Post Course Concept Maps

We now turn to consider the development in understanding as represented on the concept maps completed at the end of the course. [§4.6.] These were completed during the final lesson of the trigonometry component. First we consider P1’s map (Figure 5.14).

Now she included illustrations of trigonometrical functions and their graphs, tables of values and an illustration of the mnemonic taught within the class. It includes new facts and formulae, surd values, identities, the unit circle, a concept graph and the teacher’s table of the positive and negative values of the three trigonometric functions in each quadrant.

It is interesting to see that although the concept image graphs of sine, cosine and tangent have been drawn on the left it is evident that they have been plotted.

The final point to make about the second concept map is the complete lack of connection between the different items identified. In the first map, despite its sparsity, the different items are shown as linked; yet in the second map there are no links of any kind apart from those connecting the items to the word trigonometry. The overall impression is of many legged spider with the item at the end of each leg distinct and isolated rather than the hoped for network of connected sub-concepts, images and related facts and formulae. When questioned about the lack of connection P1 explained:

P1: Well they are sort of connected because it’s all trigonometry but at A- level trigonometry is different. It’s a lot harder and there is more to remember I: What do you think is the key to being good at trigonometry?

P1: Having a good memory.

When questioned about plotting the graphs, P1 explained:

P1: I can never remember which one is sine and which is cosine so I always work out the values to make sure.

The inclusion of so many values is a sign that was to re-emerge in the main study [§7.10.]

Figure 5.15 illustrates P2’s second concept map.

the general triangle properties but it is interesting to note that in place of the word ‘trigonometry’ she has drawn a triangle so the connection is still being made. She is the only student to include the radian sector arc and area formulae or the correlation between cos, sin and tan (-θ) and cos, sin and tan θ. She has defined tan, cos and sin (90-θ) in terms of sin, cos and tan θ but interestingly she has drawn the graphs of y= sin θ and y=cos θ incorrectly. The graph of y= sin (x-90) is also incorrect. The conclusion may be made that she has not connected the characteristics of the graph of y=sin θ to the graph of y=2sin θ or y=sin 2x etc. Of the six graphs she has shown three are incorrect. She has included two tables of trigonometric values, one for the special angles in surd form (the correct values given this time) and one for the values of 0, 90, 180, 270 and 360°. These values are correct therefore she has not connected this table of values to the graphs. This points to a lack of connection between the geometric aspects of trigonometry. P2 has included the two identities covered in this chapter and a spatial image of the CAST diagram with a note ‘for equations’. This points to an operational concept of the identities. Finally there is no indication of linkage either explicitly or implicitly in this map. This was supported by her comments in interview where she repeated the comment made above [§5.7.1] and went on to say:

You only have to learn the things that the teacher tells you at the end of the lesson. In the assessment you don’t have to explain how it is but you just use it. I don’t bother to learn it all- just the conclusions and remember those.

(P2)

P4’s second map (Figure 5.16) shows the two identities covered in this chapter, the graphs of sin, cos and tan x, a spatial image of a triangle with an angle marked θ with the ratios for sin, cos and tan defined beside it. There is a spatial image of the CAST diagram and a table of correct values for the special angles. There is also a formula for Pythagoras theorem. There is no explicit indication of connection between these different representations but the results mentioned are consistent unlike P2’s second map. In the follow- up interview, P4 said that (like P2) he just learnt the results from the lessons.

They are what is important. To pass the test you have to learn those things. It doesn’t matter if you don’t understand everything just learn the important things that are in the test.

Unfortunately the reproduction of P3’s second map (Figure 5.17) is poor.

Figure 5.16 P4’s Second Concept Map

CAST diagram in degrees and radians and the ratios for sin, cos and tan. She has also included diagrams of the special angles triangles, though the one for 30° & 60° is incorrect, and underneath are two values for the sin 60° and cos 60° clearly derived from the incorrect triangle. The graphs of sin x and cos x have been included with details of the maximum and minimum values and the angles where these occur. The main feature is the lack of connections between the items mentioned in comparison to her first map (Figure 5.4) however there is some attempt at grouping. Many of the items that were mentioned on her initial map have been omitted.

The second interview with P3 revealed that the reason she had left out items mentioned in her first map was that she knew them. She said that her understanding of trigonometry had changed because now it included the circle as well as triangles but there was still no clear indication of perceiving trigonometry as a function of angle. This extract exemplifies this:

I: Could you explain in a sentence the connection between say sine and an angle? P3: Well sin will find the angle or if you know it already it will find the length or

what ever you want to find. I: What are functions?

P3: That’s like… when you have f(x) equals something, like f(x) = 2x say I: Do functions always have to have f(x) equals something?

P3: Well the ones we’ve done have always been f(x). No sometimes we have g(x)… Yeah either f(x) or g(x) that’s what we learned. It’s another way instead of writing y equals something, just say f(x) equals it…it’s the same thing. I: Why do you think this chapter is called trigonometric functions? P3: I don’t know… it should say f(x) = sin x then it would be a function.

It appears that P3’s recognition of functions is determined by the use of function notation. In the lessons ‘trigonometric function’ is a phrase frequently used by the teacher for example, “today we are going to graph trig functions” “what happens to the graph of the sine function when we make it negative?” “the thing about these trig functions...”. The students, including P3, also used the word, for example, “it’s a trig function” “the sin and cos function oscillate between 1 and -1 but the tan function can go to infinity”, but the concept of trigonometry and the concept of functions appear quite distinct to P3. It is worth reiterating here that the relationship was not made explicit by the teacher and the students were not given any indication of the properties of function.

Functions had been part of the work covered the previous term, along with Algebraic processing skills and Equations and inequalities (see Mannall & Kenwood, (2000)

definition of a function is given as:

A mapping between two variables, usually called x, the independent variable, and y, the dependent variable. The function, or mapping, is given by f and written as f: x→y

(Ibid, p51)

This could explain P3’s reference to the notation. It may be noted however that P3 still does not appear to have an informal understanding of sin x as a function of x. She still thinks of it as operational tool or finding lengths and angles.

Figure 5.18 shown below shows P5’s second concept map.

P5’s second map (Figure 5.18) is consistent with his first (Figure 5.7) in that it appears to be a set of notes. It has maintained a grouped structure although the ratios and SOHCAHTOA have been put in a separate group this time. The triangles group is indicated and expanded indicating a strong association with triangles. The angles group now shows radians as well as degrees with notes stating that radians are ‘difficult to remember’ and its ‘easier in degrees’. Under Rules the sine rule and a version of the cosine have been defined. The graphs group shows an incorrect graph for ‘tangent’. There is no mention of the identities, the circle or the special angles; either represented spatially or as values.

after studying this component. He said:

It hasn’t really. What we learned in class was different to what we learned at GCSE, for example, CAST diagrams; I mean what is that all about. Trigonometry was always related to triangles before so when you started introducing circles I was lost. I remember the graphs from GCSE and we did some work on surds which were fine on their own but surds related to trigonometry were something else.

(P5)

The evidence of the two concept maps P5 has drawn appears to indicate that P5’s schema after studying this course on trigonometry is unchanged. This is supported by the evidence from the task questions and the follow up interview.