Methods of Mathematical Education

Content
Introduction………………………………………………………………………….page 2
Jerome bruner’s theories of cognitive development…………………………………………………………………………page 3-4
Multiple Intelligence…………………………………………………......................page 5-6
Multiple Representation………………………………………………………..….page 7-8
Deway’s social constructivism and Gagne’s information processing model ………………………………………………………………………….....................page 9-10
Developing Teaching stretergies using constructvist models……………..…….Page 11-12
Main conclusion…………………………………………………………...………..page 13
Referencess……………………………………………………………………..…..page 14

Introduction Various theories about learning have dominated the education process in different days. Invention of new theories which satisfied the disadvantages of old theories kept these changes active. Since each theorist has proved his view about the learning process, people appreciated their work through the application of the principles in the education. In this writing I will be looking into the different theories which I have learned in the class of mathematical education one. Each theory called out, its own believes and developed the less emphasized areas of the previous theory and there are also situations where the theorists argue saying his view for the theory is the best. In this portfolio I have included the ideas of different theorists and how I can make use of it while teaching. I also highlighted some of the points and examples, as well as my own views about the theories. Mainly five topics will be discussed in this writing. To conclude I have given some recommendation and reasons about the theories.

Jerome Bruner’s Theory of cognitive development
Introduction
Cognitive development is the construction of thought process, including remembering, problem solving,and decision –making, from childhood through adolescence to adulthood. Intellectual development or cognitive development is one of the aspects of development a human being goes through from his birth until the day he/she dies. Bruner does not attach ages specifically to his theory. . It is his belief that students learn concepts best through discovery learning where the students use their own previous knowledge to construct their understanding of how things operate. “To promote concept discovery, the teacher presents the set of instances that will best help learners to develop an appropriate model of the concept (Driscoll, 2005).

Bruner 's three stages of development include Enactive,Iconic and symbolic Enactive “knowledge is stored primarily in the form of motor responses.” (Alexander 2002). In this stage individuals are learning through motor skills and by experimenting with and learning to manipulate objects. An example would be teaching someone how to play the hand clapping game patty cake. In the game patty cake the motions are learned by doing the motion. It would be very difficult to try to teach someone this game by describe the actions verbally. Iconic “knowledge is stored primarily in the form of visual images ” (Alexander 2002). Human learning is generated by imagery in this stage. The individual is able to generate mental images of events.
In the iconic stage it is very important to present a number of different visual aids to students to supplement teaching material. Examples include images, videos, charts and graphs.

Classroom Example – Unit on Whales If trying to demonstrate the different sizes of the whales a teacher could show an image of a person standing next to a humpback whale which would demonstrate the size comparison between humans and humpbacks. The teacher could then show another image that would show an orca or gray whale next to a humpback. To follow up the images the teacher could then have the students graph the length of the whales in meters. This is simple and interesting
Symbolic “knowledge is stored primarily as words, mathematical symbols, or in other symbol systems ” (Alexander 2002).

An example of a teacher using the symbolic stage of instruction may be a teacher who is discussing a concept such as birds and their migration in the different seasons. The teacher may discuss flight, what the birds eat and where the birds migrate to. In this case the teacher is using language (words) to describe the birds, what they eat and their migration. Although many students may be able to understand the teachers’ lesson on birds and their migration the teacher is making an assumption that the students have a well developed symbolic system. Without preceding lessons that “hit” the enactive and iconic stage the possibility exists that the students will not have the background information to be able to fully understand the concept.
Summery with examples
From these topic of cognitive development mainly I have learn 3 things and how I can make use of it in a class room application. Here are the summery of the topic with examples.For example a grade six teacher teaching a lesson on bridges could do the following series of lessons.
1. Build a model of a bridge (enactive stage)

2. Watch a video that shows different types of bridges and then draw images of the different types of bridges. (iconic stage)

3. Research bridges on the internet and discuss the environmental impact of bridge construction. (symbolic stage)
Conclusion
Adults who are learning new concepts may have extreme difficulty learning new concepts with items taught at the symbolic stage even though their symbolic system is well developed. In order for adults to learn new concepts it may be beneficial for teachers to try a series of lessons that progress through the enactive and iconic stage. . I believe that the teachers should provide opportunities for students to build upon existing knowledge. This will help the students to concentrate the topic as well as enjoy learning.

Howard Gardner’s Multiple Intelligence
Introduction
The theory of Multiple intelligence was developed by Harvard University psychologist Howard Gardner and first appeared in Frames of Mind: The Theory of Multiple Intelligences (Gardner, 1983). In Frames of Mind, Gardner explored the ideas about the mental abilities that support the wide range of adult roles over time and across the culture.Howard Gardner’s theory proposes that people are not born with all of the intelligence they will ever have. It says that intelligence can be learned throughout life. Also, it claims that everyone is intelligent in at least seven different ways and can develop each aspect of intelligence to an average level of competency. Intelligence, as defined by Gardner, is the ability to solve problems or fashion products that are valuable in one or more cultural settings.
Theory of multiple intelligence “People are not born with all of the intelligence they will ever have it.The intelligence can be learned through a life.” (Horward Gardner 1983). Therefore what are the ways to learn? And how it help us to learn? Why it is important to follow certain theories to learn? After knowing the theory of multiple intelligence, I have found an answer for these questions. Armstrong (2000) has argued that the theory of MI has broad implications for special education. Because MI focuses on a wide spectrum of abilities, it helps place “special needs” in a broader context. I am totally agree with him because that a growth paradigm would be more appropriate for students with special needs. “Using MI as backdrop, educators can begin to perceive children with special needs as whole persons possessing strengths in many areas” (Armstrong, 2000, p. 104). I believe that Multiple Intelligences (MI) theory and strategies provide a framework and tools that can help teachers in designing classrooms, instruction, and curricula that meet the individual needs of many kinds of students. The application of multiple intelligences in the classroom can stimulate a student’s learning in new ways. Because it includes the following eight intelligences: 1. Linguistic intelligence It allows individuals to communicate and make sense of the world through language. Eg; , writers, lawyers, and public speakers. 2. Logical-mathematical intelligence : Who has this kind of intelligence is able to see cause and effect really well, they are able to identify a problem and solve it right there on the spot.Eg; Mathematicians, scientists, and engineers. 3. Spatial intelligence : It enables people to perceive visual or spatial information, to transform this information, and to recreate visual images from memory.Eg; architects, artists, surgeons, and pilots. 4. Bodily-Kinesthetic intelligence : It entails using all or part of the body to solve problems or create products.Eg; choreographers, rock climbers, and skilled artisans. 5. Musical intelligence : It allows people to create, communicate, and understand meanings made out of sound.Eg; musicians, and acoustic engineers. 6. Interpersonal intelligence: It is the capacity to recognize and make distinctions among others’ feelings and intentions, and to draw on these in solving problems.Eg; Educators, salespeople, religious and political leaders and counselors 7. Intrapersonal intelligence: It is the capacity to understand oneself, to appreciate one 's feelings, fears and motivations.Eg; planner, small business owner, psychologists, artists etc. 8. Naturalist intelligence : Allows people to solve problems by distinguishing among, classifying, and using features of the natural world.Eg; hunters, archeologists, environmental scientists, and farmers. Multiple intelligence improved standardized test scores, reduced disciplinary infractions, increased parent involvement, and increased ability to work with students with learning disabilities. Perhaps most compelling among the positive outcomes was that teachers and administrators realized the power of multiple intelligence for all students, including those students with learning differences.
Conclusion
The best learning opportunities are those that are most successful for all students. What may need emphasis, however, is the way in which lessons are specifically tailored to the needs of individual students or small groups of students. Multiple intelligence helps the students to provide the valuable knowledge and lifelong learning. It will help the students to use their valuable knowledge and past experiences while learning. Multiple intelligence also help the students to learn in different ways throughout their live. Students with special needs and their families are encouraged to exceed expectations in the school, where high-quality work is a central part of the culture.
Multiple representation Introduction An external representation is something that stands for, depicts, symbolizes or represents objects and/or processes. Examples in physics include words, diagrams, equations, graphs, and sketches. The positive role of multiple representations in student learning has been suggested by many educators. H. Simon said: “Finding facilitating representations for almost any class of problem(s) should be seen as a major intellectual achievement, one that is often greatly underestimated as a significant part of both problem solving efforts in science and efforts in instructional design Representations include all the different ways that students depict their mathematical thinking as well as the processes they use to put their thinking into those forms. Representations can include written work, oral explanations, models with manipulative materials, and even the mental processes one uses to do mathematics. Multiple representations are ways to symbolize, to describe and to refer to the same mathematical entity. They are used to understand, to develop, and to communicate different mathematical features of the same object or operation, as well as connections between different properties. Multiple representations include graphs and diagrams, tables and grids, formulas, symbols, words, gestures, software code, videos, concrete models, physical and virtual manipulative, pictures, and sounds. Representations are thinking tools for doing mathematics. To make a class room interesting we can use multiple representations because it will help the children’s to better understand the mathematics they are doing and to share their ideas with others. I have learn the same topic in developmental pscology . But I found some of the information I learned in mathematical education is different from the psycology. Real world problems do not come neatly packaged in one representation. Defining the questions and finding alternative solutions often involves reading text, searching on the internet, interpreting graphs, creating tables, solving equations, designing models, and working with others. Using Multiple Representations prepares students for the real world of problem solving. all these I have learned from mathematical education one. We have done some of the group works as well it helps us to understand the concept better.

Conclusion According to my opinion. When we solve mathematical problems, a core part of the solution process is how we represent the ideas in the problem. The form of representation we select allows us to manipulate the information (as opposed to manipulating symbols) to reach a sensible solution.With the Representation Standard, our goal is for young students to show their mathematical ideas and procedures in multiple ways. Through the use of representations, children develop their own mental images of mathematical ideas.

Social Constructivism and information processing model Introduction Constructivism is one of the views which dominated during 1930 and 1940’s. John Dewey, a believer in what he called "the audacity of imagination," was one of the first national figures in education policy. He proposed a method of "directed living" in which students would engage in real-world, practical workshops in which they would demonstrate their knowledge through creativity and collaboration. Students should be provided with opportunities to think from themselves and articulate their thoughts. Processes of instruction are unified in the degree in which they center in the production of good habits of thinking. While we may speak, without error, of the method of thought, the important thing is that thinking is the method of an educative experience. The essentials of method are therefore identical with the essentials of reflection. They are first that the pupil have a genuine situation of experience -- that there be a continuous activity in which he is interested for its own sake; secondly, that a genuine problem develop within this situation as a stimulus to thought; third, that he possess the information and make the observations needed to deal with it; fourth, that suggested solutions occur to him which he shall be responsible for developing in an orderly way; fifth, that he have opportunity and occasion to test his ideas by application, to make their meaning clear and to discover for himself their validity.
In the last ten years, many researchers have explored the role computers can play in constructivist teaching. Roger Schank, Seymour Papert and Marcia C. Linn among others have demonstrated that computers often can provide an appropriate creative medium for students to express themselves and demonstrate their acquirement of new knowledge. Online collaboration projects and web publishing have both proved to be an exciting new way for teachers to engage their students. But because constructivism can require extended periods of classroom time and individualized attention to students, some educators have resisted its adoption and continue to support a drill-and-practice approach to learning.
IN 1999 Professor Hank Becker of the University of California at Irvine began publishing the results of his 1998 Teaching, Learning and Computing Study. TLC 1998 was one of the first major national research studies to examine teachers and students who use Internet computers on a regular basis. One thing they discovered is that constructivist teachers were more likely to have students use Internet computers than traditional teachers. Those teachers who were more comfortable with constructivist teaching style were much more likely to have students use computers. I think this[computer/constructivist] relationship is perhaps due to the fact that technology provides students with almost unlimited access to information that they need in order to do research and test their ideas. It facilitates communication, allowing students to present their beliefs and products to broader audiences and also exposes them to the opinions of a more diverse group of people in the real world beyond the classroom, school and local community - all conditions optimal for constructivist learning. Conclusion
By doing this writing I have understood that If teachers want to make the best use of the Internet –then we have to utilize its interactivity, collaboration opportunities, creative opportunities - the teachers themselves must be open to interactivity, collaboration and creativity. And unless we 're able to get teachers so they 're comfortable with these constructivist concepts, chances are they won 't be comfortable having their students do these things either.

Developing teaching strategies using constructivist models Introduction Constructivism is a popular theory with mathematics educators argue that all knowledge is constructed by individuals rather than transferred. This theory is based on the assumption that humans have achieved a unique ability to recognise patterns, organise knowledge and purposively plan future actions.
Jean Piaget is a theorist who brought up a developmental theory. In his theory he emphasized that many factors like reading, listening and experiencing influences the learning process. He stated four developmental stages in his theory. The sensory motor stage is the first stage where child use the senses to gain understanding of the environment. In the preoperational stage child start using symbols. Logical thinking comes to the child in concrete operational stage and the last is the formal operational stage where child can think simultaneously. In this process, knowledge is internalized by three processes, assimilation which is the process by which new information is taken into what is already known, to make new experience. Prior understandings are to accommodate new experiences through accommodation. Equilibrium is what is said to create a balance in the process of assimilation and accommodation. This approach assumes that students comes to classrooms with ideas or opinions that need to be modified by a teacher who facilitates by devising the tasks and questions to create challenges for the students. "we cannot identify and recognize what we don 't already know."( D. Edwards and N. Mercer.1987) The social constructivist theory of Lev Vygotsky says that the child learns through interaction with adults who have a better understanding of the concept. If this term is applied in a mathematics lesson, the teacher plays a major role in the learning process.” .” Constructivist teachers seek and value students ' points of view. Knowing what students think about concepts helps teachers formulate classroom lessons and differentiate instruction on the basis of students ' needs and interests (Brooks, J. and Brooks, M. 1993).”
For example, in a lesson to find the area, the teacher needs to demonstrate the process at first. Real objects in the classroom can be used in order to generate motivation. A group work to find area can be given to allow the brighter students to help the weaker ones.
It takes time to learn: learning is not instantaneous. For significant learning we need to revisit ideas, ponder them try them out, play with them and use them. This cannot happen in 5-10 minutes (Brooks, J. and Brooks, M. 1993).
If we apply the above statement in a mathematics lesson, the lesson activities must be attractive and the students need to be active. The teacher needs to create a good environment for the students and make the class interactive. Providing the students enough time to practice is also important. If the lesson is about addition of two digit numbers, make sure that the students are familiar with the addition of one digit numbers. Instead of giving a worksheet for individuals, the teacher may try with flesh cards of questions to in groups. This will allow them to pass knowledge within them. Play a game at the closer of the lesson like “who finds the answer first”. Make them try some questions at home and check their level of understanding on the next day.
According to my opinion in a mathematics lesson of addition, the teacher needs to check what the students already know about addition. The lesson should contain activities through which students can build knowledge of addition from their prior knowledge. If they have practiced addition by drawing lines, the teacher can make them practice addition without drawing lines or objects but keeping them in mind. This might help them to get started. Since constructivism is a belief that students create their own knowledge, motivation is the key factor that will help them to create this lesson. No matter what the lesson may be, motivation is essential. In order to motivate the students’ teacher needs to make each part of the lesson interesting. This can be done by varying the techniques or the materials used to learn in the class.
Conclusion
I believe the mathematical lessons need to be interactive. Teacher needs to provide opportunity for students to share their knowledge by questioning, or giving group activities. Class discussions can also be held by the teacher to assess the knowledge level of the students. In a lesson to find an unknown side of shapes, teacher can make the students discuss on the application and the fact of using the formula for the press and if there is an easy way to find it. Make the students to try and find the area of some real objects at home with the help of someone or in a group. Comment for their answer for them to build confidence of doing it individually.
Main conclusion
Since constructivism has many advantage of applying it, it’s an inevitable approach in learning mathematics. Theorists like Piaget, Vygotsky and Bruner have stressed on their beliefs as whole and almost all the concepts of their theories are applicable in mathematics teaching. Different theorists view was different about the learning process but still there are some common similarities.
There are also some common principles of theories which come in hand for the development of a good lesson of mathematics. These principles stressed on what is involved in structuring the experience to build a bridge between present and new understanding. It may be the reason why all the theorists have commonly stressed on these points.
The overall the example suggested in the writing can be used in the classrooms of mathematics and a better result could be achieved if done well. Not only the lesson mentioned in the example but also many other lessons of mathematics can also be developed under the same concept of the theory. The information provided by these theories can be said as some golden ideas for the teachers, to develop their lessons in a meaningful way and to achieve maximum objectives especially for mathematics.

References
Brooks, J. and Brooks, M. (1993). In Search of Understanding: The Case for Constructivist Classrooms, ASCD

Edward.D and N. Mercer. Common Knowledge: The Development of Understanding in the Classroom. London: Methuen, 1987.

Davis.R.B, Constructivist Views of the Teaching and Learning of Mathematics. Washington, D.C. National Council of Teachers of Mathematics, 1991.

Gredler, M. E. (1997). Learning and instruction: Theory into practice (3rd ed). Upper Saddle River, NJ: Prentice-Hall.

Derry, S. J. (1999). A Fish called peer learning: Searching for common themes. In A. M. O 'Donnell & A. King (Eds.),

Gardner, H. (1999). Intelligence reframed: The theory in practice. New York, NY: Basic Books.

Lecture notes of Methametical Education one 2012.

References: Brooks, J. and Brooks, M. (1993). In Search of Understanding: The Case for Constructivist Classrooms, ASCD Edward.D and N. Mercer. Common Knowledge: The Development of Understanding in the Classroom. London: Methuen, 1987. Davis.R.B, Constructivist Views of the Teaching and Learning of Mathematics. Washington, D.C. National Council of Teachers of Mathematics, 1991. Gredler, M. E. (1997). Learning and instruction: Theory into practice (3rd ed). Upper Saddle River, NJ: Prentice-Hall. Derry, S. J. (1999). A Fish called peer learning: Searching for common themes. In A. M. O 'Donnell & A. King (Eds.), Gardner, H. (1999). Intelligence reframed: The theory in practice. New York, NY: Basic Books. Lecture notes of Methametical Education one 2012.