Benjamin Brink

Jonas Dreyøe

Andreas Lindenskov Tamborg

Brendan Eagan

Sara Tabatabai

David Williamson Shaffer

Morten Misfeldt

David Williamson Shaffer

Exploring teachers' assignment of mathematical competencies to planned lessons using Epistemic Network Analysis

Keywords: Epistemic network analysis, learning management systems, mathematical competencies, teacher planning

published or not. The documents may come

Introduction

There is a long history of constructing theoretical conceptual models in educational research. In mathematics education in Denmark, there is an elaborated understanding of what characterizes mathematical activity. During the last 15 years, the concept of mathematical competencies has been developed and elaborated [START_REF] Niss | Competencies and mathematical learning: Ideas and inspiration for the development of mathematics teaching and learning in Denmark[END_REF][START_REF] Niss | Kompetencer og matematiklaering -Ideer og inspiration til udvikling af matematikundervisning i Danmark (Uddannelsesstyrelsens temahaefteserie[END_REF]. In this context, these competencies are described as "... a well-informed readiness to act appropriately in situations involving a certain type of mathematical challenge" [START_REF] Brooks | The data-assisted approach to building intelligent technology-enhanced learning environments[END_REF].

The Danish Ministry of Education identifies six competencies in the mathematics curriculum for primary and lower secondary (UVM, 2014). Problem tackling is about being able to pose and solve mathematical problems, where a mathematical problem is a task where it is obvious that an answer must exist but the method of solution is unclear and can have different outcomes. The modelling competency is about being able to analyse, build and utilize mathematical models about other fields.

Reasoning is about being able to follow and assess mathematical reasoning, and being able to devise and carry out informal and formal arguments as well as understand what constitutes a mathematical proof. Furthermore, it is about being able to think mathematically, where you know what questions could be posed and what answers are expected in mathematical contexts. The representing competency, on the other hand, is about being able to utilize different representations and switch between them as well as being able to choose the most meaningful representation. It also includes being able to handle mathematical symbol-language and mathematical formalism. The communicating competency is about being able to communicate in, with and about mathematics -to be able to understand and interpret expressions and texts as well as being able to express oneself about mathematics to different audiences. The tools and aids competency concerns the ability to operate and relate to tools and aids for mathematical operation, including Information Technology. It is also about being able to meaningfully reflect upon the choice of which tool or aid to use.

Although the individual competencies can be presented separately as above, they are considered to relate. For example, the so-called flower model [START_REF] Brooks | The data-assisted approach to building intelligent technology-enhanced learning environments[END_REF][START_REF] Collins | Interactional expertise as a third kind of knowledge[END_REF] depicts competencies as separate petals that overlap towards the centre, or as a set of dimensions, which together encompass the overall mathematical competence. However, the authors of this model recognize it as a purely theoretical development when they claim that "quite obviously it is impossible to produce scientific documentation that this is theoretically and empirically the case" [START_REF] Brooks | The data-assisted approach to building intelligent technology-enhanced learning environments[END_REF]. In this paper we recognise the competencies framework as a central curriculum resource used to inform mathematics teachers in their practice. We are interested in exploring how a framework like this is understood and utilized by teachers when planning their lessons. And more specifically if it is possible to use quantitative data to develop conceptual models of how teachers relate the mathematical competencies. Hopefully such models would hold the potential to support the critical reflections of individual teachers.

Aims, data & methods

This paper describes a pilot project conducted in anticipation of large amounts of data generated from learning management systems, which have recently been adopted in Denmark. At the time of writing, this data set is not available, and instead, we use an inferior data set with the goal to explore a specific approach to visualizing data, and using these visualisations in hypothesis and heuristic building.

The data set used in this experiment was collected using the Goal Arrow software. The software was specifically designed to investigate the opportunities and challenges of digitally supporting teachers' goal-setting for teaching activities. In the Goal Arrow, ca. 70 mathematics teachers associated one or more of the six mathematical competencies with their various teaching activities. Nothing was done to ensure that the teachers were representative, and the data included both empty submissions and repeated submissions due to a lack of user input validation. The latter was corrected manually, but with a total of only 182 records remaining, the data set here is not expected to support claims to the generality of a derived conceptual model.

In this study, we employ Quantitative Ethnography [START_REF] Shaffer | Quantitative ethnography[END_REF] to provide a method where qualitative and quantitative methods blend together completely by grounding quantitative results in a qualitative understanding of data. It aims at theoretical saturation of statistical models and using quantitative results to enable thick ethnographic descriptions [START_REF] Shaffer | Quantitative ethnography[END_REF]. More specifically we used the Epistemic Network Analysis (ENA) tool which examines connections in data and uses Singular Value Decomposition (SVD) to represent it as dynamic network model that maximizes the variance in the data using a reduced number of dimensions [START_REF] Shaffer | A tutorial on Epistemic Network Analysis: Analyzing the structure of connections in cognitive, social, and interaction data[END_REF].

Quantitative Ethnography describes coding as assigning meaning to observable actions or elements of the discourse observed in one context of a specific culture. In this study, the contexts are individual teachers' teaching activities and the coding is done by the teachers themselves when they assign relevant mathematical competencies to their individual teaching activities. In the data, each competency is represented in its own column where its absence or presence in each record of teaching activity is indicated by 0 or 1.

We imported this data into ENA and defined an ENA set where we specified the codes (the competencies), the conversations (individual teaching activity records) and the units (which we do not use here, but allow us to filter and group records in the visualisations). We plotted the ENA set using default settings to produce the visualisation of the competencies' interrelations depicted in Figure 1. The plotting uses SVD, which reduces the number of dimensions while still showing the maximum variance of co-occurring competencies [START_REF] Shaffer | A tutorial on Epistemic Network Analysis: Analyzing the structure of connections in cognitive, social, and interaction data[END_REF]. The first dimension (the x axis) explains the most variance, and the second dimension (the y axis) explains the second most variance in the data. Individual teaching activities become the red plotted points. These seem fewer in number than the 182 records because they often overlap and should not be confused with the nodes in the network. The black plotted nodes in the network represent the individual competencies (codes). The lines between nodes represent co-occurrence of the competencies in teaching activities. The strength of the connections between competencies, determined by their frequency of cooccurrences, is represented using the thickness and saturation of the lines between nodes.

The positions of competencies (the nodes) in this network graph can be used to interpret the dimensions in the space and this can explain the positions of individual teaching activities (red points). This can provide a conceptual frame [START_REF] Shaffer | Epistemic Network Analysis: A worked example of theorybased learning analytics[END_REF] to interpret different combinations of mathematical competencies in teaching settings. It is our ability to do this interpretation that becomes the measure of this approach to generating empirically driven conceptual models.

Interpretation

By looking at the visualisation generated by ENA shown in Figure 1, we identified and used two different approaches to characterizing the axes.

1. For each axis, we look at the nodes with the greatest distance.

2. For each axis, we split the competencies into three groups and compare these groups.

First, we looked at the opposites on the y-axis which are problem tackling and communicating. From the definitions of these competencies, we understand communicating to be about being able to communicate in, with and about mathematics versus problem tackling which is about working in the nitty gritty with mathematical tasks that require the activation of more than routine skills. It is tempting to tentatively recognize the y-axis as being about "talking the talk" versus "doing actual mathematics", or using Collins' ( 2004) concepts, interactional expertise versus contributory expertise. But how well is this supported when we consider the other competencies that also vary on the y-axis?

With regards to splitting the competencies into three groups, we overlaid the visualisation with a grid that segregated the competencies, as shown with the dashed lines in Figure 1, such that the following groups on the y-axis were identified: Y1) communicating and representing, Y2) modelling and reasoning and Y3) tools and aids, and problem tackling.

By considering communicating and representing as a group (Y1) we see something different about the lower end of the y-axis than when we only considered communicating. Representing is not about saying the "right" things but more about getting the meaning across, by having something stand for something else. Representing is therefore concerned with getting the actual meaning across by any means and if necessary by multiple means, and not just to other people, but also to oneself. In other words, the lower end of the y-axis is not categorizable as mere interactional expertise but also requires contributory expertise. Combined with our understanding of communicating, the bottom end of the y-axis seems more precisely to be about describing and explaining mathematical issues, topics or problems in different ways in different situations.

We revisit the other end of the y-axis in the same way as above. By considering problem tackling and tools and aids as a group (Y3) we do not have to change our understanding of the top end of the y-axis, from when we only considered problem tackling. The tools and aids competency is about being able to utilize tools and aids in a mathematical context, and being able to choose the right tool for the job. Tools and aids are thus like problem tackling concerned with working on or with mathematical problems, tasks and so on efficiently. The top end of the y-axis seems to remain something to do with the actual doing of mathematics with a focus on pragmatic and efficient choices of means for mathematical work.

We must still consider the two remaining competencies, reasoning and modelling, which form the middle group (Y2). However, the interpretation of their positioning does not disturb the understanding of the y-axis discussed above. The position of reasoning near the centre of the y-axis can be explained by the perceived need for reasoning along the entire y-axis. It is needed both for describing and explaining as well as working with mathematical problems. The position of modelling near the centre of the y-axis can be explained slightly differently as a necessary bridge between talking about problems from other domains, that we want to apply mathematics to as well as being able to work on those problems with mathematics.

By looking at the placement of competencies along the y-axis, firstly with a focus on the extremes and secondly in three groups, we understand the y-axis as representing a continuum from talking about, drawing, describing and explaining mathematics over modelling to actually doing mathematical moves, actions, and solutions with tools and aids. A continuum from describing and explaining to doing mathematics.

The x-axis does not in the same way as the y-axis have a clear pair of opposites to help us interpret it, but we did attempt to compare the two competencies with the most extreme distance on the xaxis; communicating and tools and aids. It was tempting to tentatively recognize the x-axis as being about psychology and cognition versus interacting with something tangible or material. But how well is this supported when we consider the other competencies that also vary on the x-axis. We approached this by splitting the competencies into, and interpreting the three groups: X1) communicating and problem tackling, X2) representing and reasoning and X3) modelling and tools and aids.

Considering the competencies tools and aids and modelling (X3) we first identified for each of these a new concept that a) seemed to relate to its respective competency and b) seemed to relate to the other concept. The definition of tools and aids clearly defines it to be very much about technical tools and not about mental tools (e.g. about straight edges and not about theorems). This gave us the concept "materiality". We interpreted the definition of modelling to be an especially map-like portrayal in the sense that it closely resembles (although simplifies) the object being studied. Together, the concepts materiality and map-like suggested something being tangible or close to the world at the right end of the x-axis. Of course, both competencies are not physical objects in the world, they are psychological and cognitive in nature, but there seems to be an immediacy of attention to things outside of a purely mental realm; to material or non-abstract things. Loosely speaking, to things in the world.

The above immediacy of attention to things outside of a purely mental realm seemed much less necessary when we considered the two competencies problem tackling and communicating (X1). Of course, both problem tackling and communicating may relate to something in the world beyond mental or linguistic gymnastics, but the necessary immediacy does not seem to be there because communicating and problem tackling does not to the same degree have to relate to something concrete but can remain in the abstract. Although all competencies are in virtue of being competencies psychological and cognitive in nature, the competencies on the left end of the axis can be more symbolic or interpretive, and less immediately connected to the world. Loosely speaking, to things in the mind.

The remaining two competencies, representing and reasoning, which make up the middle group (X2) of the x-axis also do not (like the middle group of the y-axis) seem to disrupt our interpretation of the x-axis. Reasoning's position near the middle of the x-axis can (like its position near the centre of the y-axis) be explained in terms of it being needed across the whole axis. regardless of how directly one engages in the world, mathematical thinking requires reasoning. Representing's position near the middle of the x-axis can (like modelling's position near the centre of the y-axis) be explained as a necessary bridge between the two ends of the x-axis. To be able to apply abstract mathematical thinking or communicate to the world we need to portray the world, but those portrayals need not be as map-like as those used in modelling. Other representations are also needed. One of these representations is natural language explanations, which must be considered an important part of the communicating competency.

By looking at the placement of competencies along the x-axis in three groups, we understand the xaxis as representing a perceived continuum from working in the abstract to working more directly in the world. We have summarized this as a continuum from "in-the-mind" to "in-the-world".

Together these considerations allow us to identify the dimension of greatest variance (the x-axis) as a continuum between "in the mind" and "in the world", and the dimension with the second greatest variance (the y-axis) as a continuum between "describing and explaining" and "doing mathematics".

Limitations & discussion

We have discussed earlier that the available data is inferior. The small number of teachers and registered teaching activities, the fact that no effort was made to ensure that the involved teachers were representative, and the uncertainty due to the lack of user input validation, prevent us from drawing confident conclusions. This paper and our use of Goal Arrow data is a pilot application of the method we have described. The emphasis in this paper is on exploring an approach, and especially the extent to which we can meaningfully interpret the visualisations. Our interpretation of the current data has allowed us (as a hypothesis or heuristics building exercise) to define a space of mathematical competences. In this space, we define the dimension of greatest variance (the x-axis) as a continuum between "in the mind" and "in the world", and the dimension with the second greatest variance (the y-axis) as a continuum between "describing and explaining mathematics" and "doing mathematics".

What is most important is that we expect to be able to use ENA for visualizing massively more of the same sort of data when made available from the learning management systems currently being implemented in Denmark. We now have reason to believe that this data and their visualisation in ENA can be interpreted meaningfully. We must recognize that these interpretations are based on what teachers have expressed about their lessons, which may differ from what they have actually planned, and to a greater degree from what is actually enacted in class. However, the process of associating lessons with competencies is well established in Denmark.

Defined spaces like the one above can be thought of as adding user friendliness or simplifying an existing understanding of mathematical competencies; we are looking at a 2D space instead of a six-columned table. However, there is also a sense in which the defined space is a development of an understanding of mathematical competencies in its own right. When the teachers relate the competencies in individual teaching activities, they are adding information. They are pointing at how mathematical competencies are likely to be engaged together. They are also negotiating between the formal breakdown of the competencies and their own understanding of what mathematics requires. Because of this, we can think of these defined spaces as new conceptual models of mathematical competencies developed using leaning analytics.

We can think of these conceptual models as a complement to the competency flower model. Most obviously, they gives us another way to think about the competencies. When planning lessons, a space such as the one presented in this paper may work as a tool to support teachers' reflection and discussion on the direction of their teaching activities. Supported interactively, we can imagine teachers aiming their teaching activities at locations in the space, registering teaching activities by associating these locations to teaching activities and compare their teaching activities to those of other teachers' or their own over time. Such empirically driven conceptual models could also potentially support research and policy. For example, they could direct the attention to challenges or differences in the progression of mathematics education. Specifically, it can raise flags for individual grades, that seem to lack certain areas or have significantly different networks.

Conclusion

In this paper, we consider the context in which mathematical competencies are used in Denmark, and use data collected in the Goal Arrow software, where teachers associate competencies to their teaching activities. We describe the creating of an ENA set and the plotting of a visualisation comprising a 2D space displaying a network based on the co-occurrence of mathematical competencies in teaching activities. Notwithstanding limitations on the data, we attempt a thorough interpretation of the visualisation which allows us to explore the approach as a way of understanding how teachers use the competencies framework in their lesson planning.

Figure 1 :

 1 Figure 1: The space of mathematical competencies


Sinéad Breen
email: sinead.breen@dcu.ie

Caitríona Ní Shé
email: caitriona.nishe2@mail.dcu.ie

Ann O'shea
email: ann.oshea@mu.ie

Conjecturing tasks for undergraduate calculus students

Keywords:

We present a hypothetical learning trajectory for a sequence of tasks designed for a calculus module. The purpose of the tasks was to give undergraduates opportunities to use technology to experiment and make conjectures while developing their understanding of the effects of translations on graphs. We consider data from task-based interviews with two students. The hypothetical learning trajectory for this sequence of tasks is compared with the actual learning trajectory of the students, and we conclude there was some evidence that our learning goals were achieved.

Introduction

The types of tasks that students work on can influence the reasoning and learning processes in which they engage [START_REF] Jonsson | Learning mathematics through algorithmic and creative reasoning[END_REF]. In Ireland, recent studies have highlighted that the majority of tasks both in secondary school textbooks and in undergraduate calculus modules could be solved with imitative reasoning, that is by memorization or following a familiar algorithm [START_REF] O'sullivan | An analysis of mathematical tasks used at second-level in Ireland[END_REF][START_REF] Mac An Bhaird | A study of creative reasoning opportunities in assessments in undergraduate calculus courses[END_REF]. In this context it is important to design tasks which give students opportunities to develop higher-order mathematical thinking skills, such as conjecturing and generalizing, to move them away from rotelearning. The first and last authors (Breen & O'Shea, 2018) designed a framework of task types for undergraduate calculus modules with the aim of developing mathematical thinking skills such as those suggested by Mason & Johnston-Wilder (2004, p. 109). Subsequently, interactive versions of some of the tasks were developed using the dynamic geometry software GeoGebra. In this paper we will consider a hypothetical learning trajectory (Simon, 1994) for a set of two conjecturing tasks designed using Geogebra on the topic of graph transformations and present data from task-based interviews to explore the actual learning trajectories engendered by this sequence of tasks.

Theoretical Framework Task Design

The framework of mathematical task types used here has six task types: evaluating mathematical statements; generating examples; analysing reasoning; visualizing; using definitions; conjecturing and generalizing (Breen & O'Shea, 2018). We will focus here on the last of these; we will first review the literature to present a rationale for this task type and for using technology in the design.

The acts of conjecturing and generalizing are well-known to be part of the tools of a professional mathematician [START_REF] Bass | Mathematics and teaching[END_REF]; indeed, Bass describes the progress of most mathematical work as starting with exploration and discovery, then moving on to conjecture, and finally culminating in proof. He identifies two phases of reasoning here: reasoning of inquiry (incorporating exploring and conjecturing) and reasoning of justification (rooted in proof). The acts of conjecturing, generalising, experimenting and visualising are included in the list of processes which aid mathematical thinking given by [START_REF] Mason | Designing and using mathematical tasks[END_REF]; these authors also discuss 'natural' powers that learners possess such as the ability 'to imagine and detect patterns,..., to make conjectures, to modify these conjectures in order to try to convince themselves and others' (Mason & Johnston-Wilder, 2004, p. 34). They stress the importance of creating a 'conjecturing atmosphere', so that students can participate in inquiry and develop their mathematical thinking skills. [START_REF] Dreyfus | Advanced mathematical thinking processes[END_REF] defines generalizing as "to derive or induce from particulars, to identify commonalities, to expand domains of validity" (p. 35). He acknowledges the important role played by generalizing in the process of abstraction, in moving from a particular instance to a generality, and notes the difficulty that many students have with generalization. [START_REF] Swan | Designing a multiple representation learning experience in secondary algebra[END_REF] explains that the process of identifying general properties of a concept from particular cases is one with which a student must be able to engage in order to come to truly understand a concept. Breda and Dos Santos (2016) examined how GeoGebra tools can enable students to conjecture and provide mathematical proof, and recommended such tools be used to support the study of complex functions. The use of technology has a number of advantages: for instance, information can be gathered and processed quickly so that teachers and students can make decisions efficiently to exploit learning opportunities; moreover, the burden of computation can be removed or reduced to allow students to explore and experiment. [START_REF] Borwein | The experimental mathematician: The pleasure of discovery and the role of proof[END_REF] described specific benefits of the use of technology to mathematicians, including: to gain insight and intuition, to discover new patterns and relationships, to expose mathematical principles through graphs, to test and falsify conjectures, to explore a possible result to see if it merits formal proof, to do lengthy computations. All of these have an important role to play in responding to a conjecturing/generalizing task. [START_REF] Simon | Reconstructing mathematics pedagogy from a constructivist perspective[END_REF] introduced the notion of a hypothetical learning trajectory (HLT) as part of a model of mathematics teaching. The HLT is made up of three parts: learning goals (as set by the instructor); learning activities (designed or selected by the instructor); and the hypothetical learning process (the instructor's prediction of how student thinking will develop during the learning activities). Simon describes the symbiotic relationship between the learning activities and the hypothetical learning processesthe ideas which underlie the learning activities are based on the instructor's beliefs about student learning, and these in turn are influenced by what is observed during the learning activities. Thus theory informs practice and vice versa. [START_REF] Simon | Explicating the role of mathematical tasks in conceptual learning: An elaboration of the hypothetical learning trajectory[END_REF] advocate the use of HLTs in task and curriculum design (especially for 'problematic' topics) as a mechanism to ensure that adequate thought is given to how student learning might evolve during activities, and as a means to study the success of learning activities. The HLT construct has been used to study teaching tasks and sequences of tasks in a variety of settings including undergraduate mathematics courses. Andrews-Larson, Wawro and Zandieh (2017) note that HLTs are useful ways of tying theory to practice, and use the notion of HLT to outline how certain tasks could lead to undergraduate students developing new understanding in Linear Algebra. [START_REF] Stylianides | Facilitating the transition from empirical arguments to proof[END_REF] compared HLTs and actual learning trajectories to provide evidence that an instructional sequence of tasks had achieved the desired goals.

Hypothetical Learning Trajectory Construct

Our Task Sequence and Hypothetical Learning Trajectory

We will consider the HLT for our task sequence and the actual learning progression of two students.

Learning Goals [START_REF] Eisenberg | On understanding how students learn to visualize function transformations[END_REF] discuss the fundamental importance of developing 'function sense' with undergraduate students. They describe facets of this as including dependence, variation, covariation and the effects of operations on functions. One of the most important components of function sense is the flexibility to move between multiple representations of a function. The key to solving many problems is to think of them visually, using a graph -including problems encompassing the main facets of functions mentioned above. However, Eisenberg and Dreyfus report that many students (even those more advanced mathematically) are reluctant to do so.

The function operations that we focus on in this paper are graph transformations, specifically vertical and horizontal translations of the graph of functions from R to R. The learning goals are:

1. observing and articulating the effects of translations on graphs: in particular, describing the relationship between the graphs of and those of for (where and ); 2. observing that, in general, the functions and are different when ; 3. making conjectures, in particular generalizing data from examples observed; 4. using the technology to undertake experiments. Note that we have both local (1 and 2) and more global (3 and 4) learning goals for student reasoning and skill development arising from our task sequence. Goal 1 targets the flexibility to move between representations of a function. The goals of the task sequence do not include students providing proofs for their conjectures since our tasks deal with [START_REF] Bass | Mathematics and teaching[END_REF] reasoning of inquiry rather than reasoning of justification.

The Task Sequence

For the last number of years, we have been developing a bank of tasks using our framework. Originally these tasks were paper-based and aimed to give students opportunities to explore, spot patterns, and make conjectures based on their observations. We noticed that some students had difficulties drawing the graphs of the functions mentioned and so were not able to generalize or make a conjecture. In 2016, we redesigned these tasks using GeoGebra; we will refer to these as Tasks A and B. (Task A is shown in Figure 1 and Task B is similar except with etc.). The computational burden was thus removed from the students and we hoped that this would allow them more freedom to experiment and conjecture appropriately. In contrast with the paperbased tasks, the use of GeoGebra allowed us to enable students to quickly see graphs of the form (Task A), and (Task B), for values of a ranging over an interval. Both tasks, and others from this project, can be found at http://mathslr.teachingandlearning.ie/GeoGebra/.

Hypothetical Learning Progression

As students engage with the task sequence we expect the following activity and learning from them:

 experimenting with the sliders;  noticing how the graph of the function changes as changes, in particular noticing the difference in behaviour for positive and negative values of ;  remarking in the case of Task A (respectively Task B) on the vertical (respectively horizontal) shift of the graph and expressing the relationship between and (respectively ) mathematically;  noticing analogous relationships in the cases of and to identify a pattern;  using the data from , , and to make a conjecture about the effects of transformations of the types in Task A and B on graphs;  amalgamating the relationships observed to realise that, in general, the functions and are different when .

Data collection and analysis

In order to investigate the use and effectiveness of the tasks, the second author carried out a series of task-based interviews with a sample of students from a first-year calculus module in which some GeoGebra tasks were trialed. Four students were asked to think aloud while completing a selection of the tasks. Pre and post-tests, each consisting of the same four questions (where one question had two parts), were used at the beginning and end of the interviews in order to help determine if the students' mathematical thinking had changed as a result of completing the tasks. The interviews, which lasted about an hour, used purpose built software to record video, audio, screen and mouse movements. Each student completed between four and seven GeoGebra tasks depending on how quickly they moved through them. The interviews were transcribed by the second author using the audio recording to which she added a summary of what was happening onscreen at that time. The transcriptions were analysed for significant incidents by two of the three authors, and their results were compared and agreed on. i. What is the relationship between the pair of graphs and below? [The graphs of (with initially set at 1) and are shown as well as a slider which allows to range from -5 to 5. When the value of changes the graph changes accordingly.]

ii.

What is the relationship between the pair of graphs and below? [The graphs of (with initially set at 1) and are shown as well as a slider which allows to range from -5 to 5.]

iii.

What is the relationship between the pair of graphs and below? [The graphs of (with initially set at 1) and are shown as well as a slider which allows to range from -5 to 5.] iv.

Can you make a general conjecture about the relationship between the graphs of and from your observations about the graphs of the pairs of functions above? What happens when ? What happens when ? What happens when

Results

We will consider in some detail the responses of two of the three students who completed the pretest, then worked on Tasks A and B in the task-based interviews, and subsequently completed the post-test (see Notices that the y-intercepts of the graphs of f and h depend on the choice of a. For g, he notices that the horizontal asymptotes depend on a, (but does not use correct terminology).

Is able to make the expected conjecture.

Notices that the x-intercept varies according to choice of a. (He calls it the origin).

Is able to make the expected conjecture.

Says the statement is false and explains by referring to graph transformations.

Table 1: Student responses in task-based interview

Both Áine and Seán were able to use the sliders to obtain graphs of the different translations of the functions in question. Áine worked quickly on both tasks (she spent 2-3 minutes on each of them), she spotted the pattern and was able to verbalise it using mathematical language. For the first pair of functions on Task A she said: So as I can see here as I am taking values away from the graph shifts down the y-axis and as I add values it shifts up the y-axis. She did the same for the graphs of the translations of , predicted what would happen with , and was able to conjecture: when a >0 the graph of f moves up the y-axis and when it's less than zero it moves down.

Similarly on Task B, she was able to use the slider to generate vertical translations of the three functions, and she made a conjecture generalizing the pattern she observed.

Seán spent about 6 minutes working with the three graphs on Task A. He used the slider to examine how the functions changed for the range of values of a. He also was able to spot a pattern but focused on certain features of the graphs instead of the whole graph; for the translations of f and h he spoke about their y-intercepts (but used the term origin), and for the translations of g he noticed that the horizontal asymptotes depend on a (but did not use this term). For the general conjecture he scrolled back to the first pair of graphs, moved the slider, then looked at the other pairs of graphs:

All the graphs shift upwards in the y direction by the value whichever value a is from the original position of . When a<0 all the graphs shift down in the y direction by whatever value a is in the… by whatever value a is from wherever happens to be.

Seán worked through Task B in a similar manner for about 6 minutes. However, when asked to give a general conjecture this time Seán gave an appropriate response immediately without having to scroll back up through the three functions, as he did for the general conjecture in Task A.

Discussion

We have only presented evidence from two students who worked on a pair of conjecturing/generalizing tasks. However, from these case studies, we can draw some conclusions for these students' learning. Our data suggests that both students achieved learning goals 1, 3 and 4 as they were able to the technology to experiment with the translations, they spotted patterns and were able to verbalise them, and they were able to make conjectures based on their experiments. The students' responses to question 4(ii) on the preand post-tests give us cause to believe that the students have also achieved learning goal 2 during the task sequence. Both students gave an incorrect answer in the pre-test but in the post-test both revised their answers. Áine recognized that the statement is true for but not otherwise, and used her experiences on Tasks A and B to explain her reasoning. In the pre-test Seán considered one numeric example in order to explain his response to question 4(ii). When Seán completed the post-test question 4(ii) he immediately stated that his original response was incorrect and referred to the vertical and horizontal translations from Tasks A and B. Finally, at the end of the interview, Seán was asked if he considered any of the tasks helped him respond to the post-test questions, and he said:

[Tasks A and B] helped me to see and distinguish the differences in changing the values of a because I didn't fully grasp what it was in the beginning.

It is clear from the task-based interviews that GeoGebra took away the burden of computation; if we had asked students to draw the graphs of the three pairs of functions in Task A by hand, then it would probably have taken them a long time and they may have made mistakes. The use of the sliders in GeoGebra, allowed the students to watch how the graphs changed as the values of a changed, and they were then able to spot the pattern and then make a conjecture. [START_REF] Eisenberg | On understanding how students learn to visualize function transformations[END_REF] found that the students involved in their teaching experiment seemed to view function transformations as a sequence of two static states (the initial and final graphs) rather than as a dynamic process. They concluded that this may have been as a result of the graphing software available to them in which there was no means to see the continuous transformation developing before the students' eyes. The advent of dynamic geometry software, such as GeoGebra, means that this is no longer an issue: the students we interviewed for our study seemed to have developed an understanding of function transformation as a dynamic process.

The use of software like GeoGebra makes visualization more immediate for students and we posit that this can help with engagement. We saw, probably because of the ease of visualization in Tasks A and B, that both students felt comfortable in making a conjecture. This corresponds with [START_REF] Borwein | The experimental mathematician: The pleasure of discovery and the role of proof[END_REF] description of how mathematicians use technology in their own work, and we suggest that giving students the opportunity to use technology in this manner might encourage them to develop mathematical thinking skills [START_REF] Mason | Designing and using mathematical tasks[END_REF].

Furthermore, in the pre-test Seán seemed to see Q4(ii) as referring to a single function, but in the post-test he immediately recognizes that it is a general statement. We suggest that it is his experience of working on Tasks A and B that accounts for this change in perspective, although it may be that he is recalling earlier understanding rather than developing it during the task sequence. We note that the ability to appreciate the distinction between an instance and a generality is crucial in the development of mathematical thinking [START_REF] Dreyfus | Advanced mathematical thinking processes[END_REF]). Seán's response could also be interpreted as a move towards seeing functions as objects rather than simply actions. [START_REF] Eisenberg | On understanding how students learn to visualize function transformations[END_REF] suggested that students found transformations in the horizontal direction (e.g.

) more difficult than those in the vertical direction (e.g. ). They contended that one reason for this may simply be that more is involved in visually processing f(x+a) than f(x)+a. However, we found no evidence of this in the think-aloud interviews with our students. In fact, the students were quicker to conjecture, and more articulate in their description of, a general relationship between the graphs of f(x) and f(x+a) (Task B) than between f(x) and f(x)+a (Task A) which we supposed was due to the order in which the tasks were presented.

It may be that our students would have had more difficulties justifying their conjecture in Task B rather than Task A, but such justifications were not part of our task sequence. One might criticize our task sequence as not being cognitively challenging, but we wanted to focus on conjecturing rather than proving. The tasks could easily be modified to allow students to input other functions in order to check their hypothesis, and could be expanded to ask for justifications or proofs.

We agree with [START_REF] Simon | Explicating the role of mathematical tasks in conceptual learning: An elaboration of the hypothetical learning trajectory[END_REF] that the HLT construct is useful in task design as it highlights the importance of having clear learning goals and an informed view of how learning might take place at all stages of the design process. We feel that it can help when designing new versions of tasks if the original learning goals are not met, and furthermore provides a way of evaluating tasks by comparing the hypothetical learning process with actual learning.

We have found some evidence that conjecturing tasks can encourage students to experiment and explore. We note that [START_REF] Bass | Mathematics and teaching[END_REF] sees this exploration as the first step in most mathematical work, and therefore it is a necessary skill that students should develop to improve their mathematical thinking. With this aim in mind, we hope to continue to design and evaluate tasks of this type.

Figure

  Figure 1: Task A

Task A :

 : (GeoGebra Task) Use the slider on each graph to change the values of in the functions , , .

Table 1

 1 All three students answered Q4(ii) correctly on the post-test but two of them (given pseudonyms Áine and Seán) gave incorrect answers on the pre-test.

		below). One question which appeared on the pre-test and post-test was the
	following:
	Q4(ii) Suppose f(x) is a function defined for all real values of x. Decide if the statement is true or
	false. Explain your answer. If a is any real number then f(x+a) = f(x)+a for all values of x.
	Student Q4(ii) on pre-test	Task A	Task B		Q4(ii) on post-test
	Áine	Says 'It's a function' and writes 'true'. Later	Is able to verbalise the relationship between the	Is verbalise able	to the	Writes: if a=0 but not for other values
		she says she thought	graphs and is able to make	relationship		of a. Explains by
		that the question was	the expected conjecture.	between	the	referring to the
		asking whether the		graphs and is	GeoGebra tasks.
		expression		able to make the
		f(x+a)=f(x)+a		expected
		describes a function.		conjecture.
	Seán	Writes 'True' and gives example with
		f(x)=x, a=1, x=1. Later
		when asked what he
		thought this question
		was asking: I took it as
		a set [particular]
		function rather than an
		arbitrary function.

Acknowledgment

Nairtl and The National Forum for the Enhancement of Teaching and Learning funded this study.


Fabien Ringeval
email: fabien.ringeval@unifr.ch

Mohamed Chetouani
email: mohamed.chetouani@upmc.fr

Björn Schuller
email: schuller@tum.de

Novel Metrics of Speech Rhythm for the Assessment of Emotion

Keywords: speech rhythm, prosodic features, emotion recognition

Whereas rhythmic speech analysis is known to bear great potential for the recognition of emotion, it is often omitted or reduced to the speaking rate or segmental durations. An obvious explanation is that the characterisation of speech rhythm is not an easy task itself and there exist many types of rhythmic information. In this paper, we study advanced methods to define novel metrics of speech rhythm. Their ability to characterise spontaneous emotions is demonstrated on the recent Audio/Visual Emotion Challenge Task on 3.6 hours of natural human affective conversational speech. Emotion is assessed for the four dimensions Activation, Expectation, Power and Valence as binary classification tasks on the word level. We compare our new rhythmic feature types to the official 2 k bruteforce acoustic baseline feature set on the Audio Sub-Challenge. In the results, the rhythmic features achieve a promising relative improvement of 16% for Valence, whereas the performance is more contrasted for the three others dimensions.

Introduction

Most definitions from the literature consider rhythm as being conveyed by the perceived information during the alternation (or repetition) of events spaced over time. However, this definition considers events from various origin such as: (i) biological (e.g., food, heart, respiratory, etc.), (ii) body (e.g., choreography) or (iii) from speech [START_REF] Cummins | Speech rhythm and rhythmic taxonomy[END_REF]. Rhythm thus refers to the notion of dynamic movement in the perception of different types of phenomena. Because these phenomena are very diverse, their entanglement is evident for speech [START_REF] Tilsen | Multitimescale dynamical interactions between speech rhythm and gesture[END_REF], and since the mechanisms of human perception are also very complex [START_REF] Zwicker | Psychoacoustics: facts and models[END_REF], identifying in concrete and simple terms the characteristics of rhythm is particularly difficult. This paper investigates different metrics of speech rhythm, with the aim to study their relevance for the characterisation of spontaneous emotions from natural human conversational speech. The goal of this work is to help provide new relevant features for the analysis of affective spontaneous interactions. In the remainder of this paper we briefly introduce some rhythmic phenomena that were identified in the literature (Sec. 2), the metrics of speech rhythm (Sec. 3), including the taxonomical models (Sec. 3.1) and the new advanced ones (Sec. 3.2), the experimental setup (Sec. 4), including the database (Sec. 4.1) and the rhythmic feature set (Sec. 4.2) and present experimental results (Sec. 5) before concluding (Sec. 6).

Speech Rhythm

A literature survey on rhythm shows how difficult it is to have a precise definition of what it refers to, since many conceptual and terminological inventories are available [START_REF] Evans | Rhythm in psychological, linguistic, and musical processes[END_REF]. However, a set of pretty well established phenomena have been identified so far, such as: (i) the duality between form and structure [START_REF] Fraisse | Les structures rythmiques : étude psychologique[END_REF], (ii) some temporal distortions [START_REF] Zwicker | Psychoacoustics: facts and models[END_REF] and (iii) some preferential anchors according to the language [START_REF] Cummins | Speech rhythm and rhythmic taxonomy[END_REF].

The majority of studies that were conducted on speech rhythm were guided by a taxonomical spirit, i.e., in order to perform a classification of languages [START_REF] Cummins | Speech rhythm and rhythmic taxonomy[END_REF]. The emergence of new metrics of rhythm in the last decade brought a revival of interest in the taxonomical community. Temporal properties of consonants and vowels were used to argue for the existence of a rhythmic continuum between stress (e.g., English and German) and syllabic (e.g., Spanish and French) languages [START_REF] Gibbon | Measuring speech rhythm[END_REF].

Because emotions clearly rely on dynamic processes in both their production and perception counterparts [START_REF] Scherer | Psychological models of emotion[END_REF], new advanced models of speech rhythm would help to provide relevant features for their characterisation. Indeed, the automatic processing of both spontaneous and natural emotions still remains a challenge, specifically when it comes to real data analysis.

Metrics of Speech Rhythm

We describe in the following paragraphs several techniques that have been proposed to characterize the rhythmic information conveyed by speech. First we present the techniques used by the taxonomical community and then we describe our novel metrics of speech rhythm.

Taxonomical Models

The taxonomical models of speech rhythm use measurements of the segmental duration or the quantity of segments according to a given speech unit (e.g., vowels, consonants, words, etc.). The speech rate is, for example, often used as a unique feature of rhythm in emotion recognition systems [START_REF] Ang | Prosodybased automatic detection of annoyance and frustration in humancomputer dialog[END_REF], although many studies have shown that it is only one component [START_REF] Dellwo | The role of speech rate in perceiving speech rhythm[END_REF]. In addition, there are many metrics whose use for the characterisation of affective correlates from speech could be studied.

Vocalic and Consonantal Variability Phenomena.

Ramus et al., proposed a measure of speech rhythm based on the percentage of vocalic intervals (%V) and the standard deviation of consonantal intervals (∆C), with the aim of quantifying a rhythmic continuum between stress and syllabic languages [START_REF] Ramus | Correlates of linguistic rhythm in the speech signal[END_REF]. However, these measures would only be relevant for the study of a corpus for which the speech rate is strictly controlled. Brady et al., used circular statistical measures to study the cognitive processes of speech for Japanese [START_REF] Brady | Speech rhythm and rhythmic taxonomy[END_REF]. After a detection of the attacks of voiced syllables, a sinusoidal waveform was generated with a period set by the average interval duration between the segments. Therefore, the position of segments in time is represented by a value of phase θi in the generated sinusoid. The periodicity of segments, R, was then quantified as the sum of the vectors corresponding to the values of phase θi, divided by the number N of segments.

R = 1 N   N i=1 sin 2πθi 2 + N i=1 cos 2πθi 2   1/2
(1)

3.1.3. Variation Coefficient of Segmental Duration.
The coefficient of variation (V arco), which is defined as the ratio between the standard deviation σ and the mean µ of a given distribution, was used on vocalic and consonantal intervals by [START_REF] Dellwo | Rhythm and speech rate: A variation coefficient for ∆C[END_REF]. This measure was combined with that of %V and allowed to discriminate syllabic from stress languages. However, the differences were found to be at least as significant between the dialects of these languages.

3.1.4. Pair-wise Variability Index.

Grabe and Low proposed to measure the temporal variability of pairs of successive phonetic intervals (I k and I k+1 ) by using the rPVI [START_REF] Grabe | Durational variability in speech and the rhythm class hypothesis[END_REF] (2); a normalisation to the speaking rate has been proposed with the nPVI (3). These measures helped to strengthen the theory of rhythmic language classes outlined above.

rP V I(k) = I k+1 -I k (2) nP V I(k) = 2 I k+1 -I k I k + I k+1 (3)
Other studies suggested that the comparison of time intervals between phonetic units could be achieved by using their ratio, instead of their difference [START_REF] Gibbon | Measuring speech rhythm[END_REF]. The rhythm ratio (RR) measure (4) provided results close to the nPVI on corpora of languages.

RR(k) = I k I k+1 (4)

Advanced Models

Some authors have proposed to expand the definitions of the metrics used in the taxonomical models [START_REF] Smith | A multiresolution time-frequency analysis and interpretation of musical rhythm[END_REF]. It was suggested that the phenomena of rhythm could be generated by the dynamics in prosody. Indeed, Lerdahl et al. showed that pitch serves to distinguish the accents of music in groups of: (i) metric, (ii) phenomena and (iii) structures [START_REF] Lerdahl | A generative theory of tonal music[END_REF]. Moreover, prosodic particularities seem to be related to the strong beats of rhythm, since they fall at: (i) important changes or low values in the pitch, (ii) changes in the harmonics or (iii) changes in rates.

In the following, we first present two advanced methods of features extraction that use the repartition of speech units over time to quantify rhythm. Whereas the final two techniques use both temporal information and changes in the prosodic shape of consecutive speech units to characterise speech rhythm.

Low Fourier Frequency Analysis.

Tilsen et al. proposed a method to extract the rhythmic envelope of the speech signal [START_REF] Tilsen | Low-frequency Fourier analysis of speech rhythm[END_REF]. A low frequency (LF) signal is calculated on the speech signal by several filtering steps, which are supposed to represent the process of perception of rhythm. As the waveform of the LF signal is stationary, a Fourier transform can be used to estimate the values of entropy, centroid and the average frequency from the rhythmic envelope.

Instantaneous Frequency and Envelope.

We proposed in [START_REF] Ringeval | Hilbert-Huang transform for nonlinear characterization of speech rhythm[END_REF] to use the Hilbert-Huang Transform (HHT) for characterising speech rhythm. A speech unit interval (SUI) signal is first generated by a resampling process (cubic spline, Fs = 32 Hz) on interval durations of speech units. Because the interval duration between phonemes is known to vary from 60 ms to 1 second (i.e., from 1 Hz to 16 Hz) [START_REF] Drullman | Effect of temporal envelope smearing on speech reception[END_REF], all frequencies of speech rhythm can be captured with a sampling frequency set to 32 Hz. Empirical mode decomposition (EMD) is then applied on the SUI signal to extract the HHT derived features: instantaneous amplitude and frequency from the sum of the three first intrinsic mode functions (provided by the EMD), and the mean instantaneous frequency (MIF) obtained by the calculation proposed in [START_REF] Xie | Mean frequency derived via Hilbert-Huang transform with application to fatigue EMG signal analysis[END_REF].

Prosodic Pair-wise Variability Index.

In order to characterize the rhythm through the dynamic of prosody, we propose to extend the PVI [START_REF] Grabe | Durational variability in speech and the rhythm class hypothesis[END_REF]. The time interval measurement I k is replaced by the Varco coefficient of given prosodic low-level-descriptor (LLD), such as pitch, loudness, spectral flux, ... [START_REF] Schuller | The relevance of feature type for the automatic classification of emotional user states: low level descriptors and functionals[END_REF]. A normalisation factor α is used to take into account durations d of consecutive segments k and k + 1 and their interval duration I k in the pPVI metric:

pP V I(k) = α. (V arco k+1 -V arco k ) (5)
with α = log

d k d k+1 I k d k + d k+1 + I k and V arco = σ µ
The value of this feature is equal to zero if the dispersion of the LLD is identical on two consecutive segments of speech, which means a monotony in the prosodic component. Otherwise, the values depend on the amount of changes present in the Varco of the LLD between the speech segments. Due to the normalisation factor α, the values of pPVI also depend on both the duration and interval of the two consecutive segments; these effects are cumulative. The logarithm of the duration ratio was computed to reduce its variability. Finally, a local maximum (or minimum) in the pPVI defines prominence in a given prosodic LLD.

Prosodic Hotelling Distance.

The Hotelling distance (HD) is a measure for comparing the statistical distribution of two data sets by a calculation similar to the Mahalanobis distance. In particular, it involves a normalisation factor by the duration of the two compared speech segments. However, as the interval duration between these segments is not included in the HD calculation, we added this value using the normalisation coefficient α. This new metric is termed the prosodic Hotelling distance (PHD):

P HD(k) = α (µ k -µ k+1 ) T Σ -1 k∪k+1 (µ k -µ k+1 ) (6)
where µ k , µ k+1 denote the means of a prosodic LLD on the past (k) and new (k + 1) speech segments, and Σ k∪k+1 the covariance matrix of the joint events k and k + 1. This measure can be used for one or several LLDs, and two different techniques are available to define the matrix Σ k∪k+1 : The value of the PHD is equal to zero when the distributions of the prosodic LLD(s) are identical between pairs of consecutive segments, and positive in all other cases. It varies proportionately with the amount of change present in the statistical distribution of the LLD(s), and the normalisation factor α also influences the values of the PHD, like for the pPVI.

Experimental Setup

In this section we describe the methodology we used for the AVEC emotion recognition Audio Sub-Challenge. Our approach consisted of comparing the relevance of our rhythmic features with the official prosodic 2 k brute-forced acoustic baseline feature set.

Database

The Solid-Sensitive Artificial Listener (SAL) part of the SE-MAINE corpus was used for the AVEC challenge [START_REF] Schuller | AVEC 2011 -The first international audio/visual emotion challenge[END_REF]. In this database, participants were asked to talk in turn to four emotionally stereotyped human operators. The used language was English and all the sessions were split in three: a training, develop, and test partition. Table 1 shows the distribution of data in sessions and words for each partition. Activation, Expectation, Power, and Valence compose the annotated emotion dimensions of the Solid-SAL corpus. The binary labels of each affective dimension were obtained at the word level by a threshold on the continuous values that were rated using the tool called Feeltrace. Table 2 lists the fraction of positive instances per partition and per dimension from the Solid-SAL corpus.

Rhythmic Features

The taxonomical models of speech rhythm provide 6 LLDs at the word level: duration, interval, rPVI, nPVI and RR (for both duration and interval). As functionals are computed on these LLDs for each word, values were resampled by cubic splines (Fs = 32 Hz). 30 functionals (cf. Table 1 in [START_REF] Ringeval | Hilbert-Huang transform for nonlinear characterization of speech rhythm[END_REF] plus second order regression coefficient, mean and standard-deviation value of raising / falling values and mean number of raising / falling values) were then computed on the 6 LLDs. 4 additional features: word rate, periodicity, and Varco of durations and intervals were merged, so that 184 features of speech rhythm were returned in total by the taxonomical models.

The new advanced models provide much more features. The pPVI and PHD metrics were performed at the word level on each LLD returned by the openSMILE feature extraction's toolkit [START_REF] Eyben | The Munich versatile and fast open-source audio feature extractor[END_REF]; 25 energy and spectral related LLDs, plus 6 voicing related LLDs with delta coefficients. Obtained values were resampled by cubic splines (Fs = 32 Hz) and the 30 functionals were computed at the word level. Finally, the 30 functionals were also applied on the HHT derived features (instantaneous envelope and frequency) and 4 additional features: MIF, entropy, spectrum and mean frequency of LF filtered speech signal were merged to the feature vector; 3784 features of speech rhythm are returned in total by the new advanced models.

Results

The goal of our experiments is to evaluate the relevance of both taxonomical and new advanced models of speech rhythm for the emotion recognition. In order to compare these rhythmic feature sets with the prosodic one, we used the same classification strategy as in the challenge baseline: we used Support Vector Machine (SVM) classification with linear Kernel, Sequential Minimal Optimisation (SMO) for learning and optimised the complexity on the development partition. The SMO implementation in the WEKA toolkit was used. Because the rhythmic features are numerous and include some redundancy, especially for the new advanced models, we used a correlation-based feature selection method (CFS). This technique reduces the feature space by keeping the features that are highly correlated with the emotional classes while having low inter correlation.

When CFS is performed on all available features of speech rhythm, only 11 of them are kept for Activation and Valence, 8 for Expectation and 6 for Power. In the mean, 45% of these features are derived from the pPVI and PHD metrics, 33% from the LF signal, 12% from the HHT and the last 10% from the taxonomical models. This result shows the relevance of using dynamic to characterize emotion from speech on two different sides: local dynamic (pPVI and PHD) and global dynamic (LF). Indeed, local information on both temporal and prosodic shape changes over consecutive words are captured by the pPVI and PHD metrics, whereas the spectrum of LF signal provides global information on the envelop of speech rhythm. Results also show that using local information (i.e., interval duration between words) to estimate the envelop and the frequency of speech rhythm (HHT) appears to be much less relevant than using global information (i.e., spectrum of LF signal). Results obtained by the rhythmic features in the emotion recognition task, as well as those from the baseline (2 k brute-forced acoustic features [START_REF] Schuller | AVEC 2011 -The first international audio/visual emotion challenge[END_REF]), are given in Table 3 for each affective dimension.

In the results, the speech rhythm features outperform the baseline for Valence. The relative improvement obtained by the taxonomical features is equal to 27% for the develop partition and 16% for the test partition (UA measure); 28% and 4% respectively for the advanced features. On the other hand, results are much more contrasted for the three others dimensions. The rhythmic features achieve a lower performance than the baseline for Activitation and both develop and test partitions, excepted with the WA measure and the advanced models for test. The best scores are achieved by the rhythmic features for Expectation on the develop partition, whereas the baseline performs best on the test partition. Concerning Power, the speech rhythm features provide the best performance in all cases excepted for the test partition with the WA measure. In the mean, the baseline achieves best performance for the test partition and the speech rhythm features for develop with the UA measure. As a conclusion, results obtained by the features of speech rhythm outperform the prosodic features for emotion recognition for Valence, and achieve much more contrasted results on the others dimension. While there exists difference in the classification results between the two types of features, more detailed future investigation needs to be carried out to understand the relationship between these two types of measures and each emotional dimension. Also note that even if the taxonomical models of speech rhythm have been proposed with a rather different goal in mind, i.e., to quantify cross-linguistic difference rather than intra-language variation due to emotion, they still achieve good performance with best scores in some specific cases.

Conclusion

Both taxonomical and novel advanced models of speech rhythm were used for the spontaneous emotion recognition on the recent Audio/Visual Emotion Challenge task, which includes 3.6 hours of natural human affective conversational speech. The performance of these feature sets was compared with the usual prosodic features set. The rhythmic features achieve a promising relative improvement of 16% for Valence, whereas the performance is more contrasted for the three others dimensions. This study thus shows for the first time the relevance of using advanced models of speech rhythm for the characterisation of emotional correlates from speech, especially for Valence. Future work will use specific acoustic anchors of speech (e.g., automatically detected pseudo-phonemes [START_REF] Ringeval | A vowel based approach for acted emotion recognition[END_REF]) to provide different structural bases for the metrics of speech rhythm, as well as fusion techniques to estimate their complementarity in the emotion recognition task.

Table 1 :

 1 Overview of the AVEC dataset per partition

			Train Develop	Test
	# Sessions			31	32	32
	# Words		20 183	16 311 13 856
	Avg. word dur. [ms]	262	276	249
	Table 2: Overview of class balance: fraction of positive in-
	stances over total instances of words in the training and devel-
	oping partitions.
	Ratio	ACT.	EXP.	POWER VALENCE
	Train	0.496 0.409		0.560	0.554
	Develop 0.581 0.334		0.670	0.654
	(i) the first consists of filling the diagonal with the standard-
	deviation values of each LLD and (ii) the second technique ex-
	ploits all values from the covariance matrix.

Table 3 :

 3 Results on the AVEC 2011 Audio Sub-Challenge by the competition measure accuracy for speech rhythm and baseline features. WA stands for weighted accuracy, UA for unweighted accuracy.

	Accuracy	ACTIVATION	EXPECTATION	POWER	VALENCE		ALL
	[%]	WA	UA	WA	UA	WA	UA	WA	UA	WA	UA
					Taxonomical features
	Develop	43.5	52.4	66.7	66.8	67.7	67.2	66.9	66.9	61.2	63.3
	Test	54.8	53.1	47.9	50.0	26.2	41.4	50.7	54.8	44.9	49.8
					Advanced features
	Develop	45.2	51.0	67.3	67.3	66.9	67.1	67.2	67.9	61.7	63.3
	Test	57.6	54.1	51.1	53.0	20.0	50.0	46.5	49.2	43.8	51.6
					Baseline features
	Develop	63.7	64.0	63.2	52.7	65.6	55.8	58.1	52.9	62.7	56.4
	Test	55.0	57.0	52.9	54.5	28.0	49.1	44.3	47.2	45.1	52.0