Curricular Didactic Analysis: an experience with future teachers

Cotrado, Bethzabe; Burgos, María; Beltrán-Pellicer, Pablo

doi:10.1590/2175-6236132399vs01

Bethzabe Cotrado E-mail: bcotrado@unap.edu.pe

Bethzabe Cotrado is a doctoral candidate at the University of Granada, Spain. Professor at the Faculty of Educational Sciences, Universidad Nacional del Altiplano, Peru. Her lines of research include the training of secondary education teachers and the analysis of curricular materials on the topic of probability with the application of the theoretical tools of the Ontosemiotic Approach.

Universidad Nacional del Altiplano (UNAP), Puno – Peru

https://orcid.org/0000-0002-0071-7242

María Burgos E-mail: mariaburgos@ugr.es

María Burgos holds a PhD in Mathematical Sciences from the University of Almería. D. in Educational Sciences (Didactics of Mathematics) from the University of Granada. Professor of University, she teaches in the Degree of Primary Education and in the Master of Didactics of Mathematics of the University of Granada.

Universidad de Granada (UGR), Granada – Spain

https://orcid.org/0000-0002-4598-7684

Pablo Beltrán-Pellicer E-mail: pbeltran@unizar.es

Pablo Beltrán-Pellicer holds a PhD in Didactics from the Universidad Nacional de Educación a Distancia (UNED). Professor in the area of Didactics of Mathematics at the University of Zaragoza, where he teaches in the degree of Teaching in Primary Education and in the Master's Degree in Teaching of Compulsory Secondary Education and Baccalaureate.

Universidad de Zaragoza (UNIZAR), Zaragoza – Spain

https://orcid.org/0000-0002-1275-9976

E-mail: bcotrado@unap.edu.pe

E-mail: mariaburgos@ugr.es

E-mail: pbeltran@unizar.es

Editor in charge: Carla Karnoppi Vasques

Mathematical objects	Analysis units
Mathematical objects	DG1.1	DG1.2	DG1.3	DG1.4	DG1.5
Problem-situations
Recognizing the conditions that define a random situation	x
Expressing the value (decimal or percentages) of probability as more or less likely	x	x
Determining the probability of events using the Laplace rule or calculating their relative frequency	x			x
Interpreting information from various texts with values or descriptions of random situation			x
Formulating statements or conclusions about the probability of event occurrences					x
Languages
Verbal	x	x	x	x	x
Simbolic – numerical	x	x		x	x
Graphical		x	x
Tabular		x	x
Concepts
Random situation	x	x	x	x
Events, single events	x	x		x	x
More or less probable event	x	x
Probability	x			x	x
Frequency, relative frequency	x			x
Decimals, percentages	x			x
Bar graph, pie chart			x
Procedures
Distinguishing the conditions characterising a random situation	x
Comparison of probability expressed in decimals or percentages	x
Aplication of Laplace’s rule	x			x
Symbolic or graphical representation
Reading tables, graphs and texts with random situations			x
Application of different representations to express the value of probability		x
Calculation of relative frequency and percentage				x
Reviewing procedures				x
Drawing conclusions and correcting errors					x
Propositions
Laplace’s rule	x			x
The probability of an event is a calculable value	x	x
The relative frequency of an event varies between 0 and 1				x
Arguments
Justifying the conditions for randomness and which event is more or less likely than another event	x	x
Making assertions and drawing conclusions					x
Recognising errors in their justifications					x

Indicators by components	Fulfilment degree	Analysis units identified in the curricular programme
Problem-situation
I1. Proposing the use and formulation of problem-situations that showcase and relate different meanings of probability (intuitive, subjective, frequency-based, and classical).	Sometimes	- Every mathematical activity takes place in the context of solving problems derived from situations, which are conceived as meaningful events occurring in various settings (p. 148) - NC6, DG1.1, DG1.2, DG1.3, DG1.4 y DG1.5, DG2.1, DG2.2, DG2.3, DG2.4 y DG2.5.
I2. Emphasize the formulation of situations where the student generates, experiments with, and simulates problems involving random experiences (problematization).	Never	- The CP does not specify this indicator anywhere in the document.
Languages
I3. Promotes the use of different linguistic registers and specific representations of probability such as verbal, symbolic-numeric, tabular and graphical expressions.	Always	- Represents data with graphs and statistical or probabilistic measures: depicts the behaviour of a set of data by selecting statistical tables or graphs (p. 170). - Expresses their understanding of (…) the value of probability using various representations and mathematical language (p. 172).
I4. Appropriate linguistic level for the targeted students.	Always

Quality of analysis to identify objects	Analysis units
Quality of analysis to identify objects	DG1.1	DG1.2	DG1.3	DG1.4
Problem-situations
0 points	12	14	12	11
1 point	1	0	0	0
2 points	1	0	0	0
3 points	0	0	2	3
Languages
0 points	12	2	1	1
1 point	0	3	1	2
2 points	1	9	11	2
3 points	1	0	1	10
Concepts
0 points	10	2	2	1
1 point	3	0	8	0
2 points	1	12	4	13
3 points	0	0	0	0
Procedures
0 points	12	13	10	2
1 point	1	1	4	6
2 points	1	0	0	6
3 points	0	0	0	0
Propositions
0 points	13	14	14	2
1 point	1	0	0	12
2 points	0	0	0	0
3 points	0	0	0	0

Quality of analysis to identify objects	Analysis units
Quality of analysis to identify objects	DG2.1	DG2.2	DG2.3	DG2.4	DG2.5
Problem-situations
0 points	1	14	12	5	9
1 point	8	0	2	1	0
2 points	5	0	0	1	0
3 points	0	0	0	7	5
Languages
0 points	2	1	1	1	2
1 point	0	2	0	0	1
2 points	0	11	10	2	1
3 points	12	0	3	11	10
Concepts
0 points	1	2	2	1	4
1 point	2	2	0	2	4
2 points	11	5	1	9	0
3 points	0	5	11	2	6
Procedures
0 points	7	14	6	4	14
1 point	6	0	0	0	0
2 points	1	0	0	3	0
3 points	0	0	8	7	0
Propositions
0 points	1	14	14	3	14
1 point	11	0	0	0	0
2 points	2	0	0	0	0
3 points	0	0	0	11	0

Indicators by components	Assessment
Indicators by components	Always	Sometimes	Never	No response
Problem-situations
I1. Proposes the use and formulation of problem-situations that show and relate different meanings of probability (intuitive, subjective, frequentist, and classical).	9	1	0	0
I2. Emphasises the formulation of situations where the student generates, experiments, and simulates problems on random experiences (problematization).	6	1	3	0
Languages
I3. Promotes the use of different linguistic registers and specific representations of probability, such as verbal expressions, symbolic-numerical, tabular, and graphical.	8	0	0	2
I4. Linguistic level suitable for the target student audience.	2	4	1	3
Concepts
I5. Includes essential concepts: random and deterministic experiment, sample space, event (simple and compound, certain and impossible), favourable and possible cases, frequency, relative frequency, convergence, simulation, experimentation, variability, equiprobability, and probability.	6	2	2	0
Propositions
I6. Proposes the use of propositions and properties such as the probability of the impossible event, certain event and complementary, stability of relative frequencies c, Laplace's rule, and equiprobability.	5	5	0	0
Procedures
I7. Includes procedures for qualitative comparison of probabilities; construction of sample space, application of Laplace's rule, making predictions from observations or data, estimating probabilities, calculating and representing frequencies, using and interpreting diagrams, tables, and graphs, simulating random experiments.	7	3	0	1
Arguments
I8. Recognises the importance of argumentation as a means to demonstrate or justify propositions and solution procedures in which inductive or deductive reasoning may or may not be manifested.	5	3	0	2
Relations
I9. Presents mathematical objects (problems, definitions, etc.) related and connected to each other.	5	1	2	2
I10. Recognises and presents the articulation of the various meanings of probability (intuitive, subjective, frequentist, and classical) as an organised whole.	5	0	3	2

Indicators by components	Assessment
Indicators by components	Always	Sometimes	Never	No response
Prior knowledge
I11. Suggests progressively working on content according to the meanings of probability.	7	1	2	0
I12. Proposes achievable content with a manageable degree of difficulty in the various meanings of probability.	0	4	4	2
Cognitive conflicts
I13. Suggests posing situations to prevent and overcome errors and biases in probabilistic reasoning: representativeness and equiprobability.	3	5	1	1
Individual differences
I14. Promotes access, achievement, and support for all students.	3	2	3	2
Assessment
I15. Provides guidelines on assessment, its procedures, and the application of various techniques and instruments	6	3	1	0
I16. Proposes disseminating assessment results to make decisions.	3	2	4	1

Indicators by components	Assessment
Indicators by components	Always	Sometimes	Never	No response
Interests and needs
I17. Suggests posing probability situations based on the needs and interests of the students.	6	0	3	1
Emotions
I18. Proposes planning moments where students express emotions in the face of the proposed random situations.	3	2	4	1
I19. Suggests posing contextualised random situations and elements that can be motivating.	4	5	0	1
I20. Promotes self-esteem, avoiding rejection, phobia, or fear of posing or addressing probability situations or participating in random experiments and simulations.	3	3	4	0
Attitudes
I21. Encourages participation in activities, perseverance, responsibility, etc. to foster a positive attitude towards probability.	5	2	2	1
Beliefs
I22. Guides the teaching-learning process of probability gradually based on the beliefs of the students.	4	3	0	3
Values
I23. Takes into account and highlights the value and utility of chance and probability in the students' daily lives.	5	4	0	1

Indicators by components	Assessment
Indicators by components	Always	Sometimes	Never	No response
Teacher-student interaction
I24. Suggests appropriate communicative skills for the language of probability.	3	4	1	2
I25. Encourages the teacher to use various types of dialogue to guide communicative interaction in the classroom.	6	2	2	0
Interaction among students
I26. Suggests moments that favour dialogue and communication between students.	5	3	2	0
I27. Proposes inclusion in the group and avoids exclusion.	5	3	1	1
Autonomy
I28. Suggests autonomous work by students in solving probabilistic situations.	8	2	0	0

Indicators by components	Assessment
Indicators by components	Always	Sometimes	Never	No response
Material resources
I29. Promotes the use of manipulative materials (dice, coins, cards, balls), audiovisuals, and ICT (software, applets, and random devices) to enhance and understand the meanings of probability.	8	2	0	0
No. of students, schedule, and classroom conditions
I30. Suggests employing or prioritising an appropriate schedule to teach probability topics.	3	1	5	1
I31. Sets specific guidelines for the proper use of space, equipment, and classroom resources.	4	3	2	1
Time
I32. Proposes managing time in favour of achieving the proposed objectives for teaching probability.	3	3	3	1

Indicators by components	Assessment
Indicators by components	Always	Sometimes	Never	No response
Curriculum alignment
I33. The meanings, their implementation, and assessment of probability align with international curricular guidelines and research.	4	4	1	1
Openness to innovation
I34. Promotes the implementation of innovative activities based on research and reflective practice.	4	2	2	2
Socio-professional alignment
I35. The contemplated probability contents contribute to the socio-professional training of students	4	4	2	0
Education in values
I36. Encourages training in democratic values, inclusiveness, and equal opportunities for making inquiries (critical thinking).	4	5	1	0
Intra and Interdisciplinary Connections
I37. The probability contents relate to other intra and interdisciplinary contents.	5	1	3	1

Assessment of suitability		Accuracy in the identification of mathematical objects
		Problem-situations				Languages				Concepts				Procedures				Propositions
		0	1	2	3	0	1	2	3	0	1	2	3	0	1	2	3	0	1	2	3
*Epistemic*
Alta	2	0	2	0	0	0	0	0	2	0	0	1	1	0	2	0	0	0	2	0	0
Media	1	0	1	0	0	0	0	0	1	0	0	1	0	0	1	0	0	0	1	0	0
Baja	7	2	5	0	0	0	1	5	1	0	4	2	1	0	7	0	0	2	5	0	0
*Cognitive*
Alta	6	0	6	0	0	0	0	3	3	0	3	2	1	0	6	0	0	0	6	0	0
Media	1	0	1	0	0	0	0	1	0	0	0	1	0	0	1	0	0	1	0	0	0
Baja	3	2	1	0	0	0	1	1	1	0	1	1	1	0	3	0	0	1	2	0	0
*Affective*
Alta	2	0	2	0	0	0	0	1	1	0	0	2	0	0	2	0	0	1	1	0	0
Media	1	0	1	0	0	0	0	0	1	0	0	0	1	0	1	0	0	0	1	0	0
Baja	7	2	5	0	0	0	1	4	2	0	4	2	1	0	7	0	0	1	6	0	0
*Interactional*
Alta	1	0	1	0	0	0	0	1	0	0	1	0	0	0	1	0	0	0	1	0	0
Media	3	0	3	0	0	0	0	2	1	0	2	1	0	0	3	0	0	0	3	0	0
Baja	6	2	4	0	0	0	1	2	3	0	1	3	2	0	6	0	0	2	4	0	0
*Mediational*
Alta	3	0	3	0	0	0	1	2	0	0	0	2	1	0	3	0	0	1	2	0	0
Media	3	0	3	0	0	0	0	2	1	0	2	1	0	0	3	0	0	0	3	0	0
Baja	4	2	2	0	0	0	1	2	1	0	1	1	1	0	4	0	0	1	3	0	0
*Ecological*
Alta	5	0	5	0	0	0	0	3	2	0	2	2	1	0	5	0	0	1	4	0	0
Media	2	1	1	0	0	0	0	1	1	0	0	1	1	0	2	0	0	0	2	0	0
Baja	3	1	2	0	0	0	1	1	1	0	2	1	0	0	3	0	0	1	2	0	0

[1] Source: authors' elaboration.

Curricular Didactic Analysis: an experience with future teachers

ABSTRACT