Carlos Caetano
email: carlos.caetano@dcc.ufmg.br

Jessica Sena
email: jessicasena@dcc.ufmg.br

Franc ¸ois Brémond
email: francois.bremond@inria.fr

Jefersson A Dos Santos
email: jefersson@dcc.ufmg.br

William Robson Schwartz
email: william@dcc.ufmg.br

SkeleMotion: A New Representation of Skeleton Joint Sequences Based on Motion Information for 3D Action Recognition

Due to the availability of large-scale skeleton datasets, 3D human action recognition has recently called the attention of computer vision community. Many works have focused on encoding skeleton data as skeleton image representations based on spatial structure of the skeleton joints, in which the temporal dynamics of the sequence is encoded as variations in columns and the spatial structure of each frame is represented as rows of a matrix. To further improve such representations, we introduce a novel skeleton image representation to be used as input of Convolutional Neural Networks (CNNs), named SkeleMotion. The proposed approach encodes the temporal dynamics by explicitly computing the magnitude and orientation values of the skeleton joints. Different temporal scales are employed to compute motion values to aggregate more temporal dynamics to the representation making it able to capture longrange joint interactions involved in actions as well as filtering noisy motion values. Experimental results demonstrate the effectiveness of the proposed representation on 3D action recognition outperforming the state-of-the-art on NTU RGB+D 120 dataset.

Introduction

Human action recognition plays an important role in various applications, for instance surveillance systems can be used to detect and prevent abnormal or suspicious actions, health care systems can be used to monitor elderly people on their daily living activities and robot and human interactions.

Over the last decade, significant progress on the action recognition task has been achieved with the design of discriminative representations employed to the image and video domains on RGB data or optical flow. Such information is based on appearance or motion analysis. Due to the development of low-cost RGB-D sensors (e.g., Kinect), it becomes possible to employ depth information as well as human skeleton joints to perform 3D action recognition. Compared to RGB and optical flow, skeleton data has the advantages of being computationally efficient since the data size is smaller. Moreover, skeleton data are robust to illumination changes, robust to background noise and invariant to camera views [START_REF] Han | Space-time representation of people based on 3d skeletal data[END_REF].

Many works for 3D action recognition have focused on designing handcrafted feature descriptors [START_REF] Wang | Mining actionlet ensemble for action recognition with depth cameras[END_REF][START_REF] Yang | Eigenjoints-based action recognition using naïve-bayes-nearest-neighbor[END_REF][START_REF] Zanfir | The moving pose: An efficient 3d kinematics descriptor for low-latency action recognition and detection[END_REF][START_REF] Gowayyed | Histogram of oriented displacements (hod): Describing trajectories of human joints for action recognition[END_REF][START_REF] Wang | Mining mid-level features for action recognition based on effective skeleton representation[END_REF][START_REF] Devanne | 3-d human action recognition by shape analysis of motion trajectories on riemannian manifold[END_REF] to encode skeleton data while adopting Dynamic Time Warping (DTW), Fourier Temporal Pyramid (FTP) or Hidden Markov Model (HMM) to model temporal dynamics in the sequences. Nowadays, large efforts have been directed to the employment of deep neural networks. These architectures learn hierarchical layers of representations to perform pattern recognition and have demonstrated impressive results on many pattern recognition tasks (e.g., image classification [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF] and face recognition [START_REF] Schroff | Facenet: A unified embedding for face recognition and clustering[END_REF]). For instance, Recurrent Neural Networks (RNNs) with Long-Short Term Memory (LSTM) have been employed to model skeleton data for 3D action recognition [START_REF] Veeriah | Differential recurrent neural networks for action recognition[END_REF][START_REF] Shahroudy | Ntu rgb+d: A large scale dataset for 3d human activity analysis[END_REF][START_REF] Song | An end-toend spatio-temporal attention model for human action recognition from skeleton data[END_REF][START_REF] Zhang | View adaptive recurrent neural networks for high performance human action recognition from skeleton data[END_REF]. Although RNN approaches present excellent results in 3D action recognition task due to their power of modeling temporal sequences, such structures lack the ability to efficiently learn the spatial relations between the skeleton joints [START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF].

To take advantage of the spatial relations, a hierarchical structure was proposed by Du et al. [START_REF] Du | Skeleton based action recognition with convolutional neural network[END_REF]. The authors represent each skeleton sequence as 2D arrays, in which the temporal dynamics of the sequence is encoded as variations in columns and the spatial structure of each frame is represented as rows. Then, the representation is fed to a Convolutional Neural Network (CNN) which has the natural abil-ity of learning structural information from 2D arrays. Such type of representations are very compact encoding the entire video sequence in one single image.

To further improve the representation of skeleton joints for 3D action recognition, in this paper we introduce a novel skeleton image representation to be used as input of CNNs named SkeleMotion. The proposed approach encodes temporal dynamics by explicitly using motion information computing the magnitude and orientation values of the skeleton joints. To that end, different temporal scales are used to filter noisy motion values as well as aggregating more temporal dynamics to the representation making it being able to capture long-range joint interactions involved in actions. Moreover, the method takes advantage of a structural organization of joints that preserves spatial relations of more relevant joint pairs. To perform action classification, we train a tiny CNN architecture with only three convolutional layers and two fully-connected layers. Since the network is shallow and takes as input a compact representation for each video, it is extremely fast to train.

In the literature, many works employed or improved the skeleton image representation for 3D action recognition [START_REF] Wang | Action recognition based on joint trajectory maps using convolutional neural networks[END_REF][START_REF] Liu | 3d action recognition using data visualization and convolutional neural networks[END_REF][START_REF] Ke | A new representation of skeleton sequences for 3d action recognition[END_REF][START_REF] Li | Skeleton-based action recognition with convolutional neural networks[END_REF][START_REF] Wang | Action recognition based on joint trajectory maps with convolutional neural networks[END_REF][START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF][START_REF] Li | Co-occurrence feature learning from skeleton data for action recognition and detection with hierarchical aggregation[END_REF][START_REF] Choutas | Potion: Pose motion representation for action recognition[END_REF]. However, none of the methods model explicit motion information (i.e., magnitude and orientation) in multiple temporal scales, as the proposed approach does. Similar to our approach, the works of [START_REF] Li | Skeleton-based action recognition with convolutional neural networks[END_REF][START_REF] Li | Co-occurrence feature learning from skeleton data for action recognition and detection with hierarchical aggregation[END_REF] were the only ones that tried to encode motion on skeleton images, however they employed a naive approach by computing difference of motion joints on consecutive frames.

According to the experimental results, our proposed skeleton image representation can handle skeleton based 3D action recognition very well. Moreover, SkeleMotion representation achieves the state-of-the-art performance on the large scale NTU RGB+D 120 [START_REF] Liu | Ntu rgb+d 120: A large-scale benchmark for 3d human activity understanding[END_REF] dataset when combined with a spatial structural joint representation.

The code of our SkeleMotion representation is publicly available to facilitate future research 1 .

Related Work

In this section, we present a literature review of works that are close to the idea proposed in our approach by employing different representations based on skeleton images.

As the forerunner of skeleton image representations, Du et al. [START_REF] Du | Skeleton based action recognition with convolutional neural network[END_REF] represent the skeleton sequences as a matrix. Each row of such matrix corresponds to a chain of concatenated skeleton joint coordinates from the frame t. Hence, each column of the matrix corresponds to the temporal evolution of the joint j. At this point, the matrix size is J × T × 3, where J is the number of joints for each skeleton, T is the total frame number of the video sequence and 3 is the num-1 https://github.com/carloscaetano/skeleton-images ber coordinate axes (x, y, z). The values of this matrix are quantified into an image (i.e., linearly rescaled to a [0, 255]) and normalized to handle the variable-length problem. In this way, the temporal dynamics of the skeleton sequence is encoded as variations in rows and the spatial structure of each frame is represented as columns. Finally, the authors use their representation as input to a CNN model composed by four convolutional layers and three max-pooling layers.

After the feature extraction, a feed-forward neural network with two fully-connected layers is employed for classification.

Wang et al. [START_REF] Wang | Action recognition based on joint trajectory maps using convolutional neural networks[END_REF][START_REF] Wang | Action recognition based on joint trajectory maps with convolutional neural networks[END_REF] present a skeleton representation to represent both spatial configuration and dynamics of joint trajectories into three texture images through color encoding, named Joint Trajectory Maps (JTMs). The authors apply rotations to the skeleton data to mimicking multi-views and also for data enlargement to overcome the drawback of CNNs usually being not view invariant. JTMs are generated by projecting the trajectories onto the three orthogonal planes. To encode motion direction in the JTM, they use a hue colormap function to "color" the joint trajectories over the action period. They also encode the motion magnitude of joints into saturation and brightness claiming that changes in motion results in texture in the JMTs. Finally, the authors individually fine-tune three AlexNet [START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF] CNNs (one for each JTM) to perform classification.

To overcome the problem of the sparse data generated by skeleton sequence video, Ke et al. [START_REF] Ke | A new representation of skeleton sequences for 3d action recognition[END_REF] represent the temporal dynamics of the skeleton sequence by generating four skeleton representation images. Their approach is closer to Du et al. [START_REF] Du | Skeleton based action recognition with convolutional neural network[END_REF] method, however they compute the relative positions of the joints to four reference joints by arranging them as a chain and concatenating the joints of each body part to the reference joints resulting onto four different skeleton representations. According to the authors, such structure incorporate different spatial relationships between the joints. Finally, the skeleton images are resized and each channel of the four representations is used as input to a VGG19 [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF] pre-trained architecture for feature extraction.

To encode motion information on skeleton image representation, Li et al. [START_REF] Li | Skeleton-based action recognition with convolutional neural networks[END_REF] proposed the skeleton motion image. Their approach is created similar to Du et al. [START_REF] Du | Skeleton based action recognition with convolutional neural network[END_REF] skeleton image representation, however each matrix cell is composed by joint difference computation between two consecutive frames. To perform classification, the authors used Du et al. [START_REF] Du | Skeleton based action recognition with convolutional neural network[END_REF] approach and their proposed representation independently as input of a neural network with a two-stream paradigm. The CNN used was a small seven-layer network consisting of three convolution layers and four fullyconnected layers.

Yang et al. [START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF] claim that the concatenation process of chaining all joints with a fixed order turn into lack of semantic meaning and leads to loss in skeleton structural informa-tion. To that end, Yang et al. [START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF] proposed a representation named Tree Structure Skeleton Image (TSSI) to preserve spatial relations. Their method is created by traversing a skeleton tree with a depth-first order algorithm with the premise that the fewer edges there are, the more relevant the joint pair is. The generated representation are then quantified into an image and resized before being sent to a ResNet-50 [START_REF] He | Deep residual learning for image recognition[END_REF] CNN architecture.

As it can be inferred from the reviewed methods, most of them are improved versions of Du et al. [START_REF] Du | Skeleton based action recognition with convolutional neural network[END_REF] skeleton image focusing on spatial structural of joint axes while the temporal dynamics of the sequence is encoded as variations in columns, or encode motion information in a naive manner (difference of motion joints on consecutive frames). Despite the aforementioned methods produce promising results, they do not explicit encode rich motion information. In view of that, to capture more motion information, our approach directly encodes it by using orientation and magnitude to provide information regarding the velocity of the movement in different temporal scales In view of that, our SkeleMotion approach differs from the literature methods by capturing the temporal dynamics explicitly provided by magnitude and orientation motion information.

Proposed Approach

In this section, we introduce our proposed skeleton image representation based on magnitude and orientation motion information, named SkeleMotion.

SkeleMotion

As reviewed in Section 2, the majority of works that encode skeleton data as image representations are based on spatial structure encoding of the skeleton joints. According to Li et al. [START_REF] Li | Co-occurrence feature learning from skeleton data for action recognition and detection with hierarchical aggregation[END_REF], temporal movements of joints can also be used as crucial cues for action recognition and although the temporal dynamics of the sequence can be implicitly learned by using a CNN, an explicit modeling can produce better recognition accuracies.

Recently, a new temporal stream for the two-stream networks called Magnitude-Orientation Stream (MOS) [START_REF] Caetano | Activity recognition based on a magnitudeorientation stream network[END_REF] was developed. The method is based on non-linear transformations and claim that motion information on a video sequence can be described by the spatial relationship contained on the local neighborhood of magnitude and orientation extracted from the optical flow and has shown excellent results on the 2D action recognition problem. Motivated by such results, in this paper we propose a novel skeleton image representation (named SkeleMotion), based on magnitude and orientation of the joints to explore the temporal dynamics. Our approach expresses the displacement information by using orientation encoding (direction of joints) and magnitude to provide information regarding the velocity of the move-ment. Furthermore, due to the successful results achieved by the skeleton image representations, our approach follows the same fundamentals by representing the skeleton sequences as a matrix. First, we apply the depth-first tree traversal order [START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF] to the skeleton joints to generate a predefined chain order C that best preserves the spatial relations between joints in original skeleton structures2 . Afterwards, we compute matrix S that corresponds to a chain of concatenated skeleton joint coordinates from the frame t. In view of that, each column of the matrix corresponds to the temporal evolution of the arranged chain joint c. At this point, the size of matrix S is C × T × 3, where C is the number of joints of the chain, T is the total frame number of the video sequence and 3 is the number joint coordinate axes (x, y, z). Then, we create the motion structure D as

D c,t = S c,t+d -S c , (1)
where each matrix cell is composed by the temporal difference computation of each joint between two frames of d distance, resulting in a C × T -d × 3 matrix. By using the proposed motion structure D, we build two different representations: one based on the magnitudes of joint motions and another one based the orientations of the joint motion. We compute both representations using

M c,t = (D x c,t ) 2 + (D y c,t ) 2 + (D z c,t ) 2 (2)
and θ c,t = stack(θ xy c,t , θ yz c,t , θ zx c,t )

θ xy c,t = tan -1 D y c,t D x c,t , θ yz c,t = tan -1 D z c,t D y c,t , θ zx c,t = tan -1 D x c,t D z c,t , (3)
where M is the magnitude skeleton representation of size J ×T -d×1 and θ is the orientation skeleton representation of size J × T -d × 3 (composed by 3 stacked channels).

Since the orientation values are estimated for every joint, it might generate noisy values for joints without any movement. Therefore, we perform a filtering on θ based on the values of M as

θ c,t = 0, if M c,t < m θ c,t , otherwise , (4)
where m is a magnitude threshold value. Finally, the generated matrices are normalized into [0, 1] and empirically resized into a fixed size of C × 100, since number of frames may vary depending on the skeleton sequence of each video. Figure 1 gives an overview of our method for building the SkeleMotion representation.

Temporal Scale Aggregation (TSA)

Skeleton image representations in the literature basically encodes joint coordinates as channels. In view of that, it may cause a problem that the co-occurrence features are aggregated locally, being not able to capture long-range joint interactions involved in actions [START_REF] Li | Co-occurrence feature learning from skeleton data for action recognition and detection with hierarchical aggregation[END_REF]. Moreover, one drawback of encoding motion values of joints is the noisy values that can be introduced to the representation due to small distance d between two frames. For instance, if the computation is performed considering two consecutive frames, it could add to the representation unnecessary motion of joints that are irrelevant to predict a specific action (e.g., motion of the head joint on a handshake action).

To overcome the aforementioned problems, we also propose a variation of our SkeleMotion representation by precomputing the motion structure D considering different d distances. For each of the motion structures D, we compute its respective magnitude skeleton representation M and then stack them all into one single representation. The same is applied to compute the orientation skeleton representation θ, however a weighting scheme is applied during the filtering process explained before, as

θ c,t = 0, if M c,t < m × d θ c,t , otherwise . (5)
Such technique adds more temporal dynamics to the representation by explicitly showing temporal scales to the network. In this way, the network can learn which movements are really important for the action learning and also being able to capture long-range joint interactions.

Experimental Results

In this section we present the experimental results obtained with the proposed skeleton image representation for the 3D action recognition problem. We compare it to other skeleton representations in the literature. Besides the classical skeleton image representation of Du et al. [START_REF] Du | Skeleton based action recognition with convolutional neural network[END_REF], we compare with other representations used by state-ofthe-art approaches [START_REF] Wang | Action recognition based on joint trajectory maps using convolutional neural networks[END_REF][START_REF] Ke | A new representation of skeleton sequences for 3d action recognition[END_REF][START_REF] Li | Skeleton-based action recognition with convolutional neural networks[END_REF][START_REF] Li | Co-occurrence feature learning from skeleton data for action recognition and detection with hierarchical aggregation[END_REF][START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF] as baselines on NTU RGB+D 60 [START_REF] Shahroudy | Ntu rgb+d: A large scale dataset for 3d human activity analysis[END_REF]. We also compare our approach to sate-of-the-art methods on the NTU RGB+D 120 [START_REF] Liu | Ntu rgb+d 120: A large-scale benchmark for 3d human activity understanding[END_REF]. To isolate only the contribution brought by SkeleMotion to the action recognition problem, all other representations were tested on the same datasets with the same split of training and testing data and using the same CNN.

Datasets

The NTU RGB+D 60 [START_REF] Shahroudy | Ntu rgb+d: A large scale dataset for 3d human activity analysis[END_REF] is publicly available 3D action recognition dataset. It consists of 56,880 videos from 60 action categories which are performed by 40 distinct subjects. The videos were collected by three Microsoft Kinect sensors. The dataset provides four different data information: (i) RGB frames; (ii) depth maps; (iii) 395 infrared sequences; and (iv) skeleton joints. There are two different evaluation protocols: cross-subject, which split the 40 subjects into training and testing; and cross-view, which uses samples from one camera for testing and the other two for training. The performance is evaluated by computing the average recognition across all classes.

The NTU RGB+D 120 [START_REF] Liu | Ntu rgb+d 120: A large-scale benchmark for 3d human activity understanding[END_REF] is a large-scale 3D action recognition dataset captured under various environmental conditions. It consists of 114,480 RGB+D video samples captured using the Microsoft Kinect sensor. As in NTU RGB+D 60 [START_REF] Shahroudy | Ntu rgb+d: A large scale dataset for 3d human activity analysis[END_REF], the dataset provides RGB frames, depth maps, infrared sequences and skeleton joints. It is com-...

Implementation Details

To isolate only the contribution brought by the proposed representation to the action recognition problem, all compared skeleton image representations were implemented and tested on the same datasets and using the same network architecture. In view of that, we applied the same split of training and testing data, and employ the evaluation protocols and metrics proposed by the creators of the datasets.

The network architecture employed is a modified version of the CNN proposed by Li et al. [START_REF] Li | Skeleton-based action recognition with convolutional neural networks[END_REF]. They designed a small convolutional neural network which consists of three convolution layers and four fully-connected (FC) layers. However, here we modified it to a tiny version, employing the convolutional layers and only two FC layers. All convolutions have a kernel size of 3 × 3, the first and second convolutional layers with a stride of 1 and the third one with a stride of 2. Max pooling and ReLU neuron are adopted and the dropout regularization ratio is set to 0.5. The learning rate is set to 0.001 and batch size is set to 1000. The training is stopped after 200 epochs. The loss function employed was the categorical cross-entropy. We opted for using such architecture since it demonstrated good performance and, according to the authors, it can be easily trained from scratch without any pre-training and is superior on its compact model size and fast inference speed as well. To cope with actions involving multi-person interaction (e.g., shaking hands), we apply a common choice in the literature which is to stack skeleton image representations of different people as the network input.

To obtain the orientation skeleton image representation θ we empirically set the parameter m = 0.004, as described in Section 3.

Evaluation

In this section, we present experiments for parameters optimization and report a comparison of our proposed skeleton representation. We used a subset of NTU RGB+D 60 [START_REF] Shahroudy | Ntu rgb+d: A large scale dataset for 3d human activity analysis[END_REF] training set (considering cross-view protocol) to perform parameter setting and then used such parameter on the remaining experiments. We focused on the optimization of the number of temporal scales used on temporal scale aggregation (TSA).

Table 1. Action recognition accuracy (%) results on a subset of NTU RGB+D 60 [START_REF] Shahroudy | Ntu rgb+d: A large scale dataset for 3d human activity analysis[END_REF] dataset by applying temporal scale aggregation (TSA) on our SkeleMotion representation.

Temporal

Magnitude Orientation distances Acc. (%) Acc. (%) To set the number of temporal scales of our SkeleMotion approach, we empirically varied it from two to four temporal scales considering 20 frames in total. Table 1 shows the results obtained by such variation. We can see that the best result is obtained by using three temporal scales for both magnitude [START_REF] Gowayyed | Histogram of oriented displacements (hod): Describing trajectories of human joints for action recognition[END_REF][START_REF] Ke | A new representation of skeleton sequences for 3d action recognition[END_REF][START_REF] Liu | Ntu rgb+d 120: A large-scale benchmark for 3d human activity understanding[END_REF] and orientation [START_REF] Caetano | Activity recognition based on a magnitudeorientation stream network[END_REF][START_REF] Ke | A new representation of skeleton sequences for 3d action recognition[END_REF][START_REF] Liu | Global context-aware attention lstm networks for 3d action recognition[END_REF]. Moreover, we noticed that the performance tends to saturate or drop when considering four temporal scales. [START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF] SkeleMotion + Yang et al. [START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF] Action Number Figure 3. The complementary between SkeleMotion (Magnitude-Orientation TSA) and Yang et al. [START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF] TSSI representation on NTU RGB+D 60 [START_REF] Shahroudy | Ntu rgb+d: A large scale dataset for 3d human activity analysis[END_REF] dataset for cross-view protocol. Best viewed in color.

Table 2 presents a comparison of our approach with skeleton image representations of the literature. For the methods that have more than one "image" per representation ( [START_REF] Wang | Action recognition based on joint trajectory maps using convolutional neural networks[END_REF] and [START_REF] Ke | A new representation of skeleton sequences for 3d action recognition[END_REF]), we stacked them to be used as input to the network. The same was performed for our SkeleMotion approach considering magnitude and orientation. Regarding the cross-subject protocol, the best result was obtained by Reference Joints representation from Ke et al., [START_REF] Ke | A new representation of skeleton sequences for 3d action recognition[END_REF] achieving 70.8% of accuracy while our best result (SkeleMotion Magnitude (TSA)) achieves a competitive accuracy of 69.6%. It is worth noting that there is a considerable improvement of 12.8 (p.p.) obtained by Skele-Motion Magnitude (TSA) when compared to Li et al., [START_REF] Li | Skeleton-based action recognition with convolutional neural networks[END_REF] baseline, which also explicitly encode motion information. On the other side, the best result on cross-view protocol was obtained by our SkeleMotion Magnitude (TSA) approach achieving 80.1% of accuracy. There is an improvement of 4.5 percentage points (p.p.) when compared to the Tree Structure Skeleton Image (TSSI) from Yang et al., [START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF], which was the best baseline result. Again, there is a considerable improvement of 18.8 (p.p.) when compared to Li et al., [START_REF] Li | Skeleton-based action recognition with convolutional neural networks[END_REF] baseline. To exploit a possible complementarity of the tempo-ral (our SkeleMotion) and spatial (Yang et al. [START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF]) skeleton image representations, we combined the different approaches by employing early and late fusion techniques. For the early fusion, we simply stacked the representations to be used as input to the network. On the other hand, the late fusion technique applied was a non-weighted linear combination of the prediction scores of each method. According to the results showed in Table 3, any type of combination performed with our SkeleMotion provides better results than their solo versions. Regarding cross-subject protocol, our best results achieves 73.5% of accuracy with early fusion technique against 76.5% of the late fusion approach. Furthermore, on cross-view protocol, our best results achieves 82.4% of accuracy with early fusion technique against 84.7% of the late fusion approach. Detailed improvements are shown in Figure 3. Finally, Table 4 presents the experiments of our proposed skeleton image representation on the recent available NTU RGB+D 120 [START_REF] Liu | Ntu rgb+d 120: A large-scale benchmark for 3d human activity understanding[END_REF] dataset. Due to the results obtained on Table 3, here we employed the late fusion scheme for methods combination.

We obtained good results with our SkeleMotion representation outperforming many skeleton based methods [START_REF] Shahroudy | Ntu rgb+d: A large scale dataset for 3d human activity analysis[END_REF][START_REF] Hu | Early action prediction by soft regression[END_REF][START_REF] Hu | Jointly learning heterogeneous features for rgb-d activity recognition[END_REF][START_REF] Liu | Spatio-temporal lstm with trust gates for 3d human action recognition[END_REF][START_REF] Liu | Skeleton-based action recognition using spatio-temporal lstm network with trust gates[END_REF][START_REF] Liu | Global context-aware attention lstm networks for 3d action recognition[END_REF][START_REF] Ke | A new representation of skeleton sequences for 3d action recognition[END_REF][START_REF] Liu | Skeleton-based online action prediction using scale selection network[END_REF][START_REF] Liu | Enhanced skeleton visualization for view invariant human action recognition[END_REF][START_REF] Liu | Skeleton-based human action recognition with global context-aware attention lstm networks[END_REF][START_REF] Ke | Learning clip representations for skeleton-based 3d action recognition[END_REF][START_REF] Liu | Recognizing human actions as the evolution of pose estimation maps[END_REF]. When combining our representation with Yang et al. [START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF] we achieve state-of-theart results, outperforming the best reported method (Body Pose Evolution Map [START_REF] Liu | Recognizing human actions as the evolution of pose estimation maps[END_REF]) by up to 3.1 p.p. on cross-subject protocol and achieve competitive results on cross-setup protocol.

In comparison with LSTM approaches, we outperform the best reported method (Two-Stream Attention LSTM) by 1.7 p.p. using our skeleton image representation and 6.5 p.p. when combining it with Yang et al. [START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF] method on crosssubject protocol. Regarding the cross-setup protocol, we outperform them by 3.6 p.p. using our skeleton image representation fused with Yang et al. [START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF]. This indicates that our skeleton image representation approach used as input for CNNs leads to a better learning of temporal dynamics than the approaches that employs LSTM. [START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF] 71.7 82.4 Orientation (TSA) + Yang et al. [START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF] 69.6 78.9 Magnitude-Orientation (TSA) + Yang et al. [START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF] 73.5 82.1

Magnitude-Orientation 65.6 71.1 Magnitude-Orientation + Yang et al. [START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF] 73.2 79.5

Late

Magnitude-Orientation (TSA) 72.2 81.7 Fusion Magnitude (TSA) + Yang et al. [START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF] 75.4 83.2 Orientation (TSA) + Yang et al. [START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF] 73.6 80.6 Magnitude-Orientation (TSA) + Yang et al. [START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF] 76.5 84.7

Table 4. Action recognition accuracy (%) results on NTU RGB+D 120 [START_REF] Liu | Ntu rgb+d 120: A large-scale benchmark for 3d human activity understanding[END_REF] dataset. Results for literature methods were obtained from [START_REF] Liu | Ntu rgb+d 120: A large-scale benchmark for 3d human activity understanding[END_REF].

Cross-subject Cross-setup Approach Acc. (%) Acc. (%)

Part-Aware LSTM [START_REF] Shahroudy | Ntu rgb+d: A large scale dataset for 3d human activity analysis[END_REF] 25.5 26.3 Soft RNN [START_REF] Hu | Early action prediction by soft regression[END_REF] 36.3 44.9 Dynamic Skeleton [START_REF] Hu | Jointly learning heterogeneous features for rgb-d activity recognition[END_REF] 50.8 54.7 Spatio-Temporal LSTM [START_REF] Liu | Spatio-temporal lstm with trust gates for 3d human action recognition[END_REF] 55.7 57.9 Internal Feature Fusion [START_REF] Liu | Skeleton-based action recognition using spatio-temporal lstm network with trust gates[END_REF] 58.2 60.9 Literature GCA-LSTM [START_REF] Liu | Global context-aware attention lstm networks for 3d action recognition[END_REF] 58.3

results

Multi-Task Learning Network [START_REF] Ke | A new representation of skeleton sequences for 3d action recognition[END_REF] 58.4 57.9 FSNet [START_REF] Liu | Skeleton-based online action prediction using scale selection network[END_REF] 59.9 62.4 Skeleton Visualization (Single Stream) [START_REF] Liu | Enhanced skeleton visualization for view invariant human action recognition[END_REF] 60.3 63.2 Two-Stream Attention LSTM [START_REF] Liu | Skeleton-based human action recognition with global context-aware attention lstm networks[END_REF] 61.2 63.3 Multi-Task CNN with RotClips [START_REF] Ke | Learning clip representations for skeleton-based 3d action recognition[END_REF] 62.2 61.8 Body Pose Evolution Map [START_REF] Liu | Recognizing human actions as the evolution of pose estimation maps[END_REF] 64.6 66.9

Orientation (TSA) 52.2 54.1 SkeleMotion Magnitude (TSA) 57.6 60.4 results

Magnitude-Orientation (TSA) 62.9 63.0 Magnitude-Orientation (TSA) + Yang et al. [START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF] 67.7 66.9

Conclusions and Future Works

In this work, we proposed a novel skeleton image representation to be used as input of CNNs, named SkeleMotion. The method is based on temporal dynamics encoding by explicitly using motion information (magnitude and orientation) of the skeleton joints. We further propose a variation of the magnitude skeleton representation considering different temporal scales in order to filter noisy motion values as well as aggregating more temporal dynamics to the representation. Experimental results on two publicly available datasets demonstrated the excellent performance of the proposed approach. Another interesting finding is that the combination of our representation with methods of the literature improves the 3D action recognition outperforming the state-of-the-art on NTU RGB+D 120 dataset.

Directions to future works include the evaluation of SkeleMotion with other distinct architectures. Moreover, we intend to evaluate its behavior on 2D action datasets with skeletons estimated by methods of the literature.

Figure 1 .

 1 Figure 1. SkeleMotion representation. (a) Skeleton data sequence of T frames. (b) Computation of the magnitude and orientation from the joint movement. (c) θ and M arrays: each row encodes the spatial information (relation between joint movements) while each column describes the temporal information for each joint movement. (d) Skeleton image after resizing and stacking of each axes.

Figure 2 .

 2 Figure 2. Network architecture employed for 3D action recognition.

  Figure 2 presents an overview of the employed architecture.

Table 2 .

 2 Action recognition accuracy (%) results on NTU RGB+D 60[START_REF] Shahroudy | Ntu rgb+d: A large scale dataset for 3d human activity analysis[END_REF] dataset. Results for the baselines were obtained running each method implementation.

			Cross-	Cross-
			subject	view
		Approach	Acc. (%) Acc. (%)
		Du et al. [4]	68.7	73.0
		Wang et al. [32]	39.1	35.9
	Baselines	Ke et al. [10]	70.8	75.5
		Li et al. [14]	56.8	61.3
		Yang et al. [34]	69.5	75.6
		Orientation	60.6	65.6
	SkeleMotion Magnitude	58.4	64.2
	results	Orientation (TSA)	65.3	73.2
		Magnitude (TSA)	69.6	80.1

Table 3 .

 3 Comparison between late and early fusion techniques on NTU RGB+D 60[START_REF] Shahroudy | Ntu rgb+d: A large scale dataset for 3d human activity analysis[END_REF] dataset. Fusion Magnitude (TSA) + Yang et al.

	Cross-subject Cross-view

Chain C considering 25 Kinect joints:[2, 21,

[START_REF] Devanne | 3-d human action recognition by shape analysis of motion trajectories on riemannian manifold[END_REF][START_REF] Du | Skeleton based action recognition with convolutional neural network[END_REF][START_REF] Devanne | 3-d human action recognition by shape analysis of motion trajectories on riemannian manifold[END_REF][START_REF] Liu | 3d action recognition using data visualization and convolutional neural networks[END_REF][START_REF] Gowayyed | Histogram of oriented displacements (hod): Describing trajectories of human joints for action recognition[END_REF][START_REF] Han | Space-time representation of people based on 3d skeletal data[END_REF][START_REF] He | Deep residual learning for image recognition[END_REF][START_REF] Hu | Jointly learning heterogeneous features for rgb-d activity recognition[END_REF][START_REF] Liu | Enhanced skeleton visualization for view invariant human action recognition[END_REF][START_REF] Liu | Recognizing human actions as the evolution of pose estimation maps[END_REF][START_REF] Liu | Enhanced skeleton visualization for view invariant human action recognition[END_REF][START_REF] Hu | Jointly learning heterogeneous features for rgb-d activity recognition[END_REF][START_REF] He | Deep residual learning for image recognition[END_REF][START_REF] Han | Space-time representation of people based on 3d skeletal data[END_REF][START_REF] Gowayyed | Histogram of oriented displacements (hod): Describing trajectories of human joints for action recognition[END_REF][START_REF] Liu | 3d action recognition using data visualization and convolutional neural networks[END_REF][START_REF] Hu | Early action prediction by soft regression[END_REF][START_REF] Ke | A new representation of skeleton sequences for 3d action recognition[END_REF][START_REF] Ke | Learning clip representations for skeleton-based 3d action recognition[END_REF][START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF][START_REF] Schroff | Facenet: A unified embedding for face recognition and clustering[END_REF][START_REF] Shahroudy | Ntu rgb+d: A large scale dataset for 3d human activity analysis[END_REF][START_REF] Schroff | Facenet: A unified embedding for face recognition and clustering[END_REF][START_REF] Krizhevsky | Imagenet classification with deep convolutional neural networks[END_REF][START_REF] Ke | Learning clip representations for skeleton-based 3d action recognition[END_REF][START_REF] Ke | A new representation of skeleton sequences for 3d action recognition[END_REF][START_REF] Hu | Early action prediction by soft regression[END_REF][START_REF] Liu | 3d action recognition using data visualization and convolutional neural networks[END_REF][START_REF] Choutas | Potion: Pose motion representation for action recognition[END_REF][START_REF] Caetano | Activity recognition based on a magnitudeorientation stream network[END_REF][START_REF] Li | Skeleton-based action recognition with convolutional neural networks[END_REF][START_REF] Li | Co-occurrence feature learning from skeleton data for action recognition and detection with hierarchical aggregation[END_REF][START_REF] Liu | Ntu rgb+d 120: A large-scale benchmark for 3d human activity understanding[END_REF][START_REF] Liu | Skeleton-based online action prediction using scale selection network[END_REF][START_REF] Liu | Ntu rgb+d 120: A large-scale benchmark for 3d human activity understanding[END_REF][START_REF] Li | Co-occurrence feature learning from skeleton data for action recognition and detection with hierarchical aggregation[END_REF][START_REF] Li | Skeleton-based action recognition with convolutional neural networks[END_REF][START_REF] Caetano | Activity recognition based on a magnitudeorientation stream network[END_REF][START_REF] Liu | Skeleton-based action recognition using spatio-temporal lstm network with trust gates[END_REF][START_REF] Liu | Spatio-temporal lstm with trust gates for 3d human action recognition[END_REF][START_REF] Liu | Skeleton-based human action recognition with global context-aware attention lstm networks[END_REF][START_REF] Liu | Global context-aware attention lstm networks for 3d action recognition[END_REF][START_REF] Liu | Skeleton-based human action recognition with global context-aware attention lstm networks[END_REF][START_REF] Liu | Spatio-temporal lstm with trust gates for 3d human action recognition[END_REF][START_REF] Liu | Skeleton-based action recognition using spatio-temporal lstm network with trust gates[END_REF][START_REF] Caetano | Activity recognition based on a magnitudeorientation stream network[END_REF][START_REF] Choutas | Potion: Pose motion representation for action recognition[END_REF], as defined in[START_REF] Yang | Action recognition with spatio-temporal visual attention on skeleton image sequences[END_REF].

Acknowledgments

The authors would like to thank the National Council for Scientific and Technological Development -CNPq (Grants 311053/2016-5, 204952/2017-4 and 438629/2018-3), the Minas Gerais Research Foundation -FAPEMIG (Grants APQ-00567-14 and PPM-00540-17) and the Coordination for the Improvement of Higher Education Personnel -CAPES (DeepEyes Project).


Camille Doukhan
email: camille.doukhan1@espe-bretagne.fr

CREAD Ghislaine Gueudet
email: ghislaine.gueudet@espe-bretagne.fr

Students' difficulties at the secondary-tertiary transition: the case of random variables

Keywords: Praxeologies, Probabilities, Random variables, Registers, Secondary-Tertiary transition

The work presented here concerns the teaching and learning of random variables at the secondarytertiary transition. Our study takes place in France, where discrete random variables are studied in grade 11 and in the first post-secondary year. Comparing textbooks, we observe that random variables are introduced as tools for the modelling of random situations at grade 11, while they are presented theoretically at the post-secondary level. The answers of first year students to a test confirm this difference: they are able to use random variables to produce interpretations in a concrete situation, but are not prepared for the learning of theoretical definitions. These results can contribute to design a teaching intervention concerning probabilities for first year students.

Introduction

Difficulties encountered by students at the secondary-tertiary transition have been analyzed in many research studies in mathematics education, with different perspectives (Gueudet & Thomas, in press). Some of these studies focus on particular mathematical topics: calculus or linear algebra for example. The transition issues concerning probabilities have been less studied so far; yet probabilities are present in many tertiary courses, for non-specialists of mathematics in particular, like biologists or economists. Our study, taking place in France, addresses this issue with a focus on the concept of random variable. Our research questions are the following:

-How is the concept of random variable taught at secondary school, at the beginning of tertiary education? Which similarities or differences can be observed? -What do the students know about random variables after the end of secondary school, and does this prepare them adequately for what is taught in the first year of tertiary level?

We study these questions with an institutional perspective. We present in the next section our theoretical tools, and studies about the teaching and learning of probability which inform our research. Then we expose in Section 3 the context of our study and our methods. In Section 4 we present our results, coming from a textbook study and a test proposed to first year students. We draw conclusions from these results and present them in Section 5.

Theoretical frame and related works

In our work we consider the secondary-tertiary transition as an institutional transition: the mathematics taught is shaped by the teaching institution, and is thus different at secondary school and university. We use the Anthropological Theory of Didactics (ATD, [START_REF] Chevallard | Steps towards a new epistemology in mathematics education[END_REF], which introduces, in particular, the concept of praxeology. A praxeology has four components: a type of tasks, T; a technique,  to accomplish this type of tasks; a technology  which is a discourse explaining and justifying the technique; and a theory . Comparing the praxeologies at the end of secondary school and at the beginning of university can illustrate different kinds of changes in the mathematics taught (see e.g., [START_REF] Winsløw | An institutional approach to university mathematics education: From dual vector spaces to questioning the world[END_REF]. Some changes are very general: for a given type of tasks, a single technique is taught at secondary school, while several techniques can be expected at university; similarly, at university the theoretical block (formed by the technology and the theory [is more important than the practical block [T, ].

Other changes concern a particular type of task, for example in our case: "Determine the probability distribution of a random variable," studied below.

We also use in our work the concept of semiotic registers [START_REF] Duval | Understanding the mathematical way of thinkingthe registers of semiotic representations[END_REF]. Mathematical concepts, in particular random variables, can be described through different semiotic registers: a text, a table, or a graph. Compared with secondary school, the institutional expectations concerning the flexible use of different registers increase at university (Gueudet & Thomas, 2018). Hence observing the registers used, in particular the need to combine several registers in the same technique for a given task, can illustrate discontinuities in the transition process.

Concerning probability, several works acknowledge the specific nature of probabilistic reasoning. [START_REF] Batanero | Research on teaching and learning probability[END_REF] state for example that "probabilistic reasoning is a mode of reasoning that refers to judgments and decision-making under uncertainty and is relevant to real life" (p. 9). Such link between probability and real-life is emphasized by many authors, and in the curriculum of different countries [START_REF] Batanero | Research on teaching and learning probability[END_REF]. However, some studies indicate that an important proportion of probability problems proposed to students are not related to real life situations and that the problems involve more algebraic than probabilistic reasoning (20% in entrance tests to university in Andalucía according to [START_REF] Batanero | Characterizing probability problems posed in university entrance tests in Andalucía[END_REF]. Student difficulties are observed and connected with the teaching practices; for example [START_REF] Burrill | Fundamental statistical ideas in the school curriculum and in training teachers[END_REF] claim that a formal teaching of probabilities focusing on set theory and counting techniques can hinder the understanding of the associated phenomena.

Research in mathematics education concerning random variables is scarce. At the beginning of university, [START_REF] Amrani | Student's conceptual difficulties with respect to the notion of random variable[END_REF] observed student difficulties, after a very formal teaching about random variables. Given a random experiment, they had difficulties proposing a model with an appropriate random variable and determining the probability distribution for a given random variable. Nevertheless these authors did not observe the impact of a less formal teaching.

Context and methods

Our research takes place in France, where probability is an important part of the official curriculum at secondary school, starting at Grade 9. We focus here on Grade 11 of the scientific track (called "S section"). Students encounter random variables for the first time at Grade 11, in the context of discrete random variables. The concepts of probability distribution, expected value, and variance are defined; the Bernoulli and binomial distributions are introduced. According to the official curriculum: "The concept of probability distribution of a random variable permits the modelling of random situations and their study with the tools of probability". At Grade 12 students encounter continuous random variables, with the concept of density and the Gaussian distribution.

We chose to focus on "preparatory classes" in the tertiary level, selecting high-achieving students, who want to enter a "grande école" after two years of preparation. The classes are organized is similarly to secondary school, hence we claim that the institutional differences mainly come from the mathematics taught. Moreover, because they are high achieving, the difficulties encountered by these students should not come from an insufficient grasp of secondary school praxeologies. The official curriculum of the first year of preparatory classes (in our case an "Economics and Commercial -Scientific, ECS," preparatory class) concerning random variables is limited to discrete random variables (continuous random variables appear only in the second year). The curriculum revisits the notions taught at Grade 11: random variable, probability distribution, expected value, and introduces the study of pairs of random variables, and independent random variables. The official curriculum states that "Modelling simple random situations with random variables is a skill expected from students", meaning that the preparatory classes institution is not responsible for this modelling skill.

In the study presented here, we used two kinds of methods. First, we performed a comparative analysis of a secondary school textbook, and a preparatory class textbook. The secondary school textbook [START_REF] Barbazo | Mathématiques Barbazo 1re S[END_REF], is used by a large number of Grade 11 teachers in France. The students in the preparatory classes did have a textbook, but they often bought books to complement the course given by their teacher. The content across books was quite similar, so we chose [START_REF] Gautier | J'intègre[END_REF]. We compared the chapters concerning random variables according to five characteristics: the structure of the chapters in terms of course vs exercises (theoretical vs practical block), the definition of random variable given (considered as an important aspect of the theory), the registers of representations used, the types of tasks and associated praxeologies, and the contexts in which the exercises and examples were situated. For each chapter studied, we listed and counted the types of tasks requested in each exercise or worked example. The theoretical block appeared mainly in the content parts of the chapters (definitions, theorems, demonstrations, proposals) whereas the practical block appeared in the exercise and example parts of these chapters.

Second, we designed a test and submitted it to the students from a first year ECS class before the teaching of random variables (see Appendix). The teacher only allowed us to use 20 minutes to administer the test in class, so we had to limit its length. The test started with three questions: mathematical definition of a random variable, how to explain to a Grade 12 student what a random variable is, and give an example of random variable; then an exercise in an economical context was proposed. This exercise, taken from a website for Grade 11, was typical from what can be found in secondary school textbooks for this level. It corresponded to a random situation in a "real" context. Nevertheless the students' responsibility for modelling was reduced: the random variable to be studied was given in the first question. The types of task were: "Determine the distribution of a random variable (described by a text)"; "Compute the expected value of this random variable and interpret it"; and "Study the influence of a change in the initial data on the expected value". Fortyfour students took the test at the beginning of a course. We analyzed the students' answers using the following categories: for Questions 1 and 2: elements of theory and technology used, errors, ideas or elements related to randomness, and type of answer (theoretical, related to randomness, other).

For the Question 3 we looked at example correct or not, if not why, in what context it was stated, and which registers were used. For the exercise, the categories were: answer correct or not, the technique used, the technology (justification of the answer), use of registers, and result correctly interpreted.

Modelling "real" situations and mastering formal definitions

Comparing secondary school and preparatory classes textbooks

We noted a large difference in the proportions of courses/exercises within the chapters studied, in terms of number of pages dedicated to each: 15% for course in the Grade 11 textbook; more than 80% in the ECS textbook. We consider this as an indicator of the much greater importance of the technological-theoretical block in higher education, compared to secondary education.

Concerning the definition of random variable, in the Grade 11 textbook the students already knew the concept of sample space; a random variable X was defined as "a function from  to IR". We claim that this definition was difficult for Grade 11 students, who mostly met functions from IR to IR.

The definition of the probability distribution; as a function from IR to IR, followed immediately and it was more familiar. In the ECS textbook, the definition of random variable was situated at a much more general level. The definition of -algebra was introduced together with the notion of probability space as a triple (F, P). In this context a very general definition of a random variable was given: "an application X from  to IR such that x IR, { / X()≤ x} belongs to F".

We categorized the registers of representations present in the parts of courses, in the examples of the courses and in the exercises of the chapters studied. We observed seven different registers: symbolic, natural language, and five kinds of graphical registers: tree diagrams, function graphs, tables for representing the random variable distribution, some charts representing the random situation and various illustrations (e.g., photos). The symbolic register and the register of natural language were omnipresent in both textbooks. The graphical registers were more frequent in Grade 11 (68 examples used these registers) than in ECS (18 examples). There were, in particular, many illustrations in Grade 11, and none in ECS. There were 36 tables representing the random variable distribution in Garde 11 and only six in ECS. Overall, there were very few tree diagrams. In the secondary school textbook, random situations were often represented by charts or diagrams, but relatively few by trees.

Regarding the contexts in which the examples and exercises were posed, we distinguish two categories: exercises and examples in a non-mathematical context and in a mathematical theoretical context (what [START_REF] Batanero | Characterizing probability problems posed in university entrance tests in Andalucía[END_REF] call "problems with no context," p. 113). This choice is governed by the claim of the secondary school curriculum, about the importance of random variables for "the modelling of random situations" (Ministère de l'Education Nationale, 2010, p. 5).

In the first category, exercises and examples in a non-mathematical context, we identified real-life situations such as the lifespan of a robot or the waiting time at a doctor, but also "artificial" situations such as a flea moving on a graduated axis or a drawing of balls from an urn. There were also situations in other scientific fields such as the study of physical uncertainty of measurement. The proportion of examples and exercises in a non-mathematical context is 78% for Grade 11 and 67% for ECS (out of a total of 108 exercises and examples, respectively 66).

In spite of this difference, we observed similarities concerning the types of tasks. For both Grade 11 and ECS, the three most frequent types of tasks were: "determining the probability distribution of a random variable," "proving that a random variable admits an expected value and/or calculating it," and "proving that a random variable admits a variance and/or calculating it."

For example, for the type of task "determining the probability distribution of a random variable," the following technique was described in the Grade 11 textbook: "(1) determine the possible values x i for X; (2) compute the probabilities P (X = x i ); (3) summarize them in a table". This technique was linked with the fact that in most cases, the random variable took a limited number of values (in our test for example, X takes 4 different values), that could be presented in a table. In the Barbazo textbook, 104 exercises amongst 106 were about random variables taking a limited number of values. In the ECS textbook, in 20 exercises amongst 21 the random variable took its values between 1 and an integer n. To determine the distribution, the technique is consisted of finding a formula giving P(X=k) for 1 ≤ k ≤ n. The associated technologies in the two institutions were provided by the definitions mentioned above. Thus the praxeologies were clearly different in each textbook.

Also the differences concerning the context lead, for similar types of tasks, to different responsibilities of the students in terms of modelling. In Grade 11, 19 exercises asked students to calculate an average payout or determine whether a game would be favorable or not to the player, with no intermediate tasks. For this type of task, the technique used was to select the appropriate random variable, and then determine its distribution in order to calculate the probability of a certain event. This modelling work is totally absent for ECS, although the proportion of exercises in the context of artificial situations is high. In these exercises, the random variable is always given and the types of tasks present are "determining the probability distribution" or "calculating its expected value".

Students responses to the test

We recall here that the test started with three questions, concerning: the mathematical definition of a random variable, the explanation for a Grade 12 student of what a random variable is, and an example of random variable. These questions were directly linked with the theoretical block [.

Twenty-three of 44 students (52%) answered the first question, 33 of 44 students (75%) answered the second question, and 26 of 44 students (59%) answered the third question.

The answers were very often false or incomplete. The most common errors were confusion between the random variable and the probability distribution (which we expected, considering the definitions given in the textbook) and confusion between the definition of a random variable and its properties.

In the first question concerning the theory, six students confused the definition of a random variable with its properties, and five students confused the random variable with the probability distribution.

In the second question (in which students are expected to produce a technological discourse), three students confused the random variable with the probability distribution. In the third question, 11 students confused the random variable with the probability distribution.

The vocabulary and mathematical notations used by the students allowed us to distinguish two types of answers: theoretical answers in which we find a symbolic register with mathematical concepts and notations, and answers in relation to randomness in which we find a vocabulary in relation to randomness. Such answers contain ideas linked to chance, for example: "randomly draw a sample," "results of a random experiment," or "depends on chance."

For the first question, students mainly gave a theoretical answer (19 of 23 students); only 4 gave an answer in relation to randomness. An example of theoretical response by student S2 is:

"A random variable is a law that associates a variable entity (price, size) with X." (our translation)

The second question generated more diversity in the registers used by the students in their responses. We distinguish for this question a new type of response, in which students responded with a specific example of the use of a random variable. Almost a fourth of the students produced such a technological discourse, explaining in their answers how a random variable can be used (see the example below). These students seem to have incorporated the perspective of the official curriculum: random variables as tools for the modelling of random situations.

An example of this type of response by student S38 is as follows:

"it will be a question of insisting on the fact that a random variable is not fixed, it will allow specially to make calculations of returns, of profits. Know through expected value if it is relevant or not to play a game of chance. In order to be able to work on its expected value for example it will be necessary to draw up the law of probability (table of values)". (our translation)

Finally, on the third question, we were interested in the contexts in which students set their example. Eight of 26 students who answered this question gave an example in a real-life situation (e.g., the price of a chocolate bar, the price of a car, the life of a light bulb or the size of a plant) while four of 26 of students gave an example in an artificial context (e.g., a betting game, a die roll, or a drawing of balls from urns).

The students were more successful with the exercise than with the three preliminary questions. Thirty-seven of 44 students answered the first question that asked them to determine the distribution of a random variable (described by a text), 38 of 44 students answered the second question that asked them to compute the expected value of this random variable and interpret it, and 29 of 44 students answered the third question that asked them to study the influence of a change in the initial data on the expected value. Students who responded to these questions often offered a correct and well-justified answer.

Concerning the first question, a fourth of the students who answered used a technique involving the representation of the probabilistic situation with a diagram or a tree diagram. Questions of interpretation were often well addressed. Almost two thirds of the students who answered the second question gave a correct interpretation of the expected value.

Finally, we note that these students (who were high-achieving at secondary school) were able to use a random variable for the modelling of a concrete random situation. They interpreted the situation to find the distribution of the random variable, used it to solve a question concerning an expected value, but were unable to produce the correct theoretical/technological discourse needed at tertiary level.

Conclusion

The questions we studied here about the concept of random variable concerned on the one hand the differences between secondary and tertiary levels; and on the other hand how well prepared students were by their secondary school for the teaching proposed at tertiary level.

The textbooks' comparative study evidenced as expected that the practical block was more important at secondary school while the theoretical block was more important in the preparatory classes. The definition of random variable given at secondary school was less general, and immediately associated with the definition of the distribution, which is central in the exercises. Similar types of tasks were found in preparatory classes, but they corresponded to different praxeologies. For example, techniques were different because random variables met at secondary school had a limited number of values, while random variables in preparatory classes had values between 1 and n. At secondary school, random variables are introduced as tools to study situations and the focus was on the distribution, represented by a table. In preparatory classes, in contrast, the concept of random variable and the associated theory became central.

The student responses to the test are coherent with the textbook study: they were able to use a random variable for the study of a given situation but did not seem to handle well the theoretical and technological aspects. In particular they confounded random variable with its distribution. The textbook analysis suggests that secondary school teaching is probably not a sufficient preparation for what is expected at tertiary level, and our small sample of students seems to confirm this. Even if the concepts are the same, students at tertiary level are likely to meet difficulties for connecting the new theoretical aspects with their previous knowledge.

Beyond these observations, similar to the results obtained by studies about the secondary-tertiary transition for other topics (Gueudet & Thomas 2018), our work raises theoretical and methodological issues. Concerning the theory, the importance of modelling within the theme of probabilities leads to questions about "modelling praxeologies": how can the praxeology be described? Which kind of theory is involved in it? It is necessary to answer these questions, in particular, to design a teaching intervention in which modelling plays a central role, and theory is also present. Concerning the methods, we have observed that the analysis of textbooks can contribute to identify changes of praxeologies between different institutions. Nevertheless, identifying the theoretical and the practical blocks in a textbook is not always straightforward; different textbooks can have very different structures. We consider that this is a direction open for a further methodological work, especially important for transition studies.

Naturally the study presented here remains limited. In a further work we will consider the learning of probabilities by non-specialists in the first year of university, with the aim to propose a teaching and associated resources combining the modelling of real-life situations and theoretical aspects.


Frank Feudel
email: feudel@math.hu-berlin.de

Hans M Dietz
email: dietz@upb.de

Supporting the use of study skills in large mathematics service courses to enhance students' successone example

Keywords: Study skills, methodological support, mathematics for non-mathematics degree, coaching for study skills

Many university freshmen have difficulties in first year mathematics courses due to inappropriate study skills and learning strategies. In large mathematics service courses the effect of poor study skills is even worse because individual support is hardly possible. Past research shows that it is not sufficient to introduce new study skills because students often do not use these; they need first to be supported when working with them. In this paper we describe one possible support with a coaching program for a tool that aims to assist students in learning mathematical concepts. We describe the aims, the design, and the implementation of the coaching program supporting the students' work with the tool provided, as well as some evaluation results concerning reported students' benefit from the program, their attitudes towards the tool, and the frequency of use of the tool at the end of the coaching.

Introduction and embedding of the research

Many students have problems in their first year mathematics courses at university due to insufficient study skills [START_REF] Anthony | Factors influencing first-year students' success in mathematics[END_REF]. Empirical evidence suggests that promoting study skills in tertiary mathematics courses might lower students' difficulties in learning university mathematics (see, e.g., [START_REF] Cornick | Integrating study skills and problem solving into remedial mathematics[END_REF][START_REF] Mireles | Incorporating study strategies in developmental mathematics/college algebra[END_REF]. Teaching study skills is in particular successful if respective programs are integrated into the regular course [START_REF] Hattie | Effects of learning skills interventions on student learning: A meta-analysis[END_REF]. Hence, the second author, a mathematician, developed a methodological support program named CAT (Checklist, Ampel [=Traffic Lights], Toolbox) and, in 2010, integrated it into his course 'Mathematics for economics students' at the University of Paderborn. This program consists of several tools addressing different study skills (for details see [START_REF] Feudel | Teaching study skills in mathematics service courses-how to cope with students' refusal?[END_REF].

Past research, however, also showed that many students fail to use study skills and learning strategies taught to them or just do not use them at all [START_REF] Anthony | When mathematics students fail to use appropriate learning strategies[END_REF][START_REF] Dembo | Students' resistance to change in learning strategies courses[END_REF]. In a joint previous research project at the Centre for Higher Mathematics Education in Germany, the authors of this paper identified five additional reasons for such failure: 1) perceived time and effort required to use the methods provided, 2) difficulties in working with them, 3) perceived unhelpfulness of the methods, 4) perceived lack of benefit from them, 4) insufficient understanding of them, 5) using personal methods, and 6) no need (see [START_REF] Feudel | Teaching study skills in mathematics service courses-how to cope with students' refusal?[END_REF] for more details).

To address these problems, the authors developed a coaching program for one of the tools of the CAT program, the 'Concept Base' that aims to help students learn mathematical concepts. This tool was chosen because the authors conjectured that a suitable coaching program could address the two main reasons for not using it: 'perceived time and effort compared to the benefit' and 'difficulties when working with the methods'. These two reasons are also described (not verbatim but in terms of the content) as important reasons for not using study skills taught or optional resources provided in literature [START_REF] Dembo | Students' resistance to change in learning strategies courses[END_REF][START_REF] Guthrie | Towards greater learner control: Web supported project-based learning[END_REF]. The ideas behind the coaching program to address these reasons for the non-use of the Concept Base can be also probably transferred to other study skills or optional resources provided in mathematics courses.

Short description of the Concept Base tool

The tool was conceived as a study skill that should help students learn mathematical concepts. The 'Concept Base' is a system of various categories that students should attend to in order to make sense of a mathematical concept. These categories include: the definition of the concept, examples, non-examples, visualizations, statements involving the concept and applications. The category system was developed by the lecturer H. Dietz of the course based on his teaching experience. But these categories are also labeled as crucial in didactic literature for gaining understanding of mathematical concepts [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF][START_REF] Vollrath | Methodik des Begriffslehrens im Mathematikunterricht[END_REF].

The lecturer H. Dietz advises the students to attend to the Concept Base categories for each mathematical concept taught in the course. He also points out that the knowledge of the categories is also required in the exam. Further, he recommends the students to write them down for important concepts, for example for the ones covered in more than one lecture. These written Concept Bases may also serve as summaries when preparing for the exam later in the semester. The students can download an empty template from the course homepage as support (Figure 1). A written 'Concept Base' as a summary of aspects of a mathematical concept is also closely related to the idea of a 'concept map' by [START_REF] Novak | The theory underlying concept maps and how to construct and use them[END_REF] that organizes knowledge related to mathematical concepts. However, the latter focuses more on linking different concepts while the 'Concept Base' focuses on categories crucial for gaining an understanding of a single concept (for acquiring an adequate concept image in the sense of [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF].

Framing of the coaching program to support the use of the Concept Base

As pointed out in the introduction, the majority of the students did not use the tool 'Concept Base' after its introduction in the course in 2010, although the lecturer advised them to do so. To address two of the reasons identified in prior work on reasons for not using the Concept Base [START_REF] Feudel | Teaching study skills in mathematics service courses-how to cope with students' refusal?[END_REF], specifically 'perceived time and effort required for the creation of the Concept Base compared to its benefit' and 'difficulties when trying to create Concept Bases', the authors designed a coaching program addressing these problems in the winter semester of the academic year 2016-2017. Its goal was to help students overcome difficulties with the tool 'Concept Base' and see its benefit, so they might use it on their own. The design of the coaching program was based on the 'Decoding the disciplines'-model by [START_REF] Middendorf | Decoding the disciplines: A model for helping students learn disciplinary ways of thinking[END_REF]. The model consists of seven steps that are important for a successful implementation of subject-specific study skills and working techniques into a university course. It was developed by faculty members of different disciplines at Indiana University at Bloomington on the basis of similar problems they had experienced when trying to implement study skills and learning strategies in university courses. The seven steps are: 1) Identification of students' difficulties 2) Identification of strategies that experts of the subject (here: the lecturer as a mathematician) would use to overcome these difficulties 3) Demonstration of the strategies in the course 4) Exercise of the demonstrated strategies with the students including feedback 5) Motivation of the students to continuously use the strategies provided 6) Evaluation of the effectiveness of the strategies provided 7) Sharing of the knowledge acquired In our case the recognition of the students' difficulties by the lecturer, the development of the Concept Base and is implementation into the mathematics course were exactly steps 1-3.

Steps 4)-6) were realized in a systematic manner with the coaching program that was developed, implemented, and evaluated. Although students had practiced creating Concept Bases in the course tutorials that accompanied the lecture before 2016, the frequency and the intensity of its use varied across the tutorials because there was no mandatory guideline.

Design and implementation of the coaching program

The program consisted of two parts: I. one workshop to practice the creation of Concept Bases (Step 4 of the 'Decoding the Disciplines'-model), and II. submitting four written Concept Bases for further practice including feedback (still Step 4) and for motivating further regular use of the Concept Base (Step 5 of the model). The workshop was carried out after the Concept Base was introduced in the lecture in December 2016 by the first author and another tutor of the course who was familiar with the Concept Base. The second phase, the submission of the Concept Bases, took place afterwards during the second half of December 2016 and during January 2017, and was also conducted by the first author.

Participants of the coaching: Due to capacity restrictions only 50 participants could take part in the program (workshop + submission phase, 48 finished both). They volunteered to take part by writing an email to the first author (a randomized design was not possible due to ethical reasons, but the participants did not receive a reward). Hence, when evaluating the coaching, one has to take into account that these students were willing to work and to invest time for their study program. Afterwards, pictures of the Concept Bases that had been created in the workshop, were taken and sent to the participants via email as stimuli for creating their own Concept Bases.

These activities were meant to address three problems found out in previous research causing the non-use of the Concept Base. First, the reason 'difficulties with creating a Concept Base' was directly addressed by helping the students during the workshop. Second, the reason 'perceived lack of benefit' was addressed by communicating one main benefit of the Concept Base explicitly right at the beginning: the possibility to use it as summary when preparing for the exam. And third, the reason 'use of own methods' was addressed. It had been found out that students felt constrained by the template in Figure 1. Hence, the adaptability of the 'Concept Base' according to the students' own preferences was underlined by (1) emphasizing that the important aspect of a Concept Base is the categories and not how they are arranged on the paper, (2) by motivating the students to extend their Concept Bases by further categories if needed (e.g., by a personal concept definition, [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF], and (3) by providing lots of different examples of Concept Bases in the workshop.

Part II: Submission of four written Concept Bases with feedback:

The aim of this phase was to continuously support the students when creating Concept Bases to further reduce the difficulties with the tool. In addition, the authors wanted for the students to get more experience creating Concept Bases and become more time efficient, which could address the reason 'time effort' for not using the Concept Base. This phase was implemented as follows. The students were asked to create a Concept Base for four concepts: 1) monotonicity, 2) derivative, 3) convexity, and 4) Cartesian product of sets. The first three concepts were simultaneously taught in the lecture. The Cartesian product had already been taught at the beginning of the semester. This concept was chosen to motivate the students to also create Concept Bases for the concepts taught before the introduction of the tool in the course.

The first author provided feedback to the students on each of the four Concept Bases. The feedback had one quantitative and one qualitative component. The quantitative component was a list of aspects of the concept the authors expected the students to know. If an aspect on the list was found in the student's Concept Base it was marked with a check. Otherwise it was marked with a cross. In this way students could get an idea of the scope of the concept. In the qualitative feedback the students were given methodological hints for future Concept Bases, for example to structure the Concept Base on the basis of its categories. Mathematical mistakes in the students' Concept Bases were marked as well. The feedback gave the students ideas about what to include concretely in Concept Bases, and to help them become more efficient in their creation.

Evaluation of the Concept Base coaching

For the 6 th step of the 'Decoding the disciplines' framework we carried out an evaluation. In order to find out ( 1) to what extent the Concept Base coaching was helpful to the students, ( 2) to what extent they saw the benefit of the Concept Base before and after the coaching, and (3) whether the students used the Concept Base on their own in the following semester, several data was collected:

1. Data from a questionnaire administered to the coaching participants in January 2017 (N = 48) 2. The results of the final exams (in February 2017) of all students from the course (N = 821) 3. Data from one follow-up question administered in the next semester in June 2017 (N = 357) 1) Questionnaire for the coaching participants: The questionnaire comprised of two blocks of questions. I. To find out if the coaching program and its components (workshop, submission of four Concept Bases) was helpful to the students the questionnaire contained 10 Items with statements about the perceived helpfulness of the coaching program. An example is 'The Concept Base coaching helped me to get a better understanding of the concepts, to which a Concept Base had to be submitted.' The answers format was a 1-6 Likert-scale (1='Do not at all agree' to 6='Totally agree'). The items can form a scale 'perceived helpfulness of the coaching program' ( . Furthermore, the survey contained two items asking if the time for the participation was well invested. II. To find out if the students saw a benefit of the Concept Base before and after the coaching the questionnaire contained a question asking about the perceived benefit of the Concept Base before and after the coaching program. The students could choose from several options that had been identified in previous research [START_REF] Feudel | Teaching study skills in mathematics service courses-how to cope with students' refusal?[END_REF]. Sample options were 'better understanding of the concepts' or 'summaries for the preparation of the exam'. Multiple answers were possible.

2) Consideration of the results of the final exam:

To get an idea if the coaching might have had a positive effect on the students' performance, we also looked at the exam results. To mediate the effect that the participants were probably more motivated, we generated two special groups for the comparison of the exam results by asking all students of the course the following two questions 1 : 'Would you participate in the coaching program if it was offered next semester again?' 'How often do you use the Concept Base?' (answer format was a 1-6 Likert-scale from 1='never' to 6='every week')

We suspected that students who would agree on the first question and did not take part in the coaching were probably as motivated as the current coaching participants. With the second question we wanted to find out if the coaching really made a difference (and not just the use of the tool alone). Finally, all students were asked for their final grades in mathematics at school to find out if better results might only be due to better initial mathematical qualification.

3) Follow-up question in the next semester: In the following semester all students in the subsequent lecture "Mathematics for economics students II" were again asked the question 'How often do you use the Concept Base?', in order to find out if the participants of the coaching continued using the Concept Base on their own.

Some results of the evaluation

Results concerning the perceived helpfulness of the coaching program by the participants: Many participants found the coaching program helpful. The parameters of the 10-item scale 'perceived helpfulness' mentioned above ranging from 1='not at all helpful' to 6='very helpful' are: , ,

. The distribution of the answers to one sample item is shown in Table 1.

The Concept Base coaching helped me to get a better understanding of the concepts, to which a Concept Base had to be submitted. Moreover, the two items concerning the time effort indicate that, on average, the participants believed that the time spent for the coaching was well invested ( , , ), although there was more variation than in the scale 'perceived helpfulness'.

Results concerning the perceived benefit of the Concept Base by the coaching participants:

The coaching participants saw a greater benefit of the tool Concept Base after the coaching than before, in particular the benefit of the deep processing of mathematical concepts during the time consuming process of their creation (due to limited space details cannot be presented here).

Results concerning the success in the exam:

The pass rate in the exam of the whole mathematics course (N = 815) was 57%, which mirrors the pass rates in former years. The pass rate among the coaching participants (N = 48) was significantly higher than of the rest: 81% versus 55% (p < 0.01). These differences are not necessarily due to better initial mathematical qualification of the coaching participants because the final grades in mathematics at school did not differ significantly.

The comparison with the two comparison groups can be seen in Table 2. The differences between the coaching participants and the comparison groups are again significant (p < 0.01). Assuming that students who did not take part in the coaching program but wanted to take part in the next semester (Comparison Group 1) are as similarly motivated as the current coaching participants, Table 2 suggests that the better performance of the current participants was not only due to higher motivation. The comparison with the students of the Comparison Group 2 (students who answered the question 'How often do you use the Concept Base' with 5 or 6 on a 6-point Likert-scale, but did not take part in the coaching), suggests that the coaching program and not just the tool 'Concept Base' were important.

Results concerning the continuous use of the Concept Base:

The last important result is that most of the coaching participants continued using the Concept Base in the next semester (see Table 3). Hence, the coaching had a sustainable effect on the students' study behavior. Unfortunately, due to schedule difficulties, only 14 coaching participants took part in the mathematics course in the following semester. Hence, the result only yields a tendency; because the result is only based on one item, the interpretation also needs to be made cautiously.

Figure 1 :

 1 Figure 1: Template of a Concept Base (scaled down from one page)

  Detailed description of the two parts of the coaching program: Part I -the workshop: The workshop had several aims: 1) Clarifying the benefit of the Concept Base to the students 2) Providing different suggestions how Concept Bases can be created 3) Helping students to overcome first difficulties when creating Concept Bases It consisted of four phases: 1. Explanation of the benefit of Concept Bases by the leaders of the workshop 2. Creation of a Concept Base for the concept 'preference relation' on a poster in groups of 4-5 students (supported by the leaders of the workshop if questions occurred) 3. Comparison of the different Concept Bases in a poster session with discussion 4. Revision of the Concept Bases based on the discussions in Phase 3

Table 1 : Sample item concerning the perceived helpfulness of the Concept Base coaching

 1

	1	2	3	4	5	6
	-	-	4%	11%	40%	45%	5.26	5	0.820

Table 2 : Pass rates in the final exam of the participants and the two comparison groups

 2

Table 3 : Comparison of the use frequency of the Concept Base of the coaching participants (N = 14) with the others (N = 343) in the following semester. Answers ranged from 1='never' to 6='every week'

 3

These questions were administered in January

and were part of a larger survey about the tool 'Concept Base' not described in detail here because the focus of the paper is on the coaching.

Summary and Discussion

The major finding of the research presented here is that the coaching program aiming to support the students in working with the Concept Base was helpful and important so that the students could become familiar with it, see its benefit (even despite the necessary time effort), and become more successful in the course. It furthermore indicates that, as a consequence, they tended to continue to use the tool on their own in the next semester (with the caveat of the small sample size of coaching participants taking part in the subsequent course).

Besides the problem of the small sample size of coaching participants in the next semester two other limitations of the research need to be mentioned that may be a starting point for future research:

1) The data cannot show in which way coaching program and the tool Concept Base itself helped the students to perform better in the exam exactly. It would be important to find out if the Concept Base really helped students to acquire a solid understanding of mathematical concepts. 2) Even if we tried to mediate the effect of motivation by choosing two comparison groups, of which we expected that they were also willing to invest time for their studies, we should keep in mind that our coaching participants might nevertheless have been a group with special attributes. The results in the case of less motivated students may be different. Hence, scaling up this experiment on the whole course might be another important point for further research.

A last issue concerns the tool 'Concept Base'. Although literature like [START_REF] Vollrath | Methodik des Begriffslehrens im Mathematikunterricht[END_REF] suggests that the categories of the Concept Base are crucial for understanding a mathematical concept, we did not investigate empirically how mathematicians in general concern themselves with these categories when trying to make sense of new mathematical concepts, which would be interesting for future research as well.