Wiris Quizzes. Assessing mathematics through Moodle quizzes. M. Rosa Estela, Joel Saà, Joana Villalonga Matemàtica Aplicada III. UPC - Barcelona Tech m.rosa.estela@upc.edu, joel.saa@upc.edu, joana.m.villalonga@upc.edu October, 2009 Abstract We are offering an important improvement to an existing Moodle course based on Calculus regarding assessment through WIRISQuizzes. The contents of the course are adapted to the ones of a Calculus subject in any Engineering high degree or scientific Bachelor, but there is something that makes this course different from others. Not only do we offer a book with all the theory concepts of the subject, but also a digital version of the book with interactive exercises and laboratories. 1 WIRISQuizzes. What is it? WIRISQuizzes is a new system based on the WIRIS CAS Techonology, an online Computer Algebra System which allows to compute mathematical calculations on-line as well as produce mathematical contents, that is growing up to provide Moodle questions with random and new values. The main goal of this software is to allow the course instructors or professors to create a random family of quizz questions just by programming one through variables that can take random data. It is: every time the question is opened to be done, it displays random and new data values. Moreover, this new and random values are calculated online and in real time. It is quite easy to provide a Moodle course with this technology: it is only necessary to install the WIRIS Quizzes plug-in in the settings of the desired course. This plug-in creates a new cell at the end of the editing chart of the questions called Math Engine WIRIS Quizzes that will let us create this new contents. Depending on the question type, different WIRIS technological options are displayed in this cell but, all of them have a commom element: the WIRIS software. The WIRIS software is no more than a WIRIS CAS with a library called variables. There must be the definition of the random values that will appear in the questions, both in the wording question and in the possible answers, and the (math) program which recalculates the result answer. With this, every time the question is displayed, it gives/shows new data values, altrough the structure of the question is the same. These new and random values are called variables and they are calculated online and in real time. 1
Variables have to be marked, so the system can distinguish among variable and text or static values. The symbol that specifies/fixes a random variable is the cushion (#). So, variables are those expressions in Moodle questions that are preceded by a cushion. With this, the system knows that its real data value is given by the WIRIS software and it must not show the name of the variable. Because of the system properly works and this mechanism takes effect, when a variable is used, both in the the wording question and in the possible answers, it has to be defined in the variables box of the WIRIS software. It means that, in the WIRIS software, the variable has to be assigned to a math WIRIS method that gives it its real data value. We say a math WIRIS method because it must be a math method programmed through WIRIS language; using WIRIS commands and functions. Moreover, from this WIRIS program, one can test if it does what it is expected to do. In order to check this, just write the desired variables out of the variables box and compute the program (pressing Ctrl + Enter). So, as we can see in Figure 1, the only impact that implementing WIRIS Quizzes supposes is that a new WIRIS engine box will appear at the end of the editing chart of the questions. And the only requirement needed to use it, is to get familiar with the use of WIRIS programming language. Nothing else. Figure 1: Preview, General edition and Math Engine WIRIS Quizzes of a Moodle question with WIRISQuizzes. Currently, WIRISQuizzes works on the most used Moodle questions: Essay, Matching, MultipleChoice, ShortAnswer and True/False questions and it gives them more or less (different, instead) technologies depending on their functionality requirements. On the one hand, for example, Essay, Matching and MultipleChoice questions are just provided the WIRIS software, because they do not need anyting 2
else to properly be able to create random questions and compute answers. On the other hand, True/False and ShortAnswer questions need a little more sophisticated system so they can work properly. When working with True/False questions it will appear a blank field called Answer. This is because when values change, results can change: True can become False and False it can become True. So, to fix True or False in the question is not possible. To sort it out, the WIRIS program must evaluate if answers are true or false for every particular case and then inform Moodle about it. To do this, the WIRIS program needs a boolean variable that always gives the correct answer to Moodle. So and the Answer blank field must be filled in with this boolean variable. If this new field is not used, Moodle considers its fixed answer option and, consequently, the evaluation of the user response will be wrong. The other special question type is the ShortAnswer question. In this case three options are added to improve the answering system. These are the following ones: Formula editor, Multiple answers and WIRIS CAS. The first option sets the WIRIS Formula editor in the answer chart in orther to be able to answer with math structures and symbols, like integrals, matrices... The second one, makes it possible to answer through different fields. And, the third one, sets a WIRIS CAS in the answer so the user can compute its response from the same web page. So to sum up and as we can see, WIRISQuizzes tries to make random, more extensive and easier the creation of different questions for Moodle quizzes. To end, it is important to know that WIRISQuizzes do not corrupt the default Moodle mechanism. Despite it is implemented in a course, it only takes effect when variables are used and defined en the WIRIS Program. In other case, it does not do anything. For this reason, the cell that contains the Math Engine WIRIS Quizzes has a button that allows to show or hide these advanced options. 2 WIRISQuizzes Examples When and why can we be interested in using WIRISQuizzes? When and why could WIRISQuizzes be an interesting and useful tool? Often teachers and instructors are interested in their students getting confidence in specific (math) topics, calculations or mechanisms. Or, they want to be sure that students do exercises alone, by themselves, without copying or remembering data results. Moodle is a good system so students can get different exercises to practice and get confidence with different topics and procedures. But, in the Moodle context, that means teachers have to generate and generate enormous lists of similar exercises but with different aspects in order to: on the one hand, provide the students a lot of similar exercises with different data, properties,... so they can evaluate themselves. on the other hand, giving every student a different set of exercises, in order to evaluate them. We then notice that, creating these contents one bay one is a heavy and not rewarded work that needs an improvement. Fortunately, now, there is a new 3
and comfortable tool that makes this work easier and efficient. We are talking about integrating WIRISQuizzes inside a Moodle course. Using WIRISQuizzes, teachers only have to create one question of each type and provide it with a good WIRIS program to be sure that a big set of random exercises with different data will get to their students. From now on and within the next few paragraphs, we are going to expose some situations where WIRISQuizzes are really useful and necessary to solve this problems and how we use Moodle questions in order to obtain better academic results. We find some simple examples, for instance, in the study of complex numbers or vectors. Let s see some of them: To begin with, the study of complex numbers. We may be interested in getting the students confident with the usage of complex numbers: operating with them or just recognizing its respective polar form. In order to let students practice with it, we generate a Matching question that gives three complex numbers and their three polar expressions. The goal of this question is matching each complex number with its respective polar form. With the traditional Moodle technology, we would edit the question as follows: Title: Match each complex number with its expression in polar form. Question 1 Question: 1 Answer: {1, π} Question 2 Question: 1 + i Answer: { 2, π 4 } Question 3 Question: i Answer: {1, 3π 2 } and the student would obtain the following question: Match each complex number with its expression in polar form. 1 1 + i i 1 π 2 π 4 1 3π 2 and should match them. Moodle can even mix the options of both columns. And it is good. It is an interesting exercise where students have to transform each complex number into its polar form so they can match the answer. But there is a problem: all the students will have the same and unique exercise...this is not a problem anymore with WIRISQuizzes. We just have to add a couple of commands and through a short WIRIS program we will generate a large collection of exercises of this type. Using WIRISQuizzes, Matching question s structure will be the same. The difference, remember, is that now values are not directly given. They are given through WIRIS variables, which are defined in the WIRIS variables box and appear in the edit question preceded by a cushion. So, the aspect of the edition 4
of the question would be this: Title: Match each complex number with its expression in polar form. Question 1 Question: #c1 Answer: #C1 Question 2 Question: #c2 Answer: #C2 Question 3 Question: #c3 Answer: #C3 where #c1, #c2, #c3, #C1, #C2, #C3 are the WIRIS variables that stablish the values of the complex numbers. On the other hand, the math engine has also to be programmed. For this, and in this case, we only need to know two WIRIS commands. The first and most important one is the random command. It is the WIRIS command that generates random values, given as a real, integer or natural intervals, or a set of predefined real, integer, natural or complex values. This explains why this command is always used in all the WIRISQuizzes questions. The second command that is needed is the complex command: polar. This command returns the polar form of a complex number. So, in this first example, the math WIRIS program generates random complex values, finds out its polar form and evaluates if the given answers are correct: Math Engine WIRIS Quizzes WIRIS Program variables c1 = random ( 10, 10) c2 = random{ 1, 1} random (1, 10) + random{ 1, 1} random (1, 10) i c3 = random{ 1, 1} random (1, 10) i C1 = polar (c1) C2 = polar (c2) C3 = polar (c3) With this, two possible output questions for colleagues sitting one next to the other would be: First colleague s question: Match each complex number with its expression in polar form. Second colleague s question: 2 {2, π} 1 { } 3i 2, π { 3} 5i 5, π 2 5
Match each complex number with its expression in polar form. 10 { {10, 0} } 3 + 3i 3 2, 3 π { } 4 8i 8, π 2 Something completely different but also important as a topic regarding maths is the study of vectors. In particular, the scalar product and its consequences. Some typical examples in which we could be interested, are the following ones: Suppose we want students to discover when two vectors are perpendicular. In order to make the students learn about this topic, we generate a True/False question that asks whether two given vectors are perpendicular. With Moodle, the question would be as follows: Let u = 1i + 1j 2k and v = 1i + 1j + 1k be two vectors in R 3. Are these two vectors perpendicular? And the answer is always True. Nothing more than that. So, the effect of this exercise is very limited and we should create dozens of similar ones in order to have some randomness. However, now, we can enrich its effect and reuse this question many more times with different and unknown data. In fact, with WIRISQuizzes, new, random and different values will appear each time, and the question will be giving a more interesting effect. In this case, we only need to define the vectors coefficient as WIRIS variables so, define u and v as WIRIS vectors and ask WIRIS wether these two vectors are perpendicular, without any operation. So, the edition of the question would be: Let u = u1i + u2j + u3k and v = v1i + v2j + v3k be two vectors in R 3. Are these two vectors perpendicular? where, Math Engine WIRIS Quizzes Answer: #a (1) WIRIS Program variables u1 = random ( 10, 10) u2 = random (0, 10) u3 = random (0, 10) v1 = random ( 10, 10) v2 = random (0, 10) v3 = random (0, 10) u = [u1, u2, u3] v = [v1, v2, v3] a = perpendicular? (u, v) Notice that the Math Engine WIRISQuizzes cell has another field, the field Answer. As it is said before, in a True/False question WIRIS has to return a 6
boolean variable, which stablishes if the response is true or false. In this program, #a is this boolean variable and it advises Moodle about the new result state. If we directly ask for a in the WIRIS program window, out of the variables box, we will see if the relation is true or false every time we compute the program. With this example, a possible output would be the following one: Let u = 5i + 0j + 8k and v = 2i + 1j + 2k be two vectors in R 3. Are these two vectors perpendicular? And the answer, for this particular question, is False. But if another pupil enters in this same question he would obtain, for instance: Let u = 6i + 1j 5k and v = 3i + 8j 2k be two vectors in R 3. Are these two vectors perpendicular? Then, the correct answer is completely different. It is True. In Figure 2 there is a real example (Preview and Math Engine WIRIS Quizzes) of this type of question. Figure 2: Preview and Math Engine WIRIS Quizzes of a True/False WIRISQuizzes Moodle question. Let s see a final example. For this last situation our concerns are that students learn how to deal with the relationship between the scalar product and the angle between two vectors, and therefore we can generate a ShortAnswer that directly asks for the angle between two given vectors. For example: Find the angle between the vector u = 1i+2j+3k and the vector u = 2i+3j+4k. With this above, all of the students have the same data in the problem and it will always be the same answer. So it is not very useful because there is just one question, and once the student knows the answer, there is nothing else to do. Moreover, without WIRISQuizzes, we have to precalculate the angle between 7
them and put it in the Answer cell of the ShortAnswer so the student response could be evaluated. Apart from that, if we want students to consider the units, a possible answer and grading could be: Answer 1 Answer: 0.12 Grade: 60% Answer 2 Answer: 0.12 rad Grade: 100% But, if we don t want to precalculate the answer and we are interested in giving different vectors to the students and they getting different results, we only have to use the WIRISQuizzes technology. With it, we do not need to precalculate anything and we do not mind about vectors values. We only have to define the variables that contain the values that the question will display. Let s see: Math Engine WIRIS Quizzes WIRIS Program variables u1 = random ( 10, 10) u2 = random ( 10, 10) u3 = random ( 10, 10) v1 = random ( 10, 10) v2 = random ( 10, 10) v3 = random ( 10, 10) u = [u1, u2, u3] v = [v1, v2, v3] a = angle (u, v) In this program, we define the vectors coefficients as random WIRIS variables (in the set of numbers we prefer, now, only small integers), the vectors u and v as WIRIS vectors and we use the specific WIRIS command angle because it immediately determines the angle existing between u and v. Only and simply this. As always, we can test the math program only asking for the desired variables out of the variables box. Every time we run the math program, WIRIS returns its values. An example could be the following one: u [2, 3, 1] v [ 3, 4, 2] a 1.371 Defined this simply WIRIS program, the answer s cells are: Answer 1 Answer: #a Grade: 60% Answer 2 8
Answer: #a rad Grade: 100% where #a is the variable that contains the result, the value of the angle. So, if the value that the student gives is the same as the returned by the math WIRIS program, #a, the response is okay; otherwise it is not. In this example, students would preview the question as: Find the angle between the vector u = 2i + 3j + 1k and the vector u = 3i + 4j + ( 2)k. And the correct answer is: 1.371 rad. But, this is not at all. In this kind of questions we can add more properties or technological helps. For example, if we want the students to develop the process that gives the angle, it is possible to insert a WIRIS CAS in the Answer of the question only choosing the option Display WIRIS CAS for auxiliar computations. That appears in the Math Engine WIRIS Quizzes cell. With that, students can develop, step by step, the method that let them find out the angle between the given vectors and then insert in the response field the value obtained through the WIRIS CAS. This is shown in Figure 3. Figure 3: Math Engine WIRIS Quizzes and Preview of a ShortAnswer WIRISQuizzes Moodle question. References [1] Blanco, M. et al. Computer assisted assessment through Moodle quizzes for calculus in an Engineering Undergraduate Course. Quaderni di Ricerca in Didattica, 2009, vol. 19, núm. 2, p. 78-83. [2] Estela, M.R. et al. Teaching and Learning Calculus using WIRIS Technology in Moodle environment. In: Abstracts of the International Congress of Mathematicians, Madrid 2006. European Mathematical Society, p. 604. 9
[3] Estela M.R., Saà J.Cálculo con soporte interactivo en Moodle. Pearson Prentice Hall. 2008. [4] Estela M.R., Saà J.Cálculo con soporte interactivo en Moodle. Moodlemoot Barcelona. 2008. [5] Estela M.R., Saà J.A Calculus Course with interactive support on Moodle. International Technology, Education and Development Conference (INTED 2009) [6] Estela M.R., Saà J.Assessing calculus students in engineering schools. Active Learning in Engineering 2009. JEM Exchange Workshop Role of Technology in Formative and Summative Assessment. Barcelona. 2009. [7] Estela M.R., Saà J.Un nou model docent d acord amb l Espai Europeu d Educació Superior (EEES). Aplicació al Càlcul. II Jornada Interuniversitària d innovació docent. Universitat de Barcelona. Barcelona. 2009. [8] Estela M.R., Xambó, S. Teaching and Learning Mathematics using MapleTA and WIRIS technology in a Moodle environment. 12th Internacional Conference on Technology Supported Learning and Training. Online Educa Berlin 2006. 10