Modelling in Casyopee (alien)

Identity show tooltip helpexplode

Authors show tooltip help

M. Maracci, M.A. Mariotti

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • algebraic model of geometrical problems

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

If the PP is intended as part of an experimental design keywords could also include elements of the theoretical framework.

  • Geometry
  • Modelling
  • Algebra
  • Variable
  • Calculus
  • Function

Description show tooltip help

The PP is meant to exploit the semiotic potential of Casyopée to consolidate the notions of variable and function, through modelling activities where geometrical problems are treated with the mathematical tools of calculus. A critical reflection on the modelling process itself will be in focus as a general didactical objective.

Rationale show tooltip helpexplode

This Pedagogical Plan is centred on the use of Casyopée used as a tool for driving the modelling process of given geometrical situations.

The analysis of the semiotic potential (see Theoretical Framework) of the tool lead us to formulate the following hypothesis underpinning the Plan. Casyopée can provide a tool for mediating both: 

  1. the mathematical meaning of function as co-variation and thus consolidate (or enrich) the meanings of function that students have already appropriated;
  2. and the mathematical meaning related to the processes characterizing the algebraic modelling of geometrical situation.

These two aspects are strictly related: in fact the modelling process experienced in Casyopée should give meaning to the notion of function emerged from the process itself. The main innovative element of the Plan is the intent of constructing the idea of function as a result of the modelling experience.

Consistently with the Theory of Semiotic Mediation which inspires the design of the Plan, the teaching sequence will be structured as an iteration of didactical cycles, constituted by the following semiotic activities: activities with the artefacts, individual production of signs, collective production of signs.

Theoretical framework show tooltip help

The main theoretical framework which inspires the design of our Plan is the theory of Semiotic Mediation, drawing from a Vygotskijan paradigm. According to this theory, the use of artefacts for accomplishing a task leads the individual to the generation of personal meanings which are related to the actual use of the artefact. On the other hand, mathematical meanings may be related to the artefact and its use: in particular, in classroom context under the guidance of an expert (i.e. the teacher) the personal meanings, related to the actual use of an artefact, may be generated and evolve towards mathematical meanings. Because of this potentiality, we will say that an artefact may function as a semiotic mediator.

The semiotic mediation function of an artefact is not automatically activated; in fact such a function can be exploited by the expert (in particular the teacher) who has the awareness of the semiotic potential of the artefact both in terms of mathematical meanings and in terms of personal meanings. Such evolution is fostered by the teacher's action, guiding the process of production and evolution of signs centred on the use of an artefact.

According to the semiotic mediation theory, the complex semiotic processes of creation and evolution of personal meanings towards mathematical meanings can be developed through the design and implementation of a so called didactical cycle, that is an iterative cycle of the following kinds of activities: Activities with the artefacts, Individual production of signs, Collective production of signs.

The analysis of the semiotic potential of the artefact encompass the analysis of both the personal and mathematical meanings related to the artefact, as well as of the possible tasks which can be accomplished with the artefact. Such analysis lead us to formulate the hypothesis  that Casyopée can provide a tool for mediating both: 

  1. the meaning of function as co-variation and thus consolidate (or enrich) the meanings ot function that students have already constructed;
  2. and the meaning of the processes characterizing the algebraic modelling of geometrical situation.

In order to better specify both these hypotheses and and the meanings which can arise from the experience in Casyopee, an epistemological analysis of the modelling process is needed. Anyway, the main processes which constitute the algebraic modelling of geomterical situation seem to be:

  • recognizing the geometrical variables
  • associating numbers (numerical variables) to geometrical variables
  • associating geometrical variables (points) to numbers (numerical variables)
  • passing from moving points to mesaurable geometrical objects
  • coping with the possible too high number of geometrical variables: parametrizing
  • expressing the relation between numerical variables through formulas

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The identification of the educational goals of the Plan comes from an analysis of the semiotic potential of Casyopee which leads us to hypothesize that the use of Casyopee could mediate (in the sense of Semiotic Mediation) the meanings of function as co-variation and mathematical meanings related to the algebraic modelling of geometrical situation. 

Given Italian National Curricula and School Organization, Scientific High School Grades 12-13 seem the suitable context where aspects related to algebraic modelling of geometical situations can be introduced and dealt with. 

Theoretical framework show tooltip help

The general theoretical frame is provided by the Theory of Semiotic Mediation, already described. More specifically, the identification of educational goals is made through the analysis of the semiotic potential the artefact. The analysis of the semiotic potential of an artefact encompass the analysis of both the personal and mathematical meanings related to the artefact, as well as of the possible tasks which can be accomplished with the artefact.

Population show tooltip helpexplode

School level show tooltip help

Scientific High School, grades 12-13

Age range show tooltip help

16-18 years old

Population description show tooltip help

regular high school class

Student prerequisites show tooltip help

This PP is designed for students who already have had some formal teaching on Euclidean and Cartesian plan geometry and function. As for function, prerequisites are very basic:

  • functions as expressed by elementary formulas (e.g. Linear or quadratic) 
  • functions as expressed by a graph.

For developing the second part of the PP (Didactical cycle 2), some formal knowledge on "parameters" are needed too.

Teacher prerequisites show tooltip help

Familiarity with Casyopée and generally speaking with managing the integration of digital tools in classroom practice.

Expertise in collective discussion according the semiotic mediation theoretical frame.

Context show tooltip helpexplode

Physical context show tooltip help

Computer lab with at least one computer for each two students and a data projector.

 

Institutional context show tooltip help

Link to the Italian Ministry of Education, School Regulations (IT Online available June 4, 2007).

 

Goals show tooltip helpexplode

Curricular goals show tooltip help

The development of modelling competencies is a general didactical objective of the teaching of all the scientific discipline and of mathematics in particular.

More specifically, according to the PNI (Piano Nazionale per l'introduzione dell'Informatica) (IT, Online available June 5, 2007) curricula:

  • among the aims of the teaching of mathematics there is the development of the capability to use mathematical methods, tools and models in different situations
  • The introduction of calculus concepts has to be accompanied by a wide spectrum of possible applications in both mathematical and extra-mathematical fields

According to the curricula elaborated by the Brocca Commission (IT, Online available June 5, 2007) mathematics with its languages and models [...] represents a tool with a very high educational value. Among the aims of the teaching of mathematics there are:

  • the development of the capability to use mathematical methods, tools and models in different situations
  • dealing with problematic situations of different kind by means of mathematical models
  • provide pupils with rigorous methods of analysis, compencies  for modelling (even) complex situations

The introduction of calculus concepts has to be accompanied by a wide spectrum of possible applications in both mathematical and extra-mathematical fields

Content-epistemological goals show tooltip help

The main goals of this Pedagogical Plan is to foster the evolution of students’ personal meanings towards:

  1. the meaning of function as co-variation and thus consolidate (or enrich) the meanings ot function they have already appropriated;
  2. mathematical meanings related to the processes characterizing the algebraic modelling of geometrical situation.

More specifically,
as for the notion of function, students should consolidate or enrich:

  • the meaning of variables both geometrical and numerical,
  • the meaning of domain of a variable,
  • the meaning of function as co-variation over time of variables (even of different kind: numerical or geometrical),
  • competencies related to the passage between different representations of function (at least, algebraic and graphical ones)

as for the modelling process, students should learn to:

  • recognize geometrical variables
  • pass from unmeasurable geometrical objects (e.g. points) to measurable geometrical objects
  • associate numbers (numerical variables) to geometrical variables
  • associate geometrical variables to numbers (numerical variables)
  • express the relation between numerical variables through formulas
  • parametrize (optimize the number of variables) 

Instrumental goals show tooltip help

Use of the tools provided by Casyopee: 

  • to solve ("even complex") geometrical problems which can be approched via an algebraic modelling;
  • eventually to explore functions or families of functions. 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

The Plan is organized in two different didacticl cycles, each involving students in different kinds of semiotic activities: pair task accomplishing with Casyopee, individual writing of reports on the actvity accomplished and collective discussion based on the given tasks and the students' reports on them.

Actvities in both the didactical cycles will be centred on the use of the features of Casyopee described below, with the exception of the feature "create parameter" which will be in focus only in the latter cycle.

Theoretical framework show tooltip help

The Plan design is inspired to the construct of didactical cycle dheoy of Semiotic Mediation.

According to this theory, the complex semiotic processes of creation and evolution of personal meanings towards mathematical meanings can be developed through the design and implementation of a so called didactical cycle, that is an iterative cycle of the following kinds of activities: Activities with the artefacts, Individual production of signs, Collective production of signs.

Casyopée release September 07

Casyopée is constituted by two main environments which can “communicate” and “interact” between them: an Algebraic Environment and a Dynamic Geometry Environment (though the designers’ objective was not to develop a complete CAS or a whole complete geometric dynamic environment). Possible interactions between the two environments are supported through a third environment, the  “Geometric Calculation”.

Tool access show tooltip help

NIL

Main environment [Component]

The "main environment" of Casyopee offers functionalities for exploring and justifying properties of real functions in one-variable. It includes graphical, numeric and symbolic tools.

 

Graph of a function [Component Feature]

In Casyopée main window the graph of a function can be displayed and explored in several usual. An important features concerns the graphical representation of the graph of functions defined from the geometrical window.

If a function is defined from the geometrical environment, both the algebriac independent variable and the algebraic dependent variable depend on the position of a free bonded point in the geometrical environment: the values of the algebraic independent variable and of the algebraic dependent variables correpsond to the specific position of that point.

In the graphical representation of the graph of such a function, a dotted line displays the values of the independent variable and of the dependent variables which correspond to the actual position of the starting geometrical free bonded point.

 

Parameters [Component Feature]

In Casyopée main window, parameters can be created and manipulated.

Parameters can be created through the command "create parametrs". When creating a parameter the user has to assign a range and a step for defining its variability. Once a parameter is created, a slider for changing the value of the parameter appears.

Parameters can be exported in a sense in the geomtrical window. In fact one can create, for instance, points whose coordinates depend on parameters. In tha case, when changing the values of the parameters, the poision of the point changes accordingly.

 

Dynamic Geometry [Component]

Casyopee includes a Dynamic Geometry environment which can be activated by clicking the buttion "Dynamic Geometry" within Casyopee main window toolbar.

This environment offers some elementary functionalities usual in DGEs and functionalities that support the algebraic modelling of geometrical situation (including functionalities to export the created functions and numerical values in the "main environment" of casyopee)

 

Geometric Calculation [Component]

This environment supports the construction of an algebraic model of a "geometrical situation" reproduced in the Dynamic Geometry Environment. Within this environment one can to associate geometrical variables with numerical variables and define functions between such numerical variables.

Within the rationale of the DDA functioning, in order to build an algebraic model one has to define the desired dependent variable (trhough the command "create calculation"), the desired independent variable (throught the command "choose variable") and then the function linking them (through the command "create function"). These commande will be described below.

The whole environment could be used to mediate the meanings of variable (both geometrical and numerical), of function (in particular of "mixed" function linking geometrical and numerical variables) and the processes of associating numerical variables to geometrical ones.

 

Create Calculation [Component Feature]

This sub-environment is activated by clicking a button in the Geometric Calculation environemnt. 

It allows users to create "calculation" associated with geometrical objects created in the Dynamic Geometry window.  One can create  the distance between two poins (e.g. d(A,B)), or the lenght of a segment (e.g. AB), or the coordinates of a point  (e.g. x_A and x_B), and compound them with algebraic and elementary calculus functions (sum, product, goniometrical functions, exponential function,...).

The formal expressions of the created calculations are displayed in the Geometric Calculation environment. If one selects an expression, the corresponding current value is displayed too, and it is automatically updated when dragging objects in the Dynamic Geometry window.

This environment could be used to mediate the meanings of variable (both geometrical and numerical), of function ("mixed" function linking geometrical and numerical variables) and the processes of recognizing geometrical variables, distinguishing measurable and not-measurable geometrical variables, associating them with numerical values...

 

Choose Variable [Component Feature]

This command supports users to define three possible numerical variables which can be associated with a chosen couple of points created in the Dynamic Geometry Environment (including points o,i,j created by default by the system).
Within the rationale of the DDA functioning, the chosen variable is the intended independent variable of the function which the user is going to define.

Users can try to define variables either as distance between two points, or as difference of the x-coordinates of two points or as difference of the y-coordinates of two points.

An important aspect is that users cannot arbitrarily choose variables associated with any couple of points. In fact, when trying to define a variable associated to a chosen couple of points the user can receive messages like the following ones:
1. The variable depends on no free points (if the user chooses two fixed points).
2. The variable depends on many free points (this happens for instance when the user selects the distance between two points, one of which is a free point on the whole plane).
3. The variable depends on A (being A the label of the free point selected), not univocal dependence (this happens for instance when the user selects the distance between two points, one of which, A, is bonded on a whole straight line and the other is a fixed point)
4. The variable depends on A (being A the label of the selcted free point), it is defined upon [0,5] (this happens for instance when the user selects the distance between two points, one of which, A, is bonded on a segment 5-units long, and the other one is an extreme of the segment).
In the first three cases the variable cannot be defined. In the last case the command "validate variable" apperas and has to be clicked on to confirm the choice of the variable.

This feature could be used to mediate the meanings of variable (both geometrical and numerical), of function ("mixed" function linking geometrical and numerical variables) and the processes of associating numerical variables to geometrical ones, as well as to raise questions concerning the kind of relationship should hold between variables in order to be allowed to define a function.

Validate Variable [Component Feature]

 This command is clickable only when an admissible variable is chosen (see the feature "choose variable"), and it has to be clicked on to to confirm the choice of the variable.

Once a variable is validated and a geometrical calculation is created and selected, the system displays the following two strings:

    * <formal definition of the chosen variable> <arrow> <formal definition of the geometrical calculation created>,

    * <numerical values of the chosen variable> <arrow> <numerical value of the geometrical calculation created>.

The numerical values of both the chosen variable and the geometrc calculation created are displayed and automatically refreshed when free points in the Geometrical Window are dragged.

This command and the way inteface is organized can be used to mediated the meaning of function as co-variation.

Create a Function [Component Feature]

Once a geometrical calculation is created and selected and a variable is validated, the button "create a function" appears.

If one clicks on the button, the system checks whether there holds a functional relationship between the variable chosen and a geometrical calculation selected.

If such a relationship holds, a pop-up window is opened, where the following string is diplayed: <single letter denoting the variable chosen> <arrow> <algebraic expression of the geometrical calculation as function of the chosen variable>, in addition the domain of the choosen (independent) variable is shown and can be possibly modified.

If the "Creation of the function" is confirmed ("ok" button) the algebraic function constructed is exported in Casyopee main window.

This commands could be especially used as means to raise questions about the construction or interpretation of algebraic expressions linking (independent and dependent) variables.

 

Work plan show tooltip helpexplode

Setting show tooltip help

Activities will be performed either in computer labs or in usual classrooms, depending on the kind of the activities themselves.

Computer lab should be equipped with (at least): 

  • One computer each two students. 

  • Data projector

  • Black or whiteboard

 It woul be helpful to have one computer and one data projector also in the classroom.

Time show tooltip help

11 hours

Actors' roles show tooltip help

Actors'roles change over the different sessions which compound this PP.

What to do and how show tooltip help

The Plan is organized in two different didacticl cycles each involving students in different kind of semiotic activities: pair task accomplishing with Casyopee, individual writing of reports on the actvity accomplished and collective discussion based on the given tasks and the students' reports on them.

During the lab sessions students should work in pairs or in small groups (no student should work alone), each group working with a computer. The composition of groups may be left to students. As far as possible the composition of the groups should not change throughout all the PP

Process documentation show tooltip help

All the resources produced by students (detailed in the description of the PP "leaves") are also tools for documenting the process.

In addition screen capturer softwares could be used for recording samples of the activity within Casyopée. However such screen capturer softwares may slow down the performance of the computers hindering the correct functioning of Casyopée.

The classroom discussions could be also documented through audio or (better) video recordings.

 

Didactical cycle 1

Identity show tooltip helpexplode

Authors show tooltip help

M. Maracci, M.A. Mariotti

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • algebraic model of geometrical problems

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • Geometry
  • Modelling
  • Algebra
  • Variable
  • Calculus
  • Function

Description show tooltip help

The PP is conceived as a didactical cycle involving pupils in different kinds of activities: familiarization with the tool, problem solving through modelling supported by the tool, production of written solution and reports concerning the classroom activities, and collective discussion of results.

These semiotic activities are aimed to:

  • consolidate (enrich) pupils' meanings of the notions of variable and function, and 
  • initiate the development of modelling competencies as well as a critical reflection on the modelling process itself.

The PP has an iterative structure: the same issues can be dealt with in different moments and with different level of depth.

Rationale show tooltip helpexplode

This Pedagogical Plan is centred on the use of Casyopée used as a tool for driving the modelling process of given geometrical situations.

The analysis of the semiotic potential (see Theoretical Framework) of the tool lead us to formulate the following hypothesis underpinning the Plan. Casyopée can provide a tool for mediating both: 

  1. the mathematical meaning of function as co-variation and thus consolidate (or enrich) the meanings of function that students have already appropriated;
  2. and the mathematical meaning related to the processes characterizing the algebraic modelling of geometrical situation.

These two aspects are strictly related: in fact the modelling process experienced in Casyopée should give meaning to the notion of function emerged from the process itself. The main innovative element of the Plan is the intent of constructing the idea of function as a result of the modelling experience.

Consistently with the Theory of Semiotic Mediation which inspires the design of the Plan, the teaching sequence will be structured as an iteration of didactical cycles, constituted by the following semiotic activities: activities with the artefacts, individual production of signs, collective production of signs.

Theoretical framework show tooltip help

According to the Theory of Semiotic Mediation, the structure of a teaching sequence may be outlined as an iteration of a cycle where different kinds of activities take place, aimed to develop the complex semiotic process described above. Consistently this Plan is organized as a whole didactical cycle:

activities with the artefact: "familiarization", "problem solving"; at the end of each of which students are asked to individually write reports on the activites themselves (individual production of signs)

a class discussion starting form the given tasks and the reports written by students (collective production of signs)

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The identification of the educational goals of the Plan comes from an analysis of the semiotic potential of Casyopee which leads us to hypothesize that the use of Casyopee could mediate (in the sense of Semiotic Mediation) the meanings of function as co-variation and mathematical meanings related to the algebraic modelling of geometrical situation. 

Given Italian National Curricula and School Organization, Scientific High School Grades 12-13 seem the suitable context where aspects related to algebraic modelling of geometical situations can be introduced and dealt with. 

Theoretical framework show tooltip help

The general theoretical frame is provided by the Theory of Semiotic Mediation, already described. More specifically, the identification of educational goals is made through the analysis of the semiotic potential the artefact. The analysis of the semiotic potential of an artefact encompass the analysis of both the personal and mathematical meanings related to the artefact, as well as of the possible tasks which can be accomplished with the artefact.

Population show tooltip helpexplode

School level show tooltip help

Scientific High School, grades 12-13

Age range show tooltip help

16-18 years old

Population description show tooltip help

regular high school class

Student prerequisites show tooltip help

This PP is designed for students who already have had some formal teaching on Euclidean and Cartesian plan geometry and function. As for function, prerequisites are very basic:

  • functions as expressed by elementary formulas (e.g. Linear or quadratic) 
  • functions as expressed by a graph.

For developing the second part of the PP (Didactical cycle 2), some formal knowledge on "parameters" are needed too.

Teacher prerequisites show tooltip help

Familiarity with Casyopée and generally speaking with managing the integration of digital tools in classroom practice.

Expertise in collective discussion according the semiotic mediation theoretical frame.

Context show tooltip helpexplode

Physical context show tooltip help

Computer lab with at least one computer for each two students and a data projector.

 

Institutional context show tooltip help

Link to the Italian Ministry of Education, School Regulations (IT Online available June 4, 2007).

 

Goals show tooltip helpexplode

Curricular goals show tooltip help

The development of modelling competencies is a general didactical objective of the teaching of all the scientific discipline and of mathematics in particular.

More specifically, according to the PNI (Piano Nazionale per l'introduzione dell'Informatica) (IT, Online available June 5, 2007) curricula:

  • among the aims of the teaching of mathematics there is the development of the capability to use mathematical methods, tools and models in different situations
  • The introduction of calculus concepts has to be accompanied by a wide spectrum of possible applications in both mathematical and extra-mathematical fields

According to the curricula elaborated by the Brocca Commission (IT, Online available June 5, 2007) mathematics with its languages and models [...] represents a tool with a very high educational value. Among the aims of the teaching of mathematics there are:

  • the development of the capability to use mathematical methods, tools and models in different situations
  • dealing with problematic situations of different kind by means of mathematical models
  • provide pupils with rigorous methods of analysis, compencies  for modelling (even) complex situations

The introduction of calculus concepts has to be accompanied by a wide spectrum of possible applications in both mathematical and extra-mathematical fields

Content-epistemological goals show tooltip help

The content of this section is inherited from the parental section, in fact all these goals (with the exception of developing parametrizing competencies) are still in focus and will be in focus also in the child nodes of this HIPP (with the exception of the session "familiarization")

In fact the meaning of function as co-variation, or the meaning of modelling process should continously develop and enrich through the different activities. 

On the other hand, because the structure of this PP, one can hopefully attest  the achievement of these goals at the end of the whole didactical cycle.

The main goals of this Pedagogical Plan is to foster the evolution of students’ personal meanings towards:

  1. the meaning of function as co-variation and thus consolidate (or enrich) the meanings ot function they have already appropriated;
  2. mathematical meanings related to the processes characterizing the algebraic modelling of geometrical situation.

More specifically,
as for the notion of function, students should consolidate or enrich:

  • the meaning of variables both geometrical and numerical,
  • the meaning of domain of a variable,
  • the meaning of function as co-variation over time of variables (even of different kind: numerical or geometrical),
  • competencies related to the passage between different representations of function (at least, algebraic and graphical ones)

as for the modelling process, students should learn to:

  • recognize geometrical variables
  • pass from unmeasurable geometrical objects (e.g. points) to measurable geometrical objects
  • associate numbers (numerical variables) to geometrical variables
  • associate geometrical variables to numbers (numerical variables)
  • express the relation between numerical variables through formulas
  • parametrize (optimize the number of variables) 

Instrumental goals show tooltip help

Use of the tools provided by Casyopee: 

  • to solve ("even complex") geometrical problems which can be approched via an algebraic modelling;
  • eventually to explore functions or families of functions. 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

 

Work plan show tooltip helpexplode

Setting show tooltip help

Activities will be performed either in computer labs or in usual classrooms, depending on the kind of the activities themselves.

Computer lab should be equipped with (at least): 

  • One computer each two students. 

  • Data projector

  • Black or whiteboard

 It woul be helpful to have one computer and one data projector also in the classroom.

Time show tooltip help

8 hours

Actors' roles show tooltip help

Actors'roles change over the different sessions which compound this PP.

What to do and how show tooltip help

The activities which constitute this didactical cycle are conceived to be proposed in the given specific order: Familiarization, Optimization Problem 1, Discussion 1, Optimization Problem 2, and Discussion 2.

As for time-schedule, 2 hours for each lab session and 1 hour for each discussion seem reasonable, but obviously some deviations are possible.

Process documentation show tooltip help

All the resources produced by students (detailed in the description of the PP "leaves") are also tools for documenting the process.

In addition screen capturer softwares could be used for recording samples of the activity within Casyopée. However such screen capturer softwares may slow down the performance of the computers hindering the correct functioning of Casyopée.

The classroom discussions could be also documented through audio or (better) video recordings.

 

familiarization

Identity show tooltip helpexplode

Authors show tooltip help

M. Maracci, M.A. Mariotti

Subject domains show tooltip help

  • algebraic modelling of geometrical situations

Topics show tooltip help

  • Casyopee tools

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

The keywords below refer to those commands or environment of Casyopee which will be explored in this session and which according to our hypothesis are susceptible to mediate (a) the meaning of function as co-variation and (b) the modelling processes.

 

  • Create Geometric Calculation
  • Choose variable
  • Validate variable
  • Create Function
  • Feedbacks

Description show tooltip help

This session is meant to drive pupils towards the "discovery" of Casyopee Dynamic Geometry commands and their effects in a twofold sense:

  • the effect on the object upon which the command directly acts,
  • the effect on the environment, e.g. what commands are activated, what  ones are de-activated...

The focus is on those commands which will be used by the teacher to mediate (a) the meaning of function as co-variation and (b) the process of algebraic modelling.

Rationale show tooltip helpexplode

This session has a twofold aim: on the one hand it is meant to support students in the discovery of Casyopee Dynamic Geometry commands an their functioning, on the other hand it aims to involve students in semiotic activities related to the use of the artefact.

Also the familiarization with the DDA has to be considered in the frame of Semiotic Mediation: the aim is to make students observe and reflect upon the "effects" of their interaction with the DDA itself. Ad hoc tasks are designed for that purpose.

Attention is especially focused on those commands and subenvironments which will be used by the teacher as tools of semiotic mediation, in particular the fous is on the environment "Geometric Calculation" and on its commands: 

  • Create Calculation
  • Choose variable
  • Validate variable
  • Create a Function

In addition, pupils' attention will be drawn to informative messages provided by Casyopee. In fact we hypothesize that such messages can play a crucial role in mediating both the meaning of function, variable and of modelling process, but that at the same time they could be too easily ignored by the user.

Theoretical framework show tooltip help

In this session  the instrumentalization and instrumentation processes start. The former is directed towards the artefact and encompasses for instance the discovery and selection of the functions of the artefact. Whereas the latter is directed towards the user and is defined by her/his appropriation of schemes in order to accompish specific kinds of task.

The development of utilization schemes is but a part of the story, in fact when accomplishing a task (in pair or small group) with an artefact students are also be involved in semiotic activities.

Accordingly this session is also designed to engage students in semiotic activities related to the use with the artefact. In fact this session is designed as the first phase of a didactical cycle, as such it encompasses:

Activities with artefacts: students are faced with tasks to be carried out with the artefact. This type of activities is thought to promote the emergence of specific signs in relation to the particular use of the artefacts; in fact working in pairs fosters social exchange, accompanied by words, sketches, gestures, and the like. 

Individual production of signs (e.g. drawing, writing and the like): students are individually engaged in different semiotic activities, mainly concerning written productions.  This type of activity differs for instance from social exchanges  during the activity with the artefact, because it requires a personal contribution in order to produce written text and consequently graphic signs, which for its very nature starts to be detached from the contingency of the situated action.

These activities should promote the production of artefact-signs susceptible, because of the semiotic potential of the artefact, to evolve towards mathematical signs under the guidance of the teacher. In this senseSemiotic Mediation in the Mathematics Classroom the teacher acts as mediator using the artefact to mediate mathematical content to the students, that is the teacher uses the artefact as a tool of semiotic mediation

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

School level show tooltip help

Scientific High School, grades 12-13

Age range show tooltip help

16-18 years old

Population description show tooltip help

regular high school class

Student prerequisites show tooltip help

This PP is designed for students who already have had some formal teaching on Euclidean and Cartesian plan geometry and function. As for function, prerequisites are very basic:

  • functions as expressed by elementary formulas (e.g. Linear or quadratic) 
  • functions as expressed by a graph.

For developing the second part of the PP (Didactical cycle 2), some formal knowledge on "parameters" are needed too.

Teacher prerequisites show tooltip help

Familiarity with Casyopée and generally speaking with managing the integration of digital tools in classroom practice.

Expertise in collective discussion according the semiotic mediation theoretical frame.

Context show tooltip helpexplode

Physical context show tooltip help

Computer lab with at least one computer for each two students and a data projector.

 

Institutional context show tooltip help

Link to the Italian Ministry of Education, School Regulations (IT Online available June 4, 2007).

 

Socio-cultural context show tooltip help

Goals show tooltip helpexplode

Instrumental goals show tooltip help

Students should discover and begin to develop the utilization schemes of the commands of Casyopee Dynamic Geometry environment, though related to very specific activities.

More in details the focus is on the following features:

  • "create geometric calculation",
  • "choose variable",
  • "validate variable",
  • "create function"

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

The familiarization of pupils with Casyopee Dynamic Geometry environment is driven by a strongly structured worksheet which requires pupils to activate specific Casyopee commands, to observe their effects and report on these effects. 

The attention is specifically drawn on those tools (features of Casyopée) which according to our hypotheses can function as tools of semiotic mediation.

This task demands also an intense activity of observation and verbalization: observation of the different kind of feedbacks produced by casyopée, what messages are given and in general how the interface changes dependently on the action of the user.

The request of verbalization (firstly writing down the answers to the questions posed in Worksheet 1 and then reporting on the whole activity) is aimed to:

  • promote students' production of signs specifically related to the task;
  • "oblige" students to take into account feedbacks - such as "tutoring feedbacks" - which de facto could be ignored when working with Casyopee.

Resources show tooltip helpexplode

Familiarization Worksheet [Resource for students]

Students are presented with a written text including different activities to be accomplished  with Casyopée (in the given order). Such activities are meant on the one hand to guide students to familiarize with some commands  of the "Dinamic Geometry" environment, on the other hand to drive students' attention on the different feedbacks provided by Casyopée. In particular students are required to pay attention to:

  • the different kinds of "behaviour" of the points in the geometrical environment with respect to dragging;
  • the functioning of the"choice variable" environment, in particular to the kind of variable which can be defined, and to the kind of "informative messages" produced by the tool;
  • the functioning of the "create function" command.
Report1_request [Resource for students]

As homework, pupils are asked to individually write a report on the class activity, based on the following questions. 

Resource contents show tooltip help

Report

Re-consider the tasks accomplished with Casyopée, what did you understand about Casyopée functioning?

In your opinion, what could Casyopée help? What impressed you most (if anything)?

download .pdf

Familiarization Worksheet filled [Resource produced by students]

Students are required to accomplish the different tasks described in the Familiarization Worksheet and to solve them by written.

Report1 [Resource produced by students]

Individual reports written by students on the basis of the questions posed

Casyopee_files1 [Resource produced by students]

Students are also asked to save their work with Casyopee. The solution is saved as a .txt file which casyopee can interpret so to display the last appearance of the interface: neither a replay system is available nor the object erased by students can be displayed.

Work plan show tooltip helpexplode

Setting show tooltip help

Activities has to be performed either in a computer lab, which should be equipped with (at least): 

  • One computer each two students. 

  • Data projector

  • Black or whiteboard

 

Time show tooltip help

2 hours

Actors' roles show tooltip help

Students work in pairs all along the session, students are free to choose their own mate, they are required to accomplish the tasks described in Familiarization Worksheet, and answer by written the questions that it contains. Those who complete the Worksheet can (begin to) write the report basing on the question in Report1_request. Students who do not succeed in completing the Familiarization Worksheet are asked to produce their report at home. 

The teacher supervises the work of pupils with the aims of helping pupils to overcome possible empasse, of ensuring that everyone can proceed in the work, of ensuring that pupils do not miss some tasks and do not forget of producing answers to the posed questions. 

Although it is not explicitly foreseen, a brief discussion between teacher and pupils can take place if needed, in this case a data projector can be useful. Anyway if relevant questions are dealt with in this session, they should be reprised and systematized in the session Discussion

What to do and how show tooltip help

During the lab sessions students should work in pairs or in small groups (no student should work alone), each group working with a computer. The composition of groups may be left to students. As far as possible the composition of the groups should not change throughout all the PP

The teacher distributes the Familiarization Worksheet among students. If needed she/he could show how launching Casyopee by means of a data projector connected to a computer.

The teacher supervises the work of pupils with the aims of helping pupils to overcome possible empasse, of ensuring that everyone can proceed in the work, of ensuring that pupils do not miss some tasks and do not forget of producing answers to the posed questions. 

If some students complete the worksheet, the teachers ask them to begin to produce a written  report on the basis of Report_request.

The aim of the session is not that pupils develop a complete mastery of the functioning and meanings of Casyopee commands. Some brief discussion can take place if needed.

Process documentation show tooltip help

All the resources produced by students (see above) are also tools for documenting the process.
In addition screen capturer softwares could be used for recording samples of the activity within Casyopée. However such screen capturer softwares may slow down the performance of the computers hindering the correct functioning of Casyopée.

Optimization Problem 1

Identity show tooltip helpexplode

Authors show tooltip help

M. Maracci, M.A. Mariotti

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • algebraic model of geometrical problems

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • Geometry
  • Modelling
  • Algebra
  • Variable
  • Calculus
  • Function

Description show tooltip help

This PP is constituted by a problem-solving session, in which pupils are presented with an optimization problem in geometry. Optimization problems in geometry are quite traditional in  the last grades of italian scientific high schools, because they provide a context of application for calculus notions and tools.

The focus is on the modelling process which allows to treat geometrical problems with calculus tools rather than on calculus tools themselves.

Rationale show tooltip helpexplode

This session is part of a didactical cycle, it is especially meant to engage students with the following kind of semiotic activities: pair task accomplishing with the tool and individual writing of reports on the former activity.

More specifically students are asked to solve geometrical optimization problem. Optimization problems in geometry are quite traditional in  the last grades of scientific high schools, because they provide a context of application for calculus notions and tools.

The algebriac modelling of the given geometrical problems is supported by the tool.

Theoretical framework show tooltip help

 

This session is designed as part of the first phase of a didactical cycle, as such it encompasses:

Activities with artefacts: students are faced with tasks to be carried out with the artefact. This type of activities is thought to promote the emergence of specific signs in relation to the particular use of the artefacts; in fact working in pairs fosters social exchange, accompanied by words, sketches, gestures, and the like. 

Individual production of signs (e.g. drawing, writing and the like): students are individually engaged in different semiotic activities, mainly concerning written productions.  This type of activity differs for instance from social exchanges  during the activity with the artefact, because it requires a personal contribution in order to produce written text and consequently graphic signs, which for its very nature starts to be detached from the contingency of the situated action.

These activities should promote the production of artefact-signs susceptible, because of the semiotic potential of the artefact, to evolve towards mathematical signs under the guidance of the teacher. In this sense the teacher acts as mediator using the artefact to mediate mathematical content to the students, that is the teacher uses the artefact as a tool of semiotic mediation.

Target show tooltip helpexplode

Rationale show tooltip helpexplode

In this session students are asked to solve geometrical optimization problem - a quite usual kind of problems in italian curricula for scientific high school. The algebriac modelling of the given geometrical problems is supported by the tool.

Population show tooltip helpexplode

School level show tooltip help

Scientific High School, grades 12-13

Age range show tooltip help

16-18 years old

Population description show tooltip help

regular high school class

Student prerequisites show tooltip help

This PP is designed for students who already have had some formal teaching on Euclidean and Cartesian plan geometry and function. As for function, prerequisites are very basic:

  • functions as expressed by elementary formulas (e.g. Linear or quadratic) 
  • functions as expressed by a graph.

For developing the second part of the PP (Didactical cycle 2), some formal knowledge on "parameters" are needed too.

Teacher prerequisites show tooltip help

Familiarity with Casyopée and generally speaking with managing the integration of digital tools in classroom practice.

Expertise in collective discussion according the semiotic mediation theoretical frame.

Context show tooltip helpexplode

Physical context show tooltip help

Computer lab with at least one computer for each two students and a data projector.

 

Institutional context show tooltip help

Link to the Italian Ministry of Education, School Regulations (IT Online available June 4, 2007).

 

Goals show tooltip helpexplode

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

This session interwines activities with the tool and the individual production of signs.

The posed problem has been chosen because according to our a-priori analysis its solution could make emerge different crucial aspects concerning the modelling process, which could be discussed in the subsequent discussion (PP “Discussion 1”). Such as:

  • the possible mismatch between the number of geometrical variables characterising a geometrical situation and the number of algebraic variables characterising the algebraic model of that geometrical situation and the need to deal with such mismatch;

  • the possibility of a wide variablity of geometrical configurations: in fact if the triangle (in which the problem is instanciated) is acute-angled then there are three different classes of inscribed rectangles;

  • the variety of possible different choices for the independent variables (no choices impose over the others a-priori);

  • consequently, the variety of possible different algebraic models;

  • the possible difficulties in geometrically re-interpreting the results drawn from the algebraic treatment, which may depend on the modellling choices.

Two remarks are needed: the problem is not posed in the frame of the Cartesian Geometry, and the problem is formulated in general terms: "given a triangle...".

That raises the need for the teacher to guide a first "modelling" of the geometric situation described: from a Euclidean coordinates-free frame to the Cartesian-like frame as implemented in Casyopée.

With that respect the meaning of the expression "given a triangle..." has to be negotiated in the class, if not shared yet.

A seemingly natural choice is to approach the problem in general terms, that is to choose 3 free points in Casyopée Geometric Window as vertices of the considered triangle. The consequence of such a choice would be the construction of a model depending on at least 6 independent variables (the coordinates of the vertices), but Casyopée only support the construction of an algebraic model depending on one variable.

The teacher could:

  1. (Recommended option) anticipate this problem and initiate a class discussion for choosing 3 fixed points as vertices of the triangle (the same for anyone). This choice on the one hand limits the richness of the problem, but on the other hand makes easier the comparison among the students' solutions in the following discussion. One could orient the discussion and take a triangle with a side lying on one of the axes. This last option would prepare the terrain for the "generalization" of te problem in the PP "didactical cycle 2"
  2. anticipate the problem and ask(let) each group of students to choose 3 fixed points as vertices of its own triangle. This option could lead to a variety of geometric situation and possible models.
  3. anticipate nothing and let students possibly face the impossibility of constructing the desired algebraic model. Once students face this impossibility, one could suggest them to choose 3 fixed points as vertices of their triangle. This option would be rewarding for the richness and the complexity which would emerge, but it is also highly demanding for students and teachers and time-consuming.

Attention is needed: there are different possible constructions of a rectangle inscribed in a triangle, but some of them are not stable with respect to dragging.

 

Resources show tooltip helpexplode

General description show tooltip help

Students are presented with a worksheet (on a piece of paper) containing:

a) the text of a geometry optimization problem to be solved (using Casyopée),

b) the request of solving it, and of taking notes of their own actions with casyopee and the messages given by the tool.

Students are also asked to produce a report of the activity after each problem solving sessions, based on a couple of questions (written on a piece of paper)

 

Problem1 [Resource for students]

Pupils are given a worksheet: the content is specified below.

Resource contents show tooltip help

Solve the following:

Problem. Given a triangle, what is the maximum value of the area of a rectangle inscribed in such triangle? Find a rectangle whose area has the maximum value.

Write down your solution.

Take notes of you own actions - e.g. points created, variables chosen... - and of Casyopee messages as well.

Download .pdf

Report2_request [Resource for students]

Pupils are assigned as homework the task of producing a report on the activity performed.

 

Resource contents show tooltip help

Report

With respect to the activity accomplished with Cayopée, what did you understand about Casyopée functioning? On the contrary, what is not clear enough yet?

download .pdf

Solution1 [Resource produced by students]

Each pair of students is asked to write the solution to the problem and take notes of their interactions with casyopee (in particular the messages given by casyopee in response to their actions)

 

Casyopee_file2 [Resource produced by students]

Students are asked to save also their solution in Casyopee. The solution is saved as a .txt file which casyopee can interpret so to display the last appearance of the interface: neither a replay system is available nor the object erased by students can be displayed.

Report2 [Resource produced by students]

Students' individual written reports on the activity accomplished.

 

Work plan show tooltip helpexplode

Setting show tooltip help

Activities has to be performed either in a computer lab, which should be equipped with (at least): 

  • One computer each two students. 

  • Data projector

  • Black or whiteboard

 

Time show tooltip help

2 hours

Actors' roles show tooltip help

The teacher:

  1. at the very beginning, should quickly show how one could accomplish the tasks of the Familiarization Worksheet (PP "familiarization") using a data projector
  2. gives to each pair the worksheet containing the text of the probelm 1;

  3. reads aloud the text of the problem;

  4. if needed specifies the text of the problem - e.g. "given a triangle" or "rectangle inscribed in a triangle" (see "what to do and how" for details);

  5. should supervise students' work and possibly launch and manage "private" or "collective" discussions with students  (see "what to do and how" for details).

Students divide into smal groups (the same of the previous PP as far as possible) and then engage in accomplishing the task given, which encompasses:

  1. solving the given problem (this includes the participation to the possible discussions which will took place),

  2. producing a written solution,

  3. taking notes of the activity as required.

During the session, students and teachers can of course interact in other different ways.

What to do and how show tooltip help

At the very beginning, the teacher should quickly show how one could accomplish the tasks of the Familiarization Worksheet (PP "familiarization") using a data projector. The objectives are: 

  1. to show Casyopée features to those students who possibly did not succeed to complete the Familiarization Worksheet in the previous section (or who were not present);
  2. at the same time, to briefly remind each student the functioning of Casyopée; and
  3. to show how to display the graph of functions in Casyopée main window (there is not any specific task on that in the Familiarization Worksheet).

The teacher distributes the worksheet containing the "optimization problem 1".

Before starting the problem solving session the text of the problem is read aloud. If needed, the meaning of some terms may be discussed and clarified, e.g. "given a triangle..." or "rectangle inscribed in a triangle".

As remarked in the field "description" of the "rationale" (above): the problem is not posed in the frame of the Cartesian Geometry, and the problem is formulated in general terms: "given a triangle...".

That raises the need for the teacher to guide a first "modelling" of the geometric situation described: from a Euclidean coordinates-free frame to the Cartesian-like frame as implemented in Casyopée.

With that respect the meaning of the expression "given a triangle..." has to be negotiated in the class, if not shared yet.

A seemingly natural choice is to approach the problem in general terms, that is to choose 3 free points in Casyopée Geometric Window as vertices of the considered triangle. The consequence of such a choice would be the construction of a model depending on at least 6 independent variables (the coordinates of the vertices), but Casyopée supports only the construction of an algebraic model depending on one variable.

The teacher could:

  1. (Recommended option) anticipate this problem and initiate a class discussion for choosing 3 fixed points as vertices of the triangle (the same for anyone). This choice on the one hand limits the richness of the problem, but on the other hand makes easier the comparison among the students' solutions in the following discussion. One could orient the discussion and take a triangle with a side lying on one of the axes. This last option would prepare the terrain for the "generalization" of te problem in the PP "didactical cycle 2"
  2. anticipate the problem and ask(let) each group of students to choose 3 fixed points as vertices of its own triangle. This option could lead to a variety of geometric situation and possible models.
  3. anticipate nothing and let students possibly face the impossibility of constructing the desired algebraic model. Once students face this impossibility, one could suggest them to choose 3 fixed points as vertices of their triangle. This option would be rewarding for the richness and the complexity which would emerge, but it is also highly demanding for students and teachers and time-consuming.

If option 3 is chosen, one can hypothesize that quite soon students will begin to meet one of the following difficulties:

  1. the geometric construction of the inscribed rectangle is not stable with respect to the dragging of the vertices of the triangle;
  2. the system cannot handle the algebraic model because there are too many algebraic variables.

The teacher's intervention is needed to face and overcome these difficulties. The teachers could either interact  with each group of students separately, or start a collective discussion (preferred option, but it would require that students experience the same difficulties nearly at the same time). 

Be they "private" or "collective", the discussions with students should aim at analyzing the source of the experienced difficulties and at  constructing different strategies for overcoming them, namely to reduce the number of variables by taking a particular instanciation of the geometric situation.

The question how to deal algebraically with the generality expressed in the text of the problem will be reconsidered in the second didactic cycle.

At the end, students are assigned as homework to produce individual written reports (Report2_request). Those students who ended the problem solving session can (begin to) write down their reports in the classroom, the others will be asked to produce their reports at home.

Process documentation show tooltip help

All the resources produced by students (see above) are also tools for documenting the process.

In addition screen capturer softwares could be used for recording samples of the activity within Casyopée. However such screen capturer softwares may slow down the performance of the computers hindering the correct functioning of Casyopée.

Discussion 1

Identity show tooltip helpexplode

Authors show tooltip help

M. Maracci, M.A. Mariotti

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • algebraic model of geometrical problems

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

This session concludes the first part of the dydactical cycle. In this session a discussion should be carried out starting from students' different possible solutions of problems 1 and students' written reports as well.

The exact themes of discussion cannot be foreseen precisely, of course they should relate to the meaning of function as co-variation and to the modelling process. In this discussion, the focus could be more on aspects related to the notions of function, variables...  a more in depth reflection on modelling can be carried out in the "Discussion 2".

Anyway, in order to foresee (at some extent) a possible development of the discussion, the teacher should analyze students' productions before the discussion itself.

According to our hypotheses, the previous activities could raise questions concerning:

(as for the notion of function)

  • the meaning of variables both geometrical and numerical,
  • the meaning of domain of a variable,
  • the meaning of function as co-variation over time of variables (even of different kind: numerical or geometrical),
  • competencies related to the passage between different representations of function (at least, algebraic and graphical ones)

(as for the modelling process)

  • the recognition of geometrical variables
  • the passage from unmeasurable geometrical objects (e.g. points) to measurable geometrical objects
  • the association of numbers (numerical variables) to geometrical variables
  • the association of geometrical variables to numbers (numerical variables)
  • the parametrizing process

Rationale show tooltip helpexplode

In this session, different possible solutions to the given problem are discussed collectively, as well as students’ written texts. The aim of this discussion is to promote the dialectics between different personal meanings and the mathematical meaning related to specific signs generated in the previous semiotic activities (with or without the tool). The intention is not to make students able to use Casyopée for accomplishing the given kinf of tasks, but to foster the students’ consciousness-raising of the mathematical meanings at stake.

The role of the teacher is crucial, in fact the evolution of (individual) signs, principally related to the activity with artefacts, towards mathematical signs, is neither spontaneous nor simple.

Of course specific discussions between teacher and pupils could have taken place in the previous sessions; the main aspects addressed in those discussion should now be recalled and systematized.

Theoretical framework show tooltip help

According to the Theory of Semiotic Mediation, the structure of a teaching sequence may be outlined as an iteration of a cycle where different kinds of activities take place, aimed to develop the complex semiotic process described above. Consistently this Plan is organized as a whole didactical cycle:

activities with the artefact: "familiarization", "problem solving"; at the end of each of which students are asked to individually write reports on the activites themselves (individual production of signs)

a class discussion starting form the given tasks and the reports written by students (collective production of signs)

Target show tooltip helpexplode

Rationale show tooltip helpexplode

The identification of the educational goals of the Plan comes from an analysis of the semiotic potential of Casyopee which leads us to hypothesize that the use of Casyopee could mediate (in the sense of Semiotic Mediation) the meanings of function as co-variation and mathematical meanings related to the algebraic modelling of geometrical situation. 

Given Italian National Curricula and School Organization, Scientific High School Grades 12-13 seem the suitable context where aspects related to algebraic modelling of geometical situations can be introduced and dealt with. 

Theoretical framework show tooltip help

The general theoretical frame is provided by the Theory of Semiotic Mediation, already described. More specifically, the identification of educational goals is made through the analysis of the semiotic potential the artefact. The analysis of the semiotic potential of an artefact encompass the analysis of both the personal and mathematical meanings related to the artefact, as well as of the possible tasks which can be accomplished with the artefact.

Population show tooltip helpexplode

School level show tooltip help

Scientific High School, grades 12-13

Age range show tooltip help

16-18 years old

Population description show tooltip help

regular high school class

Student prerequisites show tooltip help

This PP is designed for students who already have had some formal teaching on Euclidean and Cartesian plan geometry and function. As for function, prerequisites are very basic:

  • functions as expressed by elementary formulas (e.g. Linear or quadratic) 
  • functions as expressed by a graph.

For developing the second part of the PP (Didactical cycle 2), some formal knowledge on "parameters" are needed too.

Teacher prerequisites show tooltip help

Familiarity with Casyopée and generally speaking with managing the integration of digital tools in classroom practice.

Expertise in collective discussion according the semiotic mediation theoretical frame.

Context show tooltip helpexplode

Physical context show tooltip help

Computer lab with at least one computer for each two students and a data projector.

 

Institutional context show tooltip help

Link to the Italian Ministry of Education, School Regulations (IT Online available June 4, 2007).

 

Goals show tooltip helpexplode

Curricular goals show tooltip help

The development of modelling competencies is a general didactical objective of the teaching of all the scientific discipline and of mathematics in particular.

More specifically, according to the PNI (Piano Nazionale per l'introduzione dell'Informatica) (IT, Online available June 5, 2007) curricula:

  • among the aims of the teaching of mathematics there is the development of the capability to use mathematical methods, tools and models in different situations
  • The introduction of calculus concepts has to be accompanied by a wide spectrum of possible applications in both mathematical and extra-mathematical fields

According to the curricula elaborated by the Brocca Commission (IT, Online available June 5, 2007) mathematics with its languages and models [...] represents a tool with a very high educational value. Among the aims of the teaching of mathematics there are:

  • the development of the capability to use mathematical methods, tools and models in different situations
  • dealing with problematic situations of different kind by means of mathematical models
  • provide pupils with rigorous methods of analysis, compencies  for modelling (even) complex situations

The introduction of calculus concepts has to be accompanied by a wide spectrum of possible applications in both mathematical and extra-mathematical fields

Content-epistemological goals show tooltip help

The main goals of this Pedagogical Plan is to foster the evolution of students’ personal meanings towards:

  1. the meaning of function as co-variation and thus consolidate (or enrich) the meanings ot function they have already appropriated;
  2. mathematical meanings related to the processes characterizing the algebraic modelling of geometrical situation.

More specifically,
as for the notion of function, students should consolidate or enrich:

  • the meaning of variables both geometrical and numerical,
  • the meaning of domain of a variable,
  • the meaning of function as co-variation over time of variables (even of different kind: numerical or geometrical),
  • competencies related to the passage between different representations of function (at least, algebraic and graphical ones)

as for the modelling process, students should learn to:

  • recognize geometrical variables
  • pass from unmeasurable geometrical objects (e.g. points) to measurable geometrical objects
  • associate numbers (numerical variables) to geometrical variables
  • associate geometrical variables to numbers (numerical variables)
  • express the relation between numerical variables through formulas
  • parametrize (optimize the number of variables) 

Instrumental goals show tooltip help

Use of the tools provided by Casyopee: 

  • to solve ("even complex") geometrical problems which can be approched via an algebraic modelling;
  • eventually to explore functions or families of functions. 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

In this session a collective discussion should take place. 

The discussion has two focuses:

  1. on the one hand, on the possible different solutions to the problem given in the previous session;
  2. on the other one, on different aspects of the modelling process, already highlighted in our a-priori analysis of the problem (see PP "Problem of Optimization 1"):
  • the possible mismatch between the number of geometrical variables characterising a geometrical situation and the number of algebraic variables characterising the algebraic model of that geometrical situation and the need to deal with such mismatch;

  • the possibility of a wide variablity of geometrical configurations;

  • the variety of possible different choices for the independent variables (no choices impose over the others a-priori);

  • consequently, the variety of possible different algebraic models;

  • the possible difficulties in geometrically re-interpreting the results drawn from the algebraic treatment, which may depend on the modellling choices.

The teacher's actions should be oriented to:

  1. facilitate the evolution of the signs produced by students referring to their with Casyopée to mathematical signs.
  2. promote students' consciousness-raising of the mathematical meanings at stake,
  3. foster students' re-appropriation of those meanings in the more global frame of modelling, and
  4. raise a shared and decontextualized formulation of the different mathematical notions in focus.

Resources show tooltip helpexplode

General description show tooltip help

During the previous session students were asked to produce a written solution to the optimization problem proposed.

The produced solutions were collected and analysed by the teacher for preparing the discussion which takes place in this PP.

Each pair of students is now given its own solution as a support for the discussion.

At the end, students are assigned as homework to produce individual written reports on the discussion.

Solution1 [Resource for students]

During the previous session students were asked to produce a written solution to the optimization problem proposed.

Each pair of students is given its own solution as a support for the discussion.

Resource contents show tooltip help
Report3_request [Resource for students]

Pupils are assigned as homework the task of producing a report on the discussion which took place in the class.

Resource contents show tooltip help

Report

Summarize the main aspects discussed in class.

Is there anything you did not understand? 

Download .pdf

Report3 [Resource produced by students]

Students' individual written reports on the discussion held.

Work plan show tooltip helpexplode

Setting show tooltip help

The discussion has to take place in a usual classrooms. Anyway, it woul be helpful to have also one computer and one data projector also.

 

Time show tooltip help

1 hours

Actors' roles show tooltip help

The teacher should initiate and orchestrate the discussion.

 That includes:

  • the choice of the solution(s) to be discussed (in particular where to start from);
  • the choice of the excerpts from students' reports to discuss, if any;
  • the orchestrations between moments of collective discussions, and moments of work with casyopee.

Students are expected to participate to the discussion and work with casyopée in accordance with the  teacher's management of the session.

What to do and how show tooltip help

The teacher should analyze students' productions and the data collected during the previous activities before this session, in order to gain some awareness of the possible meanings students constructed in relation to the notions of variable and function, and to the modelling process. In particular, main attention should be paid to specific signs produced by students to solve the problem, communicate among them or to the teacher (e.g. through reports).

The discussion aims at let students'personal meanings evolve towards mathematical meanings related to the notions of variable and function, and to the modelling process.

The exact development of the discussion cannot be foreseen, but we hypothesize that the previous activities should raise questions concerning: 

as for the notion of function

  • the meaning of variables both geometrical and numerical,
  • the meaning of domain of a variable,
  • the meaning of function as co-variation over time of variables (even of different kind: numerical or geometrical),
  • competencies related to the passage between different representations of function (at least, algebraic and graphical ones)

as for the modelling process

  • the recognition of geometrical variables
  • the passage from unmeasurable geometrical objects (e.g. points) to measurable geometrical objects
  • the association of numbers (numerical variables) to geometrical variables
  • the association of geometrical variables to numbers (numerical variables)
  • the parametrizing process

The discussion should starts from the discussion of the solutions to problem 1, provided by students - which can be possibly projected. The teacher could for instance ask students to comment a specific solution, to confront their own solution with other ones, to try to "follow" someonelse's solution. The general idea is that each student should confront with different geometrical constructions, different choices of the variables, in a word different models of the problem. In order to to that, the teacher should alternate the collective discussions and students' work with the tool.

Since other discussions will take place in the following sessions, there is no need to "say everything about modelling" in this session. The above aspects will be re-called and questioned again and again.

Process documentation show tooltip help

All the resources produced by students (see above) are also tools for documenting the process.

In addition, the classroom discussions could be also documented through audio or (better) video recordings.

Optimization Problem 2

Identity show tooltip helpexplode

Authors show tooltip help

M. Maracci, M.A. Mariotti

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • algebraic model of geometrical problems

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • Geometry
  • Modelling
  • Algebra
  • Variable
  • Calculus
  • Function

Description show tooltip help

As well as PP "Problem of Otimization", th present PP is constituted by a problem-solving session, in which pupils are presented with an optimization problem in geometry. 

Our focus is still on the modelling process which allows to treat geometrical problems with algebra tools rather than on algebra tools themselves. Accordingly, the specific task is meant to confront pupils with some specific aspects of the modelling process, which will be detailed later on. 

Many of those aspects should have already emerged during the previous activities. In the present session, o students are asked to prove what they learnt on a new situation.

This PP is constituted by a problem-solving sessions, in which pupils are presented with an optimization problem in geometry.

This session is meant to make students reinvest what they learnt in the previous sessions both in terms of specific solving strategies and in terms of more general "ideas" related to the modelling process.

 

Rationale show tooltip helpexplode

This session is part of a didactical cycle, it is especially meant to engage students with the following kind of semiotic activities: pair task accomplishing with the tool and individual writing of reports on the former activity.

More specifically students are asked to solve geometrical optimization problem. Optimization problems in geometry are quite traditional in  the last grades of scientific high schools, because they provide a context of application for calculus notions and tools.

The algebriac modelling of the given geometrical problems is supported by the tool.

Theoretical framework show tooltip help

According to the Theory of Semiotic Mediation, the structure of a teaching sequence may be outlined as an iteration of a cycle where different kinds of activities take place, aimed to develop the complex semiotic process described above. Consistently this Plan is organized as a whole didactical cycle:

activities with the artefact: "familiarization", "problem solving"; at the end of each of which students are asked to individually write reports on the activites themselves (individual production of signs)

a class discussion starting form the given tasks and the reports written by students (collective production of signs)

Target show tooltip helpexplode

Rationale show tooltip helpexplode

In this session students are asked to solve geometrical optimization problem - a quite usual kind of problems in italian curricula for scientific high school. The algebriac modelling of the given geometrical problems is supported by the tool.

 

Population show tooltip helpexplode

School level show tooltip help

Scientific High School, grades 12-13

Age range show tooltip help

16-18 years old

Population description show tooltip help

regular high school class

Student prerequisites show tooltip help

This PP is designed for students who already have had some formal teaching on Euclidean and Cartesian plan geometry and function. As for function, prerequisites are very basic:

  • functions as expressed by elementary formulas (e.g. Linear or quadratic) 
  • functions as expressed by a graph.

For developing the second part of the PP (Didactical cycle 2), some formal knowledge on "parameters" are needed too.

Teacher prerequisites show tooltip help

Familiarity with Casyopée and generally speaking with managing the integration of digital tools in classroom practice.

Expertise in collective discussion according the semiotic mediation theoretical frame.

Context show tooltip helpexplode

Physical context show tooltip help

Computer lab with at least one computer for each two students and a data projector.

 

Institutional context show tooltip help

Link to the Italian Ministry of Education, School Regulations (IT Online available June 4, 2007).

 

Goals show tooltip helpexplode

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

This session interwines activities with the tool and the individual production of signs.

The posed problem has been chosen because according to our a-priori analysis its solution could foster the emergence of different crucial aspects concerning the modelling process. It is highly likely that those aspects already emerged in the previous activities and were discussed in the PP "Discussion 1". Hence this session offers to the students the occasion to prove what they learnt in a new situation. Moreover we hypothesize that the variety of possible geometrical configurations and of possible algebraic models, and the difficulty of geometrically interpreting the algebraic results are still wider than that of problem1.

As well as problem 1, even problem 2 is formulated in general terms, in a coordinates-free frame. Hence the same dilemma for the teacher emerges: the same fixed vertices for each goup, each group choosing its own fixed vertices, or free verticesfor each goup.

The emerging aspects will be discussed in the subsequent discussion (PP “Discussion 2”). 

We list once again the aspects of the modelling process which could be raised in the session:

  • the possible mismatch between the number of geometrical variables characterising a geometrical situation and the number of algebraic variables characterising the algebraic model of that geometrical situation and the need to deal with such mismatch;

  • the possibility of a wide variablity of geometrical configurations: in fact if the triangle (in which the problem is instanciated) is acute-angled then there are three different classes of inscribed rectangles;

  • the variety of possible different choices for the independent variables (no choices impose over the others a-priori);

  • consequently, the variety of possible different algebraic models;

  • the possible difficulties in geometrically re-interpreting the results drawn from the algebraic treatment, which may depend on the modellling choices;

Resources show tooltip helpexplode

General description show tooltip help

Students are presented with a worksheet (on a piece of paper) containing:

a) the text of a geometry optimization problem to be solved (using Casyopée),

b) the request of solving it, and of taking notes of their own actions with casyopee and the messages given by the tool.

Students are also asked to produce a report of the activity after each problem solving sessions, based on a couple of questions (written on a piece of paper)

Problem2 [Resource for students]

Pupils are given a worksheet: the content is specified below.

Resource contents show tooltip help

Solve the following

Problem. Consider a given triangle and take a point of one of its medians. Through this point draw the parallel lines to the other two sides (whose medians are not drawn): such parallel lines form a parallelogram with these two sides and a triangle with the third side (whose median is drawn). What is the minimum value of the sum of the areas of the triangle and the parallelogram? Find the point of the considered median such that the value of this sum is minimum.

Write down your solution.

Did you succed in solving the problem? Summarize the main ideas of your solution.

Download .pdf

Report4_request [Resource for students]

Pupils are assigned as homework the task of producing a report on the activity performed.

 

Resource contents show tooltip help

Report

Try to explain the differences and the analogies between this last problem and that one approached and discussed in the previous sessions, and between their respective solutions.

Download .pdf

Solution2 [Resource produced by students]

Each pair of students is asked to write the solution to the problem and take notes of their interactions with casyopee (in particular the messages given by casyopee in response to their actions)

 

Casyopee_file3 [Resource produced by students]

Students are asked to save also their solution in Casyopee. The solution is saved as a .txt file which casyopee can interpret so to display the last appearance of the interface: neither a replay system is available nor the object erasen by students can be displayed.

 

Report3 [Resource produced by students]

Students'  individual written reports on the activity accomplished.

Work plan show tooltip helpexplode

Setting show tooltip help

Activities has to be performed either in a computer lab, which should be equipped with (at least): 

  • One computer each two students. 

  • Data projector

  • Black or whiteboard

 

Time show tooltip help

2 hours

Actors' roles show tooltip help

The teacher:

  1. gives to each pair the worksheet containing the text of the probelm 1;

  2. reads aloud the text of the problem;

  3. if needed specifies the text of the problem - e.g. "given a triangle"  (see "what to do and how" for details);

  4. should supervise students' work. 

Students divide into smal groups (the same of the previous PPs as far as possible) and then engage in accomplishing the task given, which encompasses:

  1. solving the given problem (this includes the participation to the possible discussions which will took place),

  2. producing a written solution,

  3. taking notes of the activity as required.

During the session, students and teachers can of course interact in different ways.

What to do and how show tooltip help

The teacher distributes the worksheet containing the "optimization problem 1". 


Before starting the problem solving session, the text of the problem is read aloud. If needed, the meaning of some terms may be discussed and clarified, e.g. "given a triangle..."

As well as problem 1, even problem 2 is formulated in general terms, in a coordinates-free frame. Hence the same dilemma for the teacher emerges: the same fixed vertices for each goup, each group choosing its own fixed vertices, or free verticesfor each goup. The teacher is given the responsability to decide whether to follow the same option of the PP "optimization problem 1" or not. At the same time, in case of differences, we recommend the teacher to discuss with students the differences of approach to the optimization problem1 and the optimization problem2 (at stake in this PP).

Students are asked to produce individual written reports on the activity (Report5_request) at the end of the problem solving session. Those students who ended the problem solving session can (begin to) write down such report in class, the others will be asked to produce their reports at home.

Process documentation show tooltip help

All the resources produced by students (see above) are also tools for documenting the process.

In addition screen capturer softwares could be used for recording samples of the activity within Casyopée. However such screen capturer softwares may slow down the performance of the computers hindering the correct functioning of Casyopée.

 

Discussion 2

Identity show tooltip helpexplode

Authors show tooltip help

M. Maracci, M.A. Mariotti

Subject domains show tooltip help

  • mathematics

Topics show tooltip help

  • algebraic model of geometrical problems

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

  • Geometry
  • Modelling
  • Algebra
  • Variable
  • Calculus
  • Function

Description show tooltip help

This session concludes the dydactical cycle.

A discussion should be carried out starting from students' different  solutions of the problem 2 and from students' written reports. What emerged in the previous discussion can also be recalled and questioned.

The aim is still fostering the evolution of students' personal meanings  towards mathematical meanings related to the notion of function and to the processes characterizing the algebraic modelling of geometrical situations.

Given that the notion of function and variables were at stake in the previous discussion, in the present one the focus should be more on the notions related to modelling processes: mathematical meanings related to the modelling process should clearly put forth.

Rationale show tooltip helpexplode

This session is the concluding phase of the first didactical cycle. 

Different possible solutions to the given problem are discussed collectively, as well as students’ written texts. The aim of this discussion is to promote the dialectics between different personal meanings and the mathematical meaning related to specific signs generated in the previous semiotic activities (with or without the tool). The intention is not to make students able to use Casyopée for accomplishing the given kinf of tasks, but to foster the students’ consciousness-raising of the mathematical meanings at stake.The role of the teacher is crucial, in fact the evolution of (individual) signs, principally related to the activity with artefacts, towards mathematical signs, is neither spontaneous nor simple.

Previous discussions between teacher and pupils have taken place in the previous sessions; the main aspects addressed in those discussions should now be recalled and re-systematized.

Theoretical framework show tooltip help

According to the Theory of Semiotic Mediation, the structure of a teaching sequence may be outlined as an iteration of a cycle where different kinds of activities take place, aimed to develop the complex semiotic process described above. Consistently this Plan is organized as a whole didactical cycle:

activities with the artefact: "familiarization", "problem solving"; at the end of each of which students are asked to individually write reports on the activites themselves (individual production of signs)

a class discussion starting form the given tasks and the reports written by students (collective production of signs)

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

Context show tooltip helpexplode

Goals show tooltip helpexplode

Curricular goals show tooltip help

The development of modelling competencies is a general didactical objective of the teaching of all the scientific discipline and of mathematics in particular.

More specifically, according to the PNI (Piano Nazionale per l'introduzione dell'Informatica) (IT, Online available June 5, 2007) curricula:

  • among the aims of the teaching of mathematics there is the development of the capability to use mathematical methods, tools and models in different situations
  • The introduction of calculus concepts has to be accompanied by a wide spectrum of possible applications in both mathematical and extra-mathematical fields

According to the curricula elaborated by the Brocca Commission (IT, Online available June 5, 2007) mathematics with its languages and models [...] represents a tool with a very high educational value. Among the aims of the teaching of mathematics there are:

  • the development of the capability to use mathematical methods, tools and models in different situations
  • dealing with problematic situations of different kind by means of mathematical models
  • provide pupils with rigorous methods of analysis, compencies  for modelling (even) complex situations

The introduction of calculus concepts has to be accompanied by a wide spectrum of possible applications in both mathematical and extra-mathematical fields

Content-epistemological goals show tooltip help

The main goals of this Pedagogical Plan is to foster the evolution of students’ personal meanings towards:

  1. the meaning of function as co-variation and thus consolidate (or enrich) the meanings ot function they have already appropriated;
  2. mathematical meanings related to the processes characterizing the algebraic modelling of geometrical situation.

More specifically,
as for the notion of function, students should consolidate or enrich:

  • the meaning of variables both geometrical and numerical,
  • the meaning of domain of a variable,
  • the meaning of function as co-variation over time of variables (even of different kind: numerical or geometrical),
  • competencies related to the passage between different representations of function (at least, algebraic and graphical ones)

as for the modelling process, students should learn to:

  • recognize geometrical variables
  • pass from unmeasurable geometrical objects (e.g. points) to measurable geometrical objects
  • associate numbers (numerical variables) to geometrical variables
  • associate geometrical variables to numbers (numerical variables)
  • express the relation between numerical variables through formulas
  • parametrize (optimize the number of variables) 

Instrumental goals show tooltip help

Use of the tools provided by Casyopee: 

  • to solve ("even complex") geometrical problems which can be approched via an algebraic modelling;
  • eventually to explore functions or families of functions. 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

As well as the previous discussion, the collective discussion which should take place in this session, has two main focuses:

  1. on the possible different solutions to the problem given in the previous session;
  2. on different aspects of the modelling process.

We already mentioned the processes characterizing the algebraic modelling of geometrical situation in a number of occasions:

  • the possible mismatch between the number of geometrical variables characterising a geometrical situation and the number of algebraic variables characterising the algebraic model of that geometrical situation and the need to deal with such mismatch;

  • the possibility of a wide variablity of geometrical configurations;

  • the variety of possible different choices for the independent variables (no choices impose over the others a-priori);

  • consequently, the variety of possible different algebraic models;

  • the possible difficulties in geometrically re-interpreting the results drawn from the algebraic treatment, which may depend on the modellling choices.

The teacher's actions should be oriented to:

  1. facilitate the evolution of the signs produced by students referring to their with Casyopée to mathematical signs.
  2. promote students' consciousness-raising of the mathematical meanings at stake,
  3. foster students' re-appropriation of those meanings in the more global frame of modelling, and
  4. raise a shared and decontextualized formulation of the different mathematical notions in focus.

Resources show tooltip helpexplode

General description show tooltip help

During the previous sessions students were asked to produce written solutions to two optimization problems (problem 1 and problem 2).

Those solution can now be used by students as a support for the discussion, which takes place in this PP.

At the end, students are assigned as homework to produce individual written reports on the discussion.

Solution1 and Solution2 [Resource for students]

These are students' solutions to the optimization problems (1 and 2).

In particular, during the previous session students were asked to produce a written solution to the optimization problem proposed.

The produced solutions were collected and analysed by the teacher for preparing the discussion which takes place in this PP.

Each pair of students is now given back its own solution as a support for the discussion.

Resource contents show tooltip help
Report5_request [Resource for students]

Pupils are assigned the task of producing a report based on the discussion which took place in the class.
Pupils are asked to explicilty put into relationship mathematical meanings and Casyopée features.

 

Report5 [Resource produced by students]

Students' individual written reports.

Work plan show tooltip helpexplode

Setting show tooltip help

The discussion has to take place in a usual classrooms. Anyway, it woul be helpful to have also one computer and one data projector also.

Time show tooltip help

1 hour

Actors' roles show tooltip help

The teacher should initiate and orchestrate the discussion.

That includes:

  • the choice of the solution(s) to be discussed (in particular where to start from);
  • the choice of the excerpts from students' reports to discuss, if any;
  • the orchestrations between moments of collective discussions, and moments of work with casyopee.
  • possibly to recall what already discussed (in particular during "discussion 1"). 

Students are expected to participate to the discussion and work with casyopée in accordance with the  teacher's management of the session.

 

What to do and how show tooltip help

As well as the PP "discussion 1", this session requires the teacher to analyze students' productions and the data collected during the previous activities. This includes also the analysis of the themes dealt with in the previous discussion and how the discussion developed. In particular, main attention should be paid to specific signs produced by students to solve the problem, communicate among them or to the teacher (e.g. through reports) and to the signs collectively produced and shared during the previous discussion.

The present session aims at still contributing to the evolution of the students' personal meanings towards mathematical meanings related to the notions of variable and function, and to the modelling process.

The main issues at stake are those already stressed:

as for the notion of function

  • the meaning of variables both geometrical and numerical,
  • the meaning of domain of a variable,
  • the meaning of function as co-variation over time of variables (even of different kind: numerical or geometrical),
  • competencies related to the passage between different representations of function (at least, algebraic and graphical ones)

as for the modelling process

  • the recognition of geometrical variables
  • the passage from unmeasurable geometrical objects (e.g. points) to measurable geometrical objects
  • the association of numbers (numerical variables) to geometrical variables
  • the association of geometrical variables to numbers (numerical variables)

The discussion should start from the discussion of the different possible solutions of problem 2, the teacher can possibly recall what emerged and was already discussed in the previous sessions. 

The teacher could for instance ask students to comment a specific solution, to confront their own solution with other ones, to try to "follow" someonelse's solution; but also (s)he can ask students to "compare" the solutions of problems 1 and 2 and try to highlight the possible common "general ideas" underlying them. 

The general idea is that each student should confront with different geometrical constructions, different choices of the variables, in a word different models of the problem. In order to to that, the teacher should alternate the collective discussions and students' work with the tool.

Process documentation show tooltip help

All the resources produced by students (see above) are also tools for documenting the process.

In addition, the classroom discussions could be also documented through audio or (better) video recordings.

Didactical cycle 2

Identity show tooltip helpexplode

Authors show tooltip help

M. Maracci, M.A. Mariotti

Subject domains show tooltip help

Topics show tooltip help

Language show tooltip help

English

Country show tooltip help

Italy

Keywords show tooltip help

Description show tooltip help

The whole PP is conceived as a didactical cycle, involving pupils in different kinds of activities: familiarization with the tool, problem solving through modelling and collective discussion of results.

The aim is mainly to drive pupils to generate mathematical meanings related to the process of "parametrizing" in the context of algebraic modelling of geometrical situation.

Rationale show tooltip helpexplode

This Plan is organized as a whole didactical cycle which alternates students' work with Casyopée and collective discussion. Aspects related to the notion of "parameter" and to the process of "parametrizing" should emerge in different activities and should be discussed at different level of depth and in different moments.

The main innovative element of the Plan is the intent of revisiting the notion of parameter, with which students are supposed to be familiar constructing, within the context of modelling. The desired effect is that students appropriate the idea of parameter as a variable whose variation is under control, in particular a variable which can be considered as a constant at a given time.

The notions of variable and of function, and the "algebraic modelling" process are still in focus.

Theoretical framework show tooltip help

According to the Theory of Semiotic Mediation, the structure of a teaching sequence may be outlined as an iteration of a cycle where different kinds of activities take place, aimed to develop the complex semiotic process described above. Consistently this Plan is organized as a whole didactical cycle:

activities with the artefact: "familiarization", "problem solving"; at the end of each of which students are asked to individually write reports on the activites themselves (individual production of signs)

a class discussion starting form the given tasks and the reports written by students (collective production of signs)

 

Target show tooltip helpexplode

Rationale show tooltip helpexplode

Population show tooltip helpexplode

Context show tooltip helpexplode

Goals show tooltip helpexplode

Curricular goals show tooltip help

The development of modelling competencies is a general didactical objective of the teaching of all the scientific discipline and of mathematics in particular.

More specifically, according to the PNI (Piano Nazionale per l'introduzione dell'Informatica) (IT, Online available June 5, 2007) curricula:

  • among the aims of the teaching of mathematics there is the development of the capability to use mathematical methods, tools and models in different situations
  • The introduction of calculus concepts has to be accompanied by a wide spectrum of possible applications in both mathematical and extra-mathematical fields

According to the curricula elaborated by the Brocca Commission (IT, Online available June 5, 2007) mathematics with its languages and models [...] represents a tool with a very high educational value. Among the aims of the teaching of mathematics there are:

  • the development of the capability to use mathematical methods, tools and models in different situations
  • dealing with problematic situations of different kind by means of mathematical models
  • provide pupils with rigorous methods of analysis, compencies  for modelling (even) complex situations

The introduction of calculus concepts has to be accompanied by a wide spectrum of possible applications in both mathematical and extra-mathematical fields

Content-epistemological goals show tooltip help

The main goals of this Pedagogical Plan is to foster the evolution of students’ personal meanings towards:

  1. the meaning of function as co-variation and thus consolidate (or enrich) the meanings ot function they have already appropriated;
  2. mathematical meanings related to the processes characterizing the algebraic modelling of geometrical situation.

More specifically,
as for the notion of function, students should consolidate or enrich:

  • the meaning of variables both geometrical and numerical,
  • the meaning of domain of a variable,
  • the meaning of function as co-variation over time of variables (even of different kind: numerical or geometrical),
  • competencies related to the passage between different representations of function (at least, algebraic and graphical ones)

as for the modelling process, students should learn to:

  • recognize geometrical variables
  • pass from unmeasurable geometrical objects (e.g. points) to measurable geometrical objects
  • associate numbers (numerical variables) to geometrical variables
  • associate geometrical variables to numbers (numerical variables)
  • express the relation between numerical variables through formulas
  • parametrize (optimize the number of variables) 

Instrumental goals show tooltip help

Use of the tools provided by Casyopee: 

  • to solve ("even complex") geometrical problems which can be approched via an algebraic modelling;
  • eventually to explore functions or families of functions. 

Specifications show tooltip helpexplode

Rationale show tooltip helpexplode

Work plan show tooltip helpexplode

Setting show tooltip help

Activities will be performed either in computer labs or in usual classrooms, depending on the kind of the activities themselves.

Computer lab should be equipped with (at least): 

  • One computer each two students. 

  • Data projector

  • Black or whiteboard

 It woul be helpful to have one computer and one data projector also in the classroom.

Time show tooltip help

3 hours

Actors' roles show tooltip help

Actors'roles change over the different sessions which compound this PP.

What to do and how show tooltip help

The activities which constitute this didactical cycle are conceived to be proposed in the given specifc order: Optimization Problem 1 - revised, Discussion on parametrization.

As for time-schedule, 2 hours for the lab session and 1 hour for the discussion seem reasonable, but obviously some deviations are possible.

Process documentation show tooltip help

All the resources produced by students (detailed in the description of the PP "leaves") are also tools for documenting the process.

In addition screen capturer softwares could be used for recording samples of the activity within Casyopée. However such screen capturer softwares may slow down the performance of the computers hindering the correct functioning of Casyopée.

The classroom discussions could be also documented through audio or (better) video recordings.

Optimization Problem 1 - revised

Identity show tooltip helpexplode

Authors show tooltip help

M. Maracci, M.A. Mariotti

Subject domains show tooltip help

    Topics show tooltip help

      Language show tooltip help

      English

      Country show tooltip help

      Italy

      Keywords show tooltip help

      "parameter" and "parametrization" to be added

        Description show tooltip help

        This PP is constituted by a problem-solving session, in which pupils are presented again with the optimization problem proposed in the PP "Optimization Problem 1", and whose possible "solutions" were discussed in "Discussion 1". 

        That problem is formulated in general terms ("Given a triangle..."), but it was solved only for specific triangles (that is specific coordinates were assigned to the triangle vertices).

        In this PP, pupils cope with the task of producing a more "general" solution.

        Rationale show tooltip helpexplode

        This session is part of a didactical cycle, it is especially meant to engage students with the following kind of semiotic activities: pair task accomplishing with the tool and individual writing of reports on the former activity.

        More specifically students are asked to produce a "more general" solution of a geometrical optimization problem, already faced in previous sessions. The notion of parameter and the parametrization process can de introduced as means to meet the request of producing more general solutions. Mathematical meanings related to the parametrizing process should emerge.

        The parametrization process is supported by the tool.

         

        Theoretical framework show tooltip help

        This session is designed as part of the first phase of a didactical cycle, as such it encompasses:

        Activities with artefacts: students are faced with tasks to be carried out with the artefact. This type of activities is thought to promote the emergence of specific signs in relation to the particular use of the artefacts; in fact working in pairs fosters social exchange, accompanied by words, sketches, gestures, and the like. 

        Individual production of signs (e.g. drawing, writing and the like): students are individually engaged in different semiotic activities, mainly concerning written productions.  This type of activity differs for instance from social exchanges  during the activity with the artefact, because it requires a personal contribution in order to produce written text and consequently graphic signs, which for its very nature starts to be detached from the contingency of the situated action.

        These activities should promote the production of artefact-signs susceptible, because of the semiotic potential of the artefact, to evolve towards mathematical signs under the guidance of the teacher. In this sense the teacher acts as mediator using the artefact to mediate mathematical content to the students, that is the teacher uses the artefact as a tool of semiotic mediation.

        Target show tooltip helpexplode

        Rationale show tooltip helpexplode

        In this session students are asked to solve geometrical optimization problem - a quite usual kind of problems in italian curricula for scientific high school. The algebriac modelling of the given geometrical problems is supported by the tool.

         

        Population show tooltip helpexplode

        School level show tooltip help

        Scientific High School, grades 12-13

        Age range show tooltip help

        16-18 years old

        Population description show tooltip help

        regular high school class

        Student prerequisites show tooltip help

        This PP is designed for students who already have had some formal teaching on Euclidean and Cartesian plan geometry and function. As for function, prerequisites are very basic:

        • functions as expressed by elementary formulas (e.g. Linear or quadratic) 
        • functions as expressed by a graph.

        For developing the second part of the PP (Didactical cycle 2), some formal knowledge on "parameters" are needed too.

        Teacher prerequisites show tooltip help

        Familiarity with Casyopée and generally speaking with managing the integration of digital tools in classroom practice.

        Expertise in collective discussion according the semiotic mediation theoretical frame.

        Context show tooltip helpexplode

        Physical context show tooltip help

        Computer lab with at least one computer for each two students and a data projector.

         

        Institutional context show tooltip help

        Link to the Italian Ministry of Education, School Regulations (IT Online available June 4, 2007).

         

        Goals show tooltip helpexplode

        Specifications show tooltip helpexplode

        Rationale show tooltip helpexplode

        This session interwines activities with the tool, the individual production of signs, and collective discussions.

        The posed problem is the same already discussed in the PPs "Problem of Optimization 1" and "Discussion 1".  

        We already highlighted that there is a mismatch between the number of geometrical variables characterising the geometrical situation described in the problem and the number of algebraic variables characterising the algebraic model of that situation, and the number of algebraic variables which one can handle. 

        In this PP such mismatch is dealt with, through introducing parameters. Parametrization is the "new" aspect of modelling which is in focus in this PP.

        Crucial aspects of the algebraic modelling process, already highlighted, are still present:

        • the possibility of a wide variablity of geometrical configurations: in fact if the triangle (in which the problem is instanciated) is acute-angled then there are three different classes of inscribed rectangles;

        • the variety of possible different choices for the independent variables (no choices impose over the others a-priori);

        • consequently, the variety of possible different algebraic models;

        • the possible difficulties in geometrically re-interpreting the results drawn from the algebraic treatment, which may depend on the modellling choices;

        Resources show tooltip helpexplode

        General description show tooltip help

        Students are presented again with the worksheet containing the text Problem of Optimization 1 (see PP "Problem of Optimization 1").

        Students are asked to approach the problem in general terms, as far as possible.

        Students are also asked to produce individual written reports of the activity.

        Problem of Optimization 1 [Resource for students]

        Students are presented again with the worksheet containing the text Problem of Optimization 1 (see PP "Problem of Optimization 1").

         

        Resource contents show tooltip help

         

        Solve the following:

        Problem. Given a triangle, what is the maximum value of the area of a rectangle inscribed in such triangle? Find a rectangle whose area has the maximum value.

        Write down your solution.

        Take notes of you own actions - e.g. points created, variables chosen... - and of Casyopee messages as well.

        Download .pdf

        Report6_request [Resource for students]

        Pupils are assigned as homework the task of producing a report on the activity performed.

         

        Resource contents show tooltip help

        Report

        1. In the last session “parameters” were introduced in Casyopée. Try to explain why.
        2. Try to explain what you mean by “parameter”.
        3. In your opinion, which features of Casyopée can be put in relationships with “parameters”? Why?

        Download .pdf

        Solution3 [Resource produced by students]

        Each pair of students is asked to write the solution to the problem and take notes of their interactions with casyopee (in particular the messages given by casyopee in response to their actions)

         

        Casyopée_files4 [Resource produced by students]

        Students are asked to save also their solution in Casyopee. The solution is saved as a .txt file which casyopee can interpret so to display the last appearance of the interface: neither a replay system is available nor the object erasen by students can be displayed.

         

        Report6 [Resource produced by students]

        Students' individual written reports

         

        Work plan show tooltip helpexplode

        Setting show tooltip help

        Activities has to be performed either in a computer lab, which should be equipped with (at least): 

        • One computer each two students. 

        • Data projector

        • Black or whiteboard

        Time show tooltip help

        2 hours

        Actors' roles show tooltip help

        The teacher 

        1. gives to each pair the worksheet containing the text of probelm1;
        2. reads aloud the text of the problem;
        3. introduces the issue of dealing with the original general problem expressed in "Problem of Optimization 1"
        4. shows how to create "parameters" in Casyopee and how to build a point (in casyopee dynamic geometry environment) whose coordinates are parameters. 
        5. launchs and coordinates a preliminary discussion aiming at estabilishing how many "casyopee-parameters" are needed to solve the general problem and to what extent of generality one can reasonably face the problem.
        6. supervises the work of students solving the problem

        Students divide into pairs and then engage in accomplishing the task given, which encompasses:

        1. solving the given problem (this includes the participation to the possible discussions which will took place),

        2. producing a written solution,

        3. taking notes of the activity as required.

        During the session, students and teachers can of course interact in other different ways.

         

        What to do and how show tooltip help

         In the PP "first didactical cycle" students were presented with "Optimization Problem 1". The original request concerned a "general" geometrical situation which students could not completely deal with in casyopee. According to our hypotheses the students and teacher should have negotiated to face a  "specific" problem through assigning particular coordinates to the vertices of the triangle.

        In this PP, the teacher introduces the issue of approaching the problem as it was originally expressed.

        The teacher can stimulate a collective discussion and orient it so to raise the following points:

        1. aiming at generalizing in some sense what done before, so to deal with a more general situation nearer to that originally expressed in the "Optimization Problem 1", one could choose a triangle with two fixed vertices and a "free" one;
        2. remembering that Casyopée cannot construct algebraic models of geometric situations depending on more than one variable, one need a way to "control" variation of the "free" vertex;
        3. basing on what students know about parameters, one can take a point whose coordinates are parameters instead of a true "free" vertex.

        The teacher shows how to create "parameters" in Casyopee and how to build a point (in casyopee dynamic geometry environment) whose coordinates are parameters. In previous PPs, the possible status of a point with respect to dragging and movement was discussed, now points with parametrized coordinates can be presented as a "new kind" of points with respect to movement.

        Starting from the discussion of the problems emerged in previous PPs when trying to solve "Problem of Optimization 1" in general terms, the teacher can present the points with parametrized coordinates as "variable points" which are anyway "controlled" and can be assumed as "fixed points" at a given moment.

        With respect to parameters and parametrized points, there are at least three issues at stake:

        1. "Parametrized points" can be displayed in the geometrical environment only if parameters are instanciated. Thus one has to question the extent to which the geometrical constructions made depend on the specific instanciations of the parameters.

        2. Quantities (measures, coordinates...) depending on parameters cannot be confronted: that can originate different obstacles during the solution of the problem.

        3. How many "casyopee-parameters" are needed to solve the general problem? At what extent of generality can one "reasonably" face the problem? Because of what highlighted at item 2, it is highly suitable to choose the smallest possible number of parameters.

        The teacher could just mentioning these issues if needed, but we think that a systematic discussion on them should be made in the next PP ("discussion on parametrization").

        Students will work in pairs to accomplish the task  "problem1" described in  the field "Resources for student".

        Students are asked to produce individually a report of the activity after the problem solving session. Those students who ended the problem solving session can (begin to) write down such report in class, the others will be asked to produce their reports at home.

        Process documentation show tooltip help

        All the resources produced by students (see above) are also tools for documenting the process.

        In addition screen capturer softwares could be used for recording samples of the activity within Casyopée. However such screen capturer softwares may slow down the performance of the computers hindering the correct functioning of Casyopée.

         

        discussion on parametrization

        Identity show tooltip helpexplode

        Authors show tooltip help

        M. Maracci, M.A. Mariotti

        Subject domains show tooltip help

          Topics show tooltip help

            Language show tooltip help

            English

            Country show tooltip help

            Italy

            Keywords show tooltip help

             parameters to be added

              Description show tooltip help

              This session concludes the last dydactical cycle. The overall organization is the same of the previous discussions.

              The reflection on modelling and on the notion of function should still progress, and  should be enriched by considerations on the parameters, the differences between parameters and variables, and on the parametrizing process, which should constitute the core of the discussion.

              Rationale show tooltip helpexplode

              This session is the concluding phase of the second didactical cycle. 

              Consistently with our theoretical hypotheses, different possible solutions to the given problem are discussed collectively, as well as students’ written texts. The aim of this discussion is to promote the dialectics between different personal meanings and the mathematical meaning related to specific signs generated in the previous semiotic activities (with or without the tool). The intention is not to make students able to use Casyopée for accomplishing the given kinf of tasks, but to foster the students’ consciousness-raising of the mathematical meanings at stake.The role of the teacher is crucial, in fact the evolution of (individual) signs, principally related to the activity with artefacts, towards mathematical signs, is neither spontaneous nor simple.

              Previous discussions between teacher and pupils have taken place in the previous sessions; the main aspects addressed in those discussions should now be recalled and re-systematized.

               

              Theoretical framework show tooltip help

              According to the Theory of Semiotic Mediation, the structure of a teaching sequence may be outlined as an iteration of a cycle where different kinds of activities take place, aimed to develop the complex semiotic process described above. Consistently this Plan is organized as a whole didactical cycle:

              activities with the artefact: "familiarization", "problem solving"; at the end of each of which students are asked to individually write reports on the activites themselves (individual production of signs)

              a class discussion starting form the given tasks and the reports written by students (collective production of signs)

               

              Target show tooltip helpexplode

              Rationale show tooltip helpexplode

              Population show tooltip helpexplode

              School level show tooltip help

              Scientific High School, grades 12-13

              Age range show tooltip help

              16-18 years old

              Population description show tooltip help

              regular high school class

              Student prerequisites show tooltip help

              This PP is designed for students who already have had some formal teaching on Euclidean and Cartesian plan geometry and function. As for function, prerequisites are very basic:

              • functions as expressed by elementary formulas (e.g. Linear or quadratic) 
              • functions as expressed by a graph.

              For developing the second part of the PP (Didactical cycle 2), some formal knowledge on "parameters" are needed too.

              Teacher prerequisites show tooltip help

              Familiarity with Casyopée and generally speaking with managing the integration of digital tools in classroom practice.

              Expertise in collective discussion according the semiotic mediation theoretical frame.

              Context show tooltip helpexplode

              Physical context show tooltip help

              Computer lab with at least one computer for each two students and a data projector.

               

              Institutional context show tooltip help

              Link to the Italian Ministry of Education, School Regulations (IT Online available June 4, 2007).

               

              Goals show tooltip helpexplode

              Curricular goals show tooltip help

              The development of modelling competencies is a general didactical objective of the teaching of all the scientific discipline and of mathematics in particular.

              More specifically, according to the PNI (Piano Nazionale per l'introduzione dell'Informatica) (IT, Online available June 5, 2007) curricula:

              • among the aims of the teaching of mathematics there is the development of the capability to use mathematical methods, tools and models in different situations
              • The introduction of calculus concepts has to be accompanied by a wide spectrum of possible applications in both mathematical and extra-mathematical fields

              According to the curricula elaborated by the Brocca Commission (IT, Online available June 5, 2007) mathematics with its languages and models [...] represents a tool with a very high educational value. Among the aims of the teaching of mathematics there are:

              • the development of the capability to use mathematical methods, tools and models in different situations
              • dealing with problematic situations of different kind by means of mathematical models
              • provide pupils with rigorous methods of analysis, compencies  for modelling (even) complex situations

              The introduction of calculus concepts has to be accompanied by a wide spectrum of possible applications in both mathematical and extra-mathematical fields

              Content-epistemological goals show tooltip help

              The main goals of this Pedagogical Plan is to foster the evolution of students’ personal meanings towards:

              1. the meaning of function as co-variation and thus consolidate (or enrich) the meanings ot function they have already appropriated;
              2. mathematical meanings related to the processes characterizing the algebraic modelling of geometrical situation.

              More specifically,
              as for the notion of function, students should consolidate or enrich:

              • the meaning of variables both geometrical and numerical,
              • the meaning of domain of a variable,
              • the meaning of function as co-variation over time of variables (even of different kind: numerical or geometrical),
              • competencies related to the passage between different representations of function (at least, algebraic and graphical ones)

              as for the modelling process, students should learn to:

              • recognize geometrical variables
              • pass from unmeasurable geometrical objects (e.g. points) to measurable geometrical objects
              • associate numbers (numerical variables) to geometrical variables
              • associate geometrical variables to numbers (numerical variables)
              • express the relation between numerical variables through formulas
              • parametrize (optimize the number of variables) 

              Instrumental goals show tooltip help

              Use of the tools provided by Casyopee: 

              • to solve ("even complex") geometrical problems which can be approched via an algebraic modelling;
              • eventually to explore functions or families of functions. 

              Specifications show tooltip helpexplode

              Rationale show tooltip helpexplode

              In this session a collective discussion should take place. 

              The discussion has two focuses:

              1. on the one hand, on the possible different solutions to the problem given in the previous session;
              2. on the other one, on parametrization within modelling process, more specifically:
              • the differences between parameters and variables;

              • the need of instanciating parameters to display graphs, diagrams, geometrical construction, etc. depending on parameters themselves;

              • the question of the extent to which the geometrical constructions made depend on the specific instanciations of the parameters;

              • the question of the extent to which the properties of the graphs displayed depend on the specific instanciations of the parameters. And thus extent to which the properties of the correpsonding functions (inferred fthrough the graph) depend on the instanciations of the parameters, too;

              • the "impossibility" of confronting (in general) quantities depending on parameters (e.g. a+b can be less, equal or greater than 0 depending on a and b; no general comparison is possible);
              • the number of parameters needed to represent a generic triangle (or any other geometrical figure) on th Cartesian Plane;
              • the difference between algebraic parameters and geometrical ones.

              Obviously the discussion can also recall and deal with other  (possibly previously discussed) aspects of the modelling process (which we highlighted in the description of the previous PPs).

              The teacher's actions should be oriented to:

              1. facilitate the evolution of the signs produced by students referring to their with Casyopée to mathematical signs.
              2. promote students' consciousness-raising of the mathematical meanings at stake,
              3. foster students' re-appropriation of those meanings in the more global frame of modelling, and
              4. raise a shared and decontextualized formulation of the different mathematical notions in focus.

              Theoretical framework show tooltip help

              The Plan design is inspired to the construct of didactical cycle dheoy of Semiotic Mediation.

              According to this theory, the complex semiotic processes of creation and evolution of personal meanings towards mathematical meanings can be developed through the design and implementation of a so called didactical cycle, that is an iterative cycle of the following kinds of activities: Activities with the artefacts, Individual production of signs, Collective production of signs.

              Resources show tooltip helpexplode

              General description show tooltip help

              During the previous session students had to face again the optimization problem 1 and were to asked to produce a written solution to  it.

              The produced solutions were collected and analysed by the teacher for preparing the discussion which takes place in this PP.

              Each pair of students is now given its own solution as a support for the discussion.

              At the end, students are assigned as homework to produce individual written reports on the discussion.

               

              Solution3 [Resource for students]

              During the previous session students were asked to produce a written solution to the optimization problem proposed.

              Each pair of students is given its own solution as a support for the discussion.

               

              Resource contents show tooltip help
              Report7_request [Resource for students]

              Pupils are assigned as homework the task of producing a report based on the discussion which took place in the class.

              In particular students are asked to explicitly put into reltaionships mathematical meanings (“function”, “variable” , “model” and “parameter”) and Casyopée features.

              Report7 [Resource produced by students]

              Students' individual written reports.

               

              Work plan show tooltip helpexplode

              Setting show tooltip help

              The discussion has to take place in a usual classrooms. Anyway, it woul be helpful to have also one computer and one data projector also.

              Time show tooltip help

              1 hour

              Actors' roles show tooltip help

              The teacher should initiate and orchestrate the discussion.

              That includes:

              • the choice of the solution(s) to be discussed (in particular where to start from);
              • the choice of the excerpts from students' reports to discuss, if any;
              • the orchestrations between moments of collective discussions, and moments of work with casyopee.
              • possibly to recall what already discussed (both during the previous PP and during "discussion 1" and "discussion 2"). 

              Students are expected to participate to the discussion and work with casyopée in accordance with the  teacher's management of the session.

              What to do and how show tooltip help

              The teacher should analyze students' productions and the data collected during the previous activities before this session, in order to gain some awareness of the possible meanings students constructed in relation to the notions of parameter and parametrization within the modelling process. In particular, main attention should be paid to specific signs produced by students to solve the problem, communicate among them or to the teacher (e.g. through reports).

              The discussion aims at let students'personal meanings evolve towards mathematical meanings related to the notion of parameter and to the parametrization process. But obviously meanings related to the notion of function and variable and to the modelling process are still in focus; aspects already discussed in previous PPs can be possibly recalled with the aim of fostering the evolution of students' personal meanings towards mathematical meanings.

              The exact development of the discussion cannot be foreseen, but we hypothesize that the previous activities should raise questions concerning: 

              • the differences between parameters and variables;

              • the need of instanciating parameters to display graphs, diagrams, geometrical construction, etc. depending on parameters themselves;

              • the question of the extent to which the geometrical constructions made depend on the specific instanciations of the parameters;

              • the question of the extent to which the properties of the graphs displayed depend on the specific instanciations of the parameters. And thus extent to which the properties of the correpsonding functions (inferred fthrough the graph) depend on the instanciations of the parameters, too;

              • the "impossibility" of confronting (in general) quantities depending on parameters (e.g. a+b can be less, equal or greater than 0 depending on a and b; no general comparison is possible);
              • the number of parameters needed to represent a generic triangle (or any other geometrical figure) on th Cartesian Plane;
              • the difference between algebraic parameters and geometrical ones;

              Together with other more generak aspects of the algebraic modelling process (which we highlighted in the description of the previous PPs).

              The discussion should starts from the discussion of the new solutions to problem 1, provided by students - which can be possibly projected.

              Process documentation show tooltip help

              All the resources produced by students (see above) are also tools for documenting the process.

              In addition, the classroom discussions could be also documented through audio or (better) video recordings.