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4ON MEASUREMENT OF EFFECTIVENESS IN TEACHING MATHEMATICS
(Об измерении эффективности преподавания математики)
Dr. Igor Ya. Subbotin, Professor, Department of Mathematics, College of Letters and Sciences,
National University, Los Angeles, USA, e-mail: isubboti@nu.edu Dr. Hassan Badkoobehi, Professor, Department of Applied Engineering School of Engineering and Technology, Technology and Health Sciences Center, San Diego, USA,
e-mail: hbadkoob@nu.edu
У cmammi продовжуеться досл1дження застосування неч1тког логти до педагог1чних ви-мгрювань у викладант математики.
КлючоеЛ слова: наечання математики, нечтка логта, педагог1чт вим1рюеання.
There are some important concepts in the theoretical ground of the learning process allowing us to develop useful applications to everyday teaching. A Vygotsky concept of the Zone of Proximate Development (ZPD) is one of fundamentals here. L. Vygotsky [VL, p. 86] defines the ZPD as "the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers". Vygotsky argued that providing the appropriate assistance (scaffolding) gives the student enough of a "boost" to achieve the task. Once the student, with the help of scaffolding masters the task, the scaffolding can be removed and the student will be able to complete the task again on his/her own. In the learning process it is important to recognize and assess a student's intellectuality capacity. "The concept of scaffolding is a process, wherein an instructor provides only the needed support by providing tasks that will enable a learner to build on prior knowledge, and once the stage of zone of proximal development has been reached, the guidance is gradually removed" (http://projects.coe.uga.edu/epltt/index.php?titl
e=Vygotsky%27s_constructivism#Vygotsky.2 7s_Theories). "The zone of proximal development defines those functions that have not yet matured but are in the process of maturation, functions that will mature tomorrow but are currently in an embryonic state. These functions could be termed the "buds" or "flowers" of development rather than the "fruits" of development" ([VL, p.86]). How to formally measure this ZPD application effectiveness? We try to formalize these ideas and make its applications measurable using the following concept developed by J. Voss in [VJ]. He argued that learning, as a specific case of knowledge transfer, consists of successive problem-solving activities, in which the input information is represented of existing knowledge with the solution occurring when the input is appropriately represented. This process implements the following states: a) representation of the input data, b) interpretation of this data, c) generalization of the new knowledge, and d) categorization of this knowledge. The states a and b could be combined in one state of interpretation the new knowledge. M. Voskogloy in the article [VM] has developed an appealing Fuzzy Logic applications based on the Voss's theory. Created
by LA. Zadeh ([Z1], [Z2]), Fuzzy Logic has been proven to be extremely productive in many applications (see, for example, [KF], [W], [BE]). There are also some interesting attempts to implement Fuzzy Logic ideas in the field of education ([VM], [EO], [PS], [SBB], [SBB1], [SMB]).
We consider in details the approach suggested in [VM], and will employ it for determining relative probabilities of the overall states in the iterative instructional model.
In [VM] the following construction has been developed. Let Ai, i =1,2,3, be the states of interpretation, generalization, and categorization respectively, and a,b,c,d,e - the linguistic variables of negligible, low, intermediate, high, and complete acquisition of knowledge respectively of each of the Ai. M. Voskoglou considers the set U = {a,b,c,d,e} and represents the A/s as Fuzzy sets in U. He denotes by nia, nib, nic, nid, nie the numbers of the students that have achieved negligible, low, intermediate, high, and complete acquisition of the state Ai respectively and defines a membership
n
function m. by m. (x) = — for each x e U
A A n
n
and, therefore, one can write Ai = {(x, —
n
):xe U}, where £m (x) = 1,i = 1,2,3. A
xeU
Fuzzy relation can be considered here as a Fuzzy set of triples, each one of which possess a degree of membership belonging [0, 1]. Consider farther the Fuzzy relation
R = mR (s): s = (x, y, z) eU3} where the membership function defined by
mR (s) = mA (x) mR2 (y)mR3 (z), for a11 s =
= (x, y, z) e U3.
This Fuzzy relation R represents all the possible profiles of student's behavior during the learning process. Further, M.Voskoglou develops the procedure of comparing few groups of students based on his ideas and supplies the article with examples showing straightforwardness of its applications. Instead of different groups of students we can consider the same class of students on different steps of learning process. M.Voskoglou in [VM] described the application of the developed proce-
dure in the following way. Let us consider a group G of n students during the process of
learning in the classroom, n E N, n > 2. Obviously, from the point of view of the teacher, there exists an uncertainty about the degree of acquisition of each state of the process from his/her students, a fact which gave us the hint to introduce the Fuzzy sets theory in order to achieve a mathematical representation of the process of learning in the classroom. For this, let us denote by Ai, i=1, 2, 3, the state of interpretation, generalization and categorization respectively, and by a, b, c, d, e the linguistic labels of negligible, low, intermediate, high and complete acquisition respectively of each of the A/s. Consider the set U = {a,b,c,d,e} , then we are going to represent the Afs as Fuzzy sets in U. In fact, if nia, nib, nic, nid and nie denote the numbers of the students that have achieved negligible, low, intermediate, high and complete acquisition of the state Ai respectively, i=1, 2, 3, we can define the membership function mAi by mAi(x)=nix/n, for each x in U and
therefore we can write Ai={(x,nix/n): x E U}. It
becomes clear then that YmMx)=1, x E U, i=l,2,3 .At this point notice that a Fuzzy relationship, like the classical ones, can be considered as a Fuzzy set of tuples each one of which possesses a degree of membership included between 0 and 1. Consider now the Fuzzy relation R={(s, mr(s)):s=(x, y, z) E U3}, where the
membership function mR is defined by mR(s)=mA(x)mA2(y)mA3(z), for all s=(x,c,z) in U3. This definition satisfies the axioms of aggregation operations in Fuzzy sets and further we have that XmR(s)=1. The Fuzzy relation R represents all the possible profiles (overall states) of the behavior of a student during the learning process. In the next, and in order to simplify our notation, we shall write ms instead of mR(s). Assume now that one wants to study the behavior of k groups of students during the learning process of the same subject, or the behavior of the same group of students during the learning process of k different subjects,
k EN, k > 2. In this case it becomes necessary
to introduce the Fuzzy variables A(t), where i=1, 2, 3 and t=1,2, ... k. Then the pseudofre-quency f (s) of the overall state s(t) is k given
by the sum Xm/t), while the probability of s(t) is t=l given by p(s)=f (s)/ fs), where Xf(s) denotes the sum of all pseudofrequences. But, since X ms=1, it becomes clear that Xf(s)=k and thereforep(s)=f(s)/k. Finally the possibility of s(t) is given by r(s)=f(s)/max f(s), where max f(s) denotes the maximal pseudofrequen-cy. The possibility of s(t) measures the degree of evidence of combined results, i.e. in other words one may say that r(s) gives the "relative probability" of s(t) with respect to the other overall states.
In the current article we accommodate the M.Voskogloy's ideas toward the measurement of efficiency of scaffolding in teaching upper division mathematics courses.
The authors of the current article teach upper division online mathematics classes for prospective mathematics school teachers and engineering students at the National University, California, USA. One of the most difficult classes here is linear algebra. The most abstract and therefore the most difficult topics in this course is the chapter dedicated to the general concept and properties of a vector space. Studying this material, students should read the corresponding book chapter, solve a significant amount of problems, answer for the board discussion questions, participate in a live format virtual synchronized classroom sessions, and take tests. Taking into account the following G. Polya's [PG] important remark: "For an effective learning the learner discovers alone the biggest possible, under the circumstances, part of the new information", we try to provide scaffolding in the most efficient way supporting the students' independent and group work. We want to offer this help at the appropriate point when this scaffolding works most efficiently. In the current article, the authors, based on the collected experimental data determined this appropriate point and the most possible profile of performance. During the instruction process, we effectively used the Iterative Instructional Model (IIM) as an efficient tool which is based on extended spirally organized iterations of the learning material, activities and skills in the learning process. One of the main goals of the learning process is long-term retention and knowledge transfer. In contrast to the traditional consecutive translation along
with the material, with polishing of all details before reaching the next step, the iterative approach suggests a holistic approach exploring all sides of a problem. This is some kind of analogy of the well-known in mathematics iteration process. Recall that the iteration is the repeated application of a mathematics procedure, where each step is applied to the output of the preceding. The physiological effectiveness of the iterative approach based on our memory properties. Our brain codes the learned information and stores it coded. To retrieve this information, the brain needs to decode it. The speed and efficiency of decoding (retrieving) depend on an amount of ways (connections) of the information decoding. The right way of learning requires creating and developing as many ways of decoding as possible. We can reach this goal by practicing, active discussions, and other activities using all possible kinds of memory (visual, motor, audio, and so on), employing the holistic approach, and raising vertical and horizontal connections between main parts of the material. A very important component here is the developing of strong connections with the already learned material.
Based on the implemented Iterative Instructional Model (IIM) (see for example [SB, SMB]), we can consider here the following tree important states of the knowledge acquisition process.
State Al. First iteration (no or very little scaffolding): Developing of main basic knowledge, first look at the "road map" of the topic (corresponds to Voss's stages of representation and interpretation of the input data). At this step, the students read and try to understand main definitions such as the concepts of the vector space axioms, an abelian group, a field, a vector space over a field, a dimension of a space, and so on. All of these concepts are quite abstract; they require very serious efforts from the beginners in order to understand it and, what is more important, to "feel" them. Actually the later is the main goal of the entire process of learning acquisition (and we hope to reach it at the state 3). At the beginning, the students try to understand these concepts and to learn by hart these definitions. They do it mostly independently, having some very brief
instructor interactive support (emails, lecture materials, internet sources, book reading). It worth nothing to understand that at this stage, the students just obtain a very superficial and formal understanding of the material. The teacher's help here is restricted and minimal. However, using the student forum option, the students can get some help from their peers. The group work at this stage is highly recommended.
State A2. Second iteration (actual scaffolding): Developing of general main knowledge, considering "a main infrastructure map" of the topic (corresponds to Voss's stage of generalization of the new knowledge). At this step, the student should operate with the terms from the paragraph above, formally manipulate with them, to learn their relationships. Solving well constructed and organized in target set of simple exercises operating with the main concepts and clarifying their general relations is extremely helpful at this stage. Again, the groups work and paired assignments are great tools at this stage. The instructor help in solving and explaining problems is welcome here.
Sate A3. Third iteration (major teacher centered help), " a detail map": Developing of completed detail knowledge in the topic (corresponds to Voss's stage of categorization of the new knowledge). The best tools here are the discussion and the summarizing lecture in which the instructor provides detail information and demonstrate proofs of the main results. For this state we successfully use well known direct explicit instruction. Explicit Direct Instruction, usually shortened to EDI, is a strategic collection of instructional practices combined together to help teachers design and deliver well-crafted lessons that explicitly teach content, especially grade-level content, to all students. EDI is based on teacher-centered, direct instruction philosophy.
At each state, the students knowledge acquisition were observed and labeled correspondingly to above described ideas from [VM]: c stands for the "below satisfactory" , b for "satisfactory", and a for "above satisfactory" level of acquisition. Of, course at each state the criteria of labeling were adapted according-
ly to students' real progress at predecessor states.
Now we compare our two classes' performances.
At one of our classes, we observed that 4, 9, and 5 students achieved below unsatisfactory, satisfactory, and above satisfactory respectively (n1c =4, nib= 18, nia= 5) at the first state. Thus An= {(c, 4/18), (b, 9/18), (a, 5/18)}. At the next state we found that A12= {(c, 3/18), (b, 11/18), (a, 4/18)}. At the final state we found that A13 = {(c, 4/18), (b, 13/18), (a, 1/18)}. Another class data set gave us the following results
A2]= {(c, 3/12), (b,7/12), (a, 2/12)}. A22= {(c, 2/12), (b, 8/12), (a, 2/12)}. A23= {(c, 1/12), (b, 10/12), (a, 1/12)}. Looking at the Ais for both sets, we can see that the higher the state iteration, the enhanced degree of acquisition of knowledge we achieve.
From the table below it turns out that the profile s=(b,b,b) has the highest pseudofre-quency for the two groups of data sets of our experiment
ms(1)=mA](b)mA2(b)mA3(b) =0.220678, ms(2) =0.324073,
and therefore f (s)= 0.54475. Thus it had also the highest probability of occurrencep(s) = 0.272375, or 27%, while its possibility is almost 1.
In [SBB] and [SBB1], another approach complementing in some sense the M. Vos-kogloy's method has been introduced. This model allows compare the performances of two sets of students. In Fuzzy Logic there is a commonly used approach in measurement of performance with the pair of numbers (xc,yc) as the coordinates of the center of mass of a cor-respondently represented of the data two dimensional figure F, which we can calculate using the following well-known formulas:
(1)
U xdxdy U ydxdy
x„ —
U dxdy Ус jj dxdy
Table
Probabilities and possibilities of profiles
Tuples Data Set I _Data Set2 ^___
A1A2A3 A11 A12 A13 A21 A22 A23 ms(1) ms(2) f(s) P(s) r(s)
aaa 0.27778 0.22222 0.05556 0.16667 0.16667 0.08333 0.00343 0.002315 0.005744 0.002872 0.010544
aab 0.27778 0.22222 0.72222 0.16667 0.16667 0.83333 0.044581 0.023149 0.06773 0.033865 0.124332
aac 0.27778 0.22222 0.22222 0.16667 0.16667 0.08333 0.013717 0.002315 0.016032 0.008016 0.02943
aba 0.27778 0.61111 0.05556 0.16667 0.66667 0.08333 0.009432 0.009259 0.018691 0.009345 0.034311
abb 0.27778 0.61111 0.72222 0.16667 0.66667 0.83333 0.1226 0.092595 0.215194 0.107597 0.395032
abc 0.27778 0.61111 0.22222 0.16667 0.66667 0.08333 0.037723 0.009259 0.046982 0.023491 0.086245
aca 0.27778 0.16667 0.05556 0.16667 0.16667 0.08333 0.002572 0.002315 0.004887 0.002444 0.008971
acb 0.27778 0.16667 0. 72222 0.16667 0.16667 0.83333 0.033437 0.023149 0.056586 0.028293 0.103875
acc 0.27778 0.16667 0.22222 0.16667 0.16667 0.08333 0.010288 0.002315 0.012603 0.006302 0.023135
baa 0.5 0.22222 0.05556 0.58333 0.16667 0.08333 0.006173 0.008102 0.014275 0.007137 0.026205
bab 0.5 0.22222 0. 72222 0.58333 0.16667 0.83333 0.080246 0.081019 0.161265 0.080633 0.296035
bac 0.5 0.22222 0.22222 0.58333 0.16667 0.08333 0.024691 0.008102 0.032793 0.016396 0.060198
bba 0.5 0.61111 0.05556 0.58333 0.66667 0.08333 0.016977 0.032406 0.049383 0.024691 0.090653
bbb 0.5 0.61111 0.72222 0.58333 0.66667 0.83333 0.220678 0.324073 0.54475 0.272375 1
bbc 0.5 0.61111 0.22222 0.58333 0.66667 0.08333 0.0679 0.032406 0.100307 0.050153 0.184134
bca 0.5 0.16667 0.05556 0.58333 0.16667 0.08333 0.00463 0.008102 0.012732 0.006366 0.023372
bcb 0.5 0.16667 0. 72222 0.58333 0.16667 0.83333 0.060186 0.081019 0.141206 0.070603 0.259212
bcc 0.5 0.16667 0.22222 0.58333 0.16667 0.08333 0.018519 0.008102 0.02662 0.01331 0.048866
caa 0.22222 0.22222 0.05556 0.25 0.16667 0.08333 0.002744 0.003472 0.006216 0.003108 0.011411
cab 0.22222 0.22222 0. 72222 0.25 0.16667 0.83333 0.035664 0.034723 0.070387 0.035194 0.12921
cac 0.22222 0.22222 0.22222 0.25 0.16667 0.08333 0.010974 0.003472 0.014446 0.007223 0.026519
cba 0.22222 0.61111 0.05556 0.25 0.66667 0.08333 0.007545 0.013888 0.021433 0.010717 0.039345
cbb 0.22222 0.61111 0. 72222 0.25 0.66667 0.83333 0.098078 0.138889 0.236967 0.118484 0.435001
cbc 0.22222 0.61111 0.22222 0.25 0.66667 0.08333 0.030178 0.013888 0.044066 0.022033 0.080892
cca 0.22222 0.16667 0.05556 0.25 0.16667 0.08333 0.002058 0.003472 0.00553 0.002765 0.010151
ccb 0.22222 0.16667 0. 72222 0.25 0.16667 0.83333 0.026749 0.034723 0.061472 0.030736 0.112844
ccc 0.22222 0.16667 0.22222 0.25 0.16667 0.08333 0. 00823 0.003472 0.011703 0.005851 0.021483
In the diagram 1 c stands for the "below satisfactory" (y1), b" for "satisfactory" (y2), and a for "above satisfactory" (y3) level of knowledge acquisition for each states A1, A2, A3.
diagram 1
In this case, formulas (1) can be easily transformed to the following simple formulas
[SBB]: (2)
1
x = — c 2
fyi + 3 ^2 + 5 yA y + y2 + y3
, y. =■
1
y + y + y yi + y + y3
2
It is easy to see that formulas (2) can be generalized for the case when our figure consists not only from three rectangles, but from n rectangles In this case we will come to the following formulas [SBB]:
(3) xc =
1
f n \
E (2i -1) y
i=1
E y
i = 1
1
E y2
i=1 n
E yi
V 1 =1
Base on our data above, we can obtain the
following tables for each corresponding state.
Ratio of the class students Set I Set II
reached the following stage of knowledge acquisition at the State I of independent student work (State A1)
c 0.22 0.25
b 0.5 0.58
a 0.28 0.17
Obviously in this case Y1 + Y2 + Гз = yi +
Consider now the State III of major instructional help.
y2 +y3 =1. Hence, formulas (2) will look like this:
(4)
xc =1 (y1 + 3 y 2 + 5 y), yc =1 (>i2 + y22 + y2).
In this case, for the Set 1 we have xc1 = 0.5(0.22+ 30.5 + 5.0.28) = 1.56; yc1= 0.5(0.222 + 0.52 + 0.282) = 0.41. For the set II we have xc2 = 0.5(0.25 + 30.58 + 50.17) = 1.42; yc1= 0.5(0.252 + 0.582 + 0.172) = 0.36 By the following rule developed in [SBB1]
Among two or more classes the class with the biggest xc performs better;
(5) If two or more classes have the same xc > 1.5, then the class with the higher yc performs better. If two or more classes have the same xc <1.5, then the class with the lower yc performs better;
we conclude that the Set I shows better performance rate at the State I. Consider now the State II.
Ratio of the class students reached the following stage of knowledge acquisition at the State II of scaffolding (State A2) Set I Set II
c 0.17 0.17
b 0.61 0.67
a 0.22 0.17
In this case, for the Set 1 we have
xc1 = 0.5(0.17+ 30.61 + 50.22) = 2.05; yc1= 0.5(0.172 + 0.612 + 0.222) = 0.23. For the set II we have xc2 = 0.5(0.17 + 30.67 + 50.17) = 1.52; yc= 0.5(0.172 + 0.672 + 0.172) = 0.28. Base on the rule (5) we conclude that the Set I demonstrates better performance rate at the State II.
Ratio of the class students reached the following stage of knowledge acquisition at the State III (State A3) Set I Set II
c 0.22 0.08
b 0.72 0.83
a 0.06 0.08
In this case, for the Set 1 we have Xd = 0.5(0.22 + 30.72 + 50.06) = 1.34;
ycl= 0.5(0.222 + 0.722 + 0.062)
0.31.
For the Set II we have xc2 = 0.5(0.08 + 3-0.83 + 5-.0.08) = 1.49;
yc1= 0.5(0.082 + 0.832 + 0.082) = 0.41.
Base on the rule (5) we conclude that the Set II shows better performance rate as the State III.
Analyzing the data at all three states we can conclude that the performance of both sets vary depending of the state. Thus, the first set demonstrated better performance at the states of no or light scaffolding. However, it performs relatively worse at the state of major instructional help. The same tendency, but with low level of variance shows the Set II. At the first state of independent work, the numbers showing the performance of each sets are quite close; the similar picture we observe at the final state of major instructional help (actually this state is the state of direct explicit instructions). However, we monitor a significant difference in the performance of these sets at the Stage II of scaffolding: for the first group of students this scaffolding showed to be much more efficient that for the second group. In any case, it is interesting to admit, that the scaffolding is an efficient tool at the stage of building the "infrastructure map", at the stage where main connections between recently learned concepts are establishing. Group work and moderate help offered by the instructor at this state create the best environment for relatively rapid progress in knowledge acquisition. This fact is a good illustration of the effectiveness of Vygotsky's ZPD concept application.
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Резюме. Субботин И.Я., Бадкубехи Хасан. ОБ ИЗМЕРЕНИИ ЭФФЕКТИВНОСТИ ПРЕПОДАВАНИЯ МАТЕМАТИКИ. В статье продолжается обсуждение применений нечеткой логики к измерению эффективности преподавания математики.
Ключевые слова: обучение математике, нечеткая логика, педагогические измерения.
Abstract. Subbotin I., Badkoobehi H. ON MEASUREMENT OF EFFECTIVENESS IN TEACHING MATHEMATICS. The article discusses some Fuzzy Logic applications to formalizing of a Vygotsky's concept of the Zone of Proximate Development and measurement of its effectiveness in teaching of upper division mathematics courses.
Key words: Learning and Transfer, Zone of Proximate Development, Mathematical modeling, Fuzzy sets, Uncertainty, Scaffolding
Стаття надйшла до редакци 04.05.2013р.
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