Geometry in Natural and Anthropic Environments and Its Teaching
A.Y. 2025/2026
Learning objectives
This class will deal with some basic questions on the learning and teaching of middle school mathematics.
The focus will be on some specific topics, the goal being the production of teaching items.
The focus will be on some specific topics, the goal being the production of teaching items.
Expected learning outcomes
Basic elements of methodologies and technologies for the teaching of geometry useful for the exercise of the teaching profession.
Lesson period: Second semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
Part A (Mathematics). The lessons will cover the main curricular topics of mathematics in lower secondary schools, with particular reference to plane and solid geometry (e.g.: problems of area and perimeter, Pythagorean Theorem, curves on the plane, isometric transformations, solids). Innovative, inclusive, and hands-on teaching methods and tools will be presented, which can be used to actively engage students, stimulating their creativity and critical thinking. Each mathematical topic will be linked to an aspect of natural sciences. An educational outing to explore geometry in natural and human-made environments is planned.
Handouts will be prepared, and bibliographic references will be provided during the lessons.
Part B. Interdisciplinary Laboratory
The lessons will be divided into 6 hours of laboratory with theoretical elements (classroom/laboratory) and 16 hours of practical laboratory work usually carried out at the Città Studi Botanical Garden, including a possible educational field trip.
Over the 22 hours of the course, various topics related to geometry found in natural environments will be covered. Main topics include: isometries in nature, phyllotaxis, floral diagrams, numerical sequences. The course aims to provide tools for using group-based and collaborative laboratory teaching methods through the development of interdisciplinary interactive learning paths for lower secondary schools.
Bibliographic references will be provided during the lessons.
Handouts will be prepared, and bibliographic references will be provided during the lessons.
Part B. Interdisciplinary Laboratory
The lessons will be divided into 6 hours of laboratory with theoretical elements (classroom/laboratory) and 16 hours of practical laboratory work usually carried out at the Città Studi Botanical Garden, including a possible educational field trip.
Over the 22 hours of the course, various topics related to geometry found in natural environments will be covered. Main topics include: isometries in nature, phyllotaxis, floral diagrams, numerical sequences. The course aims to provide tools for using group-based and collaborative laboratory teaching methods through the development of interdisciplinary interactive learning paths for lower secondary schools.
Bibliographic references will be provided during the lessons.
Prerequisites for admission
None
Teaching methods
Part A. Methodology: Hands-on, object-based learning, learning by doing, inquiry-based learning, cooperative and collaborative learning. Attendance is strongly recommended.
Part B. Laboratory with theoretical elements, practical exercises, and various types of hands-on activities.
Attendance is strongly recommended for the 6 hours of lab with theoretical elements.
Attendance is mandatory for the 16 hours of practical lab.
Part B. Laboratory with theoretical elements, practical exercises, and various types of hands-on activities.
Attendance is strongly recommended for the 6 hours of lab with theoretical elements.
Attendance is mandatory for the 16 hours of practical lab.
Teaching Resources
PDF slides and other materials used during the classroom and Botanical Garden activities will be made available on the MyAriel platform. In addition to the shared materials, students can personally explore some course topics by consulting the following recommended texts:
Parte A.
Castelnuovo E., 2017. Didattica della matematica.UTET Università.
Castelnuovo E., 2017. Pentole, ombre, formiche. In viaggio con la matematica. UTET Università.
Frigerio E., Spreafico M.L.S., 2018. Ed ora, origami. 18 laboratori di matematica con l'origami. Kangourou Italia.
Parte B.
Angelucci, G. (2024). Matematica in giardino. Quaderno di laboratorio per le classi della scuola secondaria di I grado. Milano University Press. DOI: https://doi.org/10.54103/milanoup.142
Bell D., A. (1993). La forma delle piante. Guida illustrata alla morfologia delle angiosperme (F. Bracco, Trans.; 1st.ed.). Zanichelli. (Original work published 1991).
Dedò, M. (2010). Galleria di metamorfosi. Milano. Mimesis.
Parte A.
Castelnuovo E., 2017. Didattica della matematica.UTET Università.
Castelnuovo E., 2017. Pentole, ombre, formiche. In viaggio con la matematica. UTET Università.
Frigerio E., Spreafico M.L.S., 2018. Ed ora, origami. 18 laboratori di matematica con l'origami. Kangourou Italia.
Parte B.
Angelucci, G. (2024). Matematica in giardino. Quaderno di laboratorio per le classi della scuola secondaria di I grado. Milano University Press. DOI: https://doi.org/10.54103/milanoup.142
Bell D., A. (1993). La forma delle piante. Guida illustrata alla morfologia delle angiosperme (F. Bracco, Trans.; 1st.ed.). Zanichelli. (Original work published 1991).
Dedò, M. (2010). Galleria di metamorfosi. Milano. Mimesis.
Assessment methods and Criteria
Exam includes written and oral components.
A. The written exam consists of a final test involving the solution of simple exercises similar to those covered during the course (22 points). The oral exam is a presentation (with slideshow) by the student on one of the teaching activities from Part A, enriched with a personal interdisciplinary connection to science (10 points).
The final written test can be replaced by weekly submissions of solved exercise sheets.
Attendance is not mandatory but strongly recommended. If unable to attend lessons, students must make up the missed work at home, following weekly instructions on MyAriel and submitting assignments by the deadlines.
Evaluation criteria for Part A: the written exam will assess the correctness of exercises; the oral will evaluate the structure of the presented lesson, the use of subject-specific language, and the originality of the interdisciplinary contribution.
B. The written part involves drafting a teaching project for lower secondary school, according to the format provided during the course. The project must include an original in-depth study on a topic approved by the instructor. A bibliographic review must be conducted using at least two texts suggested during the course. A slideshow (PDF or PPT format) must be submitted to the instructor at least ten days before the oral exam for potential revisions.
The oral exam includes the presentation of the teaching activity from Part A and a discussion of the teaching project from Part B, using a slideshow, in front of the instructor and classmates. Duration: 15-20 minutes.
The first slide should contain the project title and author's name; the final slide should list the bibliographic references.
If unable to attend the non-mandatory 6 hours of lab with theory elements, the missed work must be made up at home and uploaded according to MyAriel's weekly instructions.
Evaluation criteria for Part B: the project will be assessed based on understanding and clear presentation of course and reference materials, quality and quantity of bibliographic research, clarity and language proficiency, originality of the work and content, and quality of the slideshow.
Final Grade
The final grade is the weighted average of the marks for Part A and Part B (2/3 for Part A and 1/3 for Part B), expressed on a 30-point scale.
A. The written exam consists of a final test involving the solution of simple exercises similar to those covered during the course (22 points). The oral exam is a presentation (with slideshow) by the student on one of the teaching activities from Part A, enriched with a personal interdisciplinary connection to science (10 points).
The final written test can be replaced by weekly submissions of solved exercise sheets.
Attendance is not mandatory but strongly recommended. If unable to attend lessons, students must make up the missed work at home, following weekly instructions on MyAriel and submitting assignments by the deadlines.
Evaluation criteria for Part A: the written exam will assess the correctness of exercises; the oral will evaluate the structure of the presented lesson, the use of subject-specific language, and the originality of the interdisciplinary contribution.
B. The written part involves drafting a teaching project for lower secondary school, according to the format provided during the course. The project must include an original in-depth study on a topic approved by the instructor. A bibliographic review must be conducted using at least two texts suggested during the course. A slideshow (PDF or PPT format) must be submitted to the instructor at least ten days before the oral exam for potential revisions.
The oral exam includes the presentation of the teaching activity from Part A and a discussion of the teaching project from Part B, using a slideshow, in front of the instructor and classmates. Duration: 15-20 minutes.
The first slide should contain the project title and author's name; the final slide should list the bibliographic references.
If unable to attend the non-mandatory 6 hours of lab with theory elements, the missed work must be made up at home and uploaded according to MyAriel's weekly instructions.
Evaluation criteria for Part B: the project will be assessed based on understanding and clear presentation of course and reference materials, quality and quantity of bibliographic research, clarity and language proficiency, originality of the work and content, and quality of the slideshow.
Final Grade
The final grade is the weighted average of the marks for Part A and Part B (2/3 for Part A and 1/3 for Part B), expressed on a 30-point scale.
MAT/03 - GEOMETRY - University credits: 2
MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 4
MAT/04 - MATHEMATICS EDUCATION AND HISTORY OF MATHEMATICS - University credits: 4
Practicals: 16 hours
Practicals with elements of theory: 24 hours
Lessons: 18 hours
Practicals with elements of theory: 24 hours
Lessons: 18 hours
Professor:
Spreafico Maria Luisa Sonia
Professor(s)
Reception:
by appointment