Learning: "Learning is a relatively permanent change in behavior with behavior including both observable activity and internal process such as thinking, attitudes and emotion"(Burns, 1995) According to Bruner, learning is a social process, whereby students construct new concepts based on current knowledge(Thanasoulas).
There are different theories about learning that is how people learn. It is very useful to consider their application and how student learn and how teachers teach in educational programmes(Dunn, 2002). It is also very important to know that every does not learn by the same way. This understanding helps teachers to devise effective strategies for teaching(Sirhan, 2007). Burns' definition shows that learning might not manifest itself in observable behavior until some time after the educational programme has taken place.
Sensory Stimulation Theory:
According to sensory stimulation theory effective learning occurs when the senses are stimulated. According to Laird, research has proved that the majority of knowledge held by adults (75%) is learned through seeing. Hearing is the next most effective (about 13%) and the other senses - touch, smell and taste account for 12% of what we know (Laird, 1985).
Reinforcement theory was presented by B.F. Skinner. Skinner viewed that behavior is a function of consequences. The learner will repeat the desired behavior if positive reinforcement (pleasant consequent consequence) follows the behavior. (Laird 1985, Burns 1995). Competency based Training is based on this theory and although it is useful in learning repetitive tasks like multiplication tables and those work skills that require a great deal of practice, higher order of learning is not involved (Burns, 1995).
This approach emphasizes on the importance of experience, meaning, problem-solving and the development of insights (Burns, 1995, p. 112) Burns notes that this theory has developed the concept that individuals have different needs and concerns at different times, and that they have subjective interpretations in different contexts.
Holistic learning theory
The basic premise of this theory rests on that for learning to be effective require activation of 'individual personality which consists of many elements... specifically ... the intellect, emotions, the body impulse (or desire), intuition and imagination' (Laird, 1985, p. 121).
Facilitation theory (the humanist approach)
Laird (1985) discussed this theory and ascertained that learning will occur by the educator acting as a facilitator, that is by establishing an atmosphere in which learners feel comfortable to consider new ideas and are not threatened by external factors. He further characterized this theory by arguing that i) human beings have a natural eagerness to learn ii) there is some resistance to, and unpleasant consequences of, giving up what is currently held to be true and that the most significant learning involves changing one's concept of oneself.
Kolb (in McGill & Beaty, 1995) proposed an experiential learning model which is a four-stage learning process. In fact, the process can begin at any of the stages and is continuous in that there is no limit to the number of cycles you can make in a learning situation. This theory posits that without reflection one would simply continue to repeat our mistakes.
Kolb's research found that people learn in four ways with the likelihood of developing one mode of learning more than another. As shown in the 'experiential learning cycle' model above, learning is i) through concrete experience ii)through observation and reflection iii)through abstract conceptualization iv)through active experimentation. It is worth discussing Kolb's work further
Constructivism is a theory founded on observation and scientific study about how people learn. It is a method of teaching that encourages the thought process of the student and assumes that prior knowledge, attitude, motivation, and learning style affect the learning process (Spencer, 1999).It is the idea that the learners 'construct their own knowledge'. Constructing of knowledge is a personal process where the student is involved in their own learning by seeking to find meaning to their present and past experiences (Boghassiam, 2006). Constructivism may become an alternative methodology to the traditional teaching method known.
The traditional method for teaching science has its roots in what is called "behaviorism", which is the belief that an idea can be transferred intact from the mind of the instructor to the mind of the student, or that telling is teaching (Spencer, 1999). Behaviorism views the student as an unreflective responder (Boghassiam, 2006).When implementing the behaviorism method; students are expected to respond to questions in class or on a test. If their answers are correct, they receive a good grade as a way to encourage their "good behavior". If the student does not get the answer correct, they will receive a poor grade with the hope to send the message to the student that their behavior is bad and that they need to change it until they get the answer correct. There is no subjective element to learning-either in determining what to study or in how information is interpreted, used, or understood (Boghassiam, 2006). Behaviorism dominated the educational landscape 25 years ago, while the foremost learning theory today is constructivism (Boghassiam, 2006). Table 1 lists some comparative features of behaviorism and constructivism.
Table 1. Comparison of Behaviorism and Constructivism
Knowledge is constructed.
Group work promotes the negotiation of and develops a mutually shared meaning of knowledge. Individual learner is important.
The ability to answer with only one answer does not demonstrate student understanding.
Truths are independent of the context in which they are observed.
Learner observes the order inherent in the world. Aim is to transmit knowledge experts have acquired.
Exam questions have one correct answer.
(Spencer, 1999). New Directions in Teaching Chemistry: A philosophical and Pedagogical Basis. Journal of Chemical Education, 76(4), 566.
Professor Jean Piaget life's work was based on the study of the nature of knowledge and the child's psychology. Jean Piaget's theory on the mental development of children included the concept of cognitive structures, which the child himself constructs in interaction with the environment in a continuous way from birth to adolescence (Schwebel & Raph, 1973). Advocates of constructivism agree with Piaget claiming that constructivism is the individual's processing of stimuli from the environment and the resulting cognitive structures that produces adaptive behavior (Kathleen Bahr, 2008)(Brandon, 2010). These structures take the form of schemata, explanations, and experiences that a student uses to understand the world they experience around them (Todd, 2004, p. 12).
The philosophy of constructivist education operates by four major assumptions.
The first assumption is that previous constructs are the foundation of the learning process in each student. In other words, new information is transformed and interpreted based upon what a student previously learned.
Second, assimilation and accommodation processes lead to new constructions. New information may not be able to assimilate with previous knowledge, so the student accommodates the new information resulting in new zones of cognitive development or higher learning.
Third, learning is not mechanical, but a process of invention. This is the idea that students hypothesize, predict, manipulate, and construct knowledge, rather than simply memorize facts.
Finally, constructivists assume that meaningful knowledge occurs through reflection and the linking of new information to the existing framework of knowledge.(Brandon, 2010)
Piaget discusses constructivism by explaining it as "intelligence", what it means, where it comes from, and how it is developed. From Jean Piaget's point of view, intelligence is the ability to adapt to new situations. In order to adapt to a new situation, a person must comprehend the situation and come up with a solution through their understanding. In order to understand a new situation, one must assimilate it to knowledge that has already been built and brought to the situation. Piaget calls this knowledge that is brought to the new situation as "structures". Knowledge is in some way organized or structured for all ages. Structures become more elaborate as a person grows older (Schwebel & Raph, 1973, pp. 217-218). An example of this would be "adult humor". A 4 year old and their parent can go to the movies and there will be concepts in the movie that the adult will comprehend, that the child will not. Another example of richer structures of knowledge amongst older people can be tested by asking what the first thing that comes to mind when a 4, 10, and 20 year old hear the word "mommy". A 4 year old may say "she loves me" or "I love her". A 10 year old may say "she stays home and takes care of the house". A 20 year old may respond with "motherhood". Piaget would explain that because knowledge is an organized structure, no meaningful concept can exist in isolation (Schwebel & Raph, 1973). To the contrary, a behaviorist would expect for each person to have the same answer when thinking of the word "mommy".
The development of intelligence is a continuous process of constructions from birth to adolescence in a sequence that is the same for all children in all cultures (Schwebel & Raph, 1973, p. 222). A behaviorist would state that knowledge is absorbed from the outside, while Piaget would see knowledge as a building of structures from inside. Piaget supports his difference to a behaviorist by explaining that children would not have similar sequences in development if information was simply absorbed. The only major difference in development amongst children has been the rate of development and not the sequence (Schwebel & Raph, 1973, p. 222). The four factors of development are: biological factors, experiences with physical objects, social factors of inter-individual coordination and cultural and educational transmission, and factors of equilibration (Schwebel & Raph, 1973, p. 223). It is the role of the teacher to teach in such a way that all the factors are at work (Schwebel & Raph, 1973, p. 224).
Constructivists have shown that the model that is the closest to how students learn is called the learning cycle. A learning cycle is illustrated in Figure.
â€¢What did you do?
â€¢What did you find?
â€¢Is there any pattern to the data?
â€¢Predict, form a hypothesis
â€¢What does it mean?
â€¢Higher level of thinking
Figure . The Learning Cycle
(Spencer, 1999).New Directions in Teaching Chemistry: A philosophical and Pedagogical Basis. Journal of Chemical Education, 76(4), 566.
The constructivist-learning-cycle approach is also called inquiry-based and has been shown to facilitate retention of information and the transfer of thinking skills and content (Spencer, 1999). It is through the constructivist-learning-cycle that students proceed from the concrete to the abstract.(Spencer, 1999).
The constructive theory encourages learners to be active creators of their knowledge. The role of the student is to pick out and transform information, build ideas, and make decisions, while relying on cognitive structures. Rather than using the teacher's knowledge and textbooks for solving problems, the students comes up with solutions and develops knowledge in the learning process. The student must feel connected to their teacher for constructivist learning to occur. If the student feels disconnected from their teacher, they will go back to the old ways of learning they feel most comfortable with such as rote memorization, algorithmic techniques, and cramming (Todd, 2004, pp. 12-18).The job of the teacher is to design a learning format that is aligned with the student's current state of knowledge. The teacher becomes a facilitator of knowledge asking, "What do students need to learn?" rather than, "What do I want to teach?" Recognizing students' preexisting conceptions, teachers guide activities to build students' knowledge. As each new activity is taught, the student uses previous knowledge to develop more complex ideas and integrate new information (Brandon & All, 2010). The guided inquiry curriculum begins with the assumption that knowledge is not directly transferred from the instructor to the student; that is the constructivist approach is implemented and follows the learning cycle (Spencer, 1999).A closer look at guided inquiry and its practical implication will be discussed in further detail in the next section.
Application of Constructivism theory
Guided-inquiry learning is a process by which students "discover" basic concepts through active investigation (Jin & Bierma, 2011).Guided-inquiry, commonly known as POGIL (for Process Oriented Guided-Inquiry Learning), has been shown to significantly increase student comprehension of difficult-to-understand concepts (Jin & Bierma, 2011). Guided-inquiry is an "active learning" technique that focuses on concept understanding. Recent developments in classroom research results suggest that students generally experience improved learning when they are actively engaged in the classroom and when they construct their own knowledge following a learning cycle paradigm (Farrel, S.Moog, & Spence, 1999)."Active learning" techniques are used by educators who recognize that students learn better by "doing" rather than "listening" (Jin & Bierma, 2011)
Guided-inquiry is commonly used in chemistry labs. The student is guided through questions in their lab assignments until they "discover" the concepts where the mechanisms at work are too small to observe with the naked eye (Jin & Bierma, 2011). The student looks for trends and patterns in the data they collect. At that point they would form a hypothesis and then test their hypothesis. The goal is to make connections between observations and principles. This approach is based on the learning cycle: data collection, concept invention, and application (Farrel, S.Moog, & Spence, 1999). Throughout the entire lab the student would be asked guiding questions to require the student to think about the various steps rather than simply following instructions (Farrel, S.Moog, & Spence, 1999)
With the implication of guided-inquiry methods, students work in teams with specifically designed guided-inquiry materials. The materials will either provide information or guide students through experiments, so they can collect their own information. These materials can be in the form of worksheets that serve as a tutorial to the students. The worksheets can contain questions that help to construct understanding by having the student build on previous knowledge. These guiding questions would be asked for students to "discover" by their own conclusions (Jin & Bierma, 2011).A large part of guided-inquiry is the questions asked of the students. These questions can be referred to as probing questions.
Teachers who have incorporated guided-inquiry techniques into their classroom have found that their students' grades have improved because of it (Jin & Bierma, 2011);(Farrel, S.Moog, & Spence, 1999).It is important when implementing guided-inquiry for the students to be given time when working on an answer to a probing question, the teacher should observe, but once they interject their opinion, the benefit of the inquiry-based learning is compromised. The more the students discuss amongst themselves, the more likely they will come to the correct answer and learn the concept in order to retain it. Probing questions need to be constructed in a way that students can come to their predictions before the class time is complete. The downside to guided-inquiry is that it takes more time to create the learning materials and not as much information can be covered in a class period (Jin & Bierma, 2011). Because this type of curriculum takes a lot of time for the teacher to prepare, it is very important that when a teacher finds a lesson that works well, and the students are learning, they must share their lesson with other teachers. Teachers should not be spending time creating guided-inquiry curriculum that has already been created, they should be creating another lesson that has not been formatted into guided-inquiry curriculum (Jin & Bierma, 2011)
Importance of Mathematics
Ibn Khaldun, a great historian, sociologist, philosopher who born in 1332AD, in Tunis,is the Strongest personalities of Arab-Muslim culture in the period of its decline. He stressed the role of mathematics in his book 'al-Muqaddima' as
"Education should be started with mathematics. For it forms well designed brains that are able to reason right. It is even admitted that those who have studied mathematics during their childhood should be trusted, for they have acquired solid bases for arguing which become to them a sort of second nature".
According to the Britannica Concise encyclopedia, "mathematics is the science of structure, order and relation that has evolved from counting, measuring and describing the shapes of objects. It deals with logical reasoning and quantitative calculation." Since it involves quantitative calculations this makes it necessary to our everyday life as a day shall not pass without one doing some form of calculation - really or abstractly - in his or her brain. Imagine how it would be life if there was no mathematics; how were we going to count days, months and years, etc.
Mathematics plays an important role in the complex world of science and technology. The scientific and technological development since 1940 emphasized for the first time that we live in scientific age and the disciplines which support this scientific and technologist civilization such as physics, chemistry, engineering, the management sciences, economic, the biological and medical sciences and the behavioral sciences all require Mathematics for their understanding and their further development. The place of mathematics in education must be determined by analysis of the society, the culture and the civilization, which the education is designed to serve. The dominant features of our civilization today are a series of major even revolutionary scientific developments, which have occurred in the twentieth century. Mathematics by virtue of its extensive, practical applications and the aesthetic appeal of its methods and results has long held a prominent place in education.
Mathematics occupies a central position in curriculum. It is a compulsory subject from class I - X. It is being taught as an integrated subject. It covers various areas like arithmetic, algebra, geometry, trigonometry and statistics etc. Mathematics has gained the status of a special and universal language, which enables man to express his ideas about shape, quantity and relationship. Therefore knowledge of Mathematics is essential for successful life. The traditional theories of mathematics have been changed with the advent of the computer. In the past, application of mathematics was limited in extent. It is to be conceived as an integrated study of the learner's environment, which contains the element of mathematics. The development of mathematics concept in our children will be best achieved mathematics concepts from concrete situations needs perceptive teaching. Gall and Hicks (1964) described that mathematics has played a key role in science, technology, industry, business, and agriculture. Its study has been associated with habits of effective thinking, intellectual independence, aesthetic appreciation and creative expression. (Arif, 2011)
Mathematics and Science Education in Pakistan
Pakistan's Education system can be broadly divided into 'Basic Education' (primary, elementary and secondary levels) and Higher Education (post secondary and graduate levels). Both are governed by separate ministries with distinct management and financial systems. Mainstream or government schools offer primary education from class I - V (5 to 9) and then middle or elementary schooling, class VI-VIII (10-13) and finally secondary schooling, class IX - X (14 - 15). In classes IX and X students take the secondary school matriculation examination which is conducted by the Boards of Secondary Education. The medium of instruction in primary schools is Urdu, but English is introduced as a second language in class VI. The medium of instruction in the public schools is mainly Urdu, however, most of the teachers use their regional language as the language of instruction. Schools in the Provinces also teach their regional languages as a subject. However, Sindh is an exception where government schools can opt for Sindhi as a medium of instruction. As noted in the white paper and subsequently elaborated in media briefs in line with the curriculum reform to be introduced in 2007, the Ministry of Education has taken a policy decision to change the language of instruction to English in Science and Mathematics classrooms from grade VI and above (HassanAly, 2006).
Science is taught as a compulsory subject at primary and elementary level (I-VIII) in an integration of biology, chemistry, physics and earth science. In primary classes (I-V), students study science 12 % of the total school time. At middle school level (VI-VIII) students spend 13-15% of their school time in studying science. At the secondary school level (IX-X) science is optional and those who opt for it study 12-14 % on each science subject - physics, chemistry and biology. The content of the teaching in the middle classes revolves around three broad areas: living things, matter and energy, and the earth and the universe.(Halai, Razvi, & Rodrigues, 2007)
Mathematics is also taught as a compulsory subject for classes I - VIII. The curriculum content is organized mainly into five major strands, number and number operations, measurement, geometry, data handling and algebra. At the secondary level (classes IX & X) students can opt to take science group or general group. The former includes among other subjects physics, chemistry, biology and advanced mathematics (with a greater focus on algebra, functions, and trigonometry). The latter includes humanities and a course in general mathematics (with greater emphasis on arithmetic and less emphasis on algebra, functions and trigonometry). However, in 1995-96 the policy has changed according to which both groups take the same course in mathematics at the secondary level. A consequence of this policy is that school examination at class VIII serves the role as gate keepers in terms of who gets the science group and who doesn't. Those who perform well in mathematics and science at class VIII level get the opportunity of taking the science group. This bifurcation at an early stage in the academic life has implications for subsequent employment opportunities and poverty reduction.
In a historical account of science education in Pakistan, Iqbal & Mahmood (2000) maintain that until 1950 science was not taught at primary and middle level. It was the Science Education Commission established in 1959 which recommended that science education be made compulsory for classes VI-VIII. In the early sixties science education was made compulsory for I - VIII but its quality was very poor. Iqbal & Mahmood (2000) have shown that various education policies (1972, 1979, 1998-2010) have increasingly shifted the emphasis from a general education to science and technology education with a concomitant emphasis on school curriculum reforms both at primary and secondary level, to trigger students' curiosity in scientific inquiry and understanding of scientific concepts and processes(Halai, Razvi, & Rodrigues, 2007). For example, a recommendation of 1979 policy was to set Science Education Project (SEP) a six years project initiated in 1984 for promotion of science education in Pakistan at elementary and secondary and higher secondary level (6- 12) in collaboration of Asian Development Bank (Hill & Tanveer, 1990).
Relationship between mathematics and chemistry.
In all nations of the world science and mathematics are given first class attention due to numerous benefits derived from them. Chemistry which is believed to be the mother of science equips students with knowledge in areas such as drugs, diseases, pollution, food, chemicals etc which when applied to the society improves man standard of living.(udousoro, 2011)
Mathematics has generally been accepted as the foundation of science and technology and it is a very important subject in the secondary school curriculum, therefore, every nation needs it for sustained scientific and technological development.(udousoro, 2011)
Also, mathematics is considered as a service tool for the study of sciences especially chemistry. But despite its importance and usefulness, it is a subject that is most feared by students at primary, secondary, even in the tertiary levels of education. Hence student with poor mathematics knowledge cannot solve calculation problems in chemistry (Kramers-pals, Lambrechts, & Wolff, 2000).
A careful study of the senior secondary chemistry curriculum in Nigeria and West African Senior Certificate syllabus reveal that a proper understanding of the mathematical concepts on Isotopy, formula, equations, solubility, chemical kinetics, quantitative and molar ratio, radioactivity, pH and laws of chemical equilibra etc in chemistry needs a good knowledge of basic mathematics and generally involves a lot of computation which make mathematics an important aspect of it(udousoro, 2011).
The potency of mathematics as an anchor that holds the desirable achievement in the physical and biological sciences has been widely acknowledged. For instance, in chemical education, substantial studies have examined the relationship between mathematics and chemistry and found that mathematics forms the basic ingredients to the understanding of chemistry and the establishment of new facts by many researcher like Abdullahi, 1982; Johnstone, 1984; Adesoji, 1985; Carter and Brickhouse, 1989; Bello, 1990; Schmidt, 1992; Kogut, 1993; Axalonu, 1995; Adeyeye, 1999; Olayiwola, 2001; Kehinde, 2005; Adesugba, 2006(Oluwatayo, 2011).
Though earlier studies in this area conducted in Nigeria reported that many students find chemistry difficult because of its abstract nature and the mathematical skills required, the bulk of the blame has been placed on the teachers for failing to develop the needed mathematical skills in the students to tackle chemical problems with ease. It is intriguing that many chemistry teachers tactically skip some perceived difficult concepts in chemistry, especially the quantitative aspects of chemical reactions due to their inability to effectively teach them, the factor which has scared many students away from chemistry(Oluwatayo, 2011).