text:
Michiel Klaren
email: m.j.klaren@tue.nl

Birgit Pepin

Marieke Thurlings

Autism and mathematics education

Keywords: Autism Spectrum Disorder, mathematics, geometry, Van Hiele, secondary education, didactics, inclusion

Inclusive education urges educational research to deepen its understanding of students with special needs. The aim of this theoretical paper is to provide an overview of the didactical aspects in the field of autism and mathematics education. First we review literature on autism spectrum disorders (ASD) from a broad psychological perspective, and second, we focus on three cognitive theories which are used to explain the behavioral symptoms of ASD. Next, we discuss mathematics specific didactical issues that relate to these cognitive theories. Finally, we elaborate on an example of research on mathematics education for ASD students based on the Van Hiele model of thinking in geometry. In the conclusions we bring together the findings and give suggestions for future research.

Introduction

Autism is a neurodevelopmental disorder characterized by impairments in social interaction and communication, and by restricted and repetitive patterns of behavior (American Psychiatric Association, 2013). Different combinations of the impairments occur, and together form a continuum, which is called the autism spectrum of disorders (ASD; [START_REF] Wing | The continuum of autistic characteristics[END_REF]. Both the severity of the symptoms and the intellectual capacities of people with ASD vary widely. There is a growing understanding that people with autism can also provide a substantial contribution to society (e.g. [START_REF] Mottron | Changing perceptions: The power of autism[END_REF]. In most countries, and also in the Netherlands, inclusion policies require schools to develop support for children with autism. In this article we will focus on ASD students with higher intellectual abilities 1 and who require limited support, and hence can participate in inclusive secondary education.

Mathematics is a school-subject that builds on logic, is well structured, uses symbolic language with well-defined meanings and deals, at least in its purer forms, with unambiguous questions. These characteristics make mathematics a subject that is relatively easy to access for people with ASD. Research shows that autism is found more often amongst mathematics students than those of other disciplines (e.g. [START_REF] Baron-Cohen | Mathematical talent is linked to autism[END_REF]. However, other studies showed that the majority of ASD students and higher intelligence have an average mathematical ability compared to the normal population, while only some have mathematical giftedness (e.g. [START_REF] Chiang | Mathematical ability of students with Asperger Syndrome and High-Functioning Autism: A review of literature[END_REF].

In secondary schools that include ASD students, teaching mathematics is not an easy task. Based on research and our own experiences of teaching in inclusive schools, we claim that ASD students learning mathematics require specific subject-didactical support. A review of research on autism 1 In literature on autism this also indicated as High Functioning Autism and Asperger Syndrome or HFA/AS [START_REF] Whitby | Academic achievement profiles of children with High Functioning Autism and Asperger Syndrome: A review of the literature[END_REF].

Thematic Working Group 04

Proceedings of CERME10

showed that most studies are about medical or behavioral aspects, and less than 20% targets education [START_REF] Graff | Trends in autism research: A systematic journal analysis[END_REF]. Research on autism in inclusive education is mainly about pedagogic topics concerning the ASD students [START_REF] Ravet | Inclusive/exclusive? Contradictory perspectives on autism and inclusion: the case for an integrative position[END_REF]. However, there is little research on didactical aspects of autism in specific subject fields.

The aim of this theoretical paper is to provide an overview of the didactical aspects in the field of autism and mathematics education. Our research question is the following: Which didactical issues are related to ASD students learning mathematics?

Literature review

Autism spectrum disorders

After the first case studies by Kanner in 1943, andAsperger in 1944, it took until the end of the 1970's, when [START_REF] Wing | Severe impairments of social interaction and associated abnormalities in children: Epidemiology and classification[END_REF] provided their classic description of the triad of impairments of autism: first, the absence or impairment of social interaction, especially with peers; secondly, the absence or impairment of the development of verbal and nonverbal language; and third, repetitive, stereotyped activities of any kind. Confusion about different subtypes lead [START_REF] Wing | The continuum of autistic characteristics[END_REF] to the conclusion that there must be an autistic continuum, which she coined the Autism Spectrum of Disorders (ASD) with diagnoses such as classic autism, PDD-NOS (Pervasive Developmental Disorder -Not Otherwise Specified) and Asperger's syndrome. With the latest version of the diagnostic manual DSM-5 (American Psychiatric Association, 2013) the diagnosis of ASD is based on persistent deficits in social communication and social interaction, and on restricted, repetitive patterns of behavior, interests or activities. The subtypes of PDD-NOS and Asperger are no longer official diagnoses, and DSM-5 describes three levels of severity for ASD (i.e. requiring very substantial support; requiring substantial support; and requiring support).

The psychological research into autism has been dominated by three cognitive theories [START_REF] Rajendran | Cognitive theories of autism[END_REF]: the Theory of Mind deficit; Executive Dysfunction; and the Weak Central Coherence accounts.

The Theory of Mind deficit

Children with ASD experience their social environment as unpredictable and incomprehensible [START_REF] Baron-Cohen | Does the autistic child have a "theory of mind[END_REF]. They seem to treat people and things the same way. Observations showed, strikingly, that children with Down syndrome and a low intelligence developed a normal social competence, whilst children with ASD and higher intelligence did not [START_REF] Baron-Cohen | Does the autistic child have a "theory of mind[END_REF]. To explain this, Baron-Cohen, et al. did research on the Theory of Mind: neurotypical children (non-ASD) are able to impute mental states to themselves and others (in other words, they have a "theory of mind"), whilst children with ASD fail to do so. This "mind-blindness" was shown by the false belief test: a story, played out for the child with dolls, where one doll has a belief about the location of an object that is incongruous with its real location. The test subject is then asked where the doll will look for the object. To answer correctly the test subject should infer the mental state of the doll ("I think she thinks"). A large proportion (80%) of children with ASD incorrectly assumed the doll would look on the real location. To explain the 20% of children with ASD who answered correctly, second-order false belief tests ("I think she thinks he thinks") were developed, and based on those results, it was assumed that a Theory of Mind is not always lacking completely but may be not fully developed in children with ASD. Another problem was that Theory of Mind can be used to explain impairments in play, social interaction and verbal and nonverbal communication, but not for explaining the other characteristics of ASD, such as the restricted interests, obsessive desire to keep things unchanged (rigidity and inflexibility), and so on [START_REF] Frith | Autism: Beyond "theory of mind[END_REF].

Executive dysfunction

Early in the 1990's, [START_REF] Ozonoff | Executive function deficits in high-functioning autistic individuals: Relationship to theory of mind[END_REF] suggested that deficits in the executive functions could explain symptoms of autism such as narrow interests, rigidity and inflexibility.

Executive function is defined as the ability to maintain an appropriate problem-solving set for attainment of a future goal; it includes behaviors such as planning, impulse control, inhibition of pre-potent but irrelevant responses, set maintenance, organized search, and flexibility of thought and action [START_REF] Ozonoff | Executive function deficits in high-functioning autistic individuals: Relationship to theory of mind[END_REF](Ozonoff et al., , p. 1083).

The research by [START_REF] Ozonoff | Executive function deficits in high-functioning autistic individuals: Relationship to theory of mind[END_REF] showed that deficits with executive function where found in both children with classic high-functioning autism and those with Asperger's syndrome (who succeeded on the second-order false belief test). This suggested that deficits in executive function form a primary cognitive deficit in ASD.

Weak Central Coherence

In neurotypical children (non-ASD) the development of information processing is oriented towards extracting the overall meaning from the sensory input. This inclination is called 'central coherence'. [START_REF] Frith | Autism: Explaining the enigma[END_REF] described how this development is different in children with ASD, and she proposed the weak central coherence theory to explain the symptoms of autism. Psychological tests later showed that children with ASD have superior performance on local information processing, but were less inclined to global information processing [START_REF] Happé | Autism: Cognitive deficit or cognitive style?[END_REF]. In people with ASD this is also observed as a preoccupation with details and parts and a failure to understand the meaning of the whole.

Implications of ASD for mathematics education

Although the focus of this paper is on mathematics education, we first address some approaches for ASD students that apply to education in general.

General education

A general pedagogic approach for students with ASD is structured teaching (Mesibov & Shea, 2010). Structure can be provided in the physical environment (e.g. arrangement of the room and the use of visual clues), the sequence of events during the day (e.g. an understandable schedule), the individual tasks (e.g. provide specific information of the goals and the completion criteria) and the grouping of tasks into a work system. Many of these approaches in autism have not been well researched, and research is now addressing the determination of evidence-based practices [START_REF] Reichow | Development of the evaluative method for evaluating and determining evidence-based practices in autism[END_REF].

Mathematics education

Based on a review of 18 studies of mathematical abilities of ASD students with AS/HFA, [START_REF] Chiang | Mathematical ability of students with Asperger Syndrome and High-Functioning Autism: A review of literature[END_REF] found that the majority of the ASD students have average mathematical capabilities and only some ASD students have a mathematical giftedness. Based on these results, Chiang and Lin concluded that an age-appropriate mathematical curriculum can be used, but individual adjustments may be needed to support both relative strengths and weaknesses.

In a review of the literature on academic achievement profiles of ASD students, [START_REF] Whitby | Academic achievement profiles of children with High Functioning Autism and Asperger Syndrome: A review of the literature[END_REF] report that more than half (52%) of individuals diagnosed with ASD have IQs above 70 and for these children, academic goals come within reach. There is a need for appropriate interventions to allow these children to perform up to their potential and obtain meaningful employment. For mathematical abilities, Whitby and Mancil found that computational skills were intact, but applied mathematics capabilities were impaired. Issues with the application of mathematics are possibly due to executive functioning deficits with their organizational and attentional skills that have a negative effect on multi-step problem solving. Deficits in comprehension (both listening and reading) relate to contextual understanding (e.g. word problems) and conceptual understanding (e.g. abstract concepts).

With word problems, ASD students have difficulty choosing the right approach because they have, due to their weak central coherence, difficulty seeing the similarities and the common structure of different examples and exercises. An ASD adjusted didactical approach for solving word problems should address improving reading comprehension, mathematics vocabulary, computation, and everyday mathematical knowledge [START_REF] Bae | Mathematical word problem solving ability of children with autism spectrum disorder and their typically developing peers[END_REF], p. 2206).

In solving mathematical problems ("a question that exercises the mind"; [START_REF] Schoenfeld | Mathematical problem solving[END_REF] ASD students are impaired by executive dysfunction [START_REF] Ozonoff | Executive function deficits in high-functioning autistic individuals: Relationship to theory of mind[END_REF]. In mathematical problem solving ASD students have similar issues as with word problems, but are also expected to face issues with cognitive flexibility, the use of heuristics and the use of meta-cognitive strategies. Positive results have been reported on the effects of cognitive strategy instructions for students with learning disabilities [START_REF] Montague | The effects of cognitive strategy instruction on math problem solving of middle-school students of varying ability[END_REF], and these results may also be obtainable for ASD students. In classroom practice we see that some ASD students develop problem solving procedures of their own, which may work on the initial (simple) problems but cannot be generalized to later extensions. Upon receiving feedback, the rigidity of these ASD students sometimes inhibits them from accepting the time-proven approaches. Feedback is also known to play an important role in self-regulated learning [START_REF] Butler | Feedback and self-regulated Learning: A theoretical synthesis[END_REF]. Improving feedback seeking strategies of ASD students, can support their self-regulated learning, and help to overcome barriers in problem solving.

Conceptual understanding can be a challenge for ASD students: they have, due to their weak central coherence, difficulty integrating information and generalizing previously learned concepts [START_REF] Klinger | Prototype formation in autism[END_REF]. [START_REF] Minshew | Abstract reasoning in autism: A dissociation between concept formation and concept identification[END_REF] found that individuals with high-functioning autism had impaired concept formation. Furthermore, these individuals had difficulty with cognitive flexibility and showed incomplete understanding of learned concepts. Temple Grandin, diagnosed with ASD, describes (2006) how she memorizes as much facts and experiences as possible, and uses an internal search engine to retrieve visual images of prototypes to understand a concept. In general, ASD-students benefit from visualization of abstract concepts, and with their strong root memory, they remember the visualizations as prototypes of the concept.

An example: A research study on learning geometry by ASD students

To illustrate research on subject-specific didactic problems for ASD students, we describe an example from geometry and the cognitive theory of weak central coherence. This example is based on a master thesis of the first author [START_REF] Klaren | Zwakke centrale coherentie en de Van Hiele niveaus[END_REF]. The theoretical/ analytical framework is concerned with Van Hiele's theory, which defines five levels of the learning process that learners are said go through when learning geometry. The five levels are described by [START_REF] Hoffer | Geometry is more than proof[END_REF] as:

-Level 1, Recognition: the student learns some vocabulary and recognizes the shape as a whole; -Level 2, Analysis: the student analyzes properties of the figures; -Level 3, Ordering: the student logically orders figures and understands interrelationships between figures and the importance of accurate definitions; -Level 4, Deduction: the student understands the significance of deduction and the role of postulates, theorems, and proof; -Level 5, Rigor: the student understands the importance of precision in dealing with foundations and interrelationships between structures

Van Hiele posited that learners master the levels in a stepwise manner and always in the same sequence. In other words: the difficulty of questions specific to the successive levels, will rise. [START_REF] Usiskin | Van Hiele levels and achievement in secondary school geometry[END_REF] confirmed the ability of the Van Hiele theory to describe and predict the performance of students in secondary education on geometry. The fifth level was not well operationalized by Van Hiele, and left out of further analyses.

Students with weak central coherence (operationalized as students with ASD), are hypothesized to have an inversion of the difficulty of level 1 and 2. In other words: they are expected to find questions at level 1 ("shape as a whole"; see Figure 1) more difficult than questions at level 2 (analyses of properties).

To test the hypothesis, 81 children with ASD and higher intellectual abilities, age 12 to 17 years, were tested with the geometry test of Usiskin (with the texts translated in Dutch and the original diagrams). Rasch analyses was used to estimate the average difficulty (on a logit scale) of questions at each level.

The results were compared with results found for non-ASD students by [START_REF] Wilson | Measuring a Van Hiele geometry sequence: A reanalysis[END_REF] in a reanalysis of the data of Usiskin. The combined results (Figure 2) show that the estimated average itemdifficulty of questions on level 1 and 2 were indeed shifted for ASD students (level 1 more difficult and level 2 easier) compared to non-ASD students. However, level 1 was not found to be more difficult than level 2 for ASD students. Based on these findings, the (strong) hypothesis of inversion must be rejected, but the test gives some support for the anticipated differences between ASD and non-ASD students regarding the difficulty of the first two levels of Van Hiele. In the didactical practice of geometry teaching, this should raise awareness that visual recognition by ASD students is to be linked to explicit analyses of properties in order to support concept formation.

A peculiar observation was that some students who performed well on questions on higher levels, made unexpected errors in questions on level 1: on questions where the squares were to be pointed out (e.g. Figure 1), they included the rectangles. A possible explanation of this type of errors is the ambiguity of the mathematical language: in Dutch the translation of square is "vierkant" (literal translation: "four sides"). In interviews after the test, students explained their answers by stating that the rectangles had four sides. Possibly these students where not relying on the shape as a whole, but applied rule-based logic in combination with literal understanding of the mathematical concept.

Conclusions

The three cognitive theories described in this paper represent theoretical frameworks that can guide research in autism and mathematics education.

As described by the (lack of) Theory of Mind account, ASD students experience their social environment as unpredictable and incomprehensible. Research on ASD and general educational will address the development of evidence-based practices that support ASD students in their social interactions. Research on mathematics education can develop interventions that aim for using (understanding) and seeking feedback with respect to learning mathematical concepts and skills.

Related to feedback is the use of self-regulated learning. Students with ASD, with their typical weak executive functioning, can be supported by interventions that improve their metacognition and use of strategies and heuristics, especially in the field of mathematical problem solving.

Recent research on perception by people with autism (e.g. [START_REF] Pellicano | When the world becomes "too real": A Bayesian explanation of autistic perception[END_REF][START_REF] Hohwy | The predictive mind[END_REF] is deepening the neurocognitive understanding of the weak central coherence account. Research on mathematics education may benefit from these results, and improve the understanding of concept learning by ASD students.

To summarize, students with ASD have deficits in their social interaction, their contextual and conceptual understanding, and the self-regulation of their executive functioning. In order to allow successful inclusion of ASD students in education, teachers have to apply effective instruction methods to overcome the "mind-blindness" of these students. Research can help define design guidelines for instructional methods in mathematics, which are attuned to the specific needs of ASD students and allow them to see the beauty of mathematics.

Figure 1 :Figure 2 :

 12 Figure 1: Example of question on level 1 (recognition)

Proceedings of CERME10


domain: math shs

text:
Dorit Cohen
email: doritc@cet.ac.il

Anatoli Kouropatov
email: anatolik@cet.ac.il

Regina Ovodenko
email: reginao@cet.ac.il

Maureen Hoch
email: maureenh@cet.ac.il

Sara Hershkovitz
email: sarah@cet.ac.il

Geometric constructions in a dynamic environment (GeoGebra): The case of in-service teachers

Keywords: Geometric construction, dynamic geometry, GeoGebra, concept image, in-service teachers

A unit on dynamic geometric construction was included in a professional development course for inservice mathematics teachers. As a final task in that unit 28 teachers were required to construct a rhombus based on their own choice of given objects and tools, using the dynamic geometry software GeoGebra. Their responses were analysed according to: the choice of given objects; the choice of tools; the explanation and validity test; and the number of different rhombuses they claimed to have obtained. The teachers were found to have different concept images of a rhombus and different conceptions of what constitutes a valid geometric construction. While many claimed to have obtained an infinite number of different rhombuses, differences were observed in the "type of infinite". Recommendations are given for improving the task design to strengthen teachers' mathematical and pedagogical knowledge.

Introduction

From the use of a dynamic environment for geometric construction arise new frames of reference for the idea of construction. In this paper we present data accumulated during an in-service teachers' professional development course on dynamic geometric construction. We report on the data from two viewpoints: 1. The practical viewpointwe will describe teachers' construction methods in the final task in which they were required to construct a rhombus in different ways (each construction according to different given objects). We will discuss issues arising from the construction, both those concerning the nature and validity of the construction and those concerning the teachers' images of a rhombus. 2. The pedagogical viewpointwe will suggest possible ideas for improving the task design in order to hone and strengthen teachers' mathematical and pedagogical knowledge.

Theoretical background

Construction in geometry has a specific meaning: the drawing of geometric figures using only compass and straightedge without measuring angles or lengths (Hartshorne, 2000a). Geometric constructions have been a popular part of mathematics throughout history. Euclid documented them in his book entitled "Elements", which is still regarded as the authoritative geometry reference. In that work, he uses these constructions widely and extensively, and so they have become a part of the geometry field of study. Geometric constructions also provide insight into geometric concepts and give us tools to draw figures when direct measurement is not appropriate [START_REF] Hartshorne | Teaching geometry according to Euclid[END_REF]. For about a decade, geometric constructions were removed from the Israeli curriculum. In 2014 geometric constructions were again included in the geometry curriculum for middle school in Israel. Curriculum decision makers claim (INMMC, 2013) that geometric construction integrates geometric content knowledge with a deductive way of thinking that is essential for geometry proofs.

The professional literature indicates that teaching mathematics using technological tools helps in the process of constructing an abstract knowledge of mathematics, and geometry in particular [START_REF] Lagrange | Technology and mathematics education: a multidimensional study of the evolution of research and innovation[END_REF]. GeoGebra is a powerful dynamic environment that allows testing of countless number of examples; provides effective and convenient tools for confirming or contradicting conjectures; and provides a wide selection of different tools for geometric constructionsfrom digital analogs of compass and straightedge, to shortcut tools, such as automatically drawing a line parallel to a given line [START_REF] Fahlberg-Stojanovska | Compass and straightedge with GeoGebra: Using technology to master mathematics[END_REF]. GeoGebra is also a tool of assessment [START_REF] Bu | Teaching mathematics using GeoGebra; integrating pedagogy and content in teacher education[END_REF]; thinking processes can be observed by examining the construction protocol which can replay step-by-step onscreen. [START_REF] Vinner | On concept formation in geometry[END_REF] focused on the cognitive development of mathematical concepts, and proposed a model of two components: the concept definitionthe verbal description of the mathematical concept, which characterizes the concept mathematically; and the concept imagethe cognitive structure that includes all the examples and the processes related to the concept in the learner's mind. Geometric concepts have a special status: [START_REF] Fischbein | The theory of figural concepts[END_REF] coined the term "figural concepts" and argued [START_REF] Fischbein | Concepts and figures in geometrical reasoning[END_REF] that geometrical figures are characterized by both conceptual and sensorial properties. A geometrical figure is a mental abstract which is governed by a definition. At the same time, it is an image. In geometrical reasoning the two categories of properties should merge absolutely.

In the context of this research, geometric constructions, which are carried out in a GeoGebra dynamic environment, can be seen as figural concepts. We used our personal, not formal definition [START_REF] Vinner | The role of definitions in the teaching and learning of mathematics[END_REF] for geometric construction. To construct a geometric figure means to draw the figure on a computer screen using GeoGebra digital tools in such a way that essential properties of the figure remain invariant under dragging.

The purpose of the study

The aim of the present study is to:

1. Characterise the in-service teachers' geometric constructions on the final task of the unit. 2. Improve the design of the construction tasks in order to increase their contribution to the development of in-service teachers' mathematical and pedagogical knowledge.

Methodology The research population

This research was carried out in the framework of a 30-hour in-service professional development course whose aim was to acquaint middle-school and high-school teachers with a digital program for ninth-grade mathematics students. A part of this course was the geometric construction unit. Since our assumption (which subsequently proved unfounded) was that the teachers were already acquainted with traditional geometric constructions using straightedge and compass, the unit dealt mainly with the use of the dynamic geometry software GeoGebra. When necessary, traditional methods were referred to. The focus of the research was on the responses of 28 teachers to the final task of the unit.

About the final task

Throughout the course the in-service teachers tried out different geometric construction activities using GeoGebra and thus learned to create a valid construction using dynamic tools. Instructions for the final task included the following reminder.

Constructing with dynamic tools is not the same as drawing on a page since the objects (points, segments, etc.) are moveable: you can drag them and observe how other objects change accordingly. Each figure must be constructed so that it retains its characteristics even after other objects are dragged. For example, on constructing a rhombus according to its diagonals, check that after changing the lengths of the diagonals the figure remains a rhombus. This type of construction is called a valid construction.

In the final task the teachers were required to:

1. Construct a rhombus by three different methods, each construction according to different given objects (according to your choice), for example, according to its diagonals. 2. Describe each construction process and prove that it results in a rhombus. 3. State and explain how many different (non-congruent) rhombuses can be obtained by each method.

Note that no instruction was given as to whether the given object is fixed or dynamic. A fixed side will yield an infinite number of rhombuses (same side length, different angles) while fixed diagonals will yield only one rhombus. Clearly if the objects are dynamic an infinite number of rhombuses are possible, but this is a feature of the software and not of the underlying mathematics.

Data collecting and analyzing

Each teacher sent a solution which comprised a picture of the final construction of the rhombus, the GeoGebra file, and explanations and justifications for the construction process. We used interpretative methods for analyzing these data.

Findings: Characteristics of geometric constructions

A qualitative analysis of the teachers' constructions yielded four main categories.

1. The given objects on which the construction of the rhombus was based 2. The construction process itself according to tools used 3. The teacher's explanation and validity test 4. The number of different rhombuses the teacher claimed to have obtained

We present some examples of teachers' constructions, according to the above categories. The teachers' choices of given objects on which to base their construction were varied; for example, one side, two diagonals, one diagonal, an angle, a combination of the already mentioned objects, and other objects (such as area). We will present just two of these choices: one side and two diagonals.

Here are examples from three different teachers who constructed a rhombus according to its side.

Tzila's construction

1. "The construction is according to 4 equal sides, each 5 cm long." 2. Used segments of fixed length, parallel line through a point, intersection point. 3. "I created a rhombus from two adjacent sides each 5 cm long, using parallel lines; that is a parallelogram with all 4 sides equal 5 cm." 4. Infinite number. "Using a circling movement, with point A fixed, in a sort of circling round each time getting another rhombus whose diagonals are changing."

Instead of just writing that the side of the rhombus was given, Tzila added a definition of rhombus, which, like every definition, provided a sufficient condition. Possibly she misunderstood the task, and thought that she had to state the conditions for creating a rhombus, or perhaps her concept image of rhombus is a parallelogram with four equal sides, or perhaps she misunderstood the components of deductive geometry.

Anat's construction

1. A quadrilateral of equal sideseach side of length 2 cm. 2. Used segment of fixed length, circle with fixed radius.

3. "According to the theorem: a rhombus is a quadrilateral with all sides equal." 4. No answer.

Anat's explanation shows that she, like many of the teachers, was confused about the components of deductive geometry (stating she was using a theorem when in fact she was using a definition).

Yaron's first construction 1. Rhombus with side of length a. 2. Used circle with given radius, intersection point.

"

The length of the side of the rhombus is a since all the sides and the radius are length a. The construction is valid according to the dragging test. The construction relies on the principal of the length of the radius of a circle". 4. "There can be an infinite number of rhombuses since although the sides remain equal lengths the diagonals and angles can be changed." Despite Yaron's claim that the diagonals and angles can be changed, he in fact built a rhombus in which the shorter diagonal was equal to the side and so the acute angle of the rhombus was 60 o . He claimed that there are an infinite number of possible rhombuses given the length of a side, which is correct, but does not correspond to what he constructed. This error was repeated by many of the teachers. Interestingly, in another constructiondescribed below -Yaron erroneously claimed an infinite number of possible rhombuses because of the dynamic nature of the given attributes.

Here are examples of two teachers who constructed a rhombus according to its diagonals. If Nora had checked her construction she would have realised that it did not pass the validity test, as seen in figure 5.

Discussion and conclusion

In the discussion we will relate to the two viewpoints mentioned earlier: the practical and the pedagogical.

The practical viewpoint

In the context of the practical viewpoint we will characterise the findings in each of the construction categories. The first category is: Given objects for constructing a rhombus. We have presented two of the teachers' choices of given objects: one side or two diagonals. All the teachers in the course chose the size of their given objects in one of two wayseither fixed size (a number) or dynamic (a parameter). This choice seems to be related to the teacher's conception of geometric constructionwhat is permitted in such a construction. It should be noted that these teachers had not previously learned constructions in a rigorous manner, and in particular, not dynamic constructions. Therefore we relate to their answers as based on intuitions about what is a construction and on geometric knowledge relevant to the specific topic. We suggest that teachers who chose fixed numerical givens have a concrete conception of construction that is related to particular sizes during the construction process, and did not refer to the characters of the figure which is constructed with dynamic tools. Teachers who chose dynamic givens have a broader conception of construction, not connected to size. This choice of dynamic givens corresponds more closely to the instructionssee above description of the final task. In addition, there were some teachers who were confused about defining the given objects, for example, Tzila, as described above.

The second category is: Choice of construction tools. We identified three types of construction process: measuring; using the extensive range of tools supplied by GeoGebra; and using a limited range of tools (cf. [START_REF] Fahlberg-Stojanovska | Compass and straightedge with GeoGebra: Using technology to master mathematics[END_REF].

Construction by measuring involves fixing the size of the required object (side, diagonal, or angle). For example, in Nora's construction she moved two rays to form a right angle. Another teacher created four equal segments using co-ordinate axes.

An example of construction with the extensive range of tools is using the parallel line tool to ensure a parallelogram, as in Tzila's construction. Another example is using the perpendicular bisector tool, as in Yaron's construction.

Construction with a limited range of tools imitates construction with straightedge and compass. For example, Yaron built a circle of variable radius. Anat used both measurement and the limited range of tools (using a circle of radius equal to the given segment of length 2 cm).

The third category is: Explanation and validity check. None of the teachers gave formal proofs of their constructions. However their explanations provided us with an idea of their concept images [START_REF] Vinner | The role of definitions in the teaching and learning of mathematics[END_REF] of a rhombus. They explained their constructions according to their concept image and not according to the definition as given in the national school curriculum: a quadrilateral with all sides equal. A problematic concept image seems to create confusion between the different components of a deductive geometric argument, for example, between a theorem and a definition and between a theorem and a property. This confusion can be seen in the explanations of Tzila and Anat.

We discerned two problematic areas in the teachers' explanations: an incoherent concept image related to mathematical knowledge; and an intuitive conception of construction in a dynamic environment, which seem to be related to the construction process in everyday life, for example Nora's construction which produced a rhombus. This construction did not fulfill the demand that the construction will remain a rhombus also after dragging. This conception resulted in most of the teachers producing an invalid construction. There is not enough information in the data to pinpoint the main causes of these problemsare they connected to missing technological knowledge, or to missing mathematical knowledge, or to a faulty conception of construction, or to a combination of all three? More research is required to clarify this.

The fourth category is: Number of solutions. We identified three types of responses, each claiming an infinite number of solutions: relating to the dynamic nature of the given object, as seen in Yaron's second construction; relating to the mathematical properties of the given object, as seen in Tzila's construction; relating to the position of the given object in the plane. This latter type was seen in the response of a teacher who wrote: "In the plane it is possible to position the rhombus in any place you want".

The pedagogical viewpoint

In the light of these research findings we recommend investigating some aspects of task design in a dynamic environment. On one hand the task design can assist teachers to execute the task, and on the other hand it can persuade them to use this task in their classroom [START_REF] Bu | Teaching mathematics using GeoGebra; integrating pedagogy and content in teacher education[END_REF]. We suggest that it would be worthwhile to design the task on three levels: mathematical-pedagogical, technologicalpedagogical, and reflective.

The mathematical-pedagogical level requires the teacher to define the concept and to construct the figure accordingly. In line with [START_REF] Fischbein | The theory of figural concepts[END_REF] we relate to a geometrical concept as a figural concept and thus its construction has practical meaning. Subsequently the teacher is required to construct the figure based on different sufficient conditions. Such a construction task would enable identification of the teacher's mathematical knowledge and her concept image, and could help to bring that concept image closer to the concept definition.

The technological-pedagogical level requires the teacher to construct the figure at first using the limited range of GeoGebra tools (imitating straightedge and compass) and subsequently using the extensive range (enabling short cuts). We would like to investigate the connection between these two types of construction and the connection between the teacher's image of the geometric concept and her conception of geometric construction.

The reflective level requires the teacher to check the validity of her construction, while considering the meaning of validity. Giving a detailed account of the results of the validity test should be an inseparable part of the task. Such an execution of the task should strengthen the connection between the mathematical and technological aspects. For example, if the construction "collapses"after dragging the dynamic objects the properties of the required figure are not preservedit is important to understand whether the "collapse" is due to a mathematical failure or a technological failure, or a combination of both. Such an analysis should contribute to the development of mathematical knowledge, pedagogical knowledge, and technological knowledge.

We believe that the above characterisation may provide a starting point for further research and may contribute to the technology, pedagogy, and content knowledge framework [START_REF] Koehler | What is technological pedagogical content knowledge?[END_REF] to help teachers integrate technology into their teaching.

Yaron's second construction 1 .

 1 Rhombus according to two diagonals. 2. Used segment, perpendicular bisector, circle, and intersection point. 3. "I based this on the fact that in a rhombus the diagonals are perpendicular and bisect each other, and on the principal of the length of the radius of a circle. The construction is valid according to the dragging test." 4. "There can be an infinite number of rhombuses since the lengths of the diagonals are not fixed. We can lengthen or shorten them (or even only one of them). The rhombus can changeboth its sides and its angles." Yaron constructed diagonals whose length changed dynamically. He did not relate to the fact that each pair of diagonals determines only one possible rhombus. Nora's construction 1. According to two diagonals. 2. Used segments of fixed length. By trial and error changing the angles between the segments. 3. Did not check validity. 4. No answer.

Proceedings of CERME10

Acknowledgment

Our thanks to the members of the mathematics department at the Center for Educational Technology who contributed to this paper: Shoshana Gilad, Guy Hed, and Michal Fraenkel.