Anna Ida Säfström
email: anna_ida.safstrom@hh.se

Ola Helenius
email: ola.helenius@ncm.gu.se

Linda Marie Ahl
email: linda.ahl@mnd.su.se

Implementing alternative models for introducing multiplication

Keywords: Multiplication, primary school, models, implementation

The research literature provides plenty of examples of epistemological analyses of multiplication and descriptions of the complexity of the conceptual field of multiplication. Nevertheless, multiplication is often introduced as repeated addition, although decades of research have identified this pedagogical choice as leading to persistent problems in students' conceptualisation of multiplication. In this paper, we describe a teaching design that aims to implement theoretical and empirical research results regarding multiplication in classroom practice. Within our design, models in the form of iconic representations serve as a means for creating patterns that make multiplicative invariants and structures visible. The teaching we have designed is currently tested in a mid-scale randomized controlled trial and in a large-scale professional development project.

Introduction

Multiplicative structures constitute a vast part of compulsory school mathematics. The importance of the concept of multiplication for students' possibilities to develop mathematical competence cannot be overstated. When multiplication is introduced in school mathematics, children start a journey that should take them to advanced mathematical knowledge about fractions, ratio, rate, slope, similarity, trigonometry, probability and vector spaces. In addition, physics and social science are heavily reliant on multiplicative reasoning and proportional relationships, e.g., statements of equality of ratios a/b = c/d. Hence, multiplication is a fundamental concept, and it is therefore crucial to learn from educational research and to implement gained insights into classroom practice.

Up until today, research has gathered extensive knowledge on students' understanding of multiplication (e.g., [START_REF] Clark | Identification of multiplicative thinking in children in grades 1-5[END_REF][START_REF] Fischbein | The role of implicit models in solving verbal problems in multiplication and division[END_REF]Harel & Confrey, 1994;[START_REF] Larsson | Students' understandings of multiplication[END_REF][START_REF] Thompson | Fractions and multiplicative reasoning[END_REF]. We know that concepts preferably develop over long periods, through experience of a large number of situations [START_REF] Vergnaud | Multiplicative structures[END_REF]. We also know that the "whole number bias" leads students to a solid belief that multiplication makes bigger and that multiplication is synonymous with repeated addition. However, we do not yet know how to transform research results about students' understanding of the properties of multiplication into classroom practice in order to obviate the whole number bias. In this paper, we will describe an ongoing project, aiming to bridge the gap between research findings and teaching practice through an innovation [START_REF] Century | Implementation research: Finding common ground on what, how, why, where, and who[END_REF]: a carefully designed teaching sequence implementing alternative models for the introduction of multiplication. The implementation is carried out in Sweden, where multiplication traditionally is introduced in school year 2, at age 8-9 years. Research on the effects of the implementation is an integral part of the project.

The whole number bias

The whole number bias is "a tendency to use the single-unit counting scheme applied to whole numbers to interpret instructional data on fractions" (Ni & Zhou, 2005, p. 27). This bias may be a result of an unchallenged presentation of multiplication as repeated addition with whole numbers. The consequences of a biased conceptualisation have been long known. [START_REF] Bachelard | La formation de l'esprit scientifique [The formation of the scientific mind[END_REF] concluded that knowledge that has been formed and tested for validity in action tends to form a consolidated belief concerning the nature of the concept. When the concept is later challenged in new situations, assimilation might be more complex than if initial knowledge was absent. This is why an introduction of multiplication as repeated addition may become an epistemological obstacle for students. [START_REF] Fischbein | The role of implicit models in solving verbal problems in multiplication and division[END_REF] conclude that:

The initial didactical models seem to become so deeply rooted in the learner's mind that they continue to exert control over mental behavior even after the learner has acquired formal mathematical notions that are solid. (p. 16) There are two misconceptions of the properties of multiplication specifically linked to the whole number bias: conceptualising multiplication as an additive relation rather than a multiplicative relation, and a belief that a product will always be bigger than each of its factors. As long as multiplication is limited to whole numbers, these misconceptions will not constrain children's reasoning, and are therefore unlikely to be challenged. However, as soon as children are introduced to multiplication with fractions and decimals their conceptualisation will no longer be sufficient. Products such as ½ times 5 cannot be calculated as repeated addition, and the law of commutativity, which could be helpful in this situation, is difficult to explain in terms of repeated addition. Furthermore, a firm belief that multiplication makes bigger will undermine the children's possibilities to interpret situations with operations like 0.22 • 1.20 [START_REF] Bell | Choice of operation in verbal problems with decimal numbers[END_REF], since the answer is less than 1.20, which is not compatible with a conceptualisation of multiplication built on additive reasoning with whole numbers. In summary, conceptualising multiplication as repeated addition works when dealing with whole numbers but will become a misunderstanding of multiplicative properties when extended to rational and real numbers. This is a strong argument for finding alternative ways to introduce multiplication, and thereby avoiding the whole number bias.

Alternative models for introducing multiplication

The complexity of multiplication is reflected in the body of research, and might be one reason why innovative designs are scarce in both research and practice. The efforts to describe dimensions and aspects of multiplicative knowledge are diverse, and show less agreement than, for example, the literature on additive knowledge [START_REF] Nunes | Children doing mathematics[END_REF]. In [START_REF] Vergnaud | Multiplicative structures[END_REF] terms, multiplicative structures can be seen as a conceptual field, consisting of situations, invariants and symbolic representations. Compared to additive structures, multiplication embraces a much more diverse family of situations. Already in the simplest case, with two factors, multiplication unifies qualitatively different situations, which can be both one-dimensional, like scaling, and twodimensional, like one-to-many correspondence, area, intensity and Cartesian products (Harel & Confrey, 1994;[START_REF] Nunes | Children doing mathematics[END_REF][START_REF] Vergnaud | Multiplicative structures[END_REF]. In line with the theory of conceptual fields, our teaching design comprises both activities focused on situations and activities focused on invariants, with care taken to link established symbolic notation to both types of activities (see Table 1).

Week 1

Factorization (multiplicative grouping) of numbers with lattice models. Regrouping of lattices.

Week 2

Connecting lattices to symbolic notation for multiplication and division. Constructing the multiplication table by sorting lattices by factors.

Week 3

Placing numbers on number lines by measuring with skips and cubes. Discovering proportional relationships.

Week 4

Measuring number lines with halves. Equating different numeric expressions for multiple halves.

Week 5

Regrouping lattices by halving and doubling. Multiplication with half-integers.

Week 6

Exponential relationships and associativity in the context of repeated folding (inspired by [START_REF] Empson | The emergence of multiplicative thinking in children's solutions to paper folding tasks[END_REF].

Week 7

Equal grouping situations: partition, quotation and one-to-many correspondences.

Week 8

Proportions: double, half, times four and fourth.

Table 1: Overview of the sequence of multiplicative activities. Week 1-5 focus on investigating multiplicative patterns and structure, while week 6-8 focus on modelling multiplicative situations

However, since classifications of multiplicative situations, including examples of classroom activities, have been thoroughly described elsewhere, we will focus this paper on the use of models in the form of iconic representations, and how such models may aid the discovery of invariants. Since multiplicative situations can be both one-and two-dimensional, we have chosen to use both one-dimensional number lines and two-dimensional lattices as models.

A coherent conceptualisation of multiplication requires an appreciation of multiplicative invariants across situations. In other words, one has to recognise patterns in different situations as manifestations of the same structure. Within the implementation, patterns and structure [START_REF] Mulligan | Early awareness of mathematical pattern and structure[END_REF] has proven to be a useful terminology for conveying the importance of invariants to teachers, and for describing the purpose of activities where patterns are created and described. Seeing patterns calls for experience of a breadth of cases, rather than focusing on one particular situation at the time. Therefore, the first five weeks focus on exploring specific models of numbers (see Table 1).

From the perspective of embodied cognition, [START_REF] Lakoff | Where mathematics comes from: How the embodied mind brings mathematics into being[END_REF] argue that arithmetic is a conceptual blend, resting on four conceptual mappings from concrete situations. These mappings, or metaphors, each contribute to the concepts of number and arithmetic in their own way, implying that no single metaphor can reflect these concepts in full. Our lattice models are related to the first two of the grounding metaphors: object collection and object construction, while number lines are related to the latter two: measuring with a stick and movement along a path. There is a particularly close relationship between multiplication and measuring. Both simple proportions and area build on two important ideas: unitizing and norming [START_REF] Lamon | Ration and proportion: Cognitive foundations in unitizing and norming[END_REF]. Unitizing is described as the ability to construct a reference unit or a unit whole, and norming is then to interpret a situation in terms of that unit. In other words: creating a measure and then using it for measurement. This idea is also mirrored in verbal expressions on the form "two threes", where the number three becomes the unit six is measured in.

We will now describe how lattices, number lines and verbal expressions are used as models in our teaching design. In particular, we will detail how these models convey important properties of multiplication.

Lattice models

Lattice models are in many situations exchangeable with arrays, but they do have some advantages. In arrays, each unit is separated from the others and pictured as indivisible. In lattices, the units are placed in a continuum, which makes the idea of dividing units by adding new lines attainable. Furthermore, open arrays are visually closer to lattices than arrays. Within our teaching design, both interlocking cubes and drawn grids are used to create lattice models. The activities using interlocking cubes could often be done in the same way with arrays of tickers or other objects, but that would make the step to drawn grids larger. By using cubes and grids of the same size (2 • 2 cm), the two representations are easily compared, and lattices of both kinds can be used in the same systematic arrangements.

Lattices can be seen as 'bar models for multiplication', in that they provide a way to represent different multiplicative groupings of numbers, just as bars can represent different additive groupings. Within the design, the class has previously worked with finding number bonds to ten by building ten bars and naming them "_ + _". Multiplication can then be seen as a different way to group numbers (see Figure 1). Our choice of introducing multiplication by lattices hence introduce multiplication not as an additive relation, but as something different than addition. The criteria for multiplicative groupings are easily established by discussion in class, distinguishing proper lattices from other shapes. The existence of different lattices for a number calls for expressions of the form "_ • _" to be introduced as names for lattices. Finding different lattices for each number then becomes an explorative activity, leading to discussions of which lattices are the same (commutativity), and why some numbers have more lattices than others (primes and composite numbers) (Week 1, Table 1). Hence, important multiplicative invariants, which are hidden when multiplication is conceptualised as repeated addition, are made visible already in the first activity. Arranging lattices according to products reveal that multiplicative groupings do not form a pattern as simple as for additive groupings. Finding patterns in lattices therefore requires more careful instruction. Directing attention to lattices of twos leads to the discovery of even numbers and could be generalised to multiples of other numbers. This idea will inspire to a different arrangement of the lattices, focusing on factors rather than products. Building lattices of twos, threes, fours, etc. and ordering them in rows will result in the multiplication table (Week 2, Table 1). The table will spark a new exploration of patterns in the products, and the lattices will provide a means for explaining those patterns. For example, you can see two copies of the product in the second row in the corresponding product in the fourth row (a basis for the distributive law and powers), and that the products in the second column are the same as the products in the second row, because the lattices in the column are just rotated versions of the lattices in the row (commutativity).

Using interlocking cubes facilitate successive adding and subtracting of columns and rows, which are mainly additive strategies. However, they also allow for multiplicative regrouping of numbers. When comparing the two twelve lattices 6 • 2 and 3 • 4, you can split one of the lattices and rearrange the parts, in order to see how one lattice can be regrouped into the other (Week 1, Table 1). Paper grids enable generalisation of such multiplicative regroupings to half-integers (numbers of the form n + ½), since paper can be cut and arranged to illustrate e.g., 2 • 5 = 4 • 2.5. This immediately gives rise to multiplications where the product is smaller than one of the factors, such as 4 = 8 • 0.5 (Week 5, Table 1). The model hence enables the discovery of cases where multiplication does not make bigger, within the first weeks of introducing multiplication.

Number lines

Number lines are important tools for arithmetic reasoning. Approximate use of empty number lines can both aid and communicate reasoning even in complex arithmetic problems. However, for empty number lines to be meaningful, one needs to be aware of the properties of linearity, closely linked to multiplication. Within our design, the class will use a variation of number lines, which put different properties of linearity in the fore.

During choral counting, a number line with markings and numbers, placed on the wall, is often used for pointing at numbers while skip counting (e.g., 0, 2, 4, 6, etc.). In some activities, a floor number line is used. This number line has evenly spaced markings, with every fifth marking a little longer than the others, and a collection of number cards which can be placed in different ways in different activities. Every placement of two numbers raises the question of where other numbers should be and leads to discussions of equal spacing of numbers. In week 3 this is done with 0, and 1 or 2 in different positions (see Table 1). When starting with 1, the situation is a pure one-to-many correspondence, while starting with 2 requires more complex proportional reasoning. When justifying a proposed placement of numbers, or figuring out a placement, the children walk along the line, taking e.g., two-steps or three-steps. When documenting the work, teachers use verbal expressions and symbolic notation e.g., "you take three two-steps, so it's three twos, we write 3 • 2". A similar activity is done with semi-empty number lines on paper, where only markings for specific numbers are present, e.g., 0 and 3. The task is to place markings for the missing numbers between 0 and 6, using interlocking cubes as a measuring tool. This activity is also provided at different levels, with some restricted to pure one-to-many correspondences.

As in the case with the lattice model, these activities are easily extended to fractions, starting with halves. Walking in regular or half-steps to different numbers creates an opportunity to introduce multiplication of fractions in an intuitive context. Since multiple double or triple steps were called twos or threes and represented as n • 2 or n • 3, it makes perfect sense to call multiple half-steps halves and represent them as n • ½. Documenting the number of each type of step it takes to reach different numbers will result in a pattern with a proportional structure: the number of half-steps is always twice the number of regular steps (Week 4, Table 1). After whole class exploration of these patterns on the floor number line, children work in pairs to place half-integers on empty number lines, using interlocking cubes as a measuring tool, this time with 1 cube = ½.

Verbal expressions

In relation to both lattices and the number line, care is taken to use sustainable verbal expressions of the form "two threes" and "five halves". As emphasized by [START_REF] Thompson | Fractions and multiplicative reasoning[END_REF], there is a principal difference between "two threes" and "add three two times". It is possible to generalise the first expression beyond integers, to e.g., "one and a half threes" or "one and a half third", while "adding three one and a half time" is harder to conceptualise. Phrases of the first form mimic phrases used in measurement, making a number the measuring unit: "one and a half threes" -"one and a half meter". While such phrases are used to discuss both models, they relate to each model a bit differently. Each phrase, e.g., "three twos", labels a certain way of walking to a number or a series of skips to a number, which is different from what "two threes" labels. In contrast, both expressions can be linked to the same lattice, and the corresponding ways of seeing the lattice are made explicit, pointing out the three twos and the two threes in this six lattice.

Implementation of the alternative models for multiplication

Taken together, our design covers a variety of advanced aspects and properties within the first weeks of the introduction of multiplication. The handling of advanced content is made possible by the highly structured material organising the teaching of each week in a cycle with six phases, including whole class discussions as well as pair work and individual documentation. This cyclic teaching structure has been previously described and tested in preschool [START_REF] Sterner | Number by reasoning and representationsthe design and theory of an intervention program for preschool class in Sweden[END_REF]. The material developed for the implementation of the design includes explicit instructions on what the teacher should do and say in each phase, and in reaction to different student actions and reasoning. Each activity also contains a detailed description of the mathematical concepts and procedures involved, as well as the purpose of each activity in relation to children's learning. Activities, as well as instructions, were tried and evaluated by small groups of teachers during the development of the material, increasing the quality and functionality of the design. Thorough testing of the activities is important for the teachers' trust in the material, which is particularly important due to the high level of prescription in the design.

The teaching design has been implemented in an intervention research project covering 15 weeks in grade 2, where the multiplicative part covers 8-9 weeks. The project is a randomized controlled trial with 28 participating teachers in grade 2 (14 using our material, 14 in a control group). Data from the pre-and post-tests are currently being analysed. In a parallel project we are developing similar teaching sequences for grades 1 and 3. Including sequences dealing with additive structures and number, the complete material will cover most of the area of number and operations for grades 1-3. These sequences will also be tried and evaluated by small groups of teachers during development, together with auxiliary professional development material. In collaboration with 5 municipalities, we will implement the teaching design together with professional development for around 500 teachers starting in the fall of 2020.

Closing reflection

The literature on the downsides of relying on multiplication as repeated addition is substantial. Still, teaching experiments with alternative models for multiplication are sparse, one example being [START_REF] Nunes | The scheme of correspondence and its role in children's mathematics[END_REF], who report on two studies covering only a few sessions. Also among teachers and textbook authors the dominance of repeated addition as the model for introducing multiplication persists. Fishbein et al. (1985) suggest that this model is chosen because it fits the mental requirements of children in the first years of schooling. It is simply hard for teachers to resist building on addition, since it is a concept that children are already familiar with at the time when multiplication is introduced. However, as seen above, multiplication in fact has several complementary roots that are not additive, such as measuring and one-to-many correspondence [START_REF] Lamon | Ration and proportion: Cognitive foundations in unitizing and norming[END_REF][START_REF] Nunes | Children doing mathematics[END_REF]. Many of these roots are intuitive for children and can hence be used as alternative bases for introducing multiplication.

According to [START_REF] Century | Implementation research: Finding common ground on what, how, why, where, and who[END_REF], there are three key foci for implementation research: the innovation, the aligned outcomes, and the influential factors. This paper has described the innovation of an ongoing, large scale implementation project: a teaching sequence introducing multiplication by means of deliberately chosen models for making multiplicative patterns visible and comparable, which in turn inspire discussions where multiplicative structures are explicated. The analysis of test results in the randomised controlled trial will reveal to what extent this innovation achieves the desired outcomes, while the grade 1-3 project will provide opportunities to study the influential factors, as well as to further develop the innovation.

Figure 1 :

 1 Figure 1: Bars and lattices as two different ways of grouping numbers

Acknowledgment

This work was supported by the Swedish Research Council, Grant number 721-2014-2468.


Samuel Poirot
email: poirot@prism.cnrs.fr

Stefan Bilbao

Sølvi Ystad

Mitsuko Aramaki

Richard Kronland-Martinet

On the Influence of Non-Linear Phenomena on Perceived Interactions in Percussive Instruments

Keywords: Sound-Synthesis, Non-Linear Behavior, Auditory Perception, Impact Sounds, Control

In this paper, we investigate the hypothesis that perceived impact strength is strongly influenced by the non-linear behavior produced by large deformations in percussive instruments. A sound corpus is first generated from a physical model that simulates non-linear vibrations of a thin plate. The e↵ect of non-linear phenomena on the perceived strength is further quantified through a listening test. The aim of this study is to improve the expressive potential of synthesizers of percussive sounds through the development of signal transformation models. Future work will focus on the modeling of sound morphologies that correspond to non-linear behavior and the development of a transparent control strategy.

Introduction

Digital sound synthesis enables the exploration of large sound spaces. One challenge is to find perceptually consistent control strategies to explore these spaces and hereby improve user expressivity.

For a synthesizer to be "expressive", it has been conjectured that the mapping between the inputs and outputs of the device must be transparent [START_REF] Fels | Mapping transparency through metaphor: towards more expressive musical instruments[END_REF]. Transparency is here defined as an indicator of the psychophysiological distance, in the mind of the user and the audience, between the inputs and outputs of the instruments. Therefore, the control parameters and their link with the user's actions must be defined intuitively to allow for proper expression within a synthesizer. This has been investigated in several studies [START_REF] Wanderley | Gestural control of sound synthesis[END_REF] [3] [START_REF] Hoskinson | Real-time adaptive control of modal synthesis[END_REF].

In the context of environmental sound synthesis, Conan proposed an intuitive control for continuous interaction sounds [START_REF] Conan | An intuitive synthesizer of continuous-interaction sounds: Rubbing, scratching, and rolling[END_REF]. This work is a follow-up of the investigations of Aramaki et al. on the synthesis of percussive sounds with a control of the attributes of the impacted object (material, shape and size) [START_REF] Aramaki | A percussive sound synthesizer based on physical and perceptual attributes[END_REF] [START_REF] Aramaki | Thinking the sounds: an intuitive control of an impact sound synthesizer[END_REF][8] [START_REF] Kronland-Martinet | High-level control of sound synthesis for sonification processes[END_REF]. These studies have led to the development of a sound synthesis environment with an intuitive control of the evoked action (impact, rolling, scratching, rubbing) and object (material, size and shape) for solid interactions. Inspired by the ecological approach for auditory events [START_REF] Gibson | The ecological approach to visual perception[END_REF] [START_REF] Gaver | What in the world do we hear?: An ecological approach to auditory event perception[END_REF] [START_REF] Mcadams | Introduction to auditory cognition[END_REF], this environment has been developed within the action-object paradigm, which states that the sound is the result of an action on an object and that invariant structures can be independently associated to the actions and the objects. This synthesizer constitutes an experimental research environment for the PRISM laboratory that can be adapted for various uses, such as music practice, video games [START_REF] Pruvost | Perception-based interactive sound synthesis of morphing solids' interactions[END_REF], or more generally sound design.

A dynamic mapping of the user's gesture with the synthesis parameters has been set up in specific cases to improve expressive control of the synthesizer, e.g. from squeaks to self-oscillating transitions for continuous interactions with a touch pad [START_REF] Conan | An intuitive synthesizer of sustained interaction sounds[END_REF]. The perceived material has also been mapped with the strength of the impacts, and a drumstick has been instrumented to capture the gesture of the user. However, no work has so far been done to define the perceptual expectations related to the strength of the impact in a more general situation.

The objective of this study is to define expectations regarding the evolution of the sound radiated by an object when it is struck with di↵erent strengths. To this end, we study the e↵ects of non-linear behavior on the radiated sound. The idea is to apply transformations to the signal for a transparent mapping of the strength of an impact for percussive sounds. In the context of virtual music instruments, this parameter could be linked to the velocity, a central parameter for expressiveness of percussive instruments on midi controllers.

Such an idea has been investigated in the musical context for years. Most virtual instruments o↵er dynamic behavior and adapt the sound rendering to the velocity of touch on the keyboard, especially for the highest values. A simple and convincing e↵ect is to add brightness for high velocity values. For physical systems, this e↵ect is due to the behavior of the impactor, whose deformation varies non-linearly with the pressure exerted [START_REF] Sackfield | Some useful results in the classical hertz contact problem[END_REF]. For instance, the deformation of piano hammers follows a non-linear law related to their composition (wooden core surrounded by felt) [START_REF] Bensa | A hybrid resynthesis model for hammer-string interaction of piano tones[END_REF] [START_REF] Stulov | Hysteretic model of the grand piano hammer felt[END_REF]. Also, rules of thumb are proposed in various publications for a transparent mapping between the sound synthesis parameters and the user's gesture. One example is the mapping of the modulation index with the impact velocity in the context of FM [START_REF] Chowning | FM theory and applications: By musicians for musicians[END_REF], or the shapes of harmonic distortion laws to replicate a "physical" behavior [START_REF] Roads | Musique et informatique[END_REF]. In the context of environmental sound synthesis, one can also mention Warren and Verbrugge's investigations on sound morphologies responsible for breaking event recognition [START_REF] Warren | Auditory perception of breaking and bouncing events: a case study in ecological acoustics[END_REF].

The proposed approach is to model the physical system and its non-linear behavior in order to synthesize realistic sounds for various configurations. The relevance of these phenomena with respect to perceived strength of the impact can then be evaluated through listening tests. Finally, we aim to model sound morphologies responsible for the evocation of high intensity impacts.

Physical modelling allows the generation of realistic sounds, and the mapping of the user's gesture is obvious and transparent since the synthesis parameters are physically meaningful. Moreover, the ever-growing computing capacities of commercially-available hardware makes it possible to consider real-time synthesis for increasingly complex models, opening the possibility of designing virtual instruments with this type of synthesis [START_REF] Smith | Physical modeling synthesis update[END_REF][22] [START_REF] Willemsen | Real-time control of largescale modular physical models using the sensel morph[END_REF]. However, the inherent constraints of a physical model limit the possibilities of sound creation. Indeed, the design of such a virtual instrument is based on modelling the behavior of an existing element rather than on the free description of a sound. The positioning of this study is to use physical models to generate a sound corpus that is further analyzed to extract and model the sound morphologies (signal model) corresponding to the phenomena studied. In this way, we free ourselves from the constraints of physical modeling by allowing the generation of metaphorical sounds. On the other hand, this type of approach, as presented by Rocchesso et al. [START_REF] Rocchesso | Sounding objects[END_REF], is complex because it implies three levels of research: auditory perception, physics-based sounds modeling, and expressive parametric control.

This paper presents a preliminary study for the development of perceptually controlled synthesis processes. We propose here to demonstrate that non-linear phenomena play a major role in the perception of the impact strength in the case of thin plates.

Review of Notable Non-Linear Behavior for Everyday Sounding Objects

Our perceptual expectations are driven by real sounding objects that we are used to hearing and manipulating. In this section, we briefly review the nonlinear behaviors that can induce a notable transformation of the sound radiated by an object impacted with di↵erent strengths.

For this purpose we focus on percussive musical instruments, which sound timbre changes according to the excitation. The choice of the impactor (hammer, mallet, drum stick, brush, plectrum, fingers, hand palm) and the gesture (force profile, impact position) define the quantity of energy and its distribution on the resonator modes. In general, the impactor hardens as the intensity of the impact increases, inducing a brighter sound.

After excitation, the sound radiated by the instrument can be modified in case of interactions between the resonator and other elements. For example, the specific timbre of the tanpura and sitar is caused by the interaction of the strings with the bridge and the sympathetic strings. These interactions can lead to a muted sound (ghost notes, cymbal choke, palm mute), pitch modifications (natural and pinched harmonics, slide, bend, ghost notes, udu drum) and the appearance of other partial tones over time (slapping, string buzz, tanpura and sitar double bridge, snare wire, rattling elements in a cajon or a mbira).

Geometric non-linearities in the resonator itself may also occur for large amplitude deformations, resulting in time varying mode frequencies, harmonic distortion and mode coupling. These phenomena are particularly prevalent in the case of gong and cymbal crashing (wave turbulences). They can also be heard through timbre variations in steel drums, or pitch bending of strings and membranes due to large amplitude vibrations. Beyond such high amplitude vibrations, plastic deformations or ruptures can be observed.

These observations can be transposed to objects of everyday life. For example, the force with which someone knocks on the door may be recognized through the impactor (what part of the hand, and with which hardness), and the interaction with surrounding walls and rattling elements. Thin structures that admit large elastic deformations (e.g. metal sheets) behave similarly to cymbals and gongs. Weaker flexible structures deform (cans, plastic bottles), fibrous materials creak then crack (wood, composite), while more fragile structures crack then break (glass slides). Also, paper sheets, aluminum foil, plastic bags, fabrics, loose membranes and strings do not vibrate much due to their low sti↵ness. Their behavior is strongly non-linear, and the radiated sound is not tonal.

To sum up, the dynamic behavior of an impacted object may be due to the non-linear behavior of the object or non-linear interactions with other elements. A study has already been initiated to define a morphological invariant of nonlinearity related to interactions [START_REF] Poirot | Sound morphologies due to non-linear interactions: Towards a perceptual control of environmental sound synthesis processes[END_REF]. In this paper, the objective is to check whether non-linear phenomena related to large vibrational amplitudes of an object are central to the perceived intensity of the impact.

Methods

We hypothesized that the perception of the strength of an impact is correlated with the occurrence of audible non-linear phenomena in the sound radiated by an object for large vibration amplitudes. It is assumed here that the strength of the impact is expressed by the object's ability to resist this impact. This implies that the perceived strength depends on the structural characteristics of the impacted object. To verify this hypothesis, an experiment was conducted in which subjects were to judge the strength of the impact and the object's ability to resist this impact by answering the following question: In this test, we ask you to evaluate the strength of the impact and the "su↵ering" of the object for each sound (In French, Vous devrez évaluer la force de l'impact et la "sou↵rance" de l'objet pour chacun des sons). The subjects were told that the "su↵ering" of an object reflects its di culty to bear the deformation produced by the excitation. The degree of "su↵ering of the object" was used as an indicator for the perceived weakness of the object, supposed to be correlated with the occurrence of audible non-linear phenomena.

We chose to focus on the behavior of thin plates to test our hypothesis because it generates characteristic and easily recognizable sounds for large deformations. This is a first approach to identify sound morphologies linked to audible nonlinear phenomena, that can be used to define a signal transformation model that can be applied to di↵erent sound textures.

Synthesis of the Stimuli

Thin plate model. The stimuli were synthesized by numerical solving the Von Karman system, a widely used model of nonlinear vibration of plates at moderate amplitudes with a quite compact form [START_REF] Bilbao | Numerical sound synthesis: finite di↵erence schemes and simulation in musical acoustics[END_REF].

u tt = D ⇢H u + 1 ⇢H L( , u) 2 0 u t + 2 1 u t + e ⇢H F (t) (1a) = EH 2 L(u, u) (1b)
Where u(x, y, t) is the transverse plate deflection, (x, y, t) is often referred to as the Airy stress function, F (t) is the excitation force, e = (x x i , y y i ) the 2D Dirac function, 0 = 0.1 rad• s 1 and 1 = 0.001m 2 •s 1 are the damping coe cients, H is the plate thickness, in m. The material is defined as steel, with a density of ⇢ = 7800 kg•m 3 , D is set as a function of Young's modulus E = 210 GPa and Poisson's ratio ⌫ = 0.3 as

D = EH 3 12(1 ⌫ 2 ) . (2)
In Cartesian coordinates, the nonlinear operator L is defined by

L(f, g) = @ 2 x f @ 2 y g + @ 2 y f @ 2 x g 2@ x @ y f @ x @ y g (3)
for any two arbitrary functions f (x, y, t) and g(x, y, t).

We set the length and width L x = 0.6 m and L y = 0.7 m, and we synthesized the sounds for three di↵erent thicknesses H 1 = 1 mm ; H 2 = 2 mm ; H 3 = 3 mm. The boundary conditions were set free for x = 0 and

x = L x u xx + ⌫u yy = u xxx + (2 ⌫)u xyy = 0, (4)
and simply supported for y = 0 and y = L y u = u yy = 0.

(5)

These boundary conditions have been chosen to increase the occurrence of wave turbulences and limit frequency variations.

We chose to model the impact by a raised cosine rather than by a mallet model in order to be able to control the brightness independently of the strength of the impact.

F (t) = ⇢ A ⇤ ( cos(2⇡t/T ) + 1) 0  t < T 0 otherwise
A constant impact duration T is assumed for any amplitude, which corresponds to an impactor with a linear behavior. Thus, the influence of geometric non-linearities can be assessed without the non-linear behavior of the impactor interfering with this measurement. T was set to 2 ms and the amplitude A was linearly varied over 10 levels from 100 N to 1000 N.

Finite Di↵erence Scheme. Time and space are discretized. We note u n l,m the transverse displacement at the n th time step and the grid point (x = h ⇤ l; y = h ⇤ m), with h the spacing between two grid points, k = 1/f e the time step (f e = 44100 Hz). We used the following conservative scheme, as defined in [START_REF] Bilbao | A family of conservative finite di↵erence schemes for the dynamical von karman plate equations[END_REF]

tt u = D ⇢H , u + 1 ⇢H l(µ t• , u) 2 0 t• u + 2 1 t u + J ⇢H F (6a) µ t+ , = EH 2 l(u, e t+ u) (6b)
with

e t+ u n l,m = u n+1 l,m , e t u n l,m = u n 1 l,m (7a)
e x+ u n l,m = u n l+1,m , e x u n l,m = u n l 1,m (7b)
e y+ u n l,m = u n l,m+1 , e y u n l,m = u n l,m 1 (7c)
µ t+ = (e t+ + 1)/2 (7d)

l(f, g) = xx f yy g + yy f xx g 2µ x ,y ( x+,y+ f x+,y+ g) (7n)
J is the interpolation operator. J Nx/2+1,Ny/2+1 = 1 h 2 for the node in the middle of the plate (an even number of elements in length N x and width N y is defined), J l,m = 0 for the other nodes.

More details are available for the implementation of the scheme in [START_REF] Torin | Percussion instrument modelling in 3d: Sound synthesis through time domain numerical simulation[END_REF].

Loudness Equalization. A total of 30 di↵erent sounds were synthesized (3 levels for the thickness of the plate, 10 levels for the strength of the impact). The samples were normalized (their maximum value is set to 1) as follows:

s n norm = s n max(|s n |) (8)
with s n the original signal and s n norm the normalized signal at the time step n. Finally, we proceeded to a loudness equalization. There is no model for the loudness of complex sounds, so it is necessary to probe perception with pairwise comparisons. Comparing all pairs with an adaptive procedure is too timeconsuming for a pre-test (435 pairs, with about 1 minute testing time per pair). Furthermore, the task is complex because the sounds to be evaluated have a di↵erent temporal evolution. In general, sounds corresponding to a strong nonlinear behavior have less energy at low frequencies and are damped more quickly. A simplified procedure consists in choosing the median sample (H = 2mm ; A = 500N ) as a reference and compare it with the other sounds. This limits the comparison to 29 pairs and minimizes the di↵erence between the sounds to be compared.

The resulting stimuli are available for online listening3 .

3.2 Participants and Procedure.

Fourteen participants (3 female, 11 male), aged 22 to 50, were tested in this experiment. They all had normal audition and gave consent to participate in the experiment.

Participants were asked to evaluate the strength of the impact and the "suffering" ("sou↵rance" in French) of the object for each stimulus by moving two sliders on a scale without markers. The 30 sounds were randomly presented through headphones (Sennheiser HD650) in a quiet environment.

Results

ANOVAs were conducted for the perceived strength of the impact and the perceived "su↵ering" of the object. Factors were the thickness (1 mm, 2 mm, 3 mm) and the impact strength amplitude (100 N, 200 N, 300 N, 400 N, ... , 1000 N). Results are displayed in fig. 1&2.

Discussion

There is a direct correlation between the perception of the impact strength and the "su↵ering" of the object. The evaluation of the strength and the "su↵ering" are almost identical for all the sounds, although the perceived impact strength is slightly higher in general. This result is also revealed in the feedback from the di↵erent participants who said they answered almost the same value for both parameters for most sounds. This reflects the fact that the presence of non-linearities due to large object deformations is used as a perceptual cue to assess the impact strength, which is consistent with the initial hypothesis. This observation is reinforced by the significant di↵erences between the changes in the evolution of the perceived strength regarding the excitation amplitude A for each high H level (the interaction e↵ect AxH is significant p = 0.00000). Indeed, the perceived strength for H 3 (almost no non-linear phenomena for the whole range) is relatively constant regarding the level of the impact strength. Conversely, we observe a significant evolution of the perceived strength regarding A for the smallest thicknesses H 1 and H 2 , which correspond to sounds that present a progressive appearance of non-linear phenomena when the impact strength increases.

Also, the upper part of the scale remained unused for most participants. Several participants told that they were expecting stronger amplitudes of excitation that would damage or brake the plate. On the other hand, we notice that the strength is never perceived as zero, unlike the "su↵ering" of the object, and that the value of the perceived strength is always higher than the value of the "su↵ering" of the object. This is consistent with the idea that the "su↵ering" of the object only begins when non-linear phenomena begin to appear, unlike the strength that always has to be di↵erent from 0 for a sound to occur.

Conclusion & Perspectives

The purpose of this study was to investigate perceptual expectations regarding the evolution of the sound radiated by an object with respect to the strength of the impact it undergoes. To this end, we sought to evaluate the e↵ect of non-linear phenomena on the perceived impact strength. This paper focuses on the study of thin plates for moderate vibration amplitudes. A listening test was conducted to evaluate the impact strength and the "su↵ering" of the object for 30 sounds synthesized by physical modeling of the system, corresponding to 3 plates of di↵erent thicknesses impacted with an excitation amplitude ranging from 100 N to 1000 N. The evaluation results show that the perceived impact strength is directly correlated with the occurrence of non-linear phenomena in the case of thin plates.

Further, the deformations are perceived as small for the present sound corpus, since only the lower section of the evaluation scale is used by the subjects. This result encourages us to extend this experiment by modelling the e↵ects of plastic deformations and rupture on the sound radiated by an object.

The next step is to propose signal transformation models corresponding to the di↵erent non-linear phenomena, and perceptually relevant controls to improve the expressive potential of percussive sound synthesizers.

Fig. 1 .

 1 Fig. 1. Representation of responses for the perceived strength of the impact for all samples (percentage of cursor range). The level of the excitation A is presented on the horizontal axis, and each curve represents a level for the plate thickness. Dots denote the average value for all participants, vertical bars denote +/-standard errors.

Fig. 2 .

 2 Fig.2. Representation of responses for the perceived "su↵ering" of the object for all samples (percentage of cursor range). The level of the excitation amplitude is presented on the horizontal axis, and each curve represents a level for the plate thickness. Dots denote the average value for all participants, vertical bars denote +/-standard errors.

Proc. of the 14th International Symposium on CMMR, Marseille, France, Oct.[START_REF] Conan | An intuitive synthesizer of sustained interaction sounds[END_REF][START_REF] Sackfield | Some useful results in the classical hertz contact problem[END_REF][START_REF] Bensa | A hybrid resynthesis model for hammer-string interaction of piano tones[END_REF][START_REF] Stulov | Hysteretic model of the grand piano hammer felt[END_REF][START_REF] Chowning | FM theory and applications: By musicians for musicians[END_REF] 2019

https://cloud.prism.cnrs.fr/index.php/s/tx67ywnVvMg21jCProc. of the 14th International Symposium on CMMR, Marseille, France, Oct.[START_REF] Conan | An intuitive synthesizer of sustained interaction sounds[END_REF][START_REF] Sackfield | Some useful results in the classical hertz contact problem[END_REF][START_REF] Bensa | A hybrid resynthesis model for hammer-string interaction of piano tones[END_REF][START_REF] Stulov | Hysteretic model of the grand piano hammer felt[END_REF][START_REF] Chowning | FM theory and applications: By musicians for musicians[END_REF] 2019