Summary of students' thinking processes and conceptions in the control

comfortable working with functions without their algebraic expressions. Some students expressed frustrations. They mentioned that the problems would be much easier if the equations of the functions were given. Moreover, four students out six revealed

difficulties in reading and interpreting graphs and tables. Also, they have deficiencies in their graphical understanding of derivative as slope of a curve and slope of a tangent line (Question I (2) and Question IV). In addition, they have weak visualization skills. They were not able to use the graph of the derivative function and relate it to that of the original function (Question II and Question III). Finally, all students except for AC1 showed lack of an object conception of derivative. Their understanding remained at the action/ process levels.

Question I (This question corresponds to question Ion the derivative test).

Concerning question I, the graph of of and its tangent line at x= 5 are given; it requires determining ) and ). Three students in the control group demonstrated

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lack of understanding of the relationship between the derivative of a function at a point and the slope of the tangent at that point. Here is a sample of one student's response:

Researcher: What is How can you handle this problem?

CB1: (thinking...): Well, we don't have the equation of , and thus I can't find .

Researcher: What about the graph and its tangent? Do they include any useful

information?

CB1: (thinking...).No, I cannot see any relation between the graph and the

derivative. This question is hard.

The remaining three students (CA1, CA2, CB1) were successful in their answers. They mentioned that the derivative of at x = 5 is equal to the slope of the line tangent to the curve at x= 5. They used the two points (0, 1) and (5, 3) on the curve and calculated the slope ( ), thus showing an object conception. Then, when they were asked to find , only CA1 and CA2 out of the six students managed to provide an accurate answer. They used the equation of the tangent line at x= 5 to find . When asked to justify their answers, they used the concept of linearization, which approximates a function at a point using a tangent line near that point. The remaining students mentioned that should be close to 3. When asked to provide a more accurate answer, they replied they do not know since they do not have the algebraic expression of the function, thus showing an action conception of derivative.

Question II (This question corresponds to the second question on the derivative test).

In question II, the graph of + 4 is given without its equation, and the requires determining whether its derivative is increasing or decreasing. It is noticed that

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none of the students in the control group solved this question, thus showing a lack of an

object conception. Four of them failed to recognize that the derivative of a function is a

function itself which has its own properties. Also, two students demonstrated poor understanding of the relationship between the sign (positive, negative) of a function and its variation (increasing, decreasing). Here is a sample of students' responses:

Example 1

Researcher: Is the derivative of this function is increasing/decreasing or both? CB2: Well, from (-∞, 0) the derivative is increasing (going up), and from (0, ∞)

it is decreasing (going down).

Researcher: Are you aware that this graph is for and not CB2: (thinking...).So what, I think that its derivative acts the same way. Example 2

CC2: from (-∞, 0) the function is increasing, thus its derivative is positive and

hence increasing. On the interval (0, ∞), the function is decreasing, thus the derivative is negative and hence decreasing.

Researcher: So, are you saying that a positive function is the same as an

increasing function?

CB3: Yes, and a negative function means a decreasing function

Question III (This question corresponds to question III on the derivative test). In this

question, the graph of is given without its equation and three questions on are posed (intervals where is increasing/ decreasing, the critical points of and the nature of the critical points). Four students out of the six failed to answer this question, thus showing lack of an object conception of the derivative concept. Even

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though the researcher made it clear that the given graph is for ', the interviewees did not realize that is itself a function, an object that has its own properties. They

assumed that the intervals where is increasing / decreasing and the critical point of are the same for However, two students (CA1 and CA2) solved this

question successfully. It is noticed that CA2 solved this question mechanically, thus showing an action conception. First, he found the equation of , found its critical points (G'(x) = 0) and then set up the table of variation of G(x) to discuss the nature of the critical points.

Question IV (This question corresponds to the fourth question on the derivative test).In

this question, students were asked to explain geometrically why the derivatives of the two functions and , where , are equal. Four students in the control group (CB1, CB2, CB3, CC1) demonstrated an action conception for what was required. These students wrote examples of functions using polynomials, and then stated that the two functions have equal derivatives since the derivative of any constant is zero. Here is a sample of CB1's response:

Researcher: Why do the two functions have equal derivatives? CB1: Well, suppose that and .

Researcher: Do the functions have to be polynomials?

CB1: (thinking).This come to my mind when dealing with functions. I think yes. Researcher: Proceed.

CB1: The derivative of and since the derivative

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Researcher: The question asked you to explain geometrically and not

symbolically. Can you give me a graphical representation of the derivative?

CB1: I don't know

However, two students (CA1, CA2) recognized that the curve of G is the vertical translation of F along the y- axis, and that the slopes of the two curves must be equal in order for the derivatives to be equal. However, they were not able to reach out a

conclusion showing process conception for what was required.

Question V (This question corresponds to question Ion the derivative test).In this

question, a table of values for the function is given, students were asked to complete another table for ). Four students failed to answer this question showing lack of object conceptions of both the function and the derivative concepts. In other words, these students failed to use the table of values for as an object, and hence estimate the derivative at a point by calculating the average rate of change over small interval. Here is a sample of students' responses:

CB1: Well, using the information from the table, C (0) = 0.84, C (0.1) = 0.89

and so forth. Therefore, and ) =0. Same for other values.

Researcher: Why zero?

CB1: because the derivative of any constant is zero. Researcher: It is given that C(t) is a continuous function

CB1: Yes, it is given I know.

Researcher: What does a continuous function mean?

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Researcher: Okay. So, is this your final answer?

CB1: (Thinking...) This is my final answer.

Moreover, it is noticed that CA2 was not able to provide a complete correct answer, but he demonstrated an object conception of the function concept. He noticed that increases in some interval, remains constant and then decreases. Also, he mentioned that the derivative of the function must be positive, equal to zero, and negative and negative respectively. However, he failed to calculate the rate of change using two nearby points, and hence failed to find the derivative at a point numerically. Finally, two students (CA1 and CB2) solved this question correctly showing an object conception. They realized that they can take two consecutive points and calculate the slope which approximates the derivative at a point. When they were asked if there is a way to check the correctness of their answers, they did not suggest any. One way for example is to check the intervals where the function is increasing, decreasing or constant with those where its derivative is positive, negative and zero respectively.

4.5.4.2 Summary of students' thinking processes and conceptions in the experimental