Topics in Discrete Mathematics

Scholar Year: 2018/2019 - 2S

Courses

Head Teacher Responsability Ana Maria Dias Roque Lemos Boavida Head

Weekly workload Hours/week T TP P PL L TC E OT OT/PL TPL O S Type of classes Lectures Type Teacher Classes Hours Contact hours Totals 1 4,00 Ana Maria Boavida   4,00

Teaching language

Portuguese

Intended learning outcomes (Knowledges, skills and competencies to be developed by the students)

To understand topics of discrete mathematics.
To apply discrete mathematical topics to interpret, represent and investigate problems and investigations.
To reflect on the acquired knowledge to critically analyse the integration of discrete mathematical topics in the official curricular guidelines for students from 6 to 12 years old.

Syllabus

1. Systematic listing, systematic counting and reasoning: tree diagrams; arrangements and combinations; the fundamental counting principle and the addition principle; Pigeonhole principle (Dirichlet theorem); Problem solving.
2. Modelling and problem solving using graphs and trees: Definition of a graph; Vertex and edges; Valency of a vertex; paths and cycles; Simple, complete and connected graphs; Euler and semi-eulerian graphs; Hamilton graphs and trees; Hamiltonian cycle; weighted graphs and algorithms to determine optimal solutions; problem solving.
3. Iteration and recursion: patterns and iterative processes; recursive relations; problem solving.
4. Organizing and processing information: diagrams and tables; elementary notions of modular arithmetic; ciphers and codes; problem solving.
5. Algorithms and and algorithmic language: analysis, elaboration and verification of algorithms; flowcharts; problem solving.

Demonstration of the syllabus coherence with the UC intended learning outcomes

Topics of discrete mathematics have an important role in mathematical education and in the official curricular guidelines for Basic Education. It is therefore important that future teachers of 1st and 2nd cycle acquire a sound scientific knowledge on discrete mathematics topics and analyse how these topics are included in the official curriculum

Teaching methodologies

The work to be developed will be focused on the participation of students, whether in individual or in group work, looking for the deepening of knowledge related to different topics of Discrete Mathematics. The activities to be developed will include (a) the study of scientific texts related to the syllabus, (b) the exploration and critical analysis of problems and (c) the research of relevant information in order to deepen mathematical knowledge and, in particular, to develop thematic essays concerning syllabus themes.
The classes will be oriented by a problem-solving pedagogy so that the theoretical information is mobilised in situations of practical nature and vice versa. Besides, there will be tutorial sessions focused on the organization of the study, clarifying doubts and monitoring the preparation of the essay.

Demonstration of the teaching methodologies coherence with the curricular unit's intended learning outcomes

To build a solid knowledge on discrete mathematics topics and reflect on their integration in official curricular guidelines for basic education, is important that the methodologies actively involve students in the process of learning. Therefore, this CU will be organized around (1) problem solving activities and (2) oral presentations by the teacher and the students; both activities will be focused on topics of discrete mathematics. Particularly, will be explored and analysed problems and documents related to the syllabus, will be elaborated, by the students and will be presented and discussed works produced by students, namely the thematic essays.

Assessment methodologies and evidences

The evaluation elements are (a) the development and presentation of an essay and (b) an individual written test. These elements have, respectively, a weight of 60% and 40% in the final grade. Alternatively students can take a final exam.

Attendance system

continous evaluation- classroom attendence minimum of 75%

Bibliography

DeBellis, V., Rosenstein, J., Hart, E. & Kenney, M. (2011). Navigating through discrete mathematics in Prekindergarten- grade 5. Reston: NCTM.
Hert, E., DeBellis, V., & Rosenstein; J. (2008). Navigating through discrete mathematics in grades 6-12. Reston: NCTM.
Johnsonbaugh, R. (1986). Discrete mathematics. New Jersey: Prentice Hall International.
NCTM (Ed.). (2007). Princípios e normas para a matemática escolar. Lisboa: APM (tradução dos Standards 2000, publicado em 2000 pelo NCTM).
Rosenstein, J., Franzblau, D., & F., Roberts. (Eds.). (1997). Discrete mathematics in the schools. Providence, Rhode Island: American Mathematical Society, National Council of Teachers of Mathematics.