AsiaPacific Forum on Science Learning and Teaching, Volume 18, Issue 2, Article 16 (Dec., 2017) 
This section includes data obtained from themes of unit conversion and dimensional analysis collected from 66 students participating in this research.
Theme 1: Unit conversion
A unit conversion expresses the same property as a different unit of measurement. A conversion factor is a number used to change one set of units to another, by multiplying or dividing. When a conversion is necessary, the appropriate conversion factor to an equal value must be used (firefightermath.org).
On Table 2; the question "Is it correct that a constant quantity does not have a unit? Explain your answer" is given together with the analysis of responses from mathematics teachercandidates.
Table 2. Analysis of responses related to unit of constant quantity.
Comprehension Level
Encoding
f
%
FU
A constant quantity can have a unit


PU
Constant quantities have units. Accurate sampling on the subject
6
9.09
Constant quantities do not have units. Accurate sampling on the subject
1
1.52
MU
Constant quantities have units. Inaccurate sampling on the subject
45
68.18
Constant quantities do not have units. Inaccurate sampling on the subject
11
16.67
NU
Unanswered
3
4.54
None of the prospective teachers could reply to the first question. 10.61% of prospective teachers were able to answer with positive or negative responses, all of which were wrong, exemplifying their responses. The candidate teacher numbered [F40] responded with partial understanding as: "Wrong. Unit of the line constant "k" is=>F=kx^{2}=>N/m^{2}". The question was both exemplified and answered incorrectly with the rate of 84.85%. When misconceptions are cited, they are often given onesentence characterizations, as in the above (Capper, 1984). Such a sentence seems to presuppose the weaker sense of restructuring because the misconceptions are characterized simply as false beliefs that are highly resistant to tuition (Carey, 1986). Teacher candidate numbered [F23] replied the question as: “Wrong. Despite energy being a constant quantity, it has its unit which is joule.”, Teacher candidate numbered [F32] replied the question with misunderstanding as “Statement wrong. Quantities cover an area in space. Each space has a unit statement for indication." Teacher candidate numbered [F23] is in confusion about physical constants having or not having a unit, due to her response referring to conservative energy. Teacher candidate numbered [F32] displays a conceptual delusion by confusing the concept of matter with physical constants based on the word 'quantity' mentioned in the question.
Table 3 shows the analysis of the question "130 pm= ............ m".
Table 3. Analysis of responses related to conversion of basic units based on powers of ten.
Comprehension Level
Encoding
f
%
FU
13x10^{11} m
6
9.09
PU



MU
Conversion of lower units
26
39.39
Conversion of higher units
3
4.54
NU
Unanswered
31
46.97
In this question where mathematical knowledge related to unit conversion between higher and lower units by math teachercandidates, the level of success is unfortunately very low with the rate of 9.09%. Candidate teacher numbered [M7] responded the question incorrectly as “130 pm=……13.10^{8}m”; while also another candidate teacher numbered [M14] responded incorrectly as “130 pm= 13.10^{10}m”. If unit conversion information was successfully processed through working memory, it would have held in longterm memory. The transfer of schematic knowledge from controlled to automatic processing is another important factor in learning (Schneider & Shiffrin, 1977; Shiffrin & Schneider, 1977).
Table 4 demonstrates answers to the question "What is the density (g/cm^{3}) of a spherical object with a radius of 1x10^{2} µm, mass of 3x10^{19}mg?”
Table 4. Analysis of responses related to expression by a restricted unit of another unit.
Comprehension Level
Encoding
f
%
FU
(9/4π)x10^{22} g/cm^{3}


PU
Not knowledge on volume of a spherical object
1
1.52
MU
Nonconverted Units
16
24.24
Comparison of unit conversions
30
45.45
NU
Unanswered
19
28.79
It can be easily said that math teachercandidates were unsuccessful in expressing the answer to a restricted unit. The question was answered by a teachercandidate with partial understanding with correct conversion of units but calculating volume of the spherical object by using ¾πr^{3} equation. There were teachercandidates with the rate of ¼ who preferred to go ahead and calculate without converting the units. Error rate by teachercandidates during unit conversion is 45.45%.
Theme 2: Dimensional Analysis
The word dimension has a special meaning in physics. It denotes the physical nature of a quantity (Serway & Jevett, 2000). The premise of dimensional analysis is that the form of any physically significant equation must be such that the relationship between the actual physical quantities remains valid independent the magnitudes of the base units. Quantities that are clearly physically different (e.g. work and torque) may have the same dimension (Sonin, 2001).
Table 5 demonstrates the analysis of the question "A parachutist jumping from a plane is imposed to an air drag of . What is the dimension of the constant ? Explain your answer. "
Table 5. Analysis of responses related to expression by a restricted unit of another unit.
Comprehension Level
Encoding
f
%
FU
[M][T]^{1} or kg/s
3
4.54
PU
Not knowledge on Rules of Dimensional Analysis
4
6.06
MU
Not knowledge on SI Base Quantities
12
18.18
No dimension
1
1.52
Onedimensional
2
3.03
Threedimensional
1
1.52
Equal to gravitational acceleration
1
1.52
Confusion of dimension with volume
1
1.52
NU
Nonencodable or unanswered
41
62.12
Examining Table 5 shows that the question was answered with full understanding with a rate of 4.54%. During dimensional analysis, the minus in the formula was transferred and the fact symbols have no importance during dimensional analysis did confuse the teacher candidates with a rate of 6.06%. Math teachercandidates responded the question with misunderstanding with a rate of 27.29%. This question was answered by the teachercandidate numbered [M12] as "Dimension of the constant A must be equal to the volume of the parachutist", whereas by [M9] as "Dimension of the constant A is equal to gravitational acceleration and constant. Jumping parachutist cannot exceed a certain speed". It is seen that the dimensional analysis does not occur in the teachercandidate’s existing conceptual framework.
Table 6 includes the question of "Is the statement true that dimensional analysis can be used for a proportion constant in a mathematical equation? Explain your answer."
Table 6. Analysis of responses related to the knowledge on dimensional analysis and proportion constants
Comprehension Level
Encoding
f
%
FU
Cannot be used in dimensional analysis
1
1.52
PU
Incorrect explanation of reason
4
6.06
MU
Usable in dimensional analysis
46
69.70
NU
Nonencodable or unanswered
15
22.73
The teacher candidate numbered [F29] answered this question with full understanding as "Incorrect. Dimensional analysis is related to which unit should the result of a mathematical equation be expressed in". An incorrect answer from a teachercandidate is as follows [F4]: "Usable. As dimensional analysis is performed via units, it has an equivalent within the calculation of proportion constant. Everything can be revealed by dimensional analysis". The rate of unencodable responses to this question is 22.73%.
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