Misconception 2: Vertex as the Origin

A few students’ personal meanings associated the vertex as the origin. One of these, Jerry, had a personal meaning that the vertex of a quadratic function is the “origin.” Even further, he claimed that to find the vertex you “set ”, another misconception, as setting finds the -intercept. Jerry could not conceptually relate the vertex to the derivative other than the fact that “the derivative of a quadratic function can be determined” and he also did not believe that

there is more than one way to find the vertex. Jerry’s misconceptions could have prevented him from being able to correctly answer the real world problem, Question 2, during his think-aloud session. He struggled with the free fall formula and solved the equation for the variable for both finding when the rocket reached maximum height (part a) and for finding

when the rocket hit the ground (part c). He used the same procedure for two questions asking two completely different things. When prompted by the interviewer to talk about what he did to find when the rocket hit the ground, he replied:

J: Ok, so this is where everything got like really confusing….I was gonna set um either the derivative or or um the actual equation to and solve for it, but um, I don’t know, I just got really really confused…I just took the derivative and set it equal to and then I

got for seconds… and then I put into the um original.

Jerry’s mistake of seconds for the time the rocket reached the ground also created

yet the time he had found for when the rocket reached its maximum height was at seconds.

This is not possible with the way he drew his graph.

Figure 4.8 Jerry’s graphing mistake during think-aloud, Question 2

Jerry did, however, correctly find the critical points, local extreme values, and intervals of increase and decrease for part c of Question 1 during the think-aloud session, except for an arithmetic error in finding the corresponding -coordinate for the maximum point. This suggests

that Jerry is at an action level of understanding for part c of Question 1 as he could perform, yet this action level possibly keeps him from being able to apply the same concept in a different situation.

Although he proceeded through part c of Question 1 during the think-aloud session for the most part correctly, Jerry struggled on the same type question on Test 3 and the Final Exam questions. He was the only one to get all parts of both questions wrong on Test 3, including graphing the wrong shape of the quadratic function.

Jerry’s inability to graph the quadratic function suggests that he has a weak schema of quadratic functions and their corresponding graph. Furthermore, for the second question on Test 3, to find when the debris reached its maximum height, Jerry again resorted to solving for the variable from the free fall formula set equal to zero.

Figure 4.10 Jerry’s solution to the real world problem, Test 3

Not only were his solutions for Test 3 inaccurate, but on his Final Exam, he also had some erroneous responses, especially to finding the maximum of the particle. His confusing and unclear response is illustrative of the confusion and misconception he had about the maximum of the particle.

Figure 4.11 Jerry’s Final Exam response

Jerry does not even appear to be at an action level, other than his correct response to Question 1 part c during the think-aloud, and even then he made an arithmetic mistake on calculating the -coordinate of the maximum. Jerry’s misconceptions relate all the way back to

his misinterpretation of his personal meaning of the vertex of a quadratic function. His weak schema of quadratic functions possibly disables him from being able to accurately work through problems relating the vertex to the derivative in different context problems.

Nathan was another of the several students who associated the vertex as the origin.

Figure 4.12 Nathan’s misconception of the vertex

When asked to explain what a quadratic function is, Nathan wrote the quadratic formula. These two misconceptions, however, did not impact his ability to perform on Test 3, the think-aloud questions, or the Final Exam questions. Nathan correctly answered both questions during the think-aloud session and both questions on Test 3. Furthermore, Nathan had very strong

conceptual responses on the Final Exam questions. In fact, on the final exam, Nathan had some of the best conceptual responses out of all thirty participants, especially on parts g through i.

During Nathan’s group interview, when both participants were asked to explain the meaning of the derivative, Nathan was even able to connect concepts and relate the graph of the quadratic function to the graph of its derivative.

N: When on the derivative, when it’s above the -intercepts, , then on the original function it’s increasing and when it’s below its decreasing.

I: Ok, so you are showing me with your pen. So, it ah yeah, so if the derivative function is above the -axis you’re saying the original function is increasing.

N: is increasing

I: Ok, and when the derivative is below the -axis, then that means- N: -decreasing

A few minutes later, when discussing responses to the real world problem, Question 2, Nathan was also able to explain within the context of the problem why we set the derivative equal to in order to find the maximum point of a function.

N: That point where it changes from increasing to decreasing at -value , , that’s where like on the left is going up and on the right of it it’s going down so it’s changing. I: Uh huh and what happens to the rocket at that point?

N: It comes down, like I don’t know, what do you mean? It goes up then it’s slowing down and then it stops then it comes back down.

I: Ok, so you are saying it stops at ? N: Yeah, yeah it stops at .

Even though Nathan had two initial misconceptions of the quadratic function and the vertex, this did not impact his conceptual understanding of the derivative or his ability to perform on either problem during the think-aloud session or on Test 3. His ability to explain and

articulate the meanings behind the derivative concepts suggests he may be at least at a process level of understanding of the derivative.