Table of activities and their dependencies of studied process.
\\n\\n
Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\\n\\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'IntechOpen is proud to announce that 179 of our authors have made the Clarivate™ Highly Cited Researchers List for 2020, ranking them among the top 1% most-cited.
\n\nThroughout the years, the list has named a total of 252 IntechOpen authors as Highly Cited. Of those researchers, 69 have been featured on the list multiple times.
\n\n\n\nReleased this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\n\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
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He received his M.S. in the Faculty of Biology from the University of Warsaw and his PhD degree from the Institute of Experimental Biology Polish Academy of Sciences. He habilitated in the Medical University of Warsaw, and he obtained his degree of Professor from the President of Poland. Professor Szablewski is the Head of Chair and Department of General Biology and Parasitology, Medical University of Warsaw. Professor Szablewski has published over 80 peer-reviewed papers in journals such as Journal of Alzheimer’s Disease, Biochim. Biophys. Acta Reviews of Cancer, Biol. Chem., J. Biomed. Sci., and Diabetes/Metabol. Res. Rev, Endocrine. He is the author of two books and four book chapters. He has edited four books, written 15 scripts for students, is the ad hoc reviewer of over 30 peer-reviewed journals, and editorial member of peer-reviewed journals. Prof. Szablewski’s research focuses on cell physiology, genetics, and pathophysiology. He works on the damage caused by lack of glucose homeostasis and changes in the expression and/or function of glucose transporters due to various diseases. 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When studying distributed parallel programming systems, real-time systems, economic systems and many other types of systems, it plays a role modeling of the time variables associated with individual system events, the duration of the studied activities, the time history of the modeled system and many other time characteristics. Special classes of Petri nets were introduced for the modeling of these types of systems with discrete time and their properties were studied in detail. Time Petri nets and timed Petri nets [3, 4] are currently the two most important classes of low-level Petri nets that use the concept of discrete time in their definition. Other classes of low-level Petri nets with discrete time are introduced and discussed for instance in [5, 6, 7]. It can be stated that most of the currently studied classes of Petri nets with discrete time use only the relative time variables usually related to the specific marking of the given Petri net. This fact can then cause difficulties, for example, in modeling complex time-synchronized distributed systems in which an external time source is usually available and individual components of this system must be synchronized with this external time source.
Process Petri nets (PPN) [8] were primarily introduced as a special subclass of classic low-level Petri nets for their using in the area of workflow management. PPN is a continuous Petri net that include within the set of all its places the unique input place, the unique output place and a finite set of so-called resource places which may contain, along with the input place, the tokens in the entry marking of the given PPN. These tokens located in the entry marking of PPN at the resource places usually represent the permanent resources of the modeled system. The given PPN can pass into its exit marking that is reachable from its entry marking by performing the final sequence of the transition firings. The tokens of the PPN’s exit marking may be then located only at its single output place and also at its resource places.
Process Petri nets with time stamps (PPNTS) are the newly introduced class of low-level Petri nets whose definition and the properties are the main topics of this chapter. PPNTS generalize the properties of PPNs in the area of design, modeling and verification of generally parallel systems with the discrete time.
Property-preserving Petri net process algebras (PPPA) [9] were originally designed for the specification and verification of manufacturing systems. PPPA has four types of operators: extensions, compositions, refinements and reductions. All operators can preserve about 20 PPN’s properties (some of them under additional conditions), such as liveness, boundedness, reversibility, RC-property, traps, siphons, proper termination, and so on. PPPA does not need to verify composition of PPNs because all their algebraic operators preserve the specified set of the properties. Hence, if the source PPNs satisfy the desirable properties, each of the composite PPN, including the PPN that models the resulting system itself, also satisfies these properties. These original PPPA are generalized for the class of the PPNTS in this chapter and their properties of proper-formed, well-formed and pure-formed PPNTS are then newly introduced. The new COMP, SYNC and JOIN algebraic operators are defined for the class of PPNTS and their chosen properties are proved.
With the support of these operators, the PPNTS can be extended also to the areas of the project management and the determination of the project critical path with the support of the critical path method (CPM) [10]. The new CPPNET subclass of PPNTS class is then defined in this chapter to represent pure-formed time-dependent processes. It is specially designed for the generalization of the CPM activities charts and their properties. This fact is then demonstrated on the simple project example and its critical path and other properties specification.
This chapter is arranged into the following sections: Section 2 explains the base term of this chapter, that is, process Petri nets with time stamps and introduces the terms of proper-formed, well-formed and pure-formed PPNTS; Section 3 discusses algebraic operators COMP and SYNC defined over the class of PPNTS and their main properties; Section 4 then introduces the special subclass CPPNET of the PPNTS class and explains its use in the area of the project management to represent pure-formed time-dependent processes with using of PPNTSs and to find critical paths for these processes similarly as in the case of the well-known critical path method (CPM). Finally, Section 5 gives the conclusions of the research to conclude the chapter.
Let N denote the set of all natural numbers, N := {1, 2, …}; N0 the set of all non-negative integer numbers, N0 := {0, 1, 2, …}; ∅ the empty set; |A| the cardinality of the given set A;
Definition 1. Let A be a nonempty set. By the (nonempty finite) sequence σ over the set A we understand a function σ: {1, 2, …, n} → A, where n ∈ N. Function ε: ∅ → A is called the empty sequence on the set A. We usually represent the sequence σ: {1, 2, …, n} → A by the notation σ = <a1, a2, …, an > of the elements of the set A, where ai = σ(i) for 1 ≤ i ≤ n. Empty sequence ε: ∅ → A on the set A we usually represent by the notation ε = <>. We denote the set of all finite (and possible empty) sequences over the set A by the notation ASQ.
If σ = <a1, a2, …, an > and τ = <b1, b2, …, bm > are the finite sequences, where σ ∈ ASQ, τ ∈ ASQ, n ∈ N, m ∈ N, then by the concatenation of the sequences σ and τ, denoted by σ++τ, we understand the finite sequence σ++τ := < a1, a2, …, an, b1, b2, …, bm>. The following functions are defined:
length: ASQ → N0, so that: length(σ) := n, length(ε) := 0,
elements: ASQ →
prefix: ASQ × N0 → ASQ, so that:
prefix(<a1, a2, …, an>, m) := < a1, a2, …, am>, if m ≤ n,
prefix(<a1, a2, …, an>, m) := < a1, a2, …, an>, if m > n,
prefix(ε, m) := ε,
suffix: ASQ × N0 → ASQ, so that:
suffix(<a1, a2, …, an>, m) := < am + 1, am + 2, …, an>, if m < n,
suffix(<a1, a2, …, an>, m) := ε, if m ≥ n,
suffix(ε, m) := ε,
create: N × A → ASQ, so that: create(n, a) := < a, a, …, a>, where length(<a, a, …, a>) = n,
sort: (N0)SQ → (N0)SQ, so that: sort(σ) := ρ,
We use the following subsets of the set (N0)SQ:
N# := {σ ∈ (N0)SQ | (σ = ε) ∨ ((σ = <a1, a2, …, an>) ∧ (a1 ≤ a2 ≤ … ≤ an)), n ∈ N},
N0 := {σ ∈ (N0)SQ | (σ = ε) ∨ ((σ = <0, 0, …, 0>) ∧ (length(σ) = n)), n ∈ N}.
Thus, the elements of the set N# constitute the empty sequence ε and all the finite ascending ordered sequences σ consisting of non-negative integer numbers. Similarly, the elements of the set N0 then form an empty sequence ε and all the sequences in the form <0, 0, …, 0 > of any finite length.
Definition 2. Net NET is an ordered triple NET := (P, T, A), where P is finite nonempty set of places, T is finite set of transitions, P ∩ T = ∅, and A is finite set of arcs, A ⊆ (P × T) ∪ (T × P). □
The given net NET is then described with a bipartite graph containing a finite nonempty set P of places used for expressing of the conditions of a modeled process (we usually use circles for their representation), a finite set T of transitions describing the changes in the modeled process (we usually draw them in the form of rectangles) and a finite set A of arcs being principally oriented while connecting the place with the transition or the transition with the place and we usually draw them as lines with arrows.
Some commonly used notations for the nets are •y = {x | (x, y) ∈ A} for the preset and y• = {x | (y, x) ∈ A} for the postset of a net node y (i.e., place or transition). A path of a net NET := (P, T, A) is a nonempty sequence <x1, …, xk> of net nodes, where k ∈ N, which satisfies (x1, x2), (x2, x3), …, (xk-1, xk) ∈ A. A path of the net NET := (P, T, A) leading from its node x to its node y is a circuit if (y, x) ∈ A. We denote the set of all the circuits of the net NET by CIRCUITSNET. Net NET’ := (P′, T’, A’) is a subnet of the net NET := (P, T, A) if (P′ ⊆ P) ∧ (T’ ⊆ T) ∧ (A’ = A ∩ ((P′ × T’) ∪ (T’ × P′))). Net NET is connected if and only if it is not composed of two or more disjoint and nonempty subnets.
Definition 3. Process net with time stamps (PNTS) PNTS is an ordered tuple PNTS := (P, T, A, AF, TP, TI, IP, OP, RP), where
(P, T, A) is the connected net, ∀t ∈ T: (•t ≠ ∅) ∧ (t• ≠ ∅),
AF: (P × T) ∪ (T × P) → N0 is the arc function,
TP is the transition priority function, TP: T → N,
TI: (T × P) → N0 is the time interval function,
IP is the input place, (IP ∈ P) ∧ (•IP = ∅) ∧ (∀p ∈ (P \\ ({IP} ∪ RP)): •p ≠ ∅),
OP is the output place, (OP ∈ P) ∧ (OP• = ∅) ∧ (∀p ∈ (P \\ ({OP} ∪ RP)): p• ≠ ∅),
RP is the set of resource places, (RP ⊆ (P \\ {IP, OP})) ∧ (∀t ∈ T: ⌐(•t ⊆ RP)).
The class of all the PNTS will be denoted by PNTS. □
The given PNTS PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) is represented by the connected net (P, T, A) with nonempty preset and postset of each of its transitions t; the arc function AF assigning each arc with a natural number (such number has the default value of 1, if not explicitly indicated in the PNTS diagram) expressing the number of removed or added tokens from or to the place associated with that arc when firing a particular transition; transition priority function TP assigns with each transition the natural number value expressing its priority (with the default value of 1); the time interval function TI assigns to each arc of the type (transition, place) a non-negative integer d expressing the minimum time interval during which the token has to remain in the place instead of being able to participate in the next firing of some transition and it thus determines the so-called time marking of the given PNTS (the value d associated with the respective arc is given in the format +d in the PNTS diagram); the input place IP is the only one nonresource place of PNTS PNTS with no input arc(s); the output place OP is the only one nonresource place of PNTS PNTS with no output arc(s); the finite set RP of resource places is used for expressing conditions of a modeled process containing some initial resources and we use circles with the double line for their representation.
Definition 4. Let PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) be the PNTS. Then:
marking M of the PNTS PNTS is a function M: P → N0,
time marking m of the PNTS PNTS is a function m: P → N#,
where ∀p ∈ P: |M(p)| = length(m(p)),
variable τ ∈ N0 is the net time of the PNTS PNTS,
state S of the PNTS PNTS is an ordered triple S := (M, m, τ),
transition t ∈ T is enabled in the state S := (M, m, τ) of the PNTS PNTS that is denoted by t en S, if ∀p ∈ •t: (M(p) ≥ AF(p, t)) ∧ (∀n ∈ elements(prefix(m(p), AF(p, t))): n ≤ τ)),
firing of the transition t ∈ T results in changing the state S := (M, m, τ) of the PNTS PNTS into its state S′ := (M’, m’, τ) that is denoted by S [t〉 S′, where ∀p ∈ P:
M’(p) := M(p) − AF(p, t) + AF(t, p),
m’(p) := sort(suffix(m(p), AF(p, t))++create(τ + TI(t, p), AF(t, p))),
elapsing of time interval δ ∈ N results in changing the state S := (M, m, τ) of PNTS PNTS into its state S′ := (M, m, τ + δ), where ∀t ∈ T: ⌐(t en (M, m, τ)), that is denoted by S [δ〉 S′, so that:
if the transitions t1, t2, …, tn ∈ T are enabled in the state S := (M, m, τ) of PNTS PNTS (i.e., (t1 en S) ∧ (t2 en S) ∧ … ∧ (tn en S)) we say that these transitions are enabled in parallel in the state S that is denoted by {t1, t2, …, tn} en S,
finite nonempty sequence σ := t1 t2 … tn of the transitions t1, t2, …, tn ∈ T for which the following is valid in the state S1 := (M1, m1, τ1) of PNTS PNTS:
(M1, m1, τ1) [t1〉 (M2, m2, τ1) [t2〉 … [tn〉 (Mn + 1, mn + 1, τ1),
∀t ∈ T: ⌐(t en (Mn + 1, mn + 1, τ1)),
is called step σ in the given state S1 of PNTS PNTS and it is denoted by.
finite nonempty sequence ρ of steps and time intervals elapsing that represents the following finite sequence
of the state changes of PNTS PNTS is the sequence ρ := σ1 δ1 σ2 δ2 … σn δn of steps σ1, σ2, …, σn and time intervals elapsing δ1, δ2, …, δn,
we say the state S′ of PNTS PNTS is reachable from its state S if there exists the finite sequence ρ := σ1 δ1 σ2 δ2 … σn δn of steps σ1, σ2, …, σn and time intervals elapsing δ1, δ2, …, δn such that S [σ1 δ1 σ2 δ2 … σn δn〉 S′; the set of all the reachable states of PNTS PNTS from its state S is denoted by [S〉; the set of all the finite sequences ρ := σ1 δ1 σ2 δ2 … σn δn associated with all the reachable states S′ ∈ [S〉 is denoted by [S〉〉, that is,
the set of all the states S := (M, m, τ) of PNTS PNTS is denoted by S,
the set of all the markings M associated with the set S of all the states of PNTS PNTS is denoted by M, that is, M := {M | (S = (M, m, τ)) ∧ (S ∈ S)},
static state Ss := (Ms, ms, τs) of PNTS PNTS is every of its states where
the set of all the static states Ss := (Ms, ms, τs) of PNTS PNTS is denoted by Ss,
the set of all the static markings Ms associated with the set Ss of all the static states of PNTS PNTS is denoted by Ms, that is, Ms := {Ms | (Ss := (Ms, ms, τs)) ∧ (Ss ∈ Ss)},
the function ξ: M → Ms which assigns to each marking M ∈ M of a given PNTS PNTS the associated static marking Ms ∈ Ms is defined as follows:
∀ p ∈ RP: ξ(M(p)) := M(p),
∀ p ∈ P \\ RP: ξ(M(p)) := 0,
entry state Se := (Me, me, τe) of PNTS PNTS is every of its states where
∃k ∈ N: (Me(IP) = k) ∧ (length(me(IP)) = k),
∀ p ∈ P \\ (RP ∪ {IP}): (Me(p) = 0) ∧ (me(p) = <>),
∀ p ∈ RP: (Me(p) ≥ 0) ∧ (me(p) ∈ N0),
the set of all the entry states Se := (Me, me, τe) of PNTS PNTS is denoted by Se,
exit state Sx := (Mx, mx, τx) of PNTS PNTS that is reachable from its entry state Se := (Me, me, τe) is every of its states where
Sx ∈ [Se〉,
Mx(OP) = Me(IP),
∀ p ∈ P \\ (RP ∪ {OP}): (Me(p) = 0) ∧ (me(p) = <>),
the set of all the exit states Sx := (Me, me, τe) of PNTS PNTS that are reachable from its entry state Se := (Me, me, τe) is denoted by [Se〉x,
the set of all the exit states Sx of PNTS PNTS that are reachable from all its entry states Se ∈ Se is denoted by Sx. □
The above established concepts are demonstrated in a simple example of the PNTS PNTS1 := (P, T, A, AF, TP, TI, IP, OP, RP) that is shown in Figure 1, where P := {IP, P1, R1, OP}, T := {T1, T2, T3}, A := {(IP, T1), (IP, T2), (T1, P1), (T2, P1), (R1, T1), (P1, T3), (T3, R1), (T3, OP)}, AF := {((IP, T1), 1), ((IP, T2), 1), ((T1, P1), 1), ((T2, P1), 2), ((R1, T1), 1), ((P1, T3), 1), ((T3, R1), 1), ((T3, OP), 1)}, TP := {(T1, 2), (T2, 1), (T3, 1)}, TI := {((T1, P1), 3), ((T2, P1), 3), ((T3, R1), 1), ((T3, OP), 4)}, IP := IP, OP := OP, RP := {R1}.
Firing of transition T1 in PNTS PNTS1.
PNTS PNTS1 is in its entry state Se := (Me, me, τe), where marking Me := (Me(IP), Me(P1), Me(R1), Me(OP)) = (2, 0, 2, 0), time marking me := (me(IP), me(P1), me(R1), me(OP)) = (<0, 2>, <>, <0, 0>, <>) and net time τe = 0 (i.e., τe = τ). Static marking Ms ∈ Ms associated with the entry marking Me (see (xiv) and (xvii) of Definition 4) has the value Ms := ξ(Me) = (0, 0, 2, 0).
Time marking m of any PNTS expresses the current time state of the modeled system using the final (or empty) ascending ordered sequences of non-negative integers (i.e., elements of the set N#) associated with each of its places. The individual values of the time marking m associated with the arbitrary place p of the given PNTS in its state S, informally said, represent the values of the net time τ at which the respective token can first participate in the firing of selected enabled transition t of the given PNTS.
The transitions T1 and T2 are enabled in the entry state Se because (see (v) of Definition 4):
∀p ∈ •T1: (2 = M(IP) ≥ AF(IP, T1) = 1) ∧ (2 = M(R1) ≥ AF(R1, T1) = 1) ∧ (∀n ∈ elements(prefix(m(IP), AF(IP, T1))) = elements(prefix(<0, 2>, 1)) = elements(<0>) = {0}: 0 ≤ 0) ∧ (∀n ∈ elements(prefix(m(R1), AF(R1, T1))) = elements(prefix(<0, 0>, 1)) = elements(<0>) = {0}: 0 ≤ 0),
∀p ∈ •T2: (2 = M(IP) ≥ AF(IP, T1) = 1) ∧ (∀n ∈ elements(prefix(m(IP), AF(IP, T2))) = elements(prefix(<0, 2>, 1)) = elements(<0>) = {0}: 0 ≤ 0).
When enabling individual transitions of the given PNTS so-called conflicts can originate in its certain markings (or conflict transitions). At the enabling of the transitions t1 and t2 of the given PNTS in its state S the conflict occurs, if both transitions t1 and t2 have at least one input place, each of the transitions t1 and t2 is individually enabled in the state S, but the transitions t1 and t2 are not enabled in parallel in the state S (see (viii) of Definition 4) and enabling of one of them will prevent enabling of the other (i.e., (•t1 ∩ •t2 ≠ ∅) ∧ (t1 en S) ∧ (t2 en S) ∧ ⌐({t1, t2} en S)). The term of conflict transitions can be obviously easily generalized for the case of a finite set t1, t2, …, tn, n ∈ N of the transitions of the given PNTS.
The transitions T1 and T2 in the entry state Se of PNTS PNTS1 are conflict transitions because the time marking me(IP) = <0, 2 > (i.e., only one token of the entry marking Me(IP) may participate in the firing of the transition T1 or T2 in the net time τe = 0). When solving such transitions conflict we therefore follow the rule which determines, informally said, that from the set of conflict transitions the one will be enabled, whose value of the transition priority function TP is the highest. If such transition from the set of conflict transitions does not exist, the given conflict would have to be solved by other means. The transition T1 is then enabled in the entry state Se on the basis of that rule in our studied example (because TP(T1) = 2 and TP(T2) = 1).
Firing of the transition T1 changes the entry state Se := (Me, me, τe) of the PNTS PNTS1 into its state S1 := (M1, m1, τe) (i.e., Se [T1〉 S1—see Figure 1), where (see (vi) of Definition 4):
M1(IP) := Me(IP) - AF(IP, T1) = 2 − 1 = 1,
m1(IP) := sort(suffix(me(IP), AF(IP, T1))) = sort(suffix(<0, 2>, 1)) = sort(<2>) = <2>,
M1(P1) := Me(P1) + AF(T1, P1) = 0 + 1 = 1,
m1(P1) := sort(create(τ + TI(T1, P1), AF(T1, P1))) = sort(create(0 + 3, 1)) = sort(create(3, 1)) = sort(<3>) = <3>,
M1(R1) := Me(R1) - AF(R1, T1) = 2 − 1 = 1,
m1(R1) := sort(suffix(me(R1), AF(R1, T1))) = sort(suffix(<0, 0>, 1)) = sort(<0>) = <0> .
There is no enabled transition in the state S1 := (M1, m1, τ1) and it is necessary to perform the time interval elapsing with the value of δ = 2. This will change the state S1 := (M1, m1, τe) into the state S2 := (M1, m1, τ1), where τ1 := τe + δ = 2 (i.e., S1 [2〉 S2). It can be easily shown that transition T1 in the state S2 is enabled and firing of this transition changes the state S2 := (M1, m1, τ1) of the PNTS PNTS1 into its state S3 := (M2, m2, τ1) (i.e., S2 [T1〉 S3), where M2 := (M2(IP), M2(P1), M2(R1), M2(OP)) = (0, 2, 0, 0) and m2 := (m2(IP), m2(P1), m2(R1), m2(OP)) = (<>, <3, 5>, <>, <>).
It can then be easily verified that S3 [1〉 S4 [T3〉 S5 [2〉 S6 [T3〉 Sx, where:
S4 = (M2, m2, τ2) = ((0, 2, 0, 0), (<>, <3, 5>, <>, <>), 3),
S5 = (M3, m3, τ2) = ((0, 1, 1, 1), (<>, <5>, <4>, <7>), 3),
S6 = (M3, m3, τ3) = ((0, 1, 1, 1), (<>, <5>, <4>, <7>), 5),
Sx = (M4, m4, τ3) = ((0, 0, 2, 2), (<>, <>, <4, 6>, <7, 9>), 5).
There are no enabled transitions in the exit state Sx := (M4, m4, τ3) of PNTS PNTS1 that is reachable from the entry state Se := (Me, me, τe) (see (xx) of Definition 4) and there is also no time interval elapsing value δ in this state that enables any of the transitions.
The set AMs ⊆ Ms of all the allowed static markings of the given PNTS PNTS, informally said, expresses how many tokens may be located in its individual resource places if PNTS PNTS be in its (now no longer arbitrary) allowed entry state ASe ∈ ASe, where ASe ⊆ Se. For instance, the set AMs of the PNTS PNTS1 (see Figure 1) can be defined as AMs := {(0, 0, k, 0) | k ∈ N}, that is, there must be at least one token in the resource place R1 (and of course at least one token in the input place IP) in any allowed entry state ASe ∈ ASe.
Definition 5. Let PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) be a PNTS, AMs ⊆ Ms be the set of all of its allowed static markings and ASe := {ASe | (ASe = (AMe, ame, 0)) ∧ (ξ(AMe) ∈ AMs)} be the set of all of its allowed entry states. Then:
PNTS is k-bounded PNTS if
PNTS is proper-formed PNTS if
proper-formed PNTS is well-formed PNTS if
well-formed PNTS is pure-formed PNTS if
PNTS is proper-formed PNTS if for any of its state S that is reachable from any allowed entry state ASe ∈ ASe there exists its output state Sx that is also reachable from its allowed entry state ASe ∈ ASe such that the output state Sx is also reachable from the state S (i.e., ∀S ∈ [ASe〉 ∃Sx ∈ [ASe〉x: Sx ∈ [S〉). Furthermore, the cardinality of the set [ASe〉〉 of all the sequences ρ := σ1 δ1 σ2 δ2 … σn δn associated with all the reachable states S ∈ [ASe〉〉 must be finite (i.e., (∃n ∈ N: |[ASe〉〉| = n).
Proper-formed PNTS is well-formed PNTS if for any of its allowed entry state ASe ∈ ASe and for any of its exit state Sx ∈ [ASe〉x, where Sx := (Mx, mx, τx), it is true that the exit static marking ξ(Mx) of all its resource places is an element of the set AMs of all its allowed static markings if PNTS PNTS be in its allowed entry state ASe ∈ ASe (i.e., ∀ASe ∈ ASe ∀Sx ∈ [ASe〉x, Sx := (Mx, mx, τx): ξ(Mx) ∈ AMs).
Well-formed PNTS is pure-formed PNTS if for any of its allowed entry state ASe ∈ ASe, where ASe := (AMe, ame, τe), and for any of its exit state Sx ∈ [ASe〉x, where Sx := (Mx, mx, τx), it is true that the exit static marking ξ(Mx) of all its resource places is equal to the entry static marking ξ(AMe) of all its resource places that is associated with the allowed entry state ASe (i.e., ∀ASe ∈ ASe ∀Sx ∈ [ASe〉x, Sx := (Mx, mx, τx): ξ(AMe) = ξ(Mx)).
For instance, if the set AMs of the PNTS PNTS1 (see Figure 1) is defined as:
AMs := {(0, 0, k, 0) | k ∈ N} (i.e., there must be at least one token in the resource place R1 in any allowed entry state ASe ∈ ASe), then it can be shown that PNTS PNTS1 is k-bounded, proper-formed, well-formed and pure-formed PNTS,
AMs := {(0, 0, 0, 0)} (i.e., there may not be any token in the resource place R1 in any allowed entry state ASe ∈ ASe), then it can be shown that PNTS PNTS1 is k-bounded, proper-formed, but not well-formed or pure-formed PNTS (see for instance the sequence
Lemma 1. If PNTS is proper-formed PTNS then PNTS is k-bounded PNTS.
Proof. Clear. PNTS PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) is a connected net that contains the finite set T of the transitions. Then the finite number of tokens will be added to each of the places p ∈ P by firing each of the transitions t ∈ T. The number of states S ∈ [ASe〉 for any allowed entry state ASe ∈ ASe must be also finite because PNTS is proper-formed PNTS (i.e., ∃n ∈ N: |[ASe〉〉| = n). From these facts then immediately follows that in any state S ∈ [ASe〉 the finite number of tokens must be placed in any place p ∈ P, where any final number of tokens is placed in the input place IP in the entry state ASe. From these facts then immediately follows that
Definition 6. Process Petri net with time stamps (PPNTS) PPNTS is an ordered couple PPNTS := (PNTS, Se), where PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) is the PNTS and Se ∈ Se is the entry state of PNTS PNTS. The class of all PPNTSs will be denoted by PPNTS.□
We study the issue of transforming PNTS through precisely defined binary operator COMP and n-ary operator SYNC over the class PNTS and we also examine the preservation of individual PNTS’s properties when applying each of these operators. Formal enrollment of an application of generally n-ary operator OP whose operands are the PNTS PNTS1, PNTS2, …, PNTSn (n ∈ N) and whose application requires the specification of values of k formal parameters (k ∈ N) par1, par2, … park, will be denoted by the expression.
where PNTS is the resulting PNTS.
Definition 7. Let PNTS1 := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1) and PNTS2 := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2) be the PNTSs. Let AMs1 := {(AMs1(IP1), AMs1(p11), …, AMs1(p1n), AMs1(r11), …, AMs1(r1m), AMs1(OP1)) | P1 := {p11, …, p1n, r11, …, r1m}, RP1 := {r11, …, r1m}, n ∈ N, m ∈ N} be the set of all the allowed static markings of PNTS1, AMs2 := {(AMs2(IP2), AMs2(p21), …, AMs2(p2k), AMs2(r21), …, AMs2(r2h), AMs2(OP2)) | P2 := {p21, …, p2k, r21, …, r2h}, RP2 := {r21, …, r2h}, k ∈ N, h ∈ N} be the set of all the allowed static markings of PNTS2.
Cartesian product AMs1 ⊗ AMs2 is then the following set:
PNTS PNTS1 and PNTS2 are disjoint and we denote this fact by PNTS1 ∠ PNTS2 if.
Definition 8. The function COMP: PNTS × PNTS → PNTS of nets composition is defined as follows: if PNTS1 := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1) and PNTS2 := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2) be the arbitrary PNTSs, PNTS1 ∠ PNTS2, t be an arbitrary transition, where (t ∉ T1) ∧ (t ∉ T2), ti ∈ N0, then PNTS := [PNTS1, PNTS2].COMP(t, ti), where PNTS PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) fulfills the following:
P := P1 ∪ P2,
T := T1 ∪ T2 ∪ {t},
A := A1 ∪ A2 ∪ {(OP1, t), (t, IP2)},
AF := AF1 ∪ AF2 ∪ {((OP1, t), 1), ((t, IP2), 1)},
TP := TP1 ∪ TP2 ∪ {(t, 1)},
TI := TI1 ∪ TI2 ∪ {(t, IP2), ti)},
IP := IP1,
OP := OP2,
RP := RP1 ∪ RP2. □
Symbolic representation of PNTS [PNTS1, PNTS2].COMP(t, ti) can be seen in Figure 2.
Symbolic representation of PNTS [PNTS1, PNTS2].COMP(t, ti).
Lemma 2. Let PNTS1 := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1) and PNTS2 := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2) be two arbitrary PNTS, PNTS1 ∠ PNTS2, t be an arbitrary transition, (t ∉ T1) ∧ (t ∉ T2), ti ∈ N0, AMs1 and AMs2 be the sets of all the allowed static markings of PNTS1 and PNTS2. Let PNTS := [PNTS1, PNTS2].COMP(t, ti).
If PNTS1 and PNTS2 are proper-formed, resp. well-formed, resp. pure-formed, PNTS and AMs = AMs1 ⊗ AMs2 be the set of all the allowed static markings of PNTS PNTS, then also resulting PNTS is proper-formed, resp. well-formed, resp. pure-formed, PNTS.
Proof. Clear, it directly follows from Definition 5, Definition 7 and Definition 8. □
Definition 9. The function SYNC: PNTS × PNTS × … × PNTS → PNTS of synchronous nets composition is defined as follows: if PNTS1 := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1), PNTS2 := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2), …, PNTSn := (Pn, Tn, An, AFn, TPn, TIn, IPn, OPn, RPn), be the arbitrary PNTSs, ∀i, 1 ≤ i ≤ n, ∀j, 1 ≤ j ≤ n: i ≠ j ⇒ PNTSi ∠ PNTSj, where n ∈ N, pi and po be the arbitrary places, (pi ∉ P1 ∪ P2 ∪ … ∪ Pn) ∧ (po ∉ P1 ∪ P2 ∪ … ∪ Pn) ∧ (pi ≠ po), ti and to be the arbitrary transitions, (ti ∉ T1 ∪ T2 ∪ … ∪ Tn) ∧ (to ∉ T1 ∪ T2 ∪ … ∪ Tn) ∧ (ti ≠ to), af1 ∈ N, af2 ∈ N, …, afn ∈ N, ti1 ∈ N0, ti2 ∈ N0, …, tin ∈ N0, tio ∈ N0, then
where PNTS PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) fulfills the following:
P := P1 ∪ P2 ∪ … ∪ Pn ∪ {pi, po},
T := T1 ∪ T2 ∪ … ∪ Tn ∪ {ti, to},
A := A1 ∪ A2 ∪ … ∪ An ∪ {(pi, ti), (ti, IP1), …, (ti, IPn), (OP1, to), …, (OPn, to), (to, po)},
AF := AF1 ∪ AF2 ∪ … ∪ AFn ∪ {((pi, ti), 1), ((ti, IP1), af1), …, ((ti, IPn), afn), ((OP1, to), af1), …, ((OPn, to), afn), ((to, po), 1)},
TP := TP1 ∪ TP2 ∪ … ∪ TPn ∪ {(ti, 1), (to, 1)},
TI := TI1 ∪ TI2 ∪ … ∪ TIn ∪ {((ti, IP1), ti1), …, ((ti, IPn), tin), ((to, po), tio)},
IP := pi,
OP := po,
RP := RP1 ∪ RP2 ∪ … ∪ RPn. □
Symbolic representation of PNTS [PNTS1, PNTS2, …, PNTSn].SYNC(pi, po, ti, to, af1, …, afn, ti1, …, tin, tio) can be seen in Figure 3.
Symbolic representation of PNTS [PNTS1, PNTS2, …, PNTSn].SYNC(pi, po, ti, to, af1, …, afn, ti1, …, tin, tio).
Lemma 3. Let PNTS1 := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1), PNTS2 := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2), …, PNTSn := (Pn, Tn, An, AFn, TPn, TIn, IPn, OPn, RPn) be arbitrary PNTSs, ∀i, 1 ≤ i ≤ n, ∀j, 1 ≤ j ≤ n: i ≠ j ⇒ PNTSi ∠ PNTSj, where n ∈ N, pi and po be arbitrary places, (pi ∉ P1 ∪ P2 ∪ … ∪ Pn) ∧ (po ∉ P1 ∪ P2 ∪ … ∪ Pn) ∧ (pi ≠ po), ti and to be arbitrary transitions, (ti ∉ T1 ∪ T2 ∪ … ∪ Tn) ∧ (to ∉ T1 ∪ T2 ∪ … ∪ Tn) ∧ (ti ≠ to), af1 ∈ N, af2 ∈ N, …, afn ∈ N, ti1 ∈ N0, ti2 ∈ N0, …, tin ∈ N0, tio ∈ N0 and AMs1, AMs2, …, AMsn be the sets of all the allowed static markings of PNTS1, PNTS2, …, PNTSn. Let PNTS := [PNTS1, PNTS2, …, PNTSn].SYNC(pi, po, ti, to, af1, …, afn, ti1, …, tin, tio).
If PNTS1, PNTS2, …, PNTSn are proper-formed, resp. well-formed, resp. pure-formed, PNTS and AMs = AMs1 ⊗ AMs2 ⊗ … ⊗ AMsn is the set of all the allowed static markings of PNTS PNTS, then also PNTS is proper-formed, resp. well-formed, resp. pure-formed, PNTS.
Proof. Clear, it directly follows from Definition 5, Definition 7 and Definition 9. □
Critical Path Method (CPM) is a method used in modeling and project management that was developed at the end of 1950s and that is commonly used for all the types of projects including software development [10]. The CPM is the most widely used method of so-called network analysis, even though it is designed to analyze the time consumption of only deterministic projects, that is, projects where the duration of each of their activities is exactly known, including all their sub-activities.
The basis for using CPM is to create a project model that includes:
the list of all activities needed to complete the project,
the time duration of each activity that is constant,
the dependencies between the project activities,
A critical path is then a designation for a sequence of activities whose time duration directly affects the time duration of the entire project. The activities that make up the critical path are then referred to as critical activities. There may be several critical paths in the project. When managing the project, a sequence of activities within a given network chart describing this project that increases the longest total time duration of a project is called its critical path. The critical path within the network chart can be used to determine the shortest time required to complete the project. The application of the CPM method can therefore determine which activities within the studied project are “critical” (i.e., activities on the longest path in the network chart describing the project) and which activities may be delayed in the execution of the project without increasing its total time.
The special class CPNET ⊂ PNTS of PNTS is introduced in the following paragraphs to represent network chart used in the CPM method through PNTS. Special unary operator JOIN that is required in the definition of the class CPNET is introduced first.
Definition 10. The function JOIN: PNTS → PNTS of net transition joining is defined as follows: if PNTS1 := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1) be the arbitrary PNTS, p ∉ P1 be the arbitrary place, t1 and t2 be the arbitrary transitions, (t1 ≠ t2) ∧ (t1 ∈ T1) ∧ (t2 ∈ T1), ti ∈ N0, then PNTS := PNTS1.JOIN(p, t1, t2, ti), where PNTS PNTS := (P, T, A, AF, TP, TI, IP, OP, RP) fulfills the following:
P := P1 ∪ {p},
T := T1,
A := A1 ∪ {(t1, p), (p, t2)},
AF := AF1 ∪ {((t1, p), 1), ((p, t2), 1)},
TP := TP1 ∪ {(t, 1)},
TI := TI1 ∪ {(t1, p), ti)},
IP := IP1,
OP := OP1,
RP := RP1. □
Symbolic representation of the unary operator JOIN application over the PNTS PNTS1 can be seen in Figure 4.
Symbolic representation of PNTS PNTS := PNTS1.JOIN(p, t1, t2, ti).
Definition 11. Let PNTS1 := (P1, T1, A1, AF1, TP1, TI1, IP1, OP1, RP1), PNTS2 := (P2, T2, A2, AF2, TP2, TI2, IP2, OP2, RP2), …, PNTSn := (Pn, Tn, An, AFn, TPn, TIn, IPn, OPn, RPn), where n ∈ N, be the arbitrary PNTSs, ∀i, 1 ≤ i ≤ n, ∀j, 1 ≤ j ≤ n: i ≠ j ⇒ PTSNi ∠ PTSNj. The class CPNET ⊂ PNTS then contains the following PNTSs:
if p be an arbitrary place, p ∉ (P1 ∪ P2 ∪ … ∪ Pn), then PNTS BASEp ∈ CPNET, where BASEp := ({p}, ∅, ∅, ∅, ∅, ∅, p, p, ∅},
if PNTS1 ∈ CPNET, PNTS2 ∈ CPNET, t be an arbitrary transition, where (t ∉ T1) ∧ (t ∉ T2), ti ∈ N0, then also [PNTS1, PNTS2].COMP(t, ti) ∈ CPNET,
if PNTS1, PNTS2, …, PNTSn ∈ CPNET, pi and po be the arbitrary places, (pi ∉ P1 ∪ P2 ∪ … ∪ Pn) ∧ (po ∉ P1 ∪ P2 ∪ … ∪ Pn) ∧ (pi ≠ po), ti and to be the arbitrary transitions, (ti ∉ T1 ∪ T2 ∪ … ∪ Tn) ∧ (to ∉ T1 ∪ T2 ∪ … ∪ Tn) ∧ (ti ≠ to), ti1 ∈ N0, ti2 ∈ N0, …, tin ∈ N0, tio ∈ N0, then also PNTS PNTS ∈ CPNET, where
if PNTS1 ∈ CPNET, p ∉ P1 be an arbitrary place, t1 and t2 be the arbitrary transitions, (t1 ≠ t2) ∧ (t1 ∈ T1) ∧ (t2 ∈ T1), ti ∈ N0 and af ∈ N, then also PNTS ∈ CPNET, where PNTS := (P, T, A, AF, TP, TI, IP, OP, RP), such that:
PNTS := PNTS1.JOIN(p, t1, t2, ti),
⌐(∃ x1 x2 … xn ∈ CIRCUITSPNTS: (x1 ∈ T) ∧ (xn ∈ P) ∧ (n ∈ N)). □
Four simple PNTSs BASEP1, CPNET1, CPNET2 and CPNET3 that are the members of the class CPNET can be seen in Figure 5, where:
CPNET1 := [[BASEP2, BASEP3].COMP(T2, 2), BASEP4].COMP(T3, 5),
CPNET2 := [[BASEP6, BASEP8].COMP(T6, 2), BASEP7]
.SYNC(P5, P9, T5, T7, 1, 1, 1, 6, 3),
CPNET3 := [ANET2, ANET3, BASEP1].SYNC(IP, OP, T1, T8, 1, 1, 1, 4, 1, 8, 2)
.JOIN(P10, T3, T5, 4).
.JOIN(P11, T3, T6, 5).
PNTSs BASEP1, CPNET1, CPNET2 and CPNET3.
Note also that the PNTS CPNET3 does not contain any circuit as it is required in the (4) of Definition 11.
Lemma 4. Let PNTS ∈ CPNET be an arbitrary PNTS. Then PNTS is pure-formed PNTS.
Proof. Clear, it directly follows from Definition 5, Definition 7, Definition 10 and Definition 11 and from the fact that any PNTS ∈ CPNET does not contain any resource place (i.e., if the given PNTS is proper-formed PNTS, then it is also immediately well-formed and pure-formed PNTS). Furthermore, it is also clear that if we allow the existence of a circuit within the PNTS PNTS (see (iv) of Definition 11), there is always the danger of a deadlock in such a PNTS and PNTS is in this case neither a proper-formed PTSN. See, for instance, PNTS CPNET4 in its entry state Se in Figure 6, where CPNET4 := CPNET3.JOIN(P12, T7, T2, 6). It is true that CPNET4 ∉ CPNET because there exists for instance the circuit.
PNTS CPNET4 in its entry state Se.
It is also clear that after the firing of the transition T1 in the entry state Se of the PNTS CPNET4 no one transition will be enabled for any value of the net time τ in this PNTS (i.e., there exists the deadlock marking in this PNTS) and thus the CPNET4 is not even proper-formed PNTS. □
Definition 12. The class CPPNET ⊂ PPNTS contains all the PPNTSs PPNTS := (PNTS, Se) where PNTS ∈ CPNET and Se := ((1, 0, …, 0), (<0>, <>, …, <>), 0). □
An example of a simple process is presented in the following paragraphs the characteristics of which will be studied with using of the PPNTS from the CPPNET class. The studied process is described in the following table of activities (see Table 1):
Activity | Duration | Previous activities |
---|---|---|
A | 2 | — |
B | 3 | — |
C | 4 | — |
D | 2 | C |
E | 6 | B, D |
F | 5 | A |
G | 5 | C |
H | 3 | E, F |
Table of activities and their dependencies of studied process.
The CPM chart of the abovementioned process comprising the activities listed in Table 1 is shown in Figure 7 where:
selected activity of the studied process and its duration is associated with each edge of the CPM chart,
each node of the CPM chart is associated with its serial number (indicated in the upper half of the node), the earliest possible activation time of the given node (shown in the lower left quarter of the node) and the possible latest activation time of the given node (shown in the lower right quarter circle of the node),
the desired links of the individual activities are expressed through the oriented edges and their associated nodes in the CPM chart,
the critical path of the CPM chart passes through its nodes where the earliest possible activation time is equal to the latest possible activation time, that is, through nodes 1, 2, 3, 5 and 6, and it is thus formed by A, D, E and H activities (these activities are represented by dashed line graphs in the CPM chart) with a total duration of 15 time units.
CPM chart of process activities listed in Table 1.
The pure-formed PPNTS PROC that represents the process comprising the activities listed in Table 1 can be seen in Figure 8, where
PROC := [PROC1, [BASEC, BASEG].COMP(T5, 5)].SYNC(IP, OP, T1, T7, 1, 1, 0, 4, 0),
PROC1 := [[BASEA, BASEF].COMP(T3, 5), [BASEB, BASEE].COMP(T4, 6)]
PPNTS PROC of process activities listed in Table 1.
.SYNC(P1, H, T2, T6, 1, 1, 2, 3, 3).
Places A, B, C, D, E, F, G and H of PNTS PROC represent individual activities of the studied process and the appropriate values of the time interval function TI then express the time durations of relevant activities (i.e., for instance, the time duration of the activity A is represented by the value of TI(T2, A) = 2, etc.).
In order to find the critical path of the process represented by PPNTS PROC, we first perform the association of each place and transition of the PPNTS PROC with the value of the critical path function CP that is introduced in the following Definition 13.
Definition 13. Let PPNTS PPNTS := (P, T, A, AF, TP, TI, IP, OP, RP, Se), PPNTS ∈ CPPNET. The critical path function CP: (P ∪ T) → N0 is defined as follows:
CP(IP) := 0,
∀p ∈ (P \\ IP): CP(p) := CP(t) + TI(t, p), where t = •p,
∀t ∈ T: CP(t) := max({CP(p1), CP(p2), …, CP(pn)}), where •t = {p1, p2, …, pn}, n ∈ N. □
The PPNTS PROC whose each place and transition is associated with the value of the critical path function CP can be seen in Figure 9 (where, for instance, CP(E) = CP(T4) + TI(T4, E) = 6 + 6 = 12, where T4 = •E; CP(T4) = max({CP(B), CP(D)}) = max({3, 6}) = 6, where •T4 = {B, D}, etc.).
PPNTS PROC with the associated values of critical path function CP.
It follows directly from the Definition 4 that the value of the critical path function CP associated with any transition t ∈ T of the arbitrary PPNTS := (P, T, A, AF, TP, TI, IP, OP, RP, Se), where Se := ((1, 0, …, 0), (<0>, <>, …, <>), 0), then represents the net time τ value when the given transition t will be fired. The value of the critical path function CP associated with the output place OP (i.e., CP(OP)) then immediately indicates the total duration of the process critical path (i.e., the net time τ value when the transition t = •OP will be fired). The algorithm for finding the set of PPNTS nodes of which the project critical path is formed is then obvious and it is expressed by the following pseudocode in PASCAL (the set of nodes forming the critical path of the project is then contained in the CriticalPath variable):
Node := OP; CriticalPath := {OP};
WHILE (Node <> IP) DO
BEGIN
MaxValue := max({CP(X1), CP(X2), …, CP(Xn)}), where •Node = {X1, X2, …, Xn}, n ∈ N;
Node := Xi, where (Xi ∈ {X1, X2, …, Xn}) ∧ (CP(Xi) = MaxValue);
CriticalPath := CriticalPath ∪ {Node};
END;
The critical path of PPNTS PROC is after applying of the above algorithm represented by the set CriticalPath := {IP, T1, C, T5, D, T4, E, T6, H, T7, OP} (see Figure 9). It is also clear that the given PPNTS may contain more critical paths with the same total time duration.
Further research in the field of PNTSs is mainly focused on the definition of additional unary, binary and n-ary PPPA operators preserving their specified properties, for instance, the binary SUBST operator that performs the substitution of the given PNTS for the selected place of another PNTS, and so on. In the field of the project management the research is focused on modeling complex processes, which individual activities can additionally share in parallel a selected set of the resources. These resources are then represented in the given PNTS by individual tokens located in the resource places of its selected net marking. Finding the time-optimal critical path of such a process as well as verifying the properties of the given PNTS that models such a process is generally a nontrivial problem and the use of PPPAs plays a crucial role here.
Another class currently being studied is the class of multiprocess Petri nets with time stamps that represents the generalization of the class of PNTS. The given multiprocess Petri net with time stamps then represents the finite set of processes each of which is modeled by a separate PNTS that share a common set of resources modeled by its individual resource places and their tokens. Many of the studied properties of the multiprocess Petri nets with time stamps are similar or identical to those of the PNTS class and they allow for a further generalization of the concept of a critical path formed by a sequence of activities of the process modeled by the given multiprocess Petri nets with time stamps.
This chapter was supported by the SGS at the Faculty of Economics, VŠB-TU Ostrava, under Grant Evaluation of comparative applications using cognitive analysis and method of data envelopment analysis, number SP2018/146.
Policies of educational institutions, their vision of the future, and what type of students they want to raise are important (one-dimensional academics or multi-faceted academics who are artistic and knowledgeable about physical development, etc.). Through the active music education approach and methods, it will be possible for individuals, who will develop to be versatile, to discover themselves and their learning styles.
An example of this system in the world is the elementary music and movement education in Finland in which students gain experience by talking, singing, accompanying to the rhythm of words, exploring, playing games with and without rules (sound games, etc.), and dancing. The work of adapting the elementary music and movement education, the essence of which is learning through experience, in Turkey continues. Since there is still only a single type of music teacher trained in Turkey, the desired results have not yet been obtained. Programs that train music teachers have a general quality [1, 2, 3, 4]. No expert music teachers are trained for primary schools, secondary schools, high schools, fine arts high schools, and universities. Only music teachers, with general knowledge in music education who are partially specialized in their fields (experts who partly play instruments such as piano, recorder, baglama, guitar, and sing), are usually trained. The active music education experiences of these teachers are limited to special teaching methods and games, dance, and music lessons (among the courses added during restructuring by the Council Higher Education in 1996).
Firstly, since the academics, who know approaches and methods of active music education, who will apply special teaching methods and games, dance, and music lessons, are not trained in music education departments, there is also no willing and qualified instructors who will give music education and teach courses in basic, pre-school, special education, and physical education departments of universities. In the music, education departments only train a single type of teacher. According to the study data of Duru and Köse [2], in Turkey, one type of music teacher is educated for general music education; in Austria (University of Graz), two types of music teachers are trained for general music education and instrument training; and in Finland, two types of music teachers are trained for general music education and preschool music education (the University of Lahti and Lahti Polytechnic). In Denmark (The Royal Danish Academy of Music and the University of Copenhagen), two types of music teachers are trained for music schools and general music education, and in the USA (Texas University of Technology), four types of music teachers are educated for the basic fields of group music, choir, orchestra, and piano.
One of the best and most successful examples of this system in the world is the elementary music and movement education in Finland in which students gain experience by talking, singing, accompanying the rhythm of words, exploring, playing games with and without rules (sound games, etc.), and dancing. The work of adapting the elementary music and movement education, the essence of which is learning through experience, in Turkey continues. These studies were initiated under the leadership of E. Zuckmayer in what is now called Gazi University the Department of Fine Arts, Department of Music Education, and continued until the 1970s. Muzaffer Arkan has made efforts to promote Orff-Schulwerk or elementary music and movement education in what is now called Hacettepe University State Conservatory. Since there is still only a single type of music teacher trained in Turkey, the desired results have not yet been obtained. Programs that train music teachers have a general quality [2]. With this “music education program,” that has a generic quality, training people who will both be expert music teachers in their fields and can provide basic music education and who will also have conservatory level knowledge is no different from expecting a frog to turn into a prince, as in the fairytale. Is it possible, through graduate education, to train academics and fine arts high school teachers, who reflect upon and address the basic problems of music education in Turkey, with only a handful of eager and qualified graduates? Not quite. The most important thing at this point is to increase the quality of music teacher candidates during their undergraduate education. These academics and teacher candidates are required first to achieve the minimum score in foreign language test (50 points) and academic personnel and postgraduate education entrance test (50 points). For graduates of other departments (engineering, pharmacy and science, and social sciences), the minimum passing score may not constitute a problem. However, it is not easy for candidates who have been educated and graduated in the field of fine arts to achieve this minimum passing score. Candidates are required to receive special training and courses to train themselves in their area of specialization.
In 1996, changes were made to the level of undergraduate and graduate music education under the name of restructuring by CoHE. Lessons were rearranged and in the music education program, especially the courses related to the field were tried to be standardized. However, when the music programs of other countries are examined, it is seen that the same departments in different universities do not follow the same curricula. Expecting the same departments in different universities in Turkey to follow the same curricula is not a very accurate approach in terms of the conditions of the educational environment. For example, when the music departments of Selçuk University and Gazi University are compared firstly, sociological, cultural, and economic differences according to their locations, the physical conditions of the departments, the levels of the students entering the departments, the number of instructors and their interests draw attention. Although the Council of Higher Education (CoHE) attempted to establish a standard teacher profile under the concept of restructuring in 1996, even in the current program, each department continues to educate teachers with their characteristics [5].
It is important that individuals, who have graduated from music education programs focused on raising the standard, a single type of music teacher, especially those who will give basic music education at universities, conduct their masters and doctoral studies on active music education approaches and methods. This will ensure a music education with consideration for the needs of the classroom and pre-school teaching. However, music department students at the undergraduate level do not have the chance to experience the practices and methods of active music education apart from special teaching methods and games, dance, and music lessons (among the courses established as part of a restructuring by CoHE in 1996).
To enhance these experiences, students firstly need organization of classroom environment, accurate determination of instrument needs, a workshop class with Orff instruments, receive specialized training, which will enable them to experience the active music education approach (as Orff-Schulwerk, elementary music and movement education) and its methods (as Kodály, Suzuki, Dalkroze et al.), from expert educators and discover their learning patterns.
At this point, the policies of educational institutions, their vision of the future, and what type of students they want to raise are important (one-dimensional academics or multi-faceted academics who are artistic and knowledgeable about physical development, etc.). Because through the active music education approach and methods, it will be possible for individuals, who will develop to be versatile, to discover themselves and their learning styles. For this purpose, the use of the experiential learning style test developed by D. Kolb (1984) will also be valuable. In studies conducted in both music departments and other academic units, the tests developed by [6] for the determination of the learning style of individuals (Kolb 2.0 and 3.0) were used in the studies of [7, 8, 9, 10, 11], however, it has not been examined whether the individual explored the experiential learning style or not through active music education approach and methods. Unlike writers who draw attention to the genetic and biological characteristics of the learning styles, Kolb views learning styles as permanent and durable states formed by the interaction of people and the environment. Individuals prefer different learning styles or use a few at once. According to this theory, four learning stages emerge as concrete experience, reflective observation, abstract conceptualization, and active experimentation. For example, in learning through concrete experiences, it is emphasized that it is considered important to understand the current experience and solve problems rather than understanding theories or generalizations. At this stage, feeling the situation is more important than thinking about the subject [12, 13].
The aim of the study: This study has been conducted to understand how the students involved in the study internalized, or learned, their experiences of elementary music and movement education. For these reasons, in this study, an active music education approach and methods that will enable the individuals participating in the study to discover their learning styles were applied in basic sciences and music education. Besides music educators who will give basic music education, it is also important for students who will take this course to explore learning styles with an active music education approach and methods. This is because, in this study, the experience is created with both the educator and the student. In this study using elementary music and movement education, at the core of which there is experiencing and discovery learning, answers to the following questions were sought: What are the behaviors that individuals adopt through elementary music and movement training? Are individuals willing to learn by experience? Is the environment (class size, class order, etc.) appropriate for experiential learning? What kind of an impact their primary, secondary, and high school experiences have on music education and teaching? What is their perception regarding music education and teaching?
This study includes the applications conducted by the researcher alone for understanding and solving problems emerging in the Department of Basic Education, music education class. Due to the role of the researcher as a participant and also a data collection tool, the study is action research [14]
The study group consists of n = 77 students in the Department of Basic Education.
To determine the learning styles of n = 77 students participating in the study, Kolb-3 learning style test was used. For the participating students to have in-depth knowledge of elementary music education, a semi-structured interview form was applied. This interview form is a written assignment that summarizes the activities carried out for one semester. In this assignment, students were asked to define and interpret the concepts of elementary, music, Orff-Schulwerk, and movement training. Additionally, the answers to the following questions were sought: Do the activities carried out for 14 weeks of lessons, in the basic sciences program and music education course in the fall of 2017–2018 have any relation to the elementary music and movement training? Could you comment on learning styles that emerge according to the Kolb-3 learning style? The writing format of the assignment was APA 6 style, which is the preferred format in academic writing, with the font Times New Roman 12 pt, in a justified alignment format. Also, the template consists of two sections, the introduction and the conclusion. In the introduction part, the concepts of elementary, music, Orff-Schulwerk, and movement training were examined. In the conclusion part, the answers to the following questions were sought: Do the activities carried out in the basic sciences program and music education course in the fall of 2017–2018 have any relation to the elementary music and movement training? What is the experiential learning style test? How do participants view their learning styles at the end of the test?
In the Fall semester of the 2017–2018 academic year, during 14 weeks of lessons, observations, and experiences, differences in students\' experiences were noted in the individual class observation report.
A study group was created in the WhatsApp application to reinforce and recollect the study. In this group, video summaries of the practices performed every week according to the micro teaching technique were recorded and shared and discussed for 5–10 min. The participant students, who made use of the videos recorded as a result of 14 weeks of education, formed groups consisting of 1-2-4 or 6 people and made presentations about nursery rhymes and counting rhymes. They made presentations about how they could apply elementary music and movement education practices in the classroom teaching course and gained experience.
The answers to the semi-structured interview form were coded with the MAXQDA 11+ software.
To understand the ways of gaining experience in the basic education, music education course, KOLB-3 experiential learning style tests the Turkish validity and reliability study was performed by Gencel [15]. To increase the internal reliability of codes created by the MAXQDA 11+ software, the coding was repeated by the researcher after 14 weeks (after the 2017-2018 Spring semester). Based on the title of the study topic, participants\' experiential learning test results and their opinions on the test, the concept of elementary, elementary music and movement training, positive aspects of experiential music and movement practices and factors that prevent experiential music and movement practices were evaluated. The relationship of themes and codes was reviewed with an educational scientist who is experienced in computerized qualitative research and coding and unrelated codes were removed.
According to the data, regarding the Kolb learning styles, 30 of the 77-people students’ group were converging, 21 were assimilating, 16 were diverging, and 10 were accommodating. As understood from the data, 51 students had abstract conceptualization skills and 26 students had concrete experiencing skills. While 41 students were observed to learn through active experimentation, 36 students learned through reflective observation (Figure 1).
Learning styles of students participating in the study.
P37: I have a converging learning style, according to Kolb. I can learn with abstract conceptualization and active experience. That’s why I first put the actions that we learn in the course into perspective then I put them into practice. With the practices and experiments that we do in class, I try to acquire the right information. Since everyone in the classroom offers different ideas and we put these ideas into practice, I can approach the subject with different perspectives. I can participate actively in the course. P30: During our practices, we performed listening rhythm and harmonizing studies. We started from a specific topic in certain practices; it is sort of my area of expertise to concentrate on one thing, as in my learning style. In my learning style, it is essential to organize information and to focus on specific problems. P22: Converging individuals are defined as people who use abstract conceptualization and active experience learning stages and practical applier of ideas. They were expressed as deductive people with problem-solving skills, interested in technical matters. During class practices, generally, I observe first, and then give my opinions, if I have any. Since I think learning by doing is more permanent, I care about practice. So I participate in the practices and activities in class. So, I partially fit in with the converging style in the Kolb inventory. P55: I do not act without thinking. I am cautious. I\'m more inclined towards objects and symbols. I learn best with projects, practical assignments, small group discussions. My learning style helped me in class. I am learning through problem solving, decision making, logical and systematic planning of ideas so in the course so I have always asked myself the question of “How.” “How will I do it? How will it happen?” I received the answers to these questions by participating in the applications. I\'m good at organizing information, focusing on specific problems. I am successful in problem solving and technical issues rather than social and interpersonal issues. However, in the course, I participated in activities as a part of the group. I have expressed my ideas and voiced my opinions about things that concerned me. I learn by doing, experiencing. That\'s why I have always been actively involved in the course. Again, it was me who found the mistakes one by one on the board. I constantly questioned the points that I found problematic and expressed my opinions, then we reflected on them together (Figure 2).
The comments of the participants on the converging learning style.
P72: While practicing in the course, I always asked myself questions such as; Why will I learn? How will I learn? and I found the answers in practice. I like to work individually, which is a characteristic of my learning style. However, thanks to the practices we do in the course, I\'ve warmed towards group work and I have observed to be more efficient than when I worked on my own. My learning style made me ask more questions during practice. This allowed me to gain a more inquisitive identity and more awareness. P10: In fact, since observing and applying is important in most of the practices we do in class, this learning style helped me a little about learning. The fact that thinking is at the forefront in this learning style has enabled me to benefit from it in this regard since the practices we perform are related to thinking, researching, exploring. I prefer observation as it is in my learning style. I think I have developed myself in terms of learning through group work along with the course. P70: I learn by observing as in the assimilating learning style. I first examine, see then I start to practice it myself. Being preoccupied with details, I ask questions in class such as "Why are we doing these movements?" or "How and where are we going to use them?.” P61: I think the result of this test is right for me because to learn something in my life, first I would like to have information about that thing and to observe, to see how it is done. Before active participation, I like observing. If I were to interpret the Music and Movement course, which we took during the semester, in this context, I think that I have been forced to participate actively as an individual who prefers to do observation and that this has helped me improve to a certain degree in active participation. Because, in our lessons, I saw a process more appropriate for the active participation structure, rather than observation (Figure 3).
The comments of the participants on the assimilating learning style.
Sample expressions selected from several participants at the end of the study are as follows: P14: According to Kolb’s learning style questionnaire, I was in a converging style, but I see myself in the accommodating learning style. The reason for this is that I take action without thinking; I reach the truth through trial and error. I prefer active participation and learn things by doing-experiencing while learning. This is how I learned music through the music lesson we were taught. P48: To be honest, I can hardly understand the narrow definitions of visual intelligence, auditory intelligence so it was very useful to do this test and find out my learning style. The diverging learning style summarizes me very well. I\'m a serious observer. Even if I understand things, I refrain from action, think and judge within myself, ask questions of why and how but participating or sharing my ideas is not for me. That\'s why I never mentioned my ideas during practice. Things come to my mind because I pay attention to the course, it is my favorite lesson, but I always have trouble in participating in the class. After discovering my style of learning, nothing has changed. I still observe in the same way but I either participate because I have to, or I participate without sharing my ideas and thoughts. By watching, I learn in a calm, patient, elaborative, very detailed and objective way. However, I like working individually. P43: I think this lesson fits my learning style. It is because diverging learning needs sensory organs while learning. We are actively using our sensory organs in our course. Diverging learning mostly uses sight. In the course, we apply the movements by looking at our teacher. This means that we use our vision. I find the applications in the course parallel to my one-to-one learning style. P56: When shaping my ideas, I consider my feelings and thoughts. The reason why this learning style is called diverging learning is that individuals with this style show better performance when they are asked to create alternative ideas in concepts such as brainstorming. Thanks to our practices in the course, I warmed up more to group work and observed that I had more efficiency than individual study. My learning style made me ask more questions during practice. This allowed me to gain a more inquisitive identity and more awareness (Figure 4).
The comments of the participants on the diverging learning style.
P42: I think I fit the accommodating learning style. I can easily relate to people, adapt to change, I like change. I always plan before doing anything. Planning is important to me. I trust my intuition. I prefer active participation in the lessons. Because as I mentioned above, I think a student learns best through doing and experiencing. When studying, I think that the more the subject appeals to the senses, the better the student understands. P47: I have an accommodating learning style. And according to this style, I have a character that can establish easy relationships with people, and sometimes I am very impatient. I\'m open-minded and when I see that something we’re doing is lacking, wrong or unusual, I speak my mind directly. However, individuals with this style are said to be able to adapt easily to changes, but I am a little stubborn and I can\'t easily adapt to changes (Figure 5).
The comments of the participants on the accommodating learning style.
P75: Elementary means simplification of complex education. The concept of elementary that facilitates learning has an important place especially in terms of the education of children. P15: The elementary concept of Orff-Schulwerk refers to making education basic, easy and simple. Originating from the Latin word “elemetarius” means belonging to the oldest, beginning and basic elements, first-hand. We can say that it means transforming a complex structure into a structure that is easier and more solvable. P61: Breathing, tapping out the rhythm with the heartbeat, making sounds with tongue movements and giving music meaning with various body movements are the things we can frequently see in this concept. It is also possible to participate in this structure with simple instruments and simple dances that express the structure of music. In short, it is an area that individuals can use to express music by simplifying and transforming the complex theoretical teachings of music. It is expressing music by using the sounds in nature and sounds, rhythms and movements that are present in the human body (Figure 6).
The comments of the participants on the concept of elementary.
P48: Orff Schulwerk and movement education is the expression of emotions that come from inside of the individual. It teaches children to move together, and the harmony that occurs when they adapt. It allows productivity through cooperation. The outputs provide enormous benefits in the cognitive, social and emotional areas of children during the process, they provide an aesthetic sense. P39: In Orff\'s exercises, the student-centered course type is taken as a basis. In these studies, the teacher determines the principle, purpose, method, and content, informs about "What?" to do, however "How?" it will be carried out is determined together with the students. Each work process is divided into stages in itself. The main methods of this practice are improvisation, gameplay, discovery, re-finding, experiencing, transforming and discussing. Among them, especially improvisation and gameplay are predominant. Also, various social forms such as group work, circle form, U form are frequently used. P7: In the Orff system, students discover and acquire knowledge. This makes the information permanent. Lessons become more fun and enjoyable. Music teachers\' learning and using these methods well makes the lesson more enjoyable and ensures that the knowledge taught will be permanent. It will be useful to apply it to all schools. P27: Music is the means through which people express themselves. What’s special about Orff teaching is that it enables the individual to move freely, improvise, express himself or herself with music. This art does not belong to a single culture in the world and is universal. Children need to develop their imagination. The formation of music is a result of movements. Music and movement constitute a whole. Even the generation of our voice occurs as a result of the movement of our vocal cords. Children\'s creativity improves. There are no single set of straight rules, and everyone is free to make the music they want. It is out-of-the-box music education. Children engage in creative activities by singing and playing. Music and movement arise as a result of children\'s free movement in an environment where they can think freely (Figure 7).
The comments of the participants on elementary music and movement education.
P7: In music lessons in our public schools, students sit at their desks without moving and the teacher talks. There is not much participation in class. In this traditional teaching, there is information and music is taught from the information. However, in the Orff method, the student is active, interacts with other children and the teacher guides. The student is the one who thinks, puts effort, researches. Improvisation encourages students to think in different ways. It is also a useful method for mental improvement. In the Orff system, the student explores and learns the information. This makes the information permanent. Lessons are more fun and enjoyable. Music teachers\' learning and using these methods well make lessons more enjoyable and ensures that the information is permanent. It will be useful to apply it to all schools. What I do in the course will enable me to spend a lot of fun and active time with children when I start my profession. The student is completely active. With confidence, they will express themselves and their thoughts better. Even we find this lesson very different and have fun, I think children will love it. It will have an illuminating effect on their future. It is amazing… P10: Another one of its effects on learning is that what is to be learned becomes a game with music and dance. In this way, especially kids who like to play games will enjoy learning more quickly without realizing it by taking pleasure. Such learning will also last longer and be permanent. Also, since this education style is aimed at all individual s of all ages, everyone becomes active within the learning process. In other words, not only those who are skillful on that subject, but everyone gets involved. In addition to developing creativity develops, the social and mental states of individuals will also improve. Through the development of creativity, it contributes to individuals\' gaining certain skills in art. Also, since this training is carried out in free environments, through music, meaning through factors such as movement, rhythm and dance, individuals get the opportunity to express themselves. That is because music allows people to express themselves better. This type of education will improve the self-confidence of individuals since they can express themselves better. Since this style provides learning by experience through music, it will arouse curiosity and interest in individuals (Figure 8).
Positive aspects of experiential music and movement practices according to the participants.
About 20.8% of the participants stated that compared to previous experiences in music education, it may be difficult to implement music and movement practices due to crowded classes and material problems. While 11.5% of the participants have stated that long practices have a negative effect, 7.3% stated that the fact that Orff-Schulwerk or elementary music and movement practices in Turkey are not common and the classes where the practices are to be carried out are small are also preventative factors. About 10.4% of the participants stated that the division of the class into small groups of 20–25 people would reduce the factors that would adversely affect the practices.
P11: The elementary system can be applied to courses with suitable lessons and fewer people. It is not a system that can be applied in the classes where the number of people is high. Because for the movements to be carried out as a group and effectively, there should be fewer people. It is not a suitable system for every course. It is a system that must be implemented in classes since it is a system where the children express their thoughts, present their ideas, and the teacher is a guide and the students are in the center, as opposed to the memorization system. In our country, many academics and teachers are working on this subject. However, there is no institutionalization in this regard. P50: This education style is unfamiliar to me. Because I did not receive this education from the moment I started school at a young age and I got used to another style. And one of the things that are hard for people is to give up their habits. However, if this education is given to new generations, it will be easier for them than it is for us. P32: Applications with too many people in a small class sometimes lead a few students to get disconnected from the course, fail to establish one-on-one communication and to generate enough productivity. Therefore, the course environment and the class number should be taken into consideration. For that purpose, splitting into four different groups and changing the lesson times at the beginning of the semester have been much more useful. Consequently, for music and movement education, a special classroom and a limited number of students are required. P60: Since I have not had a proper music lesson until now, my skills and interest in music had become rusty. Because, since elementary school, we generally had free time during music lessons. P45: The state in particular should provide the necessary instruments for the universities and primary schools. And finally, our practices could be more energetic. The reasons why it could not be more energetic were due to both the size of the class and the unwillingness of students. Maybe a little bit of technology could have been utilized to make it more energetic; for example, a projection device could have been used to reflect on the board, we could have watched videos about Orff education, and our teacher might have wanted us to recreate what we saw similarly. Our teacher could have made an observation and joined when necessary, more fun activities could have been done in the classroom altogether. Songs could have been faster and played on the computer; we could have accompanied the song with our bodies and voices (Figure 9).
Factors preventing experiential music and movement practices according to participants.
In the preliminary interview with the students who participated in the study, they stated that their musical experiences were quite limited. The majority of the students stated that music courses in primary, secondary, and high schools were empty and meaningless, that they solved tests in music class in middle school and high school and those who studied music at the primary or secondary education stage stated that they memorized only the names of notes and rhythmic structures in the course. One student stated that their teacher would sing to them and they would sing along the song while sitting at their desks.
In light of the data obtained from the study, to ensure active participation and observation of the students, the class of 76 people was divided into four groups consisting of approximately 20 people. To provide a "U" shaped seating arrangement, the desks arranged in rows were rearranged with the students. All students participating in the study found it positive that the course was taught by dividing the class into four groups. Also, they stated that with classes limited to 20–25 people the lessons were more comprehensible and everyone participated in class. However, some students stated that since there are crowded classes in the Turkish national education system, it would not be able to attain this ideal number and therefore these practices could not be implemented.
The students who participated in the study stated that everyone\'s opinion was taken during the 1–1.5-hour studies and creating ideas together had a positive effect on them. However, the lessons were long and exhausting since it was about interpreting and re-synthesizing instead of memorizing information. This is the negative aspect of the study according to them. They stated that at the beginning of the study, listening to sounds in nature or class during practice, speaking and imitating these sounds, imitating these sounds by hitting their bodies with hands, finding the corresponding sound of imitated sounds with Orff instruments and finally, transferring them on a graph (graphical notation) to A3-sized cardboard of using EVA materials increased their interest in the sounds of nature and the duration of these sounds. They stated that the same or a similar study could be used to improve the hand skills of students in the first and second grades of primary education. In the application of the study, problems were observed while making the sounds on cartons for the duration of the sound. Therefore, the studies were continued by transferring them to nursery rhymes and counting rhymes. In similar studies on nursery rhymes, sample works similar to the aruz prosody were developed with the participants. While working on the rhythm of syllables and words, they also made sure that the syllables and letters were uttered on time, with the correct emphasis and toning. Since the works resemble the aruz prosody, some information from Divan literature was transferred to the Turkish language and literature (Figures 10–12).
Rhythmic expression of words with dots and lines: it is read by putting dots on short syllables, long lines on long syllables. (A counting rhyme similar to “Eeny, meeny, miny, moe”)
Rhythmic expression of words with lines and wavy lines: For each word, it is read with short, long, straight, upward, and downward lines and wavy lines
Rhythmic expression of words with geometric shapes: before you read the letter O, a half circle is drawn upward from the center of the circle and breath is taken. Then the letter “o” is pronounced; “pi-ti,” “karamela,” and “sepeti” syllables pronounced in such a way corresponding to each edge of the square, rectangle, and triangle for as long as the edge lengths.
In similar studies on nursery rhymes, sample works similar to aruz prosody were developed together with the participants. While working on the rhythm of syllables and words, they also made sure that the syllables and letters were uttered on time, with the correct emphasis and toning. Since the works resemble the aruz prosody, some information from Divan literature was transferred to the Turkish language and literature. Therefore, it is recommended for future studies to be carried out in similar ways.
At the beginning of the studies, listening to sounds in nature or class during practice, speaking and imitating these sounds, imitating these sounds by hitting bodies with hands, finding the corresponding sound of imitated sounds with Orff instruments could be useful.
Participants stated that transferring them on a graph (graphical notation) to A3-sized cardboard using EVA materials increased their interest in the sounds of nature and the duration of these sounds. Similar practices to the same ones can be used to improve the hand skills of students in the first and second grades of primary education.
All students participating in the study found it positive that the course was taught by dividing the class into four groups. Therefore, it is recommended to perform elementary, music, and movement practices in this way. It is recommended that the courses are taught in classes limited to 20–25 people for the studies to be more comprehensible.
For the training of music teachers who are experts in universities, high school and fine arts high schools, and primary and secondary education, a music department must be established in the first place and under this department, preschool, basic education, special education, vocational education departments must be established. In the current structure, under the fine arts department, there are only music and drawing sections.
Instead of training music teachers with general knowledge of music education who are partially specialized in their fields (experts who partly play instruments such as piano, recorder, baglama, guitar, and sing), the departments dedicated to expertise, such as those in conservatories training specialized teachers in various fields, must be established. For example, theory and sound education departments for instruments (piano, string instruments, etc.) should be created. Applications conducted between 1994 and 1998 should be examined as examples [4, 16].
The limited experience of teachers, who are trained as a single type in general music education, should be enhanced with special teaching methods and games, dance, and music lessons (among the courses established under restructuring by CoHE in 1996). To enhance these experiences, students firstly need organization of the classroom environment, accurate determination of instrument needs, and workshop classes with Orff instruments.
Students need to receive specialized training, which will enable them to experience the active music education approach as Orff-Schulwerk, elementary music and movement education and its methods as Kodály, Suzuki, Dalkroze et al., from expert educators.
The Ministry of National Education (MoNE) and universities are required to ensure that active music education is provided in a standardized way. Nowadays, online training is carried out by private firms using the names of MoNE and universities. However, there are two types of training, namely, [17] training not based on active experience and group work, and practical training [18, 19] based on active experience and group work. As the name suggests, these types of training, which must be based on an active experience, cannot be delivered online. Because, in these training sessions, body memory is in the foreground, and these training sessions are incomplete when verbal and mathematical responses are not encoded in the body. Therefore, participation in practical training should be provided.
In solving the problems experienced in music education programs, the opinions of experts in education and music education programs should be consulted; several types of music teachers should be trained instead of a single type [20, 21, 22]. Music education programs resembling each other, having common points are positive for the credit system. However, it is important to train music teachers considering the needs of regions and cities in Turkey. For example, according to this need, while training music teachers who will give basic music education in Central Anatolia, specialized music teachers who will work in universities and fine arts high schools in cities such as Ankara, İstanbul, and İzmir may be trained. In identifying these needs, academic studies and comprehensive surveys conducted by professional firms for educational institutions are required.
National and international workshops and conferences, which will allow music department students to experience an active approach and method of music education in Turkey, should be organized. These studies continue with the first International Istanbul Kodály training days, Orff-Schulwerk consultancy center and the national training and projects of Turkey [23, 24, 25, 26]. It is important to ensure that individuals who work or will work in both academic, private and public institutions participate in these training sessions with the encouragement of corporate managers. The creation of joint programs in Turkey may be enabled with the studies to be carried out with the Kodály Institute and Orff-Schulwerk forum. Significant improvements have been achieved with these studies conducted in both Europe and America.
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