Concept of Force Representations of Prospective Primary Teachers

The representation of physics concepts is essential to support teaching/learning in primary school. Investigating how future teachers represent the physical concepts has great importance. Research was carried out on concepts of force representations of 274 prospective teachers, enrolled in the Primary Education Sciences degrees. Rubrics were designed and used to analyse different dimensions involved in drawing, descriptions, didactic and disciplinary motivations. It emerged that most of the representations do not include the representation of the involved forces, but rather implies a precise didactic approach to the force concept.


Introduction
Teaching/learning concepts in the primary school require a systematic use of different iconic representation as pictures, drawings, graphs to support the educational action and the development of formal thinking [1][2][3][4][5].A wide literature investigated the learning problems on the concepts of force, but few research studied the role of representations in that field [6][7][8][9][10][11]. Visual language structures are similar to those of verbal language and the typical representations are conceptual models [3,[12][13][14][15][16].It is crucial, therefore, to study how the future teachers represent the concept of force [3,18] and how these representations change after a formative module, with respect to the conception of force, to the representation as didactic instrument, to the specific representation in physics.In the framework of the Model of Educational Reconstruction (MER) [19] we develop research on the conceptual analysis of the idea of force based on the representation involving that concept of Prospective Primary Teachers (PPT) of three Italian university (Milan, Pesaro-Urbino, Udine), before the physics module.The PPTs representations were collected in an open Form.For the PPTs of Udine, we collected also the representations produced after a formative module on the didactic of the force concept.Here we discuss the framework of research, the instruments and methods and the main results.

Theoretical framework
The theoretical framework for the research here presented is the MER [19].It involves an analysis of the disciplinary content structure to identify foundation aspects and conceptual knots to build a coherent educational purpose.Such reconstruction requires examination of textbooks and publications on the topic, reconstruction of the historical development of ideas, identification of the main spontaneous conceptions of students, analysis of the educational role of the elements chosen for the design.The construction of a grid of Foundational Nuclei (Appendix A) was the first outcome of this work and one of the cornerstones of the analysis tool developed with this research.The analysis of 118 articles [6][7][8][9][10][11][20][21][22] relative to surveys of different setting, in which spontaneous and common-sense ideas, learning difficulties with respect to the concept of force of students of different ages and geographical backgrounds were identified.The result of this work was a reference catalogue of conceptual knots (Appendix B and references therein cited).The third aspect analysed is those of representations, in particular those enhancing the understanding of educational contents [3,12,14,16].According to the Peirce's triadic conception [23], the image is a "sign" that acquiring meaning putting in relationship three entities: the represented object, the representative sign (the actual image), the observer that interpret the image.Therefore, an image in education is effective when it is meaningful with respect to the subject matter and it is resonant with the student reasoning interpreting it, that is, it is a functional integration of perceived visual and conceptual qualities as the Fischbein figural concepts [24].Studies in astronomy education [3,13,[25][26] show student's difficulties in interpreting overly complex or ambiguous images, or partial representation as for instance a three dimension phenomenon represented in two dimensions [13][14][15][25][26].There are almost two reasons for this: 1) graphical representations require specific knowledge on the topic; 2) graphical representations often conflict with the student's models [16,26].These research outcomes are relevant also in learning motions and forces.Teacher play the crucial role to help students to read and produce images building correct conceptual models.This basis was the reference for the examination of PPTs' drawings and associated description and motivations [18].

Research questions
Our research questions are:  RQ1 -How PPTs represent the concept of force before and after the instruction? RQ2 -Which phenomenological contexts are considered useful? RQ3 -Which aspects of the nature of force are included in the representations? RQ4 -Which indication emerge for PPT formation?

Instruments and methods
To answer the research questions posed, different tools here presented were developed through a multifaceted process of comparison between pedagogy specialists and physics education researchers.

The form for data collection
A pen-and-paper Form was designed to collect the PPTs representations, structured as follows: A) Master data section; B) Request section for three representations that promote understanding of the concept of force and for each one a description of the representation and comments on the effectiveness of these representation for the educational perspective and for the disciplinary one (four panes for each representation; C) Observations and Comments free-form section.The expectation was that the PPTs representation/image and description would be consistent, complement and explain each other, distinguishing the effectiveness on the educational plane and the disciplinary one, that is on the PK and PCK [27].The request before the treatment of the topic of "forces" allows for the monitoring of conceptual representations accrued in common experience and in secondary education.The collection of representations after the instruction aims to monitor how PPTs modify they conception of force.

The grids and the methods for the analysis
The analysis of PPT's representations were performed on the basis of different grids (Appendix A and C) [18].These grids were integrated in an electronic sheet, reproduced only in a little part in figure 1.The categories were distinguished according to the criteria of qualitative research [28] based on the different meanings attributed to the representations and the various descriptions accompanying each image produced.New aspects were introduced at a posteriori and a selection of examples was performed to operationally define each category.Three disciplinary efficacy groups were identified: 1) Ontological-existential aspects of forces; 2) Identification of specific aspects/elements of the force concept; 3) Introduction of specific types of forces.For the didactic efficacy see a previous paper [18].

The research context and the sample
The analysis was carried out on 816 pre-instruction illustrations of N=274 PPTs of the University of Milan Bicocca, University of Udine, University of Urbino, as specified in Table 1.For 94 PPTs of Udine, following the course of Physics Education, we collected the representation drawn by the PPTs, at the beginning and at the end of the module on force, as presented in the next subsection.

Instruction in the formative module in Udine
After acquiring initial representations, the Udine PPTs followed a Physics Education Course, including a 4-hours module on the concept of force and relative educational approaches in literature.At the start of the module, the PPT's were required to revise their drawing, including explicit representation of forces involved.Typical elements of force were represented in at least 50 % of the representations, albeit still with significant deficiencies: vectors, more often free; single forces and not forces pairs.The formation of CKs was done by analysing the fundamental nuclei of Rubric 1 (appendix A).The construction of PCKs was done by discussing the concept of force following three different research based educational perspectives [29][30]: A) Force as formal representation of interaction; B) Force and movement (using on-line sensors); C) Force in the context of levers.Students were requested to reconstruct the grid of nuclei and illustrate each of them with an adequate representation.The analysis of these representations was done for: systems involved; number of forces (vectors) represented; types of force represented.

Type of situation
The most frequent representations (figure 2) involve common situations devoid of any formal entity (256 equal to 32%-see A1.1 in figure 3) or with abstract iconic elements, such as arrows, trajectories, without representing explicitly force vectors (242 equal to 30%-see A1.2a and A1.2b in figure 3).Representations in which specific formal elements can be identified are just under 1/3 of the total number of representations (230 equal to 28%-see A1.3 in figure 3).Less than 10% (72see A1.2c fig.2) are representations of systems that metaphorically invoke the word force (muscular body, animal or character).Model situations (such as the solar system) are present in less than 2%.Less than 1% of representation included formulas (the most frequent: P=mg), graphs, concept maps.

A3. Contexts represented
Half of the representations (419 -51%) involve a person acting on an object: hanging/pushing/pulling 79/419, lifting 67/419; throwing 43/419; hitting with a foot kick/a punch (27/419).Games and sport are largely represented, with the prevalence of Tug of war (44 -The winning team is the one that "uses the greatest force").Falling objects are often represented (112/813 -14%) and about 2/3 of these are apples falling from a three and about 1/3 the falling object is left by a hand or a person.The static situations are 30% (247 situations), often looking at the equilibrium as didactic approach to the concept of force [11,20,[30][31].The non-static situation are 61% (494 representations), confirming the connection force movement well known in literature [8,11,[32][33][34][35][36], and the remaining 9% are not situation.

Number and type of systems represented
Figure 4 shows the distribution of the number of systems represented.
More than half involve two systems (53%).In most cases the force is as an action of a body on another (Earth on a body; hand on a ball).Reciprocity of interactions is represented in 5 % of the drawing, involving magnet interactions, tug-of-war, Earth/system interaction.Representation of interactions of  ) included in the representation.Legend: A5.1 -Weigh f.; A5.2 gravitational f.; A5.3elastic f.; A 5.4 friction f.; A5.5 force acting between system in contact; A5.6electric f.; A5.7magnetic f.; A5.8 nuclear f.bodies in contact never included the interacting forces.Representation with just a body/system are 25%, in 6% including also forces acting on it, compound system, as lever or rotating systems, are 6%, continuous systems 4%, with metaphoric use of the word force 9%, prevalently from Milan and Urbino.

Types of forces represented
Figure 5 represents the distribution of type of forces represented, where the forces acting between system in contact prevail.About 30% of representations included gravity and weight force (only in 3 cases mentioned jointly) and/or friction, magnetic, elastic forces.Several PPTS included the following forces (f.), added a posteriori (see section 5 and figure 6): A5.9-Archimede f.; A.5.10-11 centrifugal and centripetal f.; A5.12 apparent f.; A5.13 Muscular/body f.; 5.14 f. possessed by a system.Figure 6.Distribution of types of forces, grouped according to different perspectives.

Force of Gravity and weight force
force of gravity is more mentioned than weight force (the word weight indicating an object has not been considered here).This is probably because the former is more unambiguously identified with an attractive force or action of a massive body (the Earth, The Sun).Weight is more often confused with a property of the body, not clearly distinguished from the concept of mass, while being aware that these concepts should be distinguished.Even when it is identified as a force weight is regarded as in conjunction gravity or as an alternative to it depending on the situation.Even in some cases it is described as a force possessed by a body, or as a "force exerted daily and unnoticed on the earth", that is on the surface of the Earth, or identified tout court with the acceleration of gravity: "the acting forces are the acceleration of gravity and the force exerted by the child" (see figure 7 for examples).

Effect of a force
More than 1/3 of representations include the effects of a force: 321/813 (39 % -MIB 51/306; UD 219/283; UR 51/224.These effects are identified in most cases with a displacement or movement (75 % of the total 321 representations).In the Udine sample the weight of this aspect is more than double of the other two samples combined.Only in a few cases, deformation (10%, frequently in connection to the mass-spring system) and acceleration (5% on 321 and 2% on 813) emerge as effect of a force.Other effects, such as attraction, also have similar frequencies (10%).The following PPT's sentences involve the different implications between force and displacement/movement: "a moving object can exert its own force"; "example where forces in a certain sense cause bodies to move"; "in physics to move or push an object a force is applied"; "to make a displacement there is a need for a force"; "through a breath it is still possible to change the position of the object by making it move", "an object that was previously stationary moves as a result of the action of a subject".These reinforce the idea of a direct connection between force and motion and the easy slide from one implication to the other, evidenced in literature [8,11,[32][33][34][35][36], as opposed to the force-acceleration connection that emerged in 2% of the cases.

Aspects related to the vector nature of force
394/813 representations (48% of the total, of which 149/306 of MIB, 168/283 of UD, 77/224 of UR) include formal iconic elements: applied vectors (20%); free vectors (36%); arrows (44%).Applied vectors acts on the centre of mass in less than half of cases.Only 12 representations included the free body diagram.Free vectors are mostly used to represent vector fields, or to represent the direction of a force, separately from the represented system (see the examples of category A1.3 in figure 3).Arrows that do not have an explicit overt role as force vectors have been classified as arrows and mostly illustrate the displacements or the velocity of a system (see the example A1.2b in figure 3).Force composition was included in 7% of the representations (with a slight prevalence for the UD sample and near absence in the UR sample).In 2/3 of the cases, it is the action of two opposing forces acting on the same line; in only 1/3 of the cases, the forces represented are noncollinear.The inclined plane emerges as the most significant context in which to illustrate the concept of vector composition most adequately.

Disciplinary effectiveness
This subsection summarises the analysis of the 515 disciplinary motivations (Out of the total 816 representations).In the majority of cases (515 motivations equal to 63%) the situation represented are an occasion for introducing specific types of forces.These include gravitational force/weight (not distinguished here) and forces acting between bodies in contact (fig.8).
In the disciplinary justifications, about ¼ of the PPTs specified that the proposed drawing was effective because it allowed them to highlight the existence of a force, rather than to make them understand what a force is.The frequency of such ontological-existential elements is summarized in Table 2.In it, the force-movement connection already highlighted above, and a certain variety of aspects emerge, denoting the fragmentary nature of PPT's initial ideas about the concept of force.Here, the marginality of understanding the reciprocity of interaction of the concept of force clearly emerges.The educational motivations related to understanding specific aspects of the force concept are equally differentiated.The vector nature of the force concept (or more precisely, individual aspects related to that nature) are present in about 25 percent of the representations, with a prevalence for the intensity of a force.The compositional aspect of forces is present especially when going to situations of equilibrium of two forces.The effect of a force is also present in about ¼ of PPTs.Finally, always present in ¼ of PPTs is the motivation to introduce the specific representation to illustrate the role of mass in the concept of force, although this is almost always evoked as a slogan rather than explained in its meaning.

Data after instruction for the Udine sample
The 88% of the representation after instruction include two systems and almost a vector representing a force (cat A4.4-A4.3).In most cases two forces appear (65% -cat A4.4).The representations with only the two interacting systems were marginal (8%) and completely negligible those who represented only one system.Forces are represented as applied vectors (73%), to the centre of mass (54%), to a surface point of the interacting systems.Only in 27% of cases are forces represented as free vectors.Newton's three laws are among the aspects related to the concept of force stressed in the representations (92%), as well as the connection between force and acceleration (85%) or force and deformation (85%).Other frequently recalled aspects are the vector composition of forces (86%), the static measurement of a force with the Mass-Spring System (79%), the force torque (59%), and the force-impulse connection (55%).Among the types of forces mentioned, those defined by physics appear in about 50-70% of cases.

Conclusion
In the framework of the Model of Educational Reconstruction [19], we developed a research on the analysis of the idea of force based on the representation of that concept realized by 274 PPTs, enrolled in the Primary Education Sciences degrees of three Italian university (Milan, Pesaro-Urbino, Udine), before the physics module.For the Udine sample, the representation were collected also after instruction.
A worksheet was design to collect representation that can help pupil of primary school to understand the concept of force.For each representation, the PPTs also had to briefly describe the image/drawing/presentation and give reasons for its educational and disciplinary effectiveness.Different grids were constructed for the analysis, focused on the foundational nuclei and of the conceptual knots/learning problems of the concept of force.Such an analysis showed that, in most cases (about 70%), the pre-instruction representations include the two systems or a single system in exam, on which any forces act (RQ3).The contextualization, not accompanied by a rigorous identification of the interacting systems and types of interaction, introduces elements (i-e-the action exerted) that can divert attention from the declared objective (the force).Situations involving forces exerted in contact between two bodies (a person pushing/pulling) are prevalent.About 1/3 of the representations involved the force of gravity, and in most cases, these are distinct from those in which the PPTs speak of the weight force or of weight as a property of systems.For many PPTs emerge the need to identify the force from the effects (about 40%).These effects are mostly associated with moving or putting a system in motion.Less than 5% of PPTs focussed the representations on the concept of interaction and on the reciprocity of the forces.Some contexts seem to emerge as more familiar and effective for grounding the concept of force as interaction as the tug-of-war and the Earth/system interaction (RQ2).Looking at the set of representations made and the related motivations, we see that they underlie different didactic perspectives on the concept of force: force in situations of equilibrium; force and motion; force declined according to the different types of forces.About 1/3 of the representations concern static situation, prevailing those in which there is an equilibrium of more forces.About 2/3 of the drawings represent situations of movement in which we can recognize two approaches to the concept of force, one that looks at force as the cause of movement and the necessary cause to put in motion a body.Finally, in the perspective of the different types of force, a unified concept of force is not identified, but rather a collection of pictures in which it is the context that determine the significance of the example (RQ1).These results aim to focus the PPTs education on the concept of force, paying particular attention to the identification of the systems involved and on the representation of the acting forces (RQ4).These outcomes allowed for an intervention that dynamically adapted to the difficulties that emerged from the pre-instruction representations.Analysis of the representations made by the Udine PPT Group revealed impact on how to represent interactions between systems with pairs of forces, force as vectors applied in the centre of mass and in focusing attention on the types of forces defined by physics (RQ4).

Figure 1 .
Figure 1.Example of the implementation of the grids in the electronic sheet for the analysis (the parts in different background shade are those added to include aspects introduced by the PPT).

Figure 4 .
Figure 4. Frequency distribution of the number and type of systems represented.

Figure 7 .
Figure 7. Example of representations involving weight force and gravitational force.