Amal Houidhek

Vincent Colotte

Zied Mnasri
email: zied.mnasri@enit.rnu.tn

Denis Jouvet
email: denis.jouvet@loria.fr

Evaluation of Speech Unit modelling for HMM-Based Speech Synthesis for Arabic

Keywords: parametric speech synthesis, statistical modelling, Arabic language, speech unit modelling, vowel quantity, gemination

This paper investigates the use of hidden Markov models (HMM) for Modern Standard Arabic speech synthesis. HMM-based speech synthesis systems require a description of each speech unit with a set of contextual features that specifies phonetic, phonological and linguistic aspects. To apply this method to Arabic language, a study of its particularities was conducted to extract suitable contextual features. Two phenomena are highlighted: vowel quantity and gemination. This work focuses on how to model geminated consonants (resp. long vowels), either considering them as fully-fledged phonemes or as the same phonemes as their simple (resp. short) counterparts but with a different duration. Four modelling approaches have been proposed for this purpose. Results of subjective and objective evaluations show that there is no important difference between differentiating modelling units associated to geminated consonants (resp. long vowels) from modelling units associated to simple consonants (resp. short vowels) and merging them as long as gemination and vowel quantity information is included in the set of features.

Introduction

Making allowance for the development of human machine interaction, speech synthesis is becoming more and more widespread and omnipresent in mobile applications (e.g., mail reading service) and information delivery applications. Speech synthesis consists in the automatic generation of a natural voice from a written text; the process is called Text To Speech (TTS) synthesis [START_REF] Taylor | Text-to-speech synthesis[END_REF]. The automatic conversion of the text into a speech signal begins with a text analysis based on natural language processing. This corresponds to the front-end part, and consists in text segmentation into different levels (sentences, words...), text normalization, part of speech tagging, and conversion of the text into a sequence of phonemes. Then, a speech waveform is generated thanks to the concatenation of speech units corresponding to the desired sequence of phonemes or using speech parametric models. Developing a full speech synthesis system in a new language requires resources and knowledge. Active learning-based approaches have been proposed to reduce the need for language-specific expert knowledge [START_REF] Watts | Unsupervised and lightly-supervised learning for rapid construction of TTS systems in multiple languages from 'found'data: evaluation and analysis[END_REF]. The work presented in this paper deals with the adaptation of the speech modelling part to the Arabic language, and does not investigate the front-end part.

Amongst the TTS techniques, statistical parametric speech synthesis (SPSS) [START_REF] Tokuda | An HMM-based speech synthesis system applied to English[END_REF][START_REF] Black | Statistical parametric speech synthesis[END_REF] has been widely employed. SPSS is based on hidden Markov models (HMM) which can be automatically trained using speech data. Models depend on a representation of the speech signal with a set of parameters (e.g., duration of sounds, spectrum, and fundamental frequency (F0)). Statistics (e.g., means and variances of probability density functions) are used to describe the distribution of the speech parameter values in the training corpus. Note that, before being applied to speech synthesis, HMMs have been widely and successfully used for automatic speech recognition, thanks to several toolkits such as HTK [START_REF] Young | The HTK hidden Markov model toolkit: Design and philosophy[END_REF].

HTS, the HMM-based speech synthesis system, is based on HTK, and presents the advantage of being trainable, making changing voice characteristics possible and produces a rather good quality speech signal. It has been applied to many languages, such as Japanese [START_REF] Yoshimura | Simultaneous modeling of spectrum, pitch and duration in HMM-based speech synthesis[END_REF], English [START_REF] Tokuda | An HMM-based speech synthesis system applied to English[END_REF], and German [START_REF] Krstulovic | An HMM-based speech synthesis system applied to German and its adaptation to a limited set of expressive football announcements[END_REF]. The performance of HMM speech synthesis system depends on the parameterisation of the speech signal and on the modelling of speech units. In HTS, context dependent phoneme models are employed. This requires a description of each speech unit with a set of contextual features that includes factors affecting the pronunciation of the corresponding sound. The set of contextual features comprises linguistic as well as prosodic and phonological information to describe all the characteristics of the speech unit.

In practise, the choice of contextual features is highly prominent because they are involved in the different parts of HTS: in the training part when building decision trees for parameter sharing and in the synthesis part where the context-dependent HMM are used to predict speech parameters. Thus, they have a considerable impact on the quality of the generated speech. Although a standard set of features was proposed for English in [START_REF] Tokuda | An HMM-based speech synthesis system applied to English[END_REF], a part of contextual features is language-dependent. Hence, previous adaptation of HTS for other languages went through ignoring or adding some features to cover particularities of the target language. Le [START_REF] Maguer | Evaluation of contextual descriptors for HMM-based speech synthesis in French[END_REF] suggested a different set of features for French and evaluated the impact of adding new information on the modelling of acoustic parameters, whereas [START_REF] Silén | Analysis of duration prediction accuracy in HMMbased speech synthesis[END_REF] focused on evaluating the modelling of consonants and vowels durations for Finnish.

This paper investigates the adaptation of HTS, a widely used HMM-based speech synthesis system, for Modern Standard Arabic (MSA). The set of contextual features needs to take into account the specificities of Arabic language [START_REF] Al-Ani | Arabic phonology: An acoustical and physiological investigation[END_REF] to generate natural Arabic speech. As claimed by studies of Arabic phonology, MSA presents particular phenomena that are not considered in the standard set of contextual features, such as gemination and vowel quantity (long vs. short vowels). As stated by [START_REF] Khouja | Durée des consonnes géminées en parole Arabe : mesures et comparaison[END_REF], a geminated consonant is twice as long as its simple counterpart is, and a long vowel is twice as long as its short counterpart is. This paper proposes and evaluates several choices for modelling the speech units that differentiate, or not, units associated to long vowels from those associated to short vowels, and/or units associated to geminated consonants from those associated to simple consonants. In the experiments, information about gemination (simple or geminated consonant) and vowel quantity (short or long vowel) is always included in the set of contextual features. Arabic utterances generated by the various modelling approaches are compared to each other based on an objective evaluation of the duration of the speech segments, and on perceptive tests. Crowdsourcing approaches have been proposed for conducting subjective evaluation tests, but the detection of cheating is critical [START_REF] Buchholz | Crowdsourcing preference tests, and how to detect cheating[END_REF], and wrong decisions may bias the results. Thus, we have applied a more traditional evaluation procedure, where listeners were physically present for evaluations.

The rest of this paper is organised as follows. Section 2 presents a brief overview about speech synthesis and describes the HTS system. Section 3 describes the aspects of the Arabic speech to be taken into account in synthesis using HTS system, and introduces the proposed modelling approaches. Section 4 exposes the set of conducted experiments and discusses the results of objective and subjective evaluations.

Speech Synthesis

Brief overview

Previous works in TTS domain came up with several techniques to implement speech synthesis systems. Earlier methods relied on rule-based formant synthesis. They use phonetic units and produce speech based on rules of evolution of formants between phones [START_REF] Klatt | Software for a cascade/parallel formant synthesizer[END_REF][START_REF] Taylor | A real time speech synthesis system[END_REF]. Afterwards, a variety of concatenationbased methods have been developed. They are based on the use of a corpus, a database of recorded utterances, and they consist in joining speech units (from the corpus) which can be phonemes, diphones or syllables to obtain the desired speech signal. Among concatenation variants, one particular method called unit selection, showed improvement in terms of quality and naturalness. It consists in selecting the best sequence of speech units among many candidate units from the speech database to produce the speech signal [START_REF] Hunt | Unit selection in a concatenative speech synthesis system using a large speech database[END_REF]. This method considers both a target cost to measure similarity between selected unit and target characteristics, and a join cost to measure concatenation quality. [START_REF] Schwarz | Real-time corpus-based concatenative synthesis with catart[END_REF] has proposed an approach based on k-nearest neighbours to speed up the unit selection process.

This paper focuses on a particular approach of TTS: Statistical Parametric Speech Synthesis (SPSS), which is based on HMMs. Figure 1 presents the main blocks of a SPSS system. The process starts with text processing, then corresponding speech parameters are predicted and finally processed using a synthesis filter to generate a speech signal. In SPSS systems, the speech signal is characterized by a set of parameters extracted at regular time intervals using a sliding window; this representation combines spectral envelope parameters, fundamental frequency and aperiodic excitation parameters.

HMM-based speech synthesis system (HTS)

In HTS, speech parameters of each speech unit, i.e., spectrum features (e.g., Mel-cepstral coefficients and their dynamic features), fundamental frequency (F0), and phoneme duration are statistically modelled and generated based on the use of HMMs. First versions of HTS were using Mel-cepstral coefficients to represent the speech spectrum [START_REF] Tokuda | An HMM-based speech synthesis system applied to English[END_REF]. Later, the MGC-LSP coefficients -Mel-Generalized-Cepstrum-based Line-Spectrum-Pair - [START_REF] Koishida | Efficient encoding of mel-generalized cepstrum for CELP coders[END_REF] have been used, and associated with the STRAIGHT vocoder [START_REF] Kawahara | Restructuring speech representations using a pitchadaptive time frequency smoothing and an instantaneous-frequency-based F0 extraction: Possible role of a repetitive structure in sounds[END_REF], they led to improved speech quality [START_REF] Zen | The Nitech-NAIST HMM-based speech synthesis system for the Blizzard Challenge[END_REF]. Since then, the STRAIGHT vocoder is used with HTS. With respect to the modelling, context-dependent models are used in HTS to handle the fact the speech parameters of a speech unit are dependent on contextual features. Contextual features include information on the preceding and following segments, on the position of the segment in the syllable and in the word, and a variety of other details (cf. Section 2.2.2).

General aspects of HTS

Figure 2 describes the HTS process [START_REF] Zen | Statistical parametric speech synthesis[END_REF]. The mechanism includes two main blocks, i.e., training and synthesis. The input of the training process is a data corpus of natural speech signals. For each signal, acoustic parameters are extracted at a frame rate of 5 ms. Parameters include spectrum features (e.g., Mel-cepstral coefficients and their dynamic features), excitation (including log (F0) and its dynamic features) and aperiodicity parameters. Acoustic parameters are modelled using context dependent HMMs. Each speech segment is described with a set of contextual features. Once the training is achieved, three models are obtained: one for the duration of the sounds, one for the spectrum parameters and the last one for fundamental frequency and aperiodicity.

The synthesis process of HTS begins with converting the given text (to be synthesized) into a sequence of contextual features. The corresponding context-dependent HMMs are joined together to build the HMM of the utterance. State durations of the utterance HMM are determined in order to maximize the output probability. In HTS, state durations are predicted separately based on duration HMMs. Spectrum and excitation parameters are generated from the HMMs using a speech parameter generation algorithm, i.e., MLPG (Maximum Likelihood Parameter Generation) that maximizes the output probabilities. Ultimately, the synthesis filter, i.e., MLSA (Mel-Log Spectrum Approximation) produces the speech waveform using the generated excitation and spectrum parameters.

The high number of contextual features in HTS (about 50 features in the standard set) increases the complexity to pick essential and useful context information. Therefore, decision trees [START_REF] Jurafsky | Speech and language processing: An introduction to natural language processing, computational linguistics, and speech recognition[END_REF] are involved to group into clusters similar probability density functions. In HTS, the distributions for the spectrum, excitation and duration are clustered separately; in consequence, different phonetic decision trees are obtained for the modelling of spectrum, excitation and duration.

Contextual features

The conversion of the text into a sequence of speech segments (phones and pauses) with associated contextual features is an essential step in HTS. The contextual features should capture all factors that can affect how a speech unit is pronounced in a particular context. Therefore, they include different types of information; either linguistic, phonologic or prosodic. The set of features comprises factors at different levels, within syllables, within words and within utterances. During the synthesis phase, decision trees and contextual features are employed to predict speech parameters. Thus, the choice of contextual features is crucial and affects the speech quality. [START_REF] Tokuda | An HMM-based speech synthesis system applied to English[END_REF] suggested a list of features that influence the pronunciation of speech units:

Arabic speech synthesis

Over the last three decades, the interest in Arabic speech synthesis has been steadily rising. Therefore, most of the TTS techniques have been adapted for Arabic language. Based on formants, i.e., maxima of the spectrogram, acoustic speech is generated through formants characteristics (i.e., bandwidth and amplitude) and rules of evolution of formants between phonemes, [START_REF] Rajouani | Synthesis-by-rule of Arabic language[END_REF] proposed a rule-based Arabic speech synthesis system. Various corpus-based concatenative synthesis methods have been applied to Arabic language. These techniques consist in joining speech units of different sizes. [START_REF] Baloul | Développement d'un système automatique de synthèse de la parole à partir du texte arabe standard voyellé[END_REF] proposed an automatic speech synthesis system for Arabic based on diphone concatenation. Diphone were chosen because they are considered a stable speech unit, i.e., a diphone starts from the middle of a phoneme (stable part) until the middle of the neighbouring phoneme, thus it covers the transition phase between the two adjacent phonemes [START_REF] Moulines | A real-time French text-to-speech system generating high-quality synthetic speech[END_REF]. Later [START_REF] Cheffour | Étude et Evaluation de la Di-Syllabe comme Unité Acoustique pour le Système de Synthèse Arabe PARADIS[END_REF] brought to the fore larger speech units and came up with a concatenative speech system for Arabic based on di-syllables, i.e., units that starts from the middle of a syllable until the middle of the next syllable. Furthermore, triphones, i.e., sequences of three phonemes [START_REF] Kishore | Unit size in unit-selection speech synthesis[END_REF], were used in [START_REF] Ahmed | Réalisation d'un système hybride de synthèse de la parole Arabe utilisant un dictionnaire de polyphones[END_REF] to create Arabic speech signals. [START_REF] Abdelmalek | High quality Arabic Text-to-speech synthesis using unit selection[END_REF] applied unit selection technique for Arabic using phonemes and syllables as speech units. [START_REF] Halabi | Modern standard Arabic speech corpus[END_REF] built an Arabic speech database and used Innoetics, which is considered a high quality speech synthesiser based on unit selection [START_REF] Chalamandaris | The ILSP/INNOETICS Text-to-Speech System for the Blizzard Challenge[END_REF], to generate speech signals.

Besides, HTS has been also applied to MSA. Abdel-Hamid et al. (2006) suggested some modifications to improve speech quality. The scope of their work was on modifications of the speech parameters and on signal processing parts, a multi-band excitation model was applied and spectral parameters were extracted from spectral envelope. [START_REF] Khalil | Arabic HMM-based speech synthesis[END_REF] have used the basic HMM-based speech synthesis system to produce Arabic speech relying on phonemes speech units and on the STRAIGHT vocoder.

Arabic HMM-based speech synthesis

To apply HTS to MSA, as for any other languages, the written text must be transformed into a sequence of contextual features. For Arabic, choosing the features requires the knowledge of particularities of MSA so that all factors that may affect the pronunciation of the speech units are captured.

Arabic phonology

MSA has a set of 28 consonants, which can exist in two forms: simple and geminated, and six vowels: three short and three long vowels [START_REF] Newman | The phonetics of Arabic[END_REF]. Previous studies of Arabic phonology have analysed the following phenomena for MSA: stress, emphatic consonants, vowel quantity and gemination.

Stress

It is a prominence given to a syllable within a word and usually called lexical stress or word stress [START_REF] Black | The festival speech synthesis system[END_REF]. In Arabic language, the position of the stress in a word can be predicted through predefined rules [START_REF] Kouloughli | Contribution à l'étude de l'accent en arabe littéraire[END_REF][START_REF] Halpern | Word stress and vowel neutralization in modern standard Arabic[END_REF][START_REF] Al-Ani | Arabic phonology: An acoustical and physiological investigation[END_REF].

Emphatic consonants

It consists in the pharyngealization of consonants in Arabic [START_REF] Halabi | Modern standard Arabic speech corpus[END_REF]. Consonants of MSA can be divided into three classes; always-emphatic consonants, consonants that can never be emphatic and two-state consonants, i.e., consonants than can be emphatic in particular contexts (Laufer and Baer, 1988).

Vowel quantity

MSA has short and long vowels [START_REF] Selouani | Arabic phonetic features recognition using modular connectionist architectures[END_REF]. In spelling, long vowels are always indicated, unlike short vowels, by the following graphemes >و< /uu/, >ي< /ii/, and >ا< /aa/. When replacing a short vowel with its long counterpart, the meaning of the word changes, e.g., "هتف" /hatafa/ means, "he shouted", whereas "هاتف" /haatafa/ means, "he telephoned".

Gemination

Consonants of MSA have two forms: simple and geminated. In spelling, a geminated consonant is distinguished from its simple counterpart by adding a diacritic sign ( ّ) called shadda above the concerned consonant [START_REF] Newman | The phonetics of Arabic[END_REF]. The presence of a geminated consonant changes the meaning of the word, e.g., "درس" /darasa/ means, "he studied", whereas س" َ "در /darrasa/ means, "he taught".

Speech unit modelling

This work investigates the choice of speech units and their modelling in an HMM-based speech synthesis system for Arabic language. Thus, particularities of the language are taken into account with a focus on gemination and on vowel quantity. For what concern emphatic consonants, they are distinguished from other consonants in the data we are using; hence, they correspond to specific modelling units. With respect to stress, stress information already belongs to the set of contextual features in the standard HTS system. Previous works on Arabic speech synthesis based on HMM did not mention any explicit interest on how to model geminated consonants and long vowels. Therefore, here, several choices of unit modelling are considered to investigate whether adding gemination and vowel quantity information into the set of contextual features is enough, or should geminated consonant (resp. long vowel) be considered as fully-fledged speech units. Thus, four possible speech unit-modelling approaches are proposed below. Note that for the four proposed approaches, the vowel quantity and the gemination information are always present in the set of contextual features. It is assumed that if such information is redundant for some choice of speech units, the decision trees that are built during the training process will simply ignore it.

Single model for simple and geminated consonants, and single model for short and long vowels (C1V1)

This is the most compact system, where a geminated consonant and its corresponding simple consonants are modelled with the same unit, and similarly, a long vowel and its corresponding short vowel are modelled with the same unit. This system in called C1V1.

Differentiating only short vs. long vowels (C1V2)

In this system, a single unit models both a geminated consonant (e.g., /dd/) and its simple counterpart (/d/). Whereas a long vowel (e.g., /aa/) and its short counterpart (e.g., /a/) are modelled by two different units. This system is named C1V2.

Differentiating only simple vs. geminated consonants (C2V1)

This system uses a single unit to model both a long vowel and its short counterpart. While for consonants, two units are used, one for the simple consonant and one for its geminated counterpart. This system is named C2V1.

Differentiating simple vs. geminated consonants and short vs. long vowels (C2V2)

This is the most detailed system, where two different units are used to model a simple consonant (e.g., /d/) and its geminated counterpart (e.g., /dd/). In addition, for vowels, two different units also are used to model a short vowel (e.g., /a/) and its long counterpart (/aa/). This model is named C2V2.

Experiments

Data

In order to develop and evaluate the different systems corresponding to the speech unit modelling variants mentioned above, a set of experiments was conducted. For this purpose, an Arabic corpus was used [START_REF] Halabi | Phonetic inventory for an Arabic speech corpus[END_REF]. It contains 1806 utterances corresponding to a total of 4 hours of recorded speech; the corpus is a collection of news bulletins uttered in a neutral style by a male speaker [START_REF] Halabi | Modern standard Arabic speech corpus[END_REF]. The audio signals were recorded using a professional studio and speech signals were sampled at 48 kHz. The software Pro Tools 11 was used during recording. Besides, the Studio Microphone Neumann TLM 103 was employed, a choice justified by the fact that the device is successfully used to record human speech with high quality. When recording, the speaker was in a soundproof anechoic booth.

In the experiments reported below, a speaker-dependent modelling was used and a distinct HTS model was trained for each choice of speech unit modelling. A training corpus of 1565 Arabic utterances was considered and 30 other utterances were kept apart for the evaluation. Not all the corpus was used because there were some transcription which could affect the quality of the training process; the corresponding utterances were thus ignored. To guarantee the best possible speech quality, the vocoder STRAIGHT was used.

Contextual features and modelling

The set of contextual features proposed for MSA speech synthesis was inspired from the standard set of contextual features defined for English [START_REF] Tokuda | An HMM-based speech synthesis system applied to English[END_REF]. However, information related to tone accent and TOBI was ignored, as Arabic language does not present these characteristic rules [START_REF] Kouloughli | Contribution à l'étude de l'accent en arabe littéraire[END_REF][START_REF] Al-Ani | Arabic phonology: An acoustical and physiological investigation[END_REF]. Moreover, no additional feature about emphatic aspect was added to the set of contextual features as emphatic consonants were already distinguished from other consonants in the transcription of the corpus.

However, two additional features were introduced to take into account specificities of the Arabic language. The first feature is used to indicate the gemination characteristic (possible values are 'simple consonant', 'geminated consonant', or 'not a consonant'); the second one refers to vowel quantity (possible values are 'short vowel', 'long vowel', or 'not a vowel').

In the experiments, speech signals are generated with the four proposed systems using the same set of contextual features, which contain information about vowel quantity and consonant gemination. As mentioned, decision tree will simply ignore extra or redundant features. To measure the performance of a speech synthesis system and to compare them, subjective and objective evaluations are carried out.

Objective evaluation of phone duration

Evaluation protocol

An objective evaluation is conducted to evaluate the predicted duration of the speech units. For speech signals produced with each system (C1V1, C1V2, C2V1 and C2V2) the average, over the vowels, of the ratios between the mean duration of long vowels (LV) and the mean duration of corresponding short vowels (SV) is calculated, as well as the average ratio for geminated consonants (GC) vs. simple consonants (SC). Only phonemes with more than 10 occurrences for each class (simple/geminated consonants and short/long vowels) are considered. The calculated average ratios are compared to those obtained on natural speech.

Results and discussion

The obtained values, reported in Table 1, show that for the four systems, the duration ratios of long to short vowel and the duration ratios of geminated to simple consonant are similar to the ones calculated on natural speech. To better understand the similarity of the obtained duration ratios, the decision trees associated to the modelling of duration of the four systems have been analysed. For each system, there is only one single tree for all the units (phonemes and silence). Therefore, the four decision trees have been analysed, and Figure 3 represents the top part of the duration decision tree for the model (C2V2). In the figure, "C-Seg" refers to the current segment, "L-Seg" to the previous (left) segment, and "R-Seg" to the next (right) segment; "sil" indicates a silence, "r" and "t" are the phonemes /r/ and /t/, "GC" indicates a geminated consonant, and "LV" indicates a long vowel. Figure 3 shows that questions about the nature of speech segment such as geminated consonant (C-Seg==GC) and long vowel (C-Seg==LV) are situated at the top of the tree. This behaviour was observed for the four models.

To further investigate the prediction of phone durations, root mean square error (RMSE) between natural duration and predicted durations with HTS was calculated on generated signals for the four systems C1V1, C1V2, C2V1 and C2V2 within different phoneme classes (simple and geminated consonants, and short and long vowels). The obtained RMSE are presented in Figure 4. Results show that for each phoneme class, there is no prominent difference between the RMSE measured for the four-speech unit modelling approaches.

Subjective evaluation of global quality and naturalness

Evaluation protocol

In the present work, MOS (Mean Opinion Score) tests (ITU, 1996) were carried out to evaluate the global quality of the speech signals generated by the four systems C1V1, C1V2, C2V1 and C2V2. Two factors contribute to the global quality: naturalness and overall signal quality. The overall signal quality refers to the quality of the produced acoustic signal. The naturalness is evaluated by referring to the intonation (whether the pitch's change is natural) and the rhythm (whether length of phonemes sounds natural too).

Nine Arabic native speakers took part in this evaluation. Each participant evaluated a set of 40 stimuli, i.e., 10 stimuli from each system (C1V1, C1V2, C2V1 and C2V2) and judged the corresponding overall quality and naturalness. Listeners were asked to answer the question "How do you rate the overall quality and naturalness of what you have just heard compared to a natural speech (pronounced by a human being)?" by giving a score ranging from 1 to 5 as follows:

• Very close to natural speech (5) • Close to natural speech (4)

• A little bit far from natural speech (3)

• Far from natural speech (2)

• Very far from natural speech (1)

Results and discussion

The resulting MOS scores are presented in Figure 6 with their associated 95% confidence intervals. Results show that for the four modelling approaches, C1V1, C1V2, C2V1 and C2V2, HTS produces speech signals with similar perceived characteristics (overall quality, intonation and rhythm). In this part, a DMOS (Differential Mean Opinion Score) test (ITU, 1996) was conducted to assess the degree of perceived degradation in the HTS generated speech signals, by comparing each speech signal produced by one of the four modelling approaches to the corresponding natural speech signal. For this evaluation, twelve listeners participated; each one evaluated a set of 20 pairs, each pair consists of the same utterance produced with one of the four systems and the corresponding natural signal. Signals were presented in a defined order; first, the reference (natural signal), followed by the signal produced by one of the systems (C1V1, C1V2, C2V1 and C2V2).

Participants evaluated the degradation of signals by answering the question "How do you judge the degradation of the second signal compared to the first one?" using a five-point degradation category scale:

• Inaudible degradation (5)

• Audible but not annoying degradation (4)

• Degradation a little annoying (3)

• Annoying degradation (2)

• Very annoying degradation (1)

Results and discussion

The obtained DMOS are represented in Figure 7 with their associated 95% confidence intervals. The higher the score is, the lower the degradation is. Obtained DMOS scores show that signals produced with the four models C1V1, C1V2, C2V1 and C2V2 present a similar degree of degradation when compared to the natural speech. These results are consistent with the fact that the four systems provide a similar speech quality according to the results of global quality evaluation. A preference test (ITU, 1996) was carried out to compare the four proposed approaches for speech unit modelling with respect to the quality of produced speech. Speech signals produced with the four models (C1V1, C1V2, C2V1 and C2V2) are compared to each other. Twenty-seven Arabic native speakers participated in this evaluation. Each one evaluated a set of 23 pairs of speech signals; each pair consists of the same utterance produced with two different systems. The order of presenting the speech signals is randomly chosen for each trial. During the evaluation, participants were asked to point to the preferred signal based on the global quality of produced speech, by answering the question "How do you judge the quality of the second signal compared to the first one?" and giving a score from 1 to 7 ranging from "Much worse" to "Much better". For the analysis of the results, the scores have been grouped into three categories: 'first preferred', 'no preference' and 'second preferred', as follows:

• Much better [START_REF] Laufer | The emphatic and pharyngeal sounds in Hebrew and in Arabic[END_REF] • Best (6)

• A little better (5)

• About the same (4)

• A little worse (3) • Worst (2) • Much worse (1)

Results and discussion

Results are presented in Figure 8. For example, on the figure, the bottom line corresponds to the comparison of C1V2 with C2V1. For this line, 12% of the answers give a preference to the first system (left side, i.e., C1V2), 6% of the answers give a preference to the second system (right side, i.e., C2V1), and the middle part shows that 82% of the answers do not express any preference. The one-to-one comparison of the four models shows that listeners had no clear preference for any particular system. Therefore differentiating geminated consonants (resp. long vowels) from simple consonants (resp. short vowels) or merging them when defining the speech units, lead to a similar speech synthesis quality.

Conclusions and perspectives

This paper studied different possible approaches of speech unit modelling for Arabic speech synthesis based on HMM (HTS). Both training and synthesis part of HTS system require a description of the text with a sequence of contextual features. These contextual features are used to build the decision trees during the training of the HMM models, and to predict the acoustic parameters to generate the waveform in the synthesis phase. Accordingly, the choice of the contextual features is decisive. Part of the contextual features are language dependent, and Arabic language presents two specific phenomena that have to be considered: gemination and vowel quantity. These particularities of the Arabic language are not handled in the standard set of contextual features unlike stress. Consequently, two extra features were added to take into account these specificities. However, is it enough to add this information into the set of contextual features or should units be differentiated? In other words, should a geminated consonant (resp. a long vowel) be considered as a fully-fledged phoneme to be modelled with a specific unit, or are they the same (or very similar) phoneme as their simple (short) counterpart but with different duration (which can be modelled by the same unit)? To answer these questions, several possible modelling approaches of the speech segments have been investigated such as, the use of different units for modelling long vs. short vowels, and/or the use of different units for modelling simple vs. geminated consonants. These combinations were compared to another one, where a short vowel and its long counterpart are modelled with the same unit, and a geminated consonant and its simple counterpart are modelled with the same unit.

Objective measures show that segment durations generated with HTS for the four models are similar. This conclusion was validated by listening tests (MOS, DMOS and preference tests). Results showed that there is no prominent difference between the various modelling approaches. Thus, the identification of geminated consonants and long vowels as fully-fledged phonemes (hence modelled by specific units) is not compulsory when applying HTS for MSA, as long as this information exists in the set of contextual features.

Future work will explore Arabic speech synthesis based on deep learning approaches [START_REF] Zen | Statistical parametric speech synthesis using deep neural networks[END_REF]. Recently, different variants and architectures of Deep Neural Networks (DNN) have been introduced for speech synthesis [START_REF] Zen | Unidirectional long short-term memory recurrent neural network with recurrent output layer for low-latency speech synthesis[END_REF][START_REF] Wu | Merlin: An open source neural network speech synthesis system[END_REF], and the obtained results showed that DNN improve the quality of produced speech signals and their naturalness as well. Thus, it will be interesting to find out if DNN would, benefit from the explicit differentiation of geminated vs. simple consonants and long vs. simple vowels unlike HMM.

Figure 1 :

 1 Figure 1: Overview of SPSS

Figure 2 :

 2 Figure 2: Overview of HTS

Figure 3 :

 3 Figure 3: Duration decision tree of (C2V2) model (top part)

Figure 4 :

 4 Figure 4: RMSE between natural durations and HTS generated durations Figure 5 displays the normalized root mean square error (NRMSE) obtained by normalizing the RMSE by the mean duration values of the phoneme classes. Obtained NRMSE values are quite similar; around 25% for geminated consonants and 35% for the other phoneme classes (simple consonants, short vowels, and long vowels).

Figure 5 :

 5 Figure 5: NRMSE between natural durations and HTS generated durations

Figure 6 :

 6 Figure 6: Results of global quality evaluation and naturalness4.5 Subjective evaluation of degradation

  Figure 7: Results of degradation evaluation

Figure 8 :

 8 Figure 8: Results of preference test

  Number of phonemes within current, previous and next syllables. o Stress and accent information of current, previous and next syllables. o Vowel identity of current syllable. • Word level features o Part Of Speech tagging of current, previous and next words. o Number of syllables of current, previous and next words. o Position of current word within current phrase (forward and backward). • Phrase level features o Number of syllables of current, previous and next phrases. o TOBI end tone of current phrase (Silverman et al., 1992). • Utterance level features o Number of syllables, words and phrases in utterance.

• Phoneme level features o Current phoneme. o Previous and next phonemes. o Position of current phoneme within the current syllable (forward and backward). • Syllable level features o

Table 1 .

 1 Duration ratios.

			LV / SV	GC / SC
		Number of occurrences	262 / 884	104 / 1315
		C1V1	1.8	2.2
		C1V2	1.9	2.2
	Models	C2V1	1.8	2.1
		C2V2	1.8	2.0
		Natural	2.0	2.1

  Abdel-Hamid, O., Abdou, S. M. & Rashwan, M. (2006). Improving Arabic HMM based speech synthesis quality.In Interpseech 2006, 9 th Annual Conference of the International Speech Communication Association. Pittsburgh, Pennsylvania, USA.

			First preferrred		No preference	Second preferred
	C2V2										C1V1
	C2V2										C1V2
	C2V2										C2V1
	C1V1										C1V2
	C1V1										C2V1
	C1V2										C2V1
	0%	10%	20%	30%	40%	50%	60%	70%	80%	90%	100%
					Percentage of preference


domain: math shs

text:
Aaltje Berendina Aaten
email: a.b.aaten@rug.nl

Johan Deprez
email: johan.deprez@kuleuven.be

Gerrit Roorda
email: g.roorda@rug.nl

Martin Goedhart
email: m.j.goedhart@rug.nl

Undergraduates' reasoning while solving integration tasks: Discussion of a research framework

Keywords: Mathematical reasoning, undergraduate students, calculus

) is adequate for characterizing undergraduate students' mathematical reasoning. We conducted a small number of individual task-based think-aloud interviews in which students solved integration tasks. Several examples illustrate how we characterized reasoning types by using the framework. However, we found that some reasoning types were not covered by the framework. We propose to extend the framework by introducing a reasoning type that is mathematically founded but not creative, and as a consequence, may be intertwined with imitative reasoning.

Introduction

Students' mathematical reasoning while solving mathematical tasks is not always as well-founded as it appears, as has already been highlighted by [START_REF] Vinner | The pseudo-conceptual and the pseudo-analytical thought processes in mathematics learning[END_REF] in his article on pseudo-conceptual and pseudo-analytical thought processes in mathematics learning. Moreover, undergraduates' reasoning in the domain of calculus is found to be susceptible to ill-founded reasoning [START_REF] Lithner | Students' mathematical reasoning in university textbook exercises[END_REF]. Because of its relevance for mathematics teaching, students' mathematical reasoning demands further investigation. Different frameworks on mathematical reasoning have been reported in literature. Some frameworks focus on argumentation used while solving proving problems (e.g., [START_REF] Blanton | Understanding the role of transactive reasoning in classroom discourse as students learn to construct proofs[END_REF][START_REF] Stylianides | An analytic framework of reasoning-and-proving[END_REF]. [START_REF] Zandieh | Defining as a mathematical activity: A framework for characterizing progress from informal to more formal ways of reasoning[END_REF] constructed a framework on mathematical reasoning that distinguishes informal and formal reasoning, which seems most suited for investigating students' understanding of abstract concepts. [START_REF] Carlson | The cyclic nature of problem solving: An emergent multidimensional problem solving framework[END_REF] combined the problem solving phases Orienting, Planning, Executing, and Checking with the use of problem solving attributes Resources, Heuristics, Affect, and Monitoring. Their framework appears capable of identifying students' problem solving activities and the influence of cognitive, affective and/or metacognitive factors. However, this framework does not incorporate the foundation of mathematical reasoning. [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF] constructed a framework which does incorporate foundations of students' strategic decisions in solving mathematical tasks. This framework is often referred to when characterizing reasoning as either imitative or creative, which are the two main categories in the framework (e.g., [START_REF] Jäder | Students' mathematical reasoning and beliefs in nonroutine task solving[END_REF][START_REF] Jonsson | Learning mathematics through algorithmic and creative reasoning[END_REF]. However, the framework offers greater detail, by also defining Memorized Reasoning, and Familiar, Delimiting, and several types of Guided Algorithmic Reasoning [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF]. Since this detailed characterization of mathematical reasoning based upon its foundation appears useful for identifying students' reasoning, we selected this framework for our research. Based upon our experiences in applying this research framework, we discuss the framework's possibilities and limitations. [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF] defines reasoning as:

Theoretical framework

The line of thought adopted to produce assertions and reach conclusions in task solving. It is not necessarily based on formal logic, thus not restricted to proof, and may even be incorrect as long as there are some kinds of sensible (to the reasoner) reasons backing it (Lithner, 2008, p. 257).

Based on observations of students who solve mathematical tasks, Lithner describes various ways in which students choose a mathematical strategy to solve a task, pointing out students' 'predictive argumentation' (reasoning for choosing a strategy) and 'verificative argumentation' (reflection upon implementation of strategy), where strategy "ranges from local procedures to general approaches" and choice "is seen in a wide sense (choose, recall, construct, discover, guess, etc.)" (Lithner, 2008, p. 257). The resulting framework is visualized in figure 1. The framework distinguishes two main categories, Creative Mathematically founded Reasoning (CMR) and Imitative Reasoning. CMR1 refers to reasoning that is based on intrinsic mathematical properties, that is novel to the student (the reasoner) and for which the student has arguments [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF]. Imitative reasoning is described as reasoning in which an algorithm or answer is recalled in some way. Imitative reasoning is divided into Memorized Reasoning and Algorithmic Reasoning. Memorized Reasoning implies that the student recalls a complete answer, for example a definition or a proof that is learnt by heart. Algorithmic Reasoning occurs when a student recalls an algorithm. Lithner's framework altered over time (see [START_REF] Lithner | Students' mathematical reasoning in university textbook exercises[END_REF][START_REF] Lithner | Mathematical reasoning in calculus textbook exercises[END_REF][START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF]; in this study we applied the framework as described in [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF].

The definitions by [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF] for each of the reasoning types are listed in Table 1. In the definition of Delimiting Algorithmic Reasoning (see Table 1), the term 'set' of algorithms requires some explanation. [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF] clarifies that if no guidance is available and if the task is unfamiliar to the student, then the student must choose an algorithm from the 'set' of algorithms the student knows, based upon some kind of connection to the task.

Reasoning type Criteria

Creative Mathematically founded Reasoning Three criteria: "Novelty. A new (to the reasoner) reasoning sequence is created, or a forgotten one is re-created." "Plausibility. There are arguments supporting the strategy choice and/or strategy implementation motivating why the conclusions are true or plausible." "Mathematical foundation. The arguments are anchored in intrinsic mathematical properties of the components involved in the reasoning" (Lithner, 2008, p. 266) Imitative Reasoning

No definition is given. Imitative Reasoning is subdivided into Memorized Reasoning and Algorithmic Reasoning.

Memorized Reasoning

"The strategy choice is founded on recalling a complete answer. The strategy implementation consists only of writing it down." (Lithner, 2008, p. 258) Algorithmic Reasoning "The strategy choice is to recall a solution algorithm. The predictive argumentation may be of different kinds (see below for examples), but there is no need to create a new solution." "The remaining reasoning parts of the strategy implementation are trivial for the reasoner, only a careless mistake can prevent an answer from being reached." (Lithner, 2008, p. 259) Familiar Algorithmic Reasoning "The reason for the strategy choice is that the task is seen as being of a familiar type that can be solved by a corresponding known algorithm." "The algorithm is implemented." (Lithner, 2008, p. 262) Delimiting Algorithmic Reasoning "An algorithm is chosen from a set that is delimited by the reasoner through the algorithms' surface relations to the task. The outcome is not predicted." "The verificative argumentation is based on surface considerations that are related only to the reasoner's expectations of the requested answer or solution. If the implementation does not lead to a (to the reasoner) reasonable conclusion it is simply terminated without evaluation and another algorithm may be chosen from the delimited set." (Lithner, 2008, p. 263)

Guided Algorithmic Reasoning

Text-guided Algorithmic Reasoning: "The strategy choice concerns identifying surface similarities between the task and an example, definition, theorem, rule, or some other situation in a text source." "The algorithm is implemented without verificative argumentation." (Lithner, 2008, p. 263) Person-guided Algorithmic Reasoning: "All strategy choices that are problematic for the reasoner are made by a guide, who provides no predictive argumentation." "The strategy implementation follows the guidance and executes the remaining routine transformations without verificative argumentation." (Lithner, 2008, p. 264) It is important to note that the foundation of Creative Mathematically founded Reasoning is explicitly stated, while this is not the case for Imitative Reasoning: CMR is by definition founded in intrinsic mathematical properties, while the foundation of Imitative Reasoning is not clearly stated. The definitions of various sub-categories of Imitative Reasoning contain criteria like 'surface relations', 'surface considerations', 'surface similarities', 'no predictive argumentation', and 'without verificative argumentation'. This terminology appears to stem from earlier work: [START_REF] Lithner | Mathematical reasoning in calculus textbook exercises[END_REF] distinguished mathematically founded reasoning and superficial reasoning, where the latter was based upon surface properties and not upon mathematically relevant properties. Although in [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF], 'imitative reasoning' is not defined as founded in superficial or surface properties, many of the subtypes are (see Table 1). Moreover, all examples and explanations given by [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF] do refer to situations in which the reasoning is founded in so-called superficial properties and not in intrinsic mathematical properties, which is a criterion for CMR.

Certain studies have already used the framework to characterize students' reasoning. [START_REF] Boesen | The relation between types of assessment tasks and the mathematical reasoning students use[END_REF] employed the categories of Memorized Reasoning, Algorithmic Reasoning (without subcategories) and extended the category of Creative Mathematically founded Reasoning by defining the two subtypes Local CMR and Global CMR (in accordance with [START_REF] Lithner | Mathematical reasoning in calculus textbook exercises[END_REF]). [START_REF] Sumpter | Themes and interplay of beliefs in mathematical reasoning[END_REF] applied the framework to label episodes of students' reasoning in a study on the role of beliefs in mathematical reasoning, and showed three examples which were all labeled as Familiar Algorithmic Reasoning. [START_REF] Jäder | Students' mathematical reasoning and beliefs in nonroutine task solving[END_REF] used the framework to discern whether students' reasoning was imitative or creative. We remark that these studies have not used all sub-categories of the framework as described by [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF]. In the case of [START_REF] Sumpter | Themes and interplay of beliefs in mathematical reasoning[END_REF], only one type of reasoning was discussed. Although these studies did make use of the framework, none of them did explicitly reflect upon its applicability. Since we consider the framework a worthwhile addition to literature on mathematical reasoning, we investigate its applicability for characterizing students' reasoning. The research question we thus aim to answer is: to what extent is the framework by [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF] adequate to characterize undergraduate students' mathematical reasoning?

Methodology

The data in this study originates from interviews with three first year mathematics bachelor students, one male and two female, of varying mathematics proficiency levels, determined by previous exam scores. These students are a sub-sample of a group of 12 students participating in a longitudinal study that investigates the development of mathematical reasoning. The students are majors in mathematics at the University of Groningen (the Netherlands) or the KU Leuven (Belgium); universities which offer courses in a wide range of domains at undergraduate and graduate level. The individual taskbased think-aloud interviews lasted for approximately 1.5 hours each and are administered by the first author at the end of the students' first undergraduate year. Students were permitted to use a list with basic calculus formulas, which did not include elaborate integration formulas. The students were asked to explicate their thinking while solving tasks and, after each task, to answer the questions: "How did you come to think of using this strategy?", "How certain were you that this strategy would help you solve the problem, and why?", and "Have you seen this type of task before?". The interviews are video and audio recorded.

We used tasks to create a situation in which the students must choose a suitable strategy from a wide range of possible strategies. We considered integration tasks suitable for this purpose since the students had learnt various mathematical strategies for solving integrals, such as partial integration, substitution, partial fractions, or Euclidean division, in the courses they had taken so far. These considerations led to selection of various tasks, amongst which to solve the second integral. The students had taken courses in integral calculus in which they solved similar tasks, amongst many other types of tasks. The explicit discussion of these types of integrals had already taken place earlier in the academic year. Based upon teaching experience we expected that these tasks at the time of the interviews would be non-trivial to many students.

While integration tasks may be regarded as tasks that solely require application of procedures, these tasks can arouse various types of mathematical reasoning in students. Considering and selecting suitable procedures is a process in which Creative Mathematically founded Reasoning as well as Imitative Reasoning can become visible. Familiar Algorithmic Reasoning can be used if the student recognizes the problem type and recalls the corresponding algorithm; Delimiting Algorithmic Reasoning if the student does not recognize the task but recalls various algorithms such as partial integration or substitution of some kind; Creative Mathematically founded Reasoning can be employed if the student is not able to recall a solution strategy but instead constructs a solution or reconstructs a forgotten reasoning sequence, such as drawing a rectangular triangle and deducing a suitable substitution. We did not expect Memorized Reasoning, since the solutions to the tasks are extensive. Guided Algorithmic Reasoning also appeared improbable, since example solutions were unavailable and the interviewer would not offer any hints. The available list with formulas however could serve as inspiration.

Transcripts of the task solutions are split into episodes. An episode begins at the first consideration of a strategy (or a set of strategies) and ends when the strategy is abandoned and a new strategy is about to be considered. Using the framework to characterize parts of a solution is similar to the method of [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF] and [START_REF] Sumpter | Themes and interplay of beliefs in mathematical reasoning[END_REF]. The first author tried to characterize each of the episodes through the definitions given by [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF]. If this was unsuccessful, the difficulties were described. The findings from this analysis were discussed with the other authors until agreement was obtained.

Results

Below we describe several reasoning episodes from our data, which illustrate how we characterized reasoning using the framework and which problems we encountered.

Familiar Algorithmic Reasoning?

Example 1:   dx x 9 2 ; student A chose to rewrite the integrand by splitting it into two terms and next integrating them separately: 9

1 9 9 9 9 9 2 2 2 2 2 2          x x x x x x
. The student had applied this strategy earlier in the interview, when (unsuccessfully) solving   dx x 2 9

. The student explained: "The task was similar to the former task, so I figured I could try using the same approach." We observe that the task is recognized as a familiar type, which made the student decide to apply the same strategy as before. We characterize this reasoning as Familiar Algorithmic Reasoning. . This sub-task is of a standard form because the derivative of

9 2  x is in the numerator, making 9 2  
x t an appropriate substitution. Student A effectively chose this strategy and explained afterwards: "Because I knew that the derivative of this (points at 9 2  x in the denominator), was something like this (points at the numerator) […] So I applied substitution to this (points at

9 2  x
), because this (points at the numerator) would then be eliminated.". We observe that the student noticed the relevant mathematical characteristics of this task and knew what algorithm would solve tasks of this type. Either the task type was familiar or the student constructed the approach. The ease with which the student came to this conclusion and the fact that this type of task has been practiced extensively make us expect the student to be familiar with the task type: we characterize the reasoning as Familiar Algorithmic Reasoning. However, the student clearly founds the reasoning in intrinsic mathematical properties, which is very different from the reasoning that occurred during Example 1, where the student to base the strategy choice solely on familiarity with the task type.

We conclude that Examples 1 and 2 provide two rather distinct reasoning types while both satisfying the criteria for Familiar Algorithmic Reasoning. We observe that the student solves the task by making use of answer recall, but also reasons on the intrinsic mathematical properties of the task to be successful. In the framework, the only reasoning type that makes use of recall of an answer is Memorized Reasoning. However, the second criterion of Memorized Reasoning is not fulfilled. The strategy implementation was not just writing down the answer, since the answer was constructed and verified building upon the intrinsic mathematical properties of the components involved in the reasoning. On the other hand, the category of Creative Mathematically founded Reasoning does not reflect the important role of memory in this solution.

Memorized Reasoning or

This example shows hybrid reasoning with elements from Creative Mathematically founded Reasoning and from Memorized Reasoning.

  . […] This is of course… This is just 2 2 r r  because     2 2 sin cos … 2
r ." The student stops using this strategy. We observe that the student considered the circular coordinates (polar coordinates) since the integrand made the student think of a circle. The strategy appears to be selected based upon intrinsic mathematical properties of the task. However, the student employed the circle coordinates in an ineffective way, which shows that the student did not know why the property of the task, that it concerns a circle, implies the use of circle coordinates. The strategy of using circle coordinates are selected only because the task concerned a circle, therefore the foundation for strategy selection should be regarded as based on the task's surface properties. This example raises doubts concerning whether it is always possible to distinguish a surface property from an intrinsic mathematical property.

Conclusions and discussion

The framework by [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF] provides means to highlight foundations that underlie students' reasoning when solving a mathematical task. However, we faced several difficulties when employing the framework as an analysis instrument. Examples 1 and 2 concern rather distinct reasoning episodes, while both satisfy the definition of Familiar Algorithmic Reasoning. Whether or not the student provides mathematically founded reasons is a relevant characteristic but not included in the definitions. In Examples 2 and 3, the predictive argumentation of a strategy was imitative (based on recall of any kind), but verificative reasoning was founded in intrinsic mathematical properties of the task. These examples reveal that the framework does not cover such 'hybrid' types of reasoning. Example 4 confronted us with the more fundamental issue how to decide whether reasoning is based on 'surface properties' or on 'intrinsic mathematical properties'.

A way to improve the applicability of the framework is to include reasoning types that are mathematically founded as well as make use of some kind of imitative reasoning. This is not the same as local CMR [START_REF] Lithner | Mathematical reasoning in calculus textbook exercises[END_REF], which is reasoning that is partly Creative Mathematically founded Reasoning while the remainder is Imitative Reasoning. We propose that reasoning can be mathematically founded without being creative, and in addition, that mathematically founded reasoning can be intertwined with imitative reasoning. Whether a property is an intrinsic mathematical property or a surface property appears to depend on the student's understanding, e.g. of why a certain task property leads to a certain strategy selection. Distinguishing between the use of surface properties and intrinsic mathematical properties therefore requires a more complete picture of the students' reasoning as a whole. These suggestions are based upon difficulties faced when applying the framework on a small number of reasoning episodes within the domain of integration. To obtain a framework adequate to characterize any type of mathematical reasoning not only requires thorough investigation of specific examples, but also requires investigation of the structure of the framework such that the framework will be decisive for each reasoning episode to be characterized.

Figure 1 :

 1 Figure 1: Visualization of reasoning framework as described by Lithner (2008)

  not have to be so complicated. There must be something that I overlook. […] eh. Ah, no, wait wait, hey. sec θ times … Secant tangent? There was something about that. on my memory now, because I have solved those integrals. I know it's an integral with secant, with ln. (student calculates the derivative of is not correct) What was it like? […] wait, I think I know. ) and then you obtain indeed… I knew it was something with ln ."

Table 1 : Definitions of reasoning types, derived from Lithner (2008)

 1

	Thematic Working Group 14

Intrinsic mathematical properties or surface properties?

  That gives a nice circle. Then you have got the radius, a circle with radius 9, radius 3, I mean. It's not transformed, so you get this. Circular coordinates. Let's take a look at circular coordinates[…]

	Example 4:   dx x 2 9	; student B rewrote	y		9		x	2		to	x	2		y	2		9	and remarked it is a circle:
	"Then you get	x 	r	 cos	and	y 	r	sin		. So	r	2	cos	2		r	2	sin	2			r	2

Proceedings of CERME10

In earlier versions of the framework (e.g.,Lithner,

2004), creative mathematically founded reasoning (which was then named Plausible Reasoning) was subdivided in a global and a local subtype, but this distinction has not remained.Thematic Working Group 14Proceedings of CERME10


text:
Michael Achile Umameh
email: edmau@leeds.ac.uk

Resource systems of English teachers as they appropriate mastery approaches

Keywords: Mathematics teaching, resources, digital tools, appropriation

Introduction

I report on the resource systems [START_REF] Gueudet | Towards new documentation systems for teachers?[END_REF] of a sample of mathematics teachers' in England. An influence on these teachers' appropriation of resources for learning and teaching is current mathematics education reform that is focused on how mathematics performance in England compares to the highest attaining systems internationally. The OECD's Program for International Student Assessment (PISA, 2012) ranked England in the 25 th position in mathematics achievement, with the assessment table headed by a clutch of south-east Asian jurisdictions. PISA results also showed that England's performance in mathematics has stagnated over the years. There is a government-backed mandate to explore, adapt and embed the Singapore/Shanghai model of mastery teaching and assessment for learning approaches across England [START_REF] Hodgen | Shanghai mathematics exchange, views, plans and discussion[END_REF][START_REF] Ncetm | Mastery approaches to mathematics and the new national curriculum[END_REF]. "Mastery learning can be described as a set of group-based, individualized, teaching and learning strategies based on the premise that virtually all students can and will, in time, learn what the school has to teach" (Anderson, 1975 p.4). Although mastery learning has morphed into various adaptations, essential elements remain the feedback, corrective and enrichment [START_REF] Drury | Mastering mathematics: Teaching to transform achievement[END_REF]. Extant research points to the potential of well-implemented mastery teaching as enabling higher levels of achievement, deep understanding and confidence [START_REF] Drury | Mastering mathematics: Teaching to transform achievement[END_REF].

This study combines an activity theoretic approach with the more recent 'documentational approach' [START_REF] Gueudet | Towards new documentation systems for teachers?[END_REF] from French didactics as theoretical tools for developing an understanding of the teachers' appropriation of digital resources and building up a coherent explanation for its impacts on classroom practices. In this investigation, I document the context for the current motivation for adopting mastery teaching and examines the emergence of the resource systems of seven English mathematics teachers' as they 'resource for mastery', and the potential impact of this on classroom practices. The major aims of this study are to:  Analyze how teachers' appropriate digital resources for classroom practices.  Explore teachers' resource systems for mastery teaching.  Contribute to the discourse on teachers' appropriation of digital resources.

Research design, methodology, data collection and analysis

A qualitative case study approach [START_REF] Creswell | Qualitative inquiry and research design: Choosing among five approaches[END_REF]) was adopted. Purposive sampling was used to select seven teachers from three schools based on the use of digital resources, access, proximity and the opportunity to observe rich and real life-context of teacher practice with digital resources. Data collection was undertaken during the 2015-2016 school year through periodic whole day school visits. Data were collected through: audio-recorded semi-structured interviews; classroom observations using an adapted systematic classroom analysis notation for mathematics lessons (SCAN, [START_REF] Beeby | Systematic classroom-analysis-notation for mathematics lessons: SCAN[END_REF]; screen capture software; and collation of documents.

Thematic Working Group 15

Proceedings of CERME10

Proceedings of CERME10

Data analysis is ongoing at the time of writing and includes: coding and analyzing transcribed interviews using thematic analysis and taking into account key concepts from the literature and information emerging from the data alongside the SCAN analysis of classroom observations. Data is organized by teacher and by data type. The thematic mappings will be constantly grouped and regrouped into categories and themes for discussion.

Emergent results and implications

Preliminary results from ongoing data analysis includes:

1. Teachers' appropriation of digital tools for formative assessment (student seat-work is eanalyzed during lessons which allows the teacher to enact changes in the tasks). 2. The reality of 'emergent (in lesson) task design' afforded by access to multiple resources. 3. The emergence of Twitter as a key platform for 'massive live staffrooms' (teachers are constantly collaborating, developing task, and sharing expertise and resources on mastery approach).

I believe that this research will contribute to the ongoing discourse on issues and challenges of the integration of digital resources in classrooms and offer 'working hypotheses' for future research.


text:
Francisco Botana
email: fbotana@uvigo.es.

Tomas Recio
email: tomas.recio@unican.es.

M Pilar Vélez

Tomás Recio

The role of automated reasoning of geometry statements in mathematics instruction

Keywords: Mathematics instruction, elementary geometry, automatic reasoning with GeoGebra

  L'archive ouverte pluridisciplinaire

The tools …

By "automated proving of geometry statements" we refer to tools that mechanically output a mathematically rigorous (e.g. not based upon a probabilistic proof) yes/no answer to the conjectured truth of a given geometric statement. By "automated derivation of geometry statements" we refer to tools that, rigorously, output some/all geometric relations verified by a collection of selected elements within a geometric construction. Finally, by "automated discovery of geometry statements" we refer to tools that systematically find complementary, necessary, hypotheses for a conjectured geometric statement to become true.

The community of mathematicians and computer scientist has been working on these goals for the past 50 years, with a variety of approaches, outcomes and popularization results (cf. the pioneer work of [START_REF] Gelertner | Realization of a geometry theorem proving machine[END_REF] in the Artifical Intelligence context, or the algebraic geometry framework to automated reasoning in geometry disseminated by the book of [START_REF] Chou | Mechanical Geometry Theorem Proving[END_REF]). On the other hand, although we can mention the development of some intelligent tutorial systems designed to assist students to construct proofs in Geometry such as GRAMY [START_REF] Matsuda | GRAMY: A Geometry Theorem Prover Capable of Construction[END_REF] or GeoGebraTutor (Tessier-Baillargeon, Richard, Leduc and Gagnon, 2014), it is fair to say that, up to now, the dissemination, use and impact of these findings in the educational context is very limited. Thus, the very recent survey by [START_REF] Sinclair | Recent research on geometry education: an ICME-13 survey team report[END_REF], on geometry in education, although it includes a full section on the role of technologies and another one on "Advances in the understanding of the teaching and learning of the proving process", does no refer at all about automated reasoning tools.

…and the issues

Hence, we consider it quite relevant to address, in our poster, two issues: one, to announce the very recent implementation (2016) of tools for the automatic proving and discovery of geometric theorems over a free dynamic geometry software, with tens of millions of users worldwide, and a great impact in mathematics education. See Abánades, Botana, Kovács, Recio and Sôlyom-Gecse (2016) and [START_REF] Hohenwarter | Deciding geometric properties symbolically in GeoGebra[END_REF].

Then, recalling that the program where we have implemented our automatic reasoning tools (ART) is available over computers, tablets, smartphones, with and without internet connection, the second issue we would like to pose here is the consideration of the following questions: what could be the role, in mathematics instruction, of the ample availability of such tools? It was already 30 years ago (cf. the visionary ICMI Study "School Mathematics in the 1990's" [START_REF] Howson | ICMI Study series: School mathematics in the 1990's[END_REF] or the inspiring paper by [START_REF] Davis | The rise, fall, and possible transfiguration of triangle geometry: a mini-history[END_REF], with a section that refers to the "transfiguration" power of computer-based proofs of geometry statements) when educators started reflecting about the potential role in education of software programs dealing with automatic theorem proving (automatic discovery and derivation were inexistent at that time). But these reflections were formulated rather as considerations about the future than as proposals for the present time of their authors… Thus, in view of the current implementation of ART in well spread, dynamic geometry programs, our final goal is to make an open call to the community of math teachers and math education researchers, in order to join us preparing a research project to address the following questions: Are ART in geometry education good for anything? If yes, what are they good for? What should be the necessary changes and requirements in the educational context, if ART are to be considered good for anything?

Proceedings of CERME10