The Development of Mathematical Understanding in Students - Assignment Example

Core Mathematics Assignment Part A Q: With Particular reference to a mathematical topic of your choice, critically analyse ways in which talk may be used to promote mathematical understanding and to develop Higher Order Thinking Skills at Key stage-2. Introduction The fundamental issue for the teachers is to develop the mathematical understanding about the subject. The students are expected to remember the mathematical methods, rules, and facts by heart without grasping the basic concept of the theory and building its relationship with the previous knowledge. If they are not developing their relationship between their previous and current knowledge, then it might be possible that the students are not able to develop the habit of working independently. Under such situations, where the mathematical talks are not emphasised, the lectures are usually teacher-centred. As a result, the thinking and the concept is not supported among the students (Ofsted, 2008). To enhance the thinking ability of the students, teachers have to design the classes that are interactive and give students a chance to express their personal opinion about the subject. In a research conducted by Wickham (2008), it is suggested that in order to practice effective mathematics learning, it is not necessary to teach children about the precise or longer words of mathematics. However, if the children were given more chances to speak up about mathematics, it would surely be increasing their understanding of the subject. Later during the lessons when they would be exposed to the precise vocabulary of the subject, they can use those terms in their discussion. In this research, it is also suggested that in order to promote talks and discussions in mathematics, teachers have to design tasks in a unique way, that is, instead of solving mathematical problems in an old monotonous manner, they should introduce some changes in their teaching techniques. For example, instead of solving problems on number of worksheets using dry calculations, teachers should give them challenging tasks that would be done in the groups or on the computers. During this practise, the class would be divided into small groups, that would solve the problems by discussion. Thus, when the groups are discussing and solving a certain problem, automatically talking in mathematics is promoted. Hiebert explains in his book, Making Sense, Teaching and Learning Mathematics With Understanding, The Learning of Mathematics, Through Discussion in These Words “If you would like students to understand, then be sure they are reflecting on what they are doing and communicating” (1997). This is why it is important to value talk in mathematics. This technique helps the children to give their own reflection about the concept. Instead of accepting the ideas as they are, the sense of exploring the concepts through different ways can develop new understanding about the topic. Types of talks The development of mathematical understanding can be best conveyed through talk. In order to support this fact, the educational scientists have offered some possible progressions for talk. Few of these talking methods are used very frequently in the mathematical class whereas others are found very rarely. Altogether, three types of talks have been introduced to the mathematical class until now. Transmission Talk This is the most common type of talk. In this type, the children are mostly ‘lectured’ and the curriculum is ‘delivered’ to them. Such talks are usually teacher centred talks and the students are not allowed to give their input in the lecture. Therefore, they have to absorb everything as it is. With the use of such talks, the students mostly lose their interest in the discussed lectures, and their thoughts start wandering here and there (Joubert, 2008). Being a teacher, the drawback I felt of such lectures is that of holding of students’ attention until the end, which is very difficult task. Formal/Presentational Talk This very formal way is used by the students to express their understanding about the topic, either in spoken words or in the form of writing. Although, this talk is done by the students, but it focuses on the expectations of the audience: teachers. This process is followed in those lessons where the teachers give transmission talk and give very little thinking or discussion time to the students. Thus, if we evaluate this technique critically, the students that receive lectures in transmission talks can follow this pattern, as there is a very formal way of communication between the teachers and the students (Joubert, 2008). Informal Talk This friendly talk can be done in two ways, both amongst the students themselves and between the students and teachers. The purpose of this talk is to develop confidence in the students. Due to such type of talks, the students explore different ideas; they refine their understanding of the mathematical skills and language. This informal talk is further divided into three parts. The first is Disputational talk in which there is a tendency for the element of disagreement or individual decision-making to arise when the students are discussing the concept among each other and/or with the teacher. About the talk it is said, “The relationship [between partners] is competitive; information is flaunted rather than shared, differences in opinion are stressed rather than resolved, and the general orientation is defensive” (Joubert, 2008). This means that despite the fact that the partners are working on the same task, their approach towards the situation is different. The second talk is Cumulative Talk; in this talk, the speakers build positive and uncritical approach towards each other. The talk is being done to construct a common knowledge point. Such a scenario can be experienced in the class discussion among the teachers and the students when a new concept is promoted. The last talk is Exploratory Talk; in this talk, the students use each other’s ideas to reach the conclusion. It is the most effective talk as many problems can be solved by collaborative activities in this process. Ways in Which Talk May be Used Effectively to Promote Mathematical Understanding Mathematical understanding can be developed among the students through three ways. The first one is ‘the use of physical representation of concepts’. In this strategy, the concept is explained by the help of physical things. The second one is ‘actions aimed at building conceptual links’. Here, the concept is explained by making a linkage between the previous knowledge and the new concept. The last one is ‘the use of language base activities’. In this technique, the theory is promoted with the help of discussion that is done either among the students or between the teacher and the students (Mousley, 2004). In this discussion, we would be only catering the last type: language base activities, which is the mixture of the first two types. In this category, the teachers use different approaches to deal with the mathematical concepts. In the first approach, the teacher explains the new concept to some extent. In the second approach the students are divided into small groups; they are given an easy task related to that concept, and then they are encouraged to solve it according to their own understanding. Other than these two approaches, asking questions from the teacher can also be used as an effective tool to promote mathematical understanding (Mousley, 2004). The verbal activities that are being promoted by the teachers not only provide a platform to the students on which they can share their ideas, but it also helps them in clarifying their concept about the topic by discussing their confusion related to that topic in the class. Thus, they develop higher order thinking skills by comprehending the given problem based on their own knowledge. Ball and Bass (2000) also supported this concept in their book, ‘The collective construction of public Mathematical knowledge in the elementary classroom’, by saying that on the bases of children’s current understanding, if they are asking questions about the things that have not been understood by them, they are exploring new ideas. While adopting this idea in my class, I did the experiment in one lecture of ‘subtraction’; this particular lecture was conducted in two ways. The class was divided into two groups. For the first group I explained the concept of ‘subtraction by borrowing’ on the board, and later the students were supposed to attempt the worksheet related to this. The result of this exercise was not as satisfactory as expected. For the second group the students were not exposed to the concept taught by me. I wrote a sum on the board and let the students to think of the ways to solve it. They asked different questions about it and ultimately reached the conclusion in this way, the ratio of the understanding among the students was exceptionally high in the latter case. How to Develop Higher Order Thinking Skills In order to develop an active learning environment, and to move the young learners towards higher level of thinking, the teachers should encourage the questions rose by them because only with the help of these questions the students can excel to achieve intellectual freedom and develop critical thinking (Halpern, 1999). When the students are actively participating in the class discussion, the environment of the class can become very learnable and it can be enjoyable for both the teachers and students. Keeping this in mind, when the teachers prepare the lecture, they try to cover every aspect so that the teachers must be able to answer all the questions raised by the students. The higher order thinking skills can be developed in the students, when teachers are fully equipped with the lesson plan, and before coming to the class, they know the learning objectives of the students; what they would learn and know when lesson ends. Due to this technique, the teachers would remain focused on their lesson and can maintain the specific desired behaviour of the class. Thus, when the teachers develop a well-written objective for the class, this surely enhances the learning ability of the students and helps them build up their higher order thinking skills (Ball & Garton, 2005). In order to practice higher order thinking skills, the students should remember, understand, apply, analyze, evaluate, and create the content to some extent. ‘To remember’ means that the students should be able to recall their previous knowledge to answer the current question. ‘To understand’ indicates that the students should explain the existing idea. ‘To apply’ means that based on their previous knowledge the students should be able to use the same concept in some other situation. ‘To analyze’ refers to the answers that demonstrate to see the patterns in the study and thus the students’ ability to classify the information into parts. When the students evaluate the concept, they give justification to the decisions they are making. ‘To create’ refers to generating new ideas or ways to look at the concept (Krathwohl, 2002). As a primary teacher to build up this concept in my students for the mathematics class, I used the phenomenon of number line subtraction with them. In this situation, they had the prior knowledge of subtraction but since the approach was new to them, so they critically analyzed the situation, and on the account of their old understanding, they evaluated the problem. This was an individual task and after a specific time, I called few of the students so that they could discuss and explain their solution to the rest of the class. After this exercise, due to mutual discussion, the students were very near to the answer, thus showing they are absorbing the new concept easily. Teaching through question is another very useful technique that is used by the teachers in the classroom in order to develop higher order thinking skills. Clasen and Bonk (1990) suggest that out of many techniques that help the students to develop higher order thinking skills, questions have the greatest impact. Later in their study, they indicate that the development of the student’s mental level is directly proportional to the level of questions, so in order to enhance their level, the teachers should have maximum exchange of questions in the mathematics class. I practiced this procedure in my class; and when I asked my students to think about their proposed answers, their level of understanding increased. Using these technical approaches in my class, I used the concept of distributive law for subtraction. In order to make them understand the in-depth relationship between the numbers, I designed number of activities for my students. Therefore, on the bases of these activities and discussions, the students were able to understand and apply such a difficult concept on the 2-digit numbers mentally. This shows that after practicing talking in mathematics class, the learning ability of the students increase, and they are able to grasp some difficult task through an easy approach. PART B Q: Reflect upon the use of talk in developing mathematical understanding and promoting HIGHER ORDER THINKING SKILLS during your time on school experience, giving personal classroom examples and making clear links between theory and practice. Activity The activity was based on subtraction I opted for this concept of mathematics because according to QTS 15 it lies under the ability range for which the students are trained and they have sufficient knowledge about this area. It was a pair based activity and the objective of the activity was to subtract the three digit whole numbers. The material required for the students was two dice of different colours for each pair of children. Likewise, I tried to follow the concept of social constructivism in my class by which I adopted interesting activities in the class so that it could encourages the learning ability of the students. (ATHERTON, 2009) In the beginning of the lecture, I briefly discussed the process of subtraction with class then I wrote a three-digit sum on the board and asked the children to subtract it mentally. Once they are done with the subtraction they were suppose to explain the adopted process to the class. In this way, as discussed by Mousley (2004), through such talking in the class the children try to promote their understanding about the topic. Likewise as prescribed in the QTS standard 23, I designed the opportunity for the learners to develop their own literacy and numeric skills. After listening to their adopted methods, I revised the ‘bridging process’ and the ‘number bonds’ with the class so that all those students who were not clear about these concepts can refresh their memory. Here I tried to adopt the Higher Order Thinking Skills in the students through questions as said by Halpern (1999). Using their prior knowledge about the subtraction, the students explained their personally used methods to the class. So that based on their prior knowledge about the subtraction some skills can be developed and they can meet the learning objectives of the lecture as explained in QTS 25. Again, at this stage I tried to promote mathematics among the students through talking (Ball & Bass, 2000). In the activity, the children were supposed to roll the dice thrice so that they can draw six digits. The procedure was; for the first time when the dice were rolled, the shown numbers were 7 and 5, so for not getting negative number, the dice that had 7 was named as first number and the other was called second number. The process was repeated three times to cater for the rest of the numbers. Later the student that has been writing the numbers was supposed to subtract the two digits by using any technique. The game was repeated four times among the students, each student doing the task twice. Later, they were supposed to discuss the process among each other, hence once again the designed activity tried to promote mathematics through exploratory talking (Joubert, 2008). Once the students have been done with the activity they were suppose to present their sum and the method that lead to the solution in front of the class so that they can share their learning experience and knowledge with the class. When the students are presenting their sum, the class was open for discussion. In this way, when other students cross-question them about the method their logic and thinking developed simultaneously. Hence, as discussed by Clasen and Bonk (1990), in this way the students tried to promote Higher Order thinking skills through class discussion. After the class activity, I divided the class into four groups, keeping in mind the QTS 24 I gave them the homework related to activity. By using their reading books they were suppose to conduct the same activity at home. All the children were supposed to count the number of words of the page. Then one group had to subtract all the words that had’ T’ in it from the total. The other group subtracted all the ‘a’s and ‘the’s from the total words. The third group had to subtract the nouns and the last group was supposed to subtract the words starting with vowels from the total number of words. They were expected to do this exercise by both of the way explained in the class i.e. bridging process and the number bonds. When the children have practiced this activity in the school, they were very enthusiastic about sharing experience thus with the use of talk throughout the activity I tried to promote mathematical understanding among my students so that they can get familiar with high order thinking skills. 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Proceedings of the British Society for Research into Learning Mathematics. 28(2). Accessed on 20th May 2010 from http://www.bsrlm.org.uk/IPs/ip28-2/BSRLM-IP-28-2-20.pdf Read More