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This web site reflects the views only of the author, and the Commission cannot be held responsible for any use which may be made of the information contained therein.

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Proving theorems – intermediate level

Class

Reasoning/Geometry

- Engagement/recruiting interest

- Engagement/Sustaining efforts and persistence

- Engagement/Self-regulation

- Representation/Perception

- Representation/Language and symbols

- Action and expression/Physical action

- Action and expression/Expression and communication

- Action and expression/Executive functions

- Engagement/Sustaining efforts and persistence

- Engagement/Self-regulation

- Representation/Perception

- Representation/Language and symbols

- Action and expression/Physical action

- Action and expression/Expression and communication

- Action and expression/Executive functions

- Engineering classroom discussions

- Providing feedback

- Activating students s resources for one another

- Activating learners as the owners of their own learning

- Providing feedback

- Activating students s resources for one another

- Activating learners as the owners of their own learning

Projector or interactive whiteboard to project the power point file wth the questions.

If also the students have at disposal tablets or computers with internet connection, the polls can be administered by menas of an interactive response system (e.g. Socrative, mentimeter)

If also the students have at disposal tablets or computers with internet connection, the polls can be administered by menas of an interactive response system (e.g. Socrative, mentimeter)

60 minutes

The intervention tool is conceived to address specific difficulties related to the mathematical domain of geometry and the cognitive domain of reasoning. By means of the intervention tool, that is conceived for all the class, the students may reflect on the proving process, with specific reference to crucial steps such as understanding the text, identifying hypothesis and thesis, representing hypotheses on the figure, organizing proof as a sequence of logically connected statements.

We suggest to consider also the intervention tool “Proving – intermediate level” after this one. The tools have the same educational aim, with increasing difficulty concerning the statement to be proved.

The tool consists in a series of questions the teacher may pose to the students during a class discussion. Questions may be projected on the whiteboard. If the students have at disposal tablets or computers with internet connection, the questions can be administered by means of an interactive response system (e.g. Socrative, Mentimeter).

We suggest to consider also the intervention tool “Proving – intermediate level” after this one. The tools have the same educational aim, with increasing difficulty concerning the statement to be proved.

The tool consists in a series of questions the teacher may pose to the students during a class discussion. Questions may be projected on the whiteboard. If the students have at disposal tablets or computers with internet connection, the questions can be administered by means of an interactive response system (e.g. Socrative, Mentimeter).

By means of the intervention tool, students are guided to construct a proof, by reflecting on important steps: understanding the text, identifying hypothesis and thesis, representing hypotheses on the figure and with other representation systems (such as algebraic formulas), recalling already known geometrical facts, organizing proof in form of a deductive chain of arguments.

Balacheff N. (1982). Preuve et démonstration en mathématiques au collège, Recherches en Didactiques des Mathématiques, vol.3, pp. 261-304.

Karagiannakis, G. N., Baccaglini-Frank, A. E., & Roussos, P. (2016). Detecting strengths and weaknesses in learning mathematics through a model classifying mathematical skills. Australian J. of Learning Difficulties, 21(2), 115–141. https://doi.org/10.1080/19404158.2017.1289963

Black, P., & Wiliam, D. (2009). Developing the theory of formative assessment. Educational Assessment, Evaluation and Accountability, 21(1), 5-31.

Cusi, A., Morselli, F.,& Sabena, C. (2017). Promoting formative assessment in a connected classroom environment: design and implementation of digital resources. Vol. 49(5), 755–767. ZDM Mathematics Education.

Cusi, A., Morselli, F.,& Sabena, C. (2018). Enhancing formative assessment in mathematical class discussion: a matter of feedback. Proceedings of CERME 10, Feb 2017, Dublin, Ireland. hal-01949286, pp. 3460-3467.

Karagiannakis, G. N., Baccaglini-Frank, A. E., & Roussos, P. (2016). Detecting strengths and weaknesses in learning mathematics through a model classifying mathematical skills. Australian J. of Learning Difficulties, 21(2), 115–141.

Robotti E., Baccaglini-Frank A., (2017). Using digital environments to address students’ mathematical learning difficulties. In Innovation & Technology. Series Mathematics Education in the Digital Era, A. Monotone, F. Ferrara (eds), Springer Publisher.

Karagiannakis, G. N., Baccaglini-Frank, A. E., & Roussos, P. (2016). Detecting strengths and weaknesses in learning mathematics through a model classifying mathematical skills. Australian J. of Learning Difficulties, 21(2), 115–141. https://doi.org/10.1080/19404158.2017.1289963

Black, P., & Wiliam, D. (2009). Developing the theory of formative assessment. Educational Assessment, Evaluation and Accountability, 21(1), 5-31.

Cusi, A., Morselli, F.,& Sabena, C. (2017). Promoting formative assessment in a connected classroom environment: design and implementation of digital resources. Vol. 49(5), 755–767. ZDM Mathematics Education.

Cusi, A., Morselli, F.,& Sabena, C. (2018). Enhancing formative assessment in mathematical class discussion: a matter of feedback. Proceedings of CERME 10, Feb 2017, Dublin, Ireland. hal-01949286, pp. 3460-3467.

Karagiannakis, G. N., Baccaglini-Frank, A. E., & Roussos, P. (2016). Detecting strengths and weaknesses in learning mathematics through a model classifying mathematical skills. Australian J. of Learning Difficulties, 21(2), 115–141.

Robotti E., Baccaglini-Frank A., (2017). Using digital environments to address students’ mathematical learning difficulties. In Innovation & Technology. Series Mathematics Education in the Digital Era, A. Monotone, F. Ferrara (eds), Springer Publisher.