defined the concept of real- world as follows.
By real world, we mean the “rest of the world” outside mathematics, i.e. school, or university subjects or disciplines different from mathematics, or everyday life and the world around us. In contrast, with a purely mathematical problem, the defining situation is entirely embedded in some mathematical universe.
Albert and Antos (2000) point out that when children make connections between the real world and mathematical concepts, mathematics becomes relevant to them and as mathematics becomes relevant, students become more motivated to learn and more interested in the learning process. They offer the following suggestion for educators:
As educators, it is our responsibility to bring mathematics to life for children. For students to view mathematics with interest and enthusiasm, they must see the relevance between mathematics and everyday life. Teachers can nurture intellectual excitement by linking classroom activities with real-life experiences. As children begin to see and understand how mathematical concepts are used in their daily lives, they become more interested in learning mathematical processes. Their understanding of mathematical concepts and ideas is enhanced and strengthened. When teachers create and maintain instructional practices with which students can identify, ideas are reconstructed, confusion and conflicts are minimized, and understanding is achieved.
Galbraith (2007) distinguishes between word problems and real-world problems and argues that word problems are widely construed as close relatives of modelling problems, since both word problems and modelling problems are crouched in verbal clothes. He however, maintains that the two terms differ markedly, specifically with respect to meaningfulness and purpose, seeing that modelling problems have real-world connections, which word problems often do not have. Galbraith explains that whilst a word problem may be couched in the language of the real world there is often no sense in how one is supposed to apply the decisions to one’s real-world, since it does not show how mathematics can be applied to enhance understanding of real problems.
Boaler (1993) points out that many view the abstractness of mathematics as being a cold, detached, remote body of knowledge. She argues that this image may be broken down by the use of contexts which are more subjective and personal and also improves the ability of students to interpret events around them. Boaler explains that using real-world, local community and even individualised examples which students may analyse and interpret is thought to present mathematics as a means with which to understand reality and concludes that when learners make connections between the real world and mathematical concepts, mathematics becomes relevant to them. Bottle (2005) supports this view and states that by giving children a real context for their problem solving, gives them the best opportunity to become fluent in using mathematical skills and procedures. She concurs with Boaler and claims that if we make mathematics meaningful for children by relating it to their interests, then they will be more likely to see its relevance and use.
Mudaly (2004b) adds to this thought and states that “the direct connection between classroom mathematics and real-world mathematics is a tenuous one, because it is often difficult to relate classroom mathematics to what happens in the real world”. He goes on to say that “if the word ‘real’ in this instance is not only interpreted as a connection to the real world, but as a reference to the problem situations which appear to be real in the learner’s mind, then the relationship between real-world and classroom mathematics becomes a bearable one”.
According to De Villiers (2003), real-world situations are extremely complex and usually have to be simplified before mathematics can be meaningfully applied to them. Furthermore De Villiers (2007) explains that in the real world there are no perfectly straight lines, flat planes and spheres, nor can measurements be made with absolute precision. Learners, students and
teachers therefore must be able to make logical assumptions to simplify the original problem.
Similarly Engel and Vogel (2004) view models as an over simplification of reality where part of the available information is discarded. This loss of information due to inherent assumptions, simplifications and abstractions does not invalidate the conclusions, rather it is intended to generalise the obtained results to hold true in other similar situations. In the problems (See Appendix B) given to the participants of this research study, various assumptions had to be made. A few of the assumptions are listed below:
PROBLEM 1
• A net profit of R25 and R30 per table and bookcase is not realistic.
• The carpenter assumes that he can sell all the tables and bookcases he produces each week. In reality this may not be always possible.
PROBLEM 2
• It is assumed that the mountain peak is perfectly vertical with respect to the line chosen to be the distance from the base of the mountain peak to the two observers.
• It is assumed that the height of the person is negligible compared to the height of the mountain peak.
• It is assumed that the two observers are standing 500m apart at the same level, that is, one may not be standing at a slightly higher level than the other.
• It is assumed that the angles of inclination were accurately measured.
PROBLEM 3
• It is assumed that the lawn is perfectly rectangular, i.e. there are no flower beds that encroach on to the lawn.
When working with real-world problems where assumptions have to be made, educators need to make learners aware of these assumptions.