Connected Working Spaces: designing and evaluating modelling based teaching/learning situations Jean-baptiste Lagrange LDAR University Paris Diderot. http://jb.lagrange.free.fr

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Laboratoire de Didactique André Revuz • Groupe Technologies Numériques en éducation

• Groupe Espaces de Travail Mathématiques

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First I introduce myself I am working in the Laboratoire de Didactique André Revuz that gathers researchers in mathematices education, and also in phsics, natural sciences, geography eduction. I belong to two groups, one about technology in education, with people like Maha Abboud or Fabrice Vandebrouck, The other group deals with Mathematical working spaces, a theoretical framework that I will present during the talk.

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Working with teachers http://casyopee.eu

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With regard to field work, I work with a group of High School Teachers in the Britanny Region. We have some views about the teaching/learning of calculus at upper secondary level that I will present

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The teaching of calculus in schools (Torner, Potari & Zachariades, 2014) – far away from what the students can understand. – reduced to algorithms for algebraic calculations – teachers face difficulties to promote deeper understanding • even when innovative materials or situations are implemented.

Modelling a real life situation can – ensure smooth transition from informal towards more formal knowledge, – help students make sense of concepts by working on situations in a plurality of settings and developing connections. 4

With regard to the actual situation of teaching in school, our view is consistent with what Torner, Potari & Zachariades write in a special issue of ZDM …. For us, modelling activiies can help

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Different models of a situation

Model 1

Model 2

Real situation

Model 4 Model 3

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Thus our view of modelling is slightly different as compared to classical modelling cycles. We propose to consider for a given real situation several models belonging to distinctive scientific fields, more or less close to reality.

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All models are mathematical, some are more

Model A

Model B

Real situation

Model C Model D

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Mathematisation is then seen as a progression between models with distinctive relationship to reality and mathematics, rather than as a sharp cut between reality and mathematics.

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Working on each model is working in a specific space Model A Working space A

Model B Working space B

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An important idea is of distinctive working spaces corresponding to the work on distinctive models

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A Mathematical Working Space (MWS) • An abstract space organized to ensure the mathematical work (in an educational context). • Three dimensions of the work – Semiotic : use of symbols, graphics, concrete objects understood as signs, – Instrumental: construction using artefacts (geometric figure, program..) – Discursive: justification and proof using a theoretical frame of reference (definitions, properties…) 8

This idea is inspired by the work of Alain Kuzniak and others. This very briefly how we define a Working Space.

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A typical “innovative” situation in French high school Optimizing the area of a given surface in relationship with a « real life » situation Phase 1. Make a dynamic geometry figure. Explore and conjecture the optimal figure Phase 2. Prove the result algebraically, taking a given length for x and calculating the area as a function of x.

Juxtaposition of two phases Phase 1 : No real working space. Motivation. Instrumental-Semiotic.

Phase 2: Reduction to algebraic calculation working space Semiotic-Discursive 9

I use this idea of working space to interpret the difficulties mentioned above int the teaching of Calculus in French high school even when innovative materials or situations are implemented.. This is here a typical classroom situation of geometrical optimization. In a first phase, students have to Make a dynamic geometry figure and conjecture an optimal position of a point In a second phase, they have to build an algebraic formula for the area and use algebraic means to solve the problem. Generally it fails: students meet difficulties to build the dynamic geometry figure and spend a lot of time especailly confusing free point on a segment, free point in the plane and constructed mobile points. This is generally not expected by the teacher. Then the teacher moves more or less abruptly to the phase 2. Most students faill to build an algebraic formula. Eventually, the teacher gives the formula and the students have to solve a classical algebraic task. I interpret this First by saying tha no adequate working space has been organised for the first phase, and then the dominant working space is the ordinary algebraic calculation working space of the second phase. The work in the firs phase has no mathematical dimension. It is rather for motivating students. With regard to dimensions of work. In the first phase the semiotic and instruments associated with dynamic geometry are involved.In the second phase, the expected work is mainly discursive because the teacher wants an algebraic proof, and it involves a semiotics, which is very different as compared to dynamic geometry. Thus the two phases of work are juxtaposed rather than connected.

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Choices and Hypothesis: deeper understanding by connecting working spaces Model A

Real situation

Working space A

Model B

Careful specification of working spaces Special classroom organisation to promote connections 1.Groups of experts 2. Groups of discussion

Working space B

……………..

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Thus our hypothesis is that teaching has to specify carefully working spaces for different models, to allow students to work in each space and to make connections. Then students can get deeper understanding of calculus concepts. The hypothesis is also that there is a need for special classroom organisation, to promote connections. Thus our choice is to organise two phases of group work that I will describe.

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Second Group Work (discussion) Task: Find connections between models

First Group Work (experts) Each group works on a model (A, B, C or D)

Gr 1

Gr 2

Gr 3

Gr 4

Gr A

A1

A2

A3

A4

Gr B

B1

B2

B3

B4

Gr C

C1

C2

C3

C4

Gr D

D1

D2

D3

D4

Organization • chosen in order that each student •performs by himself key tasks related to a model, •connects different models and associated concepts, • consistent with the idea of several working spaces to model a complex reality 11

This is table to explain this organisation, for instance if there are sixteen students and four different models The sixteen students are labeled, A1, A2, A3, A4, B1, B2 and so on we make four groups of four students for a first group work, each working on a distinctive model (A, B, C or D). Thus these students are experts of a model and of the corresponding working space. These expert groups are labeled A, B, C and D, the group A is made of 4 students labelled A1 to A4 and so on for the other groups For the second group work, we mix the groups. It means that in each new group we have students from each of the previous groups, thus one expert in each model. These groups are labeled one to four. The group labeled one gathers students labeled A1 B1, C1, D1 and so on for the other groups. In this second group work, students have to find connections between models in order to prepare a synthesis. Classes in France are typically bigger than thirty, thus there are more groups or bigger groups ! This was just a simple example.

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Modelling suspension bridges 12

• Four models • Four Working Spaces • Classroom implementation (12th grade) • Observation and evaluation • Conclusion

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This is example that we implemented in 12th grade classes in order to test our hypothesis. It involves four models of a suspension bridge and the associated working spaces. I will briefy present the models and working spaces, then the Classroom implementation, some elements of Observation and evaluation, before concluding.

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• The deck is hung below main cables by vertical suspensors equally spaced. • The weight of the deck applied via the suspensors results in a tension in the main cables. • There is no compression in the deck and this allows a light construction and a long span (Golden Gate, Akashi kaikyō…) . • Not to be confused with – Catenary (deck follows the cable) – Straight cables (Kolbäck Bridge …) 13

Some basic elements about suspension bridges

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A) Physical model of tensions

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In a first model a main cable is represented by a physical mockup, that is to say weights equally spaced horizontally and suspended to a line. The static equilibrium law allows studying the sequence of tensions in the line between the suspension points: the horizontal component Hi has the same value H in all segments and the sequence of values of the vertical component (Vi) is in arithmetic progression.

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B) Model in coordinate geometry M0 and Mn the anchoring points on the pillars, and M1, M2,…, Mn-1, the points where suspensors are attached on the cable, xi, yi the coordinates of Mi. The sequence of slopes (ci) of the segment [Mi, Mi+1] is in arithmetic progression. i ci xi yi 0

-0,3

-640

163

1

-0,1

-320

53,91

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0,1

0

17,55

3

0,3

320

53,91

640

163

4 y

200 150 100 50 0 -800

-600

-400

-200

0

200

400

600

800

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The second model is a broken line in coordinate geometry. The slope of each segment is the ratio of the vertical by the horizontal components of the tension in this segment, and therefore is also in arithmetical progression. After choosing a value for n It is possible to compute iteratively the sequence of the coordinates of the points and to draw the broken line. In this model and the following, the data is taken from the golden gate bridge.

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C) Algorithmic model In the program below, the data comes from the golden gate bridge and the origin of the coordinate system is at the middle of the deck. Weight of the deck: 20 MegaNewtons Distance between two pillars: 1 280m Elevation of pillars above the deck: 163m

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The third model systematizes the construction of the second model by way of an algorithm defining a piece wise linear function. It is possible to graph the function and animate the global variables n (number of segments) and H (horizontal component of the tension) in order to visualize their influence

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D) Continuous model, using a mathematical function V(x)= P. x / 2L

f ’(x)= V(x)/H

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The fourth model is a mathematical function depending on a parameter H: the derivative is calculated as a limit of the slopes of segments in the broken line and the function is obtained by integration. Like with the algorithmic model, It is possible to adjust the parameter H in order that the curve of the function f conforms to the shape of the cable.

The third and fourth models involve a software environment : functions are created by formula and domain, and also by an algorithm; Graphs can be obtained and can be animated by way of sliders. The software Casyopée was used for the experiment (http://casyopee.eu)

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Tasks for the groups of experts • Group A: physical model – recognize horizontal component constant, compute a recurrence formula for the vertical components.

• Group B: geometrical model – compute the series of x and ycoordinates of the suspension points for a small value of n.

• Group C: algorithmic model – enter and execute the algorithm, interpret the parameter n, and adjust the parameter H.

• Group D: continuous model – find a formula for the derivative of f. Find a formula for f and adjust the parameter H.

A) Static systems working space – Semiotic: sequence of tensions – Discursive: static equilibrium law, properties of progressions – Instrumental: measurement with concrete devices B) Geometrical working space – Semiotic: sequence of points and coordinates – Discursive: analytical definition of a segment C) Algorithmic working space – Semiotic: recurrence definition of sequences expressed in the programming language – Instrumental : programming, animation of parameters D) Mathematical functions space – Semiotic: standard mathematical functions – Discursive: classical rules in calculus. – Instrumental : graphing, animation of parameters 18

This is a brief presentation of the task in each of the “expert” groups of the first Group work. The group A works on the physical model. Students have to consider the sequence of horizontal and vertical components of tensions at the connection points, recognize that the horizontal component is constant and compute a formula for the series of vertical components The Group B: works on the geometrical model. A formula for the value of the slope of each segment is given to theim, depending on a parameter H, and on the number n of segments. They have to compute the series of x and ycoordinates of the suspension points for a small value of n and a given value of H. The Group C works on the algorithmic model. An algorithm is given to them; they have to enter and execute the algorithm, interpret the parameter n, and adjust the parameter H in order that the model given by the algorithm conforms to the shape of the cable. The. Group D works on the continuous model. Students have to search for a function f whose curve models a main cable. Information is given to them about the tension along the cable. They have to find a formula for the derivative of f, taking into account that the tension is in the direction of the tangent to the curve. Then, they have to find a formula for f and adjust the parameter H in order that the curve of the function f conforms to the cable. I treid here to briefly describe the key dimensions of each corresponding working space. I will not develop the four working spaces. For the Static systems working space corresponding to the physical model, the semiotic dimension is characterized by special indexed notations for the tensions. The discursive dimension is related to reasoning and proving. We have a physical law, actually the first Newton law, and also properties of the arithmetic

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Classroom Implementation • • • •

Preparation (one hour) Groups of experts (50 mn) Groups of discussion (50 mn)) Whole class synthesis (30 mn)

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This is how it is actually implemented the situation,. The two phases of group work where prepared by a phase of activity in the physics laboratory, and of research on the web regarding different types of bridges. The session is concluded by a synthesis in whole class directed by the

teacher.

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Connections Model in coordinate geometry

Algorithmic model

Evolution of the variables x and y

Students interpret the evolution of the variables x and y in the algorithm, by connecting the body of the loop with the recurrence law of the coordinates in the geometrical model

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I made observations during the different phases and by interviewing a selection of students after the session. I report here briefly on four connections I observed. The first one is between the Model in coordinate geometry and the Algorithmic model It is about the evolution of the coordinates of the suspension points. In the geometrical model these coordinates are two sequences of numbers computed step by step by the students, while in the algorithm they are represented by iterative computer variables evolving in a loop. So students had to make the connection between the step by step recurrence of the geometrical model, and the body of the loop in the algorithmic model, getting a better understanding both of recurrence in mathematics and iteration in algorithmics.

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Connections Algorithmic model

Physical model of tensions Animation of parameters

Students recognize n as the number of suspensors and H as the horizontal tension 21

The second connection is between the Physical and the Algorithmic model. Students had to animate the parameters n and H in the algorithmic model, but this animation did not alone give a meaning to these parameters. It is only when connecting with the physical model that students recognized n as the number of suspensors (+1) and H as the horizontal tension. The notion of parameter in functions is far from obvious for secondary students, and then the connection contributed to clarify this notion for the students.

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Connections Physical model of tensions

Continuous model Gradient in a point of the curve

yi / xi = Vi / Hi

f ‘ (x) = V(x)/ H

Observer asked to explain why the gradient in a point of the curve is the quotient of V and H. Students simply wrote f '(x) = y / x = V(x) / H.

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The third connection is about the calculation of the derivative of the mathematical function in the continuous model. As an observer I asked to explain why the gradient in a point of the curve is the quotient of V and H. The expected answer was that the tension has the direction of the tangent and that it is the limit of segments in the discrete model, but the students simply wrote f '(x) = Dy/Dx = V(x)/ H. The first equality is common in the physics course, and the second derives from the definition of the components in the physical model. The connection can be seen as a short cut, avoiding considering explicitely limits, but the notion of limit is so difficult for these students to formalize, that this understanding derived from connections with physics is to be appreciated.

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Connections Identification of curves Algorithmic model

Continuous model

No show clear awareness that the function is the limit of the continuous piecewise function. From graphical evidence students thought that it was more or less the same function for big values of n. 23

There is finally a connection between the algorithmic model and the continuous model, with a similar shortcut. Students did not show clear awareness that the mathematical function of the continuous model is the limit of the continuous piecewise function of the algorithmic model. From graphical evidence they thought that it was more or less the same function for big values of n. This understanding has also to be appreciated: it involves convergence of functions. It is a difficult notion, but dynamic graphic tools now make it possible to approach visually.

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Connections

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Finally I summarize these connections in a figure on the left, and I interpret these in terms of connections between working spaces with the priviledfed dimensions for each. I have no time to detail these connections, but I stress the diversity of these connections and that two of these implys a discursive dimension linked to proof.

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Conclusion • Potential of the situation – Students understood the main aspects of the models and the connections between them. – Students understood more comprehensively concepts • in physics, • In geometry and calculus, • in algorithmics

thanks to the connections.

• Potential of the framework – Specification of adequate working spaces – Adequate classroom organisation – Evaluation of students’ modelling activity 25

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