CC BY
21
UDC 378.147
DOI: 10 .23951/1609-624X-2018-8-231-237
PRACTICAL METHODS OF ORGANIZING INTERDISCIPLINARY STUDYING IN THE FIELD OF BIOLOGY AND CHEMISTRY
A. S. Minich1, I. S. Bondarchuk2, M. V. Krivenkova3
1 Tomsk State Pedagogical University, Tomsk, Russian Federation
2 National Research Tomsk State University, Tomsk, Russian Federation
3 Tomsk school № 14, Tomsk, Russian Federation
The article provides practical examples of using an interdisciplinary approach to solving typical tasks in biology and new algorithmic approaches to determining the kinetic parameters of simple reactions in chemistry. It examines the appropriate methods to use the most common MS Excel toolkit to obtain more exact solutions of the considered tasks for the extended type values of the type of mathematical models of the desired parameters. Presents the teaching methods and practical examples of solving problems in the field of biology and chemistry which were implemented in the framework of an interdisciplinary approach, as well as their study of mathematics and computer science. The importance of interdisciplinary study with mathematical modeling of the processes that predict a change of state in biological and chemical systems is shown. These considerations contribute to the enhancement of knowledge and research, including when solving the problems by replacing numerical data with mathematical expressions. At the same time, problem analysis, solution methods, adequate assessment, visualization and interpretation of results are required. The presented practical approach effectively contributes to the implementation of the educational standard with the formation of graduates' competencies, focused on the willingness to independently solve the research problems and self-development throughout the life.
Ключевые слова: interdisciplinary studying, pedagogical activity, practical methods, MS Excel spreadsheets, mathematical models, population dynamics, predator-prey model, physical chemistry, rate constant, reaction order.
In recent times, number of publications devoted to computerized learning has significantly increased due to saturation of educational institutions with personal computers [1]. The publications discuss the computer-methodological component of the educational process and the interdisciplinary component of studying the subjects of the natural science cycle (primarily biology and chemistry) which is associated with the expansion of information accessibility. Contingently, this topic can be divided into two parts: the development of some pedagogical theories of interdisciplinary studying and the practical methods of using specialized software for simulation processes in the case of natural sciences.
Training of university students for future professionals' pedagogical activities is considered as "an object of interdisciplinary research with its relative entirety, self-sufficiency and insularity..." and "which is being trained in personnel training". Educational institutions are considered "a place of active socialization, sociocultural immersion of future teachers and increasing intellectuality, morality, civic consciousness" [2]. The use of certain "foundation technology" for "creating psychological, pedagogical, organizational and methodological conditions for actualization educational units ... with subsequent theoretical consolidation of structural units, explaining their essence, integrity and interdisciplinary studying toward to growth of professional knowledge and the teachers individu-
ality" formation are discussed [3]. Practical examples in such publications limit computer functions to the simplest calculations.
Specialized software or development of new software for the simulation processes are required as a part of course unit using it either as "black boxes" or extra time for teaching students because of improving skills in high-level programming languages [4-6]. Flatbed technologies, Mathcad computing, 3Ducation, CASE tools, etc. are considered [5, 6]. A training event for demonstrating realization of the simplest model by two-species ecological interaction "predator-prey" using system dynamics tools in the software environments ModelMaker, AnyLogic and utilizing of the agent-based "AnyLogic" tool modeling are described [6]. The training course specifics of Python programming language for solving biological tasks were considered [4].
The presented approach implements practical solutions using MS Excel spreadsheets focused on interdisciplinary studying. This widespread software is installed on all personal computers. Studied MS Excel spreadsheets are involved in computer science university curriculum. Another "interdisciplinary" component of the proposed examples is mathematics, since there is a low level of remaining "school" knowledge, especially among prospective students. It would be appropriate to recall the quote from Immanuel Kant, contained in his famous work "The Metaphysical
Foundations of Natural Science" (1786). Kant says: "In any special doctrine of Nature there is only as much genuine science as there is mathematics."
The introduction of acquired knowledge occurs when students solve practical tasks of specific subject disciplines. The used methods, when properly combined, provide successful practice in realization of interdisciplinary studying. A particular roleinthis re gard may be new practical approaches for solving typical tasks based on the available sets of personal c omputer tools [1, 7-13].
The following list of specific courses for solvmgex-amples within interdisciplinary learning approach is used:
- mathematical models in biology;
- mathematical models of chemical kinetics in physiae1 chemistiy;
- mathematical methods of solution differential equationsand ta^arinte^ation;
- methods of applied mathematics relating to statement and solutioeoptirmartion tasks;
- MS Excel spreadsheets as computational software wrth oet oe sMistic;^ fonetiont and omimizatioo "Solver" for realizinganon-linear methodof the generalized reducti on gradi ent.
A eoodaracoicalexnnplearomlheviewpoint ef iza terdisciplinarity for students of the Faculty of Biology is stuhyino vlonsioal problem of jjojtm-rtion Vyaamieo "predator-prey" models which was analyzed in the early 20tiiceotury Inrastinalrs, this observOinn lees boen correlating well with commercial information of Canadian fur Uah-nr srmoomen for a ieng lene iini S5],
This mathematical model could be defined by structural on-mme ssewn io Fib^. 1. Thia sevirm ifacrlbes the balance of available resources and their consumers; blocksdeleomioz rfvreynumZ>er5nwhicVaze reopenoil ble for population growth, predator mortalityrateand realizaVlvintercaiiony. Predatort"conzymn" pie—con-sequently, before the first block in first equation is a minurslsvu ond io the tsrhon^ ^a niuwbevauzeeaymg preys increase the generation of predators.
dG \ hC.Y
Change rate ' 1
dt | Mortality
in ]rey I = — pruy due to —■
population G predator
J
Reproduction rate of prey
Change rate in predator popuioiFoei'
.rt ~dt
u(bGX)
+
Reproduction rate of predator per prey
"Hunger" predator mortality
dG = rG -(bGX )- aG2
dX dt
= u (bGX )- sX - aX 2,
where re, G - is the predator and prey density (biomass per umt aoea); re-it speti=c gr=wtnrateofprey population tn the absense of a predator; b -isthepye-erfrc rate o= jrre datax conssimption of oren; u - io the effectivenessoO«diyertion» prey's biomare by it predt ator; 5 - is the specific rate death predator in the absence of prey.
A scrsrOetien) of mtving syatcmequationr for o time period 0 E [0; 475] is shown in Fig. iC, whesp rlie iniriol dot^(tio = (Syt xO) ; 4; ry= x=i = 0) = 2aed pas rameters oftheeqnpai°ns saitem are assigned.
tO aiooct solution oe ths systemofordmnyeifter-ential equations (ip has besn achievee using alin ex-phcits;nler ecPemr:
hx x+i Xj
dt Ac '
where At is the time integration step rnpioes din raits-lations vtability nog tin inVex r chpsacteripes titr dutev ber of time rtap.
Cunsoqueghy1 thf diifdiritncaa: roheme foe rolsmg e system of differentia 1 ermations is given by the formula:
-£=«(*, .y),
— = V(x, y),
can be written as
xi+1 xi At
y,+i~ =
■ = u
At
(u > y,)> v (x+ie —
Fig . 1. The structure of the mathematical predator-prey model
If the functional relations in this model were replace d[14, 15], we coutd obtain a mathematicalmodel of a predator of biomass changes and prey over time t, which with rrhe inclusion ofthelotiaticelament a Oave the form:
The integration eOep Ot can Or prelimiprrily esti-mated ao
Ai<minjfrG0) 1 ; (wbG0nt0)
In case of sahhittn idttabelley( the time eten ¿ad sOoold be reduced.
Fig. 2 stows a s»rea0shueO eP dalftion ucioe thU Excsl. Fysrealizationob theLoC0d-Volteyamodell students shout! have mUsiate iti Ptolohn tn tiir context of czredteoo siatemenr rnV mbhematicaimodnisoionuiai tion, computuo noience ore taema op oeghtsthm alpgota-linn aeo tde simolastskills in tomsoof numerical methods pnd uncpu sppeoUshesle fan tralitetion of egmpnfational cakuiflwns ano gtap^sL^ta^c interposta-tion of the obtd]pedrefurhs.
A
2 Lotka-Volterra equations r = 4,5
3 with a logistic amendment b = 4,0
4 5 a>0 (predator-prey model) G„ = 4,0
— — rC — (bGX) - aG2 , s = 3,0
6 7 dx — = u(bGX) - sX at u = 0,5
- uX2. u ■ b = 2,0
8 9 10 11 *0 = 2,0
t G X a = 0,12
0,00 4,00 2,00 At - 0,05
0.05 3,20 2.32
12 0.10 2,38 2,49
13 0.15 1,70 2,50
14 0,20 1,21 2,39
15 0,25 0,90 2.21
16 17 0,30 0,70 2,01
0,35 0,57 1,79
IS 0,40 0,49 1,59
19 0,45 0,45 1,41
20 0.50 0,42 1,25
21 0,55 0,41 1,10
22 23 0.60 0,41 0,97
0,65 0,42 0,86
24 0,70 0,44 0,77
25 0,75 0,47 0,68
26 0.80 0,51 0,61
27 0,85 0,56 0,55
28 0,90 0,62 0.50
4,5 4,0 3,5 3,0 2,5 2,0 1,5 1,0 0,5 0,0
X 2,5 2,0 1,5 1,0 0,5 0,0
r- ~ai\ ^hC(ee
-Ti w V /
Ch)
Fig. 2. Spreadsheetforsolving predator-prey modelequations
At the end of the lesson, students should give an interpretation obtained results (Fig. 2). In particular, it is necessaty to determine the key featime s of this problem solution and the change in the number of preda-toos mdpreysOdampefercillation-in Fig.2,a).
In addition, the phase-plane portrait of the studied system of asedatosf andpreys nombeesotSifOdene stages of the process should be analyzed (stable focus Fig. 2, b).
In this case the critical point turns into a stable fo-cusandueluiion- inta utnei^daiodingbscin^i^^c^r^s. Regardless of the entry condition, the state of the system aftebsgmerime apgroactes the sOaiiduoiy anb Sendsto it at t^-x.
Anether sxfutpIeoCMS EecsI fbnrdf nal^^ iodhe construction of mega-formulas. In this case the spread-shtbtesnbe used as an ^^^ercbcdOT combinational programming. It can be useful for solving the problem ofg^netits. in dbrtmriVs. fbranalystsof fOenotn^pas ratio frequency among descendants during dihybrid crr^tdn^[^6, tty Ios fs to detcribod. Foroxamfls, flies, taking into account two pairs of alternative traits, show the aivisioninto phenatypicciafses 9: t: 3: 1 from the intersection of dihybrid among offspring. An experimental (fexp) following ratio: 135 : 51 : 54 : 18 was obtained. The traditional method (Fig. 3, a) involves separately the calc ulation of the theoretical fre-auencies
Zcexp
J i
^ f expect
for a given number of experimental fexp
and X crit subsequent compari
X2xp= 1.721 <xlrt = 7.815 , then the
fi
- f
expect
values, the
caiculatlon of x son.
OUert^ar A exp conclusion is formulated: the null hypothesis is ac-ref-eawheoihrvaiue ff thelrve 1 a = 0.05; the number of the species ratio of the phenotypic classes is
t :f:a:t.
Almost com^etely intermediate calculations for ede ^«^at^z^et^c^o ot thd ti^t^t^ttiOTa ^[d^fO^ien
e fOsd, eci - dliy yy iicasslt() h ff ostonmhty o0tye
Md Exfeetoolkif Is 2deo for :ff^itn^o^t^tatifdt witi fitmiu-las and arrays (yig. 3,ee
Theoe2tiledureof peosonal com-ugtr sottwere Oor genetics research[ts-20]issh°rcn ]2p],
Interdisciplinary studying examples applied to physical chemtsChe prssenSed he recent pulaSC^i^^c^ys [3, 7-13, 20, 21], generally, concern the MS Excel spreadsheets. The theoretical basis tf these eeemplet is the identification of the order of a chemical reac-tionaccorting to rhe oxyecimental t^S>ee io'-;}, where i = 0,1, ..., N, N define the size of the table in compiimfe mltr the dfpendenge of (Ce taopetlptrhuy reagent C on the reaction time t.
d matOemtticalmohelo f a chemical process is described by a differential equation, f-~kCP ■ <C>=°
where k, p is a constant rate and a re action order.
à
B
Null hypothesis: the ratio of the number of individual's
2 phenotypic classes is 9 : 3 : 3 : 1 _
3
FU rexpect h |Ci:i)V Cxf
e*- cu+- 135 9 145^3 0.706«-
e"- eu cu 51 e
e e cuT- 54 3 48,38 0,654
e e cu cu _5 i 5115 U215
total 258 16 258,00 1,721-.
C$9*D5/D$9
-Xexp
~SUM(C5:C8)
df =
3.CS
XlTil
815
= !TCUM(C5 CSi-1; -onr i\Tjc 5Q5 ;-i;
=CHISQ.INV.RT (C 11;C12)
SlmefU < Xmt h em II hypothesis Is aoeptoh.Thio ratioofthe nusaber of individual's the examined phenotypic classes is 9 : 3 : 3 :1
1
2 3
5
6
7
8
9
10
11 12
13
14
15
16
A B C D E F G
Null hypothesis: the ratio of the number of individuals phenotypic classes is 9 : 3 : 3 : 1 I a = 10,05 I
uuai r. ©
Ph r ^expect h Used a name Fexp for range C6:C9 aitiemiFecpect for range D6:D9
Fe)ae 9
e+- cu cu 51 3
e e cu+- 54 3 mula I
e e cu cu 18 larrav fo
(=SUM( (Fexp-SUM(Fexp) /SUM(Fexpect) *Fexpect)"2*SUM(Fexpect)/SUM(Fexp)/Fexpect)}
Xip = -r- 1,721
Xcrit ~ 7,815 —[=CHISQ.INV.RT(D3;(COUNT(Fexp)-l))
Since Xexp < Xcrtt the null hypothesis is accepted. The ratio
offum-tr.- cf iLiiv'duil's The ex^nined |jlierioty|jK clas^i
is 9 : 3 : 3 : 1
^^c^.i^.Spreadsheet of thefrequency analcsis teckctphesotypicclasses
After separation of the cariaMes and integration, an analytical solctSon of the differenti al e quati on is used to identify the order of reaction,
Coexp(-^^z), ifp = 1,
c-n -(i-pf b ', itdd
C
and the constant; rate from the prevtous eqnntion expressed énanexplicép form.
- ln(C0lC),/fu = l, ~ /{!i-I-),ifp:t-1.h
k=l g
C(-p __ yl-: ^ 0
Thr dmuoston of th- consto1 rate k ilepentts on the dimensionality ofthec oncenhration Candlhe der gret of tiae reaekian order/n.
The fasis of Ude proposed dpproach[l,7-l30 for solviog the °iveo and tiio smulnc jdaotdloms ottjoOa;oosId;ii.l chemistryrt the; formclattona f these prorinmops crte-mization, whenthe spread ofone of PathenPifiad desire! parameteui oi a mrthemnficrl mtidel is minimized.
For agivenocdenp, r wiot ccnstants iis calcuPalrf from inifp dataaheoeO
^(Co^<Cir,um=nt l^-c^eni-p),-^,
AecoMmg to the projDosed aeproachll,7l, the identificaffnof tiiereaction order is achievedby minimiuingofshe funutionai F t) —► mCt-which de-tftmines tVe sprevd of theratevonstivts tet kt by die standdrU deviatcn (quadrnticdevmtion) divMed by the averane vfPvs k :
r N ~|0,5
F (p) = l
el
-\2
" k1
N
Np t
The netessity oornos-dimenstonality of theex-pression (*) for F (p) is due to the fact that the calculated ren ft!,- duririg tas procediae of miyiimsiygval-ues F is very large with the orders of the number, de-penixgontfiep value.
Furthermore, oopresaaonr g*r were chasen Co Cidlsesr late these functionals because of the simplicity of im-fmdemenrlng their jmrecathpre V4 g-e oMrVes0 funatiosu Pnbf Exnd fODXV.U (...t andAVORAGE t-.-d-
FSgsre 4 thoect Ithe apyacotion oS ttse oOoritfun So identify the parameters of the kinetic reaction in ac-cardance with rxperimonltal daio [20t OSt IVTIS ]Eats al Solver [7, 8, 24] by mode "Generalized Reduced Gra-dienl" id us o°. The aertais uf dVo vailntSe aolln2 [roe rction podeo) cPangeewhiie tPeaixedvfPdee"intrgec") rerul4 ^ekln in sub 05. hfe wiadew "folned-- gr d— linedCy fepncuplceI F7, into which the formula for telcutetihi11tiie funotinnahty . entered.
i= 1,2, ... ,N .
0,01000 0,00740 11,712'
P
+A 1 -P
0,00630 11,746
0,00550 11,688
0,00464 11,552
0,00363 11,699
0,00254 11,748
-L
F G H
variable cell for Solver
2 <-reaction order
<-auxiliary value
k 11,691*—|-AVERAGE(C3:C8) |
T-| 0,0056^ Objective for Solver | ~5TDEV.P(C3:C8)/F5 ]
|-IF(F$3-0; LN(B$2/B3)/A3; (B$2"F$3-B3*F$3)/F$3/A3) |
1^0.4. I\U$5 Excel epreartherf fsrceloulalien elra^^ ^e^r^synt and integer reaction order
Io scoe remove yolimit of the order of the reaction
ts an in!eger,4hen tloo s^!l]^eg foir selutions wllo l^ol^erc
^ornrod foi* fonvtional, even ^^^^li negative values. The
pto-o-ed noul1t<t^ni]enn^t4s appros^le ^o solyingpeoba
lems of physical chemistry allows to reduce signifi-
crr^^^g the oemsnMipnul iSt"it oilf)rogbammirig olrio-^
rihhuhs, to Ompeoo^ethh^ c^e^^c^hhlo^^^^rhi^c^uraoy ar^d tonx-
jrnntS rhe r^niye oo numarict^l types (for wlnolg- raiuair"1
and real, positive and negative) orders of chemical reactions for analyzing parameters of kinetic equations.
In conclusion, the teaching methods and practical examples of solving problems in the field of biology and chemistry, which were realized within an interdisciplinary approach as well as their studying mathematics and computer science, were presented. The importance of interdisciplinary studying with mathematical modeling of the processes predicting a change of state in biological and chemical systems is shown. These considerations contribute to the increase of knowledge and re-
search activities, such as, when solving problems there occures the substitution of the numerical data for mathematical expressions. In addition, tasks analysis, ways of solution, adequate assessment, visualization and interpretation of the obtained results are required.
The presented practical approach effectively contributes to the realization of the educational standard, where one of the main aim is formation of the basic skills of graduates focused on the idea of forward-looking development and the readiness of specialists for self-study throughout their lives.
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Minich A.S., Tomsk State Pedagogical University (ul. Kievskaya, 60, Tomsk, Russian Federation, 634061). E-mail: minich@tspu.edu.ru
Bondarchuk I.S., National Research Tomsk State University (pr. Lenina, 36 , Tomsk, Russian Federation, 634050). E-mail: ivanich_91@mail.ru
Krivenkova M.V., Tomsk school № 14 (ul. K. Ilmera, 11, Tomsk, Russian Federation, 634057). E-mail: allesgutmp@gmail.com
Received 01 October 2018.
DOI: 10 .23951/1609-624X-2018-8-231-237
ПРАКТИЧЕСКИЕ МЕТОДЫ ОРГАНИЗАЦИИ МЕЖДИСЦИПЛИНАРНОГО ИЗУЧЕНИЯ В ОБЛАСТИ БИОЛОГИИ И ХИМИИ
А. С. Минич1, И. С. Бондарчук2, М. В. Кривенкова3
1 Томский государственный педагогический университет, Томск
2 Национальный исследовательский Томский государственный университет, Томск
3 МАОУ СОШ№ 14 им. А. Ф. Лебедева, Томск
Приводятся практические примеры использования междисциплинарного подхода к решению типичных задач в биологии и новых алгоритмических подходов к определению кинетических параметров простых реакций в химии. Рассматриваются соответствующие методы для использования наиболее распространенного инструментария MS Excel для получения более точных решений рассмотренных задач для расширенных значений типа математических моделей желаемых параметров. Представлены методика обучения и практические примеры решения задач в области биологии и химии, которые были реализованы в рамках междисциплинарного подхода, а также изучения математики и информатики. Показана важность междисциплинарного изучения с математическим моделированием процессов, предсказывающих изменение состояния в биологических и химических системах. Все эти методы способствуют исследовательской деятельности, в том числе при решении задач заменой числовых данных математическими выражениями. При этом требуются анализ задач, способы решения, адекватная оценка, визуализация и интерпретация результатов. Представленный практический подход эффективно способствует реализации образовательного стандарта с формированием у выпускников компетенций, ориентированных на готовность к самостоятельному решению исследовательских задач и саморазвитие на протяжении всей жизни.
Ключевые слова: междисциплинарное изучение, педагогическая деятельность, практические методы, электронные таблицы MS Excel, математические модели, динамика популяций, модель хищника-жертвы, физическая химия, константа скорости, порядок реакции.
Минич Александр Сергеевич, доктор биологических наук, профессор, Томский государственный педагогический университет (ул. Киевская, 60, Томск, Россия, 634061). E-mail: minich@tspu.edu.ru
Бондарчук Иван Сергеевич, аспирант, Национальный исследовательский Томский государственный университет (пр. Ленина, 36, Томск, Россия, 634060). E-mail: ivanich_91@mail.ru
Кривенкова Мария Валерьевна, учитель, МАОУ СОШ № 14 им. А. Ф. Лебедева (ул. К. Ильмера, 11, Томск, Россия, 634057). E-mail: allesgutmp@gmail.com
Материал поступил в редакцию 01.10.2018.
