Early mathematical numerical notations developed as a need for accuracy in goods and money record transactions. Continued innovations necessitated the advancement of mathematics as a field through architecture and related geometry. The calendar arose also as a key development in mathematics and astrology through the consideration of heavenly bodies and happenings like the flooding of the river Nile for Egyptians timing. The development of mathematics has taken place through many discoveries and innovations over time. Cognitive theories have contributed to the development of mathematics in a degree since it is a subject to be studied, conceptualized, and a language to communicate ideas and information. In addition, man has been seeking to understand his environment and this happens over time. In fact, a look at innovations over time reveals the consistency in conceptualization, adoption, and accommodation of issues by the human mind which happen over time. Activities and material that contribute to the development of mathematics relate to cognitive development which takes place in stages. This paper tries to examine the developments in mathematics in the view of the cognitive concepts as well as the historical developments in the discipline, and how best these models apply as concerns learning in this discipline. The paper also looks at examples of cognitive approaches applied in some parts for the development of the teaching of this discipline.

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Four groups of cultural settings can be noted for their early contributions in the development of Mathematics, namely Arabic, Indian, Greek, and Mesopotamia. There was the use of description of stones’ measurements in weight by Babylonian students, lengths of dug canals, in addition to the development of the mathematical tables at around 2000-1600 BC by Babylon according to Melville a mathematics professor in a New York University (Melville; cited in Nosotros) and the computation of simple problems in geometry. Gibbs developed the idea of vector analysis by combining the ideas and then used this idea to determine planetary and comet orbits. He contributed also to statistical mechanics and crystallography.

According to the Cognitive Load Theory developed by Sweller, instructional material including mathematics in the scope should include problems that are goal-free and worked examples, so as there will be no heavy memory load involved. This is to allow ease of acquiring schema through allowing changes in the associated long-term memory. Exposure to learning material allows the alteration of the cognitive characteristics concerned with the said material. Such alteration is necessary for the handling of the said material by the working memory associated with the acquiring of schema. Learning therefore must be slow and in stages so must be the development of mathematics (Soloman).

Cognitive models have been used to develop teaching mechanisms or to adjust the existing practice especially for the delivery of the instruction. For example, at the end of the First World War, there arise cognitive theories were associated with Ernst Mach and were used in the teaching of calculus in Trieste city. The faculties of creativity, deductive, and intuitiveness are said to have been present in the Jacobs method. It is necessary to ensure that, according to Jacobs, the introduction of some subjects to pupils during the teaching of some disciplines be such that complications could arise as a result of using instruments from one field that are used to develop knowledge in other fields do not arise. Therefore Jacobs supports the importance of the systematic introduction of teaching methods which could also be useful in teaching and developing mathematics. Whereas differential and integral calculus are important in the teaching of physics, such ideas should be introduced after the teaching of the velocity concept in physics.

Supposedly, the development of mathematics through the cognitive sense is seen through consideration of the system by Jacobs to introduce calculus at the secondary school level. The likes of integration should be developed and introduced as inverse to differentiation, and calculation of area using integration by beginning with an intuitive method that begins with a limit of a function then goes to the derivative of a function by the conceptualization of limiting of a function, through developing of a tangent on a curve and then introducing integration as explained above. Thus the development of mathematics can be seen to be systematic and stage-wise. What may be seen as a delay in innovations in mathematics may not be justifiably a delay. Technological innovations that demand more complex attention and necessitate the development of more complex analysis arise even today. Man’s environment has never been understood in a wider scope than it is today, and this calls for more careful attention through more complex methods and systems to take care of the existing needs. For example, man has found the necessity to do things quicker like traveling and this necessitates the development of quicker machines and means than the conventional ones. This avails the need for developing mathematics as a field since he would explore options that he may never have before, or improve the existing mechanical systems and procedures.

A child develops all the way to the formal operations cognitive stage realized at about twelve to fifteen years from the concrete operational stage observed at the age of about eight to eleven years. The latter develops from the pre-operations stage experienced at the age of three to seven years, which in turn develops after the sensorimotor as the initial cognitive structure (0-2 years). In the final stage, thinking involves abstraction, while in the fourth one it involves logical but building on concrete referents, the third stage involves intuitive intelligence and in the initial stage, it resembles motor action. Ample objects should be provided to children at the initial cognitive structural method while such skills like conservation, location, and order by use of objects should be applied to pupils in the second-last stage (Genetic Epistemology (J. Piaget)). Pythagoras was one of the Greek mathematicians in the c. 570-c. 490 (Joyce) with his idea of the Pythagoras theorem and Hipparchus and his trigonometric contributions in mathematics in the c. 180-c. 125. There was the development of the Arabic numerals by the Indian mathematicians and the championing of standardization of the Indian symbols by Robertson and O’Connor. The Arabic contributed to today’s idea of algebra (al-jabr) in ca. 800-ca.847 through Musa al-Khwarizmi. Developments in mathematics can also be traced later in America and Europe. An example is the development of vector ideas by Gibbs in the twentieth century. The vector analysis’ contribution of the American mathematician and physicist was a move towards the development of algebraic systems in the ninetieth century. He revealed the developments of the vector analysis in the notes he wrote to his students at Yale University and noted that his contributions in physics would be supported by the vector analysis. His work was distributed to other scholars even in Europe and Britain. Though introduced to quaternions, the ideas were later opposed and vector analysis was later developed by the likes of Oliver Heaviside who developed his own theory in vector analysis though he spoke widely on the Gibbs notes (Vectors). Other scholars also defended the quaternions. Other scholars developed their ideas in vector analysis. Earlier on, syncopated algebra was a brand introduced by Diophantus in the c. 250 AD by mixing letters with words in a systematic way. These included such expressions like 10+z and numbers were also operated as letters (Cobb, 95). His was a concept of fixed letter value algebra and not changing letter value in functional algebra. The structural thinking in the geometrical development of algebra was available readily whereas the progress of algebra was slow. There was a need to transit to the structural analysis of algebra which led to ease of development of algebra due to its degree of complexity by the end of the sixteenth century. Abstract structure development in mathematics occurred in the 19th and the 20th centuries where there was a symbolic representation of mathematical algebra. This was the science of rings, ideals, groups, and fields as abstract structures (cobb, 98). The vector analysis which allows students to visualize the objects in mathematics was a great step towards cognitive support in the development of mathematics. In order to understand the development of mathematics in various fields like algebra, one may need to understand the structural and the operational methods. Operational methods require that any problem solving be done through taking the actual actions involved whereas there is the requirement to broaden the mind and condensation of the information as required by the structural approach. The operational is more detailed and diffuse as compared to the structural approach which is general and concise (Cobb, 99). According to this writer, the jump from operational to structural modes of thinking has contributed positively to helping the student cope with the task at hand. In solving mathematical problems like in algebra, it may be necessary that the learners be exposed to the sequence development of ideas from the simple to the most complex since ideas have evolved in a similar manner. This would help the mind to understand and conceptualize the ideas more easily through the reflection of what is being made and acted upon. Symbolic constructs and algebra approach has developed over a long time. In the development of mathematics through many scholars, there has been the requirement to understand the already developed concepts by other scholars and the need to broaden the ideas to a greater scope. Similar and identical innovations and necessities also resulted in similar mathematical developments in various or more than one part of the world. Over time, basic actions, concepts, and ideas in the mathematic development activity have served as a basis for more actions, concepts, and ideas just as in the development of ideas by scholars in this field. There appears to be mutual dependence on mathematical processes and objects. A look at the way ideas co-correlate logically, historically and cognitive processes may help explore mathematical development. The vector analysis for example does depict a development in the conceptualization of visual algebra that can boost understanding of mathematics as a discipline.

Cognitive concepts utilized in the development of mathematics have their roots in its boosting of ease of understanding, adaptability, and accommodation to the new concepts when being linked to previous concepts. Likewise, scholars in this field seem to have undergone a natural process of conceptualizing natural and man-made facts, activities, and experiences in a systematic way, which has contributed to the evolution of mathematics.

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## References

- Cobb Paul. (1994) Learning Mathematics: Constructivist and Interactionist Theories of Mathematical Development. (Kluwer Academic Publishers)
- David E. Joyce. (1995) “Chronological List of Mathematicians.” Clark University.
- Duncan J. Melville. (2001) “Chronology of Mesopotamian Mathematics.” St Lawrence University.
- Duncan J. Melville. (1999). “Old Babylonian Mathematics.” St Lawrence University.
- Genetic Epistemology (J. Piaget).
- Nosotro, Rit, (2006). An Interesting History Essay.
- Soloman Howard. Cognitive Load Theory (J. Sweller).
- Vectors. Web.