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14LANGUAGE, MATHEMATICS AND CRITICAL THINKING: THE CROSS INFLUENCE AND CROSS ENRICHMENT
(Мова, математика i критичне мислення: взаемний вплив та взаемне збагачення)
Dr. Igor Ya. Subbotin, Professor, Department of Mathematics, College of Letters and Sciences,
National University, Los Angeles, USA, e-mail: isubboti@nu.edu Dr. Michael Gr. Voskoglou, Professor, School of Technological Applications Graduate Technological Educational Institute,
Patras, GREECE e-mail: mvosk@hol.gr
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Обговорюеться вплив словарного запасу на р1вень критичного мисленню студента. Вико-ристовуючи наявну тформацт про Сдину Основну Навчальну Програму в американсьюй осв1-т1, автори на деяких конкретних прикладах за допомогою модел1 неч1тког лог1ки обгрунтову-ють свою позищю.
Ключот слова: математична лексика, критичне мислення, нечгтю моделг, оцтка знанъ.
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We will start with the very obvious thesis that for mathematics communications we use its professional language which nothing else as a regular language saturated with specially defined mathematics terms. However, beyond understanding theory and formulas, the students need to be proficient in application of math and science knowledge to different situations and challenges. "Hands-on, project-based math and science curriculum activities provide opportunities for students to think critically about the use of math and science in solving problems, deepening their knowledge of the basics. For example, in completing an engineering design project based on realistic constraints that professionals in the field may face, such as a change in federal safety requirements, students need to think critically about how to revise their design prototype to satisfy its design goals and meet its scientific requirements" [2].
That is why just the well developed reading comprehending skills are crucially important for solving mathematical content problem. Moreover, even being a skilful in the formal technical mathematics and communication of mathematics, the student,
whose reading comprehension abilities are limited, will not be able to make any progress in the application of these mathematical skills to some problems or just simple questions related to real world. This issue was one of the central themes in the current USA educational reform which is so called the Common Core Curriculum, which sets goals for K-12 classrooms emphasizing depth over breadth. It requires much better communication skills in all mathematical subjects. The student will be required not only to find the correct answer, but be ready to explain this answer and to justify and discuss all possible ways of solution. So, in other words, the student critical thinking abilities should be well developed.
The complexity of critical thinking is evident from the fact that there is no definition that is universally accepted. However, a great number of critical thinking skills as identified by are agreed upon by many authors. Some of these skills are: analysis and synthesis, making judgements, decision making, and drawing warranted conclusions and generalisations, etc. If we look at the critical thinking definitions, such as for instance "...disciplined thinking that is clear, rational,
(89)
open-minded, and informed by evidence" [3]; "...purposeful, self-regulatory judgment which results in interpretation, analysis, evaluation, and inference, as well as explanation of the evidential, conceptual, methodological, criteriological, or contextual considerations upon which that judgment is based" (see [4, p. 26]), "the skill and propensity to engage in an activity with reflective skepticism" [6]. New York: Teachers College Press.; ".disciplined, self-directed thinking which exemplifies the perfection of thinking appropriate to a particular mode of domain of thinking" [7], and so on, we can easily trace reflections of those ideas in the Common Core documents (see, for example, [1]).
So, the implementing of the standards which require serious development of critical thinking skills became extremely important in the teaching of mathematics in USA schools.
The Mathematics Standards include "..two types of standards: Eight Mathematical Practice Standards (identical for each grade level) and Mathematical Content Standards (different at each grade level). Together these standards address both "habits of mind" that students should develop to foster mathematical understanding and expertise and skills and knowledge — what students need to know and be able to do. The mathematical content standards were built on progressions of topics across grade levels, informed by both research on children's cognitive development and by the logical structure of mathematics" [1].
The Standards mandate that eight principles of mathematical practice be taught:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
None of those principles can be right implemented without well developed reading comprehensive student's skills. There are no needs to prove this obvious statement. We just want to support it by the following interesting example, which we justify using the centroid fuzzy model. For general facts on fuzzy sets we refer freely to the book [5].
A classroom application
In one of the Los Angeles Unified District inner city school having very diverse student population (Hispanic 53% , Asian 22%, Black 18%, White 7%), Algebra 2 District Assessment Test was given. The test contents can be found in the appendix attached to the article. A very professional and dedicated teacher, who conducted this test, gave it in two his Algebra 2 classes. One of them was a regular class, another was a so-called "shelter^' class, which means that waist majority of the students in this class are students for whom English is a second language, not a native tongue. It would be logical to expect that this "shelter" class test's results will be worse than in the regular class. However, the situation was opposite. Surprisingly, the "shelter" class performed better. It happened because the teacher, taking into account that the students in this class were not proficient in English, constantly worked on a daily bases on developing the students' mathematics vocabulary and comprehension in reading mathematics content problems. This training affected student's critical thinking and problem solving abilities. The results of the test are in the following chart.
Table 1
Algebra 2 (Periodic Assessment). Shelter class
% Scale Grade Amount of students % of students
89-100 A 0 0
77-88 B 5 13
65-76 C 6 16
53-64 D 9 24
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Less than 53 F 18 47
Total 38
Regular class
% Scale Grade Amount of students % of students
89-10 A 0 0
77-88 B 1 3
65-76 C 5 17
53-64 D 3 10
Less than 53 F 20 70
Total 29
The methods of assessing a group's performance usually applied in practice are based on principles of the bivalent logic (yes-no). However these methods are not probably the most suitable ones. On the contrary, fuzzy logic, due to its nature of including multiple values, offers a wider and richer field of resources for this purpose. This gave us the impulsion to compare the results of performance of the above two classes by implementing the following fuzzy model and a defuzzification technique known as the centroid method.
According to this method, the centre of gravity of the graph of the membership function involved provides an alternative measure of the system's performance. The application of the centroid method in practice is simple and evident and, in contrast to the measures of uncertainty which can be also used as alternative defuzzification techniques (for example see [12] and its references), needs no complicated calculations in its final step. The techniques that we shall apply here have been also used earlier by the authors in [9-11], [13], etc.
Given a fuzzy subset A = {(x, mfxJJ: xe U} of the universal set U of the discourse with membership function m: U —>/(), I/, we correspond to each xeU an interval of values from a prefixed numerical distribution, which actually means that we replace U with a set of real intervals. Then, we construct the graph F of the membership function y=m(x).There is a commonly used in fuzzy logic approach to measure performance with the pair of numbers (xc, yc) as the coordinates of the centre of gravity (centoid), say Fc, of the graph F, which we can calculate using the following
well-known formulas:
(1)
^xdxdy Jjydxdy
X.. = •
^dxdy j'^dxdy
Concerning the described assessment, we characterize a student's performance as very low (F) if x e [0, 1), as low (D) if x e [1, 2), as intermediate (C) if
xe [2, 3), as high (B) if x e [3, 4) and as very high (A) if x e [4, 5] respectively.
Denote by C1 the shelter class and by C2 the regular class respectively and set
U={A B, C, D, F}. We are going to represent the Ci's, i=1, 2 , as fuzzy subsets of U. For this, if niA, niB, niC, niD and niF denote the number of students of class Ci who achieved very low, low, intermediate, high and very high success respectively, we define the membership function mCi in terms of the frequencies, i.e. by
ma(x)=-n
for each x in U. Thus we can write
n
Q={(x, xe U}, i=l,2.
n
Therefore in this case the graph F of the corresponding fuzzy subset of U is the bar graph of Figure 1 consisting of five rectangles, say Ft, i=1,2,3, 4, 5 , whose sides lying on the x axis have length 1.
In this case JJ'dxdy is the area of F which
F
5
is equal to ^ v,. Also ^xdxdy
F
(S
>=1
= ^ Qxdxdy ~ ^ Ji/v | xdx = ^ v, | xdx =
;=1 Ft ;=1 о 1
I 5
-£(2/-l)v,, and
'■=1 F
5 5 У. '
jjydxdy = ^ jjydxdy = ^ J.Vi/v J =
<=1 0 /—1
" г 1 "
X J.lY6' = • Therefore formulas (1) are
'=1 0
transformed into the following form: (2)
Xc
2
1
У' = 2
1 (у1+Ъу2+5у3+1уА+9у, Л
f 2 2 2 2 2 Л
+ У2 + Уз + У4 + У5 v У1+У2+У3+У4+У5 у
ГУ
Itу?)"
* Oll
•<n(ö)
А l 8 г С Ъ r>4 f5
X
Figure 1: Bar graphical data representation
Normalizing our fuzzy data by dividing each m(x), xeU, with the sum of all membership degrees we can assume without loss of the generality that y1+y2+y3+y4+y5 = 1■ Therefore we can write: (3)
xc=~ y1+3y2+5yi+7y4+9y} ,
I 2 , 2 , 2 , 2 , 2
Ус=~ У\ У 2 +^3 +У4 + У5
mix Л with у, = '
^m(x)
xgU
2 2 2
But 0 < (yi-yz) =yi +y2-2yiy2 , therefore
2 2
yi y2 > 2yiy2 ,with the equality holding if, and only if, y1=y2.
2 2
In the same way one finds that yj +y3 >2yjy3, and so on. Hence it is easy to check that
(yi+y2+ys+y4+y5)2 < 5(yj2+y22+y32+y42+y/), with the equality holding if, and only if
yi=y2=y3=y4=y5.
Buty1+y2+y3+y4+y5 =1, therefore (4)
1 < 5(yi2+y22+y32+y42+y52), with the equality holding if and only if
y1=y2=y3=y4=y5=1 .
5
Then the first of formulas (3) gives that xc
= 1. Further, combining the inequality (4)
2
with the second of formulas (3), one finds that
1 < 10yc, or yc > _L Therefore the unique
10
minimum for yc corresponds to the centre of
gravity Fm (1 ,_L).
2 10
The ideal case is when y1=y2=y3=y4=0 and y5=1■ Then from formulas (3) we get that
xc = 9 and yc = 1 .Therefore the centre of
2 2
gravity in this case is the point
F (9, 1).
22
On the other hand, in the worst case y1=1 and y2=y3=y4= y5=0. Then by formulas (3), we find that the centre of gravity is the point
Fw (1, 1).
22
Therefore the "area" where the centre of gravity Fc lies is represented by the triangle
FwFmFiofFigure2.
*f
Figure 2: Graphical representation of the "area" of the centre of gravity
Then from elementary geometric consid-
cn>
1=1
F
erations it follows that the greater is the value of xc the better is the group's performance. Also, for two groups with the same xc >2,5, the group having the centre of mass which is situated closer to Fi is the group with the higher yc; and for two groups with the same xc <2.5 the group having the centre of mass which is situated farther to Fw is the group with the lower yc. Based on the above considerations it is logical to formulate our criterion for comparing the groups' performances in the following form:
• Among two or more groups the group with the biggest xc performs better.
• If two or more groups ha\'e the same xc> 2.5, then the group with the higher ycperforms better.
• If two or more groups have the same xc < 2.5, then the group with the lower ycperforms better.
We apply this model to the given in the table 1 case. For the shelter class, we have the following:
yc =0. 5(y12+y22+ys2+y42+y52) = 0.5(0.472 + 0.242 + 0.162 +0.132)=
0.5(0.221+ 0.058 + 0.026 +0.017)=0.16 ;
xc=0.5(0.47+3• 0.24+5• 0.16+ 7• 0.13)=0.5(0.4 7+ 0.72+0.80+0.91)=1.45.
For the regular class :
yc =0. 5(y12+y22+y32+y42+y52) = 0.5(0.70 + 0.102 + 0.172 +0.032)=
0.5(0.490+ 0.010 + 0.028 +0.001)=0.27;
xc=0.5(0.70+3• 0.10+5• 0.17+ 7• 0.03)=0.5
(0.70+0.30+0.85+0.21)=1.33.
So, according to the above stated criterion, the shelter class demonstates a better performance on this test.
Appendix. Algebra 2 Periodic Assessment [8]
Four Situations
1. Sketch a graph to model each of the following situations. Think about the shape of the graph and whether it should be a continuous line or not.
A: Candle
Each hour a candle burns down the same amount. x = the number of hours that have elapsed. y = the height of the candle in inches.
B: Letter
When sending a letter, you pay quite a lot for letters, weighing up to an ounce. You then pay a smaller, fixed amount for each additional ounce (or part of an ounce.) x = the weight of the letter in ounces. y = the cost of sending the letter in cents. C: Bus
A group of people rent a bus for a day. The total cost of the bus is shared equally among the passengers.
x = the number of passengers. y = the cost for each passenger in dollars.
D: Car value
My car loses about half of its value each year.
x = the time that has elapsed in years. y = the value of my car in dollars.
2. The formulas below are models for the situations. Which situation goes with each formula? Write the correct letter (A, B, C or D) under each one.
300 y= x
y =12-0.5x. y = 30 + 20x. y =2000 • (0.5)x.
3. Answer the following questions using the formulas. Under each answer show your reasoning.
How long will the candle last before it burns completely away?
How much will it cost to send a letter weighing 8 ounces?
If 20 people go on the coach trip, how much will each have to pay?
How much will my car be worth after 2 years?
References
[1] California Common Core State Standards Mathematics, Adopted by the California State Board of Education August 2010 and modified January 2013http://www.cde. ca.gov/be/st/ss/documen ts/ccssmathstandardaug2013.pdf
[2] Critical Thinking/Math & Science. How Can Critical Thinking & Problem Solving Skills Support Math and Science Curric-
<9D
ulum at the 9-12 Level? http://route21.p21.org/?Itemid=167&id=21 &option =com_content&view =article Accessed: February 12, 2013.
[3] Dictionary.com, "critical thinking," in Dictionary.com Unabridged. Source location: Random House, Inc. http://dictionary. reference. com/browse/critic al thinking. Available: http://dictionary. reference. com. Accessed: February 12, 2013.
[4] Facione, Peter A. Critical Thinking: What It is and Why It Counts, Insightassess-ment.com, 2011.
[5] Klir, G. J. & Folger, T. A., Fuzzy Sets, Uncertainty and Information, Prentice-Hall, London, 19
[6] McPeck, J. Thoughts on subject specificity. In S. Norris (Ed.), The generalizabil-ity of critical thinking (pp. 198-205). New York: Teachers College Press. 1992.
[7] Paul,R Teaching critical thinking in the strong sense: A focus on self-deception, world views and a dialectical mode of analysis. Informal Logic Newsletter 4(2), (1982) 2-7.
[8] Student Materials Functions and Everyday Situations © 2012 MARS, Shell
Резюме. Субботин И., Воскоглоу М. ЯЗЫК, МАТЕМАТИКА И КРИТИЧЕСКОЕ МЫШЛЕНИЕ: ВЗАИМНОЕ ВЛИЯНИЕ И ВЗАИМНОЕ ОБАГОЩЕНИЕ. В этой статье авторы обсуждают влияние словарного запаса на уровень критического мышлению студента. Используя имеющуюся информацию о Единой Основной Учебной Программе в американском образовании, авторы на некоторых конкретных примерах с помощью модели нечеткой логики обосновывают свою позицию.
Ключевые слова: математическая лексика, критическое мышление, нечеткие модели, оценка знаний.
Abstract. Subbotin I., Voskoglou M. LANGUAGE, MATHEMATICS AND CRITICAL THINKING: THE CROSS INFLUENCE AND CROSS ENRICHMENT. In this article, the authors discuss the relations between the language proficiency and mathematical thinking, and the influence of the vocabulary development on the expansion of student's critical thinking abilities. Using available information regarding the so-called Common Core Curriculum reform in USA education, some concrete examples and supporting justification based on the centroid fuzzy model were given.
Key words: mathematical vocabulary, critical thinking, fuzzy models, learning assessment.
Стаття надшшла доредакцп 16.02.2014р.
Center, University of Nottingham.
[9] Subbotin, I. Ya. Badkoobehi, H., Bi-lotckii, N. N., Application of fuzzy logic to learning assessment. Didactics of Mathematics: Problems and Investigations, 22, 38-41, Donetsk, 2004.
[10] Subbotin, I.Ya., Badkoobehi, H., Bi-lotskii, N.N., Fuzzy logic and learning assessment, Didactics of Mathematics: Problems and Investigations, 24, Donetsk, 112118, 2005.
[11] Subbotin, I. Ya, Voskoglou, M. Gr., Applications of fuzzy logic to Case-Based Reasoning, International Journal of Applications of Fuzzy Sets, 1, 7-18, 2011.
[12] Voskoglou, M. Gr., Stochastic and fuzzy models in Mathematics Education, Artificial Intelligence and Management, Lambert Academic Publishing, Saarbrucken, Germany, 2011 (for more details look at http://amzn.com./3846528218).
[13] Voskoglou, M. Gr., Fuzzy Logic and Uncertainty in Mathematics Education, International Journal of Applications of Fuzzy Sets, 1, 45-64, 2011.
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