Alana Linde

Student Interview Assignment

Solving Quadratic Equations

Overview of the Session:

During my time spent with my student that I interviewed, I was impressed with the

student’s understanding of the task and ability to articulate her thinking throughout process

behind the steps taken to solve each problem. Her ease and comfortability of working with the

task leads me to believe that either they has recently learned how to solve these types of

equations or that she learned, and retained, the information well.

After the first prompt, I think the student was confused at first. Pausing to consider if she

could indeed solve the problems in another way, she told me that because the first and third

problems could not be factored, she could not provide another solution. Looking at the second

one, she thought about it, but concluded that the approach she took was the only way she knew

how to solve for x. However, when I provided the second prompt, the student was able to make

the connections between the functions of each problem and the relationship to the graphs she

produced on her graphing calculator. Overall, I was impressed with the student that I was

working with. Her knowledge of computation and problem solving was made clear, especially in

problem B that I will expand on more later.

The student clearly demonstrated her knowledge of using the quadratic equation to solve,

as well as graphing to solve. I think the student could have been a bit more clear about the

connection between the relationship of the intersecting points on the graph. I do not quite think

that she saw that those intersections are in facts the values of x she found in A. Additionally, in C

the student was able to graph, and articulate that there was no point of intersection, but perhaps

she could have used more probing to see that there was no real solution to the functions.

Alana Linde

Transcript Excerpt and Analysis:

After I provided the second prompt to nudge the student to solve graphically and make

connections across representations, the student began to consider what the graph of each of the

problems would look like. Asking at first if it was okay to use the graphing calculator, the

student proceeded to complete A. However, moving on to B is where I was enlightened about the

student’s knowledge about cross representations and making connections.

The following is an excerpt from the interview that speaks to the student’s knowledge:

[Shifts attention back to calculator, looks at paper…]

… Well, B would be the same graph.

And how do you know that? –

Because they both equal each other. Like they both equal zero when they are set

up together. I mean they are initially written out different, but here they are the

same…

This brief encounter with the task I found interesting. The student did not actually graph

the functions in B, but rather looked at the functions, and was able to conclude that they would

be the same graph without needing to do so. I was surprised that she was able to make this

connection so quickly, especially before graphing. After asking her to elaborate more on her

finding, she articulated that because she later found that either side of the equation was the same

function after she ‘FOILed’ and distributed, that they graphs would be the same even though it

wasn’t as evident in the initial equation.

The following is an excerpt that occurred before any prompting was given to the student

about problem B:

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… So they are equal to each other… So yeah they both equal each other.

Okay, so what does that mean?

Uhm… That x is like the same value in both.

Prior to graphing, when the student was solving part B using symbolic manipulation she

concluded that x would be the same value on either side of the equation. Knowing this, I believe

she used this insight to help her later on when she was asked how the graphs of B relate to her

answer that x is the same value for both sides.

Looking back at the interview now, I wish I had probed the student to see if she could

have reached the conclusion that x is equal to all real numbers and further that when graphing,

the two functions are the same graph further confirming that x is equal to all real numbers. I

think that the student was rather close to reaching the conclusion, or maybe I just assumed that

she knew. This is great insight for my teaching that I should emphasize the importance of having

students articulate their thinking and not try to assume their thinking for them.

Analysis Results:

Looking at the student’s results in a quantifiable way to compare to the other students, I

scored the student based on three different levels. The first level that I scored the student on was

a way to measure the student’s correctness of each of the three equations that were presented in

the task. This rubric seems to be where many teachers end their analysis of student

understanding. However, it is much more enlightening when looking at the versatility and

adaptability scores for the student about their understanding of the task.

Allowing the student to begin by asking to solve for x, she (as I anticipated) started to

solve using symbolic manipulation. The student seemed comfortable with the problems at hand

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and did not have any problem using the quadratic equation to solve for x. The student was able to

articulate her thought process and providing reasoning in many instances without me having to

further question why the student went about that process.

Before any prompting, the student was able to solve all of the equations. For A, I scored

the student as a 4-Completion on the Problem-Solving Rubric. For B, the student did not

physically write down that x = x or all real numbers, but through her explanation, she was able to

articulate that x carries the same value on either side of the equation, which yields a score of 4-

Completion as well. Lastly, the student in problem C made a minor computation error (just as I

did, and called attention to in class) for her final answer. Because she followed all the steps

necessary to finding the value for x and her knowledge of imaginary numbers, this yields a 3-

Result because the problem was very nearly solved.

Based on the Rule of 4 Model, I was looking for the student to be able to represent and

analyze relationships as well as translate among the representations. Based on the readings about

this before the interview, I anticipated the student would be more versatile compared to

adaptable. In terms of the student’s versatility, I rated her at Moderate level because although she

did not correctly solve all three equations because of the small computational error in C, she was

able to solve the equations using symbolic manipulation and by graphing (after prompting).

However, for the adaptability model, I rated the student at Low because it took the second probe

for her to solve graphically and make the connections between the functions on either side of the

equation in the initial problems. Although the student was able to tell me in parts B and C that

the graph was the same, or there were no points of intersection, I am not certain that she made

the connection that the values of x were all real and no solution respectively.

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Reflections:

1. As previously mentioned the biggest thing that I have learned about student thinking and

teaching through this assignment is not to assume anything about the student’s knowledge. More

specifically, I now know not to interpret one thing that the student says as firm understanding

when in fact I do not know for certain. Looking back at the transcript from this interview, what I

thought in the moment the student understood, I am not certain that she did. At least I did not

prompt her with questions that would lead to verbal evidence of her understanding in some

instances.

In terms of the teaching aspect, I have learned that it is so important to front load the

work you put into an assignment before you implement it. I have been learning strategies of how

to do so through the 5 Practices and I hope to use the framework more in the future. When you

anticipate students’ approaches and come up with focusing questions prior to the lesson, you will

be more prepared and serve as a better guide toward your students’ understanding. I found

myself taken aback because the student was able to articulate her thoughts more clearly than I

anticipated. I think that the fact that I was so impressed with this aspect of the interview to a

point where I was not as focused on extracting more information out of the student to really get

at the heart of the problems that were assigned. From a teaching aspect, it is also crucial that you

do your work ahead of time because you need to make sure you fully understand what the

solutions mean across representations before you expect your students to be able to. Admittedly,

it has been a while since I have worked with these types of problems, so I did not have all of my

knowledge of these types of functions pulled to the front of my brain to help during the

interview.

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After doing this just once, I have a lot of personal feedback about things I would do

differently and how to better prepare myself to provide for students. Of course not having any

experience working on a task like this with a student (interview setting), it was a little foreign to

not only the student but also myself. I believe that practicing these types of things will better

prepare me in the future.

To coincide with that enlightenment, I also learned not to discount students. The student I

had was extremely bright and intentional in the way she went about solving the problems that she

was given. She also asked me for further clarification in a certain instance to be sure that she was

approaching the problem correctly. The student’s intuition about the problems that she

encountered enabled her to revisit one and catch her mistake. This could be because of what her

classroom teacher has instilled in the students. Perhaps they have a classroom environment

where they investigate math rather than just breeze through it.

Lastly, I have learned that students do not all have the same thought processes, which is

excellent. Good math instruction should enable students to approach a problem in a way that is

most comfortable to them, rather than approach it the way their teacher wants them to. However,

this means that a lot of work must be done in order for the teacher to properly assess where the

student is. I only had one student in my interview and I feel like I could have asked more. It

opens my eyes to how much work I will have to do to circulate around a room and formatively

assess students’ understanding as they are working. This is no easy task!

2. Perhaps the most revealing responses in the interview that allowed me to gauge where the

student’s understanding of the mathematics involved would be from problem B as I previously

reflected on. The student at first made an algebraic error and reached the solution of 0 = 5. She

did not trust this answer, so she went back and analyzed what she did, and as she was verbalizing

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her steps, she caught her mistake. Further, the student was able to conclude that because the

functions on either side of the equation were the same once the symbolic manipulation was

complete, the graphs would also be the same. Therefore, the values of x were the same for both

equations.

Although the student in part A and C solved for x using the quadratic equation, and

graphed after prompting. I am now seeing that she may not have made a complete connection

between these two representations. For example, when the student graphed the functions in part

A, she told me that she saw there were two points of intersection, but she wasn’t sure what I had

meant by relationship in the prompt. Steering her to speak more about the intersections that she

was just talking about, the student began to compute the x values she found from the quadratic

equation, and then used to trace function to determine that the values were really close to each

other. Wishing I would have further questioned her as to what that meant now, I thought that she

saw that connection after computing those values, but I am unsure if that connection was made

or not.

3. Looking back at the questions that I asked the student during the interview, I would

categorize them as focusing questions as opposed to funneling questions. Even when the student

was confused about my prompt and questioned me, I really tried not to give a specific answer,

but rather to lead in her a direction where she would still have to uncover what it meant for

herself. I did not feel as though I needed to provide any leading questions, besides form the

probes, during the interview because the student was proactive in her approach for solving the

problems. When I first asked the student a question pertaining to the method of her symbolic

manipulation, I wanted to see if she understood why an equation has to be set equal to zero, her

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response told me that she knew that by doing this she could use the quadratic equation for

solving for x.

Another instance where I further probed the students to explain her thinking was in part B

where she found that both functions were the same on either side of the equations. The student

seemed to hesitate with what this was telling her. By asking, ‘what does that mean?’ the student

was able to articulate that x is the same value for each equation.

When the student was asked to describe the relationship between the graphs of the

functions of each equation, she began by talking about the points of intersection, but was unsure

what I meant by ‘relationship.’ This was a difficult one for me to quickly respond to. I asked the

student if she wanted to ‘expand a little bit more about the intersection that [she] was talking

about’ to see if that would give her a step in the right direction. The student computed and found

that the intersections were close. I wish I had asked one further question along this context.

Perhaps “How does the x-value you solved for relate to the graph?” Maybe here the student

would have said that the x-values are the points of intersection.

Similarly, for part C, the student graphed and saw that there were no points of

intersection, but I did not continue to probe to ask how that related to her x-value prior graphing.

If I had redirected her back to this for instance, maybe she would have been able to articulate that

there is no solution because there is an imaginary number in her solution.

4. In order to move my student to a more robust understanding of the concepts we were

working with, I would have provided further questioning in certain instances as I spoke about

above. Working mainly with part A, I would have pressed the student to further describe how her

x values and the intersections are close in numeric value and what that means. If the student were

to continue to be unclear, perhaps posing it as, if you were only provided an image of the graph

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for this problem, how could you articulate what the solution of the problem is? Although she was

close in her articulation, I would have liked to have seen myself probe further so she could have

walked away knowing that the intersection of functions are solutions to the equation.

When grading the adaptability for parts B and C, I would like to see this student with a

higher level than what I had marked her at in the future. In doing so, I think that I could provide

the student to continue articulating what the solution of the equation is when looking at the

graph. She knew that the graphs were the same, but I did not press her to speak much about the

solution of the equation. Likewise for part C, she did find the imaginary solution to the problem

(with a small error), and noted that the graphs never intersect. However, I could have further

asked what the graph meant in terms of the solution. Maybe asking something along the lines of,

“you told me there the functions don’t intersect, but what does that mean about the solution to

the problem?”

Something that students, myself included, aren’t too familiar with is the three types of

solutions when it comes to graphing equations – a numeric solution at the point(s) of

intersection, all real numbers, and no real solution. This cause of unfamiliarity could be from

previous teaching. As I mentioned at the beginning of this analysis, teachers tend to stop their

investigation of students’ understanding based on their answers. Either it is right or wrong and

that is easy for them to quantify. These types of learning experiences could lend themselves to

only solving problems like part A where students solve for specific solution and are pleased to

find a number, even if they don’t know how that connects back to the problem.

As a follow up to this task, if students had the same issues with articulating parts B and C

or making connections between the graph and their computations, I would provide them with an

activity that gave them images of functions on graphs. We could investigate how we would

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deduce what the solutions of the equations were without solving using symbolic manipulation.

Further, after students found the solutions using the graphs, we could go back and compute as

they normally would when solving for x. I find that graphing it is much more enlightening as to

what the solutions to the equations actually means rather than just manipulating numbers and

variables.

This was a challenging task on my end. I think found that there is a lot of work that goes

into even what some may consider a straightforward task. Trying to be intentional about what I

want my future students to learn. I find it so important to highlight what the goals of the task are

and how I will bring students to that understanding. It was difficult to hold back and not jump in

to help the student when she was questioning herself in some areas of the task. I think that where

we lose the most students in math is when it becomes more and more abstract and they have not

been provided the tools to make connections across the procedures that they learn to do.

Appendix:

Includes:

Interview Transcript

Copy of Student Work

Alana Linde

Student Interview Transcript

Italicized represent when teacher is speaking, and regular depicts when the interviewee spoke.

I’m going to ask you some questions about solving equations. I’m interested in how you come up

with your answers, so I’ll ask you to talk out loud about your thought processes of how you go

about solving them. It won’t be graded… I will give you a worksheet now... Solve for x.

Okay, and just talk about how I did it after I’m done?

Mhm, or you can do it along the way.

[Student works on problem A]

… Okay… So I mean do you want me to go along and solve it out loud?

Yes. You can go step by step.

So, I subtracted 5 from both sides and set the equations equal to zero to get x2 plus 7x plus 7.

And then that’s not factorable, so I put it into the quadratic formula. So negative 7 (uhm) square

root of seven squared minus 4 times 1 times 7 over 2. And then got negative 7 plus or minus the

square root of 21 over 2.

How did you know you needed to subtract 5 when you started?

Uhm because in order to solve for x your equation needs to be equal to zero. So then by

subtracting 5 from the 12 since there’s no variable, then it would make the equation equal to

zero.

Alright. Then you can go ahead and do the other two as well.

[Students works on problem B]

… (puzzled) I don’t know if I did something wrong… Okay well, I FOILed the x minus 2

squared to get that and then that gave me x squared minus 4x plus 4. And that equation needed to

be multiplied by 2… So then I got 2x squared minus 8x plus 8. Which is equal to 2x squared

minus 8x plus 3… And then I was trying to set the equation equal to zero, and it ended up being

like 5 equals 0. So that is like… not possible.

Okay…

I think that’s how it should work out… I don’t know, that doesn’t sound right though. I don’t

know if I did something wrong.. OH! I know, yeah I forgot the.. minus 5. Okay, so 2x squared

minus 8x plus 8 and then minus 5, which equals 2x squared minus 8x plus 3. Okay. Then this

equals 3. So they are equal to each other… So yeah they both equal each other.

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Okay, so what does that mean?

Uhm… That x is like the same value in both.

Okay. Now you can move on to C.

[Student works on problem C]

… Okay so for this one, I added the x to the x squared plus x minus 2 to get x squared plus 2x

minus 2 equals negative 7. Then I added the 7 to both sides to get x squared plus 2x plus 5 equals

0, and that’s not factorable so I put it in the quadratic formula… Got negative 2 plus or minus

(uhm) in the radical then 2 squared minus 4 times 1 times 5 and got negative 2 plus or minus the

square root of negative 16 over 2. And that is... like 16 can be square rooted... Ha. I know that’s

not how you say it… Into 4, but since it’s negative you take out an ‘i’ So I got negative 2 plus or

minus 4i over 2. And you can divide each of those by 2. Or the negative 2 and 2 cancel so it’s

like negative 1 plus or minus 4i.

Okay, now since you have solved all of these. Is there another way that would could have solved

these problems, and with so can you show me.

Uhm… Not with A or C, because if they were factorable you could have factored them or done

the quadratic formula. But since they can’t factor out, I think that is the only way you can do

them. And then B… I… That’s like the only way I know how to. Just like step by step to do that.

Okay, so then my next question is what would be the relationship between the graphs of either

side of the equation?

… Am I allowed to graph it first (referring to the calculator).

Yes.

Okay so you just want me to graph like 5 and then… this part and this part or like the new

equation?

The relationship between the graph of each side of the equation.

Okay. (Begins to put into the graphing calculator) So, I have a graph going in a straight line

coming through 5 on the y axis. And then like a parabola and it intersects on 5 in 2 different

spots.. Uhm… What do you mean by like relationship?

So, do you want to expand a little bit more about the intersection that you were talking about?

I get like where they meet. Like the roots? No, just kidding… (mumbles)

[Enters values into calculator]

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Well like when you solve this out to get like the negative 7 plus radical 21 divided by 2. That’s

like very... that’s like part of the root is.. it’s like point zero 2 off from the intersection uhm

closest to the y axis, so that’s like that one. And then… (returns back to calculator to perform

functions) The other point is when negative 7 minus radical 21 divided by 2… That’s really close

to the other intersecting point between the parabola and the line across 5.

Okay… And then could you do the same for B and C as well?

[Shifts attention back to calculator, looks at paper…]

… Well, B would be the same graph.

And how do you know that? –

Because they both equal each other. Like they both equal zero when they are set up together. I

mean they are initially written out different, but here they are the same… And then….

[Back to calculator for problem C]

… Okay and then for C, negative x minus 7 is like a linear line… And x squared plus x minus 2

is like a parabola and they don’t ever meet because, x square plus x minus 2 is bounded below so

it never goes past uhm.. negative 2 and that x minus 7 won’t ever meet, so they don’t intersect.

Okay, thank you for participating in this interview.

Is that all!?

Alana Linde