# How Representations and Communication in Mathematics Support Mathematical Thinking and Understanding

Introduction

Representation is a mathematical process that is presented in the standards and principles of the School Mathematics (Santos & Semama, 2015). The processes act as basic through which the five mathematical concepts are created. As such, they give guidelines on how mathematics should be taught and how to support the students’ learning process. Learners and tutors effortlessly identify the benefits of mathematics subjects in a specific grade band. However, they are less probable to comprehend the importance of the mathematical procedure. Since the introduction of Evaluation and Curriculum Principles for School Mathematics, educators, curriculum developers, and textbook editors have usually disregarded the strong connection between process and context (Widakdo, 2017). Learners have difficulty in working out problems without understanding the mathematical concepts and using mathematical procedures. The procedure principles make available vital backing for book learning of mathematical concepts.

One of the main challenges for mathematical instruction is the failure of teachers to use the appropriate approach to develop mathematics and learning (Moschkovich, 2015). Teachers expect mathematics to be meaningful, but only a few see mathematics as a creative subject. Typical tutors are usually contented when teaching mathematics as they assume it involves the manipulation of symbols and solving routine problems (Polly, 2015). They do not ensure that the learners acquire a deeper understanding of the subject since they consider that students believe and learn what they are taught. Therefore, a shift must be made in mathematics to ensure learners critically think and conceptualize understanding (Kim & Ke, 2016). Such can be achieved by implementing written and oral communication in the mathematics class thereby improving the students’ understanding, thinking, and developing improved attitudes towards mathematics.

Representing Equality

In most cases, learners do not apprehend the representations that grown-ups take for granted, for instance, the equal symbol (Carreira, 2015). Normally, the equal (=) symbol follows the operation symbol, for example (33+22= 55) rather than before (55=33+22). Such placement triggers learners to misinterpret the symbol. As such, learners may interpret the symbol to mean an instruction to solve the mathematical problem. On the other hand, some learners may understand the equal sign to mean the solution to the mathematical problem follows next. Such understandings might hinder the shift into future mathematics. Progressing in mathematics involves proper identification and comprehension of equality and equations (Carreira, 2015). Symbols characterize mathematical concepts. As such, educators must be vigilant on the learners’ thoughts and work with them to guarantee that they understand the mathematical sign correctly.

Representing Area and Perimeter

Learners in fourth-grade created a six-centimeter by eight-centimeter quadrilateral on a centimeter grid broadsheet after which they established the expanse around the figure and used the word perimeter. The students then sketched a diagonal line from one angle of the quadrilateral to the other after which they cut the shape along the line. Afterwards the teacher told them to discover new shapes that can be formed from reconstructing the two subsequent triangular shapes along similar edges. Lastly, the teacher asked them to discover if the resultant shapes had similar areas and perimeters. However, on manipulating the triangular shapes, the learners found a new set of quadrilateral shapes and triangles. They examined the perimeters of the new figures and cogently clarified why the perimeters of the shapes were not similar. Some learners declared that the perimeters of the shapes were different but the areas stayed the same. As such, from this illustration, the tutor was competent to provide an illustration that the learners may well operate and employ in discovering new geometric concepts for themselves.

Representing Measurements

Most learners do not understand the idea of measurement tools, such as formulae and rulers (Yeo, 2018). Such is attributed to the fact that learners are not exhilarated to symbolize and use measurement elements in their original form, actually counting the units. The actions of representing procedures and units are usually ignored as learners are quickly introduced to using procedures that give answers (Moschkovich, 2015). Despite the move by the tutors, learners usually have no idea of the concepts. As a result, students usually get efficiency rather than actually learning and understanding the concept. For instance, fourth-grade learners were asked to find the area of a rectangular mat measuring 7 feet long and 8 feet wide. Most of the students got the answer wrong while only a small percentage got the correct answer. As such, the larger population of learners was not able to visualize and use counting to solve the problem instead they employed the use of a formula. Therefore, activities that require learners to measure with single units create their own measurement tools like formulae, and estimating measurements can significantly aid learners. Such can help them develop an insight that the figures achieved from a measurement activity are actually a demonstration of the object that they are quantifying (Minarni, et al., 2016). Students cannot achieve understanding and develop a critical thinking and positive attitude towards the mathematics concepts without the help of teachers and other students (Boaler, et al., 2016). Therefore, teachers must use both the oral and written communication in class to drive in mathematical concepts and bring the students to a common ground.

Oral Communication

Findings in relation to oral communication have concluded that if learners completed the assignment problems that were to be offered in class prior to the presentations, a more highly performing and richer class consultations would be created (Ponte & Chapman, 2016). As such, engaging consultations helped in deepening learners understanding as the learners discovered their weaknesses as well as that of their colleagues (Battery & Leyva, 2016). The ability of learners to communicate mathematics orally, listen, understand, and respond to the mathematical questions orally has indicated an improved understanding of mathematical concepts (Pape, et al., 2015). In addition, it is important for students to work on the mathematical problems in groups before they present their workings. Challenging learners to communicate their reasoning through group assignments presentations orally will help them come up with more than one way of solving mathematical problems (Moschkovich, 2018). Such will enable them to identify their weaknesses, thus consulting the teachers during presentations.

Written Communication

Oral communication also plays a substantial role in the nurturing of students’ understanding of mathematics. However, if written and oral communications are combined, they can effectively benefit the learners as they are intertwined and significant to one another (Kim & Ke, 2016). Written communication involves students justifying their solutions in writing before presentation. Such move developed a clear understanding and grasping of the mathematical concepts.

Conclusion

Representation is a procedure, a way to structure mathematics, a vital aspect of both learning and teaching, and a method in which learners can show their views and thoughts about mathematics. As such, tutors can employ representation and communication to explain mathematical concepts to the learners, aid learners in translating mathematical concepts, and access the learners’ mathematical thoughts.

References List

Yang, E.F., Chang, B., Cheng, H.N. and Chan, T.W., 2016. Improving pupils’ mathematical communication abilities through computer-supported reciprocal peer tutoring. Journal of Educational Technology & Society19(3), p.157.

Santos, L. and Semana, S., 2015. Developing mathematics written communication through expository writing supported by assessment strategies. Educational studies in mathematics88(1), pp.65-87.

Widakdo, W.A., 2017, September. Mathematical Representation Ability by Using Project-Based Learning on the Topic of Statistics. In Journal of Physics: Conference Series (Vol. 895, No. 1, p. 012055). IOP Publishing.

Moschkovich, J., 2018. Talking to learn mathematics with understanding: Supporting academic literacy in mathematics for English learners. In Language, Literacy, and Learning in the STEM Disciplines (pp. 13-34). Routledge.

Polly, D., 2015. Examining how professional development influences elementary school teachers’ enacted instructional practices and students’ evidence of mathematical understanding. Journal of Research in Childhood Education29(4), pp.565-582.

Kim, H. and Ke, F., 2016. OpenSim-supported virtual learning environment: Transformative content representation, facilitation, and learning activities. Journal of Educational Computing Research54(2), pp.147-172.

Carreira, S., 2015. Mathematical problem solving beyond school: Digital tools and students’ mathematical representations. Selected regular lectures from the 12th international congress on mathematical education (pp. 93-113). Springer, Cham.

Yeo, S., 2018. Investigating spatial and temporal reasoning of elementary students through gamified mathematics software. In Proceedings of the 40th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1271-1274).

Moschkovich, J.N., 2015. Academic literacy in mathematics for English learners. The Journal of Mathematical Behavior40, pp.43-62.

Minarni, A., Napitupulu, E. and Husein, R., 2016. Mathematical understanding and representation ability of public junior high school in North Sumatra. Journal on Mathematics Education7(1), pp.43-56.

Boaler, J., Chen, L., Williams, C. and Cordero, M., 2016. Seeing as understanding: The importance of visual mathematics for our brain and learning. Journal of Applied & Computational Mathematics5(5), pp.1-6.

da Ponte, J.P., and Chapman, O., 2015. 10 Prospective Mathematics Teachers’ Learning and Knowledge for Teaching. Handbook of international research in mathematics education, p.275.

Battey, D. and Leyva, L.A., 2016. A Framework for Understanding Whiteness in Mathematics Education. Journal of Urban Mathematics Education9(2), pp.49-80.

Pape, S.J., Prosser, S.K., Griffin, C.C., Dana, N.F., Algina, J. and Bae, J., 2015. Prime online: Developing grades 3-5 teachers’ content knowledge for teaching mathematics in an online professional development program. Contemporary issues in technology and teacher education15(1), pp.14-43.

Schoenfeld, A.H., and Sloane, A.H., 2016. Mathematical thinking and problem-solving. Routledge.

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