E-assessment tools for mathematical study courses
Franziska Nestler
Department of Mathematics
IWOOTI 2014 - Mittweida November 5, 2014
Goal
• mathematics is integrated in many courses of studies
• mathematics = difficult subject for students and pupils
• integrate online tests in mathematical courses at university (in addition to lectures)
→ additional possibility to exercise
→ continuous (individual) feedback about level of knowledge
→ assistance in preparation of an exam
Also interesting:
• create online courses in preparation for academic studies
• create self assessment tests
• execute electronic home works or even examinations (courses with a large number of students)
Requirement:Provide appropriate tools for mathematical exercises.
New tools for mathematical exercises in Onyx
E-Assessment-Tool ONYX:
• integrated in OLAT Campus/ OPAL
• creation/ management of single exercises and tests Cooperation with BPS (Bildungsportal Sachsen) GmbH.
Goal: implementation of tools suitable for mathematical study courses.
ProjectELMAT(E-Learning tools for mathematical topics):
Formal improvements:
• creation of formulas based on LATEX
• direct and integrated via $$. . . $$, \(. . . \),\[. . . \]
• rendering via „MathJax“
Mathematical exercises:
• parameterized/ randomized Exercises
• create possibility to enter formulas (as responses)
• create library of mathematical exercises, exchange between lecturers from different academies
LaTeX
What is LATEX?
• document preparation system and document markup language
• widely used for documentation and publication in scientific fields
• especially mathematics, computer science, natural sciences
• nice display of formulas
• example:
TEX: \( E_0 = m \cdot cˆ2 \)→ PDF:E0=m·c2
possibility to use LATEX in Onyx→increase acceptance + usability
Usage of LaTeX and MathJax
• creation of formulas with LATEX, via $$. . . $$, \(. . . \),\[. . . \]
• can be used within conceptual formulation, multiple-choice-responses, clozes, feedback text etc.
• high-grade rendering via Java script MathJax
• gives an easy way to integrate a bulk of relevant mathematical symbols into exercises
Figure:Typeset formula via LaTeX (left), rendered formula (right).
Further examples of rendered formulas
Parameterized/ randomized exercises
• creation of parameterized/ randomized exercises
• usage of computer algebra system MAXIMA (mathematical program, which can perform calculations and is able to interpret formulas)
• definition and connection of variables (possible types: integers, float, text)
• solution of exercise depends on parameter values
• creation of exercise of a certain type⇒multiple individual exercises
• individualized tests: for each attempt, another set of parameters is chosen, each student gets another test
Parameterized exercises
Exercise of a certain type: P∞ n=0
p q
n
=?
⇒multiple individual exercises:
Parameterized exercises
Exercise of a certain type: (a+bi)·(c+di) =?
⇒multiple individual exercises:
Parameterized exercises - an example
• construct exercise of a certain type:
log2 2a·4b
=a+2b
• Example: log2 23·45
=13
• define and connect according variables
Parameterized exercises - an example
Figure:Use the variables within the formulation of the exercise. (the values ofaandb are put in via {a} and {b}).
Example: aandbare randomly set toa=10,b=4. The solution isa+2b=18 (=value of the third variable {ergebnis}).
Possible: parameter dependent feedback (individual feedback).
Validation of formulas
Idea:
• classical types of exercises:
multiple choice, computation (answer is a number)
• Usage of MAXIMA→Possibility to interpret formulas
• are able to ask for formulas/ mathematical expressions as solutions
• Simple Examples:
• Solve the following equation forx: 13x=5
Correct answer: x=5/13(instead ofx=0.384615. . . )
• Solve the following equation forx>0: (2x)2=2
Correct answer: x=sqrt(2)/2(instead ofx=0.7071068. . . )
• Solve the following equation forx:ex=a.
Correct answer: x=log(a)
• Differentiate the functionf(x) =sin(x)with respect tox.
Correct answer: f0(x) =cos(x)
• Which plane is spanned by the three pointsA(1,0,0),B(0,1,0),C(0,0,1)?
Insert one possible solution.
Correct answer: x+y+z=1
Validation of formulas - an example
Correct answer (formula) is stored behind gap (as MAXIMA expression).
In our example:
Correct answer/ sample solution:
y(x) = −1 3x2+4x+8
Insert solution in MAXIMA notation:
y(x) =-1/(3*xˆ2+4*x+8) Provide necessary syntax!!!
Validation: ifsample solution = "learner response"⇒exercise was solved correctly
Syntax Validation
Insert formula→validate→save answer
Syntax is not correct
→warning
Correct syntax
→ gap is colored green,
preview of formula
Expert mode
Simple validation of answer: Exercise is solved correctly if"learner response" = sample solution.
Problem: sometimes there are more than one correct answers.
Expert mode: Exercise is solved correctly if"learner response" fulfills a certain condition.
Simple example:
Type in some functionf(x), for whichf(0) =0 and alsof(1) =0.
Possible answers:
• f(x) =x*(x-1)
• f(x) =2*x*(x-1)
• f(x) =xˆ2*(x-1)
• . . .
Exercise is solved correctly if
evaluation of inserted function in x = 0 gives zero and evaluation of inserted function inx =1 gives zero.
Expert mode - an example
classical example: representation of a plane is not unique
x+y+z=1 is the same as 2x+2y+2z=2etc.
Library of exercises
• created a library of mathematical exercises in ONYX, which is used by lecturers of different Saxon academies (TU Chemnitz, TU Freiberg, HTWK Leipzig, FH Zwickau)
• corporate administration, frequent exchange
• possibility to specify meta data (subject area, description of exercise, key words, . . . )
• exercises are already integrated in different mathematical courses
Thank you for your attention!
Contact:
• Prof. Daniel Potts
Technische Universität Chemnitz, Fakultät für Mathematik potts@mathematik.tu-chemnitz.de
• Franziska Nestler
Technische Universität Chemnitz, Fakultät für Mathematik franziska.nestler@mathematik.tu-chemnitz.de
