price, dairy cow stock price (see [START_REF] Goddard | Economics of structural change in agriculture[END_REF]Zepeda, 1995b;Karantininis, 2002). 2), is reparameterized as given by equation (3), following the classical maximum entropy formalism (Golan et al., 1996, pp. 107-110), where tk V is an H-dimensional vector of support points and tk w 3 By analogy, the GCE criterion also minimizes the divergence between the error in the form of posterior probabilities tkh w and the priors tkh u where tkh u are taken to be uniform since no prior information is available on the error term.

+ = t l
4

The alternative simpler Markov stationary problem can be obtained by simply withdrawing the covariates tn z from equation (2 The consequence is that the estimates of the transition probabilities for these size classes are likely to converge to the prior estimates and underutilize the information present in the sample data. To avoid this, variances are specified per size class, following the statistical model presented in Golan et al. (1996, pp. 182-185). By so doing, different error support bounds are specified for each Markov state relying on the 'three sigma' rule of [START_REF] Pukelsheim | The three sigma rule[END_REF] When defining the tk V vector, there are several options. One is to set

[ ] 1 , , 0 , , 1 K K = tk V
given that the Markov states are expressed in terms of proportions/shares and tk y and tl x follow in a range between zero and one. A second option is to set

[ ] T K T K tk / 1 , , 0 , , / 1 K K = V
where K is the number of states and T number of years as suggested in [START_REF] Golan | Estimation of non-stationary social accounting matrix coefficients with supply-side information[END_REF], Courchane et al. (1991), Karantininis (2002). Both options, although empirically plausible, assume a common and constant variance matrix across the different Markov states. ( )

n k n tkh tn nk tkh h n tkh tn nk tkh n tkh tn nk tkh tkh V z u V z u V z u w ẽxp ẽxp ẽxp ~ = = (6.b)
where tkh u are taken to be uniform with

H u tkh 1 = . A

condensed version of the

Lagrange problem for the IV-GCE estimator is provided in Appendix A.

The estimation procedure allows for the possibility of non-zero covariances following the one-step GCE-SUR as described by Golan et al. (1996, p. 186). In contrast to the two-stage estimation procedure usually applied in conventional estimation procedures, the unknown elements of the covariance matrix are now jointly estimated with the unknown Markov transition probabilities. The one-step GCE-SUR requires the following additional consistency constraints to be added during the estimation:

2 1 1 1 1 1 1 1 = = = = T t T t tg tg T t tk tk kg tg tk e e T e e T e e T
, for k g (7)

where

gg kk kg kg 2 2 =
. The unknown covariance correlation coefficient kg is simultaneously estimated without the need to be reparameterized with the rest of the unknowns for each pair k g, and k, g = 1,…, K.

The relative information content of the estimated parameters can be evaluated through the normalized entropy measure described in Golan et al. (1996, p.93). The measure is defined for values between zero and one, with values approaching zero in the case of no uncertainty and values approaching one in the case of perfect uncertainty (i.e. uniform distribution). Additional entropy statistics used in the paper are the so-called entropy ratio and an analogous entropy Chi-square measure, both described in Golan and Vogel (2000, pp. 454-455).

In an instrumental variable GCE (IV GCE) Markov approach, non-stationary effects can be determined by the following elasticity that determines the cumulative effects of a unit change in each covariate tn z on tk y , the vector of proportion falling in the k-th Markov state in time (t+1), as given by Karantininis (2002, p. 10):

= = l nk k lk nk l lk k n k tn tn kt y kn p x p y z y z z y 2 (8)
Appendix B recovers the probability elasticities for the IV-GCE problem from which the composite elasticity in equation ( 8) is derived.

Following the Markov formalism based on the Markov equilibrium distribution and absorbing states notions (Judge and Swanson, 1962, pp. 58-59) [START_REF] Krawiecka | Personal Communication on Data on Private Dairy Farms in Poland[END_REF]. In order to account for exit and entry, an additional size class was defined, containing the 'inactive farms' and 'potential entrants' ( 0 , = k l

). Data were normalized by a common scalar equal to the maximum number of farms contained in the aggregate transition counts. In order to capture potential non-stationary effects on the Markov transition probabilities, several explanatory variables (such as raw milk price, feeding cost, etc.) were used, but because of parsimony and the limited number of observations in the data finally only the trend variable 1 t z was kept.

The researcher may follow several principles in order to best approximate the farm size growth and to guess or estimate the probability of a farm being in a given size class. In order to avoid data mining and ensure efficiency in estimation, wherever possible the prior information should be derived from sources independent from the sample data. In this study, previous research was examined and the lessons (general patterns) drawn from this formed the basis of the prior information used (see Based on these findings in the literature, the priors on the diagonal transitional probabilities were set, moving from the top left corner to the lower right corner of the transition probability matrix from 0.80 to 0.90 (i.e.

k l p lk = = 80 . 0 f o r 4 , 3 , 2 , = k l and k l p lk = = 90 . 0 f o r 8 , 5 , K = k l
). ).

Net Shifts:

-As regarding the net shifts, one pattern observed from the literature is that farms show a tendency to develop gradually. This implies that the probability a farm will move from its current size class to an adjacent size class is generally higher than the probability it will move to more distant size classes. A second finding is that there is usually a 'switch' size class, below which farms show a tendency to decline and ultimately go out of business, whereas above this size class, farms expand their business. This finding is probably to do with the farms being predominantly family businesses and therefore with farm succession being tied to the family cycle (e.g.

ageing farmers with no successors are likely to gradually downsize their business).

Another explanatory factor might be that farms need to be a certain critical size in order to be considered 'viable', i.e. be able to finance expansion relying on internally generated savings and also be able to acquire external credit (see [START_REF] Swinnen | Agricultural privatization, land reform and farm restructuring in Central and Eastern Europe: A comparative analysis[END_REF][START_REF] Tonini | Dairy farm size restructuring in Poland and Hungary[END_REF]. Reviewing previous studies it appeared that which size class is the tipping-point size class is generally country-and casespecific (depending, for example, also on the specified number and width of size classes). Our prior estimate of the 'switch' size class is therefore based on the (conditional on prior assumptions previously made about exit for the lower size classes). The prior assumptions made so far imply that most of the lower and upper off-diagonal elements of the transition probability matrix have prior expectations equal to zero (see [START_REF] Disney | A Markov chain analysis of pork farm size distributions in the South[END_REF], Zepeda (1995) for a similar approach).

ESTIMATION RESULTS AND DISCUSSION

The IV GCE Markov model was estimated including a trend capturing for structural change. The normalized signal entropy ( ) p S for the system was 0.663 whereas the normalized noise entropy ( ) w S for the system was 0.971. The information index

( )

p Ĩ or pseudo-R 2 for the signal was 0.337. The estimated O 2 ~(K-1) statistic was 0.416, indicating that the estimated transition probabilities did not statistically differ from the priors at five per cent significance level. A similar result was obtained when computing the signal entropy ratio (i.e. only considering the signal distribution) which was equal to 2.324. The Jarque-Bera test revealed that at five percent significance level the hypothesis of normally distributed errors could not be rejected (Verbeek, 2004, p. 185). Statistical testing, at least for the signal part, was done under negative degrees of freedom, given that Kx(K-1) independent 8 transition probabilities had to be estimated, which only had K total aggregate data of finite size categories for T transitions. However, the estimates were fairly robust to changes in the prior magnitude 9 .

Even though the power of statistical tests can be weakened when there are negative degrees of freedom, several facts can be drawn from the above results. The computed statistics suggest that the data did not push the final estimates too far from the prior, which indicates either that the data signal is poor, or that the prior estimates conform to the data. This finding is also related to the negative number of degrees of freedom. Table 2 presents the estimated IV GCE Markov model (i.e. non-stationary model).

The estimated transition probability matrix itself already provides insight into the dynamic adjustment of dairy farms. For example, during the period considered there was a strong tendency for farms to persist in the same size class from one year to the next (see transition probabilities on the diagonal containing elements kk p ). The off-diagonal elements of the transition matrix provide information on the extent to which dairy farms are going to scale up or down. For example, from one period to the next, about 2 per cent of all farms with 10-29 cows will probably grow into dairy farms with 30-49 cows. In Table 2 the cumulative effects of the trend 1 t z on the 8 This is obtained by subtracting from the KxK transition probability matrix the K row adding-up condition in equation (4). 9

For a given prior configuration we carried out several estimations by changing the prior magnitude by only one digit each time. This did not change the final estimates appreciably. To save space, results are not reported here, but they are available upon request from the authors. number of dairy farms tk y in terms of elasticity is presented in the last row. The trend impact found implies that over time there is a contraction in the farms with 1-9 cows and an increase in the remaining farms. The trend also has a positive impact on the number of farms in the inactive size class (Exit). Our results fit in with Sznajder (2002, p. 253) who shows that in order to have full return from the engaged capital, including rent of the land, a Polish dairy farm needs to have a herd of at least 10-15 dairy cows. This suggests that the minimum efficient size of dairy farms, minimizing the per unit costs, or the minimum locus on the long-run average costs level for farms is a herd size of 10 or more cows.

Table 2: IV GCE-SUR Markov transition probabilities and nonstationary effects

Table 3 reports the estimated mean number of years in each transient state for each non-absorbing state (i.e. transient periods) as well as the probabilities of absorption for each non-absorbing state into the two absorbing states (i.e. absorption probabilities). These estimates provide an additional indicator of the rate of change in the number of dairy farms by herd size class. Thus for a dairy farm with 10-29 dairy cows, the mean number of years before absorption is about 50, whereas for a dairy farms with 2 cows the mean number of years before absorption is about 6. This suggests that the rate of change is faster for the small dairy farms than for the medium and large dairy farms. From the last two columns of Table 3 it also appears that in equilibrium the majority of the dairy farms with 1 and 9 cows will leave the sector, whereas the dairy farms belonging to the remaining size states will continue in dairying. More precisely, only 16 per cent of the dairy farms with 3-9 cows will persist in the dairy sector, whereas 84 per cent are expected to leave the sector. In addition, from our results it appears useful to impose some sort of prior information on the estimated Markov transition probabilities, given the relatively low projection power of the models estimated with uniform priors.

Table 4: Dairy farm size distribution: projected versus actual numbers for 2006

The estimated IV GCE-SUR model predicts the total aggregate number of dairy farms reasonably well, although the model tends to overestimate the number of farms in most of the size classes -except for the farms with 2, 30-49 or 100-199 cows, where the model underestimates the total number of farms. This is mainly attributable to the effect of net shifts from one size class to the adjacent size class.

MILK PROJECTIONS AND MILK QUOTAS IN POLAND

Based on the estimated projected dairy farm size distribution, the associated aggregate Polish milk supply was calculated. In order to do so, several simplifying assumptions were made on the average number of cows per farm of a certain size class, as well as the autonomous growth of milk yield. In addition, it was assumed that milk was being delivered by farms with more than 10 cows as well as by a proportion of the farms with 3 to 9 cows. Similarly it was assumed that the remaining milk produced from farms with 3 to 9 dairy cows was allocated to direct sales and home consumption. Milk projections were calibrated for the base year 2006. In order to compare the supply with the quota, the milk supply was corrected for the actual fat content. For a more detailed summary of the assumptions, see Appendix C. The milk projections are presented in Figure 2, which shows that direct sales will decline over time and also that milk deliveries are expected to grow slightly. This growth is attributable to restructuring in the sector as well as to genetic improvements in milk yields. Our supply projections were related to two milk quota scenarios. The first scenario represents the status quo milk deliveries and direct sales quota allocation.

The second scenario considers a gradual phasing-out of milk quotas, which could be part of a 'soft-landing strategy' before the expected removal of milk quotas in 2015 (e.g. [START_REF] Fischer-Boel | Designing the Dairy Market of the Future, Panel Discussion[END_REF]. Phasing-out is assumed to take place by a 2 percent per annum quota increase, starting from 2008 and continuing until 2015. Although hypothetical, such a scenario might well be considered in next year's 'Health Check' evaluation of the Common Agricultural Policy. Figure 3 provides the percentage overrun for milk deliveries under the two different scenarios. Whereas under the status quo the milk quotas are expected to be binding and overrun from 2008 onwards, with the 'soft-landing' scenario they appear to be only binding after 2010.

In addition under this 'soft-landing' scenario the percentage of overrun on milk deliveries is less than 10 percent at maximum, and about one third of the percentage of overrun under the status quo.

CONCLUSIONS

The projections showed that the number of dairy farms will continue to decline in the coming decade, although with an increase in the number of medium and large farms.

The size class with the largest average annual growth rate will be farms with 50-99 cows. The small dairy farms (i.e. semi-subsistence farms) will continue to exit from the sector although their relative share in the total number of dairy farms will tend to persist. It is estimated that on average, a subsistence dairy farms with 1-2 cows will persist for 7 years before absorption. In addition, only dairy farms with at least 10-29 cows and about 16 per cent of the dairy farms with 3-9 cows are expected to survive at the Markov equilibrium. Overall, our findings show that Poland is likely to be characterized by a polarized dairy farm structure with at one extreme a persistent fringe of subsistence and semi-subsistence self-employed small dairy farms and at the other extreme a growing fringe of commercially-oriented dairy farms.

The aggregated milk supply associated with the farm size distribution projections shows a slight increase of about 2 per cent per annum. Looking at the disaggregated figures for delivered and direct sales, it appears that the quantities delivered are increasing and at the same time the direct sales are decreasing. This is attributable to the restructuring of Polish dairy farms, in which there are a declining number of semi-subsistence farms producing for their own consumption and direct sales and simultaneously there is an increase in the number and scale of commercial farms focusing on deliveries. As regards the status quo scenario, the overrun of milk production makes clear that the current quota provision is likely to impede the farm size restructuring 10 . This will particularly affect the size classes with herd sizes of 10 or more dairy cows. In contrast, gradual phasing-out of the milk quota, as analysed in 10 If milk quotas are made tradable the impact might be limited or even go the other way. The value of the quota might then also act as an exit payment, inducing some farmers to leave the sector even earlier than initially planned. Moreover, evidence from [START_REF] Dawson | The post-quota performance of dairy farms in England and Wales[END_REF] on the dairy sector in England and Wales shows that even in the case of binding quota, quasi-fixed factors (i.e. labour, land, machinery, and the herd) go on to adjust, be it more sluggishly than if there are no quotas. As such, the 'temporary' quota constraint faced by Polish farmers might not have a big impact on farm restructuring. the alternative scenario, could facilitate the current restructuring process. Our findings suggest that in the latter case, an appropriate distribution scheme which allocates additional quota to the larger farms that are likely to expand might be relevant. As quota increases are likely to be accompanied by declines in milk prices, they could limit the funds available for investments and modernisation and thus slow down the speed of adjustment, although the direction of adjustment is unlikely to change.

Although the Markov chain approach appears to be flexible for handling a wide scope of dynamic factors, the predicted evolution of the Polish dairy sector might also be affected by other factors, which are not explicitly included or not sufficiently accounted for in the model. Examples are poorly functioning factor markets (hidden unemployment, dis-functioning land market) and the (vertical) integration with the downstream dairy industry (e.g. [START_REF] Petrick | The role of agriculture in Central and Eastern European rural development: An overview[END_REF]Weingarten, 2004, p. 6 and[START_REF] Latruffe | Determinants of technical efficiency of crop and livestock farms in Poland[END_REF]. For these reasons, the actual evolution might be different from the one projected in this paper, in particular for the subsistence sector. Through the gradient of the Lagrange function with respect to the unknown to be estimated, the optimal first order conditions are given by: Here the probability elasticities for the IV-GCE estimator are derived. Three types of impact elasticity are derived: the probability elasticity for an increase in lt x , the probability elasticity for increase in the tn z covariates, the cumulated probability elasticities on the total round count kt y for an increase in the tn z covariates.

F

• The marginal effect on lk p for a change in tl x is given by:

+ + $ + =

  farms had already shrunk greatly before transition: by about 25 per cent from 1981 to 1987. Dairy cow numbers declined concomitantly. At the beginning of transition,

  Figure 1: Dairy farms in Poland, 1995-2006

  , it is possible to compute several indicators such as the mean number of years for a farm being in a transient Markov state before it is absorbed in an absorbing state, as well as the probability that a transient Markov state will end up in an absorbing state. The projections of farm numbers were obtained in two steps. In the first step, the Markov transition probability matrix was multiplied by itself n times in order to obtain the transition probability matrix during n time periods. In the second step, individual elements of the transition probability matrix were multiplied by the number of farms present in their respective size class in the base year used for projections.

  We used aggregate data on the size distribution of private farms with dairy cows in Poland. The farms had been classified according to their herd size classes. The data cover the period from 1995 to 2006 and allow the recovery of the number of dairy farms in eight 6 farm size classes: 1 cow, 2 cows, 3-9 cows, 10-29 cows, 30-49 cows, 50-99 cows, 100-199 cows, > 200 cows

  Figure1). Our prior for the farms in this size class is that they have a fifty-fifty

Figure 2 :

 2 Figure 2: Milk production projections in Poland (2006=100)

Figure 3 :

 3 Figure 3:Percentage overrun for direct sales of milk

  Taking the first order condition (A.2) and bringing terms to the right hand side as a of the terms on the left and right hand side yields: the l µ ~Lagrange multiplier is lost and the IV-GCE Markov transition probabilities are finally recovered: nk Lagrange multipliers and corresponding restrictions one is redundant it is therefore convenient to normalize the expression in (A.10) by 0 ~= nk for each covariate n = 1,…, N. This provides the following scaled solutions:In a similar way it is possible to recover the proper probabilities related to the error term. Taking the first order condition (A.3) and bringing terms to the right hand side as a function () of the terms on the left and right hand side yields: normalization one constraint is lost and the IV-GCE error proper probabilities are finally recovered:

  Figure 2: Milk production projections in Poland (2006=100)

  ).

	is an H-dimensional vector of proper probabilities with	H	2	5 . Given that each
	Markov state can be characterized by a different variance, a specific definition of
	support bounds for each Markov size class is desired. In such a case, specification of
	a common and constant variance for each Markov states can lead to relatively large
	support bounds being specified for size classes where the variance is relatively small.
	F o r
	P
	e
	e r
	R
	e
	v i e
	w

Table 1 )

 1 7 . The prior information on Markov transition probability estimates may be one of three types: the probability of a farm persisting in the same farm size class (i.e. persistence), the probability of a farm entering and/or exiting the sector (i.e. entry/exit), and the probability of moving to another farm size class (i.e. net shifts).

	Submitted Manuscript

6

Nine farm size classes if the artificial entry and exit class is included.

7

A recent example neglecting this independence requirement is

[START_REF] Stokes | Entry, exit, and structural change in Pennsylvania's dairy sector[END_REF]

.

Page 12 of 36

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK period to another. More detailed analysis of these studies revealed that persistence is generally lower for small farm size classes as compared to large farm size classes.

Table 1 : Transition probability estimates: Literature overview

 1 information was known about entering farms, let alone about the probabilities of entrance in different size classes. Given this finding and the character of our data, which required us to focus on net transitions (net entry), it was decided to specify no

	Entry/Exit:
	-	As regards exit, the literature shows two basic results: small farms are more
	likely to exit than large farms (see also earlier comment), and the smaller the farm,
	the higher the probability of exit. Combining this with the already specified priors on
	persistence (which was set to 0.8 for small farms) the priors on the exit probabilities
	10 , p p	20	and 30 p were set to 0.20, 0.15 and 0.10 respectively.

-

With respect to entry, in all the studies shown in Table 1, the total number of enterprises shows a clear tendency to decline over time. Generally, very little Page 13 of 36 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 3 : Estimated transient periods and absorption probabilities

 3 Finally, the estimated Markov transition probability matrixes were used to make several projections of the number of dairy farms in the coming decade. In order to assess the predictive power of the estimated Markov models, projected values and actual values were first compared for the most recent available year (i.e. 2006). We compared two types of models: the IV GCE Markov model estimated with SUR, hereinafter called IV GCE-SUR (i.e. non-stationary model) and the similar model without the inclusion of the trend (i.e. stationary model). In addition, for each type of model we compared the model with the priors as defined in Section 4 with a model estimated using uniform (i.e. non-informative) priors. In terms of projections, the best performance was obtained for the IV GCE-SUR model with non uniform priors.

Table 5

 5 provides the projections associated with the IV GCE-SUR model. As can be seen it is predicted that by 2013 about 47 per cent of the number of dairy farms active in 2007 will have left the sector (ceteris paribus).

	F o r
	P
	e
	e r
	R
	e
	v i e
	w

Table 5 : Projected dairy farm size distribution (IV GCE-SUR)

 5

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																		Submitted Manuscript
	ẽxp tl nk x APPENDIX C: MILK PROJECTIONS -MAIN ASSUMPTIONS ẽxp $ $ k n t tl tn nk lk n t tl tn nk lk x z q x z q
										l q	1	+			lk q		ẽxp tl tn nk x z	2
														k				n	t
	tn lk p z	=	lk p	nk	tl x		lk p	k		lk p	nk	tl x	=	lk p	tl x	nk	k	lk p	nk Ẽxpressing
						the effect on lk p for a change in lt x in terms of elasticity at sample average yields:
	2 nk ~(B.4) ẽxp ẽxp ẽxp ~k 1 1 tl tn nk lk l n tn nk n t tl tn lk k n t tl tn nk lk l lt lk x z q q z x z q x z q q x p (B.1) F o r = $ k nk lk nk tl tn lk tn tn lk p x z p z p z • The cumulated effect of each covariate tn z on the total round count tk y is given by F o r
	n ẽxp t 2 ẽxp ẽxp $ tl tn nk n t tl tn nk lk tn nk x z x z q z P $ lk n tl tn q x e nk lk p 2 P e e r That in terms of elasticities translates into: 1 + l k n t nk lk q z = $ l k nk tl lk tl l tn lk x p x p z q e r	(B.5)
	lt lk p l x	= z p lk p lk $ tn	k y p lk tk tn z p for a change in tl n t = n n lk tn nk p z R tn nk z e k = l k nk lk nk tl lk k tn k p x p y z 2 n nk lk z p Ẽxpressing tn (B.6) R e the effect on lk lk n tn nk p z = tn kt tk tn tl z y y z x x in terms of elasticity at sample average yields: v i e v i e
	tl lk x p	$	lk tl p x	=	tl x	n	nk	z	tn			k		n	lk p	tn ~(B.2) nk z w w
	• The marginal effect on lk p for a change in tn z is given by:
	+ ~k + = 1 1 l k n t lk l tn lk q q q z p	ẽxp lk tl tn nk q x z	$	lk q	ẽxp tl tn nk n x z	t	2	nk	z	tn	lt x	$	+ ẽxp tl nk x (B.3)
																		n	t

Table C .1: Projection assumptions

 C

	Page
	Year	1	2	3-9 10-29 30-49 50-99 100-199 > 200 Average
	Milk Yield Annual growth
	(%)	0.00 0.00 0.25	0.50	0.75	1.00	1.25	1.50	0.66
	Milk Yield/Dairy Cow
	(Kg/Dairy Cow)	3650 3750 3850	3950	4050 4150	4250	4350	4000
	Average Number of Dairy
	Cows/Farm with Dairy Cows
	(Hd/Farm)	1	2	6	20	40	75	150	300	74

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Figures and Tables in the Text Figure 2: Dairy farms in Poland, 1995-2006

		80%											12.0%
		70%
													10.0%
		60%
													8.0%
	1-2 cows and 3-9 cows	30% 40% 50%											6.0%	> 10 cows
													4.0%
		20%
													2.0%
		10%
		0%											0.0%
		1995	1996	1997	1998	1999	2000	2001	2002	2003	2004	2005	2006
						1-2 cows		3-9 cows	> 10 cows

Note: Percentages are expressed relative to the total number of active dairy farms.

Source: Our calculations based on

[START_REF] Krawiecka | Personal Communication on Data on Private Dairy Farms in Poland[END_REF]

.

Table 6 : Transition probability estimates: Literature overview

 6 Estimates may reflect different transition period lengths, as indicated by the last column. Source: Our calculations, based on estimates from the literature.

	Authors	Year	Average	Smallest Class	Largest Class	Number of	Transition
			Estimates	Estimates	Estimates	Classes
				Dairy Studies
	Padberg	1962	0.691	0.733	0.960	4	5 years
	Hallberg	1969	0.879	0.768	0.961	5	annual
	Keane	1991	0.756	0.360	0.945	7	6 years
	Zepeda	1995	0.901	0.877	0.944	3	annual
	Stokes	2006	0.898	0.805	0.999	6	annual
				Other Studies
	Judge and
	Swanson	1962	0.511	0.412	0.672	6	annual
	Krenz	1964	0.862	0.804	1.000	6	5 years
	Lee et al.	1965	0.650	0.473	0.572	4	annual
	Ethridge et al.	1985	0.957	0.919	0.986	5	annual
	Edwards et al.	1985	0.687	0.781	0.813	8	4 years
	Garcia et al.	1987	0.836	0.930	0.929	11	annual
	Disney et al.	1988	0.605	0.400	0.732	4	5 years
	Karantininis	2002	0.531	0.386	0.768	18	annual
	Note:

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Table 7 : IV GCE-SUR Markov transition probabilities and non-stationary effects

 7 S(p i ) is the normalized entropy measure for the signal part of the estimated parameters.

	Class	Exit	1	2	3-9	10-29	30-49	50-99	100-199	> 200	S(p i )
	Entry	1.000									1.000
	1		0.118	0.882								0.727
	2		0.116	0.054	0.829							0.919
	3-9	0.063		0.044	0.872	0.021					0.722
	10-29					0.980	0.020				0.302
	30-49						0.919	0.081			0.862
	50-99							0.984	0.016		0.254
	100-199								0.989	0.011	0.183
	> 200									1.000	1.000
	z	t	1	0.011	-0.007	-0.002	-0.007	0.011	0.047	0.003	0.132	2.524
	Note: Source: Our estimates.

Table 8 : Estimated transient periods and absorption probabilities

 8 The last two columns of the table report the absorption probabilities.

	Class	1	2	3-9	10-29	30-49	50-99	100-199	0	> 200
	1	8.447							1.000	0.000
	2	2.689	5.865						1.000	0.000
	3-9	0.919	2.005	7.825	8.182	2.030	10.164	15.240	0.836	0.164
	10-29				49.980	12.402	62.087	93.091	0.001	0.999
	30-49					12.403	62.089	93.094	0.001	0.999
	50-99						62.089	93.094	0.001	0.999
	100-199							93.098	0.001	0.999
	Note:

Source: Our estimates.

Table 9 : Dairy farm size distribution: projected versus actual numbers for 2006

 9

	1	2	3-9	10-29	30-49	50-99	100-199	> 200	Total
				IV GCE-SUR
	286690	124949	148573	68203	5591	1155	140	42	635343
	2.47	-5.37	1.15	5.99	-6.43	3.34	-7.19	21.05	0.74
				IV GCE-SUR (Uniform Prior)
	183155	111209	120992	37372	4275	1184	253	69	458508
	-34.54	-15.77	-17.63	-41.92	-28.46	-15.88	51.34	82.05	-27.30
				GCE-SUR
	292110	126837	153170	67985	5564	1146	127	41	646979
	-4.40	-3.94	4.28	5.65	-6.88	-18.63	-24.15	8.85	2.59
				GCE-SUR (Uniform Prior)
	252441	154765	167159	22858	1779	1286	105	22	600415
	-9.78	17.21	13.80	-64.48	-70.23	-8.67	-37.21	-41.48	-4.79
				Actual 2006
	279791	132037	146887	64350	5975	1408	167	38	630653
	Note: Percentage deviations are reported in italics.

Source: Our estimates.

Table 10 : Projected dairy farm size distribution (IV GCE-SUR)

 10

	Year	1	2	3-9	10-29	30-49	50-99	100-199	> 200	Total
	2007	253833	115943	128116	66135	6781	1867	188	40	572902
	2008	230074	101772	111744	67492	7557	2384	216	42	521281
	2009	208359	89303	97464	68480	8298	2955	252	44	475155
	2010	188538	78335	85009	69149	8999	3576	297	47	433950
	2011	170468	68693	74146	69544	9657	4244	351	50	397153
	2012	154015	60221	64671	69703	10270	4955	415	54	364303
	2013	139049	52779	56406	69662	10837	5703	489	58	334982
	2014	125452	46245	49198	69448	11357	6485	573	62	308819
	2015	113109	40510	42911	69088	11830	7296	668	68	285480
	2016	101917	35478	37428	68603	12259	8132	774	74	264664
				Average Annual Growth Rates (%)
		-10.1	-13.23	-13.7	0.4	6.6	16.3	15.7	6.8	-8.6
	Source: Our estimates.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

ACKNOWLEDGEMENTS

We would like to thank Dr Francisco J. André for the invitation and fruitful discussions at the seminar on Economic Analysis at the Department of Economics of Universidad Pablo de Olavide, Seville, Spain on May 7 th , 2007. We also thank Dr Lucyna Krawiecka, Head of Animal Production Section, Agriculture and Environment Division, Central Statistical Office, Warsaw, Poland for providing us with the data. Dr Joy Burrough advised on the English.


Keywords: laboratory work, practical work, secondary school, teacher beliefs

.

 of the various rankings and their significance were calculated, as too were the z-scores.

The findings suggest that whilst there have been substantial changes in teachers' views about the use of practical work at Key Stages 4 and 5 (age 15-18) there have been no substantial changes at Key Stage 3 (age 11-14). Furthermore, the results are remarkably similar across subject specialism, teacher gender, and years of teaching experience although this paper will only focus on subject specialism. It appears that changes to the assessment criteria, notably the introduction of Science Investigation (Sc1) at Key Stage 4, and a growing desire amongst educational policy makers to improve the image of science, have had an effect on how those in the teaching profession perceive the value and aims of practical work particularly at Key Stages 4 and 5.

Introduction

In many countries practical work is an essential feature of the teaching and learning of science and considerable time and money is spent on teaching science through practical work and it is therefore important to be clear about the aims of practical work. In an important study in England and Wales, [START_REF] Kerr | Practical work in school science[END_REF] identified ten aims of practical work (Table 1) and subsequent national and international studies into the aims of practical work are broadly in agreement with the aims suggested by Kerr [START_REF] Johnstone | Learning in the laboratory; some thoughts from the literature[END_REF].

As in many countries, much has changed in the educational system in England and Wales in the forty-five years since [START_REF] Kerr | Practical work in school science[END_REF] reported the findings of the first published survey of teachers' views towards the aims of practical work across the full 11-18 age range in secondary schools following a traditional grammar type curriculum. Indeed, the tripartite system of grammar, secondary modern and technical schools, in which only a minority of pupils, educated in grammar schools, followed an academically orientated science curriculum whilst the majority received a more general, vocationally based, science education in secondary modern and technical schools, no longer exists. Furthermore, the introduction of widespread comprehensive education in the mid 1960s gave rise to the development of a range of new approaches to the teaching of science such as Nuffield Secondary Science in the late 1960'swith its emphasis on the use of practical work to develop scientific methods of thought and its aim to enable pupils to find out through investigation [START_REF] Gott | Investigative work in the science curriculum[END_REF] and Warwick Process Science in the 1980's with its emphasis on the use of practical work as a means of developing transferable process skills such as observing and interpreting [START_REF] Screen | A case for a process approach: The Warwick experience[END_REF][START_REF] Screen | The Warwick process science project[END_REF]. The importance of these changes, or any subject, has meant that teachers are now required to teach science to all pupils throughout the compulsory phase of their secondary school education (age 11-16). A possible consequence of this is that teachers now find themselves trying to maintain pupils' interest in a subject that, prior to the introduction of a National Curriculum, many would have chosen to drop at the end of Key Stage 3 (see Table 2 and the note at the end of the paper for additional explanation of Key Stages). It might therefore be expected that the potential affective value of practical work might be seen by teachers as more important than might otherwise have been before science became a core subject as a result of the evolutionary pressures [START_REF] Johnson | Teacher development and change in Asia and Africa: An evolutionary perspective[END_REF]) that its introduction have created in the educational environment in which science teachers operate.

Therefore whilst the Third International Mathematics and Science Survey (TIMSS, 1999) has shown performance in science, and most notably in practical work [START_REF] Kind | TIMSS performance assessment -a cross national comparison of practical work[END_REF], has improved amongst pupils in England and Wales over the last decade it is important to consider whether these improvements reflect any changes in the relative importance of the aims science teachers assign to the use of practical work. The findings of this study may be of interest to researchers, policy makers and science

The need for an enquiry

Following a grant by the Gulbenkian Foundation in 1960 Kerr undertook the first extensive survey to investigate the nature and purpose of practical work within the framework of grammar school science teaching in England and Wales. The study involved 151 schools, 56% boys' schools, 26% girls' schools and 18% co-educational, all of whom followed a common 'grammar type' curriculum and involved a total of 701 science teachers.

Despite the fact that some of the teachers appeared to be responding to the questions without those responses reflecting their actual beliefs or practices [START_REF] Abrahams | Does practical work really work? A study of the effectiveness of practical work as a teaching and learning method in school science[END_REF], the study did provide the first large-scale insight into practicing teachers' views as to the nature and purpose of school practical work in England and Wales. The findings, which involved the teachers arranging ten suggested aims (purposes) for practical work in order of perceived importance, are summarised in Table 1.

TABLE 1 GOES HERE

In conclusion [START_REF] Kerr | Practical work in school science[END_REF] However, despite all of the changes that have occurred over the last four decades there have only been two subsequent published national studies of teachers' views about the aims of practical work in England and Wales [START_REF] Lynch | Practical work in schools: An examination of teachers' stated aims and the influence of practical work according to students[END_REF][START_REF] Lynch | Practical work in schools: An examination of teachers' stated aims and the influence of practical work according to students[END_REF][START_REF] Wilkinson | A comparative study of students' and their teachers' perceptions of laboratory work in secondary schools[END_REF][START_REF] Wilkinson | A comparative study of students' and their teachers' perceptions of laboratory work in secondary schools[END_REF] carried out two similar studies in Australia). Whilst the first of these [START_REF] Thompson | Practical work in sixthform science[END_REF], focused solely on teachers' views about practical work at Key Stage 5 (age 17-18) the second [START_REF] Beatty | School science, its organisation and practical work in the 11-13 age range[END_REF] only considered views about practical work amongst teachers of pupils within part of Key Stage 3 (age 11-13).

Furthermore, both of these subsequent studies used an expanded questionnaire in which teachers were required to rank twenty aims of practical work -developed by [START_REF] Thompson | Practical work in sixthform science[END_REF] -that, whist subsuming the original ten aims of practical work used by [START_REF] Kerr | Practical work in school science[END_REF], prevented any direct comparison of either of their results with that initial study. The expanded questionnaire differed from the original in that the additional aims were designed to accommodate potential changes to teachers' aims for using practical work that, it was felt, might have arisen from the introduction of Nuffield Science.

As this study was specifically designed to examine whether there had been any changes to teachers' views about the aims of practical work across the entire secondary school age range and across all three traditional sciences, it was felt more appropriate to use the original ten aims devised by [START_REF] Kerr | Practical work in school science[END_REF] so as to enable a direct comparison with Kerr's broader findings to be made.

Plan of enquiry

The questionnaire

Whilst there are advantages and disadvantages to the use of postal surveys [START_REF] Bailey | Methods of social research[END_REF][START_REF] Cohen | Research methods in education[END_REF][START_REF] Robson | Real world research[END_REF] the decision to undertake a national study meant that a postal questionnaire survey was the most cost-effective and convenient method for collecting a relatively large amount of data from across a wide geographical area. Despite these advantages it is important to recognise two main reasons why questionnaire surveys can lead to fallacious results [START_REF] Belson | Validity in survey research[END_REF][START_REF] Cohen | Research methods in education[END_REF][START_REF] Verma | Researching education: Perspectives and techniques[END_REF] and appropriate steps were taken to prevent, or at least reduce, the impact that these might have on the findings of the study. The first reason, although not restricted solely to questionnaires, is that there is no way of ascertaining, short of receiving back all of the questionnaires, whether the returned questionnaires are truly representative of the entire sample who received them. To reduce the potential that the responses received were not representative of the sample as a whole this study sought to maximise the response rate through the use of phased follow-up letters [START_REF] Bailey | Methods of social research[END_REF][START_REF] Cohen | Research methods in education[END_REF]Hoinville & Jowell, 1978;[START_REF] Verma | Researching education: Perspectives and techniques[END_REF] achieving a final response rate of 42.5 percent. Additionally a check was made on how representative the returned questionnaires were by comparing the proportion of questionnaires sent out to those returned by school type (Table 3) which showed there to be no major discrepancy.

The second reason that a questionnaire survey, and in particular a postal questionnaire, can lead to fallacious results is that the researcher is unable to ensure that the questions are fully understood and/or answered in the way stipulated in the written instructions [START_REF] Belson | Validity in survey research[END_REF]. Whilst it was hoped that asking teachers to rank the same ten aims as those piloted and used successfully by [START_REF] Kerr | Practical work in school science[END_REF] would reduce the likelihood of teachers failing to understand what was required of them the views of a focus group of science teachers (N = 7) regarding the clarity of the questions was also sought. As a result of feedback from the focus group it was deemed necessary to make two changes to the original [START_REF] Kerr | Practical work in school science[END_REF] questionnaire to make it easier for current teachers to understand what was being asked of them and it is to these changes that we now turn.

The first of these changes was that whilst Kerr had asked teachers to respond to the questionnaire in terms of three Year group categories 1-2, 3-5 and 6 th Form the current study opted to use the three Key Stages from the National Curriculum that, it was felt by the focus group, would be more easily recognised and better understood by current teachers. Furthermore, it was felt that in England and Wales Key Stages provide natural transition points in the teaching of science at which teacher's views about the use of practical work might be more likely, if they were to change at all, to do so. Table 2 shows the changes to the way pupils are categorised by Year group in both studies.

TABLE 2 GOES HERE

Secondly, the focus group felt that because pupils were, at the time of this study, assessed on their performance of a practical investigation (Sc1), and this contributes to their grade in the General Certificate of Secondary Education (GCSE) -a public examination taken at age 16 -the aim "To fit the requirements of practical The instructions in the questionnaire asked teachers to rank the aims, in each of the three Key Stages, in order of their perceived importance by putting '1' in the appropriate column to the right of the aim which was viewed as being most important '2' alongside the next, '3' alongside the third and so on to '10' opposite the least applicable with each number from one to ten to be used one only in each column. The full questionnaire was piloted by the focus group with science teachers (N = 15) in their respective schools with no difficulties being reported.

The sample

Whilst the [START_REF] Kerr | Practical work in school science[END_REF] study was limited to teachers within 151 secondary schools following a traditional grammar type curriculum, this study used a stratified sample of 304 representative schools, selected from across the state and independent (privately funded) sectors, with three questionnaires being sent to the head of science at each school. A total of 912 questionnaires were sent out with 388 being returned (42.5%) of which 25 were excluded from analysis for various reasons. The frequency distributions of the questionnaires sent out and those returned are listed in Table 3.

There was no statistically significant difference between the frequency distributions (chi-square = 2.408, df = 3, not significant). This indicates that the responses closely reflected the stratified sample from which they were drawn.

GOES HERE

Method of analysis

This study examines whether there have been any changes in the importance teachers attached to the aims of practical work, across the entire secondary age range and across all three traditional sciences, since the [START_REF] Kerr | Practical work in school science[END_REF] study in England and

Wales. Since there are three Key Stages and three science subjects in the study, there are nine comparisons to make with the Kerr study.

A frequently used method to compare ranked data involves the calculation of the sum of ranks for each item rank. The sums are then put into rank order, and the differences between rank orders are shown diagrammatically (for examples see [START_REF] Beatty | Practical work in 11-13 science: The context, type and aims of current practice[END_REF][START_REF] Saglam | Upper high school students' understanding of electromagnetism[END_REF][START_REF] Thompson | Practical work in sixthform science[END_REF]Whitfield, 1979). Figure 1 shows such a comparison between Key Stage 3 chemistry teachers in the Kerr study and the current study. The figure also presents the mean of the sum of ranks for each aim, on which the rank order is based, since the number of the chemistry teachers at Key Stage 3 was different in each study.

FIGURE 1 GOES HERE

An initial conclusion, from a preliminary analysis of Figure 1, might be that there had been no change in the importance Key Stage 3 chemistry teachers attached to Aim 5

between both studies as it was ranked 10 th in both studies. Likewise the two-place difference between the ranks of Aim 8 suggests that there has been a change in the importance Key Stage 3 chemistry teachers ascribe to that aim having risen from being ranked 5 th to 3 rd in the rank order of aims. However, if the analysis is based on the means of the sum of ranks, instead of simply the rank orders, we find that the preliminary conclusion obscured the fact that whilst there was a small difference (0.09 points) between the means of the sum of ranks of Aim 8 there was a much greater difference (1.95 points) between the means of the sum of ranks of Aim 5.

In order to better examine the scope of any difference in teachers' views it was decided to use the means of the sum of ranks, instead of the more basic rank orders, to analyse any change in the importance teachers attached to the aims of practical work.

However, there remains one problem with using the means of the sum of ranks in any such analysis. This is that although, irrespective of the sample size, the average of the means of the sum of ranks remains 5.5 the standard deviation in both studies will differ. For example in Figure 1 it can be seen that whilst the sum of ranks in both studies is 5.5 the standard deviation is 2.15 for the Kerr study and 1.61 for the current study. In order to compare the means of the sum of ranks they need first to be expressed on the same scale and to achieve this it is necessary to convert each mean into a z-score. [START_REF] Howitt | An introduction to statistics in psychology: A complete guide for students[END_REF] make clear that 'the number of standard deviations is a universal scale of measurement' and that 'the z-score is nothing other than the number of standard deviations a particular score lies above or below the mean of the set of scores' (p. 46, italics in the original). For example, in order to calculate the z-score of Aim 9 in the current study (Figure 1), we simply subtract 5.5 In the discussions above it was assumed that the sums of ranks were representing the Key Stage 3 chemistry teachers' rank orders. Yet such an assumption requires there to be sufficient agreement between the teachers' rank orders and to ascertain whether this was the case a measure of the degree to which the teachers agree was needed. [START_REF] Kendall | Rank correlation methods[END_REF] developed the coefficient of concordance (W) as a means of calculating the degree of agreement between various rank orders. The coefficient of concordance ranges from zero, when there is no agreement between the various rank orders, to +1 when there is full agreement. The statistical significance of the coefficient of concordance can be calculated using a Chi Squared test provided there are seven or more judges doing the ranking [START_REF] Howell | Statistical methods for psychology[END_REF]. For example the coefficient of concordance for Key Stage 3 chemistry teachers in the current study was 0.28. This is statistically significant at the 1% level, suggesting that there was sufficient agreement between the Key Stage 3 chemistry teachers' rank orders to enable the sums of ranks to be considered as representative of all of the Key Stage 3 chemistry teachers' rank orders.

Having discussed the degree of agreement between rank orders and the use of z-scores in comparing the means of the sum of ranks we now illustrate the use of z-scores in our analysis of the data using a set of z-scores from the study. [The full list of z-scores can be obtained from the authors electronically].

TABLE 4 GOES HERE

We have categorised changes in z-score of between 0 and 0.333 as 'small'; 0.333 and 0.666 as 'limited'; 0.666 and 0.999 as 'moderate' and those of more than 0.999 as 'substantial'. Whilst recognising that the boundaries of such categories are, as with the assessment of the data using diagrammatic methods, essentially arbitrary we feel that they provide a useful method of comparing results from both studies.

Results

Results from the study will be presented in two sections. The first section will deal with the overall results of the study whilst the second section will discuss the changes Kendall's coefficient of concordance (W) was calculated to evaluate the agreement within the nine comparison groups. Of the nine coefficients of concordance all were found to be statistically significant at the 1% level. This suggests that the means of the sum of ranks of each group can be accepted as being representative of the importance that each group of science teachers attaches to the ten suggested aims of practical work. One notable finding was that the level of agreement in each Key Stage, and across all three science subjects, was very similar: although the level of agreement had its highest value amongst teachers at Key Stage 3. Therefore although there was agreement among science teachers about the aims of practical work, across the three Key Stages, it appears that those teaching at Key Stages 4 and 5 were less sure about those aims than teachers teaching at Key Stage 3. One possible line of future study might therefore be to investigate whether there is a similar pattern of the level of agreement in other countries, and the reason(s) for such a pattern.

Change in the aims of practical work

In this paper we will only discuss substantial changes in z-scores: where a change is said to be substantial if its magnitude is greater than or equal to 1. Table 5 shows all substantial changes in the z-score that were found.

Changes at Key Stage 3

As there were no substantial changes in the z-scores relating to teachers' views about the aims of practical work at Key Stage 3 since the original study by [START_REF] Kerr | Practical work in school science[END_REF] there are no results for Key Stage 3 in Table 5. Interestingly the most important purpose of practical work at Key Stage 3, across all science subjects, remains, as it was forty-five years ago, to arouse and maintain interest in the subject (Aim 9) whilst the least important purpose of practical work continues to be that of preparing pupils for assessed practical work (Aim 5) -which, in the current study, explicitly included Sc1

investigations.

Changes at Key Stage 4

The largest changes in z-scores observed in this study occurred at Key Stage 4 with regards Aim 5, which relates to preparing pupils for assessed practical workincluding Sc1 investigations. Although [START_REF] Kerr | Practical work in school science[END_REF] acknowledges the possible role of practical work in preparing pupils for practical examinations there was also a recognition that it was too much to expect more than a few teachers to admit to using practical work to this end. Perhaps those teaching science in the early twenty-first century are more realistic and/or pragmatic about the influence of practical examinations on the aims and values of practical work.

As [START_REF] Kerr | Practical work in school science[END_REF] notes Aims 1 and 2 refer to 'the possible effects of practical work on one's subsequent thinking and behaviour' (p. 22). The substantial negative changes in z-scores relating to these aims at Key Stage 4 (and 5) suggests that science teachers today see these aims as being less important for older pupils -although the importance of these aims for pupils at Key Stage 3 remained unchanged -than did the Whilst few teachers made any comment on this specific aim those that did mentioned the pressure of trying to ensure that their pupils did well in the Sc1 assessment, as this contributed towards their total GCSE examination grade.

Aims 9 and 10 are concerned with the 'possible affects on one's general attitude to science by increasing the desire to learn and the level of understanding' (Kerr, 1963 p. 22). Whilst there was a substantial rise in terms of the z-scores associated with physics teachers' views towards Aims 9 and 10, and a similar substantial rise amongst chemistry teachers for Aim 10, no substantial change in z-scores was found amongst biology teachers for either of these aims. However, that said, it must be emphasised that these two aims were, as at Key Stage 3, ranked 1 st and 2 nd by all teachers and the observed substantial increase in their z-score, amongst chemistry and physics teachers, simply indicates that those teachers saw them as being even more important [START_REF] Osborne | Attitudes towards science: A review of the literature and its implications[END_REF], have substantially increased the importance of practical work as a means 'to arouse and maintain interest in the subject' and that this is, and a number of physics teachers made this point, in the hope of motivating more pupils -and this does appear to be having some degree of success (Joint Council for Qualifications, 2008) -to continue with their subject post-compulsion.

Changes at Key Stage 5

Similar to the changes that occurred at Key Stage 4 there were substantial falls in the importance teachers assigned to Aims 1 and 2 and, unlike at Key Stage 4, there were no exceptions. One of the physics teachers, one who had commented on the importance of encouraging accurate observation and careful recording at Key Stage 3, made the further point, about making accurate observations and careful recordings, that "If they don't know how to do it by the time they're doing 'A' level [Key Stage 5] they shouldn't be doing 'A' level physics."

At Key stage 5 there was also a substantial fall amongst biology and physics teachers' views about the use of practical work to develop manipulative skills although this was not the case amongst chemistry teachers. Whilst only one chemistry teacher commented on this aim, so it can only be seen as anecdotal, it was interesting in so far as they claimed that "A level [Key Stage 5] chemistry requires higher order manipulative skills and it's vital that we teach these". Clearly if this is the case, and we have no evidence to support or refute this claim, it would offer a possible explanation as to why chemistry teachers continued to see this particular aim as being relatively important. Aims 9 and 10 saw the most pronounced rise in teachers' views about their importance and, unlike at Key Stage 4, this was across all the science subjects. Those teachers that commented upon this, and again there were only a few, spoke of a need to make science real and relevant in order to maintain an interest in what was a much more conceptually demanding subject than it had been at Key Stage 4. Although there has been a substantial rise in the z-scores relating to these two aims -they are now ranked 4 th and 3 rd respectively -it is unclear as to the extent to which this reflects a desire, on the part of the science teachers, to encourage Key Stage 5 pupils towards studying a science degree at university as educational policy makers and advisors (Department for Education and Employment 1996;Royal Society, 2006) suggest is vital for the UK economy.

Conclusions and Implications for Teaching and Research

The aim of this study was to see whether there had been any changes in the relative importance of the aims science teachers assign to the use of practical work, across the full secondary age range (11-18), since the last published national survey of teachers' views about practical work undertaken by [START_REF] Kerr | Practical work in school science[END_REF]. One important finding has been the fact that despite the many changes in the educational system in England and Wales during the last forty five years -including the introduction of a National Curriculum -teachers' views about the aims of practical work for pupils in Key Stage 3 have remained virtually unchanged. Indeed, our findings replicate those of [START_REF] Beatty | Practical work in 11-13 science: The context, type and aims of current practice[END_REF] who, twenty-six years ago, noted similarly, with regards KS3

(in fact only Years 7 and 8) that '[t]here is no evidence of a dramatic change in the reasons that teachers give for doing practical work since 1962, despite all the curriculum innovations in the succeeding period'. We believe that this stability, and the high degree of concordance amongst teachers regarding the aims for the use of practical work at this Key Stage in both this study and that of [START_REF] Kerr | Practical work in school science[END_REF], is a Certainly the fact that these pupils do not have to undertake public examinations involving practical work means that it is relatively unproblematic for teachers to rank the aim of preparing pupils for assessed practical work -including Sc1 investigations -very low whilst their desire to portray science as a fun and exciting 'hands-on' subject [START_REF] Abrahams | Practical work: It's better than writing but does it motivate?[END_REF] means that many teachers will rank its potential affective value i.e. to arouse and maintain interest in the subject highly.

In terms of the evolutionary model developed by [START_REF] Johnson | Teacher development and change in Asia and Africa: An evolutionary perspective[END_REF] '[c]hanges to the working environment of teachers through curricula reform, inservice provision and increased resources will lead to changes in pedagogy which in turn will alter opinions' (p. 1322). However, despite the introduction of the National Curriculum -which has clearly changed the working environment -this study has found no substantial change in current teachers' aims regarding the use of practical work at Key Stage 3 compared to those obtained by [START_REF] Kerr | Practical work in school science[END_REF] over forty years ago.

One possible way of understanding this is to propose that it is not, as [START_REF] Johnson | Teacher development and change in Asia and Africa: An evolutionary perspective[END_REF] suggest, that changes in the working environment lead to changes in pedagogy and subsequently teachers' opinions -including those on the aims and/or use of practical work -but that changes in the working environment have the potential to lead to changes in pedagogy if those changes generate pressure on (or removed it from) teachers. We would argue that the absence of any substantial change to the z-scores associated with teachers' views about the use of practical work at Key Stage 3 suggests that the National Curriculum did not generate any pressure on teachers to change their pedagogy and/or opinions, regarding practical work, in order for them to continue to operate successfully within their new working environment. The most substantial change in teachers' aims for practical work (Table 5) occurred at Key Stage 4 and it would appear to reflect the introduction of the Sc1 investigations and the associated pressure on biology, chemistry and physics teachers that this has generated in terms of needing to prepare pupils for assessed practical work. Indeed [START_REF] Donnelly | Investigations by order. Policy, curriculum and science teachers' work under the education reform act[END_REF], in a detailed study of the 'Scientific Enquiry' component of the English National Curriculum (Attainment target Sc1), found that extended, and more open-ended, investigative practical tasks were, in fact, rarely being used to teach pupils about specific aspects of scientific enquiry might have been, but almost entirely to assess their ability to conduct an empirical enquiry 'scientifically' -something that they would be required to do as part of their public examinations. It would seem, therefore, that an unintended consequence for educational policy makers of the introduction of Attainment Target Sc1 may have been that teachers' views about the use of practical work were influenced by government led changes to the assessment procedures (that generate pressure on teachers) -and that this was particularly pronounced at KS4.

Following the recent decision by educational policy makers to remove Sc1 as a means of assessment -and, as such, the pressure on teachers that was associated with it -we would predict, and this would be an interesting area for further research, that the Indeed it might be expected that if these aims were effectively met, for example, in Key Stage 3 -when their importance is ranked highly -then their relative importance at Key Stages 4 and 5 might be expected to drop as has been found in this study.

We suggest that helping those undertaking initial teacher training to recognise the way in which practicing teachers' views about the relative importance of the different aims for practical work change at different Key Stages could enable them to better reflect on their own reasons for using practical work at a particular Key Stage.

An interesting finding to emerge from our use of z-scores was the fact that some seemingly identical changes in rank order, such as the change in the rank order of Aim 9 by the biology and physics teachers at Key Stage 4, were in fact not identical in magnitude when considered in terms of changes in their respective z-scores. Indeed, in the above example, only in the case of physics teachers was the change in z-score found to be substantial: a finding that would not have emerged had the analysis relied solely upon the change in the rank order of a particular aim. For this reason we suggest that in future studies z-scores should be used instead of rank orders as a means of comparing sets of ranking data.

F

  teachers' views towards practical work in secondary schools in England and Wales

  working environments, is, in terms of the evolutionary model developed by[START_REF] Johnson | Teacher development and change in Asia and Africa: An evolutionary perspective[END_REF], that they lead to changes in teachers' practice and opinion. If this model is correct then future changes to the way science is taught and/or assessed have the potential to change the way in which teachers view the role of practical work and, as a consequence, use it. The introduction of a National Curriculum in 1989, which saw science become a 'core'

  countries where there have been substantial changes in the structure of science education.

  1963 p. 27) needed to be modified so as to make explicit that this also included the potential aim of practical work as a means of preparing pupils for the assessed practical investigation (Sc1). The modified aim, used in this study, was "To prepare pupils for assessed practical work -including Sc1 investigations".

  from 2.74, and then divide by the standard deviation of 1.61, which is -1.714. The magnitude of the z-score indicates that the mean of the sum of ranks of Aim 9 in the current study is 1.714 standard deviations away from the mean of the set of scores, and the minus sign indicates that it is below the mean of the set of scores. It is that by converting the means of the sum of ranks into z-scores we do not change their distribution but rather express them in standard deviations from the average.

  The magnitude of each difference provides a measure of the relative change in the z-score of each specific aim. The negative signs in Table4indicate that the z-scores of these aims have decreased relative to their value in the first study. For example, the change in z-score for Aim 2 was -1.997, indicating that compared with the Kerr study, the chemistry teachers in the current study gave less importance to Aim 2. On the other hand, the change in z-score for Aim 10 was 1.869, indicates that compared with the Kerr study, the chemistry teachers in the current study gave more importance to Aim 10.

  in the importance teachers attached to the aims of practical work, by Key Stage, since the[START_REF] Kerr | Practical work in school science[END_REF] study.

  the Kerr study. Comments written by some of the teachers on the questionnaires suggested that it was not that these aims -which relate to basic scientific skills -suddenly ceased to have any importance in Key Stages 4 and 5 but rather that they saw the need to prioritise different aims at different Key Stages. A frequent point made by teachers was that although the emphasis on encouraging accurate observation and careful recording along with the promotion of simple, common-sense, scientific methods of thought was seen as relatively important at Key Stage 3 -to enable pupils to begin effectively engaging with practical work -these were seen as being less important, relatively speaking, than other aims at Key Stages 4 and 5 whose importance had increased. The exception to this was that in terms of Aim 1 there was no substantial change in the z-score associated with physics teachers' views towards the importance of encouraging accurate observation and careful recording. Interestingly, although only three physics teachers commented on this point, all three emphasised the importance of accurate observation and the careful recording of data if a high mark was to be achieved in the Sc1 assessed practical.Evidently whilst teachers are aware that the National Curriculum considers observation and scientific thinking as important practical skills to be taught at Key Stage 4 -they clearly have an expectation that pupils would have developed sufficient proficiency in these skills during Key Stage 3 to enable teachers to prioritise other aims of practical work in Key Stages 4 and 5. The most substantial increase in z-score at Key Stage 4 relates to the use of practical work 'to prepare pupils for assessed practical work -including Sc1 investigations' although, as can be seen from

  than at Key Stage 3. The emphasis teachers place on the potential affective value of practical work both at Key Stage 3 and, more importantly at Key Stage 4, might help to explain the reported(Joint Council for Qualifications, 2008) rise in the number of pupils now opting to take one or more science subjects in the post-compulsory Key Stage 5. It is unclear why there was no substantial rise in z-scores amongst Key Stage 4 biology teachers regarding these two aims. One possible explanation might be that as the number of pupils taking biology at Key Stage 5 has been increasing year on year[START_REF] Osborne | Attitudes towards science: A review of the literature and its implications[END_REF] there was less pressure on biology teachers to change their views towards the use of practical work as they were evidently already effectively generating a sufficiently positive attitude towards biology amongst their pupils. Conversely the teachers of physics, which is seen as the least popular science, as measured by the number of pupils who go on to study the subject at Key Stage 5

  fact that there is less perceived competition between the aims.

TABLE 5 GOES HERE

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  Table 5, there is no substantial change at Key Stage 5.

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  -sense, scientific methods of thought' and 'to develop manipulative skills', we do not believe that this necessarily means that they no longer see these as important in an objective sense. Rather, we would claim, that such changes are more likely to reflect the fact that teachers recognise, as a result of pressures in the educational environment, that different aims are more important, relative to other aims, at different stages within secondary education if their teaching is to be effective.

	simple, common
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	including Sc1 investigations' would return to a position at, or very close to, the
	bottom of teachers' aims for using practical work at Key Stage 4.
	Whilst many science teachers at Key Stages 4 and 5 substantially decreased the status
	of the aims 'to encourage accurate observation and careful recording', 'to promote
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Table 1

 1 Teachers' ten suggested aims (purposes) for practical work in order of perceived importance(From Kerr 1963 p. 27)

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	10	To make physical phenomena more real	2	6	7
		through actual experience
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  Table 2. A comparison of the age ranges used in both studies

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	Year ranges of pupils	Youngest range	Middle range	Oldest range
	Kerr	Years 1-2	Years 3-5	6 th Form
		Ages 11-13	Ages 14 -16	Ages 17 -18
	F o Abrahams and Saglam	Years 1-3 (KS3)	Years 4-5 (KS4)	6 th Form (KS5)
	r Ages 11-14	Ages 15 -16	Ages 17 -18
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Table 3 .

 3 Number of the questionnaires by school type

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	Teachers	Comprehensive	Grammar	Independent	Specialist
		school teachers	school teachers	school teachers	school teachers
	Total sample (N = 912)	546	90	138	138
	Analysed sample (N =363)	206	46	57	54

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Table 4 .

 4 Change in the z-scores for Key Stage 5 chemistry teachers

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	Aim	The Kerr study	The current study Change in z-score
	1	-1.466	-0.105	-1.362
	2	-0.455	1.542	-1.997
	3	0.092	0.390	-0.297
	4	0.357	0.993	-0.636
	5	0.372	0.851	-0.479
	6	-1.312	-1.717	0.405
	7	0.147	-0.258	0.405
	8	-0.565	-1.004	0.438
	9	1.801	0.148	1.653
	10	1.029	-0.840	1.869

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Table 5 .

 5 Substantial changes in z-scores

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  Figure 1. A comparison between Key Stage 3 chemistry teachers in the Kerr study and the

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	current study
				The Kerr study		The current study
	Rank order of	Aim Mean of the sum	Aim Mean of the sum
	aims 1 2 3	of ranks 2.45 3.74 r F 9 o 1 10 3.79		9 10 8	of ranks 2.74 3.22 5.12
	4 5 6 7 8		2 8 6 3 7	4.05 P 5.03 e 5.57 e 5.83 r 6.72	1 6 2 3 7	5.20 5.39 5.62 5.99 6.51
	9 10		4 5	7.58 9.74	4 w e i v R e 5	7.42 7.79
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Note