My 4 units from 80 to 84.
My SEA experience at Nightingale Academy helped me to identify the types of issues that the students were struggling with. It became apparent the foundational knowledge they had was weak, therefore these topics had to be consolidated on first, before the teaching of more advanced topics could begin. From the teaching of these basic topics, it astounded me as to how students processed the concept of the number line.
Something that seemed so simple and trivial to me, became the salvation for many students. I witnessed the same responses of incredulity from my students, regardless of their ages.The number line is a horizontal straight line which has specific markers spaced evenly to represent real numbers. This is a visual tool which can be used to address a wide range of Maths topics of varying difficulty, from addition, subtraction, and manipulating negative numbers to error intervals, inequalities, quadratic inequalities and plotting graphs.In this report, I will be exploring the literature available on the number line, and how I can implement it in the manipulation of positive and negative numbers, and the entire topic of inequalities for KS3 and KS4 students respectively.The number line is a pedagogical tool which aids the learning of students by providing a visual representation. At the start of a student’s Mathematics education, the number line is primarily used to find the answer to basic arithmetic sums by counting.
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However, the learner then makes progression in their use of the number line; using it for measuring the interval between numbers instead of counting. For eg, if there is a subtraction question of 84 – 67, the learner breaks this question down into steps. 3 units from 67 to 70, 10 units from 70 to 80, and then 4 units from 80 to 84. This is a distance of 17 units altogether, and it is this kind of thinking that the students begin to develop, before incorporating this with decimals, fractions and negative numbers. The National Numeracy Strategy introduced the concept of the number line into the Primary School curriculum, and suggested for each classroom to have a “large, long number line for teaching purposes, below the board at a level which is accessible to both the teacher and students. Furthermore, table top number lines both marked and unmarked should be provided for individual use.
” (DfEE 1999 Framework for teaching mathematics)Bruner, Olver and Greenfield et al. 1966 conducted research into the course of cognitive growth of children, and concluded that forms of representations (enactive, iconic and symbolic) aided learning. Baturo and Cooper 1999 conducted a study comparing students who were in Year 6 and Year 8, where they had to divide number lines into suitable intervals and then represent improper and mixed fractions on it. “The students were chosen by their teachers to represent a cross-section of abilities in their classes, discounting extremes. The students were from schools in lower middle-class areas.” Some of the values that were required to be shown on the number line were 2 1/4, 6/3 and 11/6. The authors made several conclusions, the first being that in general, Year 6 students performed better than the Year 8 students.
From their interviews, the authors were able to find out that “the students had incomplete, fragmented or non-existent structural knowledge of mixed numbers and improper fractions.” Further findings from their study recommended that the number lines are not effective unless the teachers are aware of the problems that the students face when processing them. The results from their findings is also in agreement with the Payne’s review of research on fractions in 1976; in that students find dividing the number lines into different units problematic. Baturo and Cooper 1999 concluded that “instruction should focus on providing students with a variety of fraction representations in order to develop rich and flexible schema for all fraction types (mixed numbers, and proper and improper fractions).”