WP003 Applications of Spray Nozzles

Physics Education • October − December 2009 279

On Analysis of the Perceptions of Standard 12 Students Regarding a

Physics Concept Using Techniques of Quantum Mechanics

RAJENDRA VADNERE1 AND PRAVIN JOSHI2

1. School of Continuing Education, Y.C.M. Open University, Nasik, 422222, Maharashtra, India

E-mail: [email protected]

2. Department of Physics, H.P.T. Arts and R.Y.K. Science College, Nasik 422005, Maharashtra, India

E-mail: [email protected]

ABSTRACT

Many times it is observed in physics learning that students are not consistent in solving problems and they use contradictory ideas to answer similar questions. Also it is observed students exhibit different ‘mental model’ while answering different questions on same concept. Thus, students seem to be holding mixtures of different models on various concepts. The tools of the quantum mechanics can be used in analyzing the problem of the learning by a class of students. Some of such tools developed by Physics Education Research Group (PERG) at University of Maryland (UMd) are describe here. These techniques of analysis have been applied to a group of standard 12 Science students at Nashik (Maharashtra) district for on a concept ‘black body radiation’ from Quantum Mechanics. The results of this application have also been discussed in the present communication.

280 Physics Education • October − December 2009

It goes without saying that science has made dramatic progress in not only in bringing about revolutionary changes in our life styles but also in our understanding of the universe we live in. Particularly important in the latter aspect are the contributions of the two major developments in physics: namely the evolution of quantum mechanics and relativity theory. In this communication we would be concerned with the former.

Quantum Mechanics can be simply described as the physics of extremely small systems like molecules, atoms and nuclei. The tools of the quantum mechanics can be used in analyzing the problems of the learning by a class of students. Some of such tools1

developed by Physics Education Research Group (PERG) at University of Maryland (UMd) are described here. These techniques of analysis have been applied to a group of standard 12 Science students at Nashik (Maharashtra) district for an elementary course of Quantum Mechanics. The results of this application have also been discussed in the present communication.

The present communication is addressed to the researchers in physics communication. It shows how the novel techniques of the quantum mechanics can be used to analyze the mental models2,3 exhibited in the patterns of problem-solving. The techniques can be easily extended to the other disciplines of studies too.

TECHNIQUES OF ANALYSIS

Even though the technique of analysis using eigenvalue equations4,5 have been developed by the Physics Education Research Group at University of Maryland, the basic concepts have been described here for the benefit of those who are not familiar to the work. The understanding of basic mathematical tools

like solution of eigenvalue equations are required to understand the techniques described here.

1 Problem Space

Whenever a student is presented with a problem, (s)he goes through a complex web of mental activities.6 A simple model of these activities is summarized here.

The student reads the problem and tries to figure out which of the problem solving mechanism (which she may know) to apply. There may be more than one such problem solving mechanism available to her. In that case a random function may trigger selection of one of the competitive mechanisms. We say that the problem has taken the student to a ‘state’ of mind. We may classify such states of mind in the following scheme.

These problem solving mechanisms may involve a number of mental constructs. They are affected by the learning experience of the student. The term ‘mental model’ is used to denote such mental constructs.

The objective of education is to make the learner imitate the behavior of an expert in problem solving. This is done by presenting the necessary information on the facts and procedures of problem solving activities and is supplemented by presenting the students with a number of problem solving exercises. This is expected to invoke in the student a method or procedure which an expert in the field will resort to if challenged to undertake the given task. This situation is said to have evoked in the student an “Expert Model” state (or, E state for short).

On the other hand it is possible that the student has understood the problem, and sincerely undertakes to solve it. However, he may evoke a method of solving which may be close to the one which an expert would resort to but not exactly the same and thus lead to incorrect solution to the problem. This happens

when a number of concepts are similar in nature or when the student has not properly internalized a concept. Students of science subjects are known to use a certain models to solve the problems for a multitude of problems irrespective of the applicability of the given method due to prevalence of mental constructs described in literature7, 8 as phenomenological primitives (p-prism) and facets. In short when the student applies a mental model which is not appropriate to the given situation posed in the given problem we say that he has evoked a “misconception state” (M state) for the given problem.


It is also likely that the student has not understood the problem at all as expected by the experts and his methods are extremely removed from the methods and practices employed by the experts or he does not solve the problem at all. In such cases we say for the present communication, that he has evoked a “Null model state” (N state) for the particular problem.

In order to analyze whether a student has developed the skill of problem solving, a

number of multiple choice single response (MCSR) questions are posed to him. Each question evokes in him any one of the three states–namely, an expert state, a misconception state or a null model state. The options available to the stem are such that only one option corresponds to the expert state, while one or more choices would correspond to the misconception state or to the null model state. For a given problem the response given by the student tells us about whether he has acted like an expert or a novice or has almost not attempted it.

The analysis of the problems posed to the student gives us the probabilities of his evoking an expert model, a misconception model and a null model. The analysis may also give us the probability of mixed states. e.g., the E-M, M-N, and E-N or E-M-N mixed state. A similar analysis can also be performed on a class of student and we may find the probability of the class invoking an E, M or N model.

Figure 1

The state can be graphically represented as three-dimensional space with the three axes labeled as E, M and N. The vector drawn from origin to the a point (e, m, n) on the surface of a unit sphere (i.e. radius of unity) in the first octet of the space describes the ‘state’ such that a student or class applies E, M and N model with probabilities e2, m2 and n2 respectively (so that e2+m2+n2 =1). Thus a point (0.30, 0.40, 0.87) in Figure 1 denotes the situation for a student who may trigger E model with a probability (0.3)2 (i.e. 9%); M model with probability (0.4)2 (i.e. 16%) and N model with probability (0.87)2 (i.e. ~75%).

2 Operators and Vectors

In quantum mechanics the ‘state’ of a system is described by vector.9,10 A vector (e, m, n) described above represents a state of a student or a class of students. The state of system may get changed through operations. These may be described by a square matrix termed as an operator for the specific operation.

3 Eigenvalue Equations

Of particular importance is a transformation which does not change the direction of the vector. In such cases the transformed vector gets multiplied by a scaling factor λ where λ is a real number. For a square matrix A and vector (depicting state) ν, this is written as

Αν = λν

This equation is called as an eigenvalue equation. The solution to this equation would be a set of vectors vr which are called eigenvectors and a set of numbers (λr) called eigenvalues. In the present communication, the cases described have the real numbers as eigenvalues.

4 Student Response Vector

Let us now go back to our original problem of finding the state of a student when he attempts a problem. As described earlier, he may apply a model or method (a) which we would expect an expert to employ when faced with a similar situation (E state), or (b) mental model which is not appropriate to the given situation posed in the given problem we say that he has evoked a “misconception state” (M state), or (c) he has evoked a “Null model state” (N state) for the particular problem.

In such a case we may consider a kth student response who has been asked ‘q’ number of questions which are all multiple choice questions with ‘r’ choices for each of them. Suppose that he answers e number of questions correctly (Expert state); answers m number of questions has not properly internalized a concept (exhibiting Misconceptions) and answers n number of questions in wrong manner (exhibiting Null state). Then we can say that the probability of him in N state is n/q, while probability of finding him in E state is e/q and that for the M state is m/q. Note that q=e+ m+ n.

In quantum mechanics we speak about the probability amplitude vector such that the norm of such vector is proportion to the probability of finding the system in that state.

For a single student labeled k who has been asked q questions of multiple choices, the student state vector can be constructed as

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

qn

qm

qe

uk

The kth student model vector is represented with ku , where k = 1, 2, 3,…, N and the student ‘density matrix’ for kth student are defined as:


1 1k k k

e

D u u m e m nq q

n

⎡ ⎤⎢ ⎥

⎡ ⎤= = ⎢ ⎥ ⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

1k

e em en

D me m nq

ne nm n

⎡ ⎤⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦


m (1)

5 Class Density Matrix

We can construct class density matrix D, by taking average of student density matrix for the whole class comprising of N students as

D = 1

1 N

kk

DN =

⎛ ⎞⎜ ⎟⎝ ⎠

∑ (2)

The Student Model Density Matrix Dk, retains the structural information on individual student responses with respect to different physical models. Similarly, the class model density matrix stores important structural information about the class of students.

5.1 Three Types of Samples of Class Model Density Matrix:

In order to understand what the class density matrix tells us, we will consider three typical model conditions for a class of students. 1. Consistent one-model: When almost all the student in the class employ and exhibit same physical model (not necessary the correct one) and they are always consistent about it such situation is referred to as Consistent one-model. This situation is characterized by the class density matrix of the form

1 0 00 0 00 0 0

D⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

and with eigenvalues as 1, 0 and 0 and eigenvector are unit vectors (1, 0, 0), (0, 1, 0) and (0, 0, 1). 2. Consistent multi-model: Students in the class have several different physical models but each student has only one physical model and is consistent about it. Suppose, in a class of 100 students 60 students respond to questions exhibiting E state for all the questions asked, while 30 students respond with all questions in M state and 10 students answer all questions in a manner characteristic of N state. In such a case, the Class Model Density Matrix can be calculated by using the formulations given in the preceding section as

60 0 01 0 30 0

1000 0 10

D⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

= 0.6 0 00 0.3 00 0 0.1

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

The diagonal elements indicate the probability of finding a student with “pure” (i.e., consistent) state, in this case 0.6, 0.3 and 0.1 for E, M and N states respectively. It may be noted that the off diagonal elements are zero. The Consistent multi model situation is characterized by zero off-diagonal elements.

3 Inconsistent multi-model

This is the most pragmatic situation. The students in the class have different physical models and they are not consistent in using

these models. A typical class Density Matrix is of the form

D =

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

10.014.021.014.030.027.021.027.060.0

In this matrix, diagonal elements indicate probability of students in the ‘pure’ model i.e. 0.60, 0.30 and 0.10 corresponding to E, M and N states respectively. The off-diagonal elements indicate ‘mixing states’ (i.e., students not consistent about the states E, M or N) of the students using the different physical models in generating their responses.

5.2 Eigenvalues and Eigenvectors of Density Matrix:


Let us see how exactly the information of the individual students model states can be extracted from the class Density Matrix (D) and what the Eigenvalues and Eigenvectors of D represent.

The class model density matrix (using Eqs. 1 and 2) for a class of N students is:

1 1

1 1N N

k kk k

kD D uN N= =

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∑ ∑ u

Multiplying by vector |νμ⟩ on both side

1

1 N

k kk

D u uNμ μν ν

=

⎛ ⎞= ⎜ ⎟⎝ ⎠

∑ (3)

Let us denote the eigenvectors of D by[ν1, ν2,. . ., νμ,. . .,νw] (here w is the number of model states considered for analysis) where each of the elements here [νμ]

is a column vector and

their corresponding eigenvalues are 2μσ where

μ = 1, 2, …, w). Then eigenvalue equation vAv λ= can be replaced by

2D μ μ μν σ ν= (4)

Equating Eq (3) and (4)

2

1

1 N

k kk

D u uNμ μ μ μν ν σ ν

=

= =∑

5.3 Agreement Factor

Now let us define aμk is the agreement between

the kth student model vector and μth eigenvector.

| |k k ka u uμ μ μν ν= =

Then Eq. (3) and (4) can be written as

2

1

1 N

k k kK

D aNμ μ μν μ σ ν

=

= ⋅ = ⋅∑

21

1 1 N

k kK

aNμ μ

μ

ν μσ =

⎛ ⎞= ⎜ ⎟⎝ ⎠

∑ ⋅ (5)

Thus from Eq(5), an eigenvector νμ of D is

a weighted average of all individual student model vectors uk with weight equal to the agreements aμk between the eigenvector νμ and the single student model vector uk. Therefore the class model states represented by these eigenvectors are the set of [ν1 . . . νw] states that most resemble the salient features of all the individual student model vectors. From Eq(5), it is also obvious that the structure of | νμ⟩

will have more contribution from students

model vectors | uk⟩ that are closer to | ν Therefore if there exist a group of | u

μ⟩. k⟩’s that are very similar to each other but different

from the rest this group of | uk⟩’s will have a significant effect to make one of the eigenvectors (| νμ⟩’s) similar to them.

If we left multiply by ⟨νμ| to Eq(5) recall that

⟨νμ|νμ⟩ = 1

and

aμk = ⟨uk|νμ⟩ = ⟨νμ|uk⟩

21

1 1|N

k kk

aNμ μ μ μ

μ

ν νσ =

⎛ ⎞= ⋅⎜ ⎟⎝ ⎠

∑ ν μ

= 22

1

1 1 1N

kk

aN μ

μσ =

⎛ ⎞ =⎜ ⎟⎝ ⎠

∑

Thus

2

1

1 N

kk

aNμσ

=

⎛ ⎞= ⎜ ⎟⎝ ⎠

∑


2μ ⋅ (6)

This result indicates that the μth eigenvalue (

2μσ ) is the average of the square of the

agreements between the μth eigenvector νμ and the individual student model vector uk.

It can be concluded from Eq(6) that the eigenvalue is affected by both the similarity of the individual student model vector uk and the number of students with similar model vectors. In order to have a large eigenvalue required to have not only large 2

kaμ but also a good

number of them, which implies that more students in the class have more similar student model vectors. That is, the students in the class behave more similarly to each other (the consistency between students is high).

As we define earlier aμk is the agreement between the kth

student model vector and μth eigenvector. This agreement factor gives the degree of agreement for individual student with the class model states. Similarly it is necessary to find how the class model states itself is shifted from the physical model. Since the class model states represent the probability amplitude, the probability based agreement between a physical model and a student class model state is defined as the square of the scalar product between the two model vectors.

22 | eμη μ ην ν= (7)

where eη is the base vector (e.g., (0, 1) or (1, 0) for a two-model analysis) representing the ηth physical model.

Thus, each component of eigenvector νμη shows the agreement between the class model states and the physical models.

APPLICATION TO ANALYSIS OF QUANTUM TOPICS

The technique described above has been applied to analyze the effect of a digital multimedia package on the distribution of students’ models for a class of standard 12 students from Nashik district in Maharashtra, India.

A multimedia learning package has been developed by the authors on the quantum mechanics concept of black body radiation.11,12 It consisted of the discussion on heat transfer, electromagnetic radiation, wave principles, black body, Ferry’s black body, Lummer and Pringsheim experiment, theoretical explanation of the experiment, Wein law, Maxwell’s theory, Planck’s quantum hypothesis and equation. The use of simulation13 and use of audio was considered to be useful in the effective communication of the complex and abstract ideas.

In order to study the effectiveness of the multimedia learning package, 119 students with twelve years of schooling (i.e., standard 12 students) volunteered in the study. These students were administered a pretest. The pretest comprised of a test with the maximum possible score of 75 points. The distribution of the points with respect to the various themes, sub-themes, nature of question and objective of the question (as per Bloom’s taxonomy14) is presented in Table 1.

Table 2: The distribution of the points with respect to the various themes, sub-themes, nature of question and objective of the question (as per Bloom’s taxonomy) for the posttest

conducted on Control and Experimental Groups

A pie chart showing the distribution of the points allotted according to the objectives of the questions as per the Bloom’s taxonomy is shown in the Figure 6.

Table 1

Sr. No.

Theme Sub-theme Type of question Bloom's Taxonomy Category

Weight (Points)

1 Heat Heat transfer Select proper choice

Application 8

Tick on proper choice

Knowledge 22 2 EM Wave, Light and Sound

Medium, Energy, Velocity, Temperature Select proper

choice Knowledge 2

3 Radiation Heat transfer Describe Comprehension 16

4 Ferry's Black Body

Heat transfer Reasoning Synthesis 9

5 Black Body Absorption and Emission

Compare Analysis 12

6 Wien's law Displacement law Solve example Application 6

Total 75

Figure 5: Pie chart showing the distribution of the pretest weights

as per the Bloom’s taxonomy.


Table 2

Sr. No.

Theme Sub-theme Type of question Levels of Bloom's Taxonomy of the Cognitive domain

Weight (Points)

Select proper choice (out of 5)

Analysis 7 1 Ferry's Black Body

Conduction, Convection and Radiation Reasoning Synthesis 7

2 General Select proper choice (out of 3)

Knowledge 3

Select proper choice out of 4, 5

Analysis 4 3 Black Body Absorption and Emission

Reasoning Synthesis 4

4 Lummer and Pringsheim Expt.

Function of Apparatus

Match the pairs Applications 8

Show on the Graph Knowledge 3 5 Black Body Radiation

Nature of Graph

Select proper choice out of 2, 3

Knowledge 8

Explain and Comprehension 1 6 Black Body Radiation

Modes of Vibration Use Application 1

7 Maxwell’s Theory

Radiation Explain Comprehension 2

Displacement law State and Knowledge 2 8 Wien's law

Distribution law Explain Comprehension 2

9 Black Body Radiation

Ultraviolet Catastrophe

Explain Comprehension 1

Total 53

Figure 6. Pie chart showing the distribution of the posttest weights as

per the Bloom’s taxonomy.


The data so gathered was analyzed using the techniques described in the preceding sections. Similar to the pretest, the possible choices to the questions were labeled as E, M or N depending on whether the choice exhibit a pattern characteristic of an expert (E) or that of a person with misconception(s) (M) or that of a person exhibiting a null (N) state. The data was

analyzed using SciLab15 software. The eigenvalues and eigenvectors for the class density matrices were obtained using this analysis.

Table 3 summarizes the results of the analysis of the class density matrices for the pretest and those for posttest on Control and Experimental groups.

Table 3: Summary of the results of the analysis based on eigenvalue equations for class density

matrices for pretest and posttests for control and experimental groups.

Pretest Posttest

All students (119) Control Group (59 students) Experimental Group (60 students)

Density matrix 0.35 0.17

0.35 0.130.17 0.13

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

0.460.36

0.18


0.37 0.080.10 0.08

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

0.480.43

0.08


0.38 0.040.08 0.04

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

0.640.32

0.04

Eigenvalues

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

11.000.000.000.005.000.000.000.0

Eigenvalues

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

06.000.000.000.009.000.000.000.0

Eigenvalues

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

03.000.000.000.007.000.000.000.0

Eigenvectors

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−

93.021.036.070.009.069.0

Eigenvectors

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−

−−−

93.032.019.073.033.061.0

Eigenvectors

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

−

96.026.016.082.023.051.0

It is seen from the analysis that for the pretest, the class density matrix shows significantly large off-diagonal elements. This is characteristic of inconsistent multi-model. Thus the students, in general have inconsistent behavior. A student is likely to trigger the expert behavior or exhibit misconception state or the null state with significant probabilities. It may be seen from the diagonal elements of density matrix that, a student is likely to trigger the E state with a probability of 0.46, while the

probability of getting into M and N states are 0.36 and 0.18 respectively.

Further, the eigenvectors tell us about three states which may be the candidates exhibiting the state for a representational student. The eigenvectors for the class density matrix for the pretest are (0.72, 0.62, 0.31); (0.69, -0.70, -0.21) and (-0.09, -0.36, 0.93) with eigenvalues 0.83, 0.05, and 0.11 respectively. The maximum eigenvalue is for the first of the eigenvectors. This means that the most likely representational student is gets into the E, M



and N states with a probability of 0.52, 0.38, 0.10 respectively, which are the squares of components of (0.72, 0.62, 0.31). A measure of probability of finding this person is 0.83.

The density matrix of the posttest for the control group shows fairly significant off-diagonal elements. This means that there is significant mixing of the states. Again the density matrix indicates inconsistent multi-model. The diagonal elements of the density matrix give the probability of triggering E, M and N states as 0.48, 0.43 and 0.08 respectively. There is a significant decrease in the probability of the null state. This shows that the students have taken the tests more seriously in comparison to the pretest. The rise in the probability of triggering the E state is not very significant.

The eigenvectors for the posttest on control group indicate that the most likely candidate for a representational student is (-0.72, -0.67, -0.16) with dominant eigenvalue of 0.85. Again there is a significant decrease in the component corresponding to the null state in the dominant eigenvector.

The density matrix for the posttest for the experimental group is again indicative of the inconsistent multimodal, as the off-diagonal elements are non-zero. The diagonal elements show the probability of triggering E, M and N states as 0.64, 0.32 and 0.04 respectively. In comparison to the pretest as well as to that for the posttest (control group), the increase in the probability of triggering E (expert) state is significantly high. The decrease in the probability of triggering the null state is remarkably high with respect to that for pretest and also for the posttest for control group.

The eigenvalues and eigenvectors for the posttest in experimental group show that the dominant eigenvector is (0.83, 0.55, 0.10) with an eigenvalue of 0.90. This indicates and corroborates the improvement in performance of the students in the experimental group vis-à-vis the pretest or posttest (control group) with

significant rise in the E state component of the dominant eigenvector and also in the gain in the eigenvalue of the dominant eigenvalue.

Conclusion

The analysis of the data collected on performance of 119 student-volunteers shows that the multimedia learning package developed by the authors was successful in improving the mental models of the volunteers. It shows that the measure of the probability of triggering the expert model in the learners who underwent the multimedia package improved from 0.46 for the pretest to 0.64 for the posttest on the experimental group. This improvement is significantly large in contrast to the improvement in the control group from 0.46 to 0.48. The probability of triggering a null state in experimental group (0.04) is reduced significantly in comparison to the pretest (0.18) and the posttest for control group (0.08). Acknowledgements

We would like to thanks D. S. Wagh who encourage students to participate in the workshop.

References 1. Redish E. F. (2002), A Combined Effort of the

Physics Department and School of Education, University of Maryland Physics Education Research Group (PERG) www.physics. umd.edu/perg/ [viewed 8 Apr 2005].

2. Gentner D. and Stevens A. L., (1993) Mental Models, LEA, 1993.

3. Bao L., Jolly P. and Redish E. F., (1996) Student Mental Model of Conductivity, AATP Announcer 26(2), 70-71 (July 1996).

4. Howard A. (1987), Elementary Linear Algebra. 5th Ed. New York: John Wiley & Sons, New York, pp 301-328.


5. Wikibooks, (29/12/2006) The open-contest textbook collection http://en.wikibooks.org/wiki/Linear_Algebra/Eigenvalues_and_eigenvectors [viewed on 1/11/07].

6. Bao L., (1999) Dynamics of Student Modeling: A Theory, Algorithms and Application of Quantum Mechanics. Ph. D. Thesis, University of Maryland.

7. A. diSessa and B. L. Sherin, (1998) “What changes in conceptual change?”, Int. J. Sci. Educ. 20, 1155 (1998).

8. S. Vosniadou, (1994) “Capturing and modeling the process of conceptual change”, Learning and Instruction, (4), 45-69, 1994.

9. Claude Cohen-Tannoudji, Bernard Diu and Frank Laloe (1977), Quantum Mechanics Volume (1) (English Translation), Wiley Interscience Publication, John Wiley & Sons, Paris.

10. B. Cameron Reed (1994), Quantum Mechanics: A First Course. 2nd Ed. Canada: Wuerz Publishing Ltd.

11. Singhal S. S., Agarwal J. P. and Satya Prakash (1970), Heat and Thermodynamics, U.P., Meerut, Pragati Prakashan.

12. Michael Fowler (1999) Physics 252: Modern Physics http://galileo.phys.virginia. edu/ classes /252/ [viewed 16/4/06].

13. University of Colorado (2009) Interactive Science Simulations http://phet.colorado.edu/index.php [viewed 21/8/2009].

14. Bloom, B.S. and Krathwohl, D. R. (1956) Taxonomy of Educational Objectives: The classification of educational goals, by a committee of college and university examiners. Handbook I: Cognitive Domain. NY, NY: Longmans, Green. Last modified: February 16, 2005 The University of Mississippi. http://www.olemiss.edu/depts/educ_school2/docs/stai_manual/manual8.hmt [viewed 19/1/2006].

15. SciLab-Version 5.1.1 Consortium Scilab (DIGITEO), Copyright (c) 1989-2009 (INRIA), Copyright (c) 1989-2007 (ENPC) [Computer Program] http://www.scilab.org.

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WP003 Applications of Spray Nozzles