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Ф1ЗИКО-МАТЕМАТИЧНА ОСВ1ТА (ФМО)
випуск 3(21), 2G19
Scientific journal
PHYSICAL AND MATHEMATICAL EDUCATION
Has been issued since 2013.
Науковий журнал
Ф1ЗИКО-МАТЕМАТИЧНА ОСВ1ТА
Видасться з 2013.
http://fmo-journal.fizmatsspu.sumy.ua/
Воскоглоу М. Представлення модел'1 "5 E" за допомогою ланцюга Маркова. Ф'!зико-математична осв'та. 2019. Випуск 3(21). С. 7-11.
Voskoglou Michael Gr. A Markov chain representation of the "5 E's" instructional treatment. Physical and Mathematical Education. 2019. Issue 3(21). Р. 7-11.
DOI 10.31110/2413-1571-2019-021-3-001
Michael Gr. Voskoglou
Graduate Technological Educational Institute of Western Greece, Greece voskoglou@teipat.gr; mvoskoglou@gmail.com
A MARKOV CHAIN REPRESENTATION OF THE "5 E's" INSTRUCTIONAL TREATMENT
ABSTRACT
Formulation of the problem. The socio-constructive theories of learning have become very popular during the last decades for teaching mathematics. The "5 E's" is an instructional model based on the principles of social constructivism that has recently become very popular, especially in school education, for teaching mathematics. Each of the 5 E's describes a phase of learning which begins with the letter "E" - Engage, Explore, Explain, Elaborate, Evaluate. Depending on the student reactions, there are forward or backward transitions between the three middle phases (explore, explain, elaborate) of the 5E's model during the teaching process. The "5 E's" model allows students and teachers to experience common activities, to use and build on prior knowledge and experience and to assess their understanding of a concept continually.
Materials and methods. Probabilistic methods of analysis are used.
Results. The mathematical representation of the "5 E's" model is attempted by applying an absorbing Markov chain on its phases. A Markov Chain (MC) is a stochastic process that moves in a sequence of steps (phases) through a set of states and has a one-step memory. A finite MC having as states Si the corresponding phases Ei, i = 1, 2,.., 5, of the "5 E's" instructional model is introduced. A classroom application is also presented illustrating the usefulness of this representation in practice. The following application took place recently at the Graduate Technological Educational Institute of Western Greece for teaching the concept of the derivative to a group of fresher students of engineering.
Conclusions. The Markov chain representation of the "5 E's" model provides a useful tool for evaluating the student difficulties during the teaching process. This is very useful for reorganizing the instructor's plans for teaching the same subject in future.
KEY WORDS: constructivism, socio-cultural theories for learning, "5 E's" instructional model, absorbing Markov chains.
INTRODUCTION
The "5 E's" instructional treatment. Mathematics teaching is intended to promote the learning of mathematics. However, while theory provides us with lenses for analyzing learning, the position of mathematics teaching remains theoretically anomalous and underdeveloped. We might see one of the problems to lie in the relationships between learning, teaching and the practice of teaching. Theories help us to analyze, or explain, but they do not provide recipes for action; rarely do they provide direct guidance for practice. The application of the socio-constructive theories for learning to teaching mathematics has started during the 1980's, when the failure of the introduction of the "new mathematics" to the school curricula had already become more than evident to everybody.
The idea that knowledge is a human construction supported by the experience, first stated by Vico in the 18th century and further extended by Kant, affected greatly the epistemology of Piaget, who is considered to be the forerunner of the theory of constructivism for the process of learning. This theory appeared formally by von Clasersfeld who developed his ideas in the Piaget foundation of USA in 1975 (Glasersfeld, 1987). The constructivist approach is based on the following two principles:
- Knowledge is not passively received from the environment, but it is actively constructed by synthesizing past knowledge and experience with the new information.
- The "coming to know" is a process of adaptation based on and constantly modified by the individual's experience of the world.
On the other hand, the socio-cultural theories for learning are based on the Vygotsky's ideas claiming that knowledge is a product of culture and social interaction. Learning takes place when the individuals engage socially to talk and act about shared problems or interests (Elbers, 2003; Jaworski, 2006; Wenger, 1998). The Communities of Practice are groups of people (experts or practitioners in a particular field) who share a concern for something they do and learn how to do it better as they interact regularly, having therefore the opportunity to develop themselves personally and professionally (Goos, 2014;
ISSN 2413-158X (online) ISSN 2413-1571 (print)
Voskoglou, 2019; Wenger, 1998). The combination of the constructivism with the socio-cultural ideas is known as social constructivism (Driver et al., 1994; Jaworski, 2006).
The "5 E's" is an instructional model based on the principles of social constructivism that has become recently very popular, especially in school education, for teaching mathematics (Enhancing Education, 2019). Each of the 5 E's describes a phase of learning which begins with the letter "E". Those phases are the following:
- Engage (Ei): this is the starting phase, which connects the past with the present learning experiences and focuses student thinking on the learning outcomes of the current activities.
- Explore (E2): during this, phase students explore their environment to create a common base of experiences by identifying and developing concepts, processes and skills.
- Explain (E3): in this, phase students explain and verbalize the concepts that they have been explored and they develop new skills. The teacher has the opportunity to introduce formal terms, definitions and explanations for the new concepts and processes and to demonstrate new skills or behaviours.
- Elaborate (E4): In this, phase students develop a deeper and broader conceptual understanding and obtain more information about areas of interest by practicing on their new skills and behaviours.
- Evaluate (E5): this is the final step of the "5E's" instructional model, where learners are encouraged to assess their understanding and abilities and teachers evaluate student skills on the new knowledge.
Depending on the student reactions, there are forward or backward transitions between the three middle phases (explore, explain, elaborate) of the 5E's model during the teaching process. The "5 E's" model allows students and teachers to experience common activities, to use and build on prior knowledge and experience and to continually assess their understanding of a concept. Although it has been mainly applied in school education (Keeley, 2017) the 5 E's can be used with students of all ages, including adults (Hee et al., 2013). In this article, we shall obtain a mathematical representation of the "5 E's" model with the help of the theory of finite absorbing Markov Chains.
Finite absorbing Markov Chains. A Markov Chain (MC) is a stochastic process that moves in a sequence of steps (phases) through a set of states and has a one-step memory. That means that the probability of entering a certain state in a certain step depends on the state occupied in the previous step and not in earlier steps. This is known as the Markov property. However, for being able to model as many real life situations as possible by using MCs, one could accept in practice that the probability of entering a certain state in a certain step, although it may not be completely independent of previous steps, it mainly depends on the state occupied in the previous step (Kemeny & Snell, 1963). When the set of states of a MC is a finite set, then we speak about a finite MC. For general facts on finite MCs we refer to the book (Kemeny & Snell, 1976).
A. Markov introduced the basic concepts of MCs in 1907 on coding literal texts. A number of leading mathematicians, such as A. Kolmogorov, W. Feller, etc, developed the MC theory. However, only from the 1960's the importance of this theory to the natural, social and most of the applied sciences has been recognized (Bartholomew, 1973; Kemeny & Snell, 1963; Suppes & Atkinson, 1960).
Let us consider a finite MC with n states, say S1, S2,..., Sn, where n is a non-negative integer, n > 2. Denote by pij the transition probability from state Si to state Sj, i, j = 1, 2,., n ; then the matrix A= [p,j] is called the transition matrix of the MC. Since the transition from a state to some other state (including itself) is the certain event, we have that
pa + pi2 +... + Pin = 1, for i=1, 2, ..., n. (1)
A state of a MC is called absorbing if, once entered, it cannot be left. Further a MC is said to be an absorbing MC (AMC), if it has at least one absorbing state and if from every state it is possible to reach an absorbing state, not necessarily in one-step. Working with an AMC with k absorbing states, 1 < k < n, one brings its transition matrix A to its canonical form A* by listing the absorbing states first and then makes a partition of A* as follows
Ik I O
R I Q
(2)
In the above partition of A*, Ik denotes the unitary k X k matrix, O is a zero matrix, R is the (n - k) X k transition matrix from the non-absorbing to the absorbing states and Q is the (n - k) X (n - k) transition matrix between the non-absorbing states.
It can be shown ((Voskoglou & Perdikaris, 1991), Section 2) that the square matrix In-k-Q, where In-k denotes the unitary (n-k)x(n-k) matrix, is always an invertible matrix. Then, the fundamental matrix N of the AMC is defined to be the inverse matrix of In-k-Q. Therefore ((Morris, 1978), Section 2.4)
N = [n j] = (In - k - Q) - 1 = 1 adj (iIn_k - Q) . (3)
D (In-k Q)
In equation (3) D (In-k - Q) and adj (In-k - Q) denote the determinant and the adjoin of the matrix In-k- Q respectively. It is recalled that the adjoin of a matrix M is the matrix of the algebraic complements of the transpose matrix Mt of M, which is obtained by turning the rows of M to columns and vice versa. It is also recalled that the algebraic complement mij'of an element mijof M is calculated by the formula
mj = (-1)i++Dij, (4)
where Dj is the determinant of the matrix obtained by deleting the i-th row and the j-th column of M.
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BunycK 3(21), 2019
It is well known ((Kemeny, & Snell, 1976), Chapter 3) that the element n,j of the fundamental matrix N gives the mean number of times in state s; before the absorption, when the starting state of the AMC is Sj, where s; and Sj are non-absorbing states.
RESULTS AND DISCUSSION
The AMC model. We introduce a finite MC having as states Sj the corresponding phases Ei, i = 1, 2,..., 5, of the "5 E's" instructional model. According to the description of the "5 E's" model, performed in our Introduction, the flow diagram of this chain is that shown in Figure 1
Figure 1. The flow-diagram of the MC
From the flow-diagram of Figure 1 it becomes evident that the above chain is an AMC with S1 being its starting state and S5 being its unique absorbing state. The minimum number of steps before the absorption is 4 and this happens when we have no backward transitions between the three middle states S2, S3 and S4 of the chain. The transition matrix of the chain is the matrix
S1 S2
St S,
Si 0 1 0 0 0
S2 0 0 1 0 0
A = S 0 P32 0 P34 0
S4 0 0 P 43 0 P 45
S5 0 0 0 0 1
, with P32 + P34 = P43 + P45 = 1.
The canonical form of A is the matrix
A*=
S5 I S1 S2 S3 S4
S5 " 1 0 0 0 0 "
S1 0 0 1 0 0 "/1 I 0
S2 0 0 0 1 0 — I — R I Q
S3 0 0 P32 0 P34
S4 _ P45 0 0 P43 0
Then I4 - Q =
1 -1 0 0
0 1 -1 0
0 - P32 1 - P34
0 0 P43 1
and D(I4 - Q)=
1 -1 0
P32 1 - P34 = 1 - P34P43 - P32.
0 P43 1
Further, by equation (4), the algebraic complement of the element m11 = 1 of the transpose matrix
P^ 0
(I4 - Q)'
1 0 0 0 " 1
-1 1 - P32 0 -1
0 -1 is equal to
1 - P43 0
0 0 P34 1
1
-P43 1
: 1 - P43P34 - P32 .
In the same way we calculate the algebraic complements of all the other elements of (I4 - Q)' and replacing their values and the value of D(I4 - Q) to equation (3) we find that
S
5
1
N = [nij] ■■
1 - P34P43 - P
32
1 - P34P43 - P32 1 - P34P32 1 P34
0 1 + P34P34 -1 -P34
0 P 32 1 P 34
0 - P32 P43 -P43 -1 + P32
Since Si is the starting state of the above AMC, it becomes evident that the mean number of steps before the absorption is given by the sum
T= n11 + n12 + n13 + n14 = 3 - 2P43P34 - P32 + P34 . (5)
1- P34 P43 - P32
It is also clear that the bigger is T, the more are the student difficulties during the teaching process with the "5 E's" instructional treatment. Another factor of the student difficulties is the total time spent for the completion of the teaching process. However, the time is usually fixed in a formal teaching procedure in the classroom, which means that in this case T is the unique measure of the student difficulties.
A classroom application. The following application took place recently at the Graduate Technological Educational Institute of Western Greece for teaching the concept of the derivative to a group of fresher students of engineering. The instructor used the "5 E's" instructional treatment as follows:
Engage: The student attention was turned to the fact that the definition of the tangent of a circle as a straight line having a unique common point with its circumference does not hold for other curves (e.g. for the parabola). Therefore, there is a need to search for a definition of the tangent covering all cases and in particular of the tangent at a point of the graph of a given function.
1. Explore: The discussion in the class leads to the conclusion that the tangent at a point A of the graph of a given function y=f(x) can be considered as the limit position of the secant line of the graph through the points A(a, f(a)) and B(b, f(b)), when the point B is moving approaching to A either from the left, or from the right (see (Voskoglou, 2017b), Figure 1).
However, the slope of the secant line AB is equal to f (b) - f (a) , therefore the slope of the tangent of the graph at A is equal
b - a
to the limit of the above ratio when b tends to a.
Explain: Based on what it has been discussed at the phase of exploration, the instructor presents the formal definition
of the derivative number f'(a) at a point A(a, f(a)) of a given function y=f(x) as the limit (if there exists) of f (b) - f (a) when
b - a
a ^b , and of the tangent of the graph of y=f(x) at A as the straight line through A with slope f'(a). Some examples follow of calculating the derivative at a given point of a function and the tangent of its graph at this point. Then the definition of the derivative function y' = f'(x) of the function y=f(x) is given and suitable examples are presented to show that its domain is a subset of the domain of y=f(x).
Elaborate: Here the fact that the derivative y'=f'(x) expresses the rate of change of the function y=f(x) with respect to x is emphasized and its physical meaning is also presented connected to the speed and the acceleration at a moment of time of a moving object under the action of a steady force. The fundamental properties of the derivatives follow (sum, product, composite function, etc.) as well as a list of formulas calculating the derivatives of the basic functions and applications of them.
Evaluate: At the end of the teaching process, a number of exercises and problems analogous to those solved in the classroom was given to students on the purpose of checking at home their understanding of the subject. A week later, a written test was performed in the classroom enabling the instructor to assess the student progress.
Another important thing remaining for the instructor was to evaluate the student difficulties during the teaching process, which could help him in reorganizing properly his plans for teaching the same subject in future. This was succeeded with the help of the above developed MC representation of the "5 E's" model in the following simple way.
The instructor noticed that the student reactions during the teaching process led to 2 transitions of the discussion from state S3 (explain) back to state S2 (explore). Therefore, since from state S2 the chain moves always to S3 (Figure 1), we had 3 in total transitions from S2 to S3. The instructor also noticed 3 transitions from S4 (elaborate) back to S3. Therefore, since from state S3 the chain moves always to state S4 (Figure 1), we had 4 in total transitions from S3 to S4. In other words, we had 3+3 = 6
2 4
in total "arrivals" to S3, 2 "departures" from S3 to S2 and 4 "departures" from S3 to S4. Therefore p32 = ± and p34 = 4. In the
6 6
3 1
same way one finds that p43 = 3 and p45 = ± . Replacing the above values of the transition probabilities to equation (5) one
4 4
finds that the mean number T of steps before the absorption of the MC is equal to 14. Consequently, since the minimum number of steps before the absorption is 4, the students faced significant difficulties during the teaching process. This means that the instructor should find ways to improve his teaching procedure for the same subject in future.
CONCLUSION
The theory of MCs, being a smart combination of Linear Algebra and Probability, offers ideal conditions for the study and mathematical modelling of a certain kind of situations depending on random variables. In the paper at hands a mathematical representation of the "5 E's" teaching model was developed with the help of the theory of AMCs enabling the instructor to evaluate the student difficulties during the teaching process. This is very useful for reorganizing his (her) plans for
Ф1ЗИКО-МАТЕМАТИЧНА ОСВ1ТА (ФМО)
випуск 3(21), 2019
teaching the same subject in future. An application of this representation was also presented to teaching the concept of the derivative to engineering students. Although the development of the AMC model was proved to be quite laborious requiring the calculation of 17 in total determinants of third order (the determinant of the matrix I4- Q and the algebraic complements of its transpose matrix), its final application is very simple. The only thing needed for this purpose is the calculation by the instructor of the transitions of the AMC from S3 back to S2 and from S4 back to S3. Several other applications of MCs to education have been attempted by the present author in earlier works (e.g. see Chapters 2 and 3 of the book (Voskoglou, 2017a) and it is hoped that this research could be continued in future.
References
1. Bartholomew, D.J. (1973). Stochastic Models for Social Processes. London: J. Wiley and Sons.
2. Driver, R., Asoko, H., Leach, J.; Mortimer, E. & Scott, P. (1994). Constructing Scientific Knowledge in the Classroom, Educational Researcher, 23(7), 5-12.
3. Elbers, E. (2003). Learning and teaching mathematics in a community of inquiry, Educational Studies in Mathematics, 54, 77-99.
4. Enhancing Education. (2019). The 5 E's. Retrieved from: http://enhancinged.wgbh.org/research/eeeee.html.
5. Ernest, P. (1998). Social Constructivism as a Philosophy of Mathematics, Albany, New York, State University of New York Press.
6. Goos, M. (2014). Communities of practice in mathematics teacher education, in S. Lerman (Ed.), Encyclopedia of Mathematics Education, Dordrecht, Netherlands: Springer, 82-84.
7. Hee, J.W., Lee, E.J., Park, H.J., Chang, A.K. & Kim, M.J. (2013). Use of the 5E learning cycle model combined with problem-based learning for fundamentals of a nursing course, Journal of Nursing Education, 52 (12), 681-689.
8. Jaworski, B. (2006). Theory and practice in mathematics teaching development: Critical inquiry as a mode of learning in teaching, Journal of Mathematics Teacher Education, 9, 187-211.
9. Keeley, P. (2017). Embedding Formative Assessment into the 5E Instructional Model, Science and Children, 55 (4), 28-31.
10. Kemeny, J. G. & Snell, J. L. (1963). Mathematical Models in the Social Sciences, New York: Ginn and Company.
11. Kemeny, J. G. & Snell J. L. (1976). Finite Markov Chains, New York: Springer - Verlag.
12. Morris, A. O. (1978). An Introduction to Linear Algebra, Berkshire, England: Van Nostrand Beinhold Company Ltd.
13. Suppes, P. & Atkinson, R.S. (1960). Markov Learning Models for Multiperson Interactions, Stanford, California: Stanford University Press.
14. Glasersfeld, E. (1987). Learning as a Constructive Activity. In C. Janvier (Ed), Problems of representation in the teaching and learning of mathematics, Lawrence Erlbaum, Hillsdale, N. J., USA, 3-17.
15. Voskoglou, M. Gr. & Perdikaris, S. C. (1991). A Markov chain model in problem-solving, International Journal of Mathematical Education in Science and. Technology, 22, 909-914.
16. Voskoglou, M. Gr. (2017a). Finite Markov Chain and Fuzzy Logic Assessment Models: Emerging Research and Opportunities. Columbia, SC, USA: Createspace.com-Amazon.
17. Voskoglou, M. Gr. (2017b). A Note on the Graphical Representation of the Derivatives, Scientific Journal of Physical and Mathematical Education, 2(12), 9-16.
18. Voskoglou, M.Gr. (2019). Communities of Practice for Teaching and Learning Mathematics, American Journal of Educational Research, 7(6), 186-191.
19. Wenger, E. (1998). Communities of Practice: Learning, Meaning, and Identity, Cambridge: Cambridge University Press, UK.
ПРЕДСТАВЛЕННЯ МОДЕЛ1 "5 Е" ЗА ДОПОМОГОЮ ЛАНЦЮГА МАРКОВА Майкл Воскоглоу
Вищий технолог/'чний освiтнiй '¡нститут Зах'дно¡' Грец) Школа технолог'чних застосувань, Гре^я
Анотац'я.
Постановка проблеми. Соцiально-конструктивнi теор'й навчання стали дуже популярними у викладання математики протягом останнiх десятилть. "5 Е" - це навчальна модель, заснована на принципах соц'юльного конструктив'вму, що останшм часом стала дуже популярною при викладання математики, особливо в шкiльнiй освiтi. Кожен :з "5 Е" описуе окремий етап навчання, який починаеться з лтери "Е" - Займайтесь, Досл'джуйте, Пояснюйте, Розробляйте, Оцiнюйте. Залежно в'д реак^й учн'ю, снують переходи вперед або назад м'ж трьома середшми фазами (Досл'джуйте, Пояснюйте, Розробляйте) модел'1 "5 Е" пд час навчального процесу. Модель "5 Е" дозволяе учням та викладачам здйснювати спльну д'яльшсть, використовувати, будувати на основ': попереднх знань та досв'ду новi знання, постйно оцнювати свое розумння концепц'й
Матер/'али I методи. Використовуються ймовiрнiснi методи аналiзу.
Результати. Для математичного зображення модел '1 "5 Е" намагаемося застосувати поглинаючий ланцюг Маркова на його фазах.
Ланцюг Маркова - це стохастичний процес, який рухаеться посл'довно кроками (фазами) через наб'1р стан'ю i мае одно крокову пам'ять. Введено юнцевий ланцюг Маркова, що мае в якост'1 Б■ в'дпов'дш фази Е¡, '=1,2,...,5, навчально)'модел'1 "5 Е". Також в статт'1 представлено застосування модел'1 "5 Е" до роботи в аудиторй' з класом, що люструе кориснсть ц'е)' модел'1 на практицi. Апробац'я такого застосування вiдбулося нещодавно в Вищому технолог'чному навчальному iнститутi Зах 'дно)' Грецй для вивчення поняття пох'дно)' у групи студент 'в, майбутшх iнженерiв, на перших курсах.
Висновки. Представлення модел '1 "5 Е" за допомогою ланцюга Маркова е корисним нструментом для оцнювання труднощiв студент 'в пд час навчального процесу. Застосування ланцюга Маркова е корисним /' з позицп реорганiзацi')' планiв викладача щодо викладання того ж предмета в майбутньому.
Ключов/ слова: конструктив 'зм, соцiокультурнi теор 'й навчання, навчальна модель "5 Е", ланцюг Маркова.
