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 Perkins). From Islamic perspectives the main goals of education, according to Professor Syed Naquib Al-Attas, are to produce people with moral and ethical values, integrity, passion for knowledge, realizing that knowledge is to be acquired through the process of reasoning and thinking and the methodology of teaching is such that it will ultimately produce people who can think critically and creatively and who will be closer to God. In other words those students who are not exposed to Problem Solving will continue to learn like parrots so obsessed with trying to learn just to pass many types of examinations from school level to university examinations but fail to optimize their potentials as human beings who can play the role of problem solvers and thinkers.
Problem Solving is a form of pedagogy which helps teachers to develop HOT among learners of mathematics. In fact my HOT programs in mathematics for the training of teachers and academicians include another pedagogical approach, that is, Thinking–based learning (TBL).
Knowledge of Problem Solving and TBL by teachers will enable them to integrate or infuse HOT skills into their content instructions. Teachers could learn the various HOT skills themselves and make them aware of those skills, how to apply it and help students to make skillful decision on what strategies and mathematical procedures to apply in solving mathematics problems.
Through knowledge of problem solving and TBL teachers will be able to make thinking visible both to themselves and their students. It means that teachers are aware of the HOT skills that they are trying to integrate or infuse in every topic that they are going to teach and could prepare lessons in such a way that they can assess their students not just on their ability to produce answer based on their students content knowledge but also based on the kind of thinking skills that they apply in solving a particular problem.
Teachers will also better understand how cognitive skills and metacognitive skills are interrelated and how they could approach their teaching of mathematics so that its not about passing an examination but also to help their students to view mathematics from intellectual and philosopical perspectives such as the origin of mathematics, what is mathematics and how mathematics bring changes to teachnology and society and ways of thinking and how we lead our society through mathematics.
In fact what is lacking in the implementation of new education based on the Malaysian Blueprint is about knowledge of pedagogy among teachers. This is an apportunity for teachers to get exposed to HOT in padagogy as my experience and reserach data about pedagogy was derived from my involvement in conducting more than 800 programs of Problem Solving to students of all levels from schools to university (especially engineering students) and my experienced in conducting pedagogy related to HOT to teachers and academicians for the past 26 years.
Yes of course. It can be applied to all other subjects such as science, engineering, languages, religious studies, history and art. For example in science, teachers can develop various types of analyisis such compare and contrast, parts-whole relationship and many more. In religious studies teachers could develop critical and creative thinking through verses of Al-Quran using TBL such as asking students to elaborate on important words used in prayer such as Fatihah. Teachers could ask students to compare and contrast on certain significant words in the verses. Basically the idea is for the students to observe or gather meanings of the words and verses, making analysis and make generalization.
I would view the idea of HOT as not just about guiding students to adapt themselves to solve HOT kind of problems in examinations but also to view HOT as tools to develop a culture of thinking in schools and society. For this reason, our Center For Teaching Thinking And Innovation will embark on several projects to achieve the goal of developing a culture of thinking among all levels of society (students, teachers, parents and the laymen). Several schools were selected to participate in the project. We hope to have more schools to participate in the project and hopefully will recieve strong support not only from parents but also from GLCs and individuals who would like to see paradigm shift in our education system, that is, from traditional way of looking at education – to pass the school tests into a broader view about education, that is, to train our students and society to go through the tests of life with wisdom.
Basically the project involves parental involvement in the process of learning among children at home, the exposure of students to a series of HOT programs (initially starting with mathematics before going into more subjects) and the training of teachers in pedagogy related to HOT. Schools interested in this project are welcome to contact us at our contact number: 0132418262 or through our emails : zahari.hot@cent-ti.org.my or admin@cent-ti.org.my.
The unique thing about our Center (CenT-TI) is that we have a srong team of researchers and academicians both locally and internationally who are involved in designing, planning and reseaching in order that the project works successfully. In fact this is the first project of its kind in Malaysia where such education projects are handled by various fields of experts in HOT such as pedagogy, assessment, curicullum, neuroscience, pholosophy and IT. Our team also have various subject matter experts in engineering mathematics, engineering, science, social studies, religious studies, languages and other disciplines. As an expert myself in the field of pedagogy related to HOT, I am confident this project will be implemented successfully.