Integrating of Competences in Mathematics through ...

New Rules for Assessing Mathematical Competencies

Integrating of Competences in Mathematics through Software - Case Study H. Kulina, S. Gocheva-Ilieva, D. Voynikova, P. Atanasova, A. Iliev, A. Ivanov University of Plovdiv Paisii Hilendarski, Plovdiv, Bulgaria

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 Object of study Application of mathematical software in the training process of first-year bachelor students in order to expand their mathematical competences.

 Aim of the study To share results of a special working laboratory notebook in Software systems of mathematics course as a new kind of integrated assessment tool for mathematics competences

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 Mathematical competencies C1: thinking mathematically; C2: reasoning mathematically; C3: posing and solving mathematical problems; C4: modeling mathematically; C5: representing mathematical entities; C6: handling mathematical symbols and formalism; C7: communicating in, with and about mathematics; C8: making use of aids and tools.

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 Software systems of mathematics course (SSM) – content and implementation in the learning process From 2013, the SSM course is independent discipline for the first year of the Software Technologies and Design bachelor program in our Faculty (150 students). The aim of the SSM course: to show contemporary ways of solving mathematical problems and to motivate students to use computers, Internet and mobile devices to study mathematics. The course assists students in the other math courses such as Applied Mathematics 1, Applied Mathematics 2, Geometric Design. o Curriculum - 20 hours lectures and 20 hours labs o Content - Problems in linear algebra, analytical geometry and mathematical analysis o Software - Wolfram Mathematica, wolframalpha 4/14 EDUCON2018 17-20 April, 2018, Santa Cruz de Tenerife, Canary Islands, Spain

 Teaching methodology The lectures demonstrate Wolfram Mathematica's core capabilities in the corresponding field and various software applications for solving a variety of math problems to the students Labs are organized for individual work of the students during the classes: Each student has his own laboratory notebook to work and write the results there: o Write down the main built-in functions of the current theme in the laboratory notebook and consult the Help system of the software o Repeat the solved problems from the laboratory notebook o Solve individual problems from the notebook on their own during the class o Send their personal files to the assistant by email at the end of the class Assessment: one individual written test during the trimester + working files Final grade is calculated from the point of the 10 laboratory exercise classes (2 points per lab) via the laboratory notebook and the test (max 20 points). 5/14 EDUCON2018 17-20 April, 2018, Santa Cruz de Tenerife, Canary Islands, Spain

 Integrated learning results and development of mathematical competencies The main competence aimed to be acquired by the students is C8: making use of aids and tools in 4 domains:  algebraic equations, inequalities and systems of equations and inequalities  linear algebra (vectors and matrices)  analytical geometry  mathematical analysis

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 Solving algebraic equations, inequalities and systems of equations and inequalities s Required knowledge: Understanding the algebraic and graphical meaning of algebraic equation, system of equations, algebraic inequality and system of inequalities Acquired knowledge, skills and abilities: o how changes in the parameters of the equations affect the type of the resulting solutions of algebraic equation or system of equations; o how the type of given solution set - integer, real, complex, affects the number and types of roots of the algebraic equation (system of equations); o uses graphical representation as a confirmation of algebraic solutions; o finds the solution set of inequalities with one or more variables both algebraically and graphically (see Fig. 1); o determine different types of solution sets of systems of inequalities with two or three variables in its graphical representation. 7/14 EDUCON2018 17-20 April, 2018, Santa Cruz de Tenerife, Canary Islands, Spain

Developed competences that the student exhibits and can be stimulated: C1, C2, C3, C6, C7, C8. The use of software supports and builds on the understanding of the concepts and methods studied in the previous LAAG course and the development of the necessary competences for their implementation. For example, competences C1 and C2 are developed by the possibility for detailed examining of the influence of different parameters on the area of solutions, determination and demonstration of solutions (including their number and type), etc.

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Fig. 1 Example from the laboratory notebook for solving a system of inequalities.

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 Solving problems in linear algebra Required knowledge: Understanding the basic terms associated with matrices (diagonal, triangular, square, single, transposed, row matrix, column matrix). Acquired knowledge, skills and abilities: o performs basic algebraic operations with matrices of arbitrary order - multiplying a matrix with a number, adding matrices, multiplying matrices; o finds the rank of a matrix, the determinant of a matrix, matrix inverse, eigenvalues, etc.; o finds a solution of a system of n linear equations with n unknowns in different ways; o describes the results and presents them in an appropriate form. Developed competences that the student exhibits and can be stimulated: C1, C2, C3, C6, C7, C8. 10/14 EDUCON2018 17-20 April, 2018, Santa Cruz de Tenerife, Canary Islands, Spain

 Solving problems in analytic geometry Required knowledge: general form of equation of lines in the plane (space), normal and collinear vector of a line, general equation of a plane Acquired knowledge, skills and abilities: o graphically depicts one or more lines with Wolfram Mathematica in a Cartesian coordinate system; o finds the angle between two lines; o finds the coordinates of the intersection of two lines as a solution of a system of equations and visualizes the crossing point on the plot; o graphically builds a parallel (or perpendicular) to a line through a given point; o graphically depicts one or several planes with Wolfram Mathematica in a 3D coordinate system; o visualizes a plane defined by three points in space

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o visualizes points in three-dimensional space and verifies analytically and graphically whether they lie on the same line; o finds the angle between two planes; o visualizes a plane defined by a point and a parallel line of a given plane Developed competences that the student exhibits and can be stimulated: C1, C2, C3, C5, C6, C8

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 Solving problems in mathematical analysis Required knowledge: Conceptual understanding of functions and different ways of representing them Acquired knowledge, skills and abilities: o defines an arbitrary univariate function with Wolfram Mathematica; o monitors how the effects of changes to the parameters of a function affect its plot; o defines the domain of definition of the function; o draws a plot of the function in a suitable interval (see Fig. 3); o finds the real roots of the equation and compares them with the graphical results; o finds the limits of the function at the ends of its domain; o understands the meaning of asymptotes to the function plot; o finds the equations of asymptotes and draws them together with the plot of a function; o calculates the first and second derivative of the function o determines intervals of growth and decay, convex and concave, inflection points, local maximum and local minimum points with Wolfram Mathematica, and visualizes them on the plot of the function 13/14 EDUCON2018 17-20 April, 2018, Santa Cruz de Tenerife, Canary Islands, Spain

Developed competences that the student exhibits and can be stimulated: C1, C2, C5, C6, C8

The competencies listed not only overlap and have a common root, but also the intensive application of software enriches and develops them in areas such as: o experimentation, o visualization, o comparison, o verification of assumptions, etc.

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New Rules for Assessing Mathematical Competencies

Acknowledgment This study was supported by the RULES_MATH project, EU grant 2017-1ES01-KA203-038491

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Thank you for your attention!

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