The content from the three maps drawn by these students is shown in the tables below. The bold type indicates where content has been repeated.
Table 7.7: S1’s Concept Map Content.
Table 7.8: S2’sConcept Map Content.
S1
Map 1
Map 2
Map 3
Core
properties
Sin = opp/hyp etc
Tan x =Sin x/ Cos x Sin = opp/hyp etc Tan x =Sin x/ Cos x Sin²θ+Cos²θ=1
1/sin x=cosec x, 1/cos x=sec x, 1/tan x =cot x
Tan x=sin x/cos x
Formulae
Sin ruleCos rule Pythagoras Sin rule Cos rule L=rθ A=1/2r²θ
Values
Surd or decimal values for Sin, Cos and Tan 0, 30, 60, 90° and tan 45°π=180° Surd values for 60° and 30°
Spatial
images
Sin, Cos and Tan
graphs CAST circle Sin, Cos and Tan graphs Cosec, sec and cot graphs CAST circle
Special angles triangle
S2
Map 1
Map 2
Map 3
Core
properties
Sin =O/H etc Tan =Sin /Cos
Sin =O/H etc Sin x /Cos x =Tan x Sin²θ+Cos²θ=1
Sin =O/H etc Tan θ =Sin θ /Cos θ Sin²θ+Cos²θ=1 Cosec θ=1/sin θ etc
Formulae
Sin ruleCos rule Pythagoras Pythagoras Sin (θ+φ) Cos(θ+φ), cos (θ-φ) Tan (θ+φ)
Values
π=180° Surd values for 30°, 60°, 45°Spatial
images
Sin, Cos and Tan
graphs Sin, Cos and Tan graphs CAST circle
Sin, Cos and Tan graphs CAST circle
Special angle triangles Cosec & Sec graph
Table 7.9: S3’s Concept Map Content.
Each of the three students now added the secant, cosecant and cotangent identities and S2 and S3 added the formulae for sin, cos and tan (θ±φ). Therefore there is an increase in the core properties mentioned by each of these students.
S1 and S2 added the sec, cosec and cot graphs to their spatial images and the special angles triangles. So there is an increase in the spatial images that S1 and S2 have included since their previous maps. This is not the case with S3.
S2 and S3 have included the double angle formulae. None of the students have made any mention of RCos (θ+α).
S1 and S3 have not indicated any kind of structure but merely recorded a list of items. When S1 was asked about his map he said:
It is just what comes into my head... to help me remember.
(S1)
S2 has shown some indication of a structure but there is no evidence here of the development of an interconnected structural concept as described by Sfard (1991) [§3.1.4] or an object construction as described in the APOS theory [Dubinsky (1991) §3.1.5].
S3
Map 1
Map 2
Map3
Core
properties
SOHCAHTOA
Sin /Cos =Tan SOHCAHTOA Tan x =Sin x/Cos x SOHCAHTOA Tan x =Sin x/Cos x Cot=1/tan,Sec=1/sin (sic) Cosec =1/cos (sic)
Formulae
Sin rule Cos rule PythagorasCos (θ+φ), Cos(θ-φ)
Sin(θ+φ), Sin (θ-φ) Tan(θ+φ), Tan (θ-φ)
Values
Sin 45/Cos45=Tan 45=1 Sin& Cos 30, 60, 90° Sin 120° Cos 0°
2π rad =360°
Spatial
images
Sin graph
Incorrect Cos and Tan graphs
Sin,Cos graphs Approximate Tan graph 4 CAST circles
Sin, Cos graphs CAST circle
Other
Quadratic formulaSin 45=Cos45=√3/2 ? Cos x =1-sin x (sic) Sin x + sin (90-x)=1 (sic)
development of an interconnected schema. For example here are the first and third maps of one of the other students in the group (S6) who also studies Further maths.
Figure 7.13: S6 Concept Map 1
The 1st map is comparable with the maps drawn by the all the students initially with
regard to, both the content indicated, and the way that the items mentioned appear to be discreet and unconnected. This is the ‘spider’ style that was predominant at the start of the course.
The 3rd map shows increased content and a variety of concept images in the form of
visual spatial representations. The student has clearly attempted to organise the content as evidenced by the way in which items such as the graphs or identities are grouped together within the overall schema. However the arrows indicate that the different components are interlinked. Again some of the items originally included in the 1st map have been omitted in the 3rd map. These items are the formulae for the
sine rule and for Pythagoras theorem. When interviewed about these omissions the student said that he didn’t include the sine rule because
I know it and can use it if I need to. (S6)
This points to an interiorisation of process [§3.1.4]. He said he didn’t include Pythagoras theorem because:
Although it is used in trigonometry it’s not really about trigonometry because it’s about lengths and not angles. In the beginning
trigonometry was just about triangles, like finding lengths and angles but now it’s about angles; angles in triangles and in circles... anywhere really.
(S6)
This points to a schema reconstruction.
Another example that may indicate the development of a schema that is organised is shown here. This student started to study Further Maths but dropped it.
Figure 7.15: S7’s Concept Map 1
Again the first concept map is drawn with the items radiating from the centre and it shows little indication of a concept schema that is richly interconnected.
The third mapshows clearly an organisational structure with the words sine, cos and tan placed as centres or nodes to which their relevant graphs, identities and values are linked. However it may be observed that the content appears to be focused upon items that this student feels the need to remember. This observation was borne out by this student’s response when asked why he had dropped further maths. He said: