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Asia-Pacific Forum on Science Learning and Teaching, Volume 19, Issue 2, Article 14 (Dec., 2018) |
Findings obtained from the questionnaire regarding vectors were grouped and interpreted under five themes: (1) the length of vector, (2) vector addition, (3) subtraction, (4) scalar and (5) vector product.
Physical quantities are categorized in two as those which only have magnitude with an appropriate unit, and those which have both magnitude and direction (Adams, Bogduk, Burton and Dolan, 2006). Mass, time, volume, temperature and energy can be considered as simple examples to those which only have magnitude. Vectors are a method of expression used for quantities which have direction in addition to magnitude. Displacement, velocity, acceleration, force and momentum are examples of vector quantities. It was investigated whether teacher candidates could correctly compare the magnitude of vector with the first question, 12.12% of all the math teacher candidates gave the exact correct answer; whereas 74.24% displayed partial answer. It is also more or less equal the percentages of partial answer for female and male teacher candidates. A teacher candidate [F20] for example is in confusion about the magnitude of vectors and the vectors in terms of their illustration, stating: " A, D, F, G vector magnitudes are equal. E, H, I vectors are equal". Four vectors A, D, F and G can be defined to equal if they have the same magnitude and if they show the same direction. But, the magnitudes of vectors are equal only if they are same length, which have to represent |A| = |D| = |F| = |G|. The teacher candidate [F20] knows the magnitudes of the vectors but does not know the representation of vector magnitude. It is different to representation of vector magnitude and vector. This information that should have been successfully processed through working memory is held in long-term memory (Carlson, Chandler and Sweller, 2003). But, when vector information was presented to [F20], it could have not been successfully processed through working memory. When designing instruction, the mental load of [F20] may be exceed limits of her working memory. Then, it can be said that she has an imperfect schema for vector . 13.64% of the students gave wrong answers to this question. Teacher candidate [F 8 ] coded with gave wrong answer as " A, E, D, G, F magnitudes of vectors are equal" . A partial understanding example of the teacher candidate coded [M6] response is given in Fig.1. All teacher candidates attempted to give an answer to this question; no blank, repetitive, or unrelated answers were encountered.
Figure 1. A partial understanding example of the teacher candidate response to the length of vector.Table 1. Analysis of the question themed the length of vector
Comprehension Level
Encoding
f(%)
Female/Male (%)
FU
. For vectors whose magnitudes are equal, it has to written |A| = |D| = |F| = |G| and |E| = |H| = |I|.
8 (12.12)
14 / 6
PU
. Vectors whose magnitudes are equal are known correctly, but it is used to wrong handwriting style.
49 (74.24)
74 / 75
MU
. Vectors which are same length, sense and direction are equal.
. The magnitude of vector is defined on a line.
. When vector is turned opposite direction, the sign of it changes.9 (13.64)
12 / 19
NU
. Non-encodable or unanswered
-
-/-
Vectors may be collected graphically or analytically ( Halliday, Resnick and Walker, 1993). Both methods are being taught to each student participating in physics lectures on undergraduate levels, and it is always emphasized that it would be more practical to prefer analytical method as the number of vectors increase. In this study, a question is given about addition of two vectors using graphical method in two dimensions. Graphical methods for adding two vectors are known as the triangle and parallelogram rule of addition (Radi and Rasmussen, 2013).
While two vectors are adding, it is preferred with triangle law of vector addition by 51 math teacher candidates. 15.15% of the students were able to answer the question with partial understanding. Seven out of ten students answering the question with partial understanding added the vector with triangle law, but drew opposite direction for the resultant vector. The percentage of female math teacher candidates is bigger than the percentage of male math teacher candidates on full understanding level. All of the teacher-candidates were able to give an answer to the second question.
77.27% of the math-teacher candidates gave the correct answer to this question as showed Table 2.
Table 2. Analysis of the question themed vector addition
Comprehension Level
Encoding
f(%)
Female/Male (%)
FU
. Two vectors are added by using the triangle method of addition.
. Two vectors are added by using the parallelogram method of addition.51 (77.27)
82 / 62
PU
. The resultant vector R is drawn from the head of the second vector to the tail of the first vector.
. Two vectors are added, but the resultant vector is only drawn a line10 (15.15)
14 / 19
MU
. The tails of two vectors are superposed and then the resultant vector is drawn from the head of the first vector to the head of the second vector.
. Not given two vectors are added.
. The first vector and the negative of the second vector are added.5 (7.58)
4 / 19
NU
. Non-encodable or unanswered
-
-/-
In the process of vector subtraction, it is the method to utilize from the definition of the negative of a vector. The operation of A - B is defined as vector -B added to vector A.
A - B = A + (-B)
It is possible to take a look at vector subtraction from another perspective: The difference A - B of two vectors such as A and B, is a vector that needs to be added to the second vector in order to obtain the first. In this case the vector A - B is the vector drawn from the head of the second vector towards the tip of the first (Serway and Jevett, 2008).
Table 3. Analysis of the question themed vector subtraction
Comprehension Level
Encoding
f(%)
Female/Male (%)
FU
. It is drawn the difference vector that is from the head of the K vector to the tip of the R vector .
. The vector R and vector -K are added, and the difference vector is drawn from the tail of the vector R to the head of the negative vector K.43 (65.15)
72 / 44
PU
. The vector R and -Kvector are added, but the difference vector is drawn from the head of the vector R to the tail of the negative vector K.
. The negative vector K is named as the vector K.
. The difference vector is drawn opposite direction.
. The difference vector is drawn a line.8 (12.12)
6 / 32
MU
. Two vectors are added instead of subtract.
. The vector K and the vector R are combined head to head.13 (19.70)
22 / 12
NU
. Non-encodable or unanswered
2 (3.03)
- / 12
It is seen that the problem was answered with a rate of 65.15% with full understanding (Table 3). In partial understanding level, the rate of female math teacher candidates is quite smaller than male math teacher candidates. Eight students performed subtraction with partial understanding. The most common conceptual delusion in vector subtraction is found the negative of the vector. While 19.70% of the teacher candidates provided wrong answers to the subtraction, 3.03% of them gave unaccountable answers.
Scalar product of the vectors of A and B is expressed as A.B , this product is also known as dot product. A full understanding example of the teacher candidate coded [M16] response is given in Fig. 2. It is defined as A.B = |A||B|cos(A, B). The scalar product of two vectors is a scalar quantity. The combination of |A||B|cos(A, B) occurs frequently in physics class. The geometric significance of the inner product A.B is also similiar from the standard vector scalar product ( Hestenes, 2002).
Table 4 shows that only 6.06% of the teacher candidates answered the question with full understanding. With partial understanding, the question was answered with the rate of 22.73%. The teacher candidate numbered [M 7 ] was able to answer the question as: "S.T > S.U > S.V due to the increase of angles between them"; however, he confused the angle itself with the cosine of the angle determining the multiplication result in scalar product. Teacher-candidate numbered [M 13 ] answered the question as: "S.T > S.U > SV Adding them tail of the first one to the tip of the last one, I chose the ones with larger cross vectors" ; displaying conceptual delusion about scalar product and vector addition.
Figure 2. A full understanding example of the teacher candidate response to the scalar product.
Table 4. Analysis of the question themed scalar product of vectors
Comprehension Level
Encoding
f(%)
Female/Male (%)
FU
S.T >S.U>S.V is expressed, since S.T = |S|.|T|cos(S, T). The obtained scalar quantity is proportional with the cosine of the cos(S, T).
4 (6.06)
4 /12
PU
. The result of scalar product is ordered from the biggest to the smallest. But, the result does not explain.
. The result of scalar product is ordered from the biggest to the smallest, the result depends on the angle between two vectors.15 (22.73)
16 / 44
MU
. The magnitudes of scalar products of two vectors can not be ordered truly.
. The each value that is obtained the scalar product of any two vectors are equal the others.36 (54.54)
67 / 19
NU
. Non-encodable or unanswered
11 (16.67)
14 / 25
Table 4 shows the ratio of 54.54% incorrect answers to the question, which the teacher candidate numbered [F3] answered as: "If their magnitudes are equal, their scalar product is also equal. The scheme with scalar product or inner product in the teacher candidate coded [F3]' memory is faulty. 11 teacher candidates did not answer this question.
Given any two vectors A and B, the vector product AxB is defined as a vector, which has a magnitude of |A||B|sin(A, B), where sin(A, B) is the angle between A and B.
Table 5. Analysis of the first question themed vector product
Comprehension Level
Encoding
f(%)
Female/Male (%)
FU
The direction of the vector MxN is drawn as perpendicular to the plane that contains both M and N.
-
- /-
PU
The vector direction is determined as perpendicular, but its sense is opposite.
1 (1.52)
2 / -
MU
It is drawn perpendicular vector. But it is named as the magnitude of vector product.
. A vector is drawn in plane.
. It is calculated the magnitude of vector product.
. The vector product of two vectors is expressed that is equal to the product of the magnitudes of two vectors and the cosine of the angle between them.32 (48.48)
52 / 38
NU
. Non-encodable or unanswered
33 (50)
46 / 62
Table 5, it is seen the level of understanding and frequencies about the vector MxN, where the direction of vector is to use the right-hand rule. The four fingers of the right hand are pointed along M and then "wrapped" into N through the angle between M and N. The direction of the upright thumb is the direction of vector. None of the teacher-candidates were able to achieve the correct vector direction to be obtained from the vector product. The geometric significance of the outer product MΛN should also be familiar from the standard vector product MxN which is not commutativity (Hestenes, 2002). Only one of the teacher-candidates was able to draw the orientation of the vector, but displayed a conceptual delusion. She identified the vector direction incorrectly. The vector product is not commutative and the order in which two vectors are multiplied in across product is important. The teacher candidate numbered [F5] answered incorrectly the question given as "...draw the vector:"||M.N||.cosQ" (as shown Fig. 3). The teacher candidate coded [F5] is tried to use knowledge of algebra. This product should have also known as outer product with helping algebra curriculum. Unfortunately, both the schema of vector product and the transfer of algebra knowledge do not occur as expected. However,33 teacher candidates did not answer this question.
Figure 3. A misunderstanding example of the teacher candidate response to the vector product.
Table 6. Analysis of the second question themed vector product
Comprehension Level
Encoding
f(%)
Female/Male (%)
FU
.|SxV| > |SxU| > |SxT| is expressed, since |SxV| = |S||V|| sin(S, V). The obtained magnitude of vector product is proportional with the sine of the angle.
1 (1.51)
2 / -
PU
. The results of vector product are ordered from the biggest to the smallest. But, the result does not explain.
. The result of vector product is ordered from the biggest to the smallest, the result depends on the angle between two vectors.7 (10.61)
12 / 6 -
MU
. The magnitudes of vector products can not be ordered truly.
This operation is an inner product.
. The each value that is obtained from the vector product of any two vectors are equal the others.
The magnitude of vector product is proportional with the cosine of the angle between two vectors.38 (57.58)
62 / 44
NU
. Non-encodable or unanswered
20 (30.30)
24 / 50
Table 6, it is showed details based on level of understanding of responses given to the question that examines vector magnitude relying on angle to be obtained from vector product. The form of vector product employs the sine of the included angle instead of the cosine. It can be seen that one student answered the question with full understanding. With partial understanding, the question was answered with the rate of 10.61%. Student numbered [F7] answered the question with partial understanding as " |SxV| > |SxU| > |SxT| the angle in between is increased ", displaying a delusion about the quantity of vector product increasing with itself, not with the sine of the angle. The conventional vector product AxB is implicitly defined as the dual of the outer product (Hestenes, 2002). It is expressed outer product MΛN = i|M||N|sinθ by Hestenes (2002). Unfortunately, it could not accomplish high road transfer defined Salomon and Perkins (1989). 57.58% of the teacher-candidates gave incorrect answers to this question. Student coded [F6] gave an incorrect answer to this question as " the lengths of |SxT| > |SxU| > |SxV| would be square of the resultant vector, the one with the highest value will be the longest" 30.30% of the teacher-candidates were not able to answer this question.
