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A teacher’s primary objective is to establish a productive learning process in their classroom. In regards to mathematics, it is especially vital to ensure that students are engaged in the activities. This environment can be created by applying different communication strategies that will enable children to address any issues they are having, voice their questions and provide an instructor with feedback. This paper aims to analyze the role of talking in teaching mathematics, examine the contribution of social constructivists, and evaluate transcripts from lessons.
The ability to communicate about any subject learned in an educational setting is essential for students. This factor is especially important in mathematics because many children get confused about the terms and specific aspects of calculations. According to Campbell and Bolyard (2018), the existing process of teaching mathematics to children receives a lot of criticism. The element, however, has had positive outcomes because, through discussion, the importance of talking in mathematics classrooms became evident to educators. Campbell and Bolyard (2018) argue that communication is crucial for the subject because it promotes the need to analyze processes that contribute to solving problems. Various opportunities for enhancing the learning process arise in this case, fostering a better learning environment.
Talking to students during lessons is key to active learning and understanding. According to Anthony and Walshaw (2007, p. 54), “language plays a key part in shaping students’ mathematical experiences.” Educators that approach the process by applying terms that are used in the subjects can help their students develop their cognitive skills. Campbell and Bolyard (2018) substantiate this claim by stating that talking improves both memory skills and understanding of the subject. In addition, proper communication in the classroom can contribute to children’s confidence and interest, which affects their learning outcomes.
Receiving feedback from students during lessons is essential because it allows addressing any misunderstandings and creating strategies that would enhance learning based on children’s opinions. In this way, children can learn from each other and analyze a different perspective that another individual takes to solve a particular problem. Campbell and Bolyard (2018, para. 10) state that “learning math is not a process of acquiring a set of facts or procedures, but a process of becoming one who participates in a community that does mathematical work.” Thus, it is vital for a teacher to create a proper environment in the classroom that would encourage talking about mathematics.
Primarily, a teacher should ensure that his or her communication with students encourages students to address any issues and improve their ability to think mathematically. Anthony and Walshaw (2007) state that improper instructions can have adverse outcomes for students because they affect their perception of the subjects. However, in many cases, children do not communicate their concerns due to various fears. Several studies cited by Anthony and Walshaw (2007) provided evidence that this factor combined with a different perception of students that a teacher may have encourages children to avoid asking questions.
Thus, even when some parts of a task or a particular topic are confusing for him or her, they prefer not to address it to avoid embarrassment (Anthony & Walshaw 2007). This can lead to several consequences, including the inability to keep up with the learning process and insufficient understanding of the subject. Mathematics talk should be applied together with methods that encourage students to speak in the classroom to avoid this.
The Contribution of Social Constructivists
Social constructivists view the process of acquiring new knowledge as something affected by the external environment, more specifically the community of a particular individual. Interaction is the essential component for learning, based on their perspective because it enables a person to achieve any goal. Acar and Yilmaz (2015) conducted an experiment in an educational setting that was designed to integrate constructivists’ views of learning and identify whether the theory can be supported by evidence. Both groups were talking, and discussions were used by the authors to examine communication between children that occurs when solving mathematical problems.
One of the findings indicates that “students, who could not solve the problems individually, were able to contribute more to the problem-solving activity in the group works” (Acar & Yilmaz 2015, p. 998). Therefore, the constructivist approach can help educators endear all children in the process of learning, which will lead to their improved understanding of the matter. The constructivist theory in regards to teaching was observed by two primary researchers – Vygotsky and Bruner.
Vygotsky focuses his attention on the zone of proximal development (ZPD). The concept implies that learners can achieve different results when they receive outside help when compared to those who facilitate without additional support. The objective of the approach is to enable children to solve problems on their own by helping them develop necessary skills. According to Tomlinson and Murphy (2015), Vygotsky’s concept implies that three primary categories of achievement exist.
They are what a child can achieve without aid, what can be done with help from an educator, and what is impossible to accomplish. The second component, which is what can be learned when a teacher provides guidance, is the primary idea that guides ZPD.
Social constructivists and Vygotsky, in particular, pay specific attention to communication with adults, through which a child can learn. According to Fernandez et al. (2001), through ZPD, children can appropriate particular skills or knowledge from other members of their community. The idea is based on the opinion that cognitive development can be facilitated through two levels – instrumental and intramental.
The first one is connected to the social environment, while the second one describes individual thinking processes (Fernandez et al. 2001). Therefore, a distinct connection exists between how a person thinks on their own and in a particular social setting. This implies that a mathematics teacher can either ensure that children are capable of voicing their opinions and questions, which helps them solve problems, or the opposite. In both cases, the role of an educator is central because it affects the ZPD.
The theory of ZPD implies that teachers should provide students with challenging tasks; however, they have to facilitate proper support to ensure positive learning outcomes. According to Walker and Shore (2015, p. 2), “children, therefore, need to be challenged with learning material that they would most likely be unable to complete on their own, but, with help, could learn successfully.” This is the primary approach that Vygotsky describes with the concept of ZPD.
Vygotsky emphasizes the importance of language learning and communication in the process of learning. According to Tomlinson and Murphy (2015), language is a form of interaction that a child uses to convey ideas to a teacher. The social aspect of communication is important because, through it, a child develops his or her cognitive capabilities. It should be noted that many concepts that Vygotsky introduces contradicted the prevailing opinions in regards to education that existed in the nineteenth century. For instance, Tomlinson and Murphy (2015) state that according to this social constructivist, there is no ideal age for learning a specific subject. Thus, in some cases, an individual may be either too young or too old to acquire particular knowledge.
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The idea of relative achievement was developed by Vygotsky to compare the knowledge of children before learning a specific subject and after the end of the process. He based the approach on the idea that different types of students enter school when having varying levels of knowledge. Some of them may be studying more than the other group; however, based on the previous achievement, the brighter student may show a better result (Tomlinson & Murphy 2015).
This can be applied in the process of learning mathematics because each child has different capabilities at the beginning of classes. By remembering Vygotsky’s approach, a teacher can be able to praise each for his or her progress. Doing so will provide an understanding that the efforts of children are significant as they improve, even in cases when they do not receive the highest grades compared to other students.
Both the concept of ZPD and scaffolding were used in the process of teaching a child who receives support from an adult. This approach is valid within asymmetrical learning and, when used in symmetrical teaching, has several distinct features. Fernandez et al. (2001) argue that in peer-to-peer education, which can be facilitated in the form of a discussion, children can support each other by using different language. Therefore, teachers should pay attention to the interaction that occurs between students because those affect learning outcomes as well.
Bruner developed the theory of ZPD by introducing scaffolding as the primary approach. According to Fernandez et al. (2001, p. 43), the notion is defined as “the way an expert ‘tutor’ (such as a parent) can support a young child’s progress and achievement through a relatively difficult task.” Therefore, there is a clear connection to the concept of ZPD. It is because both ZPD and scaffolding imply that additional help can promote learning and help a child achieve better results.
In the case of scaffolding, an instructor should motivate an individual, help simplify the steps required to solve a problem, highlight essential factors that should be noted by a learner, and help deal with frustration. Additionally, the tutor should help an individual focus his or her attention on a specific task and provide an example of an ideal process (Fernandez et al. 2001).
The concept developed by Bruner implies that children can learn more by cooperating with teachers or classmates. Fernandez et al. (2001) research the idea of “scaffolding” in their study to identify evidence-based factors that can contribute to the process of learning mathematics. Thus, when comparing an independent student who does not receive outside help and the one working with his or her instructor and peers, the second learner will have better educational outcomes. It should be noted that a teacher has to be aware of the students’ ZPD to execute the approach adequately. In the opposite case, the efforts may be directed at the thing that children are currently unable to achieve, leading to demotivation and inability to progress.
Mathematical Learning Drawing
Lesson 1 focused on applying ten times tables to improve learning outcomes. In it, the teacher explained the process and used drawing cards as additional material to engage students in the educational activity. In the first dialogue, an example of peer communication was displayed. As was previously discussed in this paper, children can benefit from talking and receiving feedback from more advanced students. However, in this case, while R did attempt to provide an explanation of the thought process behind the solution, further communication indicates that M did not understand it.
Following the explored social constructivism theories, it can be argued that using additional materials can enhance learning outcomes for students. According to Harries and Spooner (2013, p. 45),” to explore these big ideas, we need to provide students with images which they can use so that they may become efficient and competent workers with numbers.” Thus, in Lesson 1, the table introduced by the teacher should provide support to children. The one aspect that should be improved is the moderation of communication between children because talking in a classroom can be very helpful for students.
When comparing the cases of N and J to that of M and R, it is evident that in the first one, children were able to interact successfully using both talk and additional drawings for understanding mathematics. In both cases, the teacher did not interrupt the process of learning through peer feedback. However, the outcomes differed; thus, it can be evaluated that in some instances, educators should help students by guiding their conversation. Due to the fact that no understanding was established between M and R, it can be concluded that R will further struggle with the task. Thus, the teacher in question should pay additional attention to M and R and help them build on each other’s ideas.
The transcript from Lesson 2 displays the use of number cards, whiteboards, and pens in the process of revising previous mathematical knowledge. In this case, children were discussing the mathematical symbols, and the teacher joined the conversation to provide an example that can be used as a reference. In this case, the talk had a crucial role for two reasons. Firstly, unlike in Lesson 1, M, J, and N were able to listen to each other, ask questions, and disagree with opinions. This led to productive work; however, it was evident that additional support from an educator is needed to enhance understanding. Thus, the second component that contributed to successful learning is the teacher’s attention to the arguments.
The children interacted and solved problems using cards with < and > symbols on them. From the transcript, it is evident that the images aided them significantly because they pointed to different cards in their conversation. Additionally, the teacher encouraged her students to use mathematical language in the second part of the activity, which promotes better learning. Mastering mathematics requires not only an understanding of basic notions and concepts but also an application of creative thinking.
This cannot be leveraged by continually solving problems and reading textbook explanations because these methods do not engage children’s imagination (Oldridge 2017). In addition, an essential factor is a difference in perception and the relative knowledge that a child has. Thus, as was displayed in Lesson 1 and 2, children that were more confident in their understanding helped others by sharing their perspective.
An important aspect that aligns with the concept of ZPD and talking for learning mathematics is the teacher’s request to explain the logic behind comparing numbers -12 and 13. While at first, R did not want to expand on the topic, he later spoke about his memories of adults comparing minus numbers by identifying which ones are closer to zero. Barnes (2008) states that new models can be easily assimilated if they fit into a person’s understanding of the world. R successfully applied this approach which helped him understand this complex idea. This serves as a great explanation of the topic and gives other students a reference that they can use. Thus, in this case, mathematics talk between peers promoted the learning outcomes of the students.
In both Lesson 1 and Lesson, two children were able to learn mathematics through images, conversation, and support from a teacher. According to Carruthers and Worthington (2004, p. 32), “children’s choices and ways to represent do not remain static since discussion modeling and conferencing alter individual’s perspective.” Therefore, by using various methods, the educator in Lessons 1 and 2 helped students explore the topic of calculation more efficiently.
In addition, it was evident that students are engaged in the process and want to gain knowledge the mathematical skills, which is a critical component of learning. Based on the fact that R enhanced the understanding of the concept behind the notions of greater than and less than, it can be argued that the approach of scaffolding and ZPD is helpful.
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