FAQ
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HOT is important in the learning process. In mathematics, for example, HOT will enable students to analyze mathematics problem so that they can understand problem situation. When a student is able to analyze problem he or she will be able to have a clearer idea on how to use suitable strategies to solve problems. Examples of questions that need HOT are:
A bakery shop baked several cakes. 1/3 of it had been sold and the remainder was packed in equal amount in 4 boxes containing 70 cakes each. Calculate the total number of cakes produced by the shop (Primary 6).
Find the area bounded by the curve y = x3 between x = -1 and +1. Solve the problem in at least 3 ways (Form 5).
Solve the equation 4×3 – x4 = 30. Say something about the problem and solution for x .Give reason for your answer (pre university – university level).
Solving societal problems such as drug addiction, drug pushers, HIV through drugs and free sex acts, lack of integrity, poor decision making and problem solving.
Those problems above need students and society as a whole, leaders and rakyat, to know critical thinking and creative thinking in mathematics and real life. Obviously learning content knowledge through the use of rote-learning is not good enough to enable students to solve such problems. HOT skills will prepare students to solve problems beyond content knowledge. It involves the application of cognitive and metacognitive skills. Learning mathematics without learning the HOT skills such as critical thinking, creative thinking and mathematical thinking is considered not mathematical learning at all. It is simply the learning of mathematical procedures which is meaningless to the learners.
Very good question. But before that let me explain what is mathematics in simple words (of course, we may view mathematics from the philosophical perspectives, its epistemology, existence and how do we bring mathematics to our life).
Basically mathematics is about, communication, reasoning and thinking, understanding technology and interpreting our world that we live in. So there is no reason why parents should send their children to learn mental arithmetic or the sort of mathematical learning that deals only with numbers. At the end of the day students have to face the reality of mathematics, that is, no examination questions will assess students on mental arithmetic. In fact mathematics examinations demand students to have the skills of analyzing problems, evaluate the outcome of the solution of the problem and reflecting on what has been learned through the solution of the problems. In otherwords, it is not about how fast a student could perform certain computation. Instead it is about what they are supposed to calculate. Hence they need to understand problem situations before selecting the right mathematical procedures or calculation. With the use of technology such as calculators and graphic calculators, the problem of not been able to compute or calculate through the use of fingers or other form of mental arithmetic does not arise.
There are several ways of learning mathematics – through rote-learning and problem solving. But mathematics, according to N.C.T.M. USA, is problem solving. So the most effective way to learn mathematics is through problem solving where learners learn several skills such as been listed below:
HOT in mathematics such as critical thinking, creative thinking and mathematical thinking. What exactly are those types of thinking and how to apply it in the process of learning mathematics and real life. Students will be made aware of the various HOT skills and could reflect and evaluate those thinking skills that they will apply in solving mathematics problem.
Learn how to use strategies of solving problems based on Polya’s model and relate those strategies to HOT skills. It must be cautioned that using Polya’s model of strategies of solving problems is not just about step by step procedures that should be followed blindly. It is a heuristic approach that requires understanding of HOT skills such as critical thinking, creative thinking mathematical thinking and how those subskills in critical thinking and mathematical thinking are related to another in certain ways.
Communication in mathematics which encompasses ability to represent problems in the form of diagrams, equations, tables and mind-map. Students would be guided on how to form effective discussion groups where everyone in the group could think skillfully and thus would be able to participate in discussion more effectively and actively. The normal group discussion without HOT approach in mathematical learning consists of participants without HOT skills. Thus they fail to conduct interdependent thinking discussion within the group and as a result the discussion would not be productive.
Management of mathematical knowledge such as how facts, concepts and the various skills and subskills of HOT could be organized through the use of two types of mind-map. One that help students to generalize what has been learned in every topic and another mind-map that could be used as journals for reflecting the contents and thinking skills on a particular topic. Our research shows that students creativity would be enhanced if they are given the freedom to create their own mind-maps instead of imposing strict procedures on how mind-maps should be created. Tony Buzan who first created the idea of mind-map never insisted that certain mind-maps could only be used for certain subject matter contents. It is up to the imagination of students to develop their own mind-maps based on their basic understanding of the basic rules in creating mind-map. In this way students will be more creative and critical not only creating their mind-maps but also in connecting one concept to another and thus develop critical and creative thinking through the use of mind-maps.
Mathematical modelling would be introduced to the learners depending on their level of academic and normally applicable to pre university students and engineering students.
Learn How To Learn. Approaching the learning of mathematics through Problem Solving (which is a form of HOT) not only helps students to solve problems systematically but also will help them to Learn How To Learn Mathematics. This is an example of 21st century learning where emphasis in learning is about critical thinking, problem solving, communication (not competition), collaboration, networking and learn how to learn.
Important Note. From the explanation above, it is obvious that problem solving is different than solving problem. Problem solving involves those processes and skills mentioned above whereas solving problem is merely about mathematical procedures and it has nothing to do with applying HOT in the solution of mathematics problems.
Based on my 41 years experience in mathematics education, students without problem solving knowledge will have the following problems:
Will not be able to think skillfully such as not being able to solve HOT questions successfully. Why? One of the most important skills in the learning of mathematics is to be able to analyze problems. Analyzing problem is not like what the motivators use to tell, that is, ‘try to understand what the problem wants’. The problem is how to understand a problem if one does not know the meaning of ‘analyze’. This skill has to be learned and can be taught. It does not come naturally even with intelligent students. As Professor Emeritus David Perkins said, ‘There is no correlation between ability to think skillfully and intelligence’. For this reason, Malaysia ranked almost at the bottom among countries who participated in TIMMS and PISA in mathematics and science even though we have many intelligent students who scored 8As in PMR examination (but not in PT3 examination) and who score 10As in SPM.
Certainly there is no learning without thinking. ‘Learning is a consequence of thinking’ (Professor Emeritus David Perkins, Harvard Graduate School Of Education). In fact the main goals of education are to teach for understanding, to help students to think critically and creatively, to solve non-routine problems and to develop a culture of thinking not only in the classrooms but also in real life (David Perk