6.6 Teaching strategies used by the mathematics teachers when teaching functions
6.6.3 Findings on the activities that were done by mathematics students during lessons on
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Figure 6.6: An extract of a lesson plan with guided discovery as a teaching strategy
Group tasks and individual tasks
Data obtained from the mathematics teachers’ lesson plans revealed that in fifty-nine (46.83%) lessons, the teachers used individual tasks to teach their students. These tasks were either given to students during lesson time or at the end of the lessons. On the other hand, in forty-seven (37.30%) of the planned lessons, the mathematics teachers allowed their students to work in groups. In some of the cases, the number of students expected to be in these groups was specified. However, the groups were kept small. Pairs or groups of three were used by the teachers. In twenty (15.87%) of the planned lessons, the teachers indicated that they used both individual tasks and group tasks.
6.6.3 Findings on the activities that were done by mathematics students during lessons
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Table 6.5: Summary of the activities that were done by the students during lessons on functions (n=126 lessons)
Activities done by the mathematics students frequency Percentage Identifying mathematical patterns and relationships 5 3.97%
Applying concepts learnt in solving real life situations 52 41.27%
Drawing ,sketching and plotting graphs of given functions 107 84.92%
Solving non- routine problems 7 5.56%
Solving project-like problems 27 21.43%
Using graphs of functions to make estimations of given variables
7 5.56%
Locating points on Cartesian planes 13 10.32%
Using their own imagination in solving real life problems 6 4.76%
Using logic or mathematical reasoning 5 3.97%
Identifying mathematical patterns and relationships
The students were given tasks that required them to identify patterns that arose from mathematical calculations. The students were also asked to identify relationships between given variables. Figure 6.6 shows a lesson plan prepared by one of the mathematics teachers with the objective of enabling students to find relationship between points on a plane.
Figure 6.7: An extract of a lesson plan with learning activities that led students into establishing relationship between points on a plane
Mr DT1’s plans of instruction had the following exercise. The researcher observed that the exercise was done by his students. It was found in the students’ exercise book.
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Draw the graphs of the following functions on graph paper using a scale of 1cm representing 1 unit on both axes for and
(i) (ii)
Comment on your observation from the two graphs.
Mr DT1 wanted his students to draw the graphs of the two functions and observe that the coefficient of x2 determines whether the graph is concave downwards or concave upwards.
Most of the students were able to notice that the graph with 3 as coefficient of x2 was - shaped and the graph with -3 as coefficient of x2 was -shaped.
Applying concepts learnt in solving real life problems
In fifty-two of the lessons planned by the mathematics teachers, the researcher found that the teachers gave their students activities that required them to apply concepts learnt on functions in solving real life problems. The following exercise was given by Ms ET1 to her mathematics students. The exercise was found in the students’ exercise books.
In order to maintain a balance on his farm, Mr Gotora made sure that the number of goats on his farm is always equal to twice the number of cattle on the farm. Take the number of cattle as x and the number of goats as y.
(i) Express the number of goats as a function of the number of cattle (ii) Find the number of goats on the farm if he had 17 cattle.
(iii)Find the number of cattle on the farm if he had 32 goats.
Drawing, sketching and plotting graphs of functions
The data obtained from the mathematics teachers’ lesson plans and from their students’
exercise books showed that in one hundred and seven lessons the teachers taught their students to draw, sketch or plot graphs of functions. The students drew graphs of linear, quadratic and cubic functions. Figure 6.7 shows one of the lessons planned by one of the mathematics teachers.
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Figure 6.8: An extract of a lesson plan prepared by a teacher who wanted students to draw a graph of a linear function
In another lesson, Mr GT2’s students were asked to perform the following task.
The following is a table of values for from x= -2 to x=3.
X -2 -1 0 1 2 3
Y -2 0
(i) Complete the table
(ii) Choose a suitable scale and draw the graph of y for the values of x given in the table.
Using graphs to estimate variables
The researcher found that in seven of the one hundred and twenty-six lessons that were analysed, the mathematics teachers instructed their students to use graphs to find estimates of given variables. The students were asked to estimate values from both the domain and the co- domain sets. The following is an example of such exercises:
Using a scale of 2 cm to represent 1 unit on both axes draw the graph of from x= -4 to x= 1. Use the graph to find
(i) The minimum value of y (ii) The value of y when x= 3.5 (iii) The value of x when y=3
Locating points on Cartesian plane
Some of the mathematics teachers who participated in this study asked their students to locate points on Cartesian plane. The students were either asked to draw graphs of functions and
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then locate given points or were given graphs of functions on a graph paper and then were asked to locate points on the graphs. The following exercise was done by Mr AT1’s students.
(a) Draw the graphs of the following functions on the same graph paper.
(b) Determine the point of intersection of the graphs on your graph paper.
Give the coordinates of the point of intersection.
(c) Hence find the solution of the following system of simultaneous equations
Solving non routine problems
Data obtained from the students’ exercise books revealed that the mathematics teachers used non routine problems as they taught concepts on functions. In seven lesson plans analysed by the researcher, the researcher found that students were given exercises on non routine problems. For instance, Mrs BT1’s lesson plan indicated that she gave the following exercise to her students in one of the lessons she taught.
Many numbers can be written in the form
For example 9= 2+3+4 where x=2 and 10=1+2+3+4 with x=1.
Find the expressions for the following numbers:
12, 13, 15 and 18
The researcher probed some of the mathematics teachers on the use of non routine problems when teaching functions. Twenty one teachers (84%) indicated that they rarely gave non- routine questions to their students. One of the teachers had the following to say:
“Non-routine and project-like questions are time consuming and challenging to our students. I do not think my students are able to solve these questions. I
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do not give such problems to my students. However, with the newly introduced curriculum, there is no way out. I have to find time for my students to solve non routine problems.”(Mr BT1, pers.comm.).
Using logic or mathematical reasoning
Although use of activities that prompted students to use logic in making conclusions was not common in the lesson plans that were analysed, the researcher found that in five of the lessons taught by the teachers, students were exposed to mathematical reasoning. The following example was obtained from exercise books for students taught by Mr CT2. The teacher was teaching sets as a prerequisite for functions.
Consider ℳ= , ℵ= and ⅅ=
.Given that ℳ=ⅅ ℵ and ⅅ ℵ , what conclusion can you make about snakes?
Solving project-like problems
Data obtained from the students’ exercise books showed that students taught by the mathematics teachers who participated in this study were sometimes given project-like problems. The students were given real life situations in which they were expected to research, for them to get workable solutions to given problems. The following task was given to students by Mr HT1.
Record sales made by a vegetable vendor at any vegetable stall for a full week.
Draw a graph to show the sales against the days of the week. Draw a trend line on the graph. Find the equation of the trend line. Use the line to decide if the vendor’s business was improving or not.