Promoting student discovery of new mathematical ideas in solving open-ended problems

JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1075.2016-0211 Educational Sci., 2016, Vol. 61, No. 11, pp. 13-20 This paper is available online at PROMOTING STUDENT DISCOVERY OF NEWMATHEMATICAL IDEAS IN SOLVING OPEN-ENDED PROBLEMS Tran Vui College of Education, Hue University, Vietnam Abstract. Recently, in Vietnam, having launched the reform mathematics curriculum, classroom mathematics teachers have learnt more about innovative teaching strategies as a means of implementingmore effective lessons focusing on mathematical thinking. The aim of this paper is to examine a specific lesson promoting new mathematical ideas in solving open ended problems. The main question that needed to be explored was how classroom teachers should create mathematical activities that would give students an opportunity to demonstrate their ability to observe, predict, rationalise and apply logical reasoning when solving open-ended problems. The results showed that well-instructed lessons with open-ended problems will support students in exploring meaningfulmathematics. Students with good basic knowledge, skills and understanding can develop their mathematical thinking to solve challenging problems. Keywords:Mathematical ideas; open-ended problems; school reform mathematics; lesson study. 1. Introduction Currently, in Vietnam, there are still some drawbacks and inadequacies in the mathematics curriculum and its accompanying textbooks. The knowledge provided in many textbook based lessons is theoretical, which does not help students practice real-life problem solving. Mathematics lessons at schools are too academic, which means that students find it difficult to understand when all mathematics results need to be proved logically. Teachers often present mathematical evidence in the form of formal deduction which applies abstract symbols in mathematical reasoning. Students then use these mathematics results when doing practice exercises and consolidating learned knowledge. Since the emphasis of the old curriculum was on procedural knowledge and memorization of algorithms, students often worked independently to complete exercises from textbooks and workbooks. When asking students questions, most teachers seek one “right” answer to the mathematical problem and will explain why that answer is correct. The reform curriculum tries to reduce the amount of basic skills and procedures in mathematics, while increasing hands-on activities that help students to grasp new ideas and develop mathematical thinking. School reforms in mathematics education aim to help students achieve the following four objectives: knowledge, skills, thinking and attitudes [1]. Received date: 12/11/2016. Published date: 17/12/2016. Contact: Tran Vui, e-mail: tranvui@yahoo.com 13 Tran Vui Pehkonen [2] stated that problems dealt with in school mathematics are usually closed problems, which will not leave much room for creative thinking. The idea of using open-ended problems to improve school mathematics teaching, to develop and foster methods for teaching problem-solving and thinking skills, has appeared in the curriculum of many countries in a form that allows teachers freedom to adopt an “open approach” [2], [3]. In [4], Nohda held the view that open-ended problems are atypical problems which should have two prerequisites. Firstly, they should suit every single student by using familiar and interesting subjects. This implies that students realize it necess ity to solve the problems, feel it possible to solve them with their own knowledge and have a sense of achievement after solving them. Therefore, the problems should be sufficiently flexible to take into account the students’ different mathematical abilities. Secondly, open-ended problems should be suitable for mathematical thinking and it should be possible to restructure them into new problems. In 2000, Foong summarized three basic criteria for an open-ended problem [3]: - It should give all students a chance to demonstrate some mathematical knowledge, skill and understanding; - It should be rich enough to challenge students to reason and think; to go beyond what they expect they can do; - It should allow the application of a wide range of solution approaches and strategies. In the process of solving an open-ended problem, students may make some mistakes, and give both correct and incorrect answers but, at the end of the problem-solving process, students will construct new ideas in mathematics. In this kind of teaching, the teacher helps students delve deeply into a textbook problem and build up a habit of questioning achieved results, encourages students to be interested in seeking alternative solutions, and promotes creativity when learning mathematics. A discovery requires creative thinking: – Which is executed on the basis of the knowledge of a rule, while the given facts have not been associated conceptually with that rule before and/or. – Which consists in the creation of a new rule. Since the discovery of new knowledge alone does not guarantee certainty, the hypothetical knowledge has to be verified. To express a discovery only means an explanatory hypothesis is becoming plausible. Nevertheless, the correctness of the rule and the case, as well as the coherence between the rule and the observed fact, could remain vague [5], [6]. The aim of this paper is to report on how open-ended problems were used in lesson study to promote the students’ mathematical ideas. 2. Content Vietnam is seeking innovation in mathematics teaching and learning strategies. The teacher ought to think of teaching in terms of several principal hands-on activities with visual representations, problematic real life situations, and open-ended problems. The innovation of teaching is to help students construct their own knowledge in an active way and to enhance their thinking through solving non-routine problems, while working cooperatively with classmates, so that their talents and mathematical competencies are developed [7], [8]. 2.1. Methodology: Lesson study There are several possibilities for innovation of mathematics education in an economy. Lesson study, which originated from Japan, is currently a central focus in the US and other 14 Promoting student discovery of new mathematical ideas in solving open-ended... economies for the professional development of teachers and the improvement of students’ mathematical ideas. For comprehensive details of Japanese lesson study [9]. In this research paper on lesson studies for developing a good lesson, a study cycle was adopted that comprised planning → implementing and observing → discussing and reflecting on the Vietnamese economy. The research focuses on lesson study as a means to innovation. The results from this lesson study showed that good teaching practices are powerful models for changing the quality of mathematics education. Research question: How classroom teachers should create mathematical activities that would give students an opportunity to demonstrate their ability to observe, predict, rationalise and apply logical reasoning when solving open-ended problems? The lesson is videotaped and analyzed using the video recording and the transcript. The actual lesson included several activities. The analysis in this paper will be conducted by dividing the actual lesson into three stages: introductory activities, activities for task 1, and activities for task 2 and task 3. Each stage will be described and analyzed. 2.2. Well-Instructed Lesson Plans Classroom teachers know how to design lesson plans and this is reflected in the ability to identify the right objectives, contents of the lesson, intended teaching strategies and aids, and appropriate allocation of time, according to lesson flow. In the teaching standards, the components for developing a well-instructed lesson plan are defined as follows [1]: - the teacher designs a lesson plan in accordance with regulations on the subject structure; - lesson plans must suitably reflect the objectives of the lessons; - lesson plans are consistent throughout the major content of the lesson; - lesson plans present a selection of teaching methods to facilitate the pupils’ learning initiative; - learning materials, aids and resources are selected and used effectively to improve teaching quality; - assessment methods are included in lesson plans; - timing for teaching activities in the classroom must be indentified in the lesson plans. In the teachers’ guidebook for mathematics there are four main activities in a lesson at each grade that teachers should follow to develop mathematical thinking: Activity 1. The teacher motivates students to work and achieves the following aims: - examines the students’ previous knowledge; - consolidates previous knowledge involved with the new lesson; - introduces the new lesson. Activity 2. The teacher facilitates the students’ exploration of mathematical knowledge and allows them to construct new knowledge by themselves. Activity 3. Students practice the new knowledge by solving exercises and problems in the textbook. Activity 4. The teacher reviews what the students have learnt in the new lesson and assigns the homework. Engaging with the lesson will give the pupils the opportunity to demonstrate their mathematical thinking through: 15 Tran Vui - the ability to observe, predict, and apply rational and logical reasoning; - knowing how to express procedures and properties through language at specific levels of generalization (by words, word formulae); - knowing how to investigate facts, situations, and relationships in the process of learning and practicing mathematics; - developing the ability to analyze, synthesize, generalize, specify, and start to think critically and creatively 10] . 2.3. Study Lesson The lesson which will be analyzed in this paper was prepared by a classroom teacher for grade five (Vietnamese students aged from 10 to 11 years old). The three main tasks and a quiz proposed in the lesson can be found in the lesson plan: Introductory Task. Use 2 cm-cards and 4 cm-cards to make a toy train of 5 wagons. Task 1. Use 2 cm-cards and 4 cm-cards to make a train with a length of 16 cm. Task2. If a train with a length of 50 cm has 20 wagons, how many of the wagons are black and how many are red? Task 3. If a train with a length of 100 cm has 36 wagons, how many are black and how many are white? Quiz (Homework). There are 33 liters of fish sauce contained in 2 liter and 5 liter-bottles. The number of bottles used is 12. Find the number of 2 liter and 5 liter-bottles used, given that all the bottles are full of fish sauce. At the end of Grade 4, students will know how to solve and express solutions of problems having three operations of natural numbers. Example. A toy train has 3 wagons with a length of 2 cm, and 2 wagons with a length of 4 cm. Find the length of the train. Answer. 3 × 2 + 2 × 4 = 14 (cm). In the second semester of grade 5, however, the problem was set in a reverse way: A toy train has two types of wagon: 2 cm-wagons and 4 cm-wagons. This train has a length of 14 cm, including 5 wagons. Find the number of 2 cm-wagons and 4 cm-wagons. The sum of the two numbers needing to be found is 5. Restricted condition: The total length of 2 cm-wagons and 4 cm-wagons is 14 cm. This reverse problem is quite different to what students have learnt in grade 4. The problem requires them to break down a natural number into the sum of two other numbers, satisfying a restricted condition logically. 2.4. Analysis of the tasks Analysis of Introductory Task. Use 2 cm-white cards and 4 cm-black cards to make a toy train of 5 wagons. 16 Promoting student discovery of new mathematical ideas in solving open-ended... This task is an introductory activity. It is an open-ended task that requires pupils to make as many trains as possible. Pupils can arrange the cards to make a train, and use the strategy "guess and check" to get many answers. To solve this task mathematically the teacher guides the students to make a systematic list of all abilities. N. of white wagons 0 1 2 3 4 5 N. of black wagons 5 4 3 2 1 0 The length of the train in cm 20 18 16 14 12 10 From the above list, students can recognize the relationship between the length and the numbers of white and black wagons. If the number of white wagons increases by one, then the length of the train decreases by 2 cm. In this task, students know that: N. of white wagons + N. of black wagons = 5 There are 6 options for this task. If the length of the train is given, then the number of white and black wagons can be calculated precisely. The length of the train is understood as a restricted condition. Students will see that the train’s longest length is 20 cm when all of the wagons are black, and its shortest length is 10 cm when all of the wagons are white. The aim of this introductory task is to help students recognize the restricted condition in finding two numbers when their sum is known. Analysis of Task 1. Make a train with a length of 16 cm. This is also an open-ended task that requires students to make a systematic list of all abilities. The restricted condition is given, but the sum of two numbers is unknown. N. of white wagons 8 6 4 2 0 N. of black wagons 0 1 2 3 4 Total of wagons 8 7 6 5 4 There are 5 answers to this task. Students know how to analyze a natural number into the sum of two natural numbers with a specific restricted condition. 16 = 8 × 2 + 0 × 4 16 = 4 × 2 + 2 × 4 16 = 0 × 2 + 4 × 4 16 = 6 × 2 + 1 × 4 16 = 2 × 2 + 3 × 4 From the list, students will see that a train with a length of 16 cm, including 6 wagons, has 4 white and 2 black wagons. The restricted condition of this problem is: Sum: N. of white wagons + N. of black wagons = 6. Restricted condition: The length of the train is 16 cm. If all the wagons are white, then the length of the train decreases: 16 - 6 × 2 = 4, then the number of black wagons: (16 - 6 × 2) ÷ 2 = 2. 17 Tran Vui Students practice this procedure to consolidate what they have learnt. The most important fact that the students need to realise is the difference of 2 cm between one black and one white wagon. Analysis of Task 2. If a train with a length of 50 cm has 20 wagons, howmany of the wagons are black and how many are white? In this task, the teacher does not ask the students to make a table, but encourages them to generalise what they have observed in the concrete situations above, to create their own procedure to solve the general problem. Students make a temporary assumption: if the train has only white wagons, the length of the train decreases: 50 - 20 × 2 = 10 The number of black wagons: (50 - 20 × 2) ÷ 2 = 5. Students review the solution by checking their answer: 15 ×2 + 5 × 4 = 50 cm. Analysis of Task 3. If a train with a length of 100 cm has 36 wagons, how many are black and how many are white? The aim of this task is to help students consolidate what they have studied. They use their procedure to solve this problem by using temporary assumption. The number of black wagons: (100 - 36 × 2) ÷ 2 = 14. Analysis of Quiz. There are 33 litres of fish sauce contained in 2-litre and 5-litre bottles. The number of bottles used is 12. Find the number of 2-litre and 5-litre bottles used, given that all of the bottles are full of fish sauce? This is an application task. Students can solve this task as homework. Students learn how to apply what they have studied from the lesson to solve a realistic problem. Students recognise that the difference between one 5-liter bottle and one 2-liter bottle is 3 liters. The number of 5-liter bottles: (33 - 12×2) ÷ 3 = 3. Thus, the answer is 9 two-liter bottles and 3 five-liter bottles. 2.5. Research Findings For introductory activities, some students made only one train of 5 wagons, as required, and then stopped working. The teacher asked the students to paste their answers on the blackboard. Most of the answers were presented except the two last options: 5 white wagons and 0 black wagons, or 0 white wagons and 5 black wagons. The teacher asked the students to arrange the data following a systematic list. For Task 1, students received a virtually blank table, although some cells contained numbers that helped the students to fill in the data more easily. For Task 2 and Task 3: - Students were asked to solve an extended problem that is difficult to guess and check. - Students were required to create a procedure to solve the task with a specific restricted condition. - Students were asked to present their answer by using mathematical operations. In these tasks, some students used mental calculations or "guess and check" strategies to find out the answers, but they could not explain the answer logically. Open-ended problems such as these are very useful for learning activities, and also as items in assessment instruments. 18 Promoting student discovery of new mathematical ideas in solving open-ended... 3. Conclusion There are multiple solutions and methods for solving open-ended problems, so they are particularly useful in all classrooms, as students have different levels of skills and understanding of concepts. They can be used to find out what students know about a concept when beginning a topic, investigating a concept, applying a skill, or ascertaining students’ understanding or skills at the end of a teaching sequence. Since there are multiple solutions and methods for finding solutions to these kinds of questions, a student’s solution may be located in the standards used in the reform curriculum and assessment documents [11]. The intention was to look at good practice as a means of improving student learning. Good practice embodied in this lesson study is based on outcomes of successful student learning, including students’ mathematical thinking, and can be used for further development or challenges. Teaching school mathematics aims to equip young pupils with basic mathematics skills and develop their mathematical thinking to solve problems. As the activities in the lesson were analysed by using videotaped recording, the teacher followed four main activities in a lesson that were suggested by the MOET to develop mathematical thinking [10]. In introductory activities, the task was open-ended: it helped students to observe many abilities, and predict some answers by "guess and check". These activities gave students opportunities to show their ability to observe, predict, rationalise and apply logical reasoning when solving open-ended problems [8]. In activities for task 1, the teacher facilitated the students to explore their mathematical knowledge and construct new mathematical ideas. Students observed and predicted answers for specific situations, suggested a procedure to solve a general problem, and invented their own procedures or algorithms to solve the problems. In activities for task 2 and task 3, students practiced the new knowledge by solving exercises and problems given by the teacher. Students applied the analysis of natural number into the sum of two other numbers with a restricted condition to systematically solve some mathematics problems, by using temporary assumption. These two tasks examined the thinking operations that occurred in the lesson such as: comparison, generalisation, and specialisation. From the analysis of the tasks, it is found that the classroom teacher knew how to create meaningful open ended problems based on students’ previous knowledge with manipulative materials that gave them an opportunity to demonstrate their ability to observe, predict, rationalise and apply logical reasoning when solving these problems. The open ended problems create a learning environment that gives students ch