Problems faced by children in understanding fractions
The concept/operation relationship is what lacks in most of the students dealing with fractions. Gould’s study highlights the problem of relating the algorithmic operations concerning a given fraction and the concept behind the operations. According to Gould, calculating ½ + ½ can appear very different to taking two cups half full of water and emptying the content of one into the other. Apart from relating the algorithmic operations to the concept underlying this operation, Gould also shows the difficulty resulting from viewing fractions from a double count context. For example, 2/5 can mean counting all the five pieces then recounting the two in relation to five. Here, the problem comes in terms of the student developing a mental framework of considering the parts that make up the unit instead of relating a part to the whole unit (Gould, 2005).
Another problem that Gould highlights is the inability of students to differentiate between partition fractions and quantity fractions. He then takes on Yoshida’s approach to assist in differentiating the two types of fractions. Dividing a unit into smaller parts and comparing the parts to the whole gives one a partition fraction. For example, dividing a mango into three equal parts and comparing one part to the whole unit which is three parts to get 1/3 gives you a partition fraction. This is contrary to quantity fractions which are based on a universal unit. It could, thus, be impossible to ask which fraction is bigger between 1/3 and ¼. A quarter of a big loaf could still be bigger than a half of a smaller one. Therefore, in quantity fractions, there has to be a universal unit (Gould, 2005).
Importance of problem posing
Most teachers consolidate their efforts to design problems for their students to solve. This might be good but Lowrie’s problem-posing study shows that problem-posing could play a very important role in assisting the students to understand the dimensions of a problem and hence facilitate his solving the problem. When properly implemented, the program of problem-posing before, during and after a mathematical activity can facilitate the understanding capability of the students. Good implementation means a deliberate organization by the teacher in terms of encouraging cooperation during solving, linking two students appropriately, encouraging students to develop more complex problems etc (Lowrie, 1999)
The importance of problem posing includes improving the student’s analytical ability. By continuously posing problems to one another, a student tries to increase the complexity of the problem. To achieve this, he will be forced to explore the different dimensions of the problem and as a result, he will understand the dimensions of the problem etc (Lowrie, 1999).
By cooperating in the solving of the problem, if the solvers are encouraged to discuss the parts of the question that proved challenging, the problem poser will have to explain and thus the solver will have an insight into the approach to such questions. In addition, as the questions continue becoming more and more complex, the problem poser will be forced to know the answer before framing the question. This is a technical part that will poke the poser’s analytical ability to stretch a little further (Lowrie, 1999).
The relationship between the study of measurements and numbers
Differences in objects can be easily noted through direct comparison. This can be through size, length, volume or mass. Using eyes can give results that are not quantified. For example, one can note without using any instrument that a double-deck bus is bigger than a saloon car or that the surface area of a desktop is bigger than the surface area of an exercise book. To express this, the student may need to use different symbols to illustrate the relationship between the two objects. For example, take b to represent the surface of the book and d to represent the surface area of the desk, the student can thus illustrate that the surface area of the book is smaller than that of the desk by using mathematical symbols. He will thus write, bb. these are graphical representations of the relationship between the two objects. (Dougherty & Venenciano, 2004)
If a third entity is added, for example, the floor of the classroom, one can make the comparison of the size of the three objects in relation to each other. If the surface area of the desk can accommodate four books, then one can express it as d=4b or d/b=4. One can also try to quantify the relationship between the surface area of the floor and the book. If the floor can accommodate 50 books, then the formula can be 20b=f (f representing the surface area of the floor). From the relationships, one can now make the relationship between the book, the desk and the floor without directly having to relate them. This can be done through the use of the relationship symbols comparisons. For example, 4b=d, 20b=f. by taking the book as a unit of comparison, one can compare the relationship between the desk and the floor without having to link them directly. This can be done by trying to ask oneself if one desk can accommodate four books and one floor can accommodate 20 books then how many clusters of four books can be found in 20 books? This will give a final result of five. This means that the surface area of five tables makes up the floor and thus 5d=f (Dougherty & Venenciano, 2004).
This is the basic principle of number work. Although its outward look does not portray the aspect of measurements, number work is purely based on the relationship of measurements. Infractions, the fraction 2/3 can be approached from a measurement context. In the given example the denominator i.e. 3 is the number of units needed to make a whole. This is to say that in 2/3, there are two units but to get the whole unit one needs 3 (Dougherty & Venenciano, 2004).
The same aspect of relationship of quantities is applied in probability. A ratio of 3:7 means that the total chance of an event is 7. It, therefore, means that the total number of units is seven but the chances of it happening is three units out of the seven (Dougherty & Venenciano, 2004).
In conclusion, the basis of numbering is directly related to relationship of measurements.
What is the definition of variation?
Variations can be termed as the differences that occur in a similar object in terms of measurement or colour. For example, the length of a string can get different measurements if done by different people independently. These different measurements of the same string are what are termed variations (Watson, 2007).
What is the definition of expectation?
In statistics, Mokros and Russell define expectation as the ‘arithmetic mean or proportion.’ This means that expectations are the available chances that can occur in case of a toss. For example, a coin has two sides, if tossed once; the expectation is that it will land as head or tail (Watson, 2007).
The importance of the study of chance
The study of chance is not just aimed at showing meaningless results by providing well-drawn graphs of data but it is an important tool in the well being of society. The outcome is not the ultimate goal but the procedure followed which eventually gives one the ability to make a balance of expectations that will eventually influence his decisions. By studying chance, society moulds individuals who are in position to put together several possible variations and expectations and thus make a founded decision. This means that chance is not only meant for the case of statistics as a career but also to every member of society. In decision making, concerning different issues of the society, the study of chance arms one with tools of trade to enable one to analyze the available data and consider the variations and thus the expectations (Watson, 2007).
In the political arena, an individual armed with knowledge of chance will be able to analyze the available variations and the possible expectations to critique or commend on the decisions of the campaigners. Due to the procedure used, one is more likely to have confidence in the decision made. Where only a single side of a situation is being shown in a case of public interest, a person armed with knowledge of chance will have to question the people responsible as to why they are not showing all the variations. For instance, the media will not have to consistently be airing one contestant and blocking others. This means that studying chance builds a responsible society in terms of providing a means of checks and balances and also a confident society because of their trust in their decision-making mechanisms (Watson, 2007).
Bobis, J., Mulligan, J.T., & Lowrie, T. (1999).Mathematics for Children: Challenging Children to think mathematically. Sydney: Prentice-Hall.
Curriculum Council (Western Australia). (1998). Curriculum Framework Learning Statement for Mathematics. Perth: Author
Dougherty, Barbara J., Fay Zenigami, Claire Okazaki, and Linda Venenciano. Measure-Up: Grade I. Honolulu: University of Hawaii Curriculum Research and Development Group, 2004.
Gould, P. Really brocken numbers: Peter Gould provides an insight into children’s thinking about fractions through their drawings and explanations. Australian mathematics classroom. Web.
Mokros, J. & Russell, S. J. (1995). Children’s concepts of average and representativeness. Journal for Research in Mathematics Education, 26, 20–39.
Watson, J.(2007). Educating for statistical literacy. The foundations of data and chance and data. Australian mathematics classroom. Web.