**Discussing technology,
classroom practice, curriculum, and content knowledge with teachers **

Universidade Federal do Ceará Faculdade de Educaçao Brasil j.castro@ufc.br |
University of Texas at Austin USA jere@mail.utexas.edu |

**Introduction**

In the movement of educational reform, the use of technology is viewed as playing a key role. Yet, the use of computers in schools is still so limited to isolated pockets that it is hard to find studies done in schools or schools systems. The tendency of schools remains on teaching technology itself for developing technical or vocational skills in students, such as the use of word processing, database or spreadsheets. Use of technology has also been used as an enrichment activity by using games, or for developing presentations not necessarily related to the curriculum. Even though these applications are valuable, recent recommendations argue for the use of technology as a tool for content learning, not only of basic skills, but more importantly of high level thinking.

Linked to that issue is the emphasis in educational reform on continuous professional development. Policy makers seldom make the mistake of developing curricula that are teacher-proof. Having teachers as one of the levelers for reform is also challenging. Ball (1994), for example, argues that teachers are the products of a system and their learning is shaped by their prior experiences with mathematics.

Teachers bring with them not only mathematical knowledge but also knowledge about what means to teach and to learn mathematics and also beliefs about what mathematics is. Those prior experiences usually reflect a very different view of what is required for them to teach now.

Ball (1996) attests that we are not going to change this overnight. We still have to cover a long way before uncovering the complex problem of changing teachers' knowledge. "While we may believe that teachers must understand subject matter in deeper ways in order to be flexible when they listen to students, we still don't know enough about how to help teachers develop such understandings" (page 505).

Like with mathematics, many teachers experiences with technology have probably been very limited and not very positive. Therefore, teachers should have experiences in using technology for their own learning before using it with students. This will help them construct models of how that technology can affect learning. Unfortunately, there is a lack of studies that address the issue of technology related to issues such as professional development and teachers content knowledge. Some studies have focused on teaching technology such as word-processing or e-mail (see for example, Hurst, 1994). Other types of applications, such as multimedia, are used to present examples of instructional possibilities together with discussions on those examples (Krajcik, Soloway, Blumensfeld, Marx, Ladewski, Bos, & Hayes, 1996; Hatfield, 1996). Even though the use of multimedia for demonstration is valuable, it doesn't connect directly to using technology for teaching. Technology in those cases, is only a medium to show examples of teaching, with the advantage of being more interactive than videotapes.

While there is agreement that math teachers need to be better prepared (Kennedy, 1990, Ball, 1991), there is not consensus on how we should proceed. In part, this is due to the fact that research in math education has concentrated mainly in how technology would affect students' learning of concepts (see Kaput, 1992 for a comprehensive review of this literature).

Very few studies address the use of technologies with teachers, especially when the focus is content knowledge and not technical skills. When they do, these studies are usually limited because they only examine one or two teachers during a short period of time. (see Nicol, 1997 and Sherin, 1997 for examples). Implications of the results for more systemic efforts or long-terms results are not discussed. Other attempts to improve teachers' subject matter knowledge have been done outside their everyday practice, in workshop or summer courses with possibly follow-up during the year (Shifter and Fosnot, 1993).

Given
the increasing use of technology, especially computers in schools, and
the growing efforts of systemic reform, we urge to realize studies on the
role of technology in professional development. We need to investigate
which technologies can be effective both for students and teachers' learning.
Our study examines the feasibility of developing teachers' knowledge within
the everyday practice and in the company of significant collegial activities.
The paper explores how the use of new approaches and technologies can be
used as an opportunity for teachers' professional development.

**Methods
for Data Generation**

Context

This research was developed at Tree High School, a large urban high school that serves a population of almost 1700 students. The students' population is predominantly Hispanic (70%), with a small group of Caucasian (20%) and African-American students (10%). About half of the students (48%) are eligible for reduced or free lunch.

Participants

All
fifteen Math teachers from Tree High school participated in the study with
varying degrees of involvement.

Eight
teachers were implementing the unit in their Algebra I classes, while the
others were participating in the discussions. Teachers were teaching the
activities and materials for the first time. (see Castro-Filho, 2000 for
more details on the study)

Setting

The
unit was implemented in all Algebra I classes during the fall of 1998 and
spring of 1999. Our study was situated during the implementation of this
unit. During the implementation, the research group and teachers met six
times.

The
first three meetings were used mostly to introduce teachers to the activities
and technologies. In the next three meetings, researchers and teachers
discussed the implementation of the unit. Topics such as teachers and students'
difficulties as well as examples of what they are learning from the experience
were discussed. All of the meetings were videotaped and later transcribed.
Other aspects were also investigated. Those are presented in Castro-Filho
(2000).

Our work with teachers had a couple of innovative aspects. First, teachers had been participating in all discussions. Researchers did not just come with a set of materials and told teachers to implement it. Teachers were actually involved in the curriculum development. They proposed modifications in some activities while others were designed with a fair amount of input from teachers. Another important aspect of our study is that we worked with the whole group of math teachers from Tree High School, instead of just one or two volunteer teachers.

The activities and the technology.

The replacement unit was divided into five topics: ratio, qualitative graphing, slope, direct variation, and linear and simultaneous equations. We will briefly describe some activities from the first three topics, called Ratio, Qualitative Graphing, and Slope. The activities present a different view of Algebra, what researchers call a functions- based approach instead of the traditional equation based approach.

The ratio topic was included to prepare students to understand the ratio and similarity relations underlying slope. It was also, in part, a response to teachers' complains that students lack an understanding of fractions. Our approach to ratio drew on Confrey's previous work on splitting (Confrey, 1995a), and it emphasized the idea of equivalence as lying alone a vector in the two-dimensional plane and as represented by a table of values with an underlying "ratio unit". Students were introduced to ratio boxes, as a compact two line display on a ratio table to permit them to see horizontal, vertical, and diagonal relations as all essential parts of a ratio-based invariance.

The second topic was called qualitative graphing. The goal was to help students develop a sense of what the steepness of a graph means in relation to one-dimensional motion. Motion detectors, a computer software called Macmotion (Thornton, Beardslee, Ravers, Nolan, & Budworth, 1994), a calculator activity called Kine-Calc (Stroup, 1994), and curricular activities (Confrey, 1997) were used in this unit.

Students investigated a series of situations involving motion. The situations included walking at different rates and in different directions in a line. The goal was to understand how motion, in terms of velocity (speed and direction) affected position versus time and velocity versus time graphs.

The next topic, on slope, focused on developing the idea of slope as a rate of change and to start the transition between qualitative and quantitative understanding of slope.

Two technologies were used. A Java Applet called Bank Account Interactive Diagram (Confrey and Maloney, 1998), was used to help students develop intuitive ideas about how to accumulate quantities to produce a balance graphs and how to analyze balance graphs for rates of change to produce transaction graphs. A calculator program, called BabyMathworlds (DeLaura and Stroup, 1996), was also used in this topic. In this activity, students built graphs of velocity vs. time and then observed how those graphs affected the movement of an object called elevator, on a calculator screen. Relations between graphs of velocity vs. time and position vs. time were also explored. Both the Bank Account ID and BabyMathworlds were intended to help students explore the idea of slope as a rate of change in the circumstance of change per single unit of time. The goal was to start preparing students to see y=mx+b, with slope (m) as a rate of change and the y intercept (b) as the initial amount at time zero.

Further
segments of the replacement unit developed other topics, but understanding
these three topic is all that is necessary for the purposes of this paper.
In the next section we report a professional development episode that typifies
the kinds of content-related discussions we see as essential to improve
mathematical instruction. Then we discuss how the use of new approaches
and technologies can be used as an opportunity for teachers' professional
development.

**Results**

The discussion about the results will be based on themes that appeared in one of the meetings between researchers and teachers. This meeting happened during the curricular transition from the Qualitative Graphing topic to the Slope topic. Teachers had just finished the lessons on Qualitative Graphing and were starting the lessons on slope using the Bank Account Interactive Diagram and the BabyMathworlds activities. For this meeting, we had selected three video-clips from the lesson on qualitative graphing. The excerpts showed some struggles that students had and also teachers' effort in helping students understand the relationship between graphs of velocity vs. time and position vs. time. After teachers saw the video, they were exploring possible reasons why students had difficulty with understanding graphs of position versus time and velocity versus time and what could be done to help students. In the following we present the most important themes that appeared during the discussion with teachers.

The distinction between velocity and speed.

In order to probe teachers' own ideas, one of the researchers, Confrey drew a velocity versus time graph with four segments: a positive constant velocity, a linearly decreasing positive velocity followed by an increasing negative velocity, and finally, a negative constant velocity.

One teacher, Felipe, stood up to interpret the graph. He accurately described that one needs to be moving when the motion detector starts, so the initial speed is different than zero. When asked, he further explained that if one does not start with a speed, the graph would start from the origin, and not from a different point. Felipe continued to interpret the graph, saying that in the next segment, the speed is going to decrease until it reaches zero, and then it will start speeding up again (we are using the term speed, because even though teachers used speed and velocity, at this point they were not making any conceptual distinction between the two).

Two
teachers questioned Felipe's interpretation, arguing that the negative
should signify something different.

One
of them argued that negative should mean going the opposite direction (an
accurate representation of negative velocity, even though she said that
the idea of negative speed seemed strange for her). Below is the discussion:

Helga:
Why would you pick up on there?[referring to the negative segment]

Michelle:
Yeah, it's negative, it's below zero...Velocity is less than zero at that
point.

Confrey:
Where, where are you pointing?

Michelle:
Well as soon as you hit below the zero. And so, you have, velocity is less
than zero, which means you have to go at a negative speed, so maybe you
have to be moving away from the detector...I never really talked about
negative speed before. Don't know what that is?

Confrey:
Can somebody else jump in?

Helga:
You turn and change directions at that point? At the zero point. Is that
a turning point? Where you then, Felipe said, speed up again [pause for
3 seconds] you are speeding back up about the same speed but you are going
the other direction. Is that what it is?...

Michelle:
You have to be moving the other direction cause if I was gonna see it,
I was gonna go like negative ten miles per hour so that would mean that
you would be moving backwards in my car. I mean you really can't literally
go to negative ten but you can signify. I mean, you are still going ten
miles per hour but if you base your velocity on a point where you are trying
to go.

This conversation revealed a couple of interesting points. At first, some teachers did not distinguished speed from velocity. This created for them a difficulty in interpreting what happens when a graph of velocity becomes negative. We see the first teacher, Felipe, using his ideas about graphs and changing speed to interpret the situation. He understood that a zero slope means no change in speed, so he started with a constant speed. He realized he needed to start moving before the motion detector to produce a graph with an initial velocity different than zero. Felipe also understood that a negative slope means that a quantity is decreasing, so speed decreases. Since he was not making any qualitative distinction between negative and positive speed, he assumed that a negative speed would just be smaller than a positive speed. Thus, he kept decreasing his speed until it was time to be constant again. Michelle and Helga admitted that they found the idea of negative speed strange. They interpreted the change from positive to negative as a change in direction. The teachers were struggling with the meaning of negative with speed which they considered a quantity but not a potential deficit (such as negative transaction or withdraw). One other teacher even said she never thought there was a distinction between the two: "It just never occurred to me, this difference between velocity and speed."

The conversation continued with teachers and researchers discussing the distinction between velocity and speed. The issue of position and distance was brought by another teacher:

Trevor: I just think that if you are going to stick with the motion detector, there is no point in talking about speed at all. You just wanna talk about velocity... because you always have direction involved. You are always talking about position instead of distance. You are not talking about how far you go, you are talking about how much you are changing your position.

Teachers then discussed the distinctions between graphs of position and distance. This led to additional insight into how velocity and speed are used differently. In the case of velocity and position, they discussed that it matters not only how much you walk but also in which direction along a one-dimensional line. They discussed how on a round trip, the average velocity is zero but the average speed is not. A graph of distance versus time would always be increasing. In the case of velocity, the graph of position will depend which direction you are going, so the graph of position can be increasing or decreasing.

Abstracting from comparing motion detectors and the Bank Account ID

Initially some teachers felt that to introduce position versus time and velocity versus time simultaneously would confuse the students. They viewed position versus time as the elementary graph and velocity versus time as derived from it. However because of their work with the "Bank Account Interactive Diagram" (Confrey and Maloney, 1998), they were able to question this simple perspective. The Bank Account program uses transactions (rates of change) to generate account balances (accumulation) and thus provide a conceptual tool where common experience reverses the elementary graph and the one derived from it.

The work with motion detectors and the Bank Account helped teachers develop a flexibility in their thinking. In two moments, while teachers were analyzing graphs of rates and accumulation, teachers were simultaneously treating them as bank account and motion graphs. Teachers very comfortably switched between interpreting a graph of rates as transactions vs. time or velocity vs. time (rates), an accumulation graph as balance vs. time or position vs. time. For example, while the group was discussing how to get an accumulation graph from a rates graph, one teacher used both Bank Account and motion contexts in his arguments:

Felipe: ... for every motion, everything that you do, there's a final point, like, for example, you add or take away from the bank account, you are at a total, some balance there. So, regardless of how you move it, there's always gonna be a point. A position, a balance, or ... whatever.

Later on the meeting, while analyzing a students' interpretation of a velocity vs. time graph, another teacher said the graph was: "showing how fast or showing how much money you are putting in." Those two examples show how teachers associated a transactions vs. time graph with a velocity vs. time graph, and a balance vs. time graph with a position vs. time graph.

Teachers also discussed the differences of working with motion and Bank Account. They pointed out that even though using motion detector is more dynamic because students walk or run, the concepts in Bank Account could be more concrete for students to understand. The use of positive and negative rates of change could be interpreted as depositing (putting in) or withdrawing (taking away). In terms of accumulation or balance versus time graphs, positive and negative are associated with having money or lacking money.

Another difference was that the Bank Account ID deals with discrete numbers while motion detector deals with continuous numbers. The comparison between these two context stimulated some fundamental acts of abstraction, where abstraction is seem as finding what is common among apparently unlike situations. (Confrey, 1995b).

Reasons for students difficulties

During the discussion, teachers recognized that there were more deeper issues than they had anticipated, that students difficulties were not just an issue of lacking basic skills, and that teachers also had to struggle with some of those ideas. A teacher raised the issue that students' struggles are also an expression of teachers' own struggles.

Helga:
...I think a lot of it really is, we are not comfortable with a lot of
this. I know, I am not... And so, until we get really comfortable with
this, it is gonna be hard for us to be able to get it across to the kids.
And I know I still got a long way to go to really feel, really secure,
and comfortable trying to teach this. And just comfortable on my own mind,
forget about trying to telling to somebody else.

This
is still a hard thing for me to really have to address, but then I've been
in the physical science. I taught the physical science stuff too so I've
got that much background too, but it is still a difficult thing.

Teachers
were divided about the implications of these deep conceptual discussions
for curricular decision-making.

Some
teachers felt that they should wait or even should not deal with those
complex issues in the Algebra I curriculum.

Other
teachers suggested that it is important to teach the more complex situations
from the beginning. Seeing only the graph of position versus time could
limit the understanding of the relations involved. One could think that
velocity is always derived from a position versus time graph. As one teacher
pointed out, in her class one student was making better sense of going
from velocity versus time graphs to position versus time graphs than the
opposite. Therefore, what is considered easy or simple may vary among students.
Even if teachers did not reach an agreement, these were really important
discussions to be having among teachers. Teachers were not only discussing
which material to use or how to use it, they were arguing about the reasons
behind the materials and the implications they have for mathematical knowledge
and for teaching mathematics.

**Discussion**

The excerpts reveal important points regarding teachers' interactions with technology and mathematical content knowledge. First, teachers had explored the materials before teaching. In the case of motion detectors, teachers undertook activities at least two times, in previous workshops with researchers, once as students and while preparing to teach the replacement unit. the discussions about velocity and speed were short and restricted to researchers or to a few of the more experienced teachers giving the definitions of speed and velocity. In contrast, the discussion presented above showed teachers bringing different examples to the conversations. Even though teachers still experienced difficulties, they engaged more in the discussions. It was when Felipe realized that his non-distinction was causing a conflict in his interpretation of the graph, that a need for distinguishing velocity and speed was created. Teachers realized that direction of motion could have a relation with the graph orientation. After the meeting even researchers admitted they expected improvement but that they were surprised by the depth of discussion and engagement among teachers. It seems that conversations started to change after teachers had to teach the materials and were confronted by issues of students' learning. Teachers admitted that they had to start thinking more in depth about those issues. The main point is they were able to bring real deep meaning to the discussions such as different interpretations of negative numbers and their relations to issues of slope.

Although the richness of the discussions can be related to teachers' previous experiences with teaching motion detector, they were not necessarily about classroom practice. As Castro-Filho and Confrey (1999) argue:

... these examples explain a common phenomenon in working with teachers. Their understanding of content deepens when they work with students. We argue this is not necessarily a response to instructional participation. It occurs only when one changes one view of the range and legitimacy of mathematical activity to include recognition and appreciation of acts of discourse and their impact on the genesis of the concept. Then, epistemological acts on students' and teachers' parts contribute to stronger and more lasting content development. (p. 18).

The discussions also reveal some important aspects of professional development. First, we cannot deny the role of researchers in creating opportunities for the discussions to emerge. The researchers had an important role in selecting excerpts and examples to be discussed as well as pointing to suggestions during the discussions. Therefore, we cannot deny the positive aspects of an intervention, such as the one executed by our research group.

Another important aspect was the use of technology. If increased student learning is to be achieved, we see the issues of appropriate technological and effective professional development as critical Tools such as motion detectors and the bank account ID are relatively simple to use. They are manageable for the teachers, but they lead to important connections between fundamental ideas in mathematics. Their design also allows for teachers (and students) to investigate their own conjectures about mathematical ideas as they provide immediate feedback in the form of real-time graphing. The action of predicting and then testing can lead to important insights and refinements of the conjecture by the learner.

The
results presented here go beyond previous reform recommendations about
giving teachers opportunity to re-learn the mathematics that they have
to teach and on the role of technology on teacher's learning (Wilson, 1994,
and Ball, 1990). Our position is that the component of learning through
practice needs to be incorporated into teachers' life.

(Sherin,
1997). We also point to the need of more curriculum and technology such
as the one used in this unit, and of professional development that could
be oriented around learning through the implementation of new curriculum.
More research with implementation needs to be conducted to investigate
learning in relation to content knowledge and the practice of teaching.

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