The concept of mathematical thinking is flexibly different from the mainstream learning techniques used by children such as rote learning or memorizing the formulae and concepts without knowing their practical implications. As it is said, “problem solving is the heart of mathematics”, mathematical thinking is an out of the box thinking process which encompasses all such ways through which we can involve mathematics in order to solve real-world problems. Not only this, it helps in the nurturance of logical and creative thinking, problem solving which can impart significant positive effects for the practical sense of worldly demands.
Characteristics of Mathematical Thinking
In 1985, Schoenfeld worked on the concept of mathematical problem solving through four factors, which comes under the heading of mathematical thinking; the resources of the mathematical knowledge, the heuristics which could be used for problem solving, the control to be exerted on problem solving process to ensure productivity and the beliefs on mathematics which affects the attempts to solve problems. Furthermore, the reasoning abilities, personal attributes such as confidence and determination and communication skills for solutions.
There are primarily four fundamental processes through which we can use mathematical thinking; the one is specialization in which we look into specific cases and examples, generalization in which we look into themes and relationships, conjecturing through which results and relationships are predicted and convincing in which the answers to ‘why’ and ‘how’ behind the concepts are communicated.
In some courses, more processes such as characterizing in which mathematical properties of the objects are identified, classifying in which groups are organized on the basis of characterization, critiquing in which comparison and evaluation is done to make cost-benefit analysis of solutions and lastly improving through which mathematical ideas are refined to develop more effective approach.
Critical Mathematical Thinking
The critical thinking ability related to mathematical concepts enable one to learn and comprehend the concepts of mathematics faster rather than simply memorizing and solving the book exercises, limited to the course requirements. The learning of mathematical thinking and reasoning skills help in proposing the new theories and generating new creative ideas to get the effective solutions to the real-world problems. Not only this, it will make the subject much easier to handle through mental math practices.
Under the umbrella of mathematical thinking, three types of mathematical reasoning are clustered such as logistic or scientific reasoning, computational reasoning and spatial or geometric reasoning. Spatial/Geometric Reasoning which is also known as spatial visualization is the ability to image or visualize the objects in the mental eye which helps in mental transformation of positions as well in examination of their specific properties. A huge contribution of mathematical research finds its significance in relating the spatial reasoning to the higher order problem solving, geometric capability as well as learner’s overall academic achievement.
Most of the Pentathlon Games focus on this domain and its integration with other two forms of reasoning. Computational Reasoning involves the time-on-task practice of arithmetic operations due to which it is widely incorporated in the game structures and is instrumental in improving learner’s performance on standardized tests. However, it is insufficient to rely solely on the arithmetic skills; therefore, logical reasoning helps in decision making and goal achievement. Logical or scientific reasoning is considered as one of the most important mathematical skills which encompasses the process of observation, organization, hypothesizing, experimentation as well as inductive or deductive reasoning to tap the logical processing of brain.
The example of logical reasoning is the strategic games which provide opportunity to boost the mechanism such as “Mathematics Pentathlon Games” are all strategic in nature in which the player is involved in investigation and selection of the required options to achieve the goal. Not only this, the scientific processing also occurs simultaneously in which learning of better observation of playing variables, hypothetical reasoning when a game is played over the course of time by analyzing the outcomes of if-then logic and concluding by linking inductive with deductive thought processing.
Development of Mathematical Thinking
In order to develop mathematical thinking, a few steps such as creating compartments; in which a mathematical sum is broken down into constituent parts and focus on each of it to understand its purpose and function of the overall aggregate. The second step involves the determination of the skills in order to solve each of the compartment and to learn the skills which are required to solve the query. Then, strategy is formulated to create and improve the footprint which requires a lot of brainstorming.
The next step is to identify the underlying patterns of the solution through which familiar problems could be foresee. The last and most essential step is to recall the examples behind the sums to identify the instances and devise the appropriate solutions.
In a nutshell, mathematical thinking focuses on conceptual learning rather than just meeting educational requirements and getting good grades. The conceptual learning, itself improves the expertise of the learner which will help in bringing the favorable outcomes. As, the problem solving is the mega goal of the subject, mathematical thinking encompasses all the important areas such as science, technology, economics and so on, which use important principles of mathematics. Through the essence of mathematical thinking, learners can make sense of ideas and create connections between different facts and concepts.
Furthermore, the ability to think and work mathematically can also be stimulated by inconsistencies, patterns and specific representations; could be improve through frequent practice, reflection, brainstorming and questioning.