Melania Bernabeu
email: melania.bernabeu@ua.es

Salvador Llinares
email: sllinares@ua.es

Mar Moreno
email: mmoreno@ua.es

The evolution of 9-year-old students' understanding of the relationships among geometrical shapes

Keywords: Geometrical thinking, primary school, geometrical shapes, relationships among geometrical figures

We conducted a teaching experiment of ten lessons with third-grade students (9-years-olds) to analyse the evolving understanding of the relationships among geometrical shapes. In the teaching experiment, pupils recognized, described, represented, built and classified two-dimensional figures. Analysis of pre-/post assessments showed some changes in students reasoning about how they began making sense of the relationships among the geometrical shapes. However, the use of formal definitions of geometrical shapes constituted a challenge for students since they imposed linguistics conditions to how the relationships among figures could be established. The results show that understanding the relationships among geometrical shapes is a slow process and depends on the logical terms used in the formal definitions.

Introduction

Students develop the comprehension of geometrical figures in a progressive way, initially, they recognise the figures by perceptual similarity, continue with the recognition of attributes and finally they base their thinking on concept based on the definition [START_REF] Satlow | When is a triangle not a triangle? Young children's developing concepts of geometric shape[END_REF]. Recognizing attributes and relating them to make classifications is considered a relevant aspect in the development of geometrical thinking [START_REF] Clements | Young children's concepts of shape[END_REF][START_REF] Elia | Young children's understanding of geometric shapes: The role of geometric models[END_REF]Levenson, Tirosh, & Psamir, 2011). Classification of geometrical shapes, links to definition and the analyse of attributes of different geometrical shapes to distinguish between critical and non-critical attributes. Classification process can be hierarchical or partitive [START_REF] De Villiers | The role and function of a hierarchical classification of quadrilaterals[END_REF][START_REF] De Villiers | Defining in geometry[END_REF].

By the term hierarchical classification is meant here the classification of a set of concepts in such a manner that more particular concepts form subsets of the more general concepts. […] In contrast to a hierarchical classification there also exists the possibility of a partition classification of concepts. In such a classification however, the various subsets of concepts are considered to be disjoint from one another (de Villier, 1994, p. 11-12).

It has been shown that students have difficulties in the transition from the recognition of the attributes and distinguish critical and non-critical aspects to perform classifications, especially hierarchical classifications of the quadrilaterals [START_REF] Bernabeu | Primary School children's (9 year olds') understanding of quadrilaterals[END_REF][START_REF] Erez | If you can turn a rectangle into a square, you can turn a square into a rectangle": young students' experience the dragging tool[END_REF][START_REF] Gal | To see or not to see: analyzing difficulties in geometry from the perspective of visual perception[END_REF][START_REF] Jones | Providing a foundation for deductive reasoning: Students' interpretations when using dynamic geometry software and their evolving mathematical explanations[END_REF]. To try to better understand this transition in students' primary school, we investigated third-grade students' evolving understanding of the relationships among geometrical shapes, especially to know how pupils understand geometrical figures and establish relationships among these geometrical figures, that is, consider whether a geometrical figure belongs or no to a figure class.

Theoretical perspectives

The relationship between levels of geometrical thinking and figure comprehension

According to Van Hiele's theory [START_REF] Clements | Geometry and spatial reasoning[END_REF][START_REF] Battista | The development of geometric and spatial thinking[END_REF], the development of students' geometrical thinking is divided into progressive levels. The first three levels describe the process up to the comprehension of relationships among geometrical shapes:

 Level 1: Visual. Students identify the figures in relation to their appearance. To recognize figures, they do so through familiar prototypes (e.g. saying that a rectangle "looks like a door"). They are not aware of the properties of geometrical figures. The reasoning of students is dominated by perception.  Level 2: Analytic/Componential [START_REF] Battista | The development of geometric and spatial thinking[END_REF]. Students identify shapes according to their properties and they can characterize the figures by their attributes. [START_REF] Battista | The development of geometric and spatial thinking[END_REF] refined this level proposing three sub-levels. He theorized that students begin with "visual-informal componential reasoning" where they begin to focus on parts of shapes and doing visual, informal and imprecise descriptions. Pupils advance to "informal and insufficient-formal componential reasoning" where they use formal terms and informal descriptions, but they make definitions with unnecessary attributes. Occasionally, students advance to "sufficient formal property-based reasoning" where they make descriptions with formal geometrical terms for all the properties of the geometrical figures. The change from insufficient to sufficient definitions is a gradual shift from seeing the figures as a whole to focusing on the parts of these to see the figure as a coherent structure [START_REF] Pegg | Interpreting student understanding in geometry: A synthesis of two models[END_REF].  Level 3: Abstract/Relational. Students identify relationships among shapes and give informal arguments to justify their classifications (e.g., a square is a special type of parallelogram). They can discover properties of kinds of figures by informal deduction. Therefore, it is considered that the hierarchical classification is at this level.

The dual nature of geometrical figures [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF] developed the ideas of concept image and concept definition. Concept image is used to describe, "the total cognitive structure that is associated with the concept, which includes all the mental images, properties, and associated processes" (Tall & Vinner, 1981, p.152). Concept image may include images that are inappropriate and contradict the definition of the concept (Hershkowitz, Bruckheimer, & Vinner, 1987;[START_REF] Vinner | The role of definitions in the teaching and learning of mathematics[END_REF][START_REF] Levenson | Preschool Geometry. Theory, Research and Practical Perspectives[END_REF]. On the other hand, concept definition is "a discursive description to specify that concept" (Tall & Vinner, 1981, page 152) accepted by the mathematical community. One relevant aspect in the relationship between concept image and concept definition is to become to recognize the attributes of the definition as a sufficient condition. This dual nature of geometrical shapes in which concept and image are interrelated is underlined by the notion of "figural concept" [START_REF] Fischbein | The theory of figural concepts[END_REF].

In the understanding of this dual nature of geometrical figures, the sufficient conditions to recognize or to represent examples of geometrical shapes can be revealed by the transformations among various semiotic representations. Duval (1995a[START_REF] Duval | Representation, vision and visualization: Cognitive functions in mathematical thinking. Basic Issues for learning[END_REF] indicates how his theory of registers of semiotic representation can be used as a tool to analyse the cognitive processes through which pupils develop the geometrical thinking. [START_REF] Duval | Understanding the Mathematical Way of Thinking -The Registers of Semiotic Representations[END_REF] indicates that we can analyse the students' productions in terms of registers regarding the coordination between different apprehensions: perceptual, discursive and operative. Perceptual apprehension is the ability to recognize a figure in a global way; the discursive apprehension is the ability to link mathematical properties with the parts of the figure; and operational apprehension is the ability to modify or construct a figure to solve the task (change orientation, decompose it). [START_REF] Duval | intellectuels [Semiosis and human thought: semiotic registers and intellectual learning[END_REF][START_REF] Duval | A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics[END_REF] defined two of such transformations: Treatment and Conversion. Treatments are transformations of representations that happen within the same register: for example, completing a figure using perceptual criteria of connectivity or symmetry. Conversions consist of transformations of representations from one register to another (such as from the visual register to the discursive one), without changing the objects that denote.

The association of a figure with mathematical statements, can be done in both directions: from the visual to the discursive (from the figure to the definition) and from the discursive to the visual (from the definition to recognize the geometrical figure) [START_REF] Duval | intellectuels [Semiosis and human thought: semiotic registers and intellectual learning[END_REF]. This translation is the one that allows linking the process of defining a geometrical figure giving a minimum set of relevant attributes. However, there are no descriptors of the cognitive processes through which students develop the geometrical thinking. Therefore, we propose to identify descriptors of the development of geometrical thinking about geometrical figures in 9-year-old pupils, and specifically,  How students understand geometrical figures and,  How reasoning about the relationships among them?

Method Context

Participants were fifty-nine third-grade pupils (9-years-old) from an elementary school located in an area of average socioeconomic status in Spain. This study was conducted as a teaching experiment designed to allow pupils establish relationships among geometrical shapes. A researcher of the research group performed the role of the teacher in both classes during the instruction.

Teaching experiment

In each class, we carried out ten sessions, two per week from May to June during the academic year 2017-2018. Pupils were asked for: recognizing and analysing attributes of different shapes, identifying critical and non-critical attributes, establishing relationships between attributes of shapes. They also had to represent and/or construct the shapes according to the given conditions or classify attending to criteria. Contents used during all sessions are shown in Table 1. During the sessions, pupils built different shapes with resources such as mecano or geoboard. For working the classification tasks, we used digital board, so students could share their resolution with the whole group. Next, students individually solved other tasks. We used hierarchical definitions. For example, we defined the isosceles triangle as a triangle with two equal sides. In this case, we consider that an equilateral triangle is an isosceles triangle since it has two equal sides. Furthermore, we defined a trapezium as a quadrilateral with one pair of parallel sides. In this case, a parallelogram is a trapezium (British version) [START_REF] Fujita | Learners' understanding of the definitions and hierarchical classification of quadrilaterals: Towards a theoretical framing[END_REF].

For this report, data come from the initial and the final test. The aim was to get evidences about pupils' understanding of geometrical figures. Students' answers were codified and agreed by the research group. For the analysis of the data and the creation of the scheme of emergent codes that we generate from the answers of the students, we rely on van Hiele levels [START_REF] Clements | Geometry and spatial reasoning[END_REF][START_REF] Battista | The development of geometric and spatial thinking[END_REF]Duval's theory (1995aDuval's theory ( , 1995bDuval's theory ( , 1999) ) that considers the transformations of the registers and the coordination of the apprehensions, the relationships between image concept and definition concept [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF], and the way that geometrical figures were related (to classify a shape in different ways and label it with different names) [START_REF] Erez | If you can turn a rectangle into a square, you can turn a square into a rectangle": young students' experience the dragging tool[END_REF].

Results

The code schemes

The emergent coding scheme changed to accommodate the features that pupils included in their work and regarding the specificity of the tasks (Table 2). We compared the responses to the initial and final test and developed the emergent coded schemes as follow:

 Students identified geometrical figures by their appearance: what that it means that the recognition of geometrical figures is purely perceptual. For example, concerning to task 5: "It is a normal and ordinary triangle"; "The same figure as the " figure a"; "All the figures are triangular"; and in the Task 6: "Yes, because it is the same".

 Students recognized some attributes of geometrical figures without establishing relationships among attributes that define the general set: that it, students could recognize some attributes of geometrical figures as number of sides, names of figures or types of figures, but they still made mistakes when they related them to its definition and perform relationships among geometrical figures. The apprehensions are not still coordinated correctly. For example, in task 5: "It has three vertices"; "It is a triangle"; and in task 6: "It has four sides"; "It is a square".

 Students were able to establish relationships among figures by recognizing that a figure belonged to a general set. That is it, pupils related the definition to the parts of plane shapes to realize relationships (hierarchical and partitive relations). They began to coordinate the apprehensions. For example, in the hierarchical relation, concerning to task 5: "Yes, because an equilateral is an isosceles" and in task 6: "Yes, because at least has 2 parallel sides". Conversely, regarding partitive relation, in task 5 a pupil said: "No, because an equilateral is not an isosceles" in task 6 another said: "No, because it has all parallel sides" (not including the parallelogram as a trapezium).

Initial and Final test: Progression of thinking about relationships among geometrical shapes

To report the progress of the understanding of plane geometrical figures, we present the answers to two tasks from the initial and final test. In both tasks we had a "Drawing Machine" that could make a set of figures with common conditions [START_REF] Battista | Cognition-Based Assessment & teaching of geometric Shapes: Building on Students' Reasoning[END_REF]. In one task, we provided a set of isosceles triangles (a triangle with two sides equals) that the machine could make and a set of scalene triangles (triangles with the three sides different) that the machine could not make. The question was that if the Drawing Machine (Figure 1) could draw an equilateral triangle (visual example). In another task, a verbal description of the shape was given: The Drawing Machine can make "quadrilaterals (polygon with four-sided) with two parallel sides" and it was asked if the parallelogram (square) (visual example) could be made by the machine (Figure 2). These tasks tried to show how students establish relationships among figures and use the logical relationships among the attributes identified in the figure . In this paper, we present the frequencies of codes of both tasks (118 answers, that is, 59 of each task) that reflect the progress from perceptual to analytic recognition, and the initial understanding of the relationships among figures (Table 2). The frequency of the students' answers has shown a development in the reasoning of elementary school pupils about geometrical figures and it has also shown how they reason about the relationships among these figures.

Codes

The development of the understanding of geometrical figures

Analysis of data has shown that students have modified their way of reasoning from the initial test to the final. First, the purely perceptual recognition decreased from 22% to 10% after the teaching experiment. Also, the recognition of attributes in the figures presented in both tests increased from 24% to 41%. On the other hand, concerning the relationships among geometrical figures, it has been possible to verify how in the final tests, these relationships have been accentuated, since in the hierarchical relations had 8% in the initial test, and this rose to 22% in the final test. Regarding partisan relationships, these results rose by 2% in the initial to 6% in the final test. On the other hand, another positive data is the decrease of the frequencies in the code of "others", in which we coded answers that did not make sense with what was demanded in the statement or this was blank. This code decreased by 44% in the initial test to 22% in the final. In conclusion, the frequencies of the final test, compared with the initial test, provide us with information to intuit that pupils begin to develop their understanding of geometrical figures.

Reasoning about the relationships among geometrical figures

Initially, 12% of the answers were linked to the relationships among geometrical figures to the general set provided in the tasks. However, after the teaching experiment this percentage increased to 33%, where it was intended to develop the recognition of the critical attributes of the figures so that they could make relationships among geometrical figures. Of this total, regarding the hierarchical relations, the answer increased from 8% to 22%. Regarding to partitive relations, it increased from 2% to 6%. These data lead us to think that students begin to develop the relationships among geometrical figures.

Discussion

The aim of this study is to know how students understand geometrical figures and how they reason about the relationships among them. We used a teaching experiment to check how geometrical thinking evolved if we introduced vocabulary and definitions related to hierarchical relationships.

As well as, we analysed two tasks, from an initial and a final test, to verify this development of the understanding of geometrical figures. Data has shown us that many pupils answered the tasks perceptually without considering the relationships of the figures or the definition of the concept, in the initial test. In contrast, after the teaching experiment, in the final test, most of pupils began to identify attributes to the plane geometrical figures and to reason about the relationships among them. These data lead us to think that the development of the understanding of geometrical figures is progressive [START_REF] Pegg | Interpreting student understanding in geometry: A synthesis of two models[END_REF][START_REF] Satlow | When is a triangle not a triangle? Young children's developing concepts of geometric shape[END_REF].

From these data, we have been able to begin to comprise the progress of the understanding of the plane geometrical figures in relation to Duval's theory (1995aDuval's theory ( , 1995bDuval's theory ( , 1999)). Because students at the beginning did not know how to coordinate the apprehensions and after the teaching experiment, they began to coordinate the apprehensions making the necessary transformations, the relationships among attributes and the use of minimum attributes of the definition for doing the relationships among geometrical figures. In this way, while developing an understanding of geometric figures and how to relate them. They began to develop the personal figural concept [START_REF] Fujita | Learners' understanding of the definitions and hierarchical classification of quadrilaterals: Towards a theoretical framing[END_REF]. Thus, we can intuit that the realization of the hierarchical classifications depends on the language that is used in the definitions of the geometrical concepts, which can help identify the minimum attributes to recognize a type of geometrical figure.

  session. Acute, right and obtuse angle Third week Fifth session. Types of triangles according to their sides: equilateral, isosceles and scalene triangle. Hierarchical relations: the equilateral is an isosceles Sixth session. Types of triangles according to their angles: acute, right and obtuse triangle Fourth week Seventh session. Types of triangles according to their sides and angles Eighth session. Types of lines: parallel and non-parallel. Types of quadrilaterals: parallelograms, trapezes and trapezoids Fifth week Ninth session. Types of parallelograms: square, rhombus, rectangle and rhomboid. Hierarchical relations: the square is a rhombus, and the square is a rectangle Tenth session. Properties of parallelograms: equal opposite sides and angles; diagonals cut in half

Figure 1 .

 1 Figure 1. Figures of Triangles in the Task 5

Figure 2 .

 2 Figure 2. Figure of Quadrilateral in the Task 6

Table 1 . Conceptual characteristics of the sessions of the Teaching Experiment

 1

Table 2 . Frequency of codes

 2

Acknowledgment

This research has been financed by Prometeo/2017/135 from Generalitat Valenciana (Spain).


Panagiotis Gridos
email: p.gridos@hotmail.com

Athanasios Gagatsis
email: gagatsis@ucy.ac.cy

Iliada Elia
email: iliada@ucy.ac.cy

Eleni Deliyianni
email: deliyianni6@hotmail.com

Mathematical creativity and geometry: The influence of geometrical figure apprehension on the production of multiple solutions

Keywords: Creativity, geometry, multiple-solution tasks (MSTs), geometrical figure apprehension

The aim of the study is to investigate the creativity demonstrated by students in multiple-solution tasks (MSTs) in geometry taking into account the role of geometrical figure. To this end, the test that was administered among 149 eleventh graders consisted of two types of problems: a. with and b. without the relevant figure. Findings revealed that fluency and flexibility were higher when the verbal description of the problem was accompanied by the relevant figure. The originality of students' solutions, though, did not differ in the two types of problems. The similarity analysis suggested distinct student profiles: (a) poor fluency and flexibility, (b) moderate fluency and flexibility and (c) high fluency and flexibility. Although the majority of students belong to the first two groups, qualitative results provide evidence that students managed to improve and gain high fluency and flexibility. Findings also indicated that geometrical figure apprehension is a prerequisite for high levels of creativity in geometry.

Introduction and Theoretical Framework

Creativity has been proposed as one of the major components to be included in Mathematics education, since "the essence of Mathematics is thinking creatively" (Mann, 2006, p. 239). [START_REF] Torrance | Creativity: Just wanting to know[END_REF] defines creativity as multidimensional; fluency, flexibility, originality (or novelty) and elaboration are aspects of creativity. Fluency is related to the flow of ideas, flexibility has to do with the ability to shift between different ideas, novelty is associated with the originality of the individual's ideas or products, and elaboration is associated with the individual's ability to describe, illuminate, and generalize those ideas. A number of researchers (e.g. [START_REF] Silver | Fostering creativity through instruction rich in mathematical problem solving and problem posing[END_REF][START_REF] Stupel | Using multiple solutions to mathematical problems to develop pedagogical and mathematical thinking: A case study in a teacher education program[END_REF] suggest that linking mathematical ideas using multiple approaches to solve problems (or prove statements) is essential for the development of mathematical reasoning and fosters better comprehension and increased creativity in Mathematics. Following Torrance, Silver (1997) claimed that in order to develop creativity by means of problem solving it is necessary to advance three components of creativity. Fluency is promoted by raising multiple ideas for the solution of one problem. Flexibility is raised when after one solution is already in hand, the individual searches for more possible solutions. Originality is developed when an individual succeeds in generating a new solution, in addition to those known to him/her at the time.

Geometry provides opportunities for investigation and proving activities that resemble the work of mathematicians [START_REF] Herbst | Establishing a custom of proving in American school geometry: Evolution of the two-column proof in the early twentieth century[END_REF]. The usefulness of geometrical figures in problem solving in geometry is beyond doubt as they provide an intuitive presentation of all constituent relations of a geometrical situation [START_REF] Duval | Geometrical pictures: Kinds of representation and specific processing[END_REF]. [START_REF] Duval | Geometrical pictures: Kinds of representation and specific processing[END_REF] distinguishes four apprehensions for a geometrical figure: perceptual, sequential, discursive and operative. To function as a geometrical figure, a drawing must evoke perceptual apprehension and at least one of the other three. Each has its specific laws of organization and processing of the visual stimulus array. In particular, perceptual apprehension refers to the recognition of a shape in a plane or in depth. Perceptual apprehension indicates the ability to name figures and the ability to recognize several sub-figures in the perceived figure. Sequential apprehension is required whenever one must construct a figure or describe its construction. The organization of the elementary figural units does not depend on perceptual laws and cues, but on technical constraints and mathematical properties. Discursive apprehension is related to the fact that mathematical properties represented in a drawing cannot be determined through perceptual apprehension. In any geometrical representation the perceptual recognition of geometrical properties must remain under the control of statements (e.g. definitions). However, it is through operative apprehension that we can get an insight into a problem solution when looking at a figure. Operative apprehension depends on the various ways of modifying a given figure: the mereologic, the optic and the place way. The mereologic way refers to the division of the whole figure given into parts of various shapes and the combination of them in another figure or subfigures (reconfiguration), the optic way is when the figure is made larger or narrower, or slant, while the place way refers to its position or orientation variation. Each of these different modifications can be performed mentally or physically through various operations. These operations constitute a specific figural processing which provides figures with a heuristic function. One or more of these operations can highlight a figural modification that gives an insight into the solution of a problem in geometry.

MSTs contain an explicit requirement for solving them in multiple ways. A number of studies (e.g. [START_REF] Levav-Waynberg | Multiple solutions to a problem: A tool for assessment of mathematical thinking in geometry[END_REF] used MSTs as an instructional tool for developing knowledge and creativity, as well as a tool for the evaluation of creativity in geometry. Moving a step forward, the present study aims to shed new light as far as MSTs are concerned on evaluating and developing creativity in geometry taking into account the role of geometrical figure. As there is a need for a multidimensional approach as far as teaching and learning in geometry is concerned, we address the following research questions (RQ):

Methodology

A written test was administered among 149 eleventh graders (77 boys and 72 girls) as follows: 147 of the students completed the test in usual classroom conditions and 2 students took part in an individual interview during the solution process of the test. The test consisted of four geometry problems that were content and face validated by two experienced Mathematics school teachers in secondary education and two professors of Mathematics Education. The students were asked explicitly to solve them in as many ways as possible. In problems 1 and 3 the wording of the problem was accompanied by the relevant figure, while in problems 2 and 4 only the wording of the problems was given. The duration of the test was 110 minutes. It is worth mentioning that the problems (without the relevant figure) were piloted among a small number of students before the final administration of the test. It was confirmed that all the problems (without the relevant figure) had the same level of difficulty.

Figure 1. The four geometry problems

The solutions provided by the students were classified in view of the strategy they used. Then, the aspects of creativity (fluency -flexibility -originality -total creativity) were calculated based on the scoring scheme that [START_REF] Leikin | Exploring mathematical creativity using multiple solution tasks[END_REF] proposed for the evaluation of performance on MSTs. The construct of solution spaces that will be explained in the next section was also used to analyze the student problem-solving performance [START_REF] Leikin | Habits of mind associated with advanced mathematical thinking and solution spaces of mathematical tasks[END_REF]. In order to answer the research questions, descriptive statistics and t-test at 95% significance level were used. Furthermore, a similarity statistical analysis was conducted using the computer software C.H.I.C. It is a method of analysis that determines the similarity connections of the variables [START_REF] Gras | Statistical implicative analysis[END_REF]. On the similarity diagram we symbolize with Pi, i=1, 2, 3, 4 the problem referred to and symbolize with Sj, j=0, 1, 2, 3, 4 the number of solutions for each problem. The interviews with the two students and one of the authors were transcribed and examined qualitatively.

Results

In order to examine the influence of geometrical figure apprehension in the development of mathematical creativity, five solutions which were provided by students in problem 2 are presented in Figure 1. These solutions are grouped into three solution spaces based on the strategies and properties that students used. In the first solution space, the main strategy is that the straight line joining the middle of the two sides of a triangle is parallel and equal to half of the opposite side, and includes two solutions (1a, 1b).The first solution was found by 8.1% of the students while the second solution which requires drawing an auxiliary line was only found by the 2.7% of students; the originality of this solution space rated 10 (10.8% <15%). In the second solution space, the main strategy is that equal corners were formed, and refers to two solutions (2a, 2b). These solutions were provided by the 28.4% of students as it did not require any auxiliary construction. Its originality rated 0.1 (56.8%> 40%). The third solution space includes one solution (3) that refers to the construction and the use of the rhomb properties. This solution which requires constructing an auxiliary line was provided only by 1.4% of the students and the originality rated 10 (1.4% < 15%). As far as the fluency for problem 2 is concerned, 41.5% of the students were unable to solve the problem, while a similar percentage of students (49%) found only one way of solving it. The percentage of students who were able to find 2 or 3 ways was quite low (9.5%). Students who produced solutions 1a, 2a and 2b possess at least three kinds of geometrical figure apprehension. In particular, the students apprehend sequentially the figure as they construct the figure from scratch by organizing its elementary units through constructive constraints and mathematical properties. Furthermore, the students have a perceptual apprehension, since they recognize the various subfigures in the circle and the inscribed triangle. They also have discursive apprehension, since they prove the solution of the problem based on theorems. In order to produce solutions 1b and 3, except the other three types of geometrical figure apprehension, operative apprehension is required as well, since the students have to construct the auxiliary lines for the construction of the median of the triangle and the construction of the rhomb to find the solution of the problem. It is evident that a geometrical figure can be constructed in many ways (mereologic modification), each of which can lead the student to a different solution of the problem. Nevertheless, from the solutions given by the students we observed that some students construct the same auxiliary line but they solve the problem differently. In fact, students apprehend operatively the figure in the same way but have different discursive figure apprehension as for the same auxiliary structure they use different properties and theorems to find the solution.

On the basis of the results of the t-test, the number of solutions (fluency) that students give to problem 1 and 3 (M = 2.16, SD = 1.42), in which the problems are accompanied by the relevant figures, are statistically bigger (t = 6.02, df = 146, p <0.001) than those that students give to problems 2 and 4 (M = 1.44, SD = 1.21) in which the figures are not given. Concerning flexibility of the solutions to the problems in which the figure is given (M = 19.88, SD = 12.65), and the problems without the corresponding figure (M = 13.93, SD = 11.37), significant differences are indicated with an apparently higher score in the problems in which the figure is given (t = 5.57, df = 146, p <0.001). Even though according to the findings students' fluency and flexibility is higher in problems in which the figure is given, this is not the case with the originality of the solutions (t = -1.87, df = 146,p> 0.05) and total creativity (t = -0.71, df = 146, p> 0.05).

In fact, there are not any significant differences as far as originality and creativity are concerned in the two types of problems. given or not. In the second sub-cluster the solutions belong to one solution space. This sub-cluster is associated with the variables P3s2, P3s3 that involve students who have solved the third problem in which the figure is given, in two or three ways, providing evidence that the figure affected its level of difficulty.

Cluster 2 contains the variables P1s2, P1s4, P2s3, and P3s4 that correspond to the students who are highly fluent and flexible in their solutions. On the one hand, these students give two, three or four solutions which mean they have the fluency to produce multiple solutions. On the other hand they are able to move from one solution to another, using different mathematical properties and representations as their solutions belong to at least two different solution spaces.

In fact, the results suggested that the participants of the study form three distinct student profiles: 172. St: Could I construct a circle as we did in the forth problem? Because the right angle ends in a semicircle, let's say that BC will be a diameter and the center will be D. So DC, DB and DA will be radiuses, so they will be equal.


It is evident that, MSTs enhance flexibility in geometric knowledge and student thinking as they encourage connections between different ways of solving them. The student, as soon as he had solved the problems, moved from zero fluency and flexibility during the solution of the first problem to high fluency and flexibility of solutions at the end of the interview. During this process he often made references to the previous problem solutions. It is worth mentioning, that the qualitative data strengthen quantitative results as the construction of auxiliary lines and reconfigurations that reflect the operative figure apprehension revealed to be the most important aspects for producing new ways of solving and achieving high levels of creativity in geometry.

Discussion

The present study explored students' creativity in MSTs using a multidimensional approach as the following dimensions are considered: (a) mathematical creativity; (b) geometrical figure apprehension; and (c) multiple-solution problem solving in mathematics education. Significant differences in the fluency and flexibility of student solutions were observed depending on whether or not the figure was given (RQb). In fact, students exhibited greater fluency and flexibility in the problems in which the verbal description of the problem was accompanied by the relevant figure. As far as the originality and the total creativity of students' solutions are concerned, they did not differ significantly in the two types of problems. On the one hand these findings are explained taking into account that even though flexibility and fluency are different aspects of creativity, they are strongly related to each other (e.g. [START_REF] Leikin | Exploring mathematical creativity using multiple solution tasks[END_REF]. On the other hand, researchers (e.g. Levav-Waynberg & Leikin, 2012[START_REF] Leikin | Evaluating mathematical creativity: The interplay between multiplicity and insight[END_REF] pointed out that originality is a more internal characteristic than fluency and flexibility. It is also more related to creativity but less dynamic.

The similarity analysis of the number of student solutions suggested three distinct profiles of students: (a) a poor fluency and flexibility group; (b) a moderate fluency and flexibility group; and (c) a group of high fluency and flexibility (RQc). The majority of students belong to the first two groups. However, qualitative results provide evidence that students managed to move from poor to high fluency and flexibility of solutions through references to previous problems they had solved. This is in line with [START_REF] Leikin | Evaluating mathematical creativity: The interplay between multiplicity and insight[END_REF] findings that fluency and flexibility are dynamic, whereas originality is a "gift". As our work deals with relative creativity [START_REF] Leikin | Exploring mathematical creativity using multiple solution tasks[END_REF], we support that mathematical creativity should be developed in all students [START_REF] Sheffield | Developing mathematical creativity -Questions may be the answer[END_REF]. As a matter of fact, creativity is influenced by the teaching, guidance and experiences of the individual [START_REF] Silver | Fostering creativity through instruction rich in mathematical problem solving and problem posing[END_REF]. Thus, the systematic exploitation in the teaching of MSTs in geometry may enhance student fluency and flexibility.

Furthermore, emphasis should be given to activities that help students overcome the perceptual way of looking at a geometrical figure and moving to operative figure apprehension. The results suggest that the students who apprehend the figure operatively are able to undertake mereologic modifications that enable them to provide different proofs in geometrical problems (RQa). An interesting finding which highlights heuristic functioning of figures is that some students use the same auxiliary construction but their solutions are based on different properties and theorems. In other words, they have the same operative apprehension but different discursive apprehension of a figure. Indeed, operative apprehension of a figure is a source of creativity in geometry as it gives insight into the solution of the problem. Nevertheless, [START_REF] Duval | Geometrical pictures: Kinds of representation and specific processing[END_REF] indicated that the mathematical way of looking at figures only results from the coordination of separate processes of apprehension over a long time. In line with this assertion, our findings revealed that geometrical figure apprehension is a prerequisite for students to provide a variety of solutions to the geometry problems and, thus, have high creativity.

Certainly, there is still need for further investigation into the teaching implications of the subject. In the future, it would be interesting and useful to examine the effects of intervention programs aiming at developing geometrical figure apprehension in view of MSTs. Furthermore, the validation of a theoretical model that examines the structure and relation of various components of geometrical figure apprehension, mathematical creativity and ability to prove in multiple ways would contribute to this domain of research too.

  (a) what is the influence of geometrical figure apprehension on the production of multiple solutions in geometry problems? (b) what kind of creativity (fluency, flexibility, originality) do students demonstrate when working with MSTs in which the geometrical figure is either given or not in the wording of the problem? (c) what are students' profiles, if any, in terms of the number of solutions in MSTs in geometry?

Problem 1 :

 1 Prove in as many ways you can that, that the median drawn to the hypotenuse of the right triangle equals half the hypotenuse. Problem 3: Let a rectangular triangle ABC (B = 90°), D midpoint of segment AB, E the midpoint of segment AC and Z the midpoint of segment BC. Prove in as many ways you can that DZ = EB. Problem 2: AB is a diameter on a circle with center O. D and E are points on circle O so that DO//EB.C is the intersection point of AD and BE. Prove in as many ways as you can that CB=AB. Problem 4: Find the center of a circle in as many ways as you can, if you only know the circumference of the circle.

1a

  Solution 8.1% DO=1/2AB, as equal radiuses in a circle. ⇒DO is a midline in triangle ABC, as parallel to BC and bisecting AB.⇒DO=1/2AB=1/2BC⇒ AB=BC. 1b Solution 2.7% D is the mean of the AC, since DO joins the medium C with D and OD//ΒC. We construct the median DB of the triangle. The angle D1 goes to a semicircle so D1 = 90 o . Thus, DB is both median and altitude of the triangle ABC. Therefore, ABC is isosceles and AB = CΒ. 2a Solution 28.4% DO=AO, as equal radiuses in a circle.⇒∠ADO = ∠A, as base angles in an isosceles triangle, ∠ADO = ∠ACB, as equal corresponding angles within parallel lines, ∠ ACB= ∠A.⇒ AB=BC, a triangle with 2 equal angles in isosceles. 2b Solution 28.4% DO=AO, as equal radiuses in a circle.⇒∠AOD=∠ABC, as equal corresponding angles within parallel lines.⇒∠A = ∠A, as shared angle⇒⊿AOD ~ ⊿ABC (2 equal angles) ⇒AB=BC (a triangle similar to an isosceles triangle is also isosceles). 3 Solution 1.4% We construct a parallel line from D to AB. Taking into account that it passes through the median D of the AC, it also passes from the median M of BC and MD = AB/2. EB // DO ⇒ΜΒ //DΟ. Therefore, MB = DO = BC / 2 and OD = ΟΒ as the radius of the circle. Then, DOBM is a rhombus⇒ OB = ΜΒ⇒ 1/2AB = 1/2ΒC⇒ AB = ΒC.

Figure 2 .

 2 Figure 2. Collective solution spaces for problem 2

Figure 3 .

 3 Figure 3. Similarity diagram of students' number of solutions In the similarity diagram, which concerns students' ability to produce various solutions, there are two significant clusters. Cluster 1 consists of two sub-clusters. The first sub-cluster includes the variables P1s0, P2s0, P3s0, and P4s0. It refers to students who dο not give any solution to any of the problems, regardless of the presence of figure or not. The second sub-cluster (P1s1, P2s1, P3s1, P4s1) is related to students who solve all problems in one way, regardless of whether the figure is

  Profile 1: Students with poor fluency and flexibility (0 solutions)  Profile 2: Students with moderate fluency (1-3 solutions) and flexibility (1 solution space)  Profile 3: Students with high fluency( 2 solutions) and flexibility( 2 solution spaces) The majority of the students have poor and moderate fluency and flexibility. In particular, fifty students (34%) have poor fluency and flexibility in at least one problem, sixty six students (45%) provide only one solution in at least one problem and twenty five students (17%) solve one of the problems in two different ways at least. Only six students (4%) have high fluency and flexibility in at least one problem. The qualitative data provided further insight in to students' potential performance. During the solution process, changes concerning fluency and flexibility of student solutions were observed. Τhe following extract from one of the student's interview (Student A) is indicative:: 10. St: I'm not so good at geometry. 17. St: Better find another student who knows them better. I do not remember them at all. 19. St: Okay, I leave the first problem and go to the third problem. (Having already found two ways in the first problem after solving the other three problems) 165. St: Let me think… I have an idea, half a minute to see something in the third problem. 166. St: Construct a parallel line from C to AB and from B to AC and create a rectangle for A = 90 o (the student is drawing the new figure). From this I will say that the diagonals of the rectangle are equal and bisected. Therefore, AD = 1 / 2BC. 170. St: I thought of something else.


Alexander Esper
email: esper@ikdg.rwth-aachen.de

Christian Lenzen

Manfred Wirsum

S Y Mpo

O Tatin

Commissioning of a Test Stand for Turbocharger Investigations at Constant Turbine Inlet Temperatures

Keywords:

In this paper, the developed turbocharger test stand design at the IKDG facility is evaluated. e specific characteristics of the test stand as well as operational test data is discussed. A test turbocharger is thermodynamically examined in regard of operation at 220 • C turbine inlet temperature. An overview of the measurement instrumentation is given and the thermodynamic parameters of the turbocharger necessary for generating a power map, are presented. e key aspect of providing a constant turbine inlet temperature nessesary for the acquisition of reliable perfomance measurements is varyfied by means of an operation point stability analysis at different rotational speeds across the whole operating range of the test turbocharger. Additionally, an uncertainty analysis of the temperature and pressure measurement instrumentation according to DIN V ENV 13005 and [ ] is conducted. Based on these results, a performance map of the compressor as well as the turbine of the turbocharger is generated and discussed. e paper is concluded by providing an outlook regarding the possibilities of upgrading the performance measurement instrumentation in the exisitng test stand design as well as upgrading certain parts of the test stand.

INTRODUCTION

Design parameters of state of the art turbochargers evolve towards slender rotating components due to cost and operation mode efficiencies. Slender rotating components increase transient response characteristics due to lower inertia and therefore, improving performance and efficiency. To evaluate the operational characteristics of a turbocharger at different turbine inlet temperatures, a test stand was developed at the Institute for Power Plant Technology, Steam and Gas Turbines (IKDG) to enable investigations of performance maps at selectable constant turbine inlet temperatures. To reproduce real time applications where turbochargers are implemented downstream the exhaust outlet of an engine, a lot of test stands use loop operation where the compressed air of the turbocharger compressor passes a combustion chamber upstream of the turbocharger turbine [ , ]. By decoupling the mass flows of turbocharger compressor and turbine, a wider range of the performance map can be investigated. Furthermore, this simplifies operating the turbine at constant inlet temperatures. erefore turbocharger test stands have been developed on which the turbocharger turbine is operated without using the compressed air mass flow from the turbocharger compressor [ , ]. A significant difference between these test stands and the test stand introduced in this paper is the use of two hot gas generators that run sequentially enabling the adjustment of the turbine inlet temperature with a maximal deviation of about +/-1.0K during steady state operation. is allows an operation of the turbocharger close to a thermodynamic equillibrium operation to maximize perfor-mance measurement accuracy compared to conventional test stands using combution chambers as hot gas generators. Further advantages in the use of an adjustable electrical heater is the simpler control of the heater, the uniform warm-up process as well as the avoidance of thermo shock stress on the measurement instrumentation due to combustion chamber ignition. On the other hand, the implementation of an electrical heater leads to slower warm-up ramps and negates a real time operation with exhaust gas. A unique feature of the IKDG test stand is the possible mass flow range for turbocharger testing since the turbine of the turbocharger is supplied by a six-stage radial compressor with a maximum air mass flow of 12 kg/s. While test stands like [ ], [ ], and [ ] also use sequential hot gas generators, the mass flow range is limited compared to the IKDG test stand setup. e commissioning of the developed test stand and its thermodynamic boundary conditions as well as the chosen operation route, lay the basis for future experimental investigations regarding transient pressure and blade vibration phenomena. . . Compressor facility e test rig is powered by a high pressure air supply system driving the turbocharger turbine at the IKDG test facility with compressed air. e flow medium is provided by a 6stage radial compressor (2) which provides a maximum air mass flow of 12kg/s at 30bara. e compressor is controlled by variable inlet guide vanes (1) at the first three stages. A security bypass valve (3) at the outlet of the compressor unit is used during the start-up procedure, for emergency shutdown as well as to adjust the required test stand mass flow. To further increase the adjustment precision a control valve (4) is installed upstream the air preheater.

NOMENCLATURE

. . Air preheater

e air exits the th stage of the compressor with a temperature of approximately 150 • C. en, the air temperature is further increased by a 10.5MW natural gas fired air preheater (5)-( 7) to a temperature of about 180 • C. For investigations requeiring higher turbine inlet temperatures the air temperature can be increased up to 500 • C.

. . Turbine section

Before the flow medium enters the turbocharger, a filter (8) cleans out any particles which may be carried along the flow path.

e mass flow entering the turbocharger turbine is measured by means of an orifice plate (9) that is designed according to DIN( -, ) . A control valve (10) is integrated into the turbine piping as a safety measure in case that the lubrication system, providing the turbocharger with oil, fails and therefore prevent damage of any rotating components. Since the NG fired heater provides temperature gradients too wide for steady turbocharger operation, an electrical preheater ( 11) is directly integrated into the test stand design thus enabling steady state parameters with a maximum turbine inlet temperature deviation of about +/-1.0K during steady state operation. e fluid enters the turbocharger turbine ( ) and exits into the stack. e position of the orifice plate upstream of the electrical preheater was chosen in order to minimize mass flow measurement inaccuracies due to hot gas streaks. e true flow temperature entering the turbine is measured at measurement plane 1.

. . Compressor section

At the IKDG test stand, the compressor and the turbine side of the turbocharger are operated independently of one another. On the induction side of the compressor the intake mass flow is measured using an orifice plate ( ). To acquire all data required to develop a performance map for the turbocharger, a thro le valve ( ) is installed at the pressure side of the turbocharger compressor ( ) allowing the adjustment of the compressor operation point from stall to surge (discharge volume: 0.008m 3 ).

e compressor is supplied with air at ambient condition and discharges the compressed air into a stack (16).

. Measurement setup e measurement instrumentation comprises the acquisition of the thermodynamic operating conditions of the turbocharger as well as the rotational speed and the lubricating oil inlet and outlet temperatures.

e static pressures are recorded by means of three holes equally distributed over the circumference of the in-and outlet planes of the turbocharger (MP1-MP4). e total inlet and outlet pressure for the turbocharger are calculated from the measured static pressure and the Mach number. Kümmel [ ] states, the pressure calculation relies on the average fluid velocity and the average Mach number (see Equation )

p t p s = 1 + κ -1 2 Ma κ κ-1
( ) e temperature at the turbine side is measured by three type K thermocouples for each measurement plane. e measurement planes are displayed in labeled MP where the flow temperature is measured.

e equipped thermocouples feature a sensor diameter of 3 mm and are distributed equally over the circumference of the measuring plane. On the compressor side the temperature at the in-and outlet are measured by means of three Pt100 resistance thermometers with the same circumferential distribution. Since the temperature is not measured by means of total temperature probes, the measured temperature does neither represent the static nor the total temperature of the fluid. e static temperature is calculated using Equation .

T s = T Probe -rec c(r) 2c p ( )
e recovery factor was approximated for turbulent flow conditions in reference to [ ]. Since the temperature dependency of the Prandtl number is negligible in the examined temperature interval from ambient temperature to 220 • C, the static temperature is calculated using a constant recovery factor.

rec = 3 √ Pr = 3 √ 0.7 = 0.8879 ( )
e total temperature is calculated using the velocity at the radial position of the thermocouple sensing head within the fluid flow path in contrast to the total pressure of the fluid that is derived by means of the average fluid velocity. For all measurement planes the temperature was measured at the equivalent same radial position for all three probes. erefore the insertion depth's influence on measurement uncertainty is not considered throughout the following investigations.

e radial velocity profile is a ained using Equation .

c(r) c m = (1 + n)(2 + n) 2 1 - r R n ( )
Kümmel [ ] suggests to set n= / as in turbulent flow profiles this value is proven to work. Performed comparative measurements with a point pressure measuring rake proved the suggested value [ ].

T t T s = 1 + κ -1 2 Ma(r) 2 ( )
.

RESULTS AND DISCUSSION

For the acquisition of turbocharger performance map data it is essential to validate the operating point stability to varify the assumption of steady state operation. Furthermore the operating point stability and the obtained thermodynamic equilibrium state of the turbocharger during measurement procedure ensures comparability of different conducted measurements. erefore, the tolerances of the steady state definition are analyzed. During data acquisition the following criteria for steady state have been formulated to ensure representative thermodynamic data parameters. e data acquisition procedure entails a sample rate of 1S/s for the measurement of temperature, pressure, mass flow and rotational speed. A total number of 60 sample points is measured and arithmetically averaged to a ain the thermodynamic parameters of the current operation point. e analysis of the operation point stability compares each sample point to the arithmetically averaged operation point thus displaying the fluctuation over the data acquisition interval of 60 sample points equivalent with 60 second measurements.

. Operation point stability

During steady state data acquisition, it is essential that neither increasing nor decreasing tendencies in turbine outlet as well as compressor outlet temperature occurs. A maximum deviation of about +/-1.0K of the turbine inlet temperature during the acquisition time interval at steady state operation was aimed for. is allows a more steadier operation of the turbocharger compared to test stands using combustion chambers as hot gas generators. A stability in rotational speed of 2Hz as well as the stability of the oil inlet temperature of +/-0.5K during acquisition procedure were also defined as required steady state criteria since fluctuations in the oil inlet temperature directly affect the stability in rotational speed. e operation point stability was analyzed for three representative rotational speeds of around 25%, 60% and 89% of the turbocharger's maximum speed. When adjusting the operation point using the thro le valve at the compressor outlet side, the temperature measured at the compressor outlet indicates, whether thermodynamic equilibrium of the operation point is achieved. e fluctuation in temperature a er adjusting the thro le valve and reaching steady state operation is shown in Figure .  A er the turbine and compressor fluid temperatures have reached thermodynamic equillibrium, the acquisition is only started when the oil inlet temperature fluctuation is within the defined limits as shown for three exemplary rotational speeds in Figure .

. Uncertainty analysis

In order to evaluate uncertainties of the acquired data, the full measuring chain has to be analyzed, since a certain amount of disturbing influences may cause deviations and errors throughout the measuring interval. In accordance with DIN V ENV the installed measurement acquisition system . . Uncertainty of temperature measurement e temperature measurement chain is modeled by means of Equation for the Type K thermocouples implemented on the turbine side. e temperature measurement chain of the compressor side is modeled by means of Equation based on the use of Pt100 resistance thermometers.

l T = T m + σ HC + σ DIN + σ Stab + σ Ref + σ Cbl + σ Vs +σ Tam + σ Lin + σ Inh + σ Lts ( ) l T = T m + σ HC + σ DIN + σ Stab + σ Shs + σ str + σ Vs

Figure .

Influencing variables for temperature measurement TC Type K e sensing element is insulated inside the compression fitting thus heat conduction is not considered. Deviations from the characteristic curves are delivered by the manufacturer of the TC. For Type K thermocouples the limiting deviation is given as . K for temperature intervals from 0 • C to 1100 • C. During data acquisition the test stand runs in steady state keeping the in-and outlet temperatures of the turbocharger constant. Because the TC reacts much faster to transients than the test stand itself, stabilization periods of the TC do not have to be considered. Errors caused by reference temperature, extension cables and linearization are accounted for by σ DIN . Errors due to a change of the ambient temperature of the MAS, inherent resistance and linearization is considered using the MAS manufactures spec sheet. For a measured sensor temperature of 300 • C an error of 0.58K has to be considered. e confidence interval of 95.5% is not used because the assumption is made that all errors would occur at the same time. erefore, the present uncertainty analysis is conducted with a confidence interval of 68.3%. is leads to a combined standard uncertainty of u TS =0.932K for the turbine temperature measurement by means of thermocouples Type K.

e Pt100 resistance thermometers are positioned in the fluid flow path in the same way as the Type K TC thus heat conduction is not considered. Deviations from the characteristic curves are again delivered by the manufacturer. For Pt100 Class A the limiting deviation is set at 0.75K for the maximum occurring temperature of 300 • C at the compressor outlet. Although internal calibration studies have shown that the error is below 0.75K, the standard manufacture value of Pt class A is used to consider the maximum error influence. In contrast to the Type K TC, errors for the Pt100 may occur due to thermo-voltage and self-heating of the sensor. Since the supply current of 1mA is too low to influence the temperature acquisition, the self-heating of the sensor is neglected. e thermo-voltage is considered according to Scheller. is leads to a combined standard uncertainty of

l Pi = V Pi + σ Mt + σ Mw + σ Ma + σ Mv + σ DPT + σ MAS ( )

Figure . Influencing variables for pressure measurement

Likewise to the temperature measurement, transient effects can be ignored. e wall influence is neglected since only the static pressure is acquired using wall pressure holes.

e influence of the Mach number has not been taken into account because occurring Ma-numbers are less than 0.3 for the measured rotational speeds. is assumption is also applicable for the viscosity impact, since it only has to be considered for Re ≤ 1000. e accuracies of the differential pressure transducers (DPT) are specified by the manufacturer and reviewed. In addition to the measurement instrumentation uncertainty, the static pressure wall tap diameters need to be included in the pressure unvertainty analysis as well. Bore diameters of 5mm were chosen for all measurement planes. e uncertainty value was calculated from the ratio of manufactured pressure hole diameter to the flow area at each measurement plane based on [ ]. Absolute uncertainty values for each measuring plane are given in Table using the diameter ratio of 0.4 to consider the hole diameter uncertainty . . . Performance map data e following section is dedicated to show the acquired data from which the performance maps of the turbine and the compressor are generated. e acquired turbocharger turbine performance map is shown in Figure . e corresponding compressor performance map is shown in Figure . e performance map is acquired by successively closing the thro le valve at the compressor outlet while maintaining a constant turbine mass flow and turbine inlet temperature. As soon as an increase in rpm occurs, the mass flow on the turbine side is readjusted. us providing the data for a characteristic curve for a defined turbine speed.

e performance data is acquired until reaching the surge line of the compressor limiting the width of the acquired rpm curve. e expected trend of the characteristic turbine rpm curves can be seen in Figure .  Due to compressibility effects the reduced mass flow bends towards an asymptote and decreases a er peaking at a turbine pressure ratio of approximately 3.5. Since the compressor side of the turbocharger is operated at ambient conditions, the inlet pressure varies depending on the rotational speed due to the orifice plate mass flow measurement upstream of the compressor inlet. e variation of the inlet pressure at compressor inlet in reference to the rotational speed is shown in Figure . To further improve performance investigation, the ambient operation should be switched to a closed-loop operation, which can be integrated into the test stand. Since in turbocharger operation the focus lies on the acquisition of the turbine performance where the compressor side is merely used to adjust the load, the closed-loop test stand version will offer a much wider range for obtaining per- Turbine pressure ratio (total-static) [-] 25 Compressor pressure ratio (total-total) [-] Normalized reduced compressor mass flow [-] 25 η eff = P pol,compressor P is,turbine ( ) where P pol,compressor is the polytropic power output of the turbocharger compressor and P is,turbine is the isentropic turbine power output calculated on the basis of an isentropic turbine outlet temperature.

∆h is,turbine = cp turbine (T bt,tot -T at,is )

( ) e calculated normalized effective efficiency is displayed in Figure in regard to the total-static turbine pressure ratio.

e analysis of the efficiency curves supports the assesment, that good thermal stability was achieved during steady state operation. . CONCLUSION AND OUTLOOK e developed turbocharger test stand provides an opportunity to operate the test turbocharger at stable conditions. Testing has shown, that the requirements of the thermodynamic parameters, like the turbine inlet temperature, are fullfilled. Hence the operation point is stable enough that data for performance map purposes can be acquired at the presented test stand. e operation of the test stand at constant turbine inlet temperatures, especially at below realtime application, can be used for investigations where the implemented measurement instrumentation is limited by maximum operation temperature. e required equilibrium state between compressor and turbine of the turbocharger was proven to be obtained during the performance map data acquisition by comparing the outlet temperatures of both sides. A stability interval of the rotational speed of 2 Hz was accomplished. e uncertainty analysis for the applied measurement instrumentation shows, that the acquired data lies within an uncertainty interval which suffices the required measurement accuracy for obtaining the needed thermodynamic parameters for generating the displayed performance maps as well as CFD validation. A constant oil inlet temperature within tolerances of +/-0.5K was also achieved during testing operation of the turbocharger.

In regard of the acquired test data the developed test stand can be modified for further investigations on turbocharger performance and efficiency.

e displayed measurement range of the turbocharger compressor can be identified as the limiting factor on the possible measurement range. Since the electrical preheater is less sensitive to low flow turbine conditions, the compressor side needs to be adjusted towards lower mass flow. Internal investigations have shown, that by adjusting the compressor side pressure losses (usage of two parallel pipes equiped with orifice plates) the measureable performance range can be increased. Furthermore, the mentioned variation in compressor inlet pressure can be adressed by implementing a closed-loop compressor cycle with integrated intercooler. us providing similar compressor inlet conditions independent from the rotational speed. In a first step Also the installed lubracating oil ciruit offers the oportunity to adjust the oil inlet temperature in a wider scope which allows a large spectrum of test turbochargers that can be investigated. Furthermore the compressor facility at IKDG provides enough mass flow to enable testing in a turbocharger scope of application from small automotive up to marine turbocharger applications.

Figure .

 . Figure . Overview of the test stand at IKDG with instrumented measurement planes

Figure .

 . Figure . Temperature fluctuation turbine inlet

Figure .

 . Figure . Temperature fluctuation compressor outlet

+σ

  Tam + σ Lin + σ Inh + σ Lts ( ) e acquired output signal l T depends on the measured temperature T m and a number of uncertainties which are explained and quantified in Figure and Figure . e thermocouples Type K are located directly in the fluid flow path of the turbine by means of compression fi ings.

Figure .

 . Figure .Influencing variables for temperature measurement Pt100

Figure

  Figure . Acquired turbocharger turbine performance map

Figure .

 . Figure . Pressure variation of the compressor inlet in reference to the rotational speed

Figure .

 . Figure . Normalized effective efficiency of the turbocharger turbine at 220 • C turbine inlet temperature

  kg/s at a certain inlet temperature of 220 • C. is allows the determination of the turbocharger behaviour and efficiency at below usual real time application. e test turbocharger is designed for turbine inlet temperature operation up to 650 • C during real time application. e test stand consists of different subsystems composed of air sup-ply infrastructure, air heater components, test turbocharger, data acquisition system and control unit. Figure shows a simplified flow diagram of the turbocharger test stand at the IKDG facility with the instrumented measurement planes (MP). e design parameters are presented in Table .

	Re	Reynolds number
	rec	Recovery factor
	s	Second
	S	Sample
	T	Temperature
	u	Uncertainty
	V	Voltage
	Symbols
	I	Output signal
	ṁ	Mass flow
	κ	Isentropic coefficient
	σ	Input signal
	Subscripts
	Cbl	Cable
	DIN	Deviation compared to DIN
	HC	Heat conduction
	Inh	Inherent resistance
	Lin	Treatment and linearization
	Lts	Long term stability
	m	Meridional / Measured
	Ma	Influence of Mach number
	Mt	Transient effects
	Mv	Viscosity impact
	Mw	Wall influence
	Ref	Reference temperature of MAS
	s	Static
	Shs	Slef heating of sensor
	Stab	Stabilization period
	t	Total
	Tam	Ambient temperature of MAS
	str	ermo-voltage
	Vs	Voltage supply of MAS
	Abbreviations
	CFD	Computational fluid dynamics
	DPT	Differential pressure transducer
	FSO	Full scale output
	MAS	Measuring acquisition system
	MP	Measuring plane
	TC	ermocouple
	. EXPERIMENTAL SETUP
	. Test stand setup
	e test turbocharger is designed for usage in marine engine
	applications and is provided by Kompressorenbau Bannewitz
	GmbH (KBB). e turbocharger provides a total compressor	c	Flow velocity (absolute)
	pressure ratio of up to 5.5 with a maximum turbine mass	c p	Specific heat capacity
	flow rate of 5.7	Ma	Mach number
			n	Mechanical exponent
			p	Pressure
			Pr	Prandtl number
			r	Radius
			R	Set radius of pipe

Table .

 . Design parameters of the turbocharger test stand at IKDG

	Turbine inlet mass flow	≤ 6kg/s
	Turbine inlet pressure	≤ 6bar
	Turbine inlet temperature currently 220 • C
	Turbine inlet temperature possible	500 • C
	Turbine wheel diameter	∼ 250mm

Input parameter Source of error Distribution Uncertainty

	σ Mt σ Mw σ Ma σ Mv σ DPT	Transient effects Wall influence Influence of Ma-number Viscosity impact DPT Zero pressure offset	Cancelled Unattended Unattended, since Ma < 0.3 Unattended, since Ma < 0.3 Unattended due to calibration
	σ MAS		Linearity/Hystere sis Temperature offset Temperature span Long-term stability Gain Error Offset Error	Rectangular Rectangular Rectangular Unattended since < 1 year 0.1% FSO 0.1mV 0.2% FSO Rectangular 160ppm of reading Rectangular 156ppm of reading
		MAS	Noise Uncertainty	Rectangular	0.62μV

Table .

 .

	Absolute uncertainty values for all turbocharger
	measuring planes
	Measuring plane Combined uncertainty 68.3%
	MP 1	3.10 mbar
	MP 2	6.05 mbar
	MP 3	6.05 mbar
	MP 4	6.40 mbar

ACKNOWLEDGMENTS

e authors would like to thank the research association Forschungsvereinigung Verbrennungskra maschinen e. V. and the Association Industrial Research (AIF) for the financial support of this research project. e authors would also like to gratefully acknowledge Kompressorenbau Bannewitz GmbH, whose support was essential for the present investigations.


Marj Horne
email: marj.horne@rmit.edu.au

Rebecca Seah
email: rebecca.seah@rmit.edu.au

A step in the development of an evidence based learning progression for geometric reasoning: focus on shape and angle

Keywords: Geometry, measurement, learning progression

come   L'archive ouverte pluridisciplinaire

Introduction.

Geometry and spatial visualization is important for STEM [START_REF] Wai | Spatial ability for STEM domains: Aligning over 50 years of cumulative psychological knowledge solidifies its importance[END_REF] and the development of number understanding [START_REF] Verdine | Spatial skills, their development, and their links to mathematics Monographs of the Society for[END_REF]). Yet in some countries the amount of geometry being taught has been reduced over many decades leading to a decline in teacher knowledge. The improvement of the teaching and learning of geometry requires both attention to the curriculum and enabling teachers to implement the curriculum more effectively.

The Australian Mathematics and Science Partnership Program (AMSPP) funded by the Australian Government Department of Industry, Innovation, Climate Change, Science, Research and Tertiary Education, was established essentially to increase the participation and expertise in STEM related areas. The Reframing Mathematical Futures Project II (RMFPII), as part of the AMSPP aimed to build a sustainable, evidence-based learning and teaching resource to support the development of mathematical reasoning in Years 7 to 10. This paper reports on part of this process with a particular focus on students' reasoning about shape and angle size.

Learning progressions, while not new, relatively recently have become the focus of systematic research efforts (see [START_REF] Siemon | Researching and using learning progressions (trajectories) in mathematics education[END_REF]. There have been different interpretations of the meaning and use of learning progressions (and the closely related learning trajectories -see the special edition of Mathematics Teaching and Learning, 6(2), 2004). The learning progression, as we are using the term, describes stages in the development of geometric reasoning which are evident sequentially in the responses of many students. It assumes that learning takes place over time, and that teaching involves recognising where learners are in their learning journey and providing challenging but achievable learning experiences that support learners' progress to the next step in their particular journey (see [START_REF] Siemon | Researching and using learning progressions (trajectories) in mathematics education[END_REF] for recent research and development on this). The learning progression is thus closely linked with teaching and learning. The research presented in this paper is in the context of a larger study which is developing a learning progression for geometric reasoning.

Geometric reasoning is the ability to critically analyse axiomatic properties, formulate logical arguments, identify new relationships and prove propositions. It is demonstrated in early stages through attention and explanation with students then moving to describing, analysing, inferring, conjecturing and deducing geometric relationships leading to them engaging in formal proof [START_REF] Brown | Developing geometric reasoning[END_REF]. Being able to invent and apply formal conceptual systems to investigate geometric relationships is part and parcel of reasoning geometrically [START_REF] Battista | The development of geometric and spatial thinking[END_REF].

In common with other developed learning progressions and trajectories a synthesis of relevant research literature was undertaken to establish a hypothetical learning progression for geometric reasoning which included a breadth of geometric concepts from the curriculum since its application was to be with classroom teachers. In constructing the progression, we acknowledge that the dominance of van Hiele levels on geometric thinking research, and their neglect of visualisation, meant that the framework we developed needed to move beyond these factors and recognise that students can reason at multiple levels, at different rates for different concepts. Our synthesis of available research findings led to the development of an initial hypothetical geometric learning progression that included both geometry and measurement concepts and which was based on [START_REF] Battista | The development of geometric and spatial thinking[END_REF] proposed four levels of thinking: visual-holistic reasoning, analytic-componential reasoning, relational-inferential property-based reasoning, and formal deductive proof. He expanded the development of property-based thinking to progress from visual-informal, to informal insufficient-formal reasoning and finally sufficient formal property-based reasoning. The hypothetical learning progression we proposed [START_REF] Seah | A Learning Progression for Geometric Reasoning[END_REF]) made a distinction between descriptive reasoning -using informal language to explain what one sees and analytic reasoningusing formal language to analyze properties of shapes. It also encompassed concepts from a range of aspects of geometry including shape, transformation and location, and measurement. This allowed us to better investigate the growth in the connectedness between visualization and mathematical discourse and contributed to the design of instructions that targeted specific thinking within a framework which was connected to, but not defined by curriculum.

The research question considered in this paper is how do students' reason about angle, angle measure in a two dimensional context of shape. This is set in the overarching project of developing a learning progression for geometric reasoning.

Method

Based on the hypothetical learning progression we designed a series of pen and paper assessment tasks, which were based on practical situations, and would allow students to demonstrate reasoning as well as other knowledge and skills. While there have been some studies (for example, [START_REF] Keiser | Struggles With Developing the Concept of Angle: Comparing Sixth-Grade Students' Discourse to the History of the Angle Concept[END_REF][START_REF] Mitchelmore | Development of angle concepts: A framework for research[END_REF], angle and angle measure is an area of geometry that has not received a lot of attention yet occupies considerable curriculum space connecting to many aspects of shape and measurement leading through to trigonometry. Typical angle questions in text books in years 7-10 require students to find the magnitude of angles from diagrams. The intention for these assessments was to seat the problems in real situations to which students could relate. An example of such a task is an angles task shown in figure 1 where students physically construct the angles through paperfolding before finding their magnitudes. The intent of this question was to focus on students' ability to use knowledge of properties of shape to make deductions about angle measure hence part a) focused on the properties of the 2D shape while part b) sought angle measures and explanation.

Figure 1: Shape and angles task

Each task was designed with a marking rubric to enable teachers to easily score the students' work. The rubric for the angles task is shown in Table 1. The scoring rubric requires clear reasoning to be given for the responses to attain the higher scores. GANG1.1 refers to a rubric score of 1 on the question GANG1 and is used in later analysis.

SCORE

DESCRIPTION For GANG1 A number of assessment forms, each with 6-7 questions, were structured covering a range of geometric content and trialled with students in years 7-10 in order to collect data on the questions and refine them as needed. On the basis of these trials and the student responses assessment forms were finalised and data collected first from a small group of trial schools in three different states and then from the larger group of project schools across Australia. This data was analysed at two levels. The first was at the level of individual questions focusing on the reasoning about specific geometric and measurement constructs while the second was at a meta-level using item response theory (IRT). For a detailed description of this IRT analysis see [START_REF] Siemon | Researching Mathematical Reasoning: Building Evidence-based Resources to Support Targeted Teaching in the Middle Years In D[END_REF].

The analysis led to the identification of eight zones based on the evidence of what these students could do. From this a learning progression based on the descriptions of these zones was created.

Results and Discussion

The teachers in all the schools had a choice of which assessment forms their students would complete. Many chose not to assess geometry. The data presented here is from 279 students in years 7-10. As can be seen many students did not attempt the question or gave irrelevant answers. The following section shows the types of responses to the two questions.

Reasoning about shape

A sample of student responses are shown here in table 3. Orientation was an issue for some students with confusion about diamond and rhombus for many students, like S2, thinking it was only a rhombus when it was positioned with a horizontal edge. Phoebe is correct when she states that the shape made from the paper is a rhombus as the opposite sides are equal to one another, opposite angles are equal to each other and the sides are parallel 3 S8 This shape is a rhombus because properties of a rhombus all sides are equal, diagonals bisect each other at right angles, all angles are equal 4 S9 I agree. This is because all the sides are the same length.

There are also two pairs of parallel lines There are two acute angles and two obtuse angles.

S10 Yes all the sides same measurement

Table 3: Student reasoning about shape

The explanations given by students such as S1-3 could be classified as "it looks like …" with either agreement that it is a rhombus or disagreement because it is a diamond. S3 goes a little further with dynamic imagery showing in the response. S4 and S5 began to use properties in their reasoning though limited and insufficient. S6 and S7 used properties but defined a parallelogram rather than a rhombus with S6 also indicating orientation. While giving a necessary and sufficient property initially, S8 went on to give further properties, the last of which was incorrect. For the highest rubric score S9 also gave extra properties but these were written as an extra while S10 correctly gave just the important property. None of the students specified that the sides had to be straight or that the shape was planar and closed but in the context this was not seen as being necessary.

Reasoning about angle sizes

Table 4 shows some of the students reasoning about the angle size. A2 demonstrated a lack of comprehension about the concept of angle using the number of lines meeting at the point to provide the angle size. Many of the students associated angle size with protractors and when asked to find the angles, the only way of which they were sure was to measure using a protractor, or for A3 and A4 imagining one, which suggests that they have had little experience of reasoning about angle sizes from given information and their knowledge of geometry. A5 began with a correct calculation using a diagram to illustrate but then became confused. A6 and A7 both calculated correctly giving reasons. The rubrics and students' responses show increasing sophistication in reasoning. However, there was little evidence of students using relational inferential property based reasoning. While the theoretical framework with which we began was useful in the design of questions and aspects of analysis the final learning progression developed is based on descriptions of what the students actually demonstrated in their geometric reasoning.

Based on responses from many students and the responses to 36 different questions on the assessment forms of which this angles question is only one, an item response analysis was done using Rasch modelling (See [START_REF] Siemon | Researching Mathematical Reasoning: Building Evidence-based Resources to Support Targeted Teaching in the Middle Years In D[END_REF]. Figure 2 show the logit map returned by the analysis, based on valid responses from 742 students. The responses to the GANG questions have been highlighted on the map. The dotted horizontal lines show where some blank rows were removed.

From the map of the item responses it can be seen that GANG1, the shape question, has responses spanning zones 3 to 7 and GANG2, the angles question, spans zones 4-7. Based on these responses the descriptions of what students could generally do at each of the zones provided the evidence based learning progression for geometric reasoning. From this question at zone 3 the students were recognizing shape because it looked like it but were not yet correctly finding any angles. At zone 4 they were starting to reason about shape using properties and were responding to angle size with general information such as acute, obtuse and right angles. In zone 6 they were correctly finding some angles and were able to explain their reasoning about some of these. By zone 7 they were able to identify shape giving necessary and sufficient properties and were able to reason about the magnitude of angles providing sound explanations. According to the curriculum these students are meant to have studied knowledge about angles in geometry and the properties and identification of quadrilaterals before year 8 so it is of some concern that almost 80% of students were operating below zone 6 with 75% unable to find the magnitude of an angle in this situation. Some students showed they were not even clear about the concept of angle.

  s are 6 because there are 6 lines finishing at the s angle. All the angles from the h are 7 because they are also reaching the end. All the angles from the f are also touching the end of the page 1 A3 30 45 20 The way I worked it out was I pictured the pretracter [sic] in my head and trying to figure the degrees of it 2 A4 45 45 -I estimated the angles by imagining a 90 degree angle on the f, sangle shown divided in two to make 45] h 180 /2  90 / 2  45 because of the 3 angles make 180 so you can divide by 3 4 A6 45, 45, 135 The half creases indicate two right angles. H and f are halves of the right angles so 90/2=45. Quadrilateral has internal sum of 360 h and the opposite is 90 total S =(360-90)/2 4 A7 45 45 135 Angle f is based on a straight line and there are 8 equal angles (180/8 = 22.5) but f is two of these 22.5 x 2 = 45 o Angle h is the same thing just positioned differently Angle s I found because a triangle equals 180 o 22.5 + 22.5 = 45 180-45 = 135 (with diagram)

Figure 3

 3 Figure 3 Logit map of responses to assessment forms

Table 1 : Rubrics for shape and angle question

 1 Disagree it is a rhombus but claim it is a parallelogram or some other quadrilateral such as kite with some Agree it is rhombus. Explanation needs to include necessary and sufficient properties, that is, it has 4 equal sides, or is a parallelogram with one of the following properties  Adjacent sides equal  Diagonals bisect each other at right angles or diagonals bisect the angles  Two lines of symmetry All angles correct with clear reasons given relating to the folding and properties. F = 45 o ; h = 45 o ; s = 135 o (e.g., Folding corner to centre creates half right angle; All angles around centre of side equal so any 2 make 45 o or Four angles of quadrilateral add to 360 o )

	0	No response or irrelevant response
	1	Disagree it is a rhombus based on appearance rather than properties
	2

properties specified, or Agree it is a rhombus with reason relying on appearance such as "if you turn it to rest on its edge it is a rhombus".

3

Agree it is rhombus specifying two properties correctly but insufficient / incorrect properties to define it.

Table 2 : Percentage of students in years 7-10 attaining rubric score

 2 Table2shows the percentage of students attaining each rubric score for the two questions.

	Rubric score	Shape (GANG1)	Angle (GANG2)
	0	39	50
	1	7	25
	2	4	8
	3	43	9
	4	7	8

Table 4 : Student reasoning about angle size

 4

Concluding comments

The students' reasoning about angle and angle measure was generally at a lower level than indicated by the curriculum. Students often did not have the basic knowledge of the properties of quadrilaterals sufficiently connected to enable them to use these in reasoning about angle size. The student responses indicate many issues in their learning about shape and angle and further investigation is needed, particularly about the understanding of angle and angle measure. The analysis of the student answers to the questions combined with the item response analysis allowed a learning progression based on students' actual responses rather than just a validation of a proposed hypothetical progression. The full learning progression is based on all the questions used in the analysis and not just the one question and while the focus is reasoning, reasoning cannot be separated from the understanding of the underlying concepts. Since the final learning progression developed is based on what students could actually do it should be of use to teachers enabling them to map students' progress and inform teaching. Further research is needed on the implementation of such a progression.


Keith Jones
email: d.k.jones@soton.ac.uk

Kotaro Komatsu
email: kkomatsu@shinshu-u.ac.jp

Task design with DGEs: The case of students' counterexamples

Keywords: Task design, dynamic geometry environment, counterexample, proof

Introduction

Geometry, being one of the main areas for proof-related activity in school mathematics, is a topic in which learners can work with counterexamples. [START_REF] Sinclair | Recent research on geometry education: An ICME-13 survey team report[END_REF] recently highlighted that more research is needed on task design with dynamic geometry environments (DGEs). We are addressing these issues by elaborating a set of theory-informed task design principles involving DGEs and using these to study students working with geometric diagrams [START_REF] Jones | Diagrams in students' proving activity in secondary school geometry[END_REF][START_REF] Komatsu | Task design principles for heuristic refutation in dynamic geometry environments[END_REF]. Given the importance of networking theoretical approaches, in this short paper we build on our studies by employing a conceptual framework on virtual manipulatives [START_REF] Osana | Manipulatives, diagrams, and mathematics[END_REF]. Our research question is how, in using Osana and Duponsel's framework, our task design helps students to produce and address what they think are counterexamples in geometry.

Virtual manipulatives

Following [START_REF] Moyer-Packenham | Revisiting the definition of a virtual manipulative[END_REF], DGE tasks where on-screen objects are draggable can be regarded as virtual manipulatives. [START_REF] Osana | Manipulatives, diagrams, and mathematics[END_REF], based on a thorough review, proposed that task design with virtual manipulatives should take into account: the surface features of representations, the pedagogical support, and students' perceptions and interpretations.

Method

We undertook a task-based interview (of 35 minutes) with two prospective teachers from a national university in Japan. Each of the participants had experience of using DGEs. They worked on the tasks in Figure 1, which were based on our design principles (for details, see Komatsu & Jones, in press). For data analysis, we used video records of the task-based interview, transcripts of the undergraduates' talk, their written work, and their DGE file. The undergraduates completed Task 1 by producing a valid proof where they showed that AE = CF and AE // CF. In our analysis, we focus on how the undergraduates worked on Task 2.

Analysis

The undergraduates produced various diagrams with the DGE, as shown in Figure 2. They considered that the cases shown in Figures 2b and2d were counterexamples to the statement (quadrilateral AECF is a parallelogram). They successfully addressed the case of Figure 2d by extending the original statement (as in Figure 2e) and proving that quadrilateral AECF in this case is a parallelogram. Here, we summarise how the set of tasks in Figure 1 embody the three elements of the virtual manipulatives framework of [START_REF] Osana | Manipulatives, diagrams, and mathematics[END_REF]:

 Task 1 contains a given diagram. This diagram includes a specific representational feature; namely a 'hidden' condition implying that perpendicular lines from points A and C always intersect with diagonal BD. By this surface feature, and the pedagogical support in the text in Task 2 explaining the DGE use (i.e. use of the DGE to check the truth of the statement), the undergraduates could discover what they viewed as 'counterexamples' to the statement (as in Figures 2b and2d).  Pedagogical support included in Task 2 was not excessive in the sense that this task gave the undergraduates the opportunity to explore the task on their own by not specifying what they were expected to do when discovering what they might consider to be 'counterexamples'.  The sequence of Task 1 and Task 2, designed by taking students' interpretations of counterexamples into account (see [START_REF] Komatsu | Task design principles for heuristic refutation in dynamic geometry environments[END_REF], stimulated the undergraduates' subsequent activity where they spontaneously dealt with the case of Figure 2d by extending the original statement with proving in Figure 2e.

Conclusion

This analysis illustrates how our theory-informed design principles are further supported with the conceptual framework on virtual manipulatives of [START_REF] Osana | Manipulatives, diagrams, and mathematics[END_REF] in terms of the surface features of representations, the pedagogical support, and students' perceptions and interpretations.

  Task 1. In parallelogram ABCD, draw perpendicular lines AE and CF to diagonal BD from points A and C, respectively. Prove that quadrilateral AECF is a parallelogram. Task 2. Construct the diagram shown in Task 1 using a DGE. Move the vertices to change the shape of parallelogram ABCD, and examine whether quadrilateral AECF is always a parallelogram.

Figure 1 :

 1 Figure 1: Tasks used in the interview