Eric J M M Arets

; Lim

Hin Fui

Discussion -working

This paper analyses the side effects of the stricter regulation on tropical hardwood or timber imports. It considers the place of Europe within the global timber market, where Europe accounts only for a very limited share. It also explains the high selectivity of European markets, with its consequences. While tropical wooden furniture and other secondary processed products are not considered as timber here, their question is also discussed. The number of empirical studies specifically dealing with the side effects of EU regulations is limited, but the results are converging, showing that these regulations have a general adverse effect, contrary to the initial aim of promoting the sustainability of tropical timbers. These side effects are essentially to divert the trade towards countries with lower standards, and to add a burden on most of the producing countries which have already a set of comparative disadvantages for the production of legal or sustainable timber. The effects are positive on a limited number of companies which markets are very dependent of Europe. The question is then analysed from a broader perspective, replacing the effects of the EU regulations as an incidental factor compared to the increasing consumption of tropical timber by the three developing giants: Brazil, India and China.

Introduction

With a long history of concerns regarding the state and the sustainability of tropical forests, the European opinion has been in the recent years particularly awaken by several spectacular activist-events and campaigns organised by environmentalist NGOs, which culminated around the years 2001 to 2003. Coincidentally, or probably as a result, several European governments decided to increase their own level of environmental requirements when purchasing tropical timbers for public markets, while some others are still pondering the opportunity of specifically addressing tropical timber trade . Denmark has issued the first policy in this matter, in 2003, followed by UK in 2004 andFrance in 2006. Netherlands is still in the process to issue definitive guidelines. At least in Danish and British policies, the tendency is to use existing ecolabels like FSC or PEFC as a reference, while the terms of French and Dutch policies are purposely not directly favouring any specific ecolabel . All of these procurement policies are unilateral measures. Conversely and at a broader level, the European Union (EU) launched an action plan to restrict the amount of illegal timber entering the EU, by increasing the legal requirements on tropical timber imports (Communication from the Commission to the Council and the European Parliament 2003). Because of a necessary compliance with WTO regulations, and the need to avoid any unfair competition with temperate timbers or other materials which may not be submitted to the same requirements nor to the same production complexity, this action plan "Forest Law Enforcement, Governance and Trade" (hereafter referred as FLEGT) is based on voluntary individual partnerships between the EU and producer countries, with assistance provided in exchange of the trade restriction, and the process is still in progress. Beside the nature of the various restriction measures, their recent and almost simultaneous development forms a continuum of actions which progressively hardens the regulation of wood imports. All the actions by governments as well by NGOs started with the legitimate concern to promote a more sustainable management of tropical forests, and to combat the illegal trade of tropical timbers. The underlying hypothesis of all actions restricting regulations of tropical hardwood imports into Europe, is that it automaticaly directly entail positive impacts on the global sustainability of tropical forests. But the reality of the impacts might be more complex, and these actions may have side effects. The purpose of this paper is not to discuss whether positive or negative aspects prevail in the impacts of stricter European regulation on tropical hardwood imports. Instead, it is to examine the global trade context of these regulations on tropical hardwood imports, and to discuss the hypothesis that they do have side effects, and if so, to analyse and discuss them.

1 The place of Europe within the world tropical timber market

Europe accounts for a very limited share of the world tropical timber consumption

After North America, Europe has traditionally dominated the world's timber trade and timber consumption, as the second largest consumer of hardwood and softwood . But this picture of Europe having a major impact on the global timber economy, is essentially due to the preeminence of temperate wood products in the European consumption, following the global pattern of the western countries. Regarding the world tropical timber market, it is a different story. Today, Europe4 represents a very limited part of the world consumption of tropical timber: less than 1% of the world tropical logs consumption, and respectively less than 7 and 6% of tropical sawn timber and plywood world consumptions. The volume of logs and plywood consumed by Europe is decreasing since 1998, and stabilising in the last years. Meanwhile, the European consumption of sawn timber is fluctuant but not declining, because it is mainly based on some specific technical requirements which only some tropical species can fulfil (Figure 1). According to the calculations based on either FAO or ITTO sources, the overall European roundwood-equivalent consumption of tropical timber is assessed between 3 to 6 % of the world consumption [computed from (International Tropical Timber Organisation 2006) and (Food and Agriculture Organization of the United Nations 2007)].

Thus the European footprint on the tropical timber production, through its own consumption, is low at the world scale. It is already decreasing for a short decade in absolute value, and it will in the future even decrease more, in relative value, because of the steady consumption increases of Brazil, India and China. These 3 countries only, account for 45 to 50% of the world consumption of tropical timber in roundwood equivalent, and their importance is increasing. Each of these countries consumes much more (4 to almost 8 times) than the whole Europe itself, while their exports of the different timber products are excessively low compared to their consumption. Two other tropical countries with a different profile, because they are more export-dependant for some products, have a major importance at the world scale. That is Indonesia and Malaysia. These two other huge consumers (each consumes more than 3 times than the whole Europe itself) along with China, Brazil and India, form a cluster of tropical countries which accounts for 60% of the world tropical timber consumption. The patterns of their overall increasing consumption, influenced by their status of developing countries, are the real driver of the requirements for tropical timbers at the world scale, and have progressively overwhelmed the European influence.

The traditional influence of Europe in some African countries is challenged

Yet European demand patterns have still some influence in a few African countries like Ghana, Côte d'Ivoire, Cameroon, Gabon and Republic of Congo, but this also is challenged.

Traditionally, these countries were exporting their tropical hardwoods almost exclusively towards Europe. But for the last decade the volume and the share of their exports to Europe is progressively decreasing, while they have diverted their sales towards Asia.

European markets are highly selective

Even if the opinion usually assumes that the European demand concerns mainly higher value timbers and thus have an impact on the economy of the timber production, its footprint in this regard is limited too. Actually, the European footprint is very high on certain specific tropical timbers, belonging to what can be called "niche markets" at the world scale, because of the specificity of the requirements (mainly regarding the delays of delivery, the logistics, the presentation and packaging of the products as well as their technical quality). These requirements make the production for European markets not so easy nor so profitable for developing economies, which explains why most of the critically export-dependant tropical countries (ie Gabon or Republic of Congo) are increasing the share of their timber sales towards Asia, because these markets are less profitable but also less requiring (Roda 2001a). While the European consumption decreases, the selectivity of the demanded products is becoming higher. As a result, the gap between the requirements of European markets and the main markets is also increasing. Conversely, this phenomenon entails a higher dependence of the decreasing number of producing firms which are specialised into the European markets. They become more sensitive to the fine tuning of this demand, and their eventual ability or impossibility to fulfil all the requirements of this demand is becoming their main comparative advantage or impediment within the international trade of tropical timbers, which has become essentially regulated by the Asian and Latin-American producers .

Imports of furniture and other secondary processed wood products have probably a higher impact than timber imports

Despite the fact that furniture and secondary processed products (hereafter named as SPWP) are beyond the scope of the timber question, it is nevertheless interesting to consider it in brief. Indeed, while the European consumption of tropical timber is slightly decreasing, the import of SPWP has increased over the last ten years, with Malaysian, Indonesian, and lately also Chinese producers being very active. Despite this progression, the tropical SPWP have been loosing market shares in Europe, because of the increase of use of PVC products and temperate SPWP (FAO 2007).

We take here into account only the quantitatively significant SPWP, which are wooden furniture (wooden seats, kitchen, office, and other furniture), wooden joinery and carpentry, wooden frames, densified wooden profiles and blocks, and various mouldings and floorings 5 . There is no statistical tool nor database allowing to assess the consumption of the tropical SPWP at the world scale. The only way to quantify the importance of these products is the international trade, which gives distorted and exaggerated proportions compared to real consumption, since the wooden SPWP are assumed to represent a greater volume in the internal consumption of the producer countries, than the actual international trade. According to the United Nations Commodity Trade database, the EU25 imported in 2005 only 15% 6 of the value world imports of the main SPWP, but 22% 7 of the value of the world imports of the main SPWP produced in tropical countries (United Nations 2007). Considering that China has become a major exporter of SPWP to Europe, the proportion could be as much as 27% 8 . Of course, one should keep in mind that these figures represent only the data registered in the UN database, not the real international trade, and that the real proportions of the consumption itself are probably much lower. This latest assumption is made on the basis of several empirical studies which suggest that in most of the tropical countries, the internal trade and consumption of forest products is much more important than the external trade . Nevertheless, this suggests that SPWP may carry the impact on tropical forests of Europe consumption, while the European regulations on tropical hardwood imports have only limited impacts on the actual evolution of the tropical forests at the global scale.

2 The evidence of negative side-effects in the context of most of the producers

From strident comments to objective studies

With a rather strident judgemental tone, Bryant [START_REF] Bryant | Environmental Labelling's Effects On The Timber Wood Industry[END_REF]) pointed out the possible sideeffects of the stricter regulations on forest products: "industrialized countries have a propensity to levy import restrictions in order to coerce developing countries into implementing sound programs […] Developing countries can easily turn to secondary sources, such as their own markets, to fulfil their immediate needs [ie need of selling their products]".

Since that time and this normative position, a few empirical studies have highlighted that, due to the relatively high gap between the relatively low demand of the countries imposing some trade standard, and the much greater demand of other countries with lower standards, the sideeffects of stricter environmental regulations on timber imports were noticeable. In the specific case of the European stricter regulations set, there is a pair of convergent empirical studies , which are also convergent with an impact study regarding stricter import regulations outside Europe, in Australia, indicating similar results and side effects (Jacko Pöyry Consulting 2005).

5 This refers to the products described by standardized international nomenclature HS1996 : Codes 440920, 4413, 4414, 4418, 940330, 940340, 940360, 940161, 940169. 6 That is approximately 8 100 000 000 USD, on 52 600 000 000 USD. 7 That is approximately 2 000 000 000 USD on 9 600 000 000 USD 8 Computed by CIRAD from the UNComTrade(United Nations 2007)

Convergence of two recent empirical works on stricter imports regulation in Europe

An impact assessment of the FLEGT process [START_REF] Indufor | Impact assessment of the EU action plan for Forest Law Enforcement, Governance, and Trade (FLEGT)[END_REF] has concluded that this generic EU regulation will have little effect on the promotion of forest sustainable forest management or against illegal activities, in the majority of tropical countries which have a very limited share of timber sales to Europe, and which constitute the majority of tropical timber producers countries. But the impacts are expected to be greater in only a few tropical countries which still largely depend on sales towards Europe. The main point is that the EU regulation on timber is likely to encourage a leakage of export towards third countries with less strict regulations, and which resell finished products to Europe [START_REF] Indufor | Impact assessment of the EU action plan for Forest Law Enforcement, Governance, and Trade (FLEGT)[END_REF]. Because the chain of custody of these finished products is much more difficult to identify than for timbers, there are currently no specific environmental regulations for their imports regarding legality of the harvests at their origin, or sustainable management of forests.

In France, an impact study of the new public procurement policy (with sustainable management criteria for timber) has shown that this regulation tool may have only a limited effect in promoting sustainable management in tropical forests At the same time it is likely to encourage producers of tropical timber to divert their sales to other parts of the world than Europe. . More in detail, this study reveals that, this regulation tool will have an important media impact. This will create a de facto segregation favouring temperate timbers as well as plantation timbers, because the offered quantities of these timbers complying with regulations is much more bigger than for complying tropical timbers from natural forests9 . Furthermore, the French study finds that this procurement policy favours the large scale industrial producers in tropical countries., but disfavours small producers from the same countries. This last effect is due to the fact that, while the large industries are usually integrated and thus get certified or can prove the legality and the origin of the material with less difficulties or relative cost, the local small and medium operators have not the economic power to force their local suppliers to meet these requirements. Some of these small scale operators are integrated, but then, do not benefit of scale economies and have a comparatively higher cost of meeting the requirements. With these two works, the adverse side effects of some stricter European regulations are identified and qualified, but no study has quantified them, so far.

The broader perspective

While the above mentioned empirical studies essentially discuss the huge gap between the level of European requirements and the actual importance of the European market, it is also interesting to consider the offer of tropical products complying with the European stricter regulations, and its potential evolutions. The share of FSC certified production forests appears to be so far the only worldwide available index able to provide, in some extent, some basic indication on the potential ability to produce timber that will comply with a minimum level of environmental or legal requirements10 . The use of such an indicator relies on the postulation that, in a given country, there is a correlation between the easiness to develop FSC certified timber, and to produce timber complying with environmental and legal European import regulations.

A recent empirical study, has confronted such FSC indicator with a range of market data (dependency of the producing countries on European markets and various timber products, market shares, and more than 200 other economic indicators (Wang 2006). It was expected that in a given country, the level of FSC certification would be related to the share of sales to markets with a high environmental sensitiveness, but the results was totally different. A factorial analysis of the extent of FSC certification showed that more than 50% of statistical inertia was explained by only two factors: firstly poverty versus high standards of living11 , and secondly the potential budget that a national society (civil or public) could theoretically allow for financing the forest12 . These results reveal a structural relationship between the extension of FSC and the economic development of a country, but not with its sales to eco-sensitive markets. It also shows that almost all the tropical countries significantly producing timber, belong to the group of the poor, with low GDP or GNI/ha, thus being structurally unable to really see an extent of FSC. This is somehow consistent with the results of the empirical studies described above, which suggest that the increase of the imports standards in Europe do not really entails any real development of the offer, having even adverse side effects, except on some companies being very dependant on the European market . By extension, the results of this factorial analysis also suggest that the stricter European regulations would hardly have an impact on the sustainability of tropical timber and forests. It seems on the contrary that the increase of European imports standards will have many unfair side effects on tropical countries, if not coming with accompanying actions to solve the problems of poverty, economic development, and comparative disadvantages, which are the main underlying factors featuring the tropical timber production legality or environmental quality [START_REF] Roda | European new trends of public procurement policies for tropical timber :lessons from the French case, and global implications[END_REF].

In the same time, Brazilian, Indian, and especially Chinese consumptions are still exponentially growing . This probably will continue to level down the relative importance of the European market for tropical timbers at a world scale. It suggests that any positive impact of the European stricter regulations is likely to be completely outnumbered by the pressure of the markets of the developing countries. But it also suggests that the adverse side effects of these regulations would be only a reinforcing factor within the global trend, mostly independent of Europe policies, which is the increasing control of tropical timber economy by the developing countries. Ultimately, these European stricter regulations could just accelerate what has already begun more than ten years ago [START_REF] Roda | Etat de l'économie des bois d'oeuvre tropicaux 2000 -State of the tropical timber economics[END_REF], that is the disconnection of Europe and of the world tropical timber economy, except for a few big timber companies installed in the tropics, with European shareholding .

Discussion

At a first level of analysis, the question of the side effects of stricter European regulations on tropical hardwoods imports appears as a quasi free competition example. With a very large demand for the products by many countries and segments of industry, an increase of market constraints imposed by one minor share of the demand (Europe) is sanctioned by a phenomenon of diversion of a part of the trade towards the larger share of the demand with lower requirements for the environmental or legal aspects of the production. The trade diversion essentially prevents the promotion of better production practices in the countries of origin. Of course this is not per se a reason for Europe to stop demanding legally and sustainably produced timber. But such a situation, if providing a clear conscience to the European stakeholders, is unfortunately of little use to improve the global sustainability of tropical hardwoods and forests.

Yet a very small part of offer and demand appears as a classical bilateral monopoly. It is the case of the few producing firms which are bound to the European market for structural reasons. These firms adapt their behaviour according to their market constraints and their production constraints (especially the difficult industrial context of developing countries). The increase of the market constraints might endanger some of these firms, but in the same time reinforce their niche, protecting them against the global competition.

At a second level of analysis, it is important to differentiate the nature of the various European regulations on tropical hardwoods imports. The unilateral regulations, as the national public procurement policies, only increase the market constraints for the firms from the tropical countries, without countermeasures to help the changes and reforms in a way that promotes economic development of these countries, or that reduces their comparative disadvantages. Thus these unilateral regulations mainly result in a diversion of the sales towards other markets, except for the few firms bound to the European market. In this regard, it is very interesting to highlight the noticeable differences between such policies of various European countries. Indeed, these differences partly correspond to the differences 13 between the clusters of hardwood producing firms having their main shareholding in the same European countries.

Conversely the bilateral regulations propose measures to help the producers to meet the market constraints. FLEGT include actions in order to support governance reforms in the targeted countries, to support law enforcement tools, verification and monitoring, to better prepare them to work towards eco-certification, to make the actors aware of the necessity to export and import timber from legal sources, etc. The action plan also mentions support to community based forestry and customary rights of indigenous people. In the same time, FLEGT will have only a limited effect because the bilateral negotiations are currently only focused on a few countries, and regarding a very small share of the world tropical hardwood consumption.

Considering the limited efficiency of the existing regulations, as well as the relative importance of their side effects, it is worth to consider the parameters of the trade. While the European demand constraints are increasing, the producing firms, which are mainly located in developing countries with comparative disadvantages, can only play on their internal production constraints (organisation, efficiency, strategy of the firm itself). The limited leverage effect of the internal constraints, compared to the huge market with low level sof requirements, facilitates the trade diversion effect. Meanwhile the critical factors of the hardwood production are the political stability of the country, governance and corruption, infrastructure and industrial services. These factors are also essentially those which usually, along with the poverty, entail a low level of legality and sustainability of the hardwood productions, They form the external production constraints, and remain generally largely out of reach of the firm action, especially when the firms are of small size. Logically, any strategy should first plan actions tackling these constraints, or helping the firms to overcome it, prior to restrict the imports the European import regulations, or in order to give sense to them 14 .

Conclusion :

The stricter European regulations on tropical timber imports presents side effects which are contradictory with their general aim, or contexts, which is to design tools favouring the sustainability of tropical timbers and forests. It essentially consists in diverting the trade from 13 Former colonies and location of the involved firms, number of the firms in the different producing countries, nature of the products and the trade, size of the corresponding firms and availability of eco-certified product, etc. 14 A complementing strategy could be to consider the question of SPWP, on which market Europe has a greater influence. But the real influence and the potential side effects, have still to be investigated.

cirad-00194212, version 1 -6 Dec 2007

Europe to other countries less demanding in terms of legality or environmental quality, and in forbidding the Europe markets to tropical countries which are already poor, still developing, or with strong comparative disadvantages. It also forbids this market to the small companies which precisely haven't the structural tools nor the economic context to produce at the required levels. Conversely, a few bigger companies with a high dependency on the European market are encouraged to improve their practices, this being even a factor of comparative advantage for them. The side effects are essentially due to the very limited importance of Europe as a world consumer of tropical timber, compared to the other consumers, creating a classical trade-diversion effect. Yet, this trade diversion effect demands further studies in order to be quantified. With the continuously growing consumption of the big developing countries, there is little prospect that this situation could change. The risk is that Europe becomes even more disconnected from the general dynamics of the tropical timber economy, except for a small quantity of products with very specific technological characteristics and uses.

Note

The literature on ecocertification side effects is also very interesting to consult, since the trade of tropical products timber markets for ecocertified products present many structural similarities with the trade of this kind of timber towards Europe. An interesting state of the art on the law aspects is given by Rehbinder, while Van Dam provides a producing-country point of view on the economics of forest certification (Rehbinder 2003;Van Dam 2003).

Figure 1

 1 Figure 1 Evolution of European consumption for the main tropical timber products (source : Faostat)

Evolution of hardwood exports to Europe of selected African countries (sum of logs, sawn timber, veneer and plywood, in Round Wood Equivalents, source : Faostat)

  The most. striking examples are the Republic of Congo, which exports to Europe of logs, sawn timber, plywood, in Round Wood Equivalent, have fallen from more than 80% to less than 15% in the recent years (Figure2). During the same period, Cameroon and Côte d'Ivoire have better resisted, and their share of export to Europe remained above 60% (Food and Agriculture Organization of the United Nations 2007). It is interesting to note that despite different political situations and economic contexts, these two countries both benefit of well developed industrial and institutional tools which allow them to process a large share of their logs, and both present a high proportion of forest industries with European shareholding.

		100%
	Share of exports to Europe (RWE)	10% 20% 30% 40% 50% 60% 70% 80% 90%						Ghana Gabon R. of Congo Côte d'Ivoire Cameroon
		0%
		1998	1999	2000	2001	2002	2003	2004
					Years
	Figure 2
	cirad-00194212, version 1 -6 Dec 2007

Countries of the European Union, hereafter referred as "Europe ".

According to , the tropical forest today only represent 5% of the total available certified areas, which entails an unbalanced offer between temperate timbers and tropical timbers.

Among all the existing forest certifications schemes, only FSC is widespread both in temperate and tropical countries, and publicly displays the characteristics of each certified timber production under its scheme

FSC certification is observed to a greater extent in countries wich present high energy consumptions and development of transports, while FSC extension is limited in countries with high poverty indicators

That is, a unknown sum of money proportional to the GNI or the GDP of the country, relatively to the forest area of the country (GNI/ha or GDP/ha)


domain: math shs

text:
Arne Svein

Øistein Sikko

Jardar Gjøvik

Maria I M Cyvin

Ragnhild L Febri

Staberg

'Walking a graph' -primary school students' experimental session on functions and graphs

Keywords: Mathematical concepts, mathematics activities, mathematics education, educational technology

In this paper we report an experimental activity involving working with functions and graphs in a grade 6 class in a Norwegian primary school. We argue that working with graph loggers in the form of an echo sound system enhances student conception of mathematical graphs.

Background and literature review

The research reported in this paper was part of the EU FaSMEd project 1 , which brings together seven European countries and South Africa, researching the use of formative assessment and technology in mathematics and science education. Part of each country's work has included case study interventions in ordinary classrooms in close cooperation with school teachers. In this paper, we focus on a particular set of lessons concerning functions and graphs, and how one particular type of technology was used to increase student activity and engagement, leading to enhanced learning.

The function concept is generally regarded as a difficult concept for students to grasp [START_REF] Dreyfus | Intuitive functional concepts: A baseline study on intuitions[END_REF][START_REF] Sajka | A secondary school student's understanding of the concept of function -A case study[END_REF][START_REF] Sierpinska | On understanding the notion of functions[END_REF]. [START_REF] Dreyfus | Intuitive functional concepts: A baseline study on intuitions[END_REF] point out that the function concept is not a single concept by itself, but has several aspects and sub-concepts associated with it. [START_REF] Demarois | Function: Organizing Principle or Cognitive Root?[END_REF] connect this to the learning of functions, saying that "for many students, the complexity of the function concept is such that the making of direct links between all the different representations is a difficult long-term task" (p.264). Also in Norway, national and international tests have shown that mathematical functions is a problematic topic area. It took humanity several thousand years of mathematical activity until functions were introduced in the 17 th century; and even then it took time to place functions on a solid foundation within mathematics. It is therefore not surprising that pupils struggle with functions and how to respond to this struggle has been addressed in various ways. During the "New Math" movement in the 1960-70s, it was believed that school mathematics should resemble research mathematics, and attempts were made to introduce functions during the first years of primary school. Eicholz, Martin, Brumfiel and Shanks (1963) did exactly that. This American textbook was translated into several languages, including Norwegian. The pendulum turned away from this with the "back to basics" movement, and, in Norway, functions were moved to the secondary school. As a consequence, most research on students' understanding of functions and graphs is conducted with secondary school students. In this paper, we report on work done in primary school. According to [START_REF] Duval | A cognitive analysis of problems of comprehension in a learning of mathematics[END_REF] we can only gain access to mathematical objects by semiotic representations. [START_REF] Janvier | Translation processes in mathematics education[END_REF] distinguished between four representations of functions: situations; graphs; tables; and formulae; and how to work with transitions between these. [START_REF] Duval | A cognitive analysis of problems of comprehension in a learning of mathematics[END_REF] stresses that "What matters is not representations but their transformation" (p. 107). That is, when learning about a mathematical concept, students deal with a representation of the object, and the main difficulty is to change between different representations of the same object. Duval distinguishes between conversions and treatments. Here, treatments take place between the same registers (e.g., changing y = 2x from one particular form to another), while conversions take place between registers (e.g., reading a table and using the numbers within it to interpret a situation). The latter seems to be far more difficult, while the former is the most common activity format in school. Consequently, Janvier's framework helps teachers to focus on the change of registers rather than algebraic manipulation alone. To support the learning of functions, many different kinds of digital tools have been developed and used. Regardless of any particular view on learning outcomes from using technological tools, it is important to realize that use of technology is more than the introduction of new tools. In a survey on mathematics teachers' use of technology in England, [START_REF] Bretscher | Exploring the quantitative and qualitative gap between expectation and implementation: A survey of English mathematics teachers' uses of ICT[END_REF] found that while ICT might contribute to change, the direction of this change was as likely to be towards "more teacher-centered practices rather than encouraging more studentcentered practices" (p. 43). The tools used in our study are a particular type of data loggers. These are mainly used in science, but we claim that inclusion of these types of tools may be beneficial also in the mathematics classroom, and may contribute to more student-centered practices. According to Newton (2010) "data-logging methods involve the use of electronic devices to sense, measure and record physical parameters in experimental settings." (p. 1247). Measurements and results of the logging can be displayed on a computer screen, either subsequently or simultaneously. We used this type of technology to study students' early understanding of graphs. The use of motion experiments in the learning of functions has been studied by several authors. Nemirovsky (2003) conjectures that "mathematical abstractions grow to a large extent out of bodily activities" (p. 106). [START_REF] Arzarello | Approaching functions through motion experiments[END_REF] claim that students can grasp mathematical concepts through meaningful sensorymotor experiences if they are encouraged to communicate and have the necessary support (p. 308). [START_REF] Arzarello | Modelling body motion: an approach to functions using measuring instruments[END_REF] point out that teachers can use new technology to design experiences for students "where graphs can be presented in a dynamical and genetic way" (p. 135). [START_REF] Robutti | Space-time representations in young children. Thinking through gestures in motion experiments[END_REF] conducted research on time-distance graphs with kindergarten children using motion sensors and calculators, finding that even very young children were able to make connections between the movements they made in front of the sensor and the graph sketched by a calculator.

The research question addressed in this paper is: How can a primary school teacher use data logger technology to enhance primary school students' engagement and conceptual knowledge about function graphs?

Method

The teaching experiment was carried out in a grade 6 class (students around 11 years old) in a primary school in Norway. The number of students in the school is close to 600, and the number of teachers around 35. The participating teacher has background from general teacher education, with specialization in mathematics and history. At the time of the experimental sessions, he had been working as a teacher for 7 years, the last three years at the school in question. He had been teaching mainly mathematics, and also some science. During his participation in the FaSMEd project he was teaching the same group of students, which began in grade 5 and continued with the same group of students into grade 6. There were 31 students in his class, 15 girls and 16 boys.

The theme of the teaching sessions was time-distance graphs. Several technological tools and software had been introduced to the participating teachers at FaSMEd meetings at the university. Teaching material from the FaSMEd toolkit that addressed time-distance graphs had also been introduced. The planning of the experimental lessons began at this meeting. The teacher would give one lesson introducing the students to graphs and to the connection between graphs and real life situations. Working with mathematical graphs connecting situations and graphical representations is usually not done in Norwegian primary schools. According to the national curriculum, functions and graphs is not a specified learning goal for students until after grade 10. This would therefore be the first time this teacher had worked with students in primary school on graphs. Because of this, he wanted to first pilot the lesson on a small group of students that he considered high achieving and with an interest in mathematics. Subsequently the lesson was repeated with a group of students considered to be lower achievers.

The technology used was two echo sounder devices developed by Pasco. This was chosen as the entry level for using it was not too high, and therefore the teacher considered it could be experimented with grade 6 students. It facilitated students to walk back and forth in relation to a logging device, such that a graph was immediately drawn on the computer screen indicating how near they were the device during a time lapse of ten seconds. The immediate live update of the graph distinguishes this activity from most regular science data logging activities. The computer was loaded with an app with premade tasks that were presented to the students. When students walked in front of the echo device, the computer would give a live display of graph in a timeposition coordinate system. The tasks2 were a mix of practical tasks: "Walk a graph", and openended questions about interpretation of the graphs from the "walks". All the results were saved and could be used by the teacher for assessment and feedback to the students. These data were e.g. used by the teacher at the end of the sessions to determine which student groups should present their work in a plenary. Students were chosen deliberately to give good examples of graphs made and how to interpret them. Data sources collected during the experiment include a) observation sheets from two sessions; b) audio recordings from two sessions, from teacher pre-and post-interviews, post-lesson reflections, interviews and q-sorting activities with students; c) video recordings from two sessions; d) transcriptions of audio and video recordings; e) photos taken at sessions and of student work; f) files and screen shots from PC during student activity; g) teacher lesson plans for two sessions.

After the "walking a graph" activity, students were interviewed in a q-sorting 3 activity; i.e. they were presented with a set of statements printed on cards and asked to sort the cards according to whether they agreed, disagreed or were undecided about, the statement on the card. This activity was carried out in groups of 3 to 4 students.

Findings

The echo sound activity made this lesson stand out from an ordinary mathematics lesson. One student said, Student A:

It was very different (…) In maths lessons we never move, we sit at our seat; except sometimes we go out to do measurements, but that is always during summer.

The tasks were also considered different to normal mathematics exercises on two accounts. First, students were not used to doing mathematics tasks using computers. Second, in the classroom they usually have to compute things, whereas in these lessons Student B: There were word problems and we had to do things.

The q-sort revealed that students generally agreed to statements that connected mathematics to real life. E.g., students agreed to the statements "Mathematics helps us to understand our surroundings" and "Mathematics is used in everyday life", whereas they disagreed with the statements "Mathematics is only for the classroom, not for real life outside school", "I can do without mathematics" and "Mathematics is not relevant for my future life". The q-sorting activities were completed around two weeks after the time-distance graph lesson. We may therefore claim that there is some evidence indicating that the lesson had made students aware of, or strengthened their awareness of, connections between mathematics in school and real life situations that can be described by mathematics or where mathematics is used. Students agreed that "Mathematics is important", claiming that Student C: We use it all the time. Everywhere. In the shop. (…) On trains. Airplanes. The bus.

It seems that these groups of students held positive attitudes towards mathematics, and that they were able to see mathematics as relevant for themselves and for real life situations. During the echo sound activity, the students had to relate what they were doing, i.e. the way they were walking in front of the sensor to the graph the software would display on the PC screen. We can see evidence that students were able to connect the pace of their walking to the slope of the graph:

Student E: It rises earlier because you walk faster.

This relates the time (horizontal axis), distance from the sensor (vertical axis) to speed (how steep the curve is), a fundamental relationship in understanding time distance graphs, and a fundamental relationship in physics, and of course in everyday life. There were several student utterances showing the same kind of understanding:

Student F: It will be more slanted the faster you walk. So you start slow, then you walk faster.

The teacher asked another student how you can find from the graph where you walked faster. The student said that Student H: You can see, because, first it is quite slanted, and then it goes straight up.

Students also developed understanding of the fact that a graph does not have to start at the origin. When trying to walk in a way that would produce a W as graph:

Student I: You have to start far away [from the sensor] because then it goes downwards and then it goes upwards and then it goes downwards.

We see here that they understand that a graph can cross anywhere on the vertical axis, and the relationship between distance from the sensor and time passed. Their descriptions and discussions did not use mathematical vocabulary or concepts. Rather, they described what they saw in everyday terms, which shows that they are able to change registers and not only operate within the same register. These examples show how such an activity helps students in the process of conversion.

The activity offers two aspects of working with graphs. On the one hand, students had to translate a given situation into a movement in front of the echo sound device, observe the graph being plotted on the computer screen and adapt their movement to change the graph as needed. On the other hand, students would interpret a graph plotted on the screen into what kind of movement that this would correspond to. In the interviews, students said the tasks in this lesson were more challenging than the mathematics tasks they normally work with, e.g., in that they had to explain how they did things. Being challenging is not really a bad thing, and students said they found the sessions to have been great fun and exciting. They claimed that they had learned a lot about graphs. During q-sorting, students who agreed to the statement "I can better understand when I use the technology tools in our mathematics lessons" also agreed that the statement referred to learning about graphs:

Student E: I learned a lot about graphs and how they change with the computers When the lesson was repeated with students that were considered to be lower achievers, the setup remained unchanged, making it more relevant for comparison. Notably, we found that it was not possible to distinguish any big differences between the first session with higher achievers and the second session with lower achievers.

Teacher: It was indeed very similar (…) maybe these were a bit slower. And I would be tougher, push the others more. (..) But I think they were clever, they were good at cooperating, learning on each other. (…) It shows that if you have open and good tasks, you have a lot of differentiation included.

The activity prompted student communication and discussion. The teacher found that students who normally keep quiet were engaged in discussion.

Teacher:

In particular, some of the girls in the last group, they were talking, usually they are very quiet. Now they talked, without me having to point at them, prompting them; now they gave their opinion (…) And I was positively surprised at how easy it was for them, to listen to each other's arguments.

It was obvious that, even if these types of activities with graphs are common in Norwegian primary schools, it had not been too difficult and that this is a topic that could easily have been done with the whole class. The teacher said that Teacher: I think, interpreting graphs, it could have been done quite easily. (…) I think this might be more fun in primary than in lower secondary school. They still find it exciting with graphs (...) they are more curious and less biased.

In the interviews, all students said that they had enjoyed taking part in the project and performing the lessons with graphs.

Student:

In my opinion everything was good (…) We learned a lot about graphs.

Discussion and conclusions

The echo sound graph plotting activity was very useful to give the students hands-on experience in using modern technology and use their own physical movements to create something to talk about mathematically. Acquiring experience with new technologies can be an educational goal in itself, and in particular, echo sound technology is not common in the classroom, but it is well known in other aspects of life. In the interviews, students claimed this was an important part of what they had learned and which distinguished these lessons from ordinary mathematics lessons. In traditional data logging experiments, students might see the data collection and the data analysis as two separate entities as these are separated in time [START_REF] Barton | Does data logging change the nature of children's thinking in experimantal work in science[END_REF]. In our experiment, the gap between the collection of the data and the displayed graph is narrowed down to practically zero. In this respect, this activity also resembles working with dynamical graph tools, like GeoGebra4 . These software tools allow students to explore graphs by manipulating parameters within designated bounds, while walking a graph changes freely the look of a graph only limited by the range of the echo sound device. This is more in line with work by e.g. [START_REF] Arzarello | Approaching functions through motion experiments[END_REF] and [START_REF] Robutti | Space-time representations in young children. Thinking through gestures in motion experiments[END_REF].

When looking at the Janvier table we see that what the students had engaged in was making a transition between a situation and a graph. However, the typical sketching activity proposed by the Janvier framework when working with functions usually has a different feel than in this experiment. Not only is the sketching part of the activity itself done in a kinesthetic manner. There is also a dual aspect in that the students continually interpret the graph whilst the graph is sketched by the program on the screen. This way we can say that students work simultaneously with two elements of Janvier's framework, giving further evidence that changing registers is a difficulty that can be overcome by giving appropriate teaching materials.

The kinesthetic part of the activity, the walking, is in itself an important aspect of the experiment. As it turned out, the designated low achievers were able to perform well and display great enthusiasm during the session. This can be related to the way learning through movement can be an alternative approach to put students in a receptive state, ready for learning. Learning through actually moving your body is rarely an aspect of mathematics lessons, but can certainly encourage engagement, as seen in this experiment.

The type of activity exemplified in this experiment is completely devoid of focus on algorithms or procedural performance in the form of computations. Students do not know in advance how to solve the problems presented, and so focus is on developing conceptual knowledge about function graphs.

From their statements, we also see that they relate mathematical concepts, like slope, to real world experiences, like speed. This is similar to findings in Robutti (2009, p. 68). A well founded conceptual understanding of functions and graphs in a time-distance setting will contribute to better understanding of functions on a more general level. When students encounter functions at higher grades, their conceptual foundation will make it easier to grasp other aspects and algorithmic approaches to functions.

The literature suggests that teachers need support of different kinds in order to conduct teaching with new technologies in a meaningful manner. For example, building on a large teacher survey in Singapore, [START_REF] Tan | Datalogging in Singapore schools: supporting effective implementations[END_REF] suggest that teachers need support from laboratory technicians, data logger training, and instructional material to use data loggers effectively. In our case, none of these were present. We do however acknowledge the collaborative effort between teacher and researchers as instrumental to the success of the sessions. It is also important to stress that learning is not an automatic outcome from playing with technological tools, no matter how sophisticated the tools are. The role of the teacher is instrumental in bringing about learning, as highlighted by Clark-Wilson, Robutti and Sinclair (2014, p.396).

Proceedings of CERME10

The echo sound activity used some tasks taken from the software bundled with the Pasco software. Instructions for using the software and tasks were translated into Norwegian by the FaSMEd team. Some additional tasks were added.

https://en.wikipedia.org/wiki/Q_methodology

http://www.geogebra.org

Acknowledgment

The FaSMEd project has received funding from the European Union Seventh Framework Programme under grant agreement n° 612337.


text:
Lina Kayali
email: l.kayali@uea.ac.uk

Irene Biza

"One of the beauties of Autograph is … that you don't really have to think": Integration of resources in mathematics teaching

Keywords: Tasks, documentational genesis, teaching triad, mathematics education software. Thematic Working Group 15

. The data comprised one interview and one lesson observation with a secondary mathematics teacher. While the documentational approach provides an overview of the set of resources being integrated to achieve a specific goal, the teaching triad offers a lens to observe teachers' considerations when implementing a task in a mathematics lesson.

Introduction

The complexity of the teaching profession imposes several factors that impact upon teachers' classroom actions, that include not only their beliefs and knowledge but also their experiences and the educational context in which they act [START_REF] Biza | Balancing Classroom Management with Mathematical Learning: Using Practice-Based Task Design in Mathematics Teacher Education[END_REF][START_REF] Speer | Issues of Methods and Theory in the Study of Mathematics Teachers' Professed and Attributed Beliefs[END_REF]. Teachers, while planning and teaching, consider the contexts they work within: their students, school environment, curriculum, etc. In other words, any study of teachers' practices should take into account the different contextual conditions in which these practices develop to include personalities, institutions, circumstances, epistemology, time issues and materials [START_REF] Herbst | Exploring the Practical Rationality of Mathematics Teaching through Conversations about Videotaped Episodes: The Case of Engaging Students in Proving[END_REF]. The study presented in this paper forms part of the PhD research of the first author and it investigates mathematics teachers' ways of balancing the different elements in their working environment, especially when using technology (in this study mathematics education software, i.e. software designed for mathematics teaching and learning purposes), by looking at their practices or intended practices within specific contexts. Furthermore, our work examines any gaps between intended technology use in mathematics classrooms and actual teachers' practices. To this aim we invite teachers' views on hypothetical classroom situations that involve teaching with technology in written responses and follow-up interviews. Then, we observe teachers' use of technology in their classroom. In this paper we present preliminary analysis from one participant, Adam, by drawing on two theoretical perspectives: the documentational approach [START_REF] Gueudet | Instrumental and Documentational Approaches: From Technology Use to Documentation Systems in University Mathematics Education[END_REF][START_REF] Gueudet | Towards New Documentation Systems for Mathematics Teachers?[END_REF] and teaching triad [START_REF] Jaworski | Investigating Mathematics Teaching: A Constructivist Enquiry[END_REF].

The documentational approach

The documentational approach looks at teachers' interactions with resources where a resource is defined as "anything that can possibly intervene in [a teacher 's] activity", it can be an artefact (e.g. a pen or a mathematical technique), a teaching material, or even a social interaction (Gueudet et al., 2014, p. 142). Adler (2000, p. 207) adds that "resource" can be also "the verb re-source, to source again or differently". During their interaction with resources, teachers develop schemes. A scheme is a set of organised procedures carried out on an artefact [START_REF] Gueudet | Towards New Documentation Systems for Mathematics Teachers?[END_REF]. It consists of "the goal of the activity; rules of action; operational invariants; and inferences" (Gueudet et al., 2014, p. 140, italics in original). Here, operational invariants are cognitive concepts established throughout the activity to be used in comparable situations [START_REF] Gueudet | Instrumental and Documentational Approaches: From Technology Use to Documentation Systems in University Mathematics Education[END_REF]. For teachers, these constitute "professional knowledge" (Gueudet et al., 2014, p. 142). The documentational approach describes a two-way influence between a resource and a teacher: a resource affects the teacher's actions and knowledge; and the teacher's perceptions and experiences impact on the way the resource is used (Gueudet et al., 2014, p.140). A management process, which [START_REF] Gueudet | Towards New Documentation Systems for Mathematics Teachers?[END_REF] call instrumental orchestration, is required in order to organise the learning environment (e.g., space, time, dialogue) by the teacher, whose responsibility is to manage the process according to the requisites of the task [START_REF] Gueudet | Instrumental and Documentational Approaches: From Technology Use to Documentation Systems in University Mathematics Education[END_REF]. When a teacher uses a set of resources according to a specific scheme for a specific goal, s/he creates a document. Such a development of a document is called documentational genesis. Thus, the documentational genesis is the process of a teacher developing schemes for adapting different sets of resources to achieve a specific target [START_REF] Gueudet | Instrumental and Documentational Approaches: From Technology Use to Documentation Systems in University Mathematics Education[END_REF]. The documentational approach studies the development of "structured documentation system[s]" that represent teachers' work and progress as a result of influencing and being influenced by different resources [START_REF] Gueudet | Instrumental and Documentational Approaches: From Technology Use to Documentation Systems in University Mathematics Education[END_REF]. In this study we have conjectured that teacher's schemes are dynamic and are being re-adapted from one situation to another and that the teaching triad [START_REF] Jaworski | Investigating Mathematics Teaching: A Constructivist Enquiry[END_REF] can help the exploration of those schemes. [START_REF] Jaworski | Investigating Mathematics Teaching: A Constructivist Enquiry[END_REF] teaching triad (TT) addresses classroom management as an act of harmony between three domains of activity: sensitivity to student (SS), mathematical challenge (MC) and management of learning (ML). These domains are evident when a teacher plans a lesson and starts to think of how to consider teaching a specific mathematical idea (MC), particular students' needs (SS), the best way to work on the task with the students (ML) (group work, individual work or classroom discussion). The same domains will be in play during lessons, but within a different context as this time the interactions with the students are happening and the teacher should respond on demand, in many cases by diverting from what was planned.

The teaching triad

As [START_REF] Jaworski | Investigating Mathematics Teaching: A Constructivist Enquiry[END_REF] and [START_REF] Potari | Tackling Complexity in Mathematics Teaching Development: Using the Teaching Triad as a Tool for Reflection and Analysis[END_REF] suggest from the "macro analysis" of classroom interactions, alongside the TT domains, teachers' plans and practices are also influenced by social factors, such as: time pressure; having to complete a set syllabus; the requirement that students know specific things for exam purposes; expectations from the teacher; school ethos; and the training provided for teachers. Such factors seem to be at the centre of teachers' considerations and they include students' social culture, teaching resources and materials, syllabus, assessment schemes, time restrictions, room constraints, and cultural considerations of what constitutes good teaching practices (Goos, 2013, p.523). Hence, the TT domains, along with these factors, reflect the range of considerations mathematics teachers have to balance. The TT can be "used as an analytical device (by researchers) and as a reflective agent for teaching development (by teachers)" (Potari & Jaworski, 2002, p. 351). We are conjecturing that the teaching triad domains are related to schemes' development regarding the use of resources. Sustaining the goals of the resource use, depends on how teaching is balanced by the teacher. Operational invariants can be derived from artefacts, mathematical concepts or social environments. All the above can be used to satisfy a specific goal and produce a proper usage and inferences, but these are flexible techniques of balancing and rebalancing of the TT domains from one lesson to another. The analysis presented here aims to investigate this conjecture.

Technology in mathematics teaching -A view through the lenses of the documentational approach and the teaching triad

Technology software and hardware devices are "artefacts" (Gueudet et al., 2014, p.141) and can be adapted to provide access to formal mathematical knowledge. They afford "opportunities for additional student actions, such as the manipulation of on-screen objects and the ability to make a range of mathematical inputs, which places an additional demand on teachers as they strive to make sense of a diversity of student activity in real-time" (Clark-Wilson & Noss, 2015, p. 95). Thus, interactions with technological resources influence teachers' documentational geneses by developing the ways they organise their classroom activities and manage learning situations with impact on the shape of the teaching process and on the way knowledge is communicated. When employing technology, resources become more complex, and so do the TT domains. Sensitivity to student becomes more evident (e.g. if students know more about technology than their teacher). Tasks can be more challenging for teachers to design, and the management of learning becomes more complicated with the higher chances of distraction. We also re-emphasise the importance of social factors when technology is used based on several premises. First, the technology use dependency on the teacher training provided (Gueudet et al., 2014, p.144;OECD, 2015, p.69). Second, the availability of hardware and internet connection (Bretscher, 2014, p. 66;OECD, 2015, p.61 & p. 146). Third, the national curriculum obligations (OECD, 2015, p.70). Fourth, and most importantly, the education policies that aim to embed technology (OECD, 2015, p.50).

Methodology

This paper reports from the first phase of a project that looks at secondary mathematics teachers' work with technology that involves participants' written responses to situation-specific tasks alongside follow up interviews and classroom observations. Situation-specific task methodology has been suggested by Biza, Nardi, & Zachariades (2007, p. 301) where tasks are given classroom situations that "are hypothetical but grounded on learning and teaching issues that previous research and experience have highlighted as seminal; are likely to occur in actual practice; have purpose and utility; and, can be used both in (pre-and in-service) teacher education and research through generating access to teachers' views and intended practices". The study is conducted in England and participants are secondary school mathematics teachers with different levels of experience and training. The work is based on providing qualitative findings established on an interpretative research methodology (Stake, 2010, p. 36).

In this paper we discuss the written response to the situation-specific task presented in Figure 1 (we call it the 3D Task), the follow-up semi-structured interview and the lesson observation (75 minutes) of one participant, Adam. Both interviews and observations were conducted by the first author. The situation described in the 3D Task regards an open investigative question to be given to the students. It did not suggest any specific use of technology, and left that to be decided by the teacher. The situation regards a geometrical problem with a potential consideration of the affordances of software available at the school where the data were collected, such as Autograph (http://www.autograph-math.com/) or Geogebra (https://www.geogebra.org/). Adam was invited to offer a written response to the 3D task and, then, he was interviewed, to clarify the answers and offer more elaboration where needed. Then, a lesson was observed of a Year 12 class (17-18 yearold 4 female and 5 male students) and was audio-recorded. The focus of the observation was Adam's use of resources, especially his use of Autograph or Geogebra, which he said he frequently used, and his classroom management. At the time of the data collection, Adam was a mathematics teacher with four years' experience, during which he taught students aged 12-18 years. He held degree in economics, a postgraduate certificate for teaching mathematics at secondary level, and was about to finish his master's degree in educational practice. The school had interactive whiteboards, a computer lab, and Geogebra and Autograph software installed on all computers. A preliminary analysis of teachers' comments and interactions during interviews and lessons was performed for the transcripts [START_REF] May | Social Research: Issues, Methods and Process[END_REF]. This was coded according to the Teaching Triad [START_REF] Jaworski | Investigating Mathematics Teaching: A Constructivist Enquiry[END_REF] into SS, ML, and MC. During each interaction, we explored the teacher's interactions with the resources, according to the documentational approach [START_REF] Gueudet | Instrumental and Documentational Approaches: From Technology Use to Documentation Systems in University Mathematics Education[END_REF].

We then reviewed the results from the interview and observation, and offered a discussion according to TT and documentational approach together.

Adam's responses to the 3D Task and follow up interview

Adam identified mathematical ideas involved in the 3D Task, such as "volume", "surface area" and "calculus". In his response to the task, he frequently repeated the word "scaffold" to state that he would try to adapt the task according to the students' needs and "prior knowledge". He emphasised that he would not use the problem as it is because it needed a lot of scaffolding and it included "too many variables". During the interview, he explained that the scaffolding would include giving hints and examples and even values to work on for weaker students. He said he will not use 1L in this problem, but would use a bigger number:

I think straight away students having to think of a length width and height that times to get 1 will be quite difficult for students… They might be able to go 1 1 1 and they might be able to go 2 ½ 1 or something like that. That will be it, they'll really struggle.

He wrote that he would use technology with the task for "gradient of curves on Autograph" and "to visualise the shapes". During the interview, when asked how and when he would use it during a lesson, Adam suggested "I think as a group activity. It wouldn't be the focus of the lesson though it would just be almost the point at the end". When asked to elaborate he said: I think it is because one of the beauties of Autograph is that it means that you don't really have to think... I want the student to be thinking about problems and how to approach problems. I think almost Autograph gives you too much, too much help and then you don't have to think about the shape of the graph because you can just plot it in Autograph. And then other, obviously other reasons a lot of my students have never used Autograph even at key stage five 1 . So, to start understanding it, it will take quite a long time and a lot of effort just to get the students to understand it to start with.

Adam said he will not use the task as it is because although it works well in "an ideal world", it does not go well with the way the syllabus is set. He anticipated problem with keeping track of calculations, prior knowledge and many involved variables (e.g. Adam suggested if a student chose to design a cylinder container, the case would be very confusing because s/he would have to think of adjusting the radius and height of the cylinder in order to find the minimum surface area).

Adam's teaching observation

The observation was on a revision lesson about solving simultaneous linear and modulus equations (i.e. equations that include absolute value). Adam started by moving a stick in the air in order to draw a specific graph, and asking the student to recognise the graph. One of these graphs was the sine graph, but the students seemed to be confused about what graphs were being drawn. Then, Adam asked his students to solve some problems that were displayed on the board. All the problems apart from one (which was designed by Adam) were chosen from the textbook. During the lesson, Adam used Autograph to check the answers given by the students, he entered the functions and the graphs were projected on the board. Then a discussion/demonstration of the algebraic solution was led by him on the white board. For example, for the simultaneous equations: y = |x + 2| and y = 3, he asked students to draw the graphs on their notebooks and see the solution before solving it algebraically: "You will get two points, you can see this graphically". Then, he started to write one of his student's algebraic answers on the board: "3 = x + 2 or -3 = x + 2". He then commented on the student's answer: "So, math says x = 1 or -1... What text book would say is y = (x + 2) or y = -(x + 2). Textbook would just say that, I'll probably do it this way". Later, with the problem that followed, he commented that: "This is GCSE grade C 2 […] This is mark C in C1 3 ". He repeatedly encouraged the students to solve another problem he displayed on the board by saying that it is an "exam question". When the students asked Adam why they should learn modulus equations, he went to his computer and googled "when to use modulus equations in real life" and gave the answers accordingly "Distance, currency exchange…". Two of the students finished with the problems on the board earlier than the rest of the class and Adam gave them an extension problem which might have been suggested spontaneously in response to the need of extra work. The extension problem was in two parts, the first asked for two different modulus functions that do not intersect, the second asked for two that intersect once. "Is that possible? Can you give me two that intersect once?", Adam asked the class, and the dialog below followed:

1 Key stage five is post-16 school education in England i.e. for students aged 16-18.

2 GCSE stands for the General Certificate of Secondary Education. It is the qualification taken by school students aged 14-16 in the UK (except Scotland). Its exams are graded on a scale of A* to U, with A* being the highest grade and U the unsatisfactory. A grade/mark C reflects an average progress (pass).

3 C1 stands for Core1 and it refers to one of the mathematics textbooks, used at Adam's school, for students aged 16-18.

Analysis

From his responses to the task and the observation we notice that Adam's resources were the textbook used at his school, help cards, a computer, Autograph, Excel, Google, interactive whiteboard, the stick he used at the beginning of the lesson observation, information about exam grades and questions, past experiences with students along with the mathematical concepts and methods. Adam's appreciation of Autograph's ease of use as a tool for visual representation was evident, so he used the software to check students' work, and present graphical solutions before going for algebraic ones. So, he would ask his students to solve graphically, check that their graphical solutions are right according to the answers on Autograph, and then ask them to find the same answers algebraically. However, Adam seemed being confused by the Autograph when it came to student B's answer on which he seemed to hesitantly agree. This might be because only one intersection point was visible within the displayed part of the graph. In this case, Adam missed the opportunity to use the full affordances of Autograph in order to improve student's B answer and to explain the correct answer to the rest of the class. There was no evidence that the rest of the class, apart from student C, realised where the problem was and how it was amended.

In terms of the TT, Adam indicated sensitivity to students "they don't know Autograph", "prior knowledge", "scaffold", "weaker students" (SS). In his teaching choices, he also showed consideration of the syllabus he had to follow, exam questions and the timeframe he had to adhere (MC and ML). The way he intended to use resources showed an attempt to balance mathematical challenges (MC) (e.g., change 1L, exercises from the textbook) with students' needs (SS) (e.g. students do not know how to use Autograph, providing extension question when needed), and management of teaching (ML) (e.g., use technology at the end of the lesson as a group activity, encouraging pair work when solving textbook exercises, graphing the equations to see the answers and then doing the algebraic solution because "Putting it in a graph might be easier") with attention to management of learning with technology (e.g. technology takes a lot of time). Now, we will look at how Adam used the available resources to design and implement his teaching. Along with the textbooks that are being used at his schools, he mentioned he would also use help cards with hints or examples. These will help him "scaffold" and build on "prior knowledge", these terms seem to be adopted during Adam's teaching practice or teacher's education courses for reflection on students' needs (SS). Also, he drew on his teaching experience (as a resource) when he mentioned in the interview that students would struggle to "keep track of their calculations" (ML and SS). In terms of Autograph as a resource, Adam would use it as a graphing software that helps visualise graphs and shapes and shows answers (ML and MC). The data showed the two-way influence between Adam and the resources. For example, Adam's belief that students do not think when using Autograph (SS) was influencing the way the resources were used, so he used Autograph to show or check answers (MC). Also, the resources available influenced the teacher's decision, so in this instance he used Autograph to show the graphical solution and then asked the students to do the algebraic solutions keeping the answers from Autograph in mind. Additionally, Adam frequently used the textbook as a source for exercises, so the textbook influenced which mathematical challenge he gave the students (MC). At the same time, Adam used the textbook exercises along with Autograph and, by doing so, Adam's way of managing the teaching affected the way the textbook was used. Adam's management of the teaching situation led us to conjecture that his use of resources is connected with a potential scheme developed in order to properly use the resources available and achieve a specific goal, which in the lesson observation was revising the topic of linear and modulus simultaneous equations. We have identified some operational invariants in Adam's schemes, for example the use of Autograph as a class activity managed by the teacher on the board. However, we believe that more data and observations are needed to further investigate Adam's schemes.

Discussion and summary

The preliminary findings we present in this paper derive from a study on mathematics teachers' practices/intended practices in relation to the used resources and especially mathematics education software.

Adam's attempt to balance the different domains of activity described by the teaching triad was evident in the interview and during the lesson observation; and his interactions with resources were influenced by considerations of these domains as well as considerations of exams' questions and grades, time management, and the syllabus. His use of technology resources was led by him on the board, because of his concern that he would be teaching mathematics and technology use if his students were to work independently or in pairs on computers. Although the teacher used Autograph frequently, his use was mainly for checking answers and displaying visual representations. This is due to his concern that Autograph offers excessive help and would stop students from thinking about the mathematical problems. The pilot observation proved that more clarification about the teacher's actions should be sought from future observations along with pre-and post-lesson interviews. This is because the pre-and post-lesson interviews will give more space for the teachers' interpretations of their classroom actions. More observations are also needed to explore the teacher's documentational work and investigate how the teaching triad helps clarify teachers' considerations when working with resources.

Figure 1 :

 1 Figure 1: The 3D Task

  x | and y = 2 | x |, shift across Adam: Oh, ya it is. Student A: Ya, you've translated it. Student B: y = | x -4 | and y = 2 | x |. Adam looked at the graphs on Autograph and nodded in what seemed like a hesitant agreement Student C: Change the slope. Adam amended the equations as student C suggested and wrote y = 2 | x -4 | and y = 2 | x | without commenting on student's B answer Adam did not follow up student's B response or student's C correction, but moved straight to a completely different activity by which he concluded the lesson.

  3D Task A group of Year 11 students are asked the following question: Design a milk container with capacity of 1L. What dimensions and which design uses less materials? Why? -What are the mathematical ideas and activities addressed in this question? -Would you use this question in class? Why or why not? What are the learning objectives for which you would use this question? Would you modify it? -Would you use technology with this question? If yes, what type of technology? If no, why? -If you were to use technology, how would you use it? -What teaching approaches and resources would you suggest for this question? -Do you anticipate any problems or challenges (either with students or resources)?

Proceedings of CERME10


text:
Nimer Baya'a
email: bayaan@qsm.ac.il

Wajeeh Daher

Rawan Anabousy

Ahlam Anabousy

The development of pre-service teachers' TPACK for the use of digital tools

Keywords: Pre-service teachers, TPACK, digital tools, professional development

The ministry of education is launching a national project to implement the use of ICT in the Israeli education system. To prepare pre-service teachers with whom we work for this kind of implementation, we designed a model, which supports them to learn to use digital tools effectively while integrating a particular pedagogy for teaching a specific mathematics or science content. The goal of the present research is to study the development of these pre-service teachers' Technological, Pedagogical and Content Knowledge (TPACK), attitudes toward computers and their ICT proficiency. For this purpose, we used and adapted questionnaires from different sources. The research results show significant improvement in the TPACK level and ICT proficiency, but no significant effect of the preparation on most of the components of the teachers' attitudes toward computers, being positively high before and after the preparation.

Introduction

 [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF]suggested the PCK (pedagogical content knowledge) model to represent the interaction of two types of teachers' knowledge: content knowledge and pedagogical knowledge. He proposed that this interaction be considered in order to understand teachers' expertise in teaching a particular subject matter. Various researchers (for example [START_REF] Koehler | What is technological pedagogical content knowledge? Contemporary Issues in Technology and Teacher Education[END_REF][START_REF] Niess | Mathematics teacher TPACK standards and development model[END_REF], built on Shulman's PCK to describe the interaction of teachers' understanding of educational technologies with their PCK that results in effective teaching with technology. Specifically, they talked about the technological pedagogical and content knowledge of teachers (TPACK), where this model describes the interactions between and among the three main components of teachers' knowledge: content, pedagogy, and technology. These interactions result in new types of teachers' knowledge, namely PCK, TCK (technological content knowledge), TPK (technological pedagogical knowledge), and TPACK. In this paper, we describe the development of pre-service teachers' TPACK as a result of preparing them in the use of digital tools over one academic year.

Technological, Pedagogical and Content Knowledge

Though some researchers consider TPACK too blunt an instrument (e.g., [START_REF] Clark-Wilson | Designing a web-based professional development toolkit for supporting the use of dynamic technology in lower secondary mathematics[END_REF][START_REF] Thomas | Teaching with digital technology: Obstacles and opportunities[END_REF], other researchers refer to it when studying mathematics teacher's professional development (e.g., [START_REF] Balgalmis | Reactions of pre-service elementary teachers' to implementing technology based mathematics lessons[END_REF][START_REF] Bowers | Using technology to explore mathematical relationships: A framework for orienting mathematics courses for prospective teachers[END_REF]. Generally speaking, TPACK is the knowledge of how to integrate technology in teaching the subject matter. This knowledge also includes the appropriation between a specific technological tool, the teaching of a specific topic and being aware of the difference between various technological tools in teaching a specific topic. Further, this knowledge means being aware of students' problems of the subject matter that could be overcome by using specific technological tools. It also includes the awareness of students' difficulties of the subject matter that result from using specific technological tools and how to overcome these problems (Koehler & Mishra, 2009). [START_REF] Robova | Developing future mathematics teachers' ability to identify specific skills needed for work in GeoGebra[END_REF] studied mathematics teachers' awareness of the specific technological skills needed for their teaching (making functions visible on the screen, changing visual appearance of graphs, interpreting numerical results, using dynamic features of a tool) and their ability to design teaching which takes the specific skills into account. Furthermore, [START_REF] Koh | Developing pre-service teachers' technology integration expertise through the TPACK-Developing Instructional Model[END_REF] described an instructional model for developing pre-service teachers' TPACK. We follow the previous attempts to suggest a preparation model for developing pre-service teachers' TPACK in utilizing digital tools in their teaching. [START_REF] Fishbein | Readings in Attitude Theory and Measurement[END_REF] defined attitude as a learned tendency to respond to an object in a consistently favorable or unfavorable way. Other researchers [START_REF] Zan | Attitude toward mathematics: overcoming the positive/negative dichotomy[END_REF] defined attitude in terms of emotions: a positive or negative emotional reaction toward a specific situation. These definitions show the possible influence of attitudes on behavior in general and on pre-service teachers' behavior in particular. Attention to attitudes has risen when ICT started to emerge as a possible tool for the improvement of teaching and learning. In this context, researchers found that these attitudes have major influence on the success and meaningful use of the ICT in their teaching [START_REF] Albirini | Teacher's attitudes toward information and communication technologies: the case of Syrian EFL teachers[END_REF].

Pre-service teachers' attitudes toward computers

In our research, attention was given to pre-service teachers' attitudes toward computers, together with the development of their TPACK and ICT proficiency, as a consequence of their preparation in the use of digital tools. We used the 'teacher's attitudes toward computers' questionnaire (TAC) as it probes teachers' attitudes toward ICT use in teaching and their intention to do so [START_REF] Baya'a | Mathematics Teachers' Readiness to Integrate ICT in the Classroom: The Case of Elementary and Middle School Arab Teachers in Israel[END_REF]. We were also interested in the pre-service teachers' proficiency level in ICT as an indicator of their intention to use ICT in their teaching as the proficiency variable is reported to affect teachers' readiness to use ICT in their teaching [START_REF] Granger | Factors contributing to teachers' successful implementation of IT[END_REF].

The research questions

The main research question is: How will the preparation of pre-service teachers in the use of digital tools, according to the model that we designed, affect their TPACK level, ICT proficiency and their attitudes toward computers?

Research context, participants and the preparation model

This current research accompanies the preparation of pre-service teachers to study how to use effectively digital tools in the mathematics or science classroom. This knowledge is the core of the TPACK model. We administered questionnaires to measure the advancement of the TPACK levels and attitudes toward computers of the pre-service teachers who implemented the model, as well as their ICT proficiency. Approximately 55 students majoring in mathematics and science teaching in intermediate schools completed the questionnaires at the beginning and end of the preparation. These students were in their third year of training alongside two courses that provided a background in the use of ICT for teaching mathematics and science.

The preparation model aimed to improve the pre-service teachers' selection of a suitable digital tool for a specific pedagogy and subject. It also tried to improve the integration of digital tools to teach some specific content. This preparation model concentrated on two aspects. First, knowing the tool technically and being able to adapt it to teach some specific content. Second, developing the ability to select and integrate suitable digital tools for some specific content and pedagogical method. In more detail, each pre-service teacher worked independently to learn to use at least two digital tools and to prepare user guides (as PDF file or digital book) that included descriptions of the most significant functions of these digital tools. Furthermore, the pre-service teachers were required to record video clips of screen shots while performing operations in these digital tools as explanations for another user. The pre-service teachers were asked to select the digital tools from a catalog of general digital tools prepared by the ministry of education in Israel. This catalog includes various digital tools that could be adapted for use in various subjects and levels, such as: Flipsnack for creating online digital books, Linoit for creating collaborative bulletin board, Socrative for personal and class assessment and Mindomo for creating mind maps.

Moreover, each pre-service teacher was required to prepare pedagogical materials on how to use the digital tools that she was engaged with in teaching mathematics or science, and then present the materials in the training workshop to receive comments from her peers and the pedagogical supervisor. Following that, the pre-service teacher reflected on her developed materials, adjusted it and uploaded all the materials to an internet site that was constructed by the pre-service teachers and the pedagogical supervisors. This internet site constituted a data bank for digital tools. In addition, each pre-service teacher was requested to prepare at least two lessons for teaching mathematics or science and pick three digital tools from the catalog (including one that she was engaged with) to use them in her teaching. These lessons had to involve also collaborative investigations that encourage the use of higher order thinking skills. Finally, each pre-service teacher picked a subject from within a digital textbook for teaching mathematics or science, and added connections to pedagogical activities based on using digital tools from the data bank site.

During the first semester, the pre-service teachers had two options: to start from the digital tool and integrate it for teaching some specific content, or starting from the content and selecting a suitable digital tool to teach that content. In the second semester, each pre-service teacher was asked to experiment with the prepared materials and lessons in her training school with at least one of the chosen tools, and reflect on the experience. This reflection was on the actual implementation of the digital tool in the classroom environment, and it was posted in the data bank for digital tools for other pre-service teachers to consider as they selected digital tools for their own use.

Research instruments

The research instruments included three questionnaires as follows: Questionnaire 1: Technological, Pedagogical, and Content Knowledge (TPACK) (revised) questionnaire, constructed on the basis of the TPACK instrument for pre-service teachers developed by [START_REF] Schmidt | Technological Pedagogical Content Knowledge (TPACK): The Development and Validation of an Assessment Instrument for Preservice Teachers[END_REF].

Questionnaire 2: Teachers' Attitudes toward Computers (TAC, v. 6.1) questionnaire: This questionnaire was tested by [START_REF] Christensen | Construct validity for the teachers attitudes toward computers questionnaire[END_REF] who concluded that it is a wellvalidated and reliable instrument for teachers' self-appraisal of their attitudes toward computers. The validity of the questionnaires was considered by giving the Arabic translations to a group of pre-service teachers who were requested to examine if the questionnaires' statements were clear to the reader. As a result, some items of the questionnaires were rephrased to clarify their meaning.

The pre-service teachers' scores in the overall constructs and their categories, before the preparation and after it, were examined for internal reliability using Cronbach alpha. The results showed high Cronbach alpha (above 0.85 for all the categories and for the overall construct) indicating adequate internal reliability for the questionnaires and their categories. These results were expected due to the extensive use of these questionnaires in the literature.

Data processing

Data was analysed using paired-samples t-test to determine if there were significant differences between scores of pre-service teachers in the various questionnaires before and after the preparation. Cohen's d (the ratio between the difference of the means and the average of the standard deviations) [START_REF] Cohen | Statistical power analysis for the behavioral sciences[END_REF] was used to compute effect sizes to assess the practical significance of results.

Results

Pre-service teachers' ICT proficiency

Table 1 shows the proficiency level of the pre-service teachers before and after the preparation (values between 1 to 5), as well as paired sample t-test between the two observations.

Before Preparation

After As displayed in Table 1, the results show that the pre-service teachers' ICT proficiency level differs significantly before and after the preparation. Large positive effect size of 0.70 was derived for the preparation on the pre-service teachers' ICT proficiency level. This advancement was mainly the result of the major improvement in their 'multimedia tools proficiency'.

Pre-service teachers' TPACK level

The TPACK level comprised the total score of the TPACK questionnaire and six other scores for each partial type of knowledge for technology, pedagogy, content and intersections between them. Table 2 shows the TPACK components' scores of the pre-service teachers before and after the preparation (values between 1 to 5), as well as paired sample t-test between the two observations. As displayed in Table 2, the pre-service teachers' scores in the components of TPACK differ significantly before the preparation and after it. Large positive effect sizes of 0.74 and more were derived for the preparation on the pre-service teachers' TPACK and its components.

Pre-service teachers' attitudes toward computers

Attitudes toward computers were assessed using 9 categories. Table 3 shows components' scores of the pre-service teachers' attitudes toward computers before and after the preparation (values between 1 to 5, except perception 1 to 7), as well as paired sample t-test between the two observations.

Before Preparation

After As displayed in Table 3, the pre-service teachers' scores in the categories of attitudes toward computers differed significantly before the preparation and after it in the categories: interaction, concern, absorption and TAC general. In these constructs, a small effect size of 0.26 was derived for the preparation on the pre-service teachers' general TAC score, and moderate effect sizes of 0.41, 0.36 and 0.37 were derived for the preparation on the interaction, concern and absorption respectively.

Discussion and conclusions

The research aimed to examine how the preparation course affected the pre-service teachers' ICT proficiency, TPACK level and their attitudes toward computers. The research results indicated several significant positive effects of the preparation model used in that preparation that related to the pre-service teachers' abilities and knowledge regarding the integration of digital tools in teaching.

Pre-service teachers' ICT proficiency

The research results indicated significant improvement in the pre-service teachers' ICT proficiency as a consequence of the preparation, especially in multimedia tools proficiency. The mathematics and science pre-service teachers usually have high ICT proficiency, but the requirements in the preparation model led to significant improvement particularly in their multimedia proficiency. These results are due to a consideration of the technology knowledge related to the digital tools in the preparation process. This resulted in the pre-service teachers increased competence in their use of digital tools for personal and professional purposes, which caused them to feel confident to utilize new digital tools independently and individually [START_REF] Prestridge | The beliefs behind the teacher that influences their ICT practices[END_REF], and thus improved significantly their ICT proficiency. This suggests that pre-service teachers need to be given the opportunities to work with technological tools in order to improve their ICT proficiency and their readiness to integrate ICT in their teaching [START_REF] Muir-Herzig | Technology and its impact in the classroom[END_REF].

Pre-service teachers' TPACK level

As a result of the preparation, the general TPACK level of the pre-service teachers, as well as its six partial types, were significantly improved. These results could be due to the attention of the prepreparation model to the ability of the pre-service teachers to appropriate the digital tools pedagogically to teaching a specific content, and vice versa. It could be said that the pre-service teachers' diverse experiences in the workshop improved their knowledge in different types of knowledge related to their teaching mathematics or science. Thus, the preparation model provided the pre-service teachers with opportunities to maintain and shift their instructional approaches enriched with innovative educational technologies [START_REF] Martin | Successful implementation of TPACK in teacher preparation programs[END_REF]. This preparation model could be implemented worldwide when taking into consideration the particular background and conditions of the pre-service teachers involved.

Pre-service teachers' attitudes toward computers

The results of this research show that following the preparation process, no significant improvement was detected in the pre-service teachers' attitudes toward computers for most of the TAC components, with exception of TAC general, interaction, concern and absorption. We should note that in both cases, before and after the preparation, the attitudes were very favorable toward computers.

As for the positive change in some attitudinal categories, such as absorption, the pre-service teachers had, during the workshop, the chance to be actually involved and improve their knowledge in computers and ICT. This might have improved their ability to solve problems related to the computer use in the classroom; which encouraged them to insist to solve these problems, even the hard ones. This influence of teachers' experience in technology on their ability to solve technological problems is supported by [START_REF] Deluca | Implementing technology education problem-solving activities[END_REF] who claims that technological knowledge overcomes technological problems in the classroom. This could improve pre-service teachers' attitudes toward computers.

Questionnaire 3 :

 3 The Use of ICT in Colleges of Education (UICT): This questionnaire was developed by The MOFET Institute (A Center for the Research, Curriculum and Program Development in Teacher Education in Israel) to track the professional development of pre-service teachers' use of ICT. We used the ICT proficiency part of the questionnaire.

Table 1 : Means, standard deviations and t-test for pre-service teachers' ICT proficiency level (n=54)

 1

	Preparation

Table 2 : Means, standard deviations and t-test for pre-service teachers' TPACK level (n=54)

 2

		Before Preparation	After Preparation
	Outcome	M	SD	M	SD	t	d
	TPACK	3.93	0.53	4.50	0.46	8.19***	1.15
	TK	3.90	0.63	4.44	0.59	6.31***	0.89
	PK	4.04	0.52	4.57	0.46	6.35***	1.08
	PCK	3.81	0.57	4.57	0.49	8.49***	1.43
	TCK	3.78	0.76	4.45	0.52	7.02***	1.05
	TPK	4.02	0.67	4.50	0.63	4.85***	0.74
	TPCK	3.88	0.78	4.51	0.68	5.19***	0.86
	*** p < 0.001

Table 3 : Means, standard deviations and t-test for pre-service teachers' TAC level (n=54)

 3

Proceedings of CERME10