EMPOWERING STUDENTS TO LEARN MATHEMATICS THROUGH JOURNAL ENTRIES: EXAMPLES FROM CANADA AND SINGAPORE Paper presented at the ERA/AARE JOINT CONFERENCE, Singapore, 25th - 29th November 1996 at the Singapore Polytechnic. Copyright ( 1996 by Dr. Sitsofe Enyonam Anku National Institute of Education Nanyang Technological University, Singapore All Rights Reserved EMPOWERING STUDENTS TO LEARN MATHEMATICS THROUGH JOURNAL ENTRIES: EXAMPLES FROM CANADA AND SINGAPORE Dr. Sitsofe Enyonam Anku (ankus@nievax.nie.ac.sg) Abstract One of the most constructive and empowering goals of mathematics education would be to equip students to monitor their own progress. Such self-monitoring can be expressed through students making journal entries. To investigate what sense students make of the mathematics they learn, I asked different groups of preservice teachers in Canada and Singapore to make journal entries. Students were to i) summarise the main points of the lesson, ii) state and explain what they understood from the day's lesson, iii) identify what they did not still understand, and iv) pass any other comments. Results showed that students from both countries found making the journal entries very useful as "it forced" them to learn harder and understand the concepts better. Introduction Currently, empowering students to learn mathematics is greatly emphasised in mathematics education. This emphasis is due largely to the impact of constructivism, with which mathematics educators view knowledge generation as an activity in which the individual must actively participate (Davis, Maher, & Noddings, 1990). Making of journal entries does not only get students to actively participate in learning, it also helps students to monitor their learning through self-assessment (Kenney & Silver, 1993). To investigate what sense students made of the mathematics concepts they learned, I asked a group of preservice mathematics students in Canada and Singapore to make journal entries. The purpose here is not to compare the two groups of students, but to document their entries and speculate on how meaningful (Anku, 1996) the students found the activity of making journal entries, despite the different learning contexts and students' background. In this paper, I provide a brief background of the students, describe the making of the journal entries, and then I discuss some results from an analysis of the journal entries. Students' Background From Canada The students (28 in all) were all first degree holders in various subjects other than mathematics. While most of them took mathematics only up to the high school level, a few took some university level mathematics courses. The last time they took a mathematics course varied from two to eight years. Some of the students registered for the course, a mathematics content course for preservice teachers, immediately after their first degree while others worked for several years before coming back to take this course. From Singapore None of the students (11 in all) was a first degree holder but they had all taken and passed a mathematics course at the Advanced Level of the General Certificate Of Education (GCE). All the students registered for the course, a mathematics methodology course for preservice teachers, immediately after their Advanced Level studies Class Context Classes in both countries were similarly organised. Classes were held once a week (3 hours per class in Canada and 2 hours per week in Singapore) and students worked in small groups of four students per group. The first few days of class were used to emphasise the need for the students to take ownership of their own learning if they were to derive maximum benefit. Also, we discussed some norms of working successfully in small groups. For example, students were to think aloud, justify their thinking, and respect the viewpoints of their colleagues. Furthermore, students were engaged in several activities, both mental and physical. However, it is my experience that many students take class activity to mean physical manipulation of objects. So, I emphasised the mental aspect of class activity as also very important. Finally, I got students to make journal entries after each class activity. Class Activities Students worked on a variety of activities. There were worksheets covering some of the mathematical concepts learned, quizzes, some mathematical investigations, development of some concepts through the solution of problems, and students' group or individual presentations of their solutions to some mathematical problems, among several others. A typical quiz for the Canadian students required them to find the probability that if any of them should give birth to 4 children, 2 would be boys and 2 would be girls. One activity for the students in Singapore was an investigation into the various conceptions of algebra. (These examples are only to give readers a feel of some of the class activities). Students from both countries were then required to reflect and make journal entries on each day's lesson, following some guidelines. Guidelines for Making Journal Entries There were some guidelines for students in both countries to follow in making the journal entries. Students were to limit each of their entries to one page. The one-page limit was imposed so as to challenge students to be brief (and to minimise verbosity) in expressing their thoughts. In fact, the one-page limit has been found to be sufficient. In making the entries, students were to i) summarise the main points of the lesson, ii) state and explain what they understood from the day's lesson, iii) identify what they did not still understand, and iv) pass any other comments. I have found these four broad areas sufficient in providing useful information on how students made sense of what they were learning. For example, letting students summarise the main points of the lesson gave me a sense of what they considered to be important to them. Stating and explaining what they understood helped me to identify any misconceptions. Being able to identify what they did not understand was itself a useful learning activity for the students and provided me information on what to re-teach. Finally, they were given the opportunity to say anything from cognitive to affective issues that might not have been covered earlier, but which might provide useful information. Results from the Journal Entries I have decided to provide two samples from the variety of students' entries. Other relevant comments from students are used to emphasise important points under the section on discussions. (All names used are pseudonyms). After a class involving the conceptions of algebra, Pui Yin from Singapore made the entry in figure 1. Main points: Johnson [the instructor] revised with us the concepts [conceptions] of algebra. For the rest of the class, we were told to explore a CD-ROM titled "The under sea world of algebra" so as to familiarise ourselves with it. This is to facilitate the future use of it in the class. What you understand: There were four concepts [conceptions] of algebra that Johnson went through with us. The first concept was algebra can be in a form of generalised arithmetic. For instance, 2 + 3 = 3 + 2, a(b + c) = ab + ac, a x b = b x a. The next concept being that algebra can be a procedure for solving problems, for example, 2x + 3 = 15. Then, algebra can also be relationships among quantities like A = (r2 where this is the formula to fine the area of a circle. A represents the area of a circle and r the radius. So, if we know either the area or radius of a circle and substitute the value into the formula, we realise that it becomes a "procedure for solving problem." Finally, algebra can be the study of structures. For example, a2 - b2 = (a + b)(a- b), y = mx + c and y = ax2 + bx + c. As for the CD-ROM, I realise that it is only a tool to aid in teaching. But, it can never take the place of a teacher in a classroom. What you do not understand: I understood the lesson, no queries. Any other comments: I used to think that once computer is introduced into classroom learning, the importance of a teacher becomes less significant. Now that I see the limitations of a computer, I realised that I was being naive. The computer is useful in teaching the basics of mathematics. For example, what is algebra and how to manipulate them. But a computer does not have emotions and cannot sense if a student is confused even though he or she may be able to follow the instructions shown on the computer and thus "solve" the problems. The higher order skills like analysis, synthesis and evaluation cannot be guided by the computer too. Only the teacher is able to provide guiding questions at the right time to facilitate such skills of the students. Finally, I want to thank you, Johnson, for all the efforts during class. I have to admit that your lessons have been very tough and sometimes, I feel very frustrated. But, I have overcome that obstacle of being dependent and able to stand on my feet, get out of the protective environment I have once created for myself into the world to explore and see things for myself. Thanks. Figure 1. Pui Yin's (Singapore) journal entry. After a lesson on classical probability and quiz (described under the section on class activity), Cindy (from Canada) made the journal entry presented in figure 2. WOW! What a quiz! We spent most of the day today talking in our groups about the quiz, talking about the importance of order, what happens when order is not important, do other examples using other things also fit this example(? On the quiz, I knew that the answer was not 1/16 because this is the answer you get if order is important. However, by the way it was phrased, order was not important in the question. Thus, the answer will be different. After going over the possibilities, I came up with a different answer, one which I believe is right. When you are told that you have 2 boys and 2 girls, order is not important. Therefore, bbgg = bgbg = bggb(. Because these orders do not matter, I came up with the of 25% chance of having two boys and two girls in any order. But I realize that this is wrong too because if you draw out the combinations you get the following: bbbb, bggg, bbgg, gbbb, gggg. These account for the possible combinations if order makes no difference. Any one of these orders equals 20% of the total. Therefore, I believe that I made mistake somewhere, but I am still sure that the answer to this problem is not 1/16 because there is more than one combination out of 16 where there will be 2 boys and 2 girls. Today's special topic of course, was the quiz, and classical probability (binomial probability because we are dealing with 2 choices - boys or girls). At the end of the class we still have mixed feelings about what the answer to the quiz might be. I may be wrong with my answer, but I am thoroughly convinced that the answer of 1/16 is wrong. Today's class was excellent, because our group got so into the discussion of probability that there was a question for almost every comment that was made, challenging that person to explain what he or she means. Also, contradiction and counter examples were used in attempts to disprove things that were being said. In sum, I left the class with a light headache because of the constant thinking about each question, but I also felt good because we covered a lot of ground that we would not have covered if we were to do this on our own. Figure 2. Cindy's (Canada) journal entry. Discussions Since what gets graded is what students perceive as valuable (Wilson, 1994), the journal entries that the students made were graded. The journal entries took 10% of the total marks for the course in Canada and 25% of the total marks for the course in Singapore. Marks were assigned according to the clarity of students comments, using the guidelines they were provided. Notice that while Pui Yin followed the guidelines "stepwise," Cindy's comments were "global" and I had to search for evidence that she addressed the issues provided in the guidelines. Pui Yin correctly identified the main point of her lesson as the "concepts [conceptions] of algebra" while Cindy's main points of the lesson were "the quiz and classical probability" and the role of "order" in computations involving binomial probability. It was useful that both students identified main points that were close to what the lessons were about. It suggests that, at least, the students could recall some useful information from the lessons. For evidence of what the students understood from the lessons, Pui Yin detailed four conceptions of algebra as generalised arithmetic, procedure for solving problems, relationships among quantities, and as the study of structures. She even gave examples in each case to demonstrate her understanding of those conceptions. Her examples provide convincing evidence that she did not only recall the main points of the lesson, but she also understood them. For Cindy, it is clear that she was convinced her answers were wrong and that contradictions and counter examples could be used to disprove things. Also, she was able to identify that order was not important in solving the problem. Although she could not solve the problem, she provided a useful account of her thinking and reasoning in grappling with the probability problem. Regarding what the students did not understand from the lessons, Pui Yin seemed to have understood what was taught and had no queries. However, Cindy was sure she made a mistake and had mixed feelings about "what the answer to the quiz might be," although she could not tell what the mistake was. Although not investigated, the extensive monitoring by Cindy of her thought processes without solving the problem might suggest some content knowledge problem. Nevertheless, I think for Cindy to have realised that she made a mistake suggests she has a metacognitive ability that is useful for problem solving. However, that metacognitive ability needs to be combined with content knowledge to find answers to problems. Notice that Cindy provided 3 answers 1/16, 20%, and 25%, but she was still not sure of the solution to the problem. Another student from Singapore, Siew Ming, commented that she was not clear on how to successfully cater to the different needs [of students] due to the difference in intelligence level, when assigning students to work in small groups. Her comment prompted a re-visit of a previous lesson on group norms (Artzt, & Newman, 1990) when I found out that other students were having similar doubts. A lot more information was provided than I was expecting when students were to pass "any other comments." Pui Yin's comments that although she found the classes very tough and sometimes frustrating, she was thankful for overcoming the "obstacle of being dependent" and getting out of the "protective environment" to stand on "her own feet," suggest a sense of satisfaction that she had derived from the lesson. Also, the comments suggest a sense of empowerment communicated through the journal entries. Similarly, Cindy provided insights into how she was benefiting from the group discussions, despite the "slight headache" resulting from the "constant thinking." She "felt good" because her group "covered a lot of ground." John (Canada) provided a more direct evidence of the benefit he derived from making the journal entries when he wrote: "I have enjoyed writing in my journal and feel that it has helped me to grasp the concepts at hand. Seeing it written out in my own words really helps me. I may continue to write them next week when we do not have to." John's comment, I believe, is evidence of the sort of empowerment that making of journal entries can provide students. Finally, a comment by David (Singapore) that "Short break in-between periods will be good (about 5 minutes)" conveyed to me the need to address some of the students affective concerns. Subsequently, the students got 10 minutes break in-between periods. Implications One of the implications of the results for classroom teachers is that whereas it is desirable to have students make journal entries, teachers will have to come up with ways of grading such entries if students are to take making of journal entries seriously. This is because students perceive as valuable what is graded. Another implication is that while students may be able to summarise the main points of a lesson, they may only be demonstrating their recall ability. Beyond the recall, students should be encouraged to provide their own examples as evidence of their understanding of concepts learned. Also, although it is useful for students to be able to monitor their thinking, it is not sufficient when it comes to solving problems. Students will have to effectively combine their metacognitive ability with their content knowledge so that they can solve problems. After all, successful problem solving includes finding answers to problems. Finally, journal entries can help teachers identify affective concerns of students. These concerns might create irritations that interfere with learning. As such, it is important for teachers to identify these concerns through journal entries by encouraging students to make any comments, cognitive or otherwise. Concluding Remarks It was useful to find, through the journal entries, how students were making sense of the mathematics they learned. The entries also provided insights into students' affective concerns. Providing grades for the entries conveyed to the students that their comments were valued. However, a lot of work was involved in reading, grading, and using information from the entries to plan subsequent instruction. Nevertheless, the benefits in terms of the empowerment of students to take control of their own learning, irrespective of the context and the students' background, are worth the efforts. I urge teachers to encourage their students to use journal entries to reflect on the mathematics they learn. The results can be rewarding. References Anku, S. E. (1996). The M3 project. Teaching and Learning, 17(1), 113-119. Artzt, A. F., & Newman, C. M. (1990). How to Use Cooperative Learning in the Mathematics Class. Reston, VA: National Council of Teachers of Mathematics. Davis, R. B., Maher, C. A., & Noddings, N. (1990). Constructivists views on the teaching and learning of mathematics. Journal for Research in Mathematics Education Monogragh 4. Kenny, P. A. & Silver, E. A. (1993). Student self-assessment in mathematics. In Norman L. Webb and Auhur F. Coxford (Eds.), Assessment in the Mathematics Classroom 1993 Yearbook, 229-238. Reston, VA: National Council of Teachers of Mathematics. Wilson, L. (1996). What gets graded is what gets valued. In Diana V. Lambdin, Paul E. Kehle, & Ronald V. Preston (Eds.), Emphasis on Assessment: Readings from NCTM'S School-Based Journals, 73-74. Reston, VA: National Council of Teachers of Mathematics. 9