Analysis of didactic transposition in teaching the concept of derivative in high schools in the case of Vietnamese textbooks in 2000, 2006 and American textbook in 2010

Abstract. Since 2000, throughout the program of Mathematics in Vietnam, the content of functions is presented and kept the central position. The derivative function is introduced at the end of 11th-grade curriculum to equip a tool to study significant knowledge of function content such as monotonous, convex, inflection point, maximum value etc. . . to support for other subjects such as chemistry, physics and so on. The research is based on didactic research model to study the differences between transposing the concept of derivative in teaching mathematics in high school in the case of Vietnamese textbooks in 2000, 2006 and American textbook (Calculus 9th edition, 2010. Ron Larson, Bruce H. Edwards, Cengage Learning). From this research, we recommend pedagogical aspects to teach these concepts. In the following part, we refer Vietnamese textbooks in 2000, 2006 and American textbook (Calculus 9th edition, 2010. Ron Larson, Bruce H. Edwards. Cengage Learning) to textbook 1, textbook 2 and textbook 3.

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HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1075.2017-0123 Educational Sci., 2017, Vol. 62, Iss. 6, pp. 10-18 This paper is available online at ANALYSIS OF DIDACTIC TRANSPOSITION IN TEACHING THE CONCEPT OF DERIVATIVE IN HIGH SCHOOLS IN THE CASE OF VIETNAMESE TEXTBOOKS IN 2000, 2006 AND AMERICAN TEXTBOOK IN 2010 Chu Cam Tho1, Nguyen Tien Dat2 1The Vietnam Institute of Educational Sciences 2Hanoi National University of Education Abstract. Since 2000, throughout the program of Mathematics in Vietnam, the content of functions is presented and kept the central position. The derivative function is introduced at the end of 11th-grade curriculum to equip a tool to study significant knowledge of function content such as monotonous, convex, inflection point, maximum value etc. . . to support for other subjects such as chemistry, physics and so on. The research is based on didactic research model to study the differences between transposing the concept of derivative in teachingmathematics in high school in the case of Vietnamese textbooks in 2000, 2006 and American textbook (Calculus 9th edition, 2010. Ron Larson, Bruce H. Edwards, Cengage Learning). From this research, we recommend pedagogical aspects to teach these concepts. In the following part, we refer Vietnamese textbooks in 2000, 2006 and American textbook (Calculus 9th edition, 2010. Ron Larson, Bruce H. Edwards. Cengage Learning) to textbook 1, textbook 2 and textbook 3. Keywords:Mathematical didactic, transposition didactic, the concept of derivative. 1. Introduction Function is an important concept in analysis, which is considered to be the main aspect of mathematic curriculum at high school [10]. The derivative function is introduced at the end of grade 11 as a tool for studying vital knowledge related to function such as monotonicity, maximum value, minimum value, and so on. The change of educational goals requires the development of different teaching institutions, within this paper we study the didactic transposition of the derivative concept in Vietnamese books in 2000, 2006 and the American textbook (Calculus 9th edition, 2010. Ron Larson, Bruce H. Edwards, Cengage Learning) to answer the question: "With the trend of teaching approaching the capacity of learners today, what is the definition of the derivative should be introduced in the high school mathematic program?" 2. Content 2.1. Introduction to the research model According to Y. Chevallard (1991) [5], Didactic is bringing the "real" society into the teaching of mathematics (Anthropological Theory). The basic relations between Institution I, Received date: 15/4/2017. Published date: 15/6/2017. Contact: Chu Cam Tho, e-mail: chucamtho1911@gmail.com 10 Analysis of didactic transposition in teaching the concept of derivative... Person X, and Object O include: Institutional relation creates constraints or lesson targets that individual teachers (students) need to transmit (acquire) the object of mathematical knowledge. It is obvious that when the educational objective changes, the teaching institution I changes to adapt that goal, resulting in the teaching content related to knowledge O changes as well. Personal relationship is an essential activity for the teacher (student) to carry out the process of transmission (acquisition) the content of that knowledge. Within this article, we are interested in three institutional relationships: R(I1,O), R(I2,O) and R(I3,O) therein I1 is the teaching institution of the derivative concept at high school in Vietnam in 2000, I2 is the teaching institution of the derivative concept at high school in Vietnam in 2006, I3 is the teaching institution the derivative concept at high school in the United States, O is the derivative concept in Grade 11. However, the question is: "Of the three institutions I1, I2 and I3: How the real life of the derivative will differ?". Bosch and Chevallard’s answer to this question is the concept of "praxeology" (praxéologie) (1999) [6]. Mathematical praxeology (mathématique praxéologie) explains the mathematical knowledge difference is resulted from the underlying theoretical foundation that institutional I applies. To be precise, I3 is based on the accuracy of theoretical knowledge in the analysis (explain the limit of the function based on language δ, ε) resulting in the notion of " the point of tangency" according to Leibniz. Meanwhile, I2 constructs the limit concept by an inductive, intuitive way (give concrete examples to form the concept of limit 0 and then lead other limits to limit 0) then come to the conception of derivative with classical mechanical mathematics (the "fluxions" theory) as Newton discovered. For I1, which introduces the derivative concept through the problem “Find the instantaneous velocity of the object” like I2 and have theoretical accuracy as I3. By the model of "mathematical organization" [T,τ ,θ,Θ] [Meaning of model: Each human mathematical activities consists of accomplishing a task t of a certain type T, by technique τ , which satisfies technology θ. At the same time, technology θ is allowed to figure out or even produce technique τ and which in turn is justifiable by theory Θ] we can compare the similarities and differences between the type of tasks that institutions I1, I2 and I3 set out to achieve the purpose of teaching the derivative concept. 2.2. Analysis of Didactic Transposition in Three Vietnamese Case Studies in 2000, 2006 and American Textbook in 2010 2.2.1. Institutional relations * The similarities General institutional objectives and regulations + Calculate the derivative of the function using definition and recognize the relationship between the continuity and the existence of the derivative. + Know how to use the basic rules of derivative function, distinguish between the derivative of the function at one point as a fixed constant while the derivative of the rule gives us corresponding number and value of the derivative (avoid repeating derivative works by defining at different values). + Determine the geometrical meaning of the derivative and the physical meaning of the derivative. From the common goal, the content included in the three textbooks is similar + How to calculate the derivative by definition, the relationship between the existence of derivative and the function continuity. + The geometric meaning of the derivative, the physical meaning of the derivative and the derivative of the function over a period. 11 Chu Cam Tho, Nguyen Tien Dat + Rules for the derivative of some common functions, derivatives of sum, difference, product, function and derivative function. + The trigonometric functions derivative, the second derivative. * The difference between the goals + I3 has additional objectives: Calculate the derivative of the implicit function [The function is defined implicitly by the equation f(x,y) = 0 such as xy =1, x2 + y2 = 1, . . . ], establish the relationship between two physical quantities which change over time. + I1, I2 give the goal of recognizing differential concepts and applying them in approximation. The difference between the content of combination (I1, I2) and I3 is + Textbook 1 [8] and Textbook 2 [6] both define the derivative of the function y = f(x) defining on interval (a, b) and x0 ∈ (a, b) then the finite limit lim x→x0 f (x)− f (x0) x− x0 is called the derivative of the function at the point. From these remark the institutions I1 and I2 only interested in finding the derivative of explicit function y = f(x) while implicit function is not mentioned. + On the geometrical meaning of the derivative, the concept of tangent to a curve is visually described as "When the secant lineM0M moves to the limit position M0T , it is called the tangent of (C) atM0". The definition of a tangent line to a curve does not cover the possibility of a vertical tangent line (which is parallel or coincident to Oy). Tangent to a circle Tangent to a curve at flexible point Tangent cuts the curve at second point + From I3: In case of vertical tangent line, the following definition can be used : If f is continuous at c and lim ∆x→0 f (c+∆x)− f (c) ∆x = ∞ then the vertical line x = c passing through (c, f(c)) is vertical tangent line to the graph of f(x). + The institution I3 introduces the definition of the implicit function for the purpose of finding the slope of a tangent line at a point of the implicit function and finding the relationship between the change quantities to solve real life problems such as In the conical tank shown from left hand side, suppose that the height of the water level in changing at a rate of -0,2 foot/min and the radius is changing at a rate of -0,1 foot/min. What is the rate of change in the volume when radius is foot and the height is feet? Does the rate of change in the volume depend on the values of and ? Explain. 12 Analysis of didactic transposition in teaching the concept of derivative... Suppose that the volume V , the radius r, the height h of the water level are all functions of time t. Knowing that these variables are related by the equation V = pi 3 r2h. Differentiate both sides of the equation with respect to , we have d dt (V ) = d dt (pi 3 r2h ) ⇒ dV dt = pi 3 [ r2 dh dt + h ( 2r dr dt )] = pi 3 ( r2 dh dt + 2rh dr dt ) + The definition of differential∆x is introduced as a change in x and the notation dy dx is read as “the derivative of y with respect to x” in textbook 3 [12]. Despite of the fact that the definition of differential is just secondary content, textbook 1 [8] and textbook 2 [6] have their own lesson about this concept and its application in approximation. The difference between I2 and combination (I1, I3) + Institution I2 uses an inductive way to construct the theory and mitigate the program in textbook 2[6] so the definition of the derivative from one side is not mentioned. It is difficult to explain theoretically “Why function f (x) = { −x2 if x ≥ 0 x if x < 0 is continuous and is not differentiable at x = 0?”. Textbook 2 [6] just describes that the graph of the function is broken at O(0,0) although the meaning of “the graph of the function is broken” is not mentioned before. In textbooks 2 and 3, “the graph of the function is broken” can be proved easily by the definition of the derivative from one side. + The concept of the tangent is only provided as intuitive visual in textbook 2 [6] that does not set definition like in textbook 1 [8] . 2.2.2. Mathematical Organization Based on the theoretical and theoretical foundations of mathematics, there is a different way of organizing mathematics (the sequence of tasks, the techniques of solving, the technology and the theory applied to explain the technique differently), resulting in different presentation in each textbook. The main tasks that textbooks put in will only study on. Task type T1: Find the derivative of y = f(x) at x0 by the limit process Technique τ1 1 Technique τ2 1 + Given x0 and ∆x Calculate ∆y = f (x0 +∆x)− f (x0) + Set the quotient ∆y ∆x + Find lim ∆x→0 ∆y ∆x then y′ (x0) = lim ∆x→0 ∆y ∆x + Given x− x0 and Calculate f (x)− f (x0) + Set the quotient f (x)− f (x0) x− x0 + Then find lim x→x0 f (x)− f (x0) x− x0 Technology θ1: Definition of derivative Theory Θ1: The limit of the function Example: Exercise 3, Textbook 2 [6], p.156 Mini task type T 1c1 : Show that y = f(x) is not differentiable at x0 Technique τ1c1 : Show that y = f(x) is not continuous at x0 Technology Θ2: The limit of the function Theory θ2: The relation between Differentiability and Continuity 13 Chu Cam Tho, Nguyen Tien Dat Example: Exercise 4, Textbook 2 [6], p.156. Mini task type T 2c1 : Find the instantaneous velocity of the object which has position s = s (t) at t = t0 Technique τ11 or technique τ 2 1 . Technology θ1: Definition of derivative. Theory Θ3: The physical meaning of the derivative Example: Exercise 7, Textbook 2 [6], p.157. + Type task T2: Find an equation of the tangent line to the graph (C) : y = f(x) at (x0, f(x0)). Technique τ2: + Evaluate f ′(x0). + Substitute into general tangent equation y − y0 = f ′ (x0) (x− x0) Technology θ1: Definition of derivative. Theory Θ4: The geometric meaning of the derivative Example: Exercise 5, Textbook 2 [6], p.156. Mini type task T c2 : Find an equation of the tangent line to the graph (C) : y = f(x) whose slope is k Technique τ2c1 : + Use technique τ1 + Solve f ′(x) = k to find solution xi. + Substitute xi into general tangent equation y − yi = f ′ (xi) (x− xi). Technology θ1: Definition of derivative. Theory Θ4: The geometric meaning of the derivative Example: Exercise 6c, Textbook 2 [6], p.156. + Task type T3: Find the derivative of y = f(x) by basic differentiation rules. Technique τ3: Use the rules of differentiation Technology θ2:Basic differentiation rules Theory Θ5: Definition of derivative. Example: Exercise 2, Textbook 2 [6], p.163. Mini task type T c3 : Find the differential of the function y = f(x) Technique τ3: Use the rules of differentiation Technology θ2: Basic differentiation rules Theory Θ5: Definition of derivative. Example: Exercise 1, Textbook 2 [6], p.171 [8] Based on the same program goal set by the institution and set out accordingly, the textbooks might have the same task types listed in the following table: Table 1: Statistics table the number of task type in textbook 1 [8] and 2 Task type Technique Textbook 1 [8] Textbook 2 [6] T1: Find the derivative of y = f(x) at x0 by the limit process τ11 or τ 2 1 3 4 τ1c1 or τ 1c 2 1 1 T2: Find an equation of the tangent line to the graph (C) : y = f(x) at (x0, f(x0)). τ2 1 3 τ2c1 1 3 T3: Find the derivative of y = f(x) by basic differentiation rules. τ3 7 13 Sum 13 24 Remark: textbook 1 [8] presented the definition of the derivative from one side so we can 14 Analysis of didactic transposition in teaching the concept of derivative... use technique τ1c2 : Calculate the derivative from the right f ′ ( x+0 ) and the derivative from the left f ′ ( x−0 ) , if f ′ ( x+0 ) 6= f ′ ( x−0 ) then y = f(x) is not differentiable at x0. * Mathematical praxeology in textbook 3 [12] + Task type T1: Find the derivative of y = f(x) by the limit process. Technique τ1: Calculate y′ (x) = lim ∆x→0 f (x+∆x)− f (x) ∆x Technology θ1: Definition of derivative Theory Θ1: The limit of the function Example: Exercise 25 – 32, Textbook 3 [12], p.104 Mini task type T 1c1 : Find the slope of the tangent line to the graph of the function y = f(x) at (x0, f (x0)) Technique τ1c1 : Use technique then substitute x = x0 Technology θ1: Definition of derivative Theory Θ2: The geometric meaning of the derivative Example: Exercise 5 - 10, Textbook 3 [12], p.104. Mini task type T 2c1 : Find an equation of the tangent line to the graph (C) : y = f(x) whose slope is k Technique : + Use technique + Solve f ′(x) = k to find solution xi. + Substitute xi into general tangent equation y − yi = f ′ (xi) (x− xi). Technology θ1: Definition of derivative Theory Θ2: The geometric meaning of the derivative Example: Exercise 25 - 32, Textbook 3 [12], p.104. Mini task type T 3c1 : Show that y = f(x) is not differentiable at x0 Technique τ3c1 : Calculate the derivative from the right f ′ ( x+0 ) and the derivative from the left f ′ ( x−0 ) , if f ′ ( x+0 ) 6= f ′ ( x−0 ) then y = f(x) is not differentiable at Technology θ1: Definition of derivative. Theory Θ1: The limit of the function. Example: Exercise 93 - 96, Textbook 3 [12], p.106. Mini task type T 4c1 : Graphical reasoning Technique τ4c1 : Use a graphing utility to graph the function and show where the function is not differentiable. Technology θ2: The relation between Differentiability and Continuity Theory Θ3: Definition of derivative Example: Exercise 99, Textbook 3 [12], p.106. Task type T2: Find the derivative of y = f(x) by basic differentiation rules. Technique τ2: Use the rules of differentiation Technology θ3: Basic differentiation rules Theory Θ3: Definition of derivative. 15 Chu Cam Tho, Nguyen Tien Dat Example: Exercise 39 - 54, Textbook 3 [12], p.115. Mini task type T c2 : Find the instantaneous velocity of the object which has position s = s (t) at t = t0 Technique τ2: Use the rules of differentiation Technology θ3: Basic differentiation rules Theory Θ3: Definition of derivative Example: Exercise 99 - 100, Textbook 3 [12], p.117. Task type τ3: Find dy dx by implicit differentiation Technique τ3: Step 1: Differentiate both sides of the equation f (x, y) = 0 with respect to Step 2: Collect all the terms involving dy dx on the left side of the equation and moveall other terms to the right side of the equation. Step 3: Factor dy dx out of the left side of the equation. Step 4:Solve for dy dx Technology θ3: Basic differentiation rules Theory Θ3: Definition of derivative Example: Exercise 1 – 16, Textbook 3[12], p.146. Mini task type T 1c3 : Determine the slope of the tangent line to the graph implicitly at the point (x0, y0) Technique τ1c3 : Use technique τ3 to solve for dy dx then evaluate dy dx when x = x0 and y = y0 Technology θ3: Basic differentiation rules Theory Θ3: Definition of derivative. Example: Exercise 29 – 32, Textbook 3 [12], p.146 Mini task type T 2c3 : Find the rate of change throughout related rates Technique τ2c3 Step 1: Identify all given quantities and quantities to be determined. Make a sketch and label the quantities Step 2: Use τ2 and τ3 to write an equation involving the changing rates either are given or are determined Step 3: Substitute into the resulting equation all known values for the variables and their rates of change. Then solve for required rate of change Technology θ3: Basic differentiation rules Theory Θ3: Definition of derivative Example: Exercise 14 – 23, Textbook 3 [12], p.154. 16 Analysis of didactic transposition in teaching the concept of derivative... 3. Conclusion From research on the content of Vietnam textbook in 2000, 2006 and the American textbook 2010 and some other researches we found that: The current textbook program in Vietnam has been reduced in comparison with textbook program 2000. The notion of derivative has been presented in a modeling teaching approached to form the physical, geometric meaning, affine approximation meaning of the derivative concept. However, the fundamental link these twomeanings of derivative concept does not exist in the textbook 1 and 2 [2]. Student feels confusedly to reach the theory in textbook 1 and 2 with their tasks type (mathematic exercises) [9] that does not actually offer many type of tasks that apply in real life to help students orient their knowledge as close to future occupations. In contrast, the American textbook has built seamless knowledge from theory to practical application. Despite the similar tasks type which are find the derivative of y = f(x) by the limit process, find the derivative of y = f(x) by basic differentiation rules and so on, otherwise, textbook 3 has differently type of tasks including: Determine the slope of the tangent line to the graph implicitly, graphical reasoning, ... etc to solve problems in real life. At present, the Vietnam’s education is changing with the tendency of learner activation [7]. We recommend the definition of derivative in Vietnamese textbook will be approached like the way in American textbook and modeled after the following task type: Task type T: Find the slope of the tangent line to the graph of the function y = f(x) at given point by tangent approximation procedure in GeoGebra software. Technique τ : Step 1: Draw the graph of the function y = f(x) and the secant line through the point of tangency A