Introduction
This essay looks into the theme of effective mathematics teaching and learning in a primary school setting, drawing upon theory, practice, and pedagogy examined in taught sessions. The discussions revolve around the knowledge quarter, specifically with a focus on connections, transformations, and contingency as important aspects in shaping teaching approaches. In addition, the effect of a teacher’s subject knowledge on children’s learning will be discussed. The main objective is to identify good practices and understand the impact of theory and policies on teaching and learning in mathematics.
Transformations
One key element of the Knowledge Quarter is transformations. Transformations refer to the ability to see how diverse mathematical concepts and procedures are related. For instance, an instructor who understands transformations can aid students in seeing how addition and subtraction are related or how multiplication and subtraction are related. In teaching and learning mathematics, several theories are involved in understanding how students progress and how teachers can appropriately support their learning using transformation. Such a theory is constructivism. Constructivism is a learning theory that posits that individuals actively construct their knowledge and understanding based on their experiences and interactions with the learning environment. In the case of transformations, students build their understanding of spatial relationships and geometric concepts by engaging in hands-on activities and examining real-world examples. Another important theory is Piaget’s theory of cognitive development. Jean Piaget’s theory emphasizes how children’s cognitive abilities develop over time. In the case of transformations, Piaget’s theory stipulates that as children grow, they move from concrete operational thinking to more abstract reasoning, allowing them to grasp more complex geometric transformations.
There are diverse ways teachers can incorporate transformations in their teachings. One approach is to incorporate materials to aid students to see the connections between different concepts. For instance, students can use blocks to represent addition and subtraction, or they can incorporate place value blocks to represent multiplication and division. Another approach to incorporate transformation is to offer students opportunities to solve problems that need them to make connections between diverse concepts. For example, students might be asked to solve a problem entailing conversion between diverse units of measurement, or they might be asked to solve a problem that entails using different mathematical strategies.
The National Curriculum for Mathematics stipulates that students need to be taught to understand the relationship between diverse mathematical concepts and to use these relationships to solve problems. This entails understanding the transformations that can be integrated into mathematical ideas. The Curriculum also stipulates that students should be taught on how to use mathematical concepts in real-world contexts. This entails using mathematics to solve issues that entail making connections between different concepts (National Curriculum, 2021). Nonetheless, Ofsted’s School Inspection Handbook (2017) states that inspectors will look for evidence that teachers “help pupils to make connections between different mathematical ideas and to use these connections to solve problems” (p. 145)
Connections
Another key aspect of Knowledge Quarter is connections. This refers to one’s ability to see how mathematics is linked to other subjects like technology and science. For instance, teachers who understand connections can aid students in seeing how mathematics is incorporated to solve issues in the real world. There are diverse theories of learning that emphasize the significance of connections. One of the primary theories is constructivism. This is a theory of learning that stipulates that learners actively construct their own knowledge by making connections between new data and their existing knowledge. Another theory of learning that emphasizes the significance of connections is schema theory. Schema theory stipulates that learners store data in their minds in the form of schemas. Schemas are mental contexts that aid us to organize and understand data. When an individual learns new information, we connect it to existing schemas in our minds. This helps us to make sense of the new information and to remember them more easily.
Children progress in mathematics by making connections between different mathematical concepts. For instance, a child who understands the concept of addition can make connections to the concept of subtraction. This allows the child to see how the two concepts are related and to incorporate them to solve issues. There are several ways that teachers can aid children in making connections in mathematics. One way is to incorporate concrete materials. Concrete materials can aid children in visualizing mathematical ideas and making connections between them. For instance, a teacher might incorporate blocks to ad children understand the addition concept. Another approach to aid children makes connections in mathematics is to use real-world problems. Real-world issues can aid children to see how mathematics is used in the world around them. This can aid them in making connections between mathematical ideas and seeing how they can be incorporated to solve real-world issues. Finally, teachers can aid children in making connections by offering them opportunities to discuss their thinking. This can aid children in articulating their understanding of mathematical ideas and making connections between them.
The National Curriculum for Mathematics stipulates that students need to be taught to understand the relationship between diverse mathematical concepts and to incorporate these relationships to solve issues (National Curriculum, 2021). This entails understanding the connections between mathematical ideas and how these connections can be incorporated to solve issues. Other legislation and policies about the use of connections is Ofsted’s School Inspection Handbook (2017) which stipulates that instructors will look for evidence that teachers aid pupils to make networks among different mathematical concepts and to use these connections to solve problems.
Contingency
The final element of the Knowledge Quarter is contingency. This refers to the ability to adapt to teaching to meet the necessities of individual students. For instance, a teacher who understands contingency might offer diverse support to students struggling with a particular concept. There are a number of theories of learning that emphasize the significance of contingency and using misconceptions to address learning. These theories include constructivism and schema theory, already discussed above.
Children can progress in mathematics through connections between diverse mathematical concepts and correcting their misconceptions. Teachers can aid children’s progress in mathematics by recognizing their misconceptions and offering them opportunities to correct them. The first approach is using wait time. Wait time refers to the duration a teacher waits after asking a question prior to providing an answer. When instructors use wait time, they offer students adequate time to think about the question and try to answer on their own, thus establishing relevant connections. Another approach is the use of scaffolding. Scaffolding refers to the process of offering students with hints and suggestions but not giving them an outright answer. This aids students in learning to solve problems on their own and correct their misconceptions.
The National Curriculum for Mathematics does not specifically mention the use of contingency in mathematics. The Curriculum stipulates that teachers need to showcase a secure knowledge of the mathematics they are teaching (National Curriculum, 2021). The NCETM’s Mathematics Matters (2008) report states that teachers need to be conscious of common misconceptions and how to address them. This implies that teachers should be aware of the common misconceptions that students have in mathematics and should be able to address them effectively.
Conclusion
Subject knowledge is essential for effective mathematics teaching. The Knowledge Quartet provides a framework for teachers to develop their subject knowledge and ensure that all students progress. By using the Knowledge Quartet, teachers can help students see the connections between different mathematical concepts, see how mathematics is used in the real world, and adapt their teaching to meet the needs of individual students. Teachers who understand transformations, connections, and contingency can better help students progress in mathematics.
References
National Curriculum (2021). http://www.appgmathsnumeracy.org.uk/wp-content/uploads/2016/01/APPG-Session-Ofsted-write-up-Dec2015.pdf
NCETM (2008) Mathematics Matters https://oggiconsulting.com/wp-content/uploads/2018/05-Further-Reading/Mathematics-Matters-Final-Report.pdf
Ofsted’s School inspection handbook. (2017). https://dera.ioe.ac.uk/id/eprint/30206/1/School_inspection_handbook_section_5.pdf
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