Introduction
It is a common occurrence for a mathematics class to consist of students with different development and degree of preparedness, varied academic performance, attitudes to learning, interests, and health status. The teacher, in this case, has to diversify their approach to teaching in order to ensure everyone’s engagement; however, it might become quite difficult in the traditional organization of education. More often than not, the teacher is forced to conduct training in accordance with the average level – of development, knowledge, and performance. In other words, the teacher has to design education plans focusing on an image of an “average student.” This inevitably leads to the fact that “strong” students are artificially restrained in their development, while “weak” students are doomed to chronic lag. Thus, this essay draws on the problem of how a teacher can cater for the different student levels in the mathematics classroom, ensuring equal educational opportunities for every student.
Literature Review
The topic of differentiated teaching has accumulated a significant body of research over the years, as it has been discussed in length from various perspectives. The concept can be used for teaching any subject; however, the importance of differentiated learning has been specifically emphasized in mathematics, which relies heavily on the specific sequence of knowledge. Still, there remain barriers to overcome before differentiated teaching can be successfully implemented in the educational practice.
Research shows that the topic of issues in implementing differentiated teaching is rather relevant today. Suprayogi et al. (2017) discuss in their study the technicalities of using the method in the classroom, and what influences the possibilities of its success. The study emphasizes that the growing diversity of the students in schools is the main driving force behind the popularity of differentiated teaching. However, while the method is considered the most efficient solution, it still poses a challenge for implementation for the teachers. According to the authors (2017), “actual differentiated instruction (DI) implementation of teachers is linked to a complex set of variables: teachers’ DI self-efficacy, teaching beliefs, teaching experience, professional development, teacher certification, and classroom size” (p. 291). Moreover, Suprayogi et al. (2017) also add that “research implication calls to invest teacher professional development programs” (p. 291). The study states that, in order to apply differentiated teaching properly, one needs to actually understand and agree with its concept.
It can be said that the actual success of the method’s usage depends the most on the teacher’s professional competence and willingness to accommodate, as well as on the provision of development programs for teaching staff. van Geel et al. (2018) support that claim, stating that “differentiated instruction is an important but complex teaching skill which many teachers have not mastered and feel unprepared for” (p. 51). This study also draws on the importance of offering the teachers opportunities to enhance their understanding of differentiated teaching. However, van Geel et al.’s (2018) research focuses more on the processes and tools for assessing the teachers’ professional skills in relation to differentiated teaching to determine their quality. According to van Geel et al. (2018), “the international literature on assessing teachers’ differentiation qualities describes the use of various instruments, ranging from self-reports to observation schemes and from perceived-difficulty instruments to student questionnaires” (p. 51). The study claims that in order for the teachers to be able to develop their differentiating skill further, they require well-designed self-evaluation tools to track their progress.
It is also interesting to discuss how differentiated teaching affects students’ performance, especially in mathematics. Prast et al. (2018) conducted a large-scale study to determine the effect of the method on students and their academical proficiency. According to the authors (2018), “teacher professional development about differentiation raised student achievement, as they learned how to adapt mathematics education to diverse educational needs” (p. 22). Teachers were offered a professional development program, and the effects of it were observed in the students. The authors (2018) emphasize that with the help of the program, “teachers learned how to adapt mathematics education to diverse educational needs” (p. 22). This research offers a significant body of evidence for the efficiency of differentiated teaching: students from thirty primary schools were observed for two consecutive years. Prast et al.’s (2018) study is similar to van Geel et al.’s and Suprayogi et al.’s researches as it also focuses on teachers and their differentiated teaching skills. However, the differences lie in the fact that Prast et al. (2018) linked teacher development directly to the students’ mathematical performance, providing supporting data on the point that differentiated teaching positively affects academical success.
The student performance and inclusive educational practices are other aspects of differentiated teaching that pose interest to researchers. Porta and Todd (2022) discuss in their study how teachers in senior secondary classrooms use differentiated instruction to cater to the needs of diverse learners. The authors (2022) state that “students with learning difficulties often enter secondary schooling with knowledge deficits and weaknesses in areas such as literacy, numeracy, writing, reading and comprehension, placing them behind their peers” (p. 2). The teacher must create optimal conditions for the development of every student in order to overcome the constantly arising contradictions between the mass nature of education and the individual way of mastering knowledge and skills. This leads to the need to use level differentiation in the process of teaching – especially with a subject such as mathematics (Porta and Todd, 2022). In conditions of differentiated learning, strong and weak students feel comfortable, and the school treats each student as a unique, inimitable personality. Remaining within the framework of the class-lesson system and using the differentiation of teaching, teachers are able to approach the personal orientation of the educational process.
Similar ideas and concepts are discussed in the study conducted by Woolcott et al. (2021), drawing further on the development of practice framework for teaching mathematics in secondary education. The research (2021) states that “a teacher’s ability to effectively differentiate instruction in the classroom is crucial in catering for student individuality and diversity, especially in the context of inclusive learning” (p. 1). Differentiated teaching allows to transition from homogenous groups of students that consist of moderately evenly matched people to heterogeneous groups that can accommodate a wide variety of learners. This approach offers students more diverse settings, experiences, and opportunities for development. According to the authors (2021), “differentiated instruction offers elements that foster positive learning environments, embracing diversity in a way that provides for individual growth in learning based on a student’s ability, interest and readiness levels” (p. 2). This study, however, not only draws on the implications of implementing differentiated learning like Porta and Todd’s (2022), but also discusses how mathematical educators can overcome barriers to adopting the method. Developing a practice framework for differentiated instruction can help teachers grow their confidence in the approach, as well as adjust it specifically to their classrooms to ensure best performance.
Equal opportunities in learning in the context of Australasian mathematics education are also discussed by Anthony et al. (2019). Here, the authors take a review approach, offering insights into various frameworks for differentiated teaching and how they promote equity. Anthony et al. (2019) also offer a critique of the concept, stating that the challenges associated with it are mostly due to its vagueness in the relevant literature.
Teachers need to have a solid understanding not only of the general umbrella term of differentiated instruction, but also of its specific models and application directions. Moreover, Antony et al. (2019) emphasize that “teaching in culturally and academically diverse classrooms requires addressing socio-political, psychological, and instructional factors that influence students’ success and social inclusion” (p. 123). Teachers need an advanced set of skills, knowledge, and various competencies to apply differentiated teaching correctly if they want to improve student satisfaction and performance. Still, the authors (2019) report that, overall, “approach that values students’ strengths raises teacher expectations, with frequent teacher reports in the first year of professional learning of ‘surprises’, particularly with regards the quality of thinking” (p. 122). While similar to other studies on the topic of differential instruction, this research takes into account details that might seem irrelevant in teaching mathematics, and reveals them to be as important as other features.
Finally, the study conducted by Cevikbas and Kaiser (2020) offers a rather specific approach to catering to diverse student needs. The research discusses a particular method of differentiated teaching – flipped classroom. According to the authors (2020), “innovative methods can change the paradigm of teaching mathematics and inspire teachers to espouse new ideas and gain new experiences” (p. 1291). Thus, the study reviews specifically how teaching of mathematics can transform if a flipped classroom technology is applied, finding that this method proves to help support students’ mathematical thinking (Cevikbas and Kaiser, 2020). The students benefit from the flipped classroom because it allows them to work in their own tempo and with the tools that are most comfortable for them. Moreover, in a flipped classroom, a disabled student would no longer feel the pressure of needing to be on par with their classmates who are used to traditional learning methods.
Task Recommendation
Task 1
Basic proportional reasoning and ratios: determining the amount of partially filled blue and red paint cups to create a specific number of purple paint cups.
Task 2
Properties of ratios and fractions: determining the school president by the percentages of overall, 6th, and 7th grade votes.
Task 3
Rates and equations: determining the amount of stock shares two companies need to exchange to get equal profits.
Each of the three tasks proposes a problem on the theme of ratios, proportional reasoning, and fractions; they are placed in a sequence from less to more complex. Task 1 requires the use of visuals and/or real paint cups to demonstrate the process of mixing paint in specific proportions. The use of supplementary equipment will help a teacher ensure that the students understand the problem correctly and recognize how the concept of proportioning and ratios applies to it. Moreover, the teacher may group students with different levels and offer them an opportunity to mix the paints themselves to show them how proportioning works. Tasks 2 and 3 will require group roleplay: the class will need to be divided into respective alliances mentioned in these tasks and play out the problems. With this approach, students will be able to grasp the real-life concept of voting and recognize the patterns of votes distribution in ratios for the Task 2. For the Task 3, this method will provide the students with better understanding of the concept of stocks and how companies operate their assets. Additionally, during the group exercises, “strong” students will be asked to offer assistance to “weak” students by explaining processes of solving the problems.
The current stage of development of school mathematical education is characterized by the desire for a student-oriented educational process. This involves the construction of individual educational trajectories, taking into account the subjective experience of the individual, their preferences and values, the actualization of the personal functions of students in the learning process. The proposed sequence draws on the concepts that not only are familiar to students, but are also recognizable in the real world. It offers opportunities to apply mathematical thinking to the processes outside the classroom, and understand complex terms and topics through simple and association-based explanations and problems. Moreover, the possibility to approach a mathematical issue from the point of roleplay generates interest and enhances engagement in the students, serving as motivational tool. Differentiated teaching requires the teacher to be creative with their educational practices, and this sequence reflects this principle.
There is also the aspect of group work: during collective activities, students learn to interact with others, communicate their ideas, and apply communal effort to problem solving. By combining students of different levels in groups to work together, the teacher offers more developed students opportunities to improve their skills by helping less developed ones. Children think in unusual patterns, and they can explain a concept in several different ways to their peers to get them to understand it properly. Moreover, “weaker” students will not feel excluded from the process during group activities, as small sizing of groups will not only give them an opportunity to engage, but actually require participation from them. Implementing this sequence into the curriculum offers a teacher a way to observe students during independent work to determine their strengths, weaknesses, and unique thinking abilities. With task gradually rising in complexity from one to another, but maintaining relations to one general topic, it will be easier for students to rely on the existing knowledge and develop it further.
Reflection
One of the ways to implement a student-centered approach in teaching mathematics is the differentiation of teaching. This method allows each student to receive mathematical training at different levels in accordance with their individual characteristics, abilities, and background. Thus, in order for the use of differentiated learning technology to bring positive results, it is necessary to perform an assessment of the educational activities of students. Education should be variable to the individual characteristics of the students, and assist them in their development through different means and measures. It is necessary for the teacher in their practice to make sure that the student works to the best of their ability, experiences the joy of educational work, and consciously and firmly assimilates the program material. With a properly organized educational process, the progress of students increases, which indicates a positive rise in their learning opportunities. The task of the teacher is to closely monitor these changes and make adjustments to the organization of the process of education to ensure that diverse needs of the students continue to be met.
Dividing the class into groups helps to organize students and offer them independency to mutually check each other’s work, which raises communal responsibility for completing tasks. Thus, a differentiated approach to students helps to prepare low-performing students for the perception of new material and fill the gaps in knowledge in time. Finally, it can help the teacher make a wider use of the cognitive abilities of students, especially more developed ones, and constantly maintain interest in the subject. While it is, indeed, a complex task to develop an appropriate differentiated teaching strategy, with the help of relevant academic literature and existing frameworks, a teacher can design a curriculum that fits their specific classroom.
References
Anthony, G., Hunter, J., & Hunter, R. (2019). Working towards Equity in Mathematics Education: Is differentiation the answer? In G. Hine, S. Blackley, & A. Cooke (Eds.), Mathematics Education Research: Impacting Practice (Proceedings of the 42nd annual conference of the Mathematics Education Research Group of Australasia) (pp. 117–124). essay, MERGA.
Cevikbas, M., & Kaiser, G. (2020). Flipped classroom as a reform-oriented approach to teaching mathematics. ZDM, 52(7), 1291–1305.
Illustrative Mathematics. (2022a). Dueling candidates. Illustrative Mathematics.
Illustrative Mathematics. (2022b). Perfect purple paint II. Illustrative Mathematics.
Illustrative Mathematics. (2022c). Stock swaps, variation 2. Illustrative Mathematics.
Porta, T., & Todd, N. (2022). Differentiated instruction within senior secondary curriculum frameworks: A small‐scale study of teacher views from an independent South Australian school. The Curriculum Journal.
Prast, E. J., Van de Weijer-Bergsma, E., Kroesbergen, E. H., & Van Luit, J. E. H. (2018). Differentiated instruction in primary mathematics: Effects of teacher professional development on student achievement. Learning and Instruction, 54, 22–34.
Suprayogi, M. N., Valcke, M., & Godwin, R. (2017). Teachers and their implementation of differentiated instruction in the classroom. Teaching and Teacher Education, 67, 291–301.
van Geel, M., Keuning, T., Frèrejean, J., Dolmans, D., van Merriënboer, J., & Visscher, A. J. (2018). Capturing the complexity of differentiated instruction. School Effectiveness and School Improvement, 30(1), 51–67.
Victorian Curriculum and Assessment Authority. (2022a). Victorian Curriculum – Learning in Mathematics. Victorian Curriculum.
Victorian Curriculum and Assessment Authority. (2022b). Victorian Curriculum – Mathematics Level 7 Content Descriptions. Curriculum.
Woolcott, G., Marks, A., & Markopoulos, C. (2021). Differentiating instruction: Development of a practice framework for and with Secondary Mathematics Classroom Teachers. International Electronic Journal of Mathematics Education, 16(3). Web.