http://google.com, pub-7771400403364887, DIRECT, f08c47fec0942fa0 The learning process

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The learning process

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LECTURE-3
LEARNING THEORIES
Learning styles are unique to learners and it is important for teachers to be conversant with different theories about learning and instructional theories to enable them make effective progress in their classroom. Therefore, if the students have to be taught mathematics effectively then the mathematics teachers have to clearly understand how children learn mathematics and what capacities for learning our students’ possess from early years in mathematics learning. Psychologists have provided us with a lot of information about the nature of our learners and how they learn mathematics.
The learning process
Learning is a change in behavior of an organism or a change in mental associations. For example, a form two teacher prior to going to class to teach may prepare an instructional objective such as:
    ‘By the end of the lesson students will be able to solve simultaneous equations by substitution method’
Let us assume that prior to teaching students will have no idea on how to solve simultaneous equation such as;   3x +2y = 4
         x -  y = 3
 If however after the lesson the lesson the students can solve simultaneous equation of the type given above, then they are now behaving differently. They can now do what they were unable to do previously. We say that there has been a change in their behavior. So once a student shows a certain behavior which he was unable to exhibit before we say that learning has occurred. Well there are many ways of learning which will be discussed as we go by.
Approaches to mathematics teaching:
Understanding of theories about how people learn and the ability to apply these theories of teaching are prerequisites for effective mathematics teaching. In the past, many educators and mathematics teachers neglected the application of theories about the nature of learning and centered their teaching methods around knowledge of the subject.
Recent finding in learning theory, better understanding of mental development and new applications of theory to classroom teaching now enable teachers to choose teaching strategies according to information about the nature of learning. The following are some of the theories which the teacher can adapt.
Behaviorist theory
Olivier (1988) noted that behaviorism assumes that students learn what they are taught, or at least some subset of what they are taught because it is assumed that knowledge can be tranfered intact from one persn to another. The student is viewed as a passive recipient of knowledge, an ‘empty vessel’ to be filled, a blank sheet on which the teacher can write (Watson&Pavlor).Behaviorist, therefore, believe that knowledge is taken directly from experience and that a students current knowledge is unnecessary to learning.
This theory sees learning as conditioning, whereby specific responses are linked with specific stimuli. According to Thorndike’s law of exercise, the more times a stimulus – induced response is elicited the longer the learning (response) will be retained. The law of effect states that appropriate stimulus-response (S-R) bonds are strengthened by success and reward (positive reinforcement)and inappropriate S-R bonds are weakened by failure (negative reinforcement). Consequently the organization of learning must proceed from simple to the complex; short sequences of small items of knowledge and exercise of these in turn through drill and practice. That is one learns by stock pilling, by accumulation of ideas (Bouvier, 1987).
Constructivist theory
According to Piaget (1970) and Skemp(1979) the student here is not seen as passively receiving knowledge from the environment; it is not possible that knowledge can be transferred ready-made and intact from one person to another. Therefore, although instruction clearly affects what children learn, it does not determine it, because the child is an active participant in the construction of his own knowledge. This construction activity involves the interaction of a child’s existing ideas and new ideas, i.e new ideas are interpreted and understood in the light of that child’s own current knowledge, build up out of his previous knowledge. Children do not only interpret knowledge, but they organize and structure this knowledge into large units of interrelated concepts. We shall call such unit of interrelated ideas in the child’s mind a schema. Such schemas are valuable interrelated tools, stored in memory, which can beretrieved and utilized. Learning then basically involves the interaction between the child’s schemas and new ideas
Defination
Schema is a unit of interrelated ideas or concepts in a child’s mind.  
Gestalt psychology;
In Gestalt psychology, the German word Gestalt is interpreted as "pattern" or "configuration". Gestalt psychologists emphasized that organisms perceive entire patterns or configurations, not merely individual components. The view is sometimes summarized using the adage, "the whole is more than the sum of its parts." Gestalt psychologists  invested in understanding how our mind goes about making sense of our chaotic surroundings.
Through observation, Wertheimer, Koffka and Kohler theorized that we tend to group elements, recognize patterns, and simplify complex imagery. Gestalt principles, proximity, similarity, figure-ground, continuity, closure, and connection, determine how humans perceive visuals in connection with different objects and environments. The Gestalt psychologists believed that the most fruitful way to view psychological phenomena is as organized, structured wholes. They argued that the psychological "whole" has priority and that the "parts" are defined by the structure of the whole, rather than vice versa. Gestalt theories of perception are based on human nature being inclined to understand objects as an entire structure rather than the sum of its parts. That is, “what takes place in each single part already depends upon what the whole is" 
Wertheimer defined a few principles that explain the ways humans perceive objects. Those principles were based on similarity, proximity, continuity. The Gestalt concept is based on perceiving reality in its simplest form. The various laws are called laws or principles, depending on the paper where they appear—but for simplicity's sake, this article uses the term laws. These laws took several forms, such as the grouping of similar, or proximate, objects together, within this global process. These laws deal with the sensory modality of vision. However, there are analogous laws for other sensory modalities like auditory and others. The visual Gestalt principles of grouping were introduced in Wertheimer (1923). Through the 1930s and '40s Wertheimer, Kohler and Koffka formulated many of the laws of grouping through the study of visual perception.
Law of proximity
The law of proximity states that when an individual perceives an assortment of objects, they perceive objects that are close to each other as forming a group. 
Law of similarity
The law of similarity states that elements within an assortment of objects are perceptually grouped together if they are similar to each other(i.e  when things appear to be similar to each other, we group them together. And we also tend to think they have the same function). This similarity can occur in the form of shape, colour, shading or other qualities ( i.e objects can be grouped together in terms of similarity in colour , shape etc). In algebra, for example, students’ misinterpretations of expanding (a + b)2as a2 + b2 or 3(a + b)2 as 3a2 + 3b2  or log (x + y) as log x + log y is viewed as emanating from the application of the distributive law. The formal distributive property of multiplication over addition is deeply deposited in their mind so that they intuitively misapply the rule in similar situations. The examples for these categories again emanated from the overgeneralization of the distributive law Luka (2013).
OR students incorrectly misapply into expressions like  to get  . This is an application of a known rule to an inappropriate situation by incorrectly perceiving the similarities of the two situations.
Law of closure
Gestalt psychologists believed that humans tend to perceive objects as complete rather than focusing on the gaps that the object might contain(i.e when we look at a complex arrangement of visual elements, we tend to look for a single, recognizable pattern). In other words, when you see an image that has missing parts, your brain will fill in the blanks and make a complete image so you can still recognize the pattern.
. For example, a circle has good Gestalt in terms of completeness. However, we will also perceive an incomplete circle as a complete circle. That tendency to complete shapes and figures is called closure. The law of closure states that individuals perceive objects such as shapes, letters, pictures, etc., as being whole when they are not complete. Specifically, when parts of a whole picture are missing, our perception fills in the visual gap. Research shows that the reason the mind completes a regular figure that is not perceived through sensation is to increase the regularity of surrounding stimuli. For example, the figure that depicts the law of closure portrays what we perceive as a circle on the left side of the image and a rectangle on the right side of the image. However, gaps are present in the shapes. If the law of closure did not exist, the image would depict an assortment of different lines with different lengths, rotations, and curvatures—but with the law of closure, we perceptually combine the lines into whole shapes. 
In mathematics students, more often than not, simplify 5x + 4 as 9x  when 5x + 4 is actually an object (i.e a final answer).The student perceives that the answer should not contain an operator symbol. The student also perceives that the “+” sign “as an invitation to do something” and the student goes ahead to do it (Chow, 2011). The students perceive open algebraic expressions as ‘incomplete’ and try to ‘finish’ them by oversimplifying as in the case above where the student simplified 5x+4 as 9x because they consider an answer such as 5x+4 as incomplete (yes cos humans tend to perceive objects as complete rather than focusing on the gaps that the object might contain i.e sth is missing, our perception fills in the visual gap).
Law of Past Experience
The law of past experience implies that under some circumstances visual stimuli are categorized according to past experience. If two objects tend to be observed within close proximity, or small temporal intervals, the objects are more likely to be perceived together. For example, the English language contains 26 letters that are grouped to form words using a set of rules. If an individual reads an English word they have never seen, they use the law of past experience to interpret the letters "L" and "I" as two letters beside each other, rather than using the law of closure to combine the letters and interpret the object as an uppercase U.
On a similar note, students also often misunderstand the meaning of operational symbols when paired with variables. For instance, since students are used to joining two terms when they see the addition symbol (i.e. 2 + ½ = 2 ½), they will mistakenly believe that 2 + x is the same as 2x(past experience) 
The problem-solving approach
Gestalt psychology contributed to the scientific study of problem solving.] In fact, the early experimental work of the Gestaltists in Germany marks the beginning of the scientific study of problem solving. Later this experimental work continued through the 1960s and early 1970s with research conducted on relatively simple (but novel for participants) laboratory tasks of problem solving. 
Given Gestalt psychology's focus on the whole, it was natural for Gestalt psychologists to study problem solving from the perspective of insight, seeking to understand the process by which organisms sometimes suddenly transition from having no idea how to solve a problem to instantly understanding the whole problem and its solution. In a famous set of experiments, Köhler gave chimpanzees some boxes and placed food high off the ground; after some time, the chimpanzees appeared to suddenly realize that they could stack the boxes on top of each other to reach the food. 
Max Wertheimer distinguished two kinds of thinking: productive thinking and reproductive thinking. Productive thinking is solving a problem based on insight—a quick, creative, unplanned response to situations and environmental interaction. Reproductive thinking is solving a problem deliberately based on previous experience and knowledge. Reproductive thinking proceeds algorithmically—a problem solver reproduces a series of steps from memory, knowing that they will lead to a solution.




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