Christine Vial Kayser

, which was unsuccessful, because it did

Outer space intermingles with individuals inner/imaginary space that can be a) synchronised with the outer space (in situations either culturally familiar, emotionally pleasing or by frequent usage) 2 or b) desynchronized (in unfamiliar, ecstatic, aesthetic situations), creating unique "experiences". Thus the environment interacts with the ontology of the individual that enters it (see Jedrewski, 2017 on the notion of ontological field) while individual's inner space affects the perception of the outer space (see [START_REF] Pasqualini | Seeing"and"feeling"architecture: how bodily self-consciousness alters architectonic experience and affects the perception of interiors[END_REF] for the binary interaction with architecture). This interaction is dynamic because the observation of a work of art has a duration and is to be understood as a succession of moments [START_REF] Bachelard | La poétique de l'espace[END_REF] more or less fractioned or fluid in immersive artworks. The reunification of time and space is materialised through the motion of the body, its gestures, its standing (Leroi-Gourhan, 1964). Thus bodily posture and behaviour are a sign of the cognitive and emotional reactions to the aesthetic experience.

We aim to tackle this dynamic by capturing proprioception, eye movement, trajectories, pauses, facial expression, words production, emotional, cognitive arousal, post-experience memory), as well as group behaviour in four different situations of experiencing art:

1) an experience in Virtual Reality in which the viewer's body is virtually transported into another space (that of an artwork); 2) an experience in Augmented Reality in which the viewer wears a device on his head that makes him/her believe he enters with his body into another realm; 3) an experience in an actual museum, to compare with the situation 1) and 2) 4) AR can be experienced in different contexts and will allow us to test the experience in hospitals and prisons in particular.

Our case studies will be: 1) the immersive experience of Virtual Reality, of Van Gogh Paintings in Atelier des Lumières, Paris, or of Impressionist paintings, at Château d'Auvers-s-Oise (TBD) . It will compare to a visit at the Musée d'Orsay (of the same artworks).

2) the experience of Virtual Mona Lisa, an Augmented Reality device 3 , followed by a visit of the actual Mona Lisa in the Louvre and a comparison with its use in specific contexts (hospitals, prisons).

3) the experience of Danish-Icelandic artist Eliafur Eliasson's VR digital sceneries, compared with that of actual landscapes (sunset, fog…).

What are VR and AR? (see Invaluable, 2019). Both VR and AR produce virtual spaces and offer an encounter with the artworks markedly different from traditional museums: there are no original works to be seen. Both settings can include music and an explanatory sound-track. Some AR applications use smartphones or tablets, and others use headsets thanks to which the experience is as immersive as in VR. In both cases, the auratic space of the original work is erasedor possibly extended to the entire virtual space in which the visitor wonders. We feel as if we enter the painting, going behind its surface (as Alice in Wonderland). N. B.: The attractivity and sense of the marvellous which those devices seem to generate might be due to the trespassing of the auratic space of the artwork. This is one of the questions to be addressed.

Yet there still are differences between the two technologies:

 VR immerses the viewer in a projected space, where the viewer enters as a group and can share his impressions with others, move his body freely according to his emotional reactions (lie on the floor, stand up in a corner, move around..).  The visit using AR such as the Virtual Mona Lisa is achieved by wearing a device on the eyes in which the body is erased: the visitor sees the virtual space moving around him/her as if he/she was moving within it and seeing changes with his eyes but without a bodily feeling.

Only his/her hand is still in the real world, pressing on remote control (see Christophe Lopez studies on the matter). He/ she enters alone in this virtual space.  In VR the visitor can walk away from the space but cannot consciously alter it (even though his/her body movement may interact with the flow of images, provoking changes, of which he/she is unaware). In AR, the visitor can press on remote control to move forward or backwards and choose several options. The experience seems to retain actively conscious and cognitive features (to be verified).  In VR the actual architecture of the site is used as a screen, it is thus both present and absent:

for example, l'Atelier des Lumières is a former foundry4 . The architecture is typical of the 19 th century and still very present: there is a chimney; multiple metal staircases and pillars; a canal runs through it. The round chimney twists the image; metal pillars interact with Van Gogh views of Paris, adding an industrial touch but are sometimes at odds with the image projected when it consists in flowers (fig. 1); the canal dramatises the image projected on the water which is seen to flow. Yet these features are at times covered by the image and somewhat "erased". The rhythm of the architecture waxing and waning is an crucial factor of the overall experience, in as much as the rhythm of the images themselves. What are they evocative of?

The vague nostalgia of the 19 th -century industrial ruins?

In the case of Auvers, the exhibition is located in a 18 th -century castle, which architecture is partially visible. At times images are projected on elaborate chimney and window frames.

In Eliasson's installations, the architecture is similarly both present and absent. It  In AR, all images are digital. Yet in some case AR content is shown in confrontation with an actual space (fig. 2). As shown here the virtual space can be more enticing than the real space (it has more depth, is brighter, richer in content). Part of the reason is that virtual visits present themselves as "spectacles". Furthermore, the physical qualities of lightness, anti-gravity, bird's view, swiftness permitted by the technology affect the "flow experience" which is an essential aspect of experiencing art [START_REF] Falk | Learning from Museums: Visitor Experiences and the Making of Meaning[END_REF][START_REF] Kirchberg | The Museum Experience: Mapping the Experience of Fine Art[END_REF][START_REF] Cattaneo | A TMS study on the contribution of visual area V5 to the perception of implied motion in art and its appreciation[END_REF]. Last, the immersive character produces a shield from the real and an affective bubble. It affects the impact of the environment.

The use of Virtual technologies in artistic spaces thus offers a significant experimental case study, in which questions of the relation between imagination, representations, emotions, the body and the material can be grasped.

So we shall ask:  What contributes to the emotional value and cognitive content of the experience, how cognitive content and emotional value are related?  Is there a feeling of "wellness", of wonder, and what is it based on? How does it manifest itself?  What is the role of the senses in the experience of virtual images (vision, audition, tactility); the role of the perception of one's bodily gravity, enclosure, expansion, verticality, orientation, relative size, eye's height; do we value the feeling of infinity, of fluidity, of lightness? Do we feel the same on other occasions? In the scenery? In other non-immersive artworks?  What is the role of our motor activities [START_REF] Gallese | Neuroesthetics: The Body in Esthetic Experience[END_REF] such being able to lie on the floor, being immerse in very large images? What is the role of our biological mechanisms -induced by the virtual presence of natural elements-in the triggering of positive or negative affordances? (For such previous studies on biological factors in the aesthetic experience in films see [START_REF] Grodal | Embodied Visions: Evolution, Emotion, Culture and Film[END_REF]Azavedo and Tsakiris, 2017).  What is the influence of music, of light, of hues?  How does it compare with the influence of "slow visit devices", i.e. special preparation made by the visitor, that forces him/her to relax, appreciate the "here and now", open up his imagination and emotional modessuch as the practice of yoga or sophrology before the visit?  Is there a difference in feeling when using an immersive AR device compared to a smartphone which provides an image inserted in the real environment?  What is the effect of being in a group or alone?  What is the medium/ long-term benefit of this experience?  Are viewers more sensitive to viewing the original artwork/ original scenery after this virtual visit? WHY These questions have been those of the phenomenology of art for the last 50 years (Dufrenne, [START_REF] Taminiaux | Notes sur une phénoménologie de l'expérience esthétique[END_REF][START_REF] Talon | La dimension affective du sentir dans l'expérience esthétique[END_REF]. Similarly extensive research by artists and art theorists has explored: the role of the frame of the artwork (the parergon of Jacques Derrida), of the symbolic space inside the work, of the space surrounding the artwork and the way the artwork occupies the gallery, the relation of the viewer's body to sculpture versus painting. This concern was especially that of Minimalism, Installation art, Relational aesthetics [START_REF] Krauss | Sculpture in the Expanded Field[END_REF][START_REF] Bourriaud | Relational aesthetics[END_REF][START_REF] Metais | Faire l'expérience de l'autre : intersubjectivité, altérité et installation artistique[END_REF][START_REF] Bowen | Spatial Encounters, co-authored with[END_REF]. It led to recent museum studies [START_REF] Falk | The Museum Experience[END_REF]Kulkik, Stock, Kashtan, 2011). Artists are creating immersive installations since the 1970s (see Barbara Hammer, Evidentiary Bodies fig. 10 and Bill Viola's videos fig. 11) but are becoming more prevalent.

-Yet the complex physical, cognitive and emotional interaction of the viewer's bodyunderstood itself as a trusted environment -with both the artwork, the gallery space, other visitors, has not been much studied in cognitive science. The literature on it is recent [START_REF] Tröndle | An Integrative and Comprehensive Methodology for Studying Aesthetic Experience in the Field: Merging Movement Tracking[END_REF][START_REF] Vartanian | Preference for curvilinear contour in interior architectural spaces: Evidence from and nonexperts[END_REF][START_REF] Seamon | Architecture and phenomenology, In LU, Duanfang, The Routledge Companion to Contemporary Architectural History, Routledge. Towards an Enactive Approach[END_REF] and limited. Several projects are now being conducted in a multi-disciplinary way, such as: "E-motion mapping museum experience" of the Swiss Centre for research; "Body & Image in Arts & Sciences Project" (BIAS) from UCL/Warburg Institute. Yet they do not appear to interrogate the use of VR as immersive space, the role of affects, the concept of space interaction.

-The growing use of virtual reality and Artificial Intelligence in our daily life adds another dimension of space to our experience and is infringing on the artistic experience (through VR, AR, as well as information provided through AI devices such as smartphone, robots, beacons to register and monitor people behaviours). Their cognitive and emotional qualities are crucial to understanding the new forms of subjectivity and intersubjectivity that might explain their success, their addictivity, and what may come out of it. The intimate and performative character of smartphones may have an element of trust, which acts as a shield from the alien environment and might be understood as affective scaffolding [START_REF] Colombetti | Scaffoldings of the affective mind[END_REF]. It leads people to isolate themselves from the environment to bend on their smartphones (fig. 3). The use of AR and AI has been the topic of a limited number of cognition and affective studies [START_REF] Savva | Continuous Recognition of Player's Affective Body Expression as Dynamic Quality of Aesthetic Experience[END_REF][START_REF] Clowes | The Cognitive Integration of E-Memory[END_REF][START_REF] Clowes | Immaterial engagement: human agency and the cognitive ecology of the internet[END_REF]. These positive emotional affordances have not been taken fully into account. The intangible, immaterial and a-corporeal semantic associated with virtual images and AI might be overstated. Indeed we think that the body is still actively involved in these interactions. Neuroscientists have attempted to measure the role of emotions in aesthetic experience (Vartanian and Goel, 2004b;[START_REF] Chatterjee | The assessment of art attributes Empirical studies of the arts[END_REF]). Yet the emotional potential of AI, AR and VR devices has so far been quite neglected [START_REF] Clowes | Immaterial engagement: human agency and the cognitive ecology of the internet[END_REF][START_REF] Clowes | The Cognitive Integration of E-Memory[END_REF].

Multi-disciplinary and collaborative projects on the topic are raredespite a few notable exceptions: the GDR Esthétique Arts & Sciences-Esars conceived and led by Zoi Kapoula; the Swiss E-motion project and UCL/Warburg "Body & Image in Arts & Sciences Project" (BIAS). They have produced several multi-disciplinary publication (see [START_REF] Freedberg | Motion, emotion and empathy in esthetic experience[END_REF][START_REF] Ticini | Enhancing aesthetic appreciation by priming canvases with actions that match the artist's painting style[END_REF]Scarinzi, 2015;Kapoula, Volle, Renoult, Andreatta, 2018). Yet many neuroscientists understand aesthetics as a search of beauty and harmony, while art theorists know the importance of the "ugly" and dystopia in acquiring knowledge of the self/environment. The contributions of art history and visual art theory in these endeavours are scarce [START_REF] Onians | Neuroarthistory : from Aristotle and Pliny to Baxandall and Zeki[END_REF] and contested [START_REF] Fimiani | Simulations incorporées et tropismes empathiques. Notes sur la neuroesthétique, Images Re-vues[END_REF].

HOW

The methodology is based on theoretical premise and an experimental protocol.

Theoretical premise

We shall use cognitive sciences (psychology and neuroscience) as well as the humanities (phenomenology, pragmatics, cognitive philosophy, cultural and museum studies). We posit that the experience of art, like any action of a subject, is supported by an intention involving our "living body" because, as wrote William James "the world we experience is presented at all times with our body as centre, centre of vision, centre of action, centre of interest"6. The artistic experience is thus linked to a search of satisfaction/ conatus, which is at least of three types: cognitive/educational; immersive/meditative/sensory; social [START_REF] Falk | Learning from Museums: Visitor Experiences and the Making of Meaning[END_REF]Kirchberg, Tröndle, 2015). During the visit, the viewer is not as if in apnoea, suspending his life, body and culture to engage transcendentally with the artwork, as assumed by Kantian aesthetics. The experience is largely determined by the environment, which presents itself as a set of possibilities of action/ interaction leading to positive or negative perception (affordances) [START_REF] Gibson | The Ecological Approach to visual perception[END_REF][START_REF] Leroi-Gouran | Les affordances : de James Jerome Gibson aux formalisations récentes du concept[END_REF][START_REF] Menninghaus | The Distancing-Embracing model of the enjoyment of negative emotions in art reception[END_REF]. We engage with the environment through our vertical/horizontal bodily coordinates, frontal view, lateral hearing, which all concur to the shaping our emotions, imagination and even superior cognitive capacities (Johnson 1987[START_REF] Johnson | The Meaning of the Body[END_REF][START_REF] Scarinzi | Grounding Aesthetic Preference in the Bodily Conditions of Meaning Constitution[END_REF]. Thus our experience is embodied, i.e., it uses both body and mind, sensations and representations, emotions and analytical abilities. It is also enacted, built on exchanges between the body-mind and its environment in an active process; embedded, taking place in a physical, social and cultural environment and moment, and extended using external objects to constructs the exchange between the body-mind and the perception [START_REF] Newen | The Oxford Handbook of 4E Cognition[END_REF].

The artist, as a mediator between the collective and the individual, transmits the tensions between the individual and the social, the conscious and the unconscious to forms and materials. They are then communicated to viewers through complex mechanisms involving intersubjectivity (Durt, Fuchs, Tewes, 2017), empathy and mirror neurons [START_REF] Freedberg | Motion, emotion and empathy in esthetic experience[END_REF][START_REF] Onians | Neuroarthistory : from Aristotle and Pliny to Baxandall and Zeki[END_REF].

The aesthetic experience complexifies perceptive mechanisms because an artwork does not allow for direct, sub-conscious reading of what it is about, what cognitive or emotional benefit it may provide, how to handle it. It is a form of dissent from habitual, conventional, and socially accepted affordances. During the artistic experience, the viewer's representations and belief about the world, drawn from daily experience, are thus confronted with alien objects (the artworks). An obvious example is the painting This is Not a Pipe by Magritte, 1929. Another is the Campbell soup can by Andy Wharhol, 1962.

Semir Zeki has remarked that fauvist paintings, which do not have "natural" colours are analysed in a visual area of the cortex (V5) which is not that of natural scenes (V2), while figurative paintings are interpreted in that area (Zeki, 1999, p. 200-202.).

Even figurative paintings before the 20 th c. are never a copy of the real but a specific interpretation of it, as shown in the unfathomable presence of Leonardo Da Vinci's Mona Lisa, which cannot be grasped merely as the portrait of a famous lady. In the case of ethnographic museums, artworks are the signs of hidden (magical, religious) presence or manifestations of things gone.

Artistic spaces are manmade environments. They are cultural, affective and adaptative niches, suitable for a particular social activity, sharing elements of familiarity (trust) and habitus (transparency) that evolve with technology itself. The process of grasping what the artwork is about goes through several stages, more or less complex depending on the nature of the work, its closeness to the real. It is also influencedpositively or negatively-by the environment [START_REF] Sterelny | Minds: Extended or Scaffolded?[END_REF] and the viewer's expectations, and prior knowledge.

Contrary to a self-centred perception proposed both by Cartesian philosophy and phenomenology we insist here on the role of the shared public space, with its artworks, language signs, architecture and people, in creating a common, intersubjective and personal experience reshaped by object perception, as expressed by Bruno Latour: "It's clear that each object -each issue-generates a different pattern of emotions and disruptions, of disagreements and agreements. There might be no continuity, no coherence in our opinions, but there are a hidden continuity and a hidden coherence in what we are attached to. … Objectstaken as so many issuesbind all of us in ways that map out a public space … a hidden geography" [START_REF] Latour | From Realpolitik to Dingpolitik -or How to Make Things Public[END_REF].

WHAT FOR (expected outcome)

The expected outcome will be theoretical but also applied.

At a theoretical level

 The objective is to confront philosophical intuitions (in particular of phenomenologists), empirical observations -by artists, curators, art historians-and neuroscientific data potentially supporting these intuitions, regarding: the cognitive (sense-making) power of art; the importance of the contextarchitectural, cultural and socialin this process; and the role of the "living body" of the viewer/s, with its implicit inner and peripersonal space. The experience involves a horizontal dimension (the viewer looking at artworks and other people), a vertical dimension (the sense of one's body presence, verticality, gravity and the architectonic volume) and a dynamic process of encounters between at least 5 components, which are both material and immaterial:  1 the physical space in which the viewers and the artwork are immersed;  2 the implicit space of the artwork (surrounding it, its virtual inner space);  3 the virtual space of digital technologies;  4 the intersubjective space of social encounters and practices  5 the inner space of the viewer (internal image of self, which is both physical and imagined); 1 Physical, architectural/urban space in which the viewers and the artwork are immersed The visible spatial context that includes both the viewer and the artwork, such as the outdoor (the urban, natural space) or indoor (museum architecture, stage). It can have various qualities, of which we may want to test the effect on the experience. In most cases, the space around the work is considered as being of aesthetic quality ie separated from the ordinary, but it is not always the case (see Street art, graffiti). In our case study the presence of industrial remains is significant of the machine and contrasted with artworks evoking natural spaces. The presence of other people in these spaces plays a role in qualifying themwhether they are "art lovers", watch, or unrelated (passers-by…).

2 Implicit space of the artwork The space around/within the work of art has been studied amply by artists especially in the 1960s (minimal art, Land art) but also in Installation art and in the performing arts. It has also been theoretically examined by anthropologist regarding ethnic, ritual, magical art supposed to have a vicarious agency [START_REF] Gell | L'art et ses agents, une théorie anthropologique, Presses du Réel (trans. of Art and agency: an anthropological theory[END_REF]. This space has for long been restricted to a small vacuum around the work until it became the entire room (in "Site-specific Installation art"). Critical studies also indicate that the larger space (the gallery space, its corridors, its windows) affects the experience of the artwork (as revealed by some artists such as Richard Serra), and so does the political and infrastructural context [START_REF] Bowen | Spatial Encounters, co-authored with[END_REF].

3 The virtual space of digital technologies It appears to be both social and internalised. It is present in the art gallery, through smartphones and immersive digital images. We would ask what "regimes of desire" they answer, what quality of experience they provide (cognitive, affective, aesthetic).

Intersubjective space

Phenomenology informs us about this concept, but mostly presents it as existing in a 2D space, a "plan of immanence" in which one "ego" meets another, without taking into account the 3D space in which the two subjectivities are contained. This 3D space can be actual (yet virtual) or mental. When mental, it can be memorial, cultural, affective, cognitive. It is probably the product of encounters between "alter egos", but also of the environment they are immersed in (Métais, 2019).

5 The inner space of the viewer The spectator's inner space is built over time through an encounter between his body and mind, self and environment. It may interact with the outer space through a peripersonal space. This space is probably the crucial space to study as it screens the accessibility, desirability (or threat) of other spaces according to their physical or symbolic distance, and affordances.

On an applied level, this project will contribute to the design of artistic environments (actual and virtual) and add knowledge to the benefits of using AR, VR and Artificial intelligence to complement the experience. It will help improve the quality of the experience of visiting museums, biennales and outdoor art and lead to new interdisciplinary strategies to design art spaces as systems (SMART environments) 7 . It can also provide information on the benefits of art in stressful situations (hospitals, retirement homes, prisons, homeless shelters, business districts, schools, etc.) and help conceive integrative, adaptive environments. This research may yield results useful to the field of museum studies, art mediation, art therapy.

  Fig. 1.1 Eliafur Eliasson, the Weather project, 2003, Tate, London

 Fig. 1

 1 Fig.2AR used in a museum to add information (a missing painting)

Fig. 3

 3 Fig. 3 Shanghai Biennale, visitors in the interactive space, engaging with their phone and not the interactive experience, December 2018. Photo C Vial Kayser

Fig. 4

 4 Fig. 4 Visitor with a robot Pepper at the Smithsonian

Fig. 5

 5 Fig. 5 Cooper Hewitt Museum, New York, « Immersion room »

Fig. 6

 6 Fig. 6 Chateau d'Auvers sur Oise, Virtual visit on Impressionism

Fig. 7

 7 Fig. 7 Barbara Hammer, Evidentiary Bodies (still), 2018 (detail). Three-channel HD video installation (colour, sound). Dimensions variable; running time: 9 mins. 30 secs. Courtesy of The Barbara Hammer Estate, COMPANY, New York, and KOW, Berlin. © The Barbara Hammer Estate.

Fig. 8

 8 Fig. 8 Two films combine fire and water. 2015 Courtesy Bill Viola Studio. Photo: Kira Perov. Fire performer: Robin Bonaccorsi. Water performer: John Hay.

https://esrc.ukri.org/funding/funding-opportunities/open-research-area-ora-for-the-social-sciences-2019/

https://www.atelier-lumieres.com/fr/latelier-lumieres-dhier-aujourdhui

https://news.artnet.com/art-world/cleveland-museum-art-studied-digital-engagement-visitors-results-encouragin-1567123?utm_content=from_artnetnews&utm_source=Sailthru&utm_medium=email&utm_campaign=Europe%20Afternoon%20Ju ne%207&utm_term=New%20Euro%20Newsletter%20List%20%2890%20Day%20Engaged%20Only%29

William James, « L'expérience de l'activité », dans Id., Essais d'empirisme radical.Marseille, Agone (2005). Cité par Richard Shusterman, Le corps et les arts : le besoin de soma-esthétique, Diogène, 2011/1 (n° 233-234), p. 9-29. En ligne http://www.cairn.info/revue-diogene-2011-1-page-9.htm.

This project was elaborated in view of answering a call for International Research Project in Social Science, launch by the ESRC (Economic and Social Research Council) not meet the criteria of ESRC (i.e., 50% of the concept and aims should be related to Social Sciences). The project will nevertheless be conducted in its theoretical dimension by the author. The experimental aspect intended in partnership with psychologist is at the moment suspended.


Luciana Vellinho Corso¹
email: l.corso@terra.com.br

Évelin Assis¹
email: evelin_assis@hotmail.com

Intervention in counting principles: teachers' perceptions on students' learning profile

Keywords: Teachers' perceptions, Counting principles, Learning profile

The research field about teachers' perceptions on students' performance highlights the importance of understanding this approach in order to help students to improve their learning outcomes. This study aims to contribute to this discussion. It involved 136 first-grade students, from 10 classes of 3 Brazilian public schools. The teachers participated by filling out a questionnaire about the students' learning profile, including aspects such as: behavior, attention, social interaction and knowledge by area (mathematics, reading and writing). The teachers' answers were related to the students' achievement in a counting principles task. The results show that teachers have correct perceptions about the students who master the counting principles, but the same is not true for the ones who did not consolidate or were still developing the principles. The educational implication of the study is pointed out.

Teachers' role in the learning process of the students is often recognized by researchers in the mathematics field, especially when they present the research outcomes and reflect on its educational implications [START_REF] Hoge | Teacher-based judgments of academic achievement: a review of literature[END_REF][START_REF] Südkamp | Accuracy of teachers' judgments of students' academic achievement: A meta-analysis[END_REF]. Studies focusing on this perspective provide important data concerning teachers' perceptions on students' learning which brings implications for the way teachers plan their pedagogical actions [START_REF] Hoge | Teacher-based judgments of academic achievement: a review of literature[END_REF][START_REF] Südkamp | Accuracy of teachers' judgments of students' academic achievement: A meta-analysis[END_REF]. In this sense, the knowledge provided by this field of research allows to better comprehend and favor the teaching and learning processes. This research involves the investigation of teachers' perceptions about the learning profile of first year students and seeks to verify if there is a relation between the teachers' perceptions and the students' actual performance in a counting principles task. The counting principles, established by Gelman and Gallistel (1978), are the basis for children's learning how to count with success. The principles are: stable order, one to one correspondence, cardinality, abstraction and order irrelevance. As evidenced by Stock, Desoete & Roeyers (2009), children with good achievement in counting by the end of kindergarten presented good performance in arithmetical abilities in the first year, pointing to the predictive value of the counting principles.

In 1989, Hoge and Coladarci conducted a systematic review on teacher judgments about their students' achievement. Initially, the authors contextualized teacher judgments through two perspectives: the cognition of teachers and their assessment. The first one addresses the decisionmaking of educators (in relation to planning, for example), noting that this process is influenced by their judgment on their students. The second involves thinking about the importance of the accuracy of teacher judgments in the context of assessment: educational decisions are made on the basis of the educators' assessment and their judgment is an important source of information about the students [START_REF] Hoge | Teacher-based judgments of academic achievement: a review of literature[END_REF].

In 2012, 23 years after the review mentioned, Südkamp, Kaiser & Möller also studied the precision of teachers' judgment about the performance of their students, conducting a meta-analysis on this subject. The authors pointed out that teacher judgment is one of the primary sources of information about students' academic performance, as [START_REF] Hoge | Teacher-based judgments of academic achievement: a review of literature[END_REF] highlighted. An accurate assessment of students' performance is a prerequisite for educators to be able to adapt their pedagogical practices, make decisions and support the students' development of an appropriate academic self-concept [START_REF] Südkamp | Accuracy of teachers' judgments of students' academic achievement: A meta-analysis[END_REF].

The literature showed studies involving teachers' judgment/perception with different objectives: to verify predictors of performance [START_REF] Teisl | The utility of kindergarten teacher ratings for predicting low academic achievement in first grade[END_REF], to identify students with difficulties in mathematics [START_REF] Nelson | A comparison of methods to screen middle school students for reading and math difficulties[END_REF], and to observe the relationship between teacher perceptions and students' performance in mathematics, specific math skills, student feeling, motivational factors, and activities choices (Eds & Potter, 2013;Martinez, Stecher, & Borko, 2009, Schappe, 2012).

Although these works show divergences in some aspects, they generally point to a positive relation between the teachers' perceptions on students' performance and the students' actual learning outcomes. The authors note the need for research on this relationship, highlighting the importance of teacher judgment for students' evaluation, pedagogical planning and decision making. The present study seeks to contribute to this discussion. The purpose of this study is to investigate the relationship between teachers' perceptions about the learning profile of their 1st year students and the students' actual performance in a counting principles task.

Method Participants

The study included 136 Brazilian students (63 girls and 73 boys), aged between 6 and 7 years old, from 10 groups of 1st year of Elementary Schools. It involved 3 public schools. This group of children was participating in an intervention research, entitled "Intervention in counting principles for elementary school students", submitted as a paper to TWG2, this year. Students were assessed in a counting principles task and their teacher filled out a questionnaire focusing on their perceptions about the students' academic achievement. Therefore, besides the students, their teachers (n=10) also took part in this study. All of them were graduated in Pedagogy and 9 had post-graduation in the Education area. Regarding the teaching experience with first year of Elementary School: 1 teacher had 6 months, 1 teacher had 2 years, 2 teachers had 3 years, 1 teacher had 5 years, 1 teacher had 9 years, 1 teacher had 18 years, 1 teacher had 28 years, 1 teacher had 31 years and 1 teacher had 36 years. The parents of all participating students gave their authorization through the signing of the Free and Informed Consent Form and the Dissent Form. Approval for this study was obtained from the Research Ethics Committee of Universidade Federal do Rio Grande do Sul.

Procedures

The researcher administered the counting principles task, between September and December 2017, individually to each child in a room provided by the schools. The teachers received the questionnaire at the beginning of the data collection and returned it according to their availability.

Instruments 1) Counting principles task: The task is conducted in an interview format. There are specific questions to assess each principle. The questions were asked in the following order: stable order ("Until how much can you count?" "Count out loud up to the number you said, please"); one to one correspondence (The experimenter shows 10 chips in a row and asks "how many chips are there?" After, the researcher shows 10 chips misaligned and asks "how many chips are there?". The same procedure is repeated with 15 chips. This activity will be called "A"); cardinality (by the end of counting 15 items, the experimenter asks "how many are there in total? Can you give me 10?" This activity will be called "B"); abstraction (the experimenter asks "if you were counting 15 candies, would you count the same way (in the same order) you counted the chips?" This activity will be called "C"); order irrelevance (the experimenter asks the child to count the 15 chips, arranged in a row, in a different order, that is, starting from another chip. Then, the child is asked to say how many chips would remain if the row was undone. After, the experimenter asks the child to count 8 chips of the set, separating them aside, and then, to count the remaining 7 chips of the row. By the end, the child has to say how many chips there are in total. This activity will be called "D"). For each principle, the subjects were classified into three groups: principle consolidated (Yyes); in construction (IC); not consolidated (N). Some criteria were established to classify the children in each group. Some of the questions (2, 3 and 5) are formed by more than one query. In this case, the child's answer was classified as "Y" if he/she answered all the solicitations in each question correctly; "IC" if only one part of the question was correctly answered; "N" if the student did not answer any part of the question correctly.

2) Teachers' questionnaire: The teachers received a questionnaire, adapted by the author, based on the scale of "Strengths and Weaknesses of Attention-Deficit / Hyperactivity-symptoms and Normalbehaviors -SWAN" [START_REF] Swanson | Categorical and dimensional definitions and evaluations of symptoms of ADHD: The SNAP and the SWAN Ratings Scales[END_REF], containing questions about attention, behavior, knowledge by area (mathematics, writing and reading) and social interaction. The objective of the questionnaire, presented in the table below, was to investigate the teachers' perceptions about the general learning profile of their students. For each statement, teachers could select one of four frequency options: often, sometimes, rarely, or never.

Results

The relationship between the questionnaires of the teachers and the performance of the children in the counting principles task was investigated. In order to do the analyses, it was necessary to make an adjustment in the questions of the questionnaire. Although all of them counted with the same frequency for answers, there were distinct kinds of affirmation, which meant that the same frequency attributed to two different questions resulted in opposite answers. For example: the "often" frequency is a positive answer to question number 1 ("shows interest for the proposals"), but negative for the question number 20 ("shows difficulty in understanding instructions"). Considering this fact, before running the analyses, it was necessary to transform the answers so they kept the same direction of association. They were classified as positives (1 to 9, 18, 21, 21 and 25), which had maintained the frequency (often, sometimes, rarely, never), and negatives (10, 12 to 17, 19, 20, 22 and 24), which suffered the transformation (never, rarely, sometimes, often). In this way, the frequency "often" for the positive question number 7 ("engages in activities") had the same effect of the frequency "never" for the negative question number 23 ("demonstrate slowness to finish the task")". The box-plots, shown below, were constructed to highlight the frequency of distribution of the teachers' answers in relation to the students' constructs. It can be seen that most of the graphs below shows increasing frequency throughout the answers, with few occurrence in "never" and a gradual increase in the following answers. The only graph that does not follow this pattern is related to the construct "IC" (in construction principle) in activity C: this may be due to the fact that only 6 children presented this construct, making the analysis difficult due to the scarcity of data for this variable. Considering the information presented in the graphs, the Kruskal-Wallis test was applied to verify if the median of performance of the children in the counting principles task, between the four groups of frequencies (never, rarely, sometimes and often), was different, that is, if the oscillations in the graphs were statistically significant (p <0.05). The samples were independent, so if one group has a high median, it does not interfere in the median of another group (high or low). As observed above in figure 2, in the one to one correspondence principle the medians of the "N"(not consolidated) construct were the same, since their p-value was greater than 0.05. In the "IC" (in construction) and "Y" (consolidated) constructs there was a difference between the medians, indicated by p-value less than 0.05, the medians of each response being different from each other. Although a numerical difference was observed between the medians of "N" construct responses, a statistical significance was not found (at 5% level) due to the low sample size. The number of students with the "IC" and "Y" constructs, which presented a significant difference, was 50 and 57 respectively, while the "N" construct had 29 students.

In the cardinality principle, it was possible to observe difference between the four medians only for the "Y" construct (p <0.05), with the "N" and "IC" constructs not showing statistically significant differences. Again, this can be attributed to the size of the sample of each construct, since 113 children showed mastery of the cardinality principle, while only 17 were constructing it and 6 did not demonstrate its consolidation. In the abstraction principle, the "N" and "IC" constructs, with 14 and 6 subjects, respectively, did not present differences between the medians of the responses. The "Y" construct, with 116 students, showed to be statistically significant (p <0.05) with the four medians differing from each other. In the irrelevance of the order principle, the construct "N", with 63 subjects, was shown to be statistically significant (p <0.05), with the four medians being different. The "Y" construct, although with a small number of students, 27, was also evidenced as statistically significant (p <0.05). The answers "Never" and "Rarely" presented median equality, as indicated by the letters in the table. The construct "IC", in turn, even with a larger sample than the previous one, with 63 subjects, was not statistically significant, showing equal medians.

Discussion

It was expected that the results of the data analysis, relating the teachers' perceptions about their students' general learning profiles and the constructs presented by the students in the assessment of the counting principles, would show that children who did not consolidate a certain principle would present a very frequent occurrence of "Never" or "Rarely" answers in the questionnaires; students who were constructing the principles would have "Sometimes" answers more frequently; children who demonstrated a "yes" construct for the counting principles, supposedly, would present a large occurrence of "Often" answers in the teaching questionnaires. In general, it is possible to consider that there was little differentiation in the occurrence of responses for almost all constructs: both those who did not consolidate the principles and those who were constructing them, or who already mastered them, obtained the same types of answers in the questionnaires, varying in the number of times they received them.

The results raise some possibilities of analysis. First, the relationship between teachers' perceptions about the students' learning profiles was in agreement with the performance of the students only in the cases of those who showed consolidation of the counting principles. Such a result converges with other studies demonstrating that teachers' judgements, through rating scales, are good predictors of students' academic performance, more specifically, in relation to students who did not develop learning difficulties in mathematics [START_REF] Teisl | The utility of kindergarten teacher ratings for predicting low academic achievement in first grade[END_REF]. Another possibility concerns how teachers answered the questionnaires: once they received the documents and were able to fill them out without the presence of the researcher, it is not known exactly how they did it, what criteria they used to choose a response in to the detriment of another, whether they filled out quickly or dedicated time to think about the issues raised there. A third point refers to the number of students per class. The classes involved in this study had a maximum of 25 students. Although it seems a small number, it is complex for a teacher, alone, to have complete knowledge of all his/her students. Even though the questionnaire was delivered in the second half of the year, a period of time when teachers would have had time to get to know their students, it is difficult to know, precisely, how much and what each teacher knew about their pupils.

Limitations

The research results described in this paper should be considered in the context of some limitations. First, the counting principles task used was selected because it is a short and practical instrument, however, we believe that a broader task could elucidate more aspects to be analyzed and discussed. The same must be said in relation to the Swan questionnaire. Another limitation concerns the choice of statistical analysis that was conducted. It is possible that different findings would have emerged if the study included the association of different statistical tests, which could have made it possible to reinforce or question some of the results found or even to elucidate other possible outcomes.

Conclusion

This study showed that the teachers have correct perceptions of their students when it comes to students with good performance, that is, those with "Y" construct in the counting principles. Students who do not present consolidation or who are constructing the principles were perceived by the teachers in a way that did not correspond to the students' performance in the counting task used in this study.

This work points to the need and importance of deepening and expanding the development of teacher studies and their perceptions. More importantly, it is extremely relevant to investigate how teachers form their perceptions, what criteria they consider when evaluating students and how they see their students.

This study shows that there are still many aspects that need to be discussed between researchers and teachers, seeking to establish a significant link between theory and practice. It is imperative that teachers, especially those responsible for the early years of Elementary School, acquire knowledge about the factors involved in learning the initial mathematics: knowing how the learner learns will help them to identify aspects related to not learning and, therefore, perhaps their perceptions are clearer and they can recognize children who are facing problems. It is imperative that the link between researchers and teachers be built and used in favor of student learning, allowing the two sides to dialogue and understand how academic conceptions can contribute to pedagogical practices and vice versa.

  1 -Shows interest for the proposals 14 -Shows difficulties in writing 2 -Pays attention to the explanations 15 -Shows difficulties in reading 3 -Keeps the materials organized 16 -Shows difficulties in mathematics 4 -Complies with the teacher's requests 17 -Avoid tasks involving numbers 5 -Interactes well to colleagues 18 -Shows counting knowledge 6 -Manages well the time attributed to accomplish a task 19 -Shows difficulties in socializing 7 -Engages in activities 20 -Shows difficulty in understanding instructions 8 -Knows to take turns: when to speak and when to listen 21 -Plays together with colleagues at playtime 9 -Questions when in doubt 22 -Complaints about the class proposals. 10 -Requires a lot of attention from the teacher 23 -Demonstrates slowness to finish the tasks 11 -Moves a lot during class 24 -Is easily distracted 12 -Forgets about making the homework 25 -Expresses him/herself well 13 -Does not comply with the rules

Figure 1 :

 1 Figure 1: Box-plot graphs

Table 1 : Teachers' questionnaire

 1

Figure 2: Kruskal-Wallis test results

		One to one correspondence					Cardinality
	Construct	Number					Construct	Number
	of the	of	Answers of				of the	of	Answers of
	principle	students	the teachers Median p-value	sig	principle	students	the teachers Median p-value	sig
	N	29	Never	0	0,0843		N	6	Never	1	0,6615
			Rarely	2					Rarely	4,5
			Sometimes	7					Sometimes	5,5
			Often	12					Often	9,5
	IC	50	Never	2	0,0188	a	IC	17	Never	0	0,089
			Rarely	4		b			Rarely	3
			Sometimes	6		c			Sometimes	6
			Often	12		d			Often	12
	Y	57	Never	1	0,0446	a	Y	113 Never	1	0,0003	a
			Rarely	4		b			Rarely	3		b
			Sometimes	6		c			Sometimes	6		c
			Often	11		d			Often	12		d
			Abstraction						Order Irrelevance
	Construct	Number					Construct	Number
	of the	of	Answers of				of the	of	Answers of
	principle	students	the teachers Median p-value	sig	principle	students	the teachers Median p-value	sig
	N	14	Never	1	0,3114		N	63	Never	1	0,0061	a
			Rarely	6					Rarely	4		b
			Sometimes	6,5					Sometimes	6		c
			Often	9					Often	11		d
	IC	6	Never	4,5	0,2873		IC	46	Never	0,5	0,2982
			Rarely	8,5					Rarely	3
			Sometimes	4,5					Sometimes	6
			Often	4					Often	12
	Y	116	Never	1	0,0001	a	Y	27	Never	2	0,0096	a
			Rarely	3		b			Rarely	3		a
			Sometimes	6		c			Sometimes	5		b
			Often	12		d			Often	11		c


Sandra Gleißberg
email: sandra.gleissberg@ph-gmuend.de

Klaus-Peter Eichler
email: klaus-peter.eichler@nord.no

The offer of tasks to work on multiplication in grades 2 and 3

Keywords: Elementary school mathematics, multiplication, textbook analysis

In this paper, we present an analysis of four teaching materials of grades 2 and 3, which are widely used in Germany. We analyzed the tasks in these textbooks and workbooks with a focus on developing and consolidating the operational understanding of multiplication. For this purpose, we present a framework developed by us. Its foundations are the levels of representation according to Bruner and the fact that mathematical terms are concepts. The analysis shows a homogeneous picture across all four textbooks and workbooks: Already in class 2, the vast majority of tasks require working on the non-verbal-symbolic level alone. Only a small part of the tasks promotes connections to the iconic, to the enactive or to the verbal-symbolic level and challenges students to intermodal translations from one level of representation to another. In grade 3, compared to grade 2, an even larger proportion of tasks are limited to the non-verbal-symbolic level.

Context of the study.

Multiplication is a core content of teaching mathematics in primary school. The learning objectives in regard to multiplication are that students firstly should acquire a conceptual understanding of the operation multiplication, secondly, should be able to solve any task using different strategies and finally they should be able to retrieve more and more basic multiplication facts directly from memory (e.g., [START_REF] Kling | Three Steps to Mastering Multiplication Facts[END_REF]CCSSI, 2010 p. 23;KMK, 2004, p. 9).

There seems to be an international consensus nowadays that students should both acquire a sound conceptual understanding of multiplication and eventually solve all basic tasks accurately and effortlessly (Gaidoschik, 2017, p. 2).

However, three problems can very often be observed up to the secondary school level. First, there are deficits in the mastery of the basic facts1 , i.e. in their retrieval as fact knowledge. Often students have forgotten basic multiplication facts they have once memorized and are unable to reconstruct this knowledge. This can be seen, for example, if they are unable to reduce fractions in secondary school because they cannot identify common factors.

Second, while students are often able to reproduce the basic task equations from memory, they are unable to solve multiplication tasks with a factor greater than 10.

Third, we repeatedly observe difficulties with the conceptual understanding of multiplication: students know that 5 × 4 = 20, but they are unable to justify this because they do not know the meaning of this term. Often, they just argue that 5 × 4 is a shortened notation for the addition task 4 + 4 + 4 + 4 + 4.

All three problems have the same reason: A lack of the conceptual understanding of multiplication. In light of these observations, it is necessary to investigate the causes and to provide information to improve the teaching and its results. We are sure that no teacher works with the intention of bad results. There are obviously factors with a negative influence on teaching and its results. Such a factor is, for example, practicing with spontaneously chosen tasks, in which subjectively important, especially familiar or widespread tasks dominate, and the exercise then does not meet the need. In our opinion, teachers are obviously not aware of the negative influence of such factors.

Undoubtedly, the success of teaching depends on the selection of tasks and the way teachers work with them (cf. [START_REF] Fanghänel | Arbeiten mit Aufgabenein wesentliches Mittel zur Gestaltung eines modernen Mathematikunterrichts[END_REF]. One essential source of such tasks is the teaching material2 . Therefore, the impact of teaching materials such as textbooks and workbooks is given. That is why we investigated four teaching materials of grades 2 and 3 which are widely used in Germany. The objective of our analysis is the suitability of the tasks contained in the materials in regard to promote understanding of the operation multiplication.

About the matter: Understanding multiplication.

Terms like 3 × 4 or 6 × 8 are non-verbal-symbolic representations of concepts. They are mental reflections of objective reality and describe classes of situations. In this sense, the product m × n can be represented both by the union of m equal and pairwise disjunctive sets with the cardinality n or by a combinatorial representation: Both possibilities of representation can be spatial-simultaneous as well as temporal-successive. This is consistent with the findings of [START_REF] Anghileri | Arithmetic operations on whole numbers: multiplication and division[END_REF]. They identify six key aspects of multiplication (and division): equal grouping, allocation/rate, number line, array, scale factor, Cartesian product. Regarding the so-called "axiomatic" definition of multiplication as a repeated addition, Park and Nunes (2001, p. 771) emphasize "that the origin of children's understanding of multiplicative relations is in their schema of one-to-many correspondence." and that the "repeated addition is only a procedure for solving multiplication problems, not its conceptual basis" (ibid.).

Content-related understanding of a concept means to be able to identify, realize and systematize it. When identifying a term like 3 × 4, an appropriate example is found in an enactive, an iconic, or a verbal-symbolic representation. Realizing a term such as 3 × 4 means to create an enactive representation (e.g., an action), an iconic representation (e.g., a picture) or a verbal-symbolic representation (e.g., a suitable story, a word problem), which is consistent with this term. This is illustrated in Figure 1. When students systemize terms, they find relationships between two terms. This can be referenced with an enactive or an iconic representation.

Often this relationship is created without any references only on the nonverbal-symbolic level.

The importance of the conceptual understanding of the operation multiplication consists on the one hand in the fact that students can apply this knowledge to describe reality, e.g., to solve word problems. On the other hand, it enables students to solve every multiplication task, especially those with factors greater than 10, by drawing e.g., dots and skip counting, laying e.g., cubes and skip counting and deriving answers using reasoning strategies based on known facts. [START_REF] Baroody | Why children have difficulties mastering the basic number combinations and how to help them[END_REF][START_REF] Kling | Three Steps to Mastering Multiplication Facts[END_REF].

Mastery of the basic facts of multiplication must be distinguished from fluently multiplying: Fluently multiplying means to solve tasks using strategies, properties of the operation, means determine the value of a term by calculating at skill level (Kling & Bay-Williams, 2015, p. 550). Mastery of all the basic tasks of multiplication means to be able to retrieve from memory all products of two one-digit numbers. In our opinion, mastery is indispensable at the latest in middle of grade 3 because then this knowledge is needed as a tool for solving tasks with factors greater than 10.

Even if the final goal here is to retrieve the facts from memory, the ability to solve the basic tasks of multiplication is the basis for this and must consequently be secured before memorizing the basic tasks of multiplication. Only if students have a conceptual understanding of the operation multiplication and are able to determine the value of the terms, they can reconstruct forgotten basic facts. The reconstruction of forgotten facts can be done by using connections to other facts, e.g., by using properties of multiplication, by activities with manipulatives or by using mental representations.

Framework for the classification of tasks for the consolidation of arithmetic operations.

Because a suitable work with tasks in the sense of Fanghänel (cf. 2000) is essential for the quality of teaching and its results, we examined this work with tasks and the selection of tasks in more detail. The choice of tasks undoubtedly depends on the range of tasks offered by published programs (textbooks, workbooks etc.). To assess this offer of tasks, we analyzed four widely used programs4 . We decided to analyse textbooks and workbooks for both grade 2 and grade 3 because teachers usually use these materials anyway. Supplementary materials such as software or online resources were therefore not analyzed.

Many frameworks exist for analyzing textbooks regarding different aspects. A short overview gives for example O' [START_REF] Keeffe | A framework for textbook analysis[END_REF]. A specific framework which focusses the appropriation of arithmetic operations and takes in consideration that terms like 3 × 4 are concepts was not available until now. We developed the following framework to classify tasks for the appropriation of multiplication. It is generally usable for all arithmetic operations. This framework is not limited to analyzing printed materials. It can also be used to analyze educational software etc. The understanding of the operation multiplication must be promoted. Therefore, we were looking for tasks which emphasize the identification, realization, and systematization of concepts, e.g., of terms. In ggrades 2 and 3, mastering the basic number sentences by memory should be supported. Finally, we determined the number of tasks which aim to determine the value of the term. So, we classified the tasks in the programs as follows:

ICE:

SCW:

Tasks for Systematizing Concepts Without reference to the enactive or to the iconic level of representation. Such tasks are, for example, 'You know 10 × 8. How can you then calculate 9 × 8?' Tasks like those shown below are also tasks for systematization without a reference to the enactive or to the iconic level of representation:

1 0 × 7 = 5 × 8 = 2 × 3 = 5 × 7 = 1 × 7 = 2 × 8 = 4 × 3 = 7 × 5 = 9 × 7 = 7 × 8 = 8 × 3 =
VT: All tasks where the Value of a Term must be determined without any reference.

Our framework offers the possibility to validly classify all tasks in teaching materials, as we implemented all possible forms of representation as well as all steps of the process of learning concepts, as described above.

We are well aware that the determination of term values is indispensable in math lessons. Similarly, it is essential to establish relations between equations without always referring them in enactive or iconic mode. Mastering both requirements -determining the value of a term and establishing the connection between two terms -at skill level is crucial for successful work in the next grades. We use the term skill level as described in the IDMT assessment framework (cf. Brendefur et al., 2016, pp. 177). Nevertheless, these skills can only be used meaningfully if students can represent nonverbal-symbolic expressions at the enactive or the iconic level. The present study, therefore, focuses on those tasks that challenge students to an intermodal translation between the levels of representation.

Findings.

The findings of our analysis are presented in more detail in Table 1 shows the results of our analysis. It shows that the offer of tasks across all four analyzed programs is approximately equal, one-sided and qualitatively unsatisfactory.

Findings of our analysis of second grade programs.

The result of our analysis shows that tasks requiring translations between the levels of representation are significantly under-represented in all four programs. The only tasks to be found are those in which the students are challenged to draw a matching picture for a given term (RCI). Not every program contains tasks that require the students to lay or build something with material suitable to a given term (RCE). In three of the programs, there are 5, 16, and 6 tasks. The total number of these tasks is far too small.

Tasks that challenge students to represent a term verbal-symbolically (RCV) are extremely rare.

Only two of the programs offer such tasks at all and this only seven respectively three times. This finding explains the well-known shortcomings of the students in solving word problems.

The four programs contained 0, 0, 3, 0 tasks for systematization with reference at the iconic level. Not one of the four programs meets our expectations. The systematization of the basic number sentences on a purely non-verbal-symbolic level plays a major role in all four programs: 279, 82, 212 and 318 tasks are represented here. The tasks of systematization on a purely non-verbalsymbolic level are wrapped in different manners. According to the mathematical core, they are tasks like the ones shown below.

3 × 8 = 24  6 × 8 = 48 (doubling), 5 × 8 + 1 × 8 = 6 × 8 or 10 × 7 -1 × 7 = 9 × 7 (so-called neighbourhood tasks), 5 × 8 = 8 × 5 (so-called commutativity tasks).

Such tasks are undoubtedly essential and indispensable. In solving these tasks, however, the students do not have to think about the meaning of the operation. They can undoubtedly use these tasks to acquire networked knowledge, the meaning of which they do not necessarily have to grasp: New equations are developed from equations. For quite a few students, this is exactly the picture of mathematics lessons: lessons in which new strings are created from existing strings. Hence it is not surprising that some students consider multiplication as a trick of mathematicians to shorten long equations of addition.

Findings of our analysis of third grade programs.

First of all, it is apparent that the range of tasks offered for multiplication in third grade is considerably less than in second grade: In total there are 368, 585, 430, and 731 tasks for multiplication. This is not even half of the amount that is offered in second grade. The rate of tasks that promote the understanding of the content of the operation by establishing references to the enactive, iconic or verbal-symbolic level drops markedly: the four programs contain only 27 (7.3%), 30 (5.1%), 27 (6.3%) and 34 (4.7%) of such tasks. Tasks that support an active construction of meaning are thus almost completely missing.

The emphasis is on the determination of term values. With regard to networked memorization, it should be noted that, as in second grade, the systematization of equations is always carried out only at the non-verbal-symbolic level. Tasks that promote the development of memorizing techniques, thematize such techniques, etc. are in fact not included: In all four programs there are altogether only 4 (!) such tasks.

Discussion.

The operation multiplication is acquired in grade 2. Here, the focus is first of all on the conceptual understanding of multiplication. This is usually developed on the basis of contexts. Therefore, it is to be expected that the programs of this grade will offer a qualitatively and quantitatively wide range of tasks with reference to the enactive or iconic level, to ensure the understanding of the meaning of the operation. We expected, that also the grade 3 programs would contain a variety of tasks to systematize and to consolidate the meaning of the operation. We hoped to find a lot of tasks for identification, realization or systematization with reference to the enactive or iconic level.

Accordingly, there areas expectedin grade 2 many tasks in all the programs in which students have to identify a suitable term, particularly to pictures andmore rarelyto actions.

Because the cognitive process that takes place in mathematics lessons only finds a preliminary, relative completion when applying the acquired knowledge, we expected to see many tasks in which the students have to realize the term. We expected that students would be challenged to draw a picture that represents a term, to represent the term using manipulatives or to tell a story fitting the term. The evaluation of the four programs in grade 2 shows that in the programs are only 67 (5%), 181 (12%), 43 (4%), and 63(4%) such tasks and that the tasks are mostly from the type ICI.

The range of tasks that support a conceptual understanding is insufficient in both grades, while the dominance of tasks which aim determining the value of a term is remarkable. It starts in second grade and dominates in third grade. In each of the investigated programs, the teacher must use additional tasks that she/he chooses from other sources or that she/he independently develops.

Right from the development of the first basic facts onwards, it makes sense to network these and build up connected, systematic knowledge. Therefore, we expected already in second grade many tasks that promote the systematization of the basic number sentences with reference to images (iconic level -SCE) or actions (enactive level -SCI). The analysis of the programs shows that such systematization, in fact, did not take place. Not one of the four programs contained a task with a suggestion for systematization with an enactive reference.

If the teacher does not supplement the range of tasks offered by the programs, the students have too little opportunity to grasp the meaning of multiplication and to consolidate this knowledge.

The analysis of the grade 3 programs shows that if a child in grade 2 has not understood the meaning of the operation multiplication, the materials in grade 3 hardly offer any chance for building up this understanding.

Based on our findings we cannot draw conclusions regarding classroom practice. But, we can assume that the examined materials are a basis for learning multiplication in math classrooms. The teacher always can compensate deficits of the materials. Therefore, further research is needed, especially the teachers' choice of tasks has to be analyzed.

  Def.: If a, b  N and a = |A|, b = |B|, than is a × b = |A × B| the product of a and b 3 . Def.: If a, b  N and |B 1 | = |B 2 | = |B 3 | = … = |B a | = b and  i, k mit i ≠ k: B i  B k = , than is a × b = | B 1  B 2  B 3  …  B a | the product of a and b.

Figure 1 :

 1 Figure 1: Levels of representation according Bruner (1964)

  Tasks for the Identification of the Concept starting from the Enactive level of representation. These are tasks such as 'Place five times three cubes and write a matching task.' (The word 'task' here means a term in the mathematical sense of the word.) ICI: Tasks for the Identification of the Concept starting from the Iconic level of representation. In these tasks, a picture is given and the students should give a task that matches the picture. (The word 'task' here means a term in the mathematical sense of the word.) ICV: Tasks for the Identification of the Concept starting from the Verbal-symbolic level of representation. These are tasks such as 'Three rows of five knights each come out of the castle. Write a suitable task' or 'Clown August has five trousers and three jackets. How many different possibilities does he have to be dressed?' RCE: Tasks for the Realization of Concepts to the Enactive level of representation. These are tasks such as 'Lay with cubes matching 4 × 3'. RCI: Tasks for the Realization of Concepts to the Iconic representation level. These are tasks such as 'Draw a picture matching 4 × 3'. RCV: Tasks for the Realization of Concepts to the Verbal-symbolic level of representation. These are tasks like 'Write a story matching 4 × 3.' SCE: Tasks for Systematizing Concepts with reference to the Enactive representation level. These are tasks such as 'How do you go from 5 × 4 to 6 × 4? Lay with cubes and explain'. SCI: Tasks for Systematizing Concepts with reference to the Iconic representation level. These are tasks such as 'How do you go from 5 × 4 to 6 × 4? Draw and explain'.

table 1

 1

	below:

Table 1 : Tasks of multiplication offered in four different programs (textbooks and workbooks)

 1

Basic fact means in this paper an equation of multiplication in which both factors are less than 10 or equal 10.

The terms "teaching material" and "program" mean in this paper the unity of textbook and workbook.

|M| means the cardinality of the set M.

There were the programs "Fredo Mathematik" and Super M (both Cornelsen Schulbuchverlag) as well as "Denken und Rechnen" and "Welt der Zahl" (both Bildungshaus Schulbuchverlage).