Using tests for promoting interactions in exercise lessons of mathematics education for students in the faculty of mathematics Hanoi National University of education

JOURNAL OF SCIENCE OF HNUE 2011, Vol. 56, N◦. 1, pp. 66-76 USING TESTS FOR PROMOTING INTERACTIONS IN EXERCISE LESSONS OF MATHEMATICS EDUCATION FOR STUDENTS IN THE FACULTY OF MATHEMATICS HANOI NATIONAL UNIVERSITY OF EDUCATION Nguyen Phuong Chi Hanoi National University of Education E-mail: nguyenphuongchivn@yahoo.com Abstract. This article is intended to justify the possibility for promoting interactions in exercise lessons of Mathematics Education by using tests, and to clarify manners for doing it. The procedures of using tests in exercise lessons were mentioned and illustrated by specific examples. A pedagogical experiment was also conducted and its results were clearly analyzed. Keywords: interactive teaching method, impact, response, interaction, testing exercises. 1. Introduction Because of the conflict between the requirements of society with current in- adequate teaching methods, a campaign to innovate teaching methods at all levels in the education system has been raised and promoted for the past several years. The dominant ideology of this campaign is stated in many different forms, but the common focus is the impact on the basic elements of teaching methods, namely self-conscious, active, initiative and creative activities of the learner, conducted in- dependently or in interaction. Furthermore, a system consisting of three main factors in the pedagogical activities, which are learners, teachers and the environment along with the rela- tionship between these factors, was considered. Since then a new teaching method called interactive teaching method in which learners, teachers and the environment constantly interact has been started [2;16]. This article is also based on the theory of interactive teaching method. In the process of teaching, it is always difficult to organize interactions. Be- cause teachers often work with a large amount of students, it is hard to create interactions between teacher and each individual student, or interactions among students themselves. However, the author of this article found that testing exercises can also be used to enhance interactions, so that the quality of teaching can be improved. 66 Using tests for promoting interactions in exercise lessons of mathematics education... Therefore, this paper researches the use of testing exercises during exercise lessons of Mathematics Education to promote interactions, innovate methods of teaching mathematics, contributing to improving the quality of teaching this subject in the Hanoi National University of Education. 2. Content 2.1. Interactive teaching method A disadvantage of traditional teaching methods is that it is aimed at teachers rather than students, focusing on teaching rather than learning. To overcome this disadvantage, a campaign to innovate teaching methods has been promoted by paying due attention to students and learning. Furthermore many educators have initiated a new teaching method called in- teractive teaching method. In this method, three main pedagogical activity factors learner, teachers and the environment in turn constantly interact by three functions learning, supporting and influencing on the basis of the impact, response and in- teraction concepts [2;16]. An activity, an effect or a certain element of an agent is considered to be an impact on other agents when it causes the response (causes the change) of this agent. This response sometimes becomes an impact to react to the initial agent, and it may also impact on other agents. Interaction is the impact between two or more agents. In this paper there is the development of some theoretical aspects of the in- teractive teaching method: Firstly, with regard to the learners, we need to concentrate the following cases: - Interaction between a teacher and an entire group or class which is considered as a whole, - Interaction between a teacher and an individual in a group or in a class, - Interaction among individual learners in each group or in a class. The last case is more and more developed with cooperation learning forms such as group learning, project work, ... Secondly, we are especially concerned about a significant part of the environ- ment which directly and continually relates to the teaching process; that is a system which faces learners, and has impact on the process of adaptation (assimilation and accommodation) of learners. So it is possible to understand that is the environment in a narrow sense [4;210]. Thirdly, the paper extended the concepts of impact, response and interaction for the indirect cases, when agents are certain tools which teachers use for replacing themselves to impact on learners or respond to the impact of learners. Those are the indirect impacts and responses predicted by teachers to prepare helpful situations for learners. Thus the article concerns also indirect impacts, responses and interactions. These indirect cases usually occur in teaching processes using some tools such as 67 Nguyen Phuong Chi testing exercise, software or documentation of programming teaching, information and communication technologies . 2.2. Use tests to promote interactions in the teaching pro- cess Many people use the term ”test” in a very broad sense which may also refer to an essay. That is why they distinguish ”objective testing” from ”essays”. This paper uses the term ”test” in a narrower sense, as objectivity is a necessary condition. Hence, the term test used in this article mean objective test in many other books and journals. Testing is a scientific method which allows using a series of routines for study- ing one or more personal features (these features can be distinguished by experi- ments) with the purpose of going to most possible highly quantified statements with regard to a relative expression level of a feature to be studied [3;7]. It is common to distinguish between standardized tests and tests self-created by teachers, referred to as self-created tests [1;258-260]. Unlike standardized tests which are carefully built and experimented on by experts of evaluation and testing, and can last for many years, tests created by teachers are prepared for a particular content with a specific group of students at a particular moment. This particular moment could be during or at the end of a lesson, each week, in half months, monthly, or in a quarter or yearly semester. Effectiveness of using testing exercises in examination and evaluation have been studied in many scientific works and referred to in many books and magazines in our country and throughout the world. This article does not go into the above function, but intends to use tests to enhance interactions in the learning processes. Illustrating is specifically for the subject of Mathematics Education. This idea originated from potentials which can be exploited from tests to enhance interactions in learning processes, as will be shown below: 2.2.1. Use testing exercises to indirectly interact between teachers and learners and to increase interactions among learners First of all we consider a usual exercise lesson (without using tests or any other special means). When a teacher gives an exercise to students, he has caused an impact on the students. If the teacher asked the whole class to do the exercise on draft paper, that is, the students have responded, but almost no one has reacted to the teacher during the time that they do the exercise. On the other hand, if the teacher asks one student to go to the blackboard to do the exercise instead of asking all students to work separately, then there is only interaction between the teacher and the student, there is no interaction between the teacher and other students. In the case of using testing exercises together with answer sheets including analysis of wrong answers, besides interactions between the teacher and learners 68 Using tests for promoting interactions in exercise lessons of mathematics education... in the usual way of teaching, other interactions can be enhanced: learners can indirectly react to the teacher (by writing or marking to the exercise sheet), the teacher can also react to every student (indirectly impact through the answer sheets). When each student reads the answers and muses about those, that means they are responding. When comparing their answers with the answer sheets, students know the results, but they may be still interested in other options analyzed in the answer sheet. Each time a student moves to another option of answer, it is regarded as an indirect reaction to the teacher. Reading and thinking the answer means that the student has responded to the indirect reaction of the teacher. Thus, testing exercises together with answer sheets including analysis of wrong answers can cause interactions between the teacher and each student over and over again. Such testing exercises together with answers sheets including analysis of wrong answers can also be given to students to do in groups, so that many interactions among group’s members can be enhanced. Therefore, using testing exercises together with answers sheets including anal- ysis of wrong answers will increase interactions between teachers and learners, in particular: - Indirect interactions between teachers and learners, and - Interactions between learners and learners. 2.2.2. Use testing exercises to create mechanisms for timely feedback in the process of teaching By definition mentioned at the beginning of Section 2.2, an important feature of testing is that it is assessed based on routines, so on the one hand it can quickly give the result and on the other hand it can be performed by technical means or by people who are not teachers. These two advantages allow making timely feedback mechanisms (as in programming teaching or in using testing exercises) in links of chains needed for both learners and teachers in the teaching process. That increases interactions between teachers and learners. 2.2.3. Use various types of testing exercises to attract learners Experience has shown that all common types of testing exercises such as multi- ple choice questions (special case is 2 possibilities right or wrong), fill in the blanks, rearrange the orders, pairing or tripling have effects on consolidating knowledge. Such various types of exercises easily cause interest to learners. Even some types of tests which reveal several disadvantages in examination and evaluation can and should be still used to consolidate knowledge in the process of teaching, such as: - Questions which have only two options right or wrong: the probability of random correct answer for this type of question is 50%. Because of this, if we use these questions to evaluate then the test results may be not exact, but we can still 69 Nguyen Phuong Chi use them to reinforce knowledge, because in this case it is not necessary to score, moreover this type of exercise is easily composed and also attractive for learners. - Exercises which ask to select multiple options (not necessarily choose only one correct option) in a series of given questions are rarely used because they often cause difficulties for getting a realistic score. However, to consolidate knowledge, it is not necessary to score. Furthermore this kind of exercise is often very interesting because it requires to muse on more than one case which need to pick out only one correct options. Thus, because various types of testing exercises can attract learners, they can help to promote interactions in learning processes. 2.3. How to use testing exercises in the teaching and learn- ing process? 2.3.1. Use testing exercises together with answer sheets including anal- ysis of wrong answers In answer sheets of testing exercises today, they often just write the correct answer. However, this article intends to provide answer sheets including analysis of wrong answers at the appropriate time. Answer sheets including analysis of wrong answers have many effects: Firstly, they help learners to deepen knowledge, to see clearly not only the right answer but also analyze the wrong ones; they indicate the wrong place and if possible point out the reasons. Secondly, answer sheets including analysis of wrong answers can be used to enhance interactions in the teaching and learning process to contribute to overcoming a one-sided way of teaching that only teachers cause impacts. This effect has been clarified in Section 2.2.1 and 2.2.2. The following is an example of such testing exercises with answers sheets including analysis of wrong answers: Exercise: Choose the correct sentences among the ones below: a) The relationship between mathematics and the reality has universality, entirety and multi-level features. b) It is necessary for students to see clearly how each mathematical theorem and concept is applied in real life. c) Studying mathematics needs abstract thinking without specific thinking. d) Teaching students suitably for their power at different moments means suitably increasing demands at these different moments. e) Students should self-consciously, actively, initiatively and creatively study and not follow the teacher’s control in the learning process. Solution: 70 Using tests for promoting interactions in exercise lessons of mathematics education... a) True. See textbook p. 69. b) False. The relationship between mathematics and the reality has entirety. It is impossible to require students to clearly see how each theorem or each mathe- matical concept is applied in real life. c) False. Each scientific knowledge in general and mathematical knowledge in particular is a unity between specific things and abstract things. To dominate an abstract content, we need to illustrate it by specific things. On the other hand, when working with specific things, we need to look towards suitable abstract contents, so that we can dismiss non-essential signs to hold essential ones, and can dismiss individual things to hold the rules. d) True. An ability of a student, or his/her qualification, is not a constant. It continually changes during the learning process, it increases in general. So teaching students suitably with their power at different moments means suitably increasing demands at these different moments. e) False. Need to ensure the consistency between the control activities of teachers and learning activities of students. The proper teacher’s control do not lose the conscious, active, initiative and creative characters of students in the learning process. 2.3.2. Use testing exercises with coordination of essays in teaching ex- ercises Besides the advantages of testing exercises, such as mentioned in 2.2.1, 2.2.2, 2.2.3, not including the limitations when they are used for examinations and evalua- tion, they also have some other limitations in the process of teaching. A considerable limitation is that testing exercises are not useful in training skills and developing high intellectual capacity. Hence, in the teaching process teachers should supplement essays, not use only testing exercises. In general, testing exercises and essays have their own advantages and they can complement each other. Both types are needed to be used in teaching exercises. Essays should be exploited to practice skills and develop high intellectual capac- ity, while testing exercises help learners to consolidate, deepen knowledge, clarify mistakes that may arise, and increase interactions in the learning process. 2.3.3. Procedures of using testing exercises in exercise lessons of Math- ematics Education Based on the idea mentioned in 2.3.1, this section proposes three procedures of using groups of testing exercises together with answer sheets including analysis of wrong answers during exercise lessons of Mathematics Education. These three ways (denoted in turn by procedure 1, procedure 2 and procedure 3) only differ in the second stage, so they can be presented in the same diagram below (Figure 1). Procedure 2 is different from procedure 3 only in not using information and 71 Nguyen Phuong Chi Figure 1. Producedures of using testing exercises communications technology. The following two exercise lessons which use testing exercises to consolidate knowledge are examples respectively for procedure 1 and procedure 3: Example 1: Exercise lesson on Content of the subject Mathematics (performed according to procedure 1). Activities of teacher and students in the class: (1) Teacher gives each student a copy of the assignment. (2) The teacher divides the class into groups (each table is a group). (3) The teacher asks each group to read carefully the topic of exercise number 4, and then calls up a student to repeat four goals of teaching mathematics in front of the whole class. If this student does not a full answer then teacher tells others to give a supplement. This activity helps students to review the theory and understand the requirement of the assignment. (4) The teacher asks the students in groups to discuss the answer, and then write down the result of the whole group. (5) Each group selects a representative student to read out the solution of the group. If all groups have inconsistent results in some content then the teacher let the groups have more discussions. During the discussion some groups may recognize that their answers are incorrect. (6) The teacher hands out the answer sheets (attached separately). Each group compares the correct answers with the results of group discussions, exchanges ideas of wrong answers. Finally, if the group still has different opinions from the teacher in some answers then they record them so that teacher can explain immediately or 72 Using tests for promoting interactions in exercise lessons of mathematics education... at another time later. (7) The teacher asks each student to add one work item or a series of work items to each purpose of teaching mathematics in school. The group confirms and records its opinions (after the whole group has adjustments if necessary). (8) The teacher gives some essay exercises. (9) End of lesson. Example 2: Exercise lesson on Methods of teaching mathematics (performed according to Procedure 3). Activities of teachers and students in the class: (1) Teacher gives each student a copy of the assignment which consists of 15 testing questions. (2) Teacher asks students to answer those questions within 5-7 minutes. (3) After the time for reflection, teacher and the class discuss the answer of each question. (4) Teacher in turn projects onto the screen and reads aloud to the class the content of each question. After projecting and reading each question, the teacher calls up several students to give their answers. (5) The teacher does not give the correct answer yet. For a number of questions that students answered incorrectly in many different ways (the class only has time to work carefully with a number of such questions), the teacher asks students to explain their answers. If students have many different controversial opinions for a question, the lesson will be more interesting and the students can deepen knowledge that they have learned. After all necessary debates and opinion exchanges for that question, the teacher projects on-screen its solution and moves on to the next question. (6) After finishing answers all 15 testing questions, the teacher hands out the answer sheet including analysis of wrong answers (attached separately). Students compare the correct answers with their answers, see the explanation in the answer sheet to know the reason why there is such an answer. They can discuss in groups about ideas which are different from the answer sheet. Finally, if the group still has different opinions from the teacher in some answer