Proposal for a European Curriculum for lower secondary school teacher training
Editing a unitary proposal for a European curriculum for secondary school mathematics teacher training is not an easy task, due to the deep differences between the participant countries’ national training systems. In fact, it is enough to compare educational systems and teacher training systems of the project’s partner countries (CZ, DK, FR, IT, SK) to understand how hard is to even think about a single European curriculum. One striking aspect concerns the different ages of involved trainee teachers: in some countries teacher training starts at the same moment as university courses, while in other countries it involves either graduates or people having a higher education degree.
More promising is the analysis, carried out in the five partner countries, of two extremely relevant and interesting aspects related to mathematics teacher training:
 Lower secondary school curricula are basically similar as concerns mathematics. They generally share a long series of topics, except for some meaningful aspects. However, not the same can be said about the attainment of comparable objectives in terms of knowledge and competencies reached at the end of this school level.
 Great similarities can be found in educational methods for teacher training. Going beyond differences in levels and formative educational paths, the different systems offer a number of proposals including both traditional lessons and activities involving trainees directly within an active approach to the different topics. There seems to be a shared attempt to guide students to perceive links between mathematics and reality as well as between mathematics and other disciplines.
Starting from these remarks, it seems possible to set up a shared proposal centred at the following two aspects^{[1]}:
 A set of topics to be dealt with, embracing all the topics characterising the different Lower Secondary School curricula in each of the participant countries. This list can be the basis for training activities to be proposed to trainee teachers. When trainees show to have attained competencies related to these topics in their previous education, the trainer is allowed to focus on epistemological, historical and educational reflections. Otherwise, these topics will be explicit object of teaching.
 A meaningful method for outlining proposals, allowing preservice teachers not only to acquire (or reflect upon) the different topics, but also helping them understand the potential difficulties their pupils will meet. The aim is to outline possible interventions or teaching strategies that may help pupils overcome these difficulties and increase their understanding.
 This method, constructed on the basis of shared meaningful practices, was elaborated, experimented and documented by the Project’s Team.
The Project’s Team chose some of the topics we referred to earlier, and identified some aspects of practice they considered to be positive and effective: then they carried out teaching experiments according to modalities we will illustrate later. Good teaching practice described in the next chapters sheds a light on methodology and, at the same time, offers the opportunity to evaluate potentialities and possible limitations.
Topics to be dealt with
We mean to identify a shared set of topics to be dealt with within a Lower Secondary School mathematics teacher training course. As we mentioned earlier, the best option was to refer to the different mathematics curricula for this school level. Since teachers are obviously supposed to know the topics they will later teach, trainers will avail of a solid basis for the educational activities they will propose to trainee teachers.
As to trainee teachers’ competencies related to these topics, they need to be complete enough for trainees to master them, even when the correspondent school curriculum requires a rather superficial knowledge by pupils^{[2]}. As before, when trainees show they have attained competencies related to these topics in their past education, the trainer is allowed to focus on epistemological, historical and educational reflections. Otherwise, these topics will be explicitly taught.
It is anyway desirable to make trainees aware that different levels of competencies are required (to them and to pupils), through a metacognitive examinations of what they have learned, in view of their forthcoming teaching activity.
The Project’s Team did not meet major difficulties in choosing topics and making a list, because they referred to the topics already listed in the different national curricula: topics are actually the same ones, in the great majority of cases. In fact, when clear differences came out, the team decided to follow a criterion of “majority” and meaningfulness in the choice, taking into account opinions made explicit by components of the Project’s Team. This is also shown by the choice of activities for teaching experiments.
Major differences in curricula are those related to suggestions about ways of dealing with the different topics (respecting the different national approaches) as well as to abilities that should be attained with relation to the different topics. For this reason, we thought that a deep comparison was not appropriate and we only listed the topics, referring to the single national curricula for further details.
In the table below, for each topic there is a note highlighting the main similarities and differences. In cases where topics were dealt with at very different levels, we tried to identify those aspects that could provide a common ground for a discussion involving all the different partner countries.
Required Knowledge 
Notes 
Arithmetic
Integer numbers and operations; divisibility. Least common multiple and greatest common divisor.
Relative numbers and operations.
Fractions and operations. Decimal representation of numbers.
Rational numbers and Operations. Powers and Roots. Real numbers
Percent, ratios, proportions.
Scientific notation in powers of 10. 
This is probably the topic with less difference: curricula agree on both required knowledge and abilities. Another common aspect is the suggestion about working with calculators, which become an object of suitable educational remarks. 
Algebra
Using letters in formulas. Expressions with variables.
Linear equations and inequalities.
Examples of algebraic calculus. 
The level of both knowledge and competencies required for this topic is extremely variable in the different countries, also depending on pupils’ age. Some curricula introduce quadratic equations and inequalities and systems of equations.
However the topic is generally dealt with through links to meaningful examples of applications. 
Geometry
Point, straight line, plane. Halfline, line segment, halfplane, angle.
Circle, circumference of a circle. Polygons. Triangle. Quadrilateral.
Congruence and similarity of geometric figures. Isometry. Point symmetry. Line reflection. Translation.
Basic geometric constructions: angles, triangles, quadrilaterals, regular polygons.
Main 3D figures: polyhedron; cube; cuboid; prism; pyramid; circular cone and cylinder.
Coordinate systems: the Cartesian plane and reference systems. 
This topic is not in the Slovakian curriculum, which nevertheless considers most of the abilities listed here as already attained at previous levels. Another variable is the required level of “theoretical” competence (knowledge of definitions, classification of theorems, Thales’ or Pythagoras’ theorems…), although it is generally agreed that it should be, at least partially, promoted explicitly.
Some countries’ curriculum promotes the use of suitable software together with classical instruments, for geometrical constructions. However, this is a widespread practice, even where it is not explicitly requested. 
Functions
Functions and graphs. Linear and quadratic functions.
Direct and Inverse ratios; their representation. 

Sizes and Measurements
Measurements: meaning and computation. Units
Measures by formulas: surface of plane regular figures; side surface and volumes of some solids.
Angle sum of polygons. Length of circle. Π
Scales. 
Not all curricula explicitly list this topic, but they all require the acquisition of competencies related to measure.

Representation and organization of data
Data collection; representations and readings.
Frequencies. Bar charts, piecharts. Averages. 
This topic is not included in all curricula, but references to it can be spotted around in different sections. For this topic, it is generally recommended to refer to data related to meaningful real life examples and there is an explicit suggestion about using spreadsheets and calculators.

Problem solving
Translating from natural into formal language; Using induction, generalization, deduction. Conjecturing, discussing and proving about observations in different contexts. Examples and counterexamples.
Recognizing problems, data and goals. Formulating problems, describing procedures and giving solutions in an understandable manner, both in writing and oral.
Critically evaluating the different strategies to solve a problem. 
This topic is explicitly listed by the Italian and Danish curricula only, but it is implicitly used by many others and it was considered as particularly interesting by the Project’s Team. The topic should be dealt with in an interdisciplinary way, making links to the study of other disciplines, both scientific and linguistic/humanistic. 
Topics: similarities and differences
Method for outlining proposals
One fundamental issue for lower secondary school mathematics teacher training is that European countries may share a common teaching method for outlining proposals, providing a common basis for understanding, meeting and exchanging, beyond the differences characterising educational and training systems. This method must enable trainee teachers to acquire or reenforce their knowledge about the considered topics, but also prepare them to face possible future didactical obstacles in the classroom context.
In order to reach these aims, trainee teachers should not be presented with the various topics merely following the “transmission of knowledge” model. In this case, not only would the acquisition of knowledge related to the topics be more difficult, but also the trainee teacher might become convinced that a similar approach works with students as well, with possible serious consequences. If the trainer calls for trainees’ intellectual abilities, the outcome might be a focus on theoretical aspects of the proposal by trainees themselves and a consequent oversight of meanings linked to reality.
In actual fact, in all standards it is shown how mathematics gives tools to act, choose and decide in daily life. They promote the development of logical thought, abilities of abstraction and both bidimensional and tridimensional visualisation, using formulas, models, graphics and diagrams.
The matter is to give pupils a scientific education necessary for a consistent representation of the world and for understanding their daily environment; they have to realise that complexity can be expressed with basic laws.
The detachment between mathematics and reality in students’ perception is always a source of difficulties: at the school level we are dealing with, pupils are going through a developmental phase that leads them to the acquisition of abstract and rational thinking. Mathematical concepts are understandable as much as they are rooted in real life, by means of meaningful examples of possible applications.
The same teaching proposals offered to trainee teachers should have as many as possible common points with the activities they will actually propose to their pupils in class, including all aspects related to interdisciplinary connections and links to real life, when compatible with the topic under consideration.
Therefore, we can talk about “learning by modelling”. Trainers communicate their own conceptions of mathematics teaching by putting them into practice in the sessions they deliver. In turn, trainees are then expected to implement in their own classrooms the sessions they have experienced as pupils. Modelling strategies differ from cultural strategies (where the trainer passes on a piece of information), from demonstration strategies (where the trainer transmits a teaching practice by implementing it effectively in his/her classroom) and from transfer strategies (where the trainer transmits referential knowledge about teaching and tries to harness the transfer phenomenon carried out by the trainees).
In this way the trainee teacher is not only led to reconsider mathematical concepts in depth, increasing his/her theoretical understanding and appreciating their relevance: he/she also has the opportunity to experience, at least partially, critical points, obstacles and solutions that are likely to come up in their future classroombased activities.
A collective discussion should follow the phase involving the acquisition of knowledge of the topic. In this way trainees are allowed to share opinions, difficulties and discoveries and to outline meaningful teaching strategies and concrete ideas about the examined topic.
A subsequent phase must test what trainees discovered: the same activity is proposed in one or more pilot classes, reporting on pupils’ reactions as well as on successful or unsuccessful learning, possibly making use of video recordings. The final step is a collective discussion where, under the trainer’s guidance, trainees start from the teaching experiments carried out in pilot classes to compare hypotheses, reflections and initial teaching choices with what came out from piloting: this offers trainee teachers the chance to systematise the discoveries made throughout their work.
^{1} The word curriculum’s etymology includes two aspects: the proposal’s content and the instrument (currus) that enables one to both propose and make the content accessible.
^{2} For instance, it is fundamental for trainee teachers to be able to deal with and solve equations and inequalities of first and second degree, whatever the curriculum’s request about this topic. Therefore trainee teachers will practice with symbolic calculations and will be able to better “master” problems, due to their abilities in approaching them algebraically. Also, it seems advisable that the main plane geometry theorems be dealt with, (for instance Thales’, Euclid’s and Pythagoras’ theorems), although the curriculum might include only some or any at all: this will enable us to work on the concept of theorem and its relevance.