CASE STUDIES

This section presents case studies created from three students in each class. Details are included from interviews to describe the attitudes they had toward mathematics as well as their experiences within the instructional period. The students were interviewed about their views of mathematics, their mathematics educational experiences, and their responses to a graph they could read and interpret for the interviewer.

Students in this section were selected from the larger classroom groups. The classroom instructor assisted in selecting students to represent a range of achievement within the classes, and backgrounds. Four students from each class were chosen to participate in interviews before instruction using the technology and after the period of instruction. The students were asked questions designed to allow them to express their feelings toward school mathematics, previous mathematics experiences, and calculator technology. A graph was included in the interviews to allow the students to share their version of conceptual understanding. Students were expected to use this graph to describe what they interpreted about the graph. The students were audiotaped and were allowed to draw or write out their ideas with colored pens and paper.

Themes and trends emerging from the interviews were analyzed to determine information about the students' view of mathematics and the instructional period.

- Student pseudonym and ID number
- General description
- Observations and quotes from interviews, videotapes, and classroom observances pertaining to;
- Indicators of identification with mathematics
- Indicators of concept development
- Indicators of attitude toward mathematics
- Conclusions regarding this student

The classroom teacher reported that the students enrolled in this class did not have classroom experience working with the concepts of rate, speed, or graphing prior to the instructional period. None of the students had used a graphing calculator before the instructional period. The classroom teacher expressed concerns about the students’ ability to maintain the equipment in operating condition. The students had not worked cooperatively very much prior to the instructional period and the regular teacher had often found that it was difficult to keep the students on-task during group work.

The three students selected for case studies from the basic mathematics class were Ashley, Jennifer, and Michael.

__Ashley ID#21__

__General description.__ Ashley was a black female student in
the basic mathematics class. The regular teacher described her to be
a low performing student. She was not disruptive in during
instruction. She was pleasant to interview and seemed very excited
about participating in the instruction with technology.

__Observations and interviews.__ Ashley's responses in the
initial interview suggested that she had a view of mathematics that
was limited to consumer mathematics. When asked if she thought
mathematics was relevant to her life she responded:

"Going to the store, yes. Like seeing if the person gives you the right amount of change."

This response implies that Ashley understands the importance of mathematics as a calculation tool necessary for consumer affairs. No mention was made of additional mathematics concepts that may be relevant beyond the classroom.

She indicated that math should be easier and more exciting. When asked how she felt about mathematics she responded:

"I don’t like it. It takes too much time to write out the problems. It’s boring; there’s no fun about it. No excitement."

This statement suggests that she may not identify with the calculation-based mathematics emphasized by lower tracked mathematics. Her use of technology was limited to a typical four-function calculator for efficiency. When asked if she used a calculator she responded with the following.

"I do sometimes but I think it’s better to work out the problems. Sometimes they take too long with the thousands and millions to add up so I use a calculator"

"Better" is an important word in her description as she conveys the importance of doing mathematics without an aid. Her description of the graph suggested that she thought it depicted an event but was unsure exactly what it was depicting.

Interviewer: "What about this part of the graph?" [Horizontal Line] Ashley: "It was repeating. It was doing the same thing over the time period"

Interviewer: "What do you think it was repeating?"

Ashley: "Someone selling something…."

Interviewer: "Like what?"

Ashley: "Someone was probably racing or jumping."

She described the graph as someone selling something that implies a need to relate math to a consumer environment. When she was asked to explain more she then changed her mind and described the graph as someone racing or jumping. The "jumping" suggested that she held a misconception of the graph as a picture or map (Mevarech, 1997). She did recognize the horizontal line as a continuous event and described someone repeating something. When asked to explain why, she described:

"The graph was doing the same thing over the time period."

This suggests recognition of time passing but the y-axis (in this case, but not mentioned by Ashley, distance) is not changing.

"Jumping the same length or going the same speed or something."

The mention of speed her is important as it implies that she has an intuitive notion about the meaning of the graph's representation.

Interviewer: "Why did you say speed?" Ashley: "Because it got greater or something"

Interviewer: "What happened between here and here"

Ashley: "They probably slowed down"

She continued with this idea by explaining that any interval directed upward means they are going faster.

Interviewer: "How about over here?" Ashley: "They was gradually going faster."

Interviewer: "Where at and does it stop?"

Ashley: [nods] Affirmatively

Interviewer: "Where?"

[graph writing]

[shows the plateau interval]

The videotapes and observation notes offered evidence that she worked very well with the members of her group and generally operated the technology more than the other members did. She seemed comfortable with the equipment and continually prompted others to generate a graph while she operated the equipment. She explained how a graph she had made could represent her trip to school on the bus (Presented in the Chapter 4). This presentation was important for two reasons.

- Ashley transferred the units of the distance sensor (seconds and meters) to units of the scenarios and graphs she created and described (minutes and miles).
- Ashley made general descriptions of scenarios outside the classroom using terms and ideas from the distance sensor experiments.

She explained that each horizontal interval was the bus stopping and getting someone else and each interval that sloped up meant the bus was going toward the school. This suggested that a developing ability of interpreting information from a graph. She was recalling her personal experiences and applying the mathematics she learned to them.

Ashley's final interview responses indicated that she enjoyed the time of instruction and the technology:

"It was fun and I think the school should get some of those calculators."

She was informed that the school does indeed have this equipment and she expressed interest in using them more with the classroom teacher. She described the math covered in the instructional time as both "easy" and "hard" and when asked to explain how it was easy or hard she explained difficulty when she encountered instances of having to make decisions and describe what was going on:

"We had to try and figure what we were doing in certain spots. I had never done anything like that before."

She suggests that the novelty of the situation was difficult. It was hard because she had not done anything like that before. This required more from her than the typical routine involved within the basic math class. She described the math as being relevant to her life by describing her experience with the bus route graph again.

She also displayed evidence of being able to interpret the graph and the global attributes of the graph as opposed to viewing the graph as a literal picture of up and down movement. She mentioned specific ideas related to the representation of speed in the graph

Interviewer: "How is the distance and time related?" Ashley: "The graph is going straight up, it went up quicker."

__Conclusions.__ Ashley is presented as a low performing
student who encountered and seemed to learn some mathematical ideas
that were relevant to her. I learned quite a bit from her responses
and in turn chose to use her as a case study. She exhibited more
positive responses toward mathematics during and after the period of
technology intensive instruction. This type of instruction that
challenges low performing students to explore more and relate the
mathematics they are learning to events in their lives could be
beneficial for their learning of mathematics and their attitudes
toward mathematics. The period of technology intensive instruction
was new and exciting to Ashley and additional periods like this one
could help Ashley continue to grow in her experiences and using
mathematics.

__Jennifer ID#14__

__General description.__ Jennifer was a white female student in
the basic mathematics class. Jennifer became an interesting case
because she transferred out of the basic class involved in the
instruction to another basic mathematics class during the
instructional period. This transfer meant that she was not included
in the activities for more than one-half of the instructional
time.

__Observations and interviews.__ Jennifer was interesting given
her absence through most of the activities that the class
participated in. Her responses to the final interview reflected that
she had not been present through the instructional period where most
students began to develop sophisticated terminology to explain or
interpret graphs. Most students made connections between the sample
graphs during the interview and their ideas with the motion detector
however; Jennifer was unable to describe graphs using sophisticated
terms like speed and slope.

Jennifer understood some concepts but she was unsure as to how she should explain them. The other students were confident in their decisions about interpreting graphs and determining information from them as they related to the technology used in the classroom. Jennifer seemed hesitant when talking about speed and slope, and she frequently questioned the interviewer.

She was present for the first three days and she understood that a person moving in front of the motion detector could create a graph similar to the sample graph presented during the interview. She did not seem to understand the relationship between the attributes of the graph the person's speed or position. She also did not appear to have conceptualized the relationship between time and distance that is represented in the graphs. Her responses to questions about the graph suggested that she recognized only one aspect of the relationship. She recognized that a longer line on a graph represented more distance covered while not stating that the distance was covered in less time.

__Conclusions.__ Jennifer did not seem to understand many of
the more sophisticated ideas that were covered during the
instructional period. She was unable to accurately describe the
behavior of the graph in terms other than the motion detector. The
graphs did not have much relevance to her outside of the few days she
spent working with the equipment and in essence she did not explain
where she would need to use any of this knowledge outside of the
classroom. Jennifer is included as a case study as her absence helped
me consider the value of the technology-intensive instruction as well
as the form of infusion that this instruction utilized. Without this
infusion Jennifer was not as able to describe events on graphs
accurately and was not able to generalize information as well as many
of the other students particularly the low performing students in
attendance throughout the instruction.

__Michael ID# 25__

__General description.__ Michael was a white male student in
the basic mathematics class. He was frequently shy and quiet sitting
in the back reading, as reported by the classroom teacher. But with
the introduction of the technology, and collaborative learning he
became very animated and involved in classroom activities. The most
notable of these occurrences is presented as a snapshot in the
previous section.

__Observations and interviews.__ Michael's initial interview
responses suggested that he thought mathematics was important and
relevant beyond schooling. He described specific situations in which
mathematics would be valuable.

"It’s not the easiest class but math, ya gotta learn for your future. You’re gonna use math for everything, almost every job. Even if you don’t have a job that uses math, you know like you need to balance your checking account. Teachers use math, mathematicians, Architect-Geometry. Everybody uses adding subtracting, multiplication and division-basic stuff."

He seemed to have a notion that mathematics was important but he was frustrated with mathematics that he perceived and described as useless or a review.

"We learned those [NUMBER PALINDROMES] and that’s new learning, when in your life are you going to use that? Same thing like…What is that called…in high school you do lots of algebraic word problems or something, and some of that you’ll never use in your life."

Michael had an intuitive notion of the concept of rate, and reading graphs prior to the instructional period. This is suggested by his discussion of a horizontal line on a graph.

"You are standing still, you are not moving but time is expiring. It’s like a rest or something."

He already recognized the relationship between the components of the graph-distance and time. He knew that one component (time) is changing but the other is not. These responses were all made prior to the instruction.

During the interview after the instructional period Michael demonstrated strong understanding of these concepts and the ability to generalize them to more areas.

"The reason a car goes faster, the way you can tell is that is covers a larger amount of distance in a shorter amount of time. That’s how speed is."

This brief mention of speed indicates that he is able to identify the concepts and relationships in the graph with an idea beyond the motion sensor and give the relationship an appropriate name, in this case speed.

He explained these two graph intervals in terms of the relationship between distance and time as it related to the motion sensor experience.

VS.

__Figure 17__. Michael's attempt to depict more distance in
less time.

His explanation focused on the distance that was covered was the same, roughly 2 meters but the time that passed while that distance was covered was quite different.

"Shallow is that you are taking the same amount of distance but covering it over more time. They start here and it takes about five seconds to move two meters. "

Michael makes a sophisticated statement about this graph. He is able to determine the notion of distance changing over time.

The observation notes and videotape analyses indicate that Michael was very animated during the instructional period. He assumed a leadership role in his group and routinely explained concepts and topics to his group members. One particular observation early in the instructional period was Michael explaining his ideas to his group on a chalkboard (presented in one of the snapshots).

__Conclusions.__ Michael's attitude toward school mathematics
in the initial interview was not very positive as suggested by his
belief that math was a "review and useless", yet he knew that math
was important and relevant. By the end of the instructional period he
seemed to be more positive about mathematics especially mathematics
through technology suggested by responses to survey items and
responses to interview questions.

The algebra student case studies are presented in the same format as the basic mathematics students. The three algebra students selected for case study analysis were Lauren, Bob (female-student), and Kent. The three students were selected based on the same criteria as the basic students.

__Lauren ID#7__

__General description.__ Lauren was a white female student in
the algebra class. Lauren was a quiet student and like most of the
algebra students was content to work individually to complete the
tasks required by the instructional activities after completing the
group experiments.

__Observations and interviews.__ Lauren's initial interview
indicated that right now she was experiencing some difficulty with
word problems but she feels she does well otherwise in
mathematics.

"In general it’s ok but some things like word problems are a little bit more hard for me. Overall it’s o.k. Word problems-train problems, some are easy but when they get complicated they just go through one ear and out the other."

She knows that math is important and relevant to her

"Yeah because it’s important and everything for you to know what you are doing in business or in college."

Her responses to the interview indicate that she had experience calculating rate before from the formula D=R*T but she has trouble identifying characteristics of a graph of that relationship.

Lauren: "Keeping a consistent speed or getting faster because this dropped then it leveled off then it got way higher." Lauren: "I don’t know the time though."

Interviewer: "When it dropped what happened?"

Lauren: "I guess it just slowed down."

Lauren seems to misconceive of the graph as a picture. She describes the graph in terms of ups and downs. When it dropped down she says it slowed down when in fact the object moved back toward its original position.

In the final interview Lauren still described difficulty with her word problems but she became much more adept at detailing the information from the graph. She enjoyed using the calculators in the instructional time and enjoyed the mathematics. Here she describes how she was finally able to use the trace button to give her the information she wanted on the initial graph.

"Yeah, the first graph you showed us I had no idea how it could go up and down and what that meant then after a while we learned that. We learned how to use the trace button to find the slope and find the speed and how fast it is going per meter."

She used specific points on the graph to give her measure of the average speed over an interval and then the slope. This implies that she combined what she learned before with what she learned during the instructional period. She explains the graph in terms of the equipment.

"Well, they started out here and using one of the CBRs this means that they are getting further away from the detector. Then they stop for brief moment and then go back toward the detector. Then they wait for a bit then here they start moving again. And they get to a slow down and then they stop."

This showed that she had a very good transfer of the class activities with technology to the paper version.

__Conclusions.__ Lauren was not observed on any of the
videotapes but judging by her results on the pre and post test and
her interview responses it looks as though the instructional period
helped her solidify and conceptualize some things she learned prior
to the instructional period. She was a high performing student who
performed very well on both achievement tests and performed well
prior to the instructional period.

__Bob ID 13 (female)__

__General description.__ Bob was a __female__, black student
in the algebra class. She was a quiet student who did not participate
very often but was considered by the regular teacher to be one of the
higher achieving students in the class.

Her responses after the instruction seemed to show that the experience with the equipment was somewhat less than enjoyable.

"There are a couple of things I don’t understand like using the TRACE button. I’ve always thought that doing mathematics was important and doing this stuff doesn’t make it any less important."

This implies a belief that traditionally mathematics is done by paper and pencil without technology and that the stuff covered during the instructional period of this research was extra. She did however mention that it was relevant or rather it did not detract from the way she was taught.

__Observations and interviews.__ The interview results from Bob
indicate that she felt positively toward mathematics and her
experiences in mathematics education. She felt that math is important
and yet she recognized that some topics are difficult for her.

"I think it is important, sometimes I don’t like it. If I don’t understand something then I don't like it at all. I know I am going to need it in college and throughout my life. If you don’t know math then you cannot buy or do anything. If you get the wrong change then you will not know."

Bob's attitude toward mathematics was fairly positive. She recognized that she was pretty good at it and she described the necessity for studying mathematics.

"If you don’t know math then you cannot buy or do anything. If you get the wrong change then you will not know I’ve always been in the higher math classes. I’m not perfect but I guess I ‘m good at it."

Her anticipation of the calculator instruction was that it would be something new and she was not concerned about challenges this instruction may pose.

"Excited, not so much that we will be doing something challenging but something new."

Bob described a possible situation depicted through the initial graph:

"We just learned D=RT so that graph might be the rate or something. The rate at which something is going like 34 miles/hour or something." [she divided graph into increments and showed where 30,40,50,100 miles were and the hours.]

She wanted to apply a formula (D=R*T) to the graph she was given. Numerical units were not represented on the graph. She tried to establish numerical units of her own to calculate what she thought the graph was showing. Even after she quantified the axes she was unable to use them to accurately describe the graph.

"If their distance is going this way and it goes down they could have gone backwards. Is this the whole graph?"

During the interview after the instructional period Bob was able to explain in more detail what was happening in the graph especially in terms of speed as she explained a steep interval and some of the details associated with interpreting that segment of the graph.

"Definitely faster because covering more distance in less time."

She recognized this fact and applied this fact while making her own graphs yet she was unable to determine exact values or read specific points on the graph to determine speed.

Bob was frustrated with the equipment she could not figure out how to analyze specific points on the graphs that were created. This is interesting and it highlights her frustration with not being able to use her familiar D=R*T formula. Bob consistently wanted to work with specific numbers and discrete points. This offers evidence of a misconception described by Mevarech and Kramarsky (1997) of students confusing intervals and discrete points when describing graphs and graphical situations. She still sees math as really important but no more important than before the instruction.

__Conclusions.__ Bob was an interesting case study of a high
performing student. She exhibited frustration with concepts
especially when confronted with a situation she was unfamiliar with.
In this study Bob found that working with global attributes of the
graph and the intervals of the graph to be more difficult for her and
she frequently sought to examine the discrete points on the graph. In
many ways the technology did not offer Bob the opportunity to
dynamically explore the graphs within her group. I learned high
performing students can benefit from this type of instruction but it
would require more guidance through the explorations.

__Kent ID#4__

__General description.__ Kent was a male, Asian student in the
algebra class. He was interesting from a number of perspectives
concerning this research. He was a high performing student in his
classes and he described support from home concerning his
performance. He described in detail how many-and what kinds of fields
required proficiency in mathematics.

"81 or 83 major field use mathematics."

Kent’s interviews outline a view of mathematics that is very positive.

__Observations and interviews.__ Kent indicated that
mathematics is very important and necessary. This was evident by
responses from both interviews.

"Math’s Great, I like it a lot, It’s a necessary part of life. I get it a lot in my family. It’s easy, it’s fun."

Yet, he does not view specific graphing concepts as very relevant even after the instructional period.

"Basically everything involves mathematics it doesn’t matter if it is common sense or not common sense. Eighty-three major fields need mathematics. I think its eighty-one or eighty three. We can’t really use the graphing calculators in our daily routine, we don’t have to graph in our daily routine."

His family places a great deal of importance on success in mathematics and they encourage him in his mathematics education. He describes a belief about the relationship of technology and mathematics. He looks at mathematics as important to his understanding of technology, specifically computer technology. On the other hand, he offers the belief that using calculators in mathematics is akin to cheating.

"Mathematics requires you to use your brain. With a calculator you’re just entering numbers."

This is another example of both low performing students and high performing students’ view of technology for efficiency and ease in mathematics. Although he described mathematics as easy and fun, it is clear that he recognized the difficulty and elitist nature of knowing mathematics.

Kent had an intuitive notion about distance v. time graphs prior to the technology-intensive instructional period. He was able to explain most of what was going on in the graph. He immediately noticed that the first change in slope of the graph from an incline to a decline signified a change in direction for the object creating the graph. He realized that this showed an object coming back the way it had come, or retracing their steps.

"Well it’s a distance and time graph, showing someone’s motion. They are going along confidently at first. With no doubt. Then they slow down. 0 they begin going back to where they started."

His terminology about the rate that the person was traveling seems to show less sophistication. He describes the second incline as a faster trip solely based on the appearance that more distance was covered and not specifically more in distance in less time.

"Because this line is longer and covers more distance than the first line. Almost like they covered double."

However, in the post interview he gave responses that were much more sophisticated in terms of the equipment and in the features of the graph.

"You are staying at a constant distance from the detector that creates the graph and that creates a flat line."

This understanding of a constant distance is a sophisticated way to explain what the other students explained as no change in distance but time is passing.

__Conclusions.__ Kent seemed very content with his experiences
in mathematics and suggested that technology should not be used to
learn math. He did however display strong positive responses to
technology after the instructional period and he even seemed to enjoy
the instruction. He was very knowledgeable about the concept of rate
but had difficulty interpreting graphs before the instruction. After
the instruction he applied what he knew before the instruction to the
interpretation of graphs and reading information from them explored
during the instruction.

The case studies and snapshots work in unison to create an image of the experiences the students in each class have. The snapshots presented in the previous chapter included some students that were involved in the case studies. These students are notably Ashley and her experiences transferring concepts to her experiences outside of class. Michael is described while presenting his experiences cooperating and experimenting with his group members. Kent is quoted describing a very important point about horizontal lines.

By including these students in two areas I intended to present two facets of the experiences these students had:

- The case study students shared personal experiences and ideas with me through the interviews and personal observations as well as the general interactions and experiences they were engaged in during the daily instruction with the technology represent opportunities for looking at.
- The snapshots represent opportunities for examining the students' experiences with the specific activities while interacting with one another and engaging in experimentation with the equipment.

The case studies represented high performing and low performing students from the algebra and basic mathematics classes. They helped demonstrate the effects that the technology intensive instruction had on individual students in the class. There were different effects on these students and they suggest that the instruction affected their attitudes and conceptual development differently. The algebra students had encountered the formula for calculating distance from rate and time. They had not however, as reported by the classroom teacher, encountered graphical representations of distance and time. The algebra students frequently attempted to apply numerical and computational methods to interpret the graph during the interviews and instruction. They expressed difficulty interpreting what the global aspects of the graph were. The basic mathematics students had no experience with interpreting graphs or the distance and rate formula. However, Michael and Ashley both exhibited intuitive notions about distance and time graphs.

The basic mathematics students expressed frustrations with the boredom they experienced in mathematics in the initial interviews compared with the final interview description of the instructional period as exciting and they expressed more desire to perform activities like these. The algebra students thought it was fun and it probably would not be any less important to their future in mathematics. The basic mathematics students expressed overwhelming enthusiasm at doing something different and something that related the mathematics to them and they mentioned that they would probably encounter technology like this later.

When these case studies are examined together with the snapshots it appears that the period of instruction was very beneficial for both low performing students and high performing students especially considered individually within the classroom. This is evident in increased transfer of concepts, increased interaction and classroom participation due to the dynamic nature of the instruction; increased positive attitudes toward the mathematics learned and explored during the instructional period.

The results from the achievement data, survey data, snapshots, and case studies have been presented to portray aspects of this research and the influence of technology-intensive education on high performing and low performing students. After examining the students from these results and data and the experiences and explorations that they have engaged in I will present a summary discussion and recommendations for mathematics education in the next chapter.