Hallie Bennett
Mathematical Problem Solving
Case Study
3 May 2012
For my case study, I picked two different students and had them work on the same
problem. The two students, who are 5th graders and at different ability levels are
students that I do not work with on a regular basis. I had to pick other individuals
than the ones I work with because my students are all in grades K-4 on IEPs and do
not have the ability to complete the problems that were in the book. The problem
that the individuals worked on was # 1 Worm Journey in chapter 1. For the problem
and each student, I recorded what the individuals said as they worked through the
problem, so that I could go back and review how they thought it through, how they
showed their work, and if there was any problems or misconceptions that we had to
go through at the end of the problem. After both girls solved the problem, we took
the time to go over the problem and discuss the correct way to solve the question.
The first student was on the right track for solving the problem. The second student
had a little more trouble solving the problem and needed more guidance to work
towards the answer. Below are the scripts for each student and the problem they
did as well as their pictures, my explanations, notes, and analysis on how they did
on the problem.
First Problem:
Student #1 –
H: I explained to the student what they needed to do for this project.
S: The student read the problem and then drew a picture of a ruler on the paper she
was using to show her work.
S: What do you call this? (Pointing to the ruler on her page)
H: You could call that a ladder or a ruler
S: Um you could put the ruler with 11 lines I mean 12 lines and then you need to…
(Reads the problem again and then starts drawing her picture with more lines and
detail)
S: The worm crawls up 3 feet and then goes down 2 feet everyday and then crawls 3
up and then down 2. So, 3-2 = 1. Then you go 1 + 3 is, 1 + 3 is 4 and then minus 2 is 2
and then the next day he goes up three again and then minus 2 is 3 and plus 3 is 6 -2
is 4. Up to 12 days and then on the 12 it will be all the way up the wall because
everyday it goes up 3 and than down 2.
H: So, how many days will it take him to reach the top of the wall?
S: 12 days
H: Ok, good job. Thank you for working hard.
Analysis and notes for student 1, problem 1: The student was on the right track
when solving the problem. She understood that the worm went up 3 feet everyday
and than fell 2 feet every night, so in the end the worm went up 1 each day. She was
drawing a diagram and doing the proper math. However, she did not finish her
diagram before coming up with an answer, which is why I believe she got the
answer incorrect. Also, the student was not recording the math that she was doing
in her head. So, in my conclusion she guessed that the answer was 12 days because
each day it went up one, which means 1 + 1, 12 times would give you 12 days. I think
if she would have finished her work and picture she would have seen that the
answer was 11 days. After she gave me the answer of 12, we went over the answer
and discussed how she was on the right track, but she needed to continue to work
on her picture and record her math facts. I also showed her the way I solved the
problem when I had to do it and told her that I too guessed 12 days when I did the
problem the first time. After we finished our discussion, I encouraged her to keep
drawing pictures when she is solving math problems because it helps both her self
and others visualize what she is doing. I also instructed her to record the math she
does in her head so that she does not lose track of what she is trying to do.
Student # 2H: I explained to the student the project I was working on and let the student know
that she was going to help me out with it. She was excited. I also let her know the
problem we were working on and explained that she was going to try and solve it
for me, there was no penalty for getting the answer wrong.
S: The student read the problem.
S: (The student recaps what the problem says): Today I, um my thing is the worm is
trying, the worm is on the bottom and he is on the wall and he is trying to go on top
of it, it is 12 feet of wall, so everyday he crawls up 3 feet and then when it is 2 feet he
will sleep, when it is night time and maybe it will take him 6 days to get up the wall.
S: …and he is a tiny worm, and he is trying to go up a wall and when he is done he
will get through it and then when he is done he might see the world, up and up. He
might see a crow that will eat him. He might see a squirrel that will put him in the
nest and eat him or maybe he might have some jazz dancing.
H: Could you possibly show me some work on how you came up with 6 days or a
picture to show me that you came up with 6 days?
S: Like, this…
H: Ok
S: I am going to draw a wall (student begins to draw picture)… I am going to make
lines and then make lines here. So, the worm is right there and its going to be 2 feet,
2 feet and the worm is smiling and maybe he is wearing a hat and shades because it
is very, very, very hot out. So, he got his hat and his shades. So, he is going to take his
time because it’s a small worm, so maybe 1, 2, 3, 4, 5, 6. 6 days to the top (pointing at
her picture on her paper). 2 feet and when he makes 2 feet he will go nana.
H: Nana?
S: Yes, nana.
H: Does nana mean sleep in Yupik?
S: Yes.
S: …and the worm goes on top and its beautiful. He may see beautiful cities and he
may see a worm lady and then he may see the beautiful town, so beautiful. He will
see crows, maybe he will see a squirrel or the perfect, perfect, perfect lady or he
might see different people. He may think wow, why do those people have legs. I
wonder if they are not soft? Do they have hair? Yeah, they have hair. I wonder why I
don’t have hair. So, every 2 days, 2 feet, sleep, 2 feet, sleep. That will make him have
energy and if he didn’t have the cap and shades that would make him very hot and
he may have tan lines. It will keep you protected from the sun if you have shades
and a hat. You have to keep hydrated, hydrated, hydrated by drinking water and you
need exercise and lots of water to give you energy.
H: So, your answer is 6?
S: Responds yes with her eyebrows.
H: Ok, thank you.
Analysis and notes for student 2, problem 1: At the beginning of the problem, it
seemed like the student was on the right track, however you can quickly see that the
student got off track. I allowed her to go on for a little bit to see if she would get back
to the problem, but I had to guide her by asking her if she could show some work.
When she drew her picture it was just a picture with 6 lines and then she
demonstrated drawing lines that were 2 feet long. Her thinking was not clear on the
problem, but I did not want to give away too much when she was working on the
problem. When she continued again working on the problem she kept
demonstrating that the worm fell 2 feet, but never mentioned anything about the
worm going up 3 feet. So, again it was hard to follow her train of thought. The only
thing I could think of for how she got her answer was 3 x 2 = 6. So, after she gave her
answer and I finished noting how she worked out the problem I asked her why she
got the answer she did and she responded, “I do not know”. From there, I
demonstrated for her how she could have solved the problem with either a picture
or simple addition and subtraction facts. When I completed the explanation she said
that makes sense.
For this student, it is important for her to learn how to stay on track when working
on problems and how to be confident in herself. I believe she went off topic from the
problem because she was unsure how to solve the problem and needed to find a
different way to fill her time, so she made up stories about the individuals in the
problem. For instruction for this student, I would teach her how to look at a problem
and pull out the important information from it. I would also show her how to draw a
picture or diagram to represent the information in the problem and how to record
her math on paper, so that she does not forget what she is doing and so she can
double check her work at the end.
My students had so much fun with the first problem that they asked if they
could do another one. So, we worked on problem #3 Free Concert Tickets in
chapter 2 as well. Below are the scripts and pictures from this problem.
Student #1H: I picked out a second problem that the students could work on.
S: The student read the problem
S: So, we have Alex (the student is writing down on a piece of paper the name of the
individuals calling into the radio show)… Blake, Chuck, and um…what does this say?
H: We can call her Dariah.
S: …and we put them in order in the order that they called. We could go Blake,
Chuck, Dariah, and Alexis (recording this on paper).
S: And then it could go Chuck, Dariah, Blake, Alexis (recording on paper) Alexis?
H: Yes, Alexis. You are doing well.
S: And the next one could go Dariah, Blake, Alexis, and Chuck. Is it done?
H: Do you think it is done?
S: Yes.
H: Than okay, how many different ways?
S: 4
H: 4, ok good job.
S: Well actually 16 because 4 X 4 = 16
H: Ok, why did you do that multiplication problem, using those numbers?
S: I do not know; it just seemed like the right problem to do.
H: Ok, we will discuss this when we discuss the answer to the problem.
S: Okay, sounds good.
Analysis and notes for student 1, problem 2: The student had a good start to this
problem. She was recording her thinking on a piece of paper and she was heading in
the right track. However, I noticed as she was finishing the problem, she may have
believed that she could only have one person call in first. For example, Alexis could
only be first once. At least, this is what her worked showed. However, she later said
well actually its 16 because 4 x 4 = 16, but when I asked her why she did that
multiplication she was unable to explain why. Another note to make, is that the
student’s classroom teacher explained to me that they have not worked on making
systematic lists to solve problems, so that could affect her ability to have solved this
problem. If she would have had that background knowledge she would have been
able to complete the problem correctly because when I began to show her how I
made my list to solve the problem, she was able to help me complete the list with no
confusion about how to do it.
Student # 2
H: I picked out a second problem that the students could work on.
S: Student reads the problem.
S: Hi, this is Shea Matthias and I am going to work on a problem in chapter 2 about
free concert tickets. There is Alex, Blake, Chuck, and Dariah. They are trying to win
free tickets so that they can go to the concert. So, they listen to the radio so they can
get free tickets. So, maybe they will answer some questions or ask some questions.
So, this is how they will get free tickets…they will listen on the radio, they will be on
the radio, and then Alex, Blake, Chuck, and Dariah they will listen to the radio. So,
they went on the phone and asked if they could get free tickets so they could go to
the concert. What is a radio show?
H: A radio show…it is kind of like a talk show you see on television, but on the radio.
They play music and those in charge of the radio talk show talk about different
topics and then people can call in and ask questions or they can answer questions to
win prizes.
S: Oh, ok. Well, when they were calling they might answer questions. So, maybe they
want free tickets because they really want to go to the concert for a date. So, they
call very quickly so they can get free tickets. So, they call. (drawing on her paper….)
And they say can I get free tickets please and maybe they will say yes or no and they
say yes and maybe when they say yes, they will get free tickets they can get ready
and they can dress up very nice to go to the concert.
H: So in the question it says, list all the possible orders in which their calls could
have been received, so how many different ways could they have called? For
example, Alex calls first, then Blake, then Chuck then Dariah. What is a different
order that they can call in?
S: Um, maybe they can call in different numbers when they heard it on the radio and
then they will answer different questions and different people will answer.
H: Ok, so do you know how we sequence, when we put different events or things in
order? So, we do Alex, Blake, Chuck and Dariah. Let me show you an example. We
will use A, B, C and D (writes this down on paper). Now, you can switch the order up
and Blake can call first than Alex, than Chuck then Dariah. So, how many different
ways can we switch the order or names around to give us different orders?
S: C, D, A B
S: D, B, C, A
S: C, B, D, A
S: A, D, B, C
S: And um, (she looks through her other answers to see what she already has and to
see what she may be missing)
S: A, C, D, B
S: Do I have BD?
S: B, D, A, C
S: C, A, B, D
S: D, A, C, B
S: A, B, D, C
S: I will do one more and I will be done.
H: You are doing a good job. I like how you are working it out.
S: Student is looking through her other answers to see what she is missing
S: B, C, D, A
H: So, how many ways do you have?
S: Counts and says 11.
H: I count one more.
S: 12.
H: Circle your answer so we know what it is. Good job.
Analysis and notes for student 2, problem 2: As you can see, the student started
off with describing a story using the people in the problem. She began the problem
off track and had to be redirected to the problem. I believe she began with a story
because she was unsure what a radio show was and how it worked, so instead of
trying to solve the problem, she went to telling a story about the problem which is
an area she is comfortable in. To get her to understand what the problem was, I had
to give her an example of what I was looking for. I did not only explain it to her in
words, but I also wrote it down on her paper to give her an idea. From there, she
seemed to get the point of the problem because she began recording additional
combinations that could be used. Also, she was looking back at combinations she
previously made and was able to build new combinations based on the information
she already had and she was making a lot of progress. I believe that if she would
have had the background knowledge, like I said about the previous student then she
would have been able to solve this problem with no issues. Having that background
knowledge and previous experience is very important for the success of these
students.