Sally Fitzgibbons Foundation

Beginning the Academic Essay

A study by Francisco (2013) reported that problem solving in mathematics is enhanced by communication and collaboration in the classroom. Consistent with this previous study, Cai & Lester (2010) indicate, learning mathematics in the classroom can be enriched by including different types of discourse between both the teacher and students and among students. Thus, communicating ideas and verbalizing mathematical thinking can improve comprehension and allow students to explore and analyze other perspectives (Weber, Maher, Powell, & Stohl, 2008). In addition, making mathematical connections using different points of view enables students to expand their own schemas to formulate new knowledge (Skemp, 1987). Active learning and problem solving with a focus on communication is emphasized in the following Communication Standards from preschool to twelfth grade (National Council of Teachers of Mathematics, 2000, p. 12):
• Organize and consolidate students’ mathematical thinking through communication;
• Communicate students’ mathematical thinking coherently and clearly to peers, teachers, and others;
• Analyze and evaluate the mathematical thinking and strategies of others;
• Use the language of mathematics to express mathematical ideas precisely.
Many studies on improving mathematical learning include group collaboration and discourse as a means to promote mathematical engagement (Slavit, Bornemann, & Haury, 2009; Staples, 2007; Weber et al., 2008). The term collaborative “implies joint production of ideas, where students offer their thoughts, attend and respond to each other’s ideas, and generate shared meaning or understanding through their joint efforts” (Staples, 2007, p. 4). In one study, students working in learning groups were encouraged to justify their answers through mathematical reasoning, and to challenge other students’ thought processes and problem solving techniques. As a result, the reorganization of the learning communities to foster mathematical thinking and collaboration contributed to increased achievement levels and mathematical engagement (Slavit et al., 2009).
An important factor to consider for collaboration to be initiated in a mathematics learning environment is the role the teacher plays. According to Slavit et al., (2009) and Staples, (2007), the teacher’s guidance plays a key role in generating discourse and participation. These researchers found that the teacher was responsible for creating a culture where inquiry enables discussion and mathematical reasoning. Additionally, a collaborative environment was found to foster respectful discourse where students could challenge each other’s justifications without using negative criticisms. By promoting a collaborative learning community, the teacher not only guides the conversations, but also actively facilitates the discussions to enable the learning of the mathematical objectives (Pugalee, 2001).
Another important element to consider in the development of a collaborative learning environment is the students’ level of self-efficacy. For example, many of the students who have passed my previous math courses with low grades have gaps in their fundamental knowledge and have struggled to achieve success in their future mathematics courses. These low grades combined with difficulty understanding the concepts affect not only their academic achievement, but also their intrinsic educational ambition. Motivation is critical to learning and is generally formulated cognitively (Bandura, 1993). Students who have high levels of self-efficacy or confidence seem to work more diligently at understanding the concepts and are more inclined to participate in collaborative group discussions.
Theoretical Perspectives
The theoretical perspectives framing the current problem of practice include Bandura’s social cognitive theory, especially self-efficacy and Vygotsky’s constructivist theory. The principals behind these theories will inform my intervention through the design, implementation and observation of the intervention in practice.
Social cognitive theory. Personal, behavioral and environmental factors all contribute to students’ learning. “In social cognitive theory, people must develop skills in regulating the motivational, affective, and social determinants of their intellectual functioning as well as the cognitive aspects,” (Bandura 1993, p.136). As such, students need to believe in themselves and their capabilities for cognitive strategies to be effective. Furthermore, self-efficacy and self-regulation skills are both important elements in the development of mathematical environments conducive to learning. According to Bandura (2009), a person’s level of efficacy affects their self-motivation and behavior as it is impacted by their goals and aspirations. Similarly, Ormrod (2008) defined self-efficacy as an individual’s introspective view on their own ability to accomplish certain actions and/or reach their goals. Students with high self-efficacy tend to engage in difficult tasks and will likely persist at problem solving to achieve mastery (Bandura, 1989; Ormrod, 2008). Additionally, individuals with high confidence and self-efficacy will keep challenging themselves even after a setback or failure (Bandura, 1989). On the other hand, students who have low self-efficacy or lack confidence in their mathematical problem solving abilities are more likely to lose motivation and decrease their efforts in the face of a setback or failure. Students who have performed poorly in past mathematics classes might be capable of developing better problem-solving skills, but due to past negative experiences in mathematics they may lose interest and/or resign themselves to failure. This suggests that the lack of student engagement and low mastery levels could be resolved by increasing their self-efficacy.
Students’ self-efficacy beliefs can be linked to self-regulation to further improve their effort and performance. Bandura (1993) stated, “Much human behavior, which is purposive, is regulated by forethought embodying cognized goals. Personal goal setting is influenced by self-appraisal of capabilities” (p.118). Setting goals, assessing cognitive processes, and reflecting on success and/or failure are all aspects of self-regulation (Ormrod, 2008). Similarly, Zimmerman (2002) considers self-regulation the act of being self-directive where students are able to convert their mental capabilities into academic performance skills. Therefore, a self-regulated learner can take responsibility for his or her academic outcomes by understanding the learning targets, setting goals, and taking action to achieve success (Young, 2005). Additionally, by monitoring behavior related to their goals, students can self-reflect and increase their success, (Zimmerman, 2002). Improving the students’ cognitive skills can generate motivation, which could then influence self-regulation to promote effort and understanding of the mathematical concepts.
Learning and using self-regulatory skills seems to entail having a strong sense of self-efficacy where setbacks could be viewed as merely a challenge and are quickly taken up as a way to increase knowledge. However, if students have low self-efficacy, they discard the skills they learned when they fail, (Bandura, 1989). The ways people behave, feel, motivate themselves and think are all impacted by efficacy beliefs, (Bandura, 1993). In order to build self-assurance in students with lasting effects, the previous failures and challenges should be recognized. Thus, students who are performing below average in mathematics should feel more confident in their understanding of the mathematical content and to improve their problem-solving skills.
Bandura (1989, p735) has recommended four principal sources for inducing self-efficacy:
• Direct mastery experiences;
• Observing people similar to oneself succeed by perseverant effort;
• Social persuasion that one possesses the capabilities to succeed;
• Judgments of bodily states and various forms of somatic information.
The first principle, mastery experiences are not easy successes but rather opportunities to use cognitive, behavioral and self-regulatory skills in a sustained effort, (Bandura, 1995). For instance, Zimmerman (1995) recommends educational aids to impart knowledge and strategies in incremental steps. Another example is giving the students the opportunity to challenge themselves with difficult problems allowing them to achieve success through scaffolding and structure.
The second principle which Bandura (1995) called “vicarious experiences provided by social models” (p. 3), involves observing other students similar to them achieve successes. Competent models can be inspirational as students acquire and develop higher levels of self-efficacy. Peer teaching and collaborating with other students provides opportunities for mirroring interactions.
Social persuasion is the third principle Bandura described which could strengthen students’ self-efficacy by encouraging self-improvement and conveying positive feedback. The teacher is instrumental in this principle. The teacher not only praises the students, but they also help deepen the students’ understanding by using structured situations to encourage inquiry and reflection. Additionally, the teacher’s instructional competency is fundamental in creating positive experiences that provide guided practice and feedback for the students’ cognitive development.
The last of Bandura’s principles of self-efficacy consists of the students’ “physiological and emotional states in judging their capabilities.” (Bandura, 1995, p. 4). For example, maintaining a positive learning environment in the classroom could help to reduce stress and increase student engagement. Hence, the teacher’s talents and self-efficacy are important in creating an atmosphere conducive to learning (Bandura, 1993).
The four principals described above can be strengthened with the use of self-regulated learning. Zimmerman, (1995), encourages self-directed learning to help students focus on self-referential processes to allow for self-assessment of their own efficacy. An example of such an effort could be peer-teaching, where students self-regulate their own learning in order to teach the content to their peers.
Constructivist theory. Constructivism is grounded in creating a learning environment where students, with guidance from the instructor, actively explore/construct solutions and develop knowledge. The main goal of constructivist teaching is to foster and facilitate subject matter understanding (Henderson & Gornik, 2007). It is not the transmission of information by passively absorbing it in a quiet environment where the teacher lectures and the students take notes. Construction of knowledge provides students with the ability to actively explore the content, to make new meanings and to connect their knowledge in a social collaborative atmosphere. Dewey (1916) stated, “education is a constant reorganizing or reconstructing of experience,” (p. 89). Similarly, Piaget a constructivist theorist proposed that students create their own knowledge about new content and relate it to prior experiences to formulate original ideas and understandings (Ormrod, 2008).
Vygostky’s believed that learning is a social and collaborative process where knowledge is formed by interactions with other individuals (Schreiber & Valle, 2013). Additionally, he considered speech and discourse a fundamental element in constructing a social culture (Smagorinsky, 2007). Discourse is a way to interact by representing different perspectives by communicating and actively engaging in the instructional tasks (Cai & Lester, 2010). A social learning environment can incorporate problem solving, critical thinking, discourse, collaboration, and creativity in the learning process.
Social constructivists, such as Dewey and Vygostky, argue that knowledge is formed through interactions (Dewey, 1916; Schreiber & Valle, 2013). Students contribute different backgrounds, knowledge, and unique experiences that help them to engage in multiple learning opportunities. Skemp (1987), suggests “mathematics is about 5% rote learning, and 95% intelligent learning” (p.103). Where rote learning is memorization of facts, intelligent learning allows students to apply their mathematical knowledge to build on their understanding and intelligence. Additionally, the knowledge is actively processed and internalized to make meaning and connections from existing cognitive information (John-Steiner & Mahn, 1996). As students construct meaning and make connections from prior knowledge and experiences they generate shared ideas from multiple perspectives. Since each student possesses different worldviews and background knowledge, social interactions give students the opportunity to learn from numerous depictions of reality (Schreiber & Valle, 2013). Students can benefit from insights and different models to represent their thinking.
As Vygostky studied the learning and development of individuals, he found that interactions with more capable adults or peers contributed to the level of possible development (John-Steiner & Mahn, 1996). This concept of learning is known as the Zone of Proximal Development (ZPD) illustrated in Figure 1. This theory is best explained by visualizing concentric circles. The center circle represents the level of the students’ knowledge and what he or she can do without external assistance. The next ring represents the ZPD. This area is the knowledge and content understanding that can be learned and mastered through meaningful collaboration and social interactions. Learning in this ring consists of both subject matter (e.g., mathematical thinking, justifying answers…) and making connections through discourse (e.g., inquiry, communication…). For example, Goos (2004) used Vygotsky’s Zone of Proximal Development as a framework to analyze collaborative mathematical inquiry with three elements:
1. Teacher-Student Interaction: The ZPD as Scaffolding
2. Student-Student Interaction: The ZPD as Collaboration
3. Everyday and Scientific Concepts: The ZPD as Interweaving
Mathematical collaborative inquiry requires students to learn by making sense of and participating in activities that involve problem solving and making connections to other contexts and ideas. It includes mathematical justifications and challenging one another’s answers. NTCM (2000, p. 13), has developed Mathematical Connection Standards for grades prekindergarten to twelfth grade which engage students to:
• Recognize and use connections among mathematical ideas;
• Understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
• Recognize and apply mathematics in contexts outside of mathematics.
Reorganization of teaching practices and incorporating active interactions are essential to empower students to feel confident about their mathematical successes and to enable them to self-regulate as they continue to learn/justify their findings. “To organize education so that natural active tendencies shall be fully enlisted in doing something, while seeing to it that the doing requires observation, the acquisition of information, and the use of a constructive imagination, is what most needs to be done to improve social conditions” (Dewey, 1916, p. 161).

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