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Many primary schools throughout New Zealand are changing their current teaching mathematics methods, to incorporate the new organizational practice used in developing mathematical inquiry community classrooms. These community-based classroom teaching methods strive towards equitable learning for diverse learners, to further enhance their mathematical proficiency, by developing risk taking and problem-solving skills. The purpose of this essay is to therefore, analyze the teaching methods of a junior maths teaching video on stream, that captures the key learning and teaching methods by the teacher, relevant to teaching this diverse group of young children. A lesson of fractions is being taught, where the teacher incorporates real life experiences into her lesson plan. Also evident in her teaching methods, were the four elements of (Cobb’s 2007) pedagogical system that incorporated the following into her lesson such as a non-threatening classroom atmosphere, the use of instructional tasks, classroom discourse and tools and representations. Patient, nurturing and encouraging teaching methods were evident throughout the video, that resulted in a positive, learning and development experience for all the children. Cobb (2007) sees teaching as a coherent system rather than a set of discrete, interchangeable strategies, it encompasses four elements namely: a non-threatening classroom atmosphere, instructional tasks, classroom discourse and tools and representations.
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In order for children to learn and do mathematics, they also need to learn to talk about mathematics, which then becomes, very much a social activity. This would also involve sense making, mathematical communication/language, developing rich connected mathematical understandings and developing socio group participatory norms that support these practices in order to become successful learners. The socio group participatory norms evident were: being respectful, active listening and active contribution. The following socio group participatory norms are indicated in mathematical practices as being, respectful communication, leaving no one behind, taking no passengers, shared understanding, asking questions, active listening, active contribution, collaboration, arguing and disagreeing where necessary with regards to math practices, valuing contributions and equal status. (Massey University, Mathematical Practices and Social Group Norms – A revision, Topic 1a). Continual repetition, questioning and asking for responses, was also frequently evident during the lesson, and this also enabled the children to make connections to the big ideas, in order to learn new strategies. According to Bright and Joyner (2008) state that good questioning helps students become more flexible thinkers and become more confident in with multiple ways of using important mathematical ideas. The use of a non-threatening classroom atmosphere was very evident from the start of video 1, where the new entrant class, sat happily on the floor, fully engaged, listening attentively, as their teacher began the lesson on fractions. (Junior Math Lesson: Video 1). As the lesson progressed, the children were able to follow the instructions given by the teacher, by being respectful, attentive and working together using the buddy system. Many of the groupings had been done previously and were effective, where the teacher either put in pairs, one introverted and extroverted child or at times, two quiet children, in order to get them to talk. Often the more outgoing child would be happier doing the mathematical explaining verbally, as the child would use mathematical language, justification, connecting to the big ideas and the other child would do the writing. According to Hunter (2007) researchers indicate that through small group interactions, these students are provided with opportunities to participate in and contribute to productive mathematical discourse without being in the public eye. During the lesson, the teacher often used revoicing, to get the children to repeat what she had just said, so that they could relay by further reinforcing the concept.
Instructional tasks help the children and teacher to become actively engaged during the lesson to ensure that, the entire lesson is directed towards the development of a particular skill, concept or new idea. The task given to the children was to learn about fractions by cutting naan into half, then into quarters and then finally counting all the pieces, to see how many they had altogether. Counting numbers – there is a number word and a matching symbol that tell exactly how many items are in a group. (Massey University: Key/Big ideas in Math). For this lesson the teacher incorporated the use everyday life experiences, for example, Esha’s parents had recently made naan for Esha’s brother’s first birthday. According to Averill and Harvey (2010) states that from their early years children encounter the language and representations of halves and quarters, along with the concept of fair sharing, and these continue to develop in a more structured way in the classroom. The math task given and questions used by the teacher, suited this new entrant year group, as they were able to grasp the concepts of addition, and the use of fractions, such as the ½ and the ¼ by all utilizing the same strategy to work out the problem. According to Averill and Harvey (2010) state that it is widely accepted that learning occurs within a sociocultural context, in which learners build their understanding based on prior knowledge gained through experience and discussion (eg., Ball, 1993; Hiebert & Carpenter, 1992; Yackel & Cobb, 1996).
The big idea that the teacher used in this lesson of fractions, was to emphasize that, when the denominator gets bigger, the fraction gets smaller. According to Small (2009) if a teacher understands this is the big idea behind the expectation/outcome, he or she can teach the concepts and skills involved in conversion much more meaningful instead of a more superficial skills-based approach. For a new entrant, cutting up numbers and by saying that ½ is bigger than a ¼, is a huge concept for them to grasp, when they have spent the past five years of their lives, only knowing that, the bigger the number, the more monetary value it holds.
According to Lotan (2003) tasks require problem solving as they are open ended, they provide students with multiple carry points to the task and multiple opportunities to show intellectual competence in order to evaluate and devise strategies. The math task given and questions used by the teacher, suited this new entrant year group, as they were able to grasp the concepts of addition, and the use of fractions such as the ½ and the ¼ by all utilizing the same strategy to work out the problem. According to Sullivan, Aulert, Lehmann, Hislop, Shepard and Stubbs (2013) state that positive classroom culture is not a matter of rules and procedures but the ongoing and interactive support teachers provide that encourages students to take up the challenge of tasks. The big idea that the teacher used in this lesson of fractions, was to emphasize that, when the bottom fraction gets bigger, the pieces are getting smaller. According to Small (2009) if a teacher understands this is the big idea behind the expectation/outcome, he or she can teach the concepts and skills involved in conversion much more meaningful instead of a more superficial skills-based approach. For a new entrant, by cutting up numbers and by saying that ½ is bigger than a ¼ is a huge concept for them to grasp, when they have grown up, only knowing that, the bigger the number the more value it holds. In order for the children to grasp the concept of the current task, they would need to rely on their previous knowledge of fractions and strategies in order to solve the current problem or task. According to Lotan (2003) group tasks enable students to share their experiences, justify their beliefs and opinions in order to analyze and evaluate tasks.
Classroom discourse is defined as whole-class discussion in which students talk about mathematics in such a way that reveal their understanding and concepts which also allow them to engage in mathematical debate. During the lesson, as the teacher went step by step, as she asked the children to tell their buddy about the mathematical concept that they were working on. This enabled each buddy to listen to what the other buddy was saying, that enabled each child to understand each step. According to Sullivan, Aulert, Lehmann, Hislop, Shepard and Stubbs (2013) the processes for managing sharing of student strategies are a critical component of the establishment of an overall culture of support that create opportunities for student learning. During the lesson the teacher continued to use many questions, which encouraged the children to listen and to be actively involved and to value each contribution and to build on it. The teacher responded positively and encouraging at all times that made the children relax and feel that they could make an active contribution to the lesson. According to Bright and Joyner (2008) the use of many different questions during a lesson, can enhance effectiveness of different components of instruction so that a greater percent of students will remain engaged in conversation about the mathematical idea being used. The teacher was able to listen attentively and could identify that the children were on track with the task. The teacher also asked Sammy and Travis to show and explain their workings to the class, to further encourage others to participate within the class setting, that they could all achieve the final step together. The teacher also asked Pippa to share her work by showing the additional strategies she had used, in order to get the 14 such as adding 3 + 3 = 6 and that 4 + 4 = 8 to make 14 pieces altogether. (Junior Math Lesson: Video 6). By doing this showed the class that there is more than one strategy to solve a problem. Most of the class did not understand this concept but the teacher said that students also need to struggle at times, to give themselves more time, to process and understand new strategies. (Junior Math Lesson: Video 6). “Teachers can foster a classroom culture that values and promotes productive struggle by providing students with challenging tasks that are designed in such a way that they are all accessible to all students and the expectation is that everyone will persist when solving challenging mathematical tasks”. (Livy, Muir and Sullivan, 2018).
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The use of tools such as a whiteboard/whiteboard marker and representations in the forms of diagrams or numbers, and fractions written in number form, all enable the child to visually see the problem more clearly in order to gain a better understanding of the mathematical concept.
Representations can also be used in the form of symbols, word problems and rich tasks to further broaden problem solving skills and deepen mathematical content. According to Goldin (2014) mathematical representations can be defined as visible or tangible productions such as diagrams, number lines and equations. The use of diagrams in this lesson, further assisted all students in the understanding of mathematical concepts as they were able to draw the same diagrams into their books and work through each problem systematically.
In conclusion, the use of the DMIC math model in the primary school sector, is greatly significant because it not only encourages, all students to become actively engaged and active learners, it also encourages, a diverse group of students to acquire mathematical proficiency, so that no child is ever left behind. This model greatly encourages group work too, so that students can work collaboratively, facilitated by the teacher, to come together, to discuss their solutions and big ideas within the problem, to further encourage confidence, self-belief and most importantly, to be valued member of the classroom.
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