Teaching Thinking through Mathematical Processes: A Review

Teaching Thinking through Mathematical Processes: A Review

Prof. Mbuthia Ngunjiri

Laikipia University, Nakuru, Kenya

DOI: https://dx.doi.org/10.47772/IJRISS.2023.701161

ABSTRACT

This paper focuses on seven mathematical processes that describe the action of doing mathematics. It is clear that doing mathematics means engaging in processes. These include: problem solving, reasoning and proving, tools and computational strategies, connecting, representation, and communication. The mathematical processes can be seen as the processes through which students think, acquire, and apply mathematical knowledge and skills in everyday use of mathematics. In mathematics instruction, students are largely unaware of the processes involved in mathematics which requires surface and deep thinking and addressing this issue is important. Therefore, mathematics teachers need to understand how the mathematical processes can be taught to improve student thinking and understanding of the subject. It is also recommended that a study be done on how self-efficacy beliefs of students in mathematics can be influenced by teaching of the mathematical processes.

Keywords: Problem Solving, Reasoning, Proving, Reflecting, Tools, Computational Strategies, Connecting, Representing, Communicating.

INTRODUCTION

Critical thinking plays a significant role in mathematics. When faced with problems to solve, students routinely make reasoned judgments about what, and how to think. While thinking about mathematical concepts, procedures, strategies, tools, representations, and models, decisions are made through the use of criteria and appropriate evidence. To think like a mathematics student is to think critically through the mathematical processes (NCTM, 2005). By promoting, teaching and assessing critical thinking through   the processes, teachers, not only help students to think like a mathematician, they also ensure students think to learn about mathematics. Therefore, it is through mathematical processes that teachers promote thinking. Students must learn how to problem solve, communicate, reason and prove, reflect, represent, connect, and select tools and computational strategies to be able to share ideas, observations, and problem-solving processes in or outside mathematics classrooms (Reys et al.,2001).

REVIEW OF LITERATURE

The mathematical processes that appear to support effective learning   in mathematics are as follows: problem solving, reasoning and proving, reflecting, selecting tools and computational strategies, connecting, representing, and communicating. In the context of teaching: (i) Mathematical processes develop through different grades and support lifelong learning, (ii) they are taught in ways that address the different needs of different types of learners, and (iii) a variety of groupings and instructional strategies help students improve their mathematical processes (Fernandez – Cezar et al., 2020). The following literature addresses the seven mathematical processes.

The first mathematical process is problem solving which is central to the learning of mathematics (NCTM, 2000, 2005, 2020; Reys et al., 2001; The Ontario Curriculum, 2020). Garfola and Lester (1985) suggested that students are largely unaware of the processes involved in problem solving, and addressing this issue within problem solving instruction may be important. According to Kantowski (1977), to become a good problem solver in mathematics, a student must develop a sound base of domain specific knowledge in mathematics.

In their view, how effective one is in organizing that domain specific knowledge also contributes to successful problem solving. This argument is supported by Schoenfeld and Herrman (1982) who posited that novices appear to attend to surface features of problems, whereas, experts categorize problems on the basis of fundamental principles involved in the problems. According to Schoenfeld (1985), providing explicit instruction and use of heuristics should enhance problem solving performance. In their view, heuristics are kinds of information available to students in making decisions during problem solving that are aids to the generation of a solution. For example, tendency to “write an equation”, to “set sub- goals”, “restate a problem”, and to “draw a figure” are heuristic in nature.

Schoenfeld (1985) and Kantowski (1997) have provided two requirements for successful problem solving. These are: (i) use of heuristics, and (ii) possession of domain specific knowledge. However, Mayer (1987) offered more specific types of knowledge needed for success in mathematical problem solving. In Mayer’s view, students’ performances differ because they possess differing amounts and kinds of knowledge.

First, a student’s needs to be able to translate each statement of a problem into some internal representation. Furthermore, this translation requires linguistic and factual knowledge. Second, a student needs to be able to integrate each of the statements of a problem into a coherent representation. The integration process requires that a student should be able to recognize problem types and also be able to distinguish between information that is relevant for a solution from that is which is not, which is called schematic knowledge. Third, a student needs to devise and monitor solution plan which is called strategic knowledge, and finally, a student should be able to apply the rules of arithmetic (i.e., computational skills) which is called procedural knowledge. In all, according to Mayer (1987), a student may be unable to generate a correct answer due to lack of linguistic and factual knowledge, in particular difficulties in comprehending sentences that express relations among variables.

This implies that students need practice to represent each sentence in a problem. Further, students who are unable to generate the correct answer in a problem may be lacking knowledge of problem types. If a problem does not fit into one of the student’s existing categories, the problem is likely to be misinterpreted. This implies that students need practice in recognizing those problems that go together and those that do not. Moreover, student may fail to solve a problem due to lack of appropriate strategies. This implies that students need instruction and practice in determining which strategies to apply and when to apply. Lastly, a student may fail to generate the correct answer due to lack of appropriate computational skills. This implies that students need practice in solving computational problems.

According to Polya (1965), there is a four-step procedure of solving problems. These are: (i) reading and understanding the problem, (ii) devising a solution plan, (iii) carrying out the plan, and (iv) looking back at the solution process. The first Polya’s step is similar to Mayer’s (1987) argument for students’ ability in linguistic and factual knowledge. The Polya’s second step is similar to Mayer’s argument for the student’s ability in knowledge of problem types to be able to come up with a solution plan. The third Polya’s step is similar to Mayer’s arguments for students’ ability in knowing solution strategies. These strategies include: (i) drawing a picture or a diagram, (ii) looking for a related problem, (iii) restating the problem, (iv) trying to think about other information appropriate to determine the unknown, and trying to look backwards (Polya, 1965).

The second mathematical process in the learning of mathematics is reasoning and proving (NCTM, 2000, 2005, 2020; The Ontario Curriculum, 2020; Rey et al. ,2001). According to the Ontario Curriculum (2005), in mathematics, students will develop and apply reasoning skills (i.e., recognition of relationships, generalization through inductive reasoning, use of counter-examples) to make guesses or mathematical conjectures, assess conjectures, justify conclusions, and plan and construct organized mathematical arguments. This process of reasoning and proving involves exploring phenomena, developing ideas, making mathematical conjectures, and justifying results.

According to Nyein and Thein (2018) there are four important points about mathematics reasoning: (1) reasoning is about making generalizations, (2) reasoning leads to a web of generalizations, (3) reasoning leads to mathematical memory built on relationships, and (4) learning through reasoning requires making mistakes and learning from them. In their view, students should be encouraged to reason from the evidence they find in their exploitations and investigations or from what they already know to be true, and to recognize the characteristics of an acceptable argument in the mathematics classroom. Therefore, the reasoning process supports a deeper understanding of mathematics by enabling students to make sense of the mathematics they are learning.

The third mathematical process is reflecting (The Ontario Curriculum, 2005, 2020). In their view, good problem solvers regularly and consciously reflect on and monitor thought processes. By doing so, they are able to recognize when the technique they are using is not fruitful, and to make conscious decision to switch to a different strategy, rethink the problem, and search for related content knowledge that may be helpful.

According to the Ontario curriculum (2005), one of the most valuable processes for students is to reflect in their learning with others. One way of doing this is to bring students together following an investigation to share out strategies and solutions. In the process, students are expected to share their thinking, defend and justify the strategies they used and solutions they reached, and talk about any challenges that faced. Exposing them to this information from a variety of students allows them to compare their own thinking and process to those of others, and evaluate and deepen their own understanding of the mathematical concept (The Ontario Curriculum, 2005).

The fourth mathematical process is selecting tools and computational strategies (Brumbaugh & Rock,2006: The Ontario Curriculum, 2005, 2020). Brumbaugh and Rock (2006) posits that students will always select a variety of concrete, visual, and electronic tools, and appropriate computational strategies to investigate mathematical ideas and to solve problems. For example, the mathematics teacher can encourage students to use technology to solve problems when the focus is on problem solving rather than on paper –and- paper skills (e.g., calculators, spreadsheets, and geometer’s sketch pad). According to the Ontario Curriculum (2005), students need to develop the ability to select the appropriate electronic tools, manipulatives, and computational strategies to perform mathematical tasks, to investigate mathematical ideas, and to solve problems.

The fifth mathematical process is connecting. Successful mathematical thinking means noticing how ideas are related. Costa and Kallick (2000) posits that it is making higher level connections that allows the students to draw forth a mathematical event and apply it to a new context in a way that connects familiar ideas with new concepts or skills. Furthermore, making good connections means seeing how mathematical concepts are connected to others and to the real world.

The Ontario Curriculum (2005), states that students need to see the connections and the relationships between mathematical concepts and skills from one topic of mathematics to another. As they continue to make such connections, students begin to see that mathematics is more than a series of isolated skills and concepts and they can use their learning in one area of mathematics to understand another. Moreover, seeing connections among procedures and concepts also helps to deepen students’ mathematical understanding.

The sixth mathematical process is representation (NCTM, 2000, 2005, 2020; Reys, et al., 2001; The Ontario Curriculum, 2020). According to the Ontario Curriculum (2005), students represent mathematical ideas and relationships and model situations using concrete materials, pictures, diagrams, graphs, tables, numbers, words and symbols. For example, a teacher can introduce new concepts using concrete materials. The various forms of representation help students to make connections and develop flexibility in their thinking about mathematics.

Reys et al. (2001) gives three major goals for representation as a process in mathematics. (1) Creating and using representations to organize, record, and communicate mathematical ideas, (2) selecting, applying, and translating among representations to solve problems, and (3) using representations to model and interpret physical, social, and mathematical phenomena.

The seventh mathematical process is communication (NCTM, 2000, 2005, 2020; The Ontario Curriculum, 2020; Rey et al., 2001). A student who is poor at communicating cannot explain his/her thinking which means there is no ability to justify with examples and does not see feedback as important. Students who are successful at mathematical communication, however, seek clarification (Reys et al., 2001). In their view, communication allows interaction and enable students to question, criticize, and clarify.

According to the Ontario Curriculum (2020), a student who is successful in mathematical communication is able to: (i) explain his/her thinking clearly and concisely, (ii) seeks clarification, (iii) realizes that it is normal to make mistakes, and (iv) when others come up with new ideas, he/she asks them to explain those ideas or tries to figure out why that makes sense.

CONCLUSION

Mathematics plays a key role in shaping how individuals deals with the aspects of social, scientific and technological life, but today, as in the past, many students struggle with mathematics and become discontented with the subject as they progress in their schooling. Therefore, it is important for mathematics teachers to understand how mathematical processes can be taught effectively to improve students’ knowledge and understanding of concepts and skills. The seven mathematical processes highlighted in this paper (i.e., problem solving, reasoning and proving, reflecting, use of tools and computational strategies, connecting, representing, and communicating) can support the acquisition and the use of mathematical knowledge and skills, and should form the base of practice in mathematics classrooms.

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