# Using web videos and virtual learning environments to help prospective teachers construct meaning about children's mathematical thinking.

The use of web-based videos of children illustrating their
understanding of mathematics concepts and problems is the focus of this
article. Interviews with 23 children in grades K-5 were video taped
concerning their understandings of mathematics content identified in
major standards recommendations. These videos were placed on the web in
approximately five-minute segments and categorized according to the
child's pseudo name, grade level, and task/content area.
Prospective teachers used the videotapes to analyze behaviors,
illustrating certain theoretical concepts. Samples of the prospective
teachers responses are provided. The article also discusses the
potential of such use of videos. By having the videos on the World Wide
Web (WWW or Web), users can watch them in their homes, or other places
of convenience, as well as the videos can be used for distance learning.
A number of related references are provided.

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This study is an investigation of the potential use of videos on the Web to help prospective teachers construct meaning about children's mathematical thinking. While prospective teachers need experience working directly with children, an alternative initial method of creating such an experience is the focus of this study. The intent of the videos is not to replace actual experience with children but rather to bring deepened insights into such expenences. The reason for putting the videos online is so that prospective teachers can easily access them from home, on campus, at work, or wherever, and to be able to access the videos at a time that is convenient for the participants, as well as provide a model for distance learning.

THE PARTICIPANTS

The participants for the study have been prospective teachers, mostly graduate but some undergraduate students, who are working on teacher certification in elementary education or special education. The participants were taking a course in topics in elementary mathematics at the time of the study.

BACKGROUND AND RELATED LITERATURE

Many times prospective teachers in teacher education programs often have had little experience with students prior to their internship. Alternatively they spend a lot of time in classrooms but do not have direct connections with related theory. It is difficult to make connections between the abstract theories and the real world of practice. The bridge between theory and practice varies greatly, and sometimes the bridge is a big gap. To treat theory in the absence of experiences with children will likely result in the theory quickly being abandoned or forgotten. At the same time building a class based heavily on field experiences has its problems as well. The experiences are often random, may or may connect with the underlying theories, and risk the possibility that the students never make the links between theory and practice. Furthermore, consequences of actions can affect real students and sometimes do damage to the students, and the supervision of field experiences often is labor intensive and/or not easy to do. In addition, the actions in a classroom are multidimensional and often difficult to isolate and control one variable at a time.

In 1972-73, the author had a vision of building a simulated teaching laboratory. Initial efforts occurred with a PLATO program, which she created, that allowed perspective teachers to instruct simulated students (Flake, 1973). The results were very encouraging. Participants in the study commented that they began to think of these simulated students as real; they also liked the notion that they were not doing permanent damage to the simulated students; and after going out into the schools the participants reported back to the author that the real students behaved just like the simulated students. In addition, a very popular button that was added at the request of the students was the restart button, when at times participants were not happy with the way things were going and they just wanted to throw out that class and start all over again--a phenomenon that does not normally happen with real classes. Furthermore, the simulation provided a common experience for class discussion of alternatives for interacting with the simulation. Brown (1999) found similar results.

Today we have the potential of online and/or distance learning. These have been studied by a number of people (e.g., Bialac & Morse, 1995; Gilbert, 1999; Huang, 1996-97).

A simulation is a working analogy of a life-like situation; for characteristics and examples (Flake, 1996-2000a). Virtual reality and virtual learning environments are a special type of simulation and have been studied for creating positive learning environments (e.g., Ferrington & Loge, 1992; Hsu, 1999; Klingenstein, 1998; McGonigle & Eggers, 1998; McLellan, 1998; Pearson, 1999). Virtual learning involves learning through telecommunications with user control and possibly using video streaming; for characteristics and examples (Flake, 1996-2000b). Virtual learning environments can include: (a) the use of telecommunications technologies that transcend barriers of time and place, associated with the traditional lecture format; (b) provide educational services through technologies which can be accessed from homes, institutions, communities, and workplaces; (c) allow users to have more control of their learning; and (d) provide for more flexibility in the course structure form. Klingenstein (1998) described a nu mber of factors to take into consideration for creating a virtual learning environment, including the use of video streaming, atmosphere, the nurturing of inquiry, and the building of a community of learners. Pearson (1999) used an electric network to facilitate discussion between perspective teachers and built a virtual community. Klingenstein also reported the importance of high-speed video servers and data lines, such as DSL and ADSL and other infrastructure considerations. Further, he discussed the evolution of Internet2, which should help address many of the current problems with the Internet.

Uses of video and multimedia have been investigated for helping prospective teachers gain a better understanding of students' behaviors (Kagan & Tippins, 1991; Fisher, Deshler, & Schumaker, 1999; O'Hagan, 1995). Kagan and Tippins (1991) studied the use of videos in helping interns to interpreting students' verbal and nonverbal behavior during instruction. Fisher, Deshler, and Schumaker (1999) studied teachers' understanding when using an interactive multimedia program and an inclusive practice.

Ten easy steps for designing a virtual classroom created by Hsu (1999) included the following steps:

1. assess the needs and the necessary conditions to satisfy them;

2. estimate the development cost, effort, and implications;

3. plan the virtual classroom;

4. design the virtual classroom;

5. prepare and distribute contents;

6. enable communication;

7. implement online student assessment methods;

8. implement class management procedures;

9. set up the system; and

10. maintain and update the virtual classroom.

Identified stages of virtual learning include the following. (a) Instructors' stages of virtual learning are: excited; apprehensive; questioning; determined; over stimulated; questioning revisited; and exhausted. (b) Students' stages are: confused; shocked; timid; frustrated; and "Eureka" (McGonigle & Eggers, 1998). Major paradigm shifts take time to bring about changes.

DESIGN

Careful thought went into the design of this study. Elements of design have included the content of the videos, the design of the technology, and the design of the experiences of the participants.

Content Design of the Videos

The content of the videos was determined from state and national standards in elementary mathematics. Twenty-three real children from grades kindergarten through fifth grade were interviewed about their concepts/ problems in mathematics in the areas of: patterning and algebraic thinking, space and geometry, measurement, early number concepts, number and operations, rational numbers, and probability and statistics. Two hundred and sixty eight videos were then created of approximately five minutes each. The videos were organized so that they could be accessed and examined by child, grade level, and/or content. Each child chose his or her own pseudo name that is used in the videos.

Technology Design of the Videos

Interviews with the 23 children were videotaped using a digital video camera. The videos are streamed to allow for a variety of accesses. The videos are stored on a RealServer G2 (see http://www.real.com/products/), which allows for high-speed access. The videos should be used with Realplayer G2 which can be downloaded from: http://www.real.com/products/. An up-to-date browser such as either Netscape, Version 4.7 or higher, or Internet Explorer, Version 5.0 or higher.

Regretfully the only microphone that was used for video tapings was the microphone from the video camera. Sometimes people have trouble with the audio portion of the videotapes. They have been given the following directions:

There are several different places where audio can be adjusted, including:

* On the task bar there may be a megaphone or other audio icon. Participants can place the mouse on that icon and see if a volume control appears. If so adjust it.

* In the control panel of the computer there should be either an audio or a multimedia option. If so, participants can adjust the volume there.

* The monitor, earphones, or audio output may have capabilities of adjustment, as well.

* The audio or multimedia software itself often has the ability to adjust the audio volume.

* The audio can be heard but sometimes caution needs to be used to be sure that optimal volume is available.

Design of Prospective Teachers' Experiences

The prospective teachers were asked to watch the videos in a variety of settings, involving various children, grade levels, and content. They initially viewed videotapes along content strands to see how the content evolved from grade level to grade level. Then they responded to more pedagogically oriented questions to help bring out clearer connections between theory and practice. The following are sample questions that prospective teachers have responded to:

Using the videotapes of interviews illustrating children's mathematical understandings, discuss the following (when responding to the following questions to provide a case, provide the child' grade level, name and the task, and then give specific incident within the task):

1. Identify three cases, if possible, of children having relational understanding in the videotapes and justify.

2. Examine Dan (3rd Grade, Blocks) and Art (4th Grade, Addition). What are your observations about these two cases? Discuss.

3. Describe three cases, if possible, where the interviewer uses scaffolding to help the student solve the problem at hand.

4. Do you feel that these students have come from a "rule based" curriculum or a "model based/problem-solving" curriculum? Discuss with specific incidences.

5. Identify and describe three cases above, if available, where the students seemed to be using dynamic imagery and describe nonverbal cues that suggest such.

6. Describe the potential evolution of a content strand, e.g., division concepts, from K through 5th grade. Identify and describe levels of understanding.

7. Select a grade level and describe the range of understandings within that grade.

As the prospective teachers were analyzing the videotapes, they were also working with manipulatives and technology, themselves. After these experiences the prospective teachers worked directly with real children.

SAMPLE RESPONSES OF PROSPECTIVE TEACHERS

While working through the videos and answering questions, the prospective teachers needed to think through the relationship between theory and practice. The following are sample responses of the prospective teachers concerning the videos.

Relational Understanding

Relational understanding involves the students making connections between related ideas and/or concepts. Sample participant responses included:

* In answering this question, I used the definition from page II of the Flake text (Flake, in progress) for relational understanding, which is "making connections or relationships from one situation to another." Based on this definition, Lisa (second grade, addition) was able to take the addition problem 23 + 18 from paper and convert it into blocks. She was able to represent 23 as two rows of ten and 3 individual blocks. She was also able to represent 18 blocks as one row of ten blocks and 8 individuals. She added the two groups and represented 31 as three rows of ten and one individual block. This shows her ability to connect the relationship of blocks or actual things to the values of the numbers.

* Relational understanding refers to "webs of ideas." Whereby one idea is associated with another, or other ideas, thereby enriching understanding (van de Walle, 1998). I believe one example of relational understanding is Matthew (fifth grade, division) who seemed to readily be able to solve the problem using rules (although from listening to him I am not sure what the rules are), but equally he was able to associate the numbers on the page to models (the blocks). Further, in the latter minutes of the tape he seemed to be able to overcome the limitations of the blocks by constructing a decimal system.

* Justin (kindergarten, division) did not have the need to use the blocks when "sharing fairly." He did most of his problem solving in his head. I feel he must have a clear understanding since he was able to make connections of the sharing fairly concept without having to use a manipulative to "verify" his answer.

* Rose (kindergarten, division & number sense) demonstrated a good relational understanding with division and number sense. With regard to number sense, she gave a "good estimate" for the number of blocks; she was able to answer correctly "the number left when 1 was taken away" and "another taken away" and finally "add another back." Rose also demonstrated good relational understanding with division when asked to share fairly; "one person leaves, how many cookies would each person receive?" The fact that she was able to base her answers on the understanding that if one person left, there would be more cookies for the remaining folks reflects comprehension of relational understanding.

* Lisa (second grade, feely bag) impressed me very much with the connections she made between the shapes in the bag and her drawings on paper. She was correct every time and was very confident in her answers. She found all the shapes in the bag as well. She knew what the shapes were and then she found relationships in order to draw them on her paper.

* Relational understanding is displayed by Fog in his transformations video (fourth grade, transformations/space sense). The interviewer presented the student with a shape that was transformed in six different ways. The student was asked to describe what was done to the shape to make it look different. Fog observed that the shape was sometimes flipped (after the term was provided by the interviewer) and sometimes turned. He also observed that the size changes. Through his responses, it was apparent that Fog understood that the shape remained the same, though it was transformed in some way. He was displaying his understanding of what has happened to the shape to make it appear different.

* Nathan (fourth grade, addition) first did addition on paper and then with blocks with three-digit-numbers. He did very well with blocks. He easily counted the blocks, while he was adding and put them together. He counted tens, ten by ten, to 100 and said this is one hundred and put it near other hundreds. I think he had relational understanding.

Through these experiences the prospective teachers have been able to make the connections between the theory of relational understanding and the reality of the children's understandings. By giving specific examples they address their concepts of the theory.

Specific Cases

A couple of specific cases of children were identified for the prospective teachers to study. In these cases things did not go exactly as expected and illustrated pitfalls that a number of children make. The following are samples of responses to these specific cases.

* Dan (third grader, blocks) worked with blocks to solve mathematical problems. The interviewer initially place blocks out on the table. He was asked to estimate how many were there, and he guessed 20. In actuality there were 26 blocks present, so his estimation was fairly reasonable. He was asked what the 2 in the 26 meant, and he said it was 20. He then thought that the 6 meant a 60, but he was then led to see that it was 6. Next he put the blocks into groups of 3 then counted them, then put the blocks in groups of 10 and recounted them. When given the subtraction problem 26-18, he couldn't solve it on paper; however, he knew that he couldn't take 8 from 6. He solved the problem using blocks and came up with the correct answer of 8. He believed the blocks more than he did his own answer (on paper), displaying proof that he could learn well using manipulatives and by using visualization. He didn't display understanding of concepts on paper, as well as the borrowing rule in subtraction.

* Art (fourth grade, addition) was given a paper problem with 234 + 389 =?. The interviewer asked Art to solve the problem (after stating he had seen problems like it). Art used his fingers to count and answered it (solved the problem quickly, using his fingers). He was then asked to demonstrate the problem with blocks. He didn't have any difficulty showing 234 and 389 with blocks. He used 2 flats (100/each flat), 3 longs (10/ each long), and 4 singles and 3 flats, 8 longs, and 9 singles. He actually counted some of the longs to make sure there were 10. He counted 4 + 9 (pulled out 10 singles and exchanged for I long); then stated 1 + 3 = 4 and 4 + 8 12 (used fingers); exchanged 1 long and 10 singles (combining the paperwork problem-carrying numbers-with blocks).

* Art continued trying to "carry" with the blocks-but he tried to carry a long into the one hundred's place rather than a flat or 10 longs-illustrating some confusion about the place value system. The interviewer suggested he exchange for different size blocks by asking, "what could you trade for a big flat?" (Leading question) Art knew 10 would trade so he then exchanged for a big flat giving 6 flats.

While it is nice to have prospective teachers see examples of students doing well, it is perhaps even more instructive for these prospective teachers to see cases where the students make common errors. These two, cases particularly illustrate some confusion on the part of these students.

Scaffolding

Scaffolding is the notion of a more experienced person helping students, by asking questions, probing, doing things that the students might not figure out completely on their own. Ultimately the notion is that the scaffolding is gradually removed and the students function independently. The following are sample participant responses to uses of scaffolding:

* In answering this question, I based my understanding of scaffolding on the Flake text from Chapter 2, which states that scaffolding can be accomplished by, "other students, a mentor, a computer, manipulative materials, or other models." These are tools that can aid in the students' sense making. The teacher's role can be to ask focus questions, when necessary, to help the students redirect their attention." An example of scaffolding was with Brittany (first grade, transformations). Brittany was asked how to make the cat on the left side of the paper look like the cat on the right side of the paper. When she was unable to perform this task, the interviewer picked up the paper and showed Brittany how the picture could be turned to match the other paper and showed Brittany how the picture could be turned to match the other picture. When Brittany was presented with the second problem, she was able to say, "Maybe I could turn it."

* As it relates to education, the word "scaffolding" generally encompasses the notion that students sometimes need "support" or assistance when attempting to solve a problem or achieve a goal that may be too difficult without the help of a skilled partner (often the teacher.) As the student begins to understand the concept, the skilled partner gradually withdraws assistance so that the student can begin to solve the problem unaided. In the videos, the interviewer often uses the technique of scaffolding to help the student solve problems. Scaffolding is used in Rosie's measurement video (first grade, estimating length/measurement). The interviewer first asked Rosie to estimate the width of a poster board without actually using the measuring apparatus (difficult to see, looked like cubes linked together). After she completed that task, the interviewer asked that she actually measure the width. After Rosie was done, the interviewer suggested that she measure again, and this time, overlap the measuring device "li ke footstep, one over the other..." When Rosie still had difficulty measuring, the interviewer physically assisted Rosie by placing her finger where the measuring tool should go next. Again scaffolding was used because some elements within the task may have been too difficult for Rosie to complete by herself.

* The first case of scaffolding I observed was with Andy (second grade, division) and the area of division. Andy was given the paper problem of 23 divided by 4 but did not know how to proceed working the problem. She began by moving the blocks randomly (distributing more blocks in some groups and less in other groups). The interviewer mentioned that it appeared some groups did not have as many (or were "missing some"). Andy counted the blocks and "discovered" the 2 of the groups had more blocks that the 2 groups. She then proceeded to redistribute some of the blocks. She counted and recounted to make sure she had the same number in each group. The interviewer then "assisted" Andy with transferring the concept of the blocks to the paper problem. I couldn't hear the response to determine if she was correct with paper problem.

* Mark (fifth grade, missing values) illustrated that he already knew the basic concept of missing values when he said, "I do this all the time." The first problem he was asked to solve he did the quickness and ease, but the successive problems increased gradually in difficulty, while still testing the same skill. The interviewer seemed to assess at what level does he seem to have the most difficulty and that's when she intervened, she was working at his Zone of Proximal Development.

From using the videos the participants were more clearing able to understand the role of scaffolding at a very concrete level. They were able to identify ways that the interviewer helped the children bridge to the concepts.

"Rule-based" Curriculum or "Model-based/problem-solving" Curriculum

A more traditional view of mathematics instruction is to provide the students a rule followed by a number of practice problems, a "rule-based curriculum." However, a more current view of mathematics learning is to engage students in problem-solving activities and give the students models that will allow them to figure out the problem themselves, a "model-based! problem-solving curriculum." Sample participant responses to whether they thought the children had come from a "rule-based" curriculum or "model-based/problem-solving" curriculum.

* A rule-based curriculum is a curriculum based upon giving students the rule followed by practice problems, reasoning that the more practice, the more proficient. However, a problem-solving curriculum encourages students to use models or develop their own, to see patterns and relationships. The video featuring Louisa has sufficient clues to infer the students come from a rule-based curriculum. First, she got confused early in the process. Second, I see no dynamic imagery used. Third, she did not seem comfortable with the manipulatives. As discussed elsewhere, Louisa did not have any familiarity with the base 10 blocks.

* I can honestly say that I feel, for the majority, that these students have been learning in a rule-based curriculum. They are dependent on working out mathematical solutions by using rules and formulas rather than by thinking of alternative routes to solving. I myself think I am a victim of a rule-based curriculum, and at times find it difficult to think of alternative ways to solve problems. Math has always been a source of stress in my educational career, and I think that had I been taught differently, with a more model-based/problem solving strategy. I truly hope to incorporate a more model-based/problem solving strategy in my classroom, using blocks, tangrams, and candy (to make it fun).. .the sky is the limit.

* I feel the majority of these kids, especially the younger ones, come from a model-based/problem solving curriculum. For the most part, they seem quite able to use the manipulatives without too much direction from the interviewer. There weren't too many questions about "how do you want me to show you?" or "how can I do that with the blocks?" I believe a child might have a difficult time demonstrating the manipulatives for a mathematical problem if they were only accustomed to "doing it on paper."

* My overwhelming response to this question is that the majority of these students have come from a "rule-based" curriculum with some exposure to "model-based" ideas. Though some students such as Lisa and Art performed problems using manipulatives with relative ease, most of the other students either could not understand the relationship between the blocks and their problems on paper or they were very uncomfortable using the blocks. Even Lisa and Art, however, worked their problems on the paper and tried to count on fingers and in the air rather than using the blocks to answer. My ideas are based on the following specific examples:

--Felicia (Kindergarten, division) did not grasp the idea that these blocks were useful in helping her figure out the problem. She started playing with the blocks as if they were people and had no idea of their use in dividing. This led me to believe she had liffle or no exposure to "model-based problem solving.

--Lisa (second grade, addition) was able to show what she did with the blocks when asked by the interviewer, but she did not initially use the blocks to solve the problems. She relied on what she already knew about solving addition problems, the rules, to come up with a solution to 23 + 18.

--Art (fourth grade, addition), when trying to solve the problem 623 - 234, tried to use his fingers and looked into space rather than getting the blocks to try to solve his problem first. Just like Lisa, he was able to show how the blocks relate to the numbers, which lead me to believe he has been exposed to some "model-based" ideas, but he still attempted to solve his problem by rules first.

--Louisa (third grade, division), when given a division problem, rather than using the blocks to "share fairly," tried to use multiplication tables to find what would be closest to the number she is dividing. This is a very "rule-based" idea and when Louisa finally used the blocks, she easily found the answer to the problem.

As these examples indicated, these students tried to solve these problems by rules first and then, either because they were encouraged or asked to by the interviewer, they used the manipulatives in the "model-based" style to solve the problems, often getting results much easier and faster. It is as a result of these observations that I feel these learners are learning under a "rule-based" curriculum.

* In my opinion, these students have come from a "rule-based" curriculum. Because I can say most of the students can do addition-subtraction and the other processes on paper correctly. But they do not know what that means in real life. For example, they can do multiplication on paper easily but they could not do it with blocks, and they could not understand multiplication has the same process with addition.

* I feel that these children have come from a "rule-based" curriculum. The rule-based curriculum is based on giving students the rule followed by practice problems. The model-based curriculum is based on using models that students apply for problem solving and to figure out patterns and relationships. The children are not familiar with the use of models. The smaller children don't seem to have a very good connection between what they are doing on paper and a mathematics model. Examples: Brittany (first grade, place value)-When she was asked what the 1 in 16 meant, she said, "One cookie." Brandon (first grade, blocks) counted out 19 blocks and wrote the number 19 on a sheet of paper. When asked what the one meant, he said it meant one block. The older children can work problems on paper, but they really don't seem to know why. They are working the "rules" and coming up with an answer. Examples: Art (fourth grade, addition) was able to solve the problem easily on paper, but was very unsure with the blocks. He k new there were number places, but I think he had trouble visualizing them and actually physically manipulating them. Fog (fourth grade, multiplication) was able to solve the problem quickly by using the rules and even talked the problem through. When he was asked to solve the problem using blocks, however, he had a hard time. I don't think he had had much experience with manipulatives. He didn't seem to understand at first that "four times," meant that the same number would be added together four times.

Although the participants saw different behaviors and justified their responses using different data and came up in varying cases with different conclusions, they did see the importance of how the children were taught. By observing these children's behaviors it was much clearer to the participants that they should help children learn to reason their way through mathematical ideas rather than simply giving the children a rule and asking them to follow the rule.

Dynamic Imagery and Nonverbal Cues

Underlying many areas of mathematics learning is the concept of dynamic imagery, where students can manipulate their images by taking images of objects apart, turning them around stretching or shrinking them, and so forth. The following were participant responses to the children using dynamic imagery and nonverbal cues:

* My answers to this question are based on my understanding of dynamic imagery from the Flake text, which stated that dynamic imagery is "the ability to mentally manipulate images in one's mind." The best example of dynamic imagery in the videos were in the transformations section where learners were literally asked to tell how they would manipulate or "transform" one image to make it look like another. One example of the use of dynamic imagery is Sam (fifth grade, transformations). Sam used his hands to give clues to the processes taking place in his head when explaining how the images on the page flip or turn to become the next image.

* Fog (fourth grade, algebraic & missing values) concentrated hard when given the 3 missing values problem. With 43 + N = 67, Fog looked around, eyes were moving as though he was picturing something: he got the correct answer and the interviewer asked him how? He replied with "I counted." When given the next problem, 40 x N = 80, Fog whistled, looked all around, again trying to picture the answer (or at least trying to picture how he should work the problem). Again, he got the correct answer: 20. When asked how, he replied that he knew (4) 10's = 40, so (4) 20's would = 80. He also worked both problems on paper, but I was unable to see his work. The final problem 4 x N - 3 = 97, seemed to concern Fog at first. He made the comment "Holy Cow!" After spending a few moments looking around, eyes and lips moving, hitting his pencil on the table he seemed to visualize and then verbalize the problem. He got the correct answer "25" by saying he knew (4) x 25 100 and if he subtracted "3" from 100 he would have 97. You could almost "feel" Fog working/thinking the problem in his mind!

* Within the realm of mathematics, dynamic imagery refers to a student's ability to transform, reverse, or change perspective when solving a problem. Some nonverbal cues that suggest a student is using such imagery include a student's hand motion and/or a student looking up while thinking about a problem. One example of dynamic imagery is found in Fog's volume video (fourth grade/volume/measurement). The interviewer provided the student with a clear container filled with cubes and asked the student to estimate how many cubes were in the container. Fog looked at the container and then briefly looked up as though he were imaging the number of cubes. After he asked for a piece of paper, Fog began working out an answer. Again, as he wrote, he looked around using dynamic imagery to visualize the answer. Through the use of dynamic imagery, it is possible for a student to envision what needs to occur m order to solve a specific problem.

* Louisa (third grade, feely bag) used imagery while drawing the objects on paper. She felt the shapes in the bag and drew them very small on the paper. I would say it is absolutely related to using imagery in this process.

* I used Sarah's video to represent a video for dynamic imagery. Sarah was working with transformations. Dynamic imagery is exemplified in her instruction as she was working to solve how her shapes must be manipulated. She did not have a physical block to work with but she used hand gestures to develop an image of what must be done to get the shapes she was working with to move it into new positions.

* I selected Matthew (fifth grade, drawing a house on a hill). In this process every student tried to draw a house on a hill. Some of the students paid attention to the slope of the hill while drawing the house, but turning the paper, by turning their heads or by turning their bodies with the hill. Some of the students did not pay any attention. But I can say all of the students were using dynamic imagery. Why I selected Matthew for this question is that we can easily understand and see his using dynamic imagery from his picture. He paid the most attention for the hill, and he thought it was better to draw a house like this on this hill.

By actually examining videos of the children, the prospective teachers can begin to develop a concrete understanding of student's possible thinking. This, in turn, can help them learn to develop experiences that will help students develop and use such a dynamic imagery.

Evolution of a Content Strand from K through Fifth Grade

By being able to look at children's performances at each grade level the participants were more concretely able to see the evolution of ideas developed from grade to grade and varying from child to child. In this particular case they examined a content area (division) that traditionally is not dealt with in Kindergarten through second or third grade but the interviews illustrated that children could deal with such concepts, and that there were transition steps that could help the children evolve into the standard views. For example, with division, the younger children did understand "sharing fairly" as a means to dividing in a case such as, "There are nine cookies and four children, how many cookies does each child get if they share fairly?" From that concept a transition move can be made to, "How many cookies would it take to go around once?" This in turn shifts to repeated subtraction, which then leads to the more traditional view of division of multiple repeated subtractions. These ideas do not develop qui ckly with children. It takes a lot of experiences, appropriate transitions, and maturation on the part of the children.

Range of Understandings within that Grade

Again it was constructive for the participants to look carefully at the ability ranges of the children within one grade. Many classroom teachers say that their classes often have about a five-grade range in abilities within a single grade level. When the children were selected for this project there were deliberate efforts to get a wide ability range. As the participants examined the videos they were able to see that ability range.

PROSPECTIVE TEACHERS' EVALUATION OF EXPERIENCE

The participants found the videotape experiences very useful. When they moved to the actual experiences with the children they felt much better prepared than earlier groups because of what they had gained from the video experiences.

At the same time there were some concerns. The videotape viewing took a considerable amount of time. However, it is likely that it would have taken much longer for them to view the same behaviors in the schools with actual children. Also there were some technical problems.

Some of the participants complained about the audio portion, but the author was not sure that the participants understood the variety of ways that the audio could be adjusted. Other students seemed to function just fine with the audio. This would suggest that some students needed to explore more ways to adjust their audio on their computers. Posted on the web site now are multiple ways of adjusting the audio so that future classes can make better use of the audio. Also in future tapings each child will have his/her own microphone to help improve the audio.

Most of the students were using telephone connections rather than high-speed lines, such as ADSL, DSL, or cable lines. As a matter of fact, when these participants took part in this study, ADSL and DSL were not available in this area, now they are available in limited parts of the area. Video streaming is a major help in this direction. The author believes that the high-speed lines will become far more dominant in time.

DISCUSSION

While there were some technical problems, the author believes that this method holds much potential for the future. The videotaping took time, but it helped to have the interviews carefully conceptualized so the variety of experiences did appear as they did with these children. For prospective teachers, the awareness of a variety of behaviors and abilities is very important for helping them understand the challenge that is ahead of them.

The technical piece will greatly improve as these capabilities become more and more integrated into our society. The author anticipates that within five years, television, telephone, and the Internet will be quite integrated. Greater access to online video will exist and allow for prospective teachers to easily sort for examples of a variety of levels of understanding in the comfort of their own home or alternative sites. Further, for those at a great distance from campus, the reality base of online videos can greatly enhance other forms of communication in terms of understanding children's levels of understanding.

SUMMARY AND CONCLUSION

This study investigated the use of online videos of children for helping prospective teachers understand how children think and the children's mathematical understandings. While this is an early stage of investigation, the author believes that it presents enormous potential for teacher education or other forms of education. While it is not quite at the form of a full-blown simulation where the participants "instruct" the students and make changes based upon their previous behaviors, it does provide them with experiences that help bridge between the theory and practice and allows them a transition between academic class instruction and the real world of schools and children. Research and development in this area should continue to discover how to effectively use the technology that we have available and that will be continuing to develop.

Acknowledgements

The author gratefully acknowledges the children, their teachers, and the prospective teachers who have willingly participated in this study and given permission to use their information. Florida State University is also enthusiastically acknowledged for providing the technical capabilities and support for this project.

References

Bialac, R.N., & Morse, G.E. (1995). Distance learning: Issues for the experienced teacher as a novice in the virtual classroom. ED, Education at a Distance, 9(2), J3-J6.

Brown, A.H. (1999). Simulated classrooms and artificial students: The potential effects of new technologies on teacher education. Journal of Research on Computing in Education, 32(2), 307-318.

Ferrington, G., & Loge, K. (1992). Virtual reality: A new learning environment. Computing Teacher, 19(7), 16-19.

Fisher, J.B., Deshler, D.D., & Schumaker, J.B. (1999). The effects of an interactive multimedia program on teachers' understanding and implementation of an inclusive practice. Learning Disability Quarterly, 22(2), 127-42.

Flake, J.L. (1973). The use of interactive computer simulations for sensitizing mathematics methods students in questioning behaviors. Doctoral dissertation, University of Illinois, Urbana-Champaign. (ERIC Document Reproduction Service No. ED 107 503).

Flake, J.L. (1996-2000a). Computer simulation. Retrieved from the World Wide Web on Janice Flake's Home Page, Florida State University on March 27, 2002, from: http://mailer.fsu.edu/~jflake/CompSim.html

Flake, J.L. (1996-2000b). Virtual learning/virtual reality. Retrieved from the World Wide Web on Janice Flake's Home Page, Florida State University, on March 27, 2002, from: http://mailer.fsu.edu/~jflake/virtual.htm

Flake, J.L. (In-Progress). Elementary and middle school mathematics learning in the technological age. In-class materials presently in use at Florida State University, Tallahassee, FL.

Gilbert, J. (1999). But where is the teacher? Learning and Leading with Technology, 2 7(2), 42-44.

Hsu, S. (1999). How to design a virtual classroom 10 easy steps to follow. THE Journal (Technological Horizons In Education), 27(2), 96.

Huang, A.H. (1996-97). Challenges and opportunities of online education. Journal of Educational Technology Systems, 25(3), 229-247.

Internet2 [Online]. Available: http://www.internet2.edu/

Kagan, D.M., & Tippins, D.J. (1991). Helping student teachers attend to student cues. Elementary School Journal, 91(4), 343-56.

Klingenstein, K.J. (1998). The technical realities of virtual learning: An overview for the non-technologist. CAUSE/EFFECT, 21(1), 24-27, 32-34, & 47. Retrieved from the World Wide Web March 27, 2002, from: http://www.educause.edu/ir/library/html/cem9815.html

McGonigle, D., & Eggers, R.M. (1998). Stages of virtuality: Instructor and student. TechTrends, 43(3), 23-26.

McLellan, H. (1998). The Internet as a virtual learning community. Journal of Computing in Higher Education, 9(2), 92-112.

O'Hagan, C. (1995). Custom videos for flexible learning. Innovations in Education and Training International, 32(2), 131-138.

Pearson, J. (1999). Electronic networking in initial teacher education: Is a virtual faculty of education possible? Computers & Education, 32(3), 221-238.

van de Walle, J.A. (1998). Elementary and middle school mathematics: Teaching developmentally. 3rd edition. New York: Longman.

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This study is an investigation of the potential use of videos on the Web to help prospective teachers construct meaning about children's mathematical thinking. While prospective teachers need experience working directly with children, an alternative initial method of creating such an experience is the focus of this study. The intent of the videos is not to replace actual experience with children but rather to bring deepened insights into such expenences. The reason for putting the videos online is so that prospective teachers can easily access them from home, on campus, at work, or wherever, and to be able to access the videos at a time that is convenient for the participants, as well as provide a model for distance learning.

THE PARTICIPANTS

The participants for the study have been prospective teachers, mostly graduate but some undergraduate students, who are working on teacher certification in elementary education or special education. The participants were taking a course in topics in elementary mathematics at the time of the study.

BACKGROUND AND RELATED LITERATURE

Many times prospective teachers in teacher education programs often have had little experience with students prior to their internship. Alternatively they spend a lot of time in classrooms but do not have direct connections with related theory. It is difficult to make connections between the abstract theories and the real world of practice. The bridge between theory and practice varies greatly, and sometimes the bridge is a big gap. To treat theory in the absence of experiences with children will likely result in the theory quickly being abandoned or forgotten. At the same time building a class based heavily on field experiences has its problems as well. The experiences are often random, may or may connect with the underlying theories, and risk the possibility that the students never make the links between theory and practice. Furthermore, consequences of actions can affect real students and sometimes do damage to the students, and the supervision of field experiences often is labor intensive and/or not easy to do. In addition, the actions in a classroom are multidimensional and often difficult to isolate and control one variable at a time.

In 1972-73, the author had a vision of building a simulated teaching laboratory. Initial efforts occurred with a PLATO program, which she created, that allowed perspective teachers to instruct simulated students (Flake, 1973). The results were very encouraging. Participants in the study commented that they began to think of these simulated students as real; they also liked the notion that they were not doing permanent damage to the simulated students; and after going out into the schools the participants reported back to the author that the real students behaved just like the simulated students. In addition, a very popular button that was added at the request of the students was the restart button, when at times participants were not happy with the way things were going and they just wanted to throw out that class and start all over again--a phenomenon that does not normally happen with real classes. Furthermore, the simulation provided a common experience for class discussion of alternatives for interacting with the simulation. Brown (1999) found similar results.

Today we have the potential of online and/or distance learning. These have been studied by a number of people (e.g., Bialac & Morse, 1995; Gilbert, 1999; Huang, 1996-97).

A simulation is a working analogy of a life-like situation; for characteristics and examples (Flake, 1996-2000a). Virtual reality and virtual learning environments are a special type of simulation and have been studied for creating positive learning environments (e.g., Ferrington & Loge, 1992; Hsu, 1999; Klingenstein, 1998; McGonigle & Eggers, 1998; McLellan, 1998; Pearson, 1999). Virtual learning involves learning through telecommunications with user control and possibly using video streaming; for characteristics and examples (Flake, 1996-2000b). Virtual learning environments can include: (a) the use of telecommunications technologies that transcend barriers of time and place, associated with the traditional lecture format; (b) provide educational services through technologies which can be accessed from homes, institutions, communities, and workplaces; (c) allow users to have more control of their learning; and (d) provide for more flexibility in the course structure form. Klingenstein (1998) described a nu mber of factors to take into consideration for creating a virtual learning environment, including the use of video streaming, atmosphere, the nurturing of inquiry, and the building of a community of learners. Pearson (1999) used an electric network to facilitate discussion between perspective teachers and built a virtual community. Klingenstein also reported the importance of high-speed video servers and data lines, such as DSL and ADSL and other infrastructure considerations. Further, he discussed the evolution of Internet2, which should help address many of the current problems with the Internet.

Uses of video and multimedia have been investigated for helping prospective teachers gain a better understanding of students' behaviors (Kagan & Tippins, 1991; Fisher, Deshler, & Schumaker, 1999; O'Hagan, 1995). Kagan and Tippins (1991) studied the use of videos in helping interns to interpreting students' verbal and nonverbal behavior during instruction. Fisher, Deshler, and Schumaker (1999) studied teachers' understanding when using an interactive multimedia program and an inclusive practice.

Ten easy steps for designing a virtual classroom created by Hsu (1999) included the following steps:

1. assess the needs and the necessary conditions to satisfy them;

2. estimate the development cost, effort, and implications;

3. plan the virtual classroom;

4. design the virtual classroom;

5. prepare and distribute contents;

6. enable communication;

7. implement online student assessment methods;

8. implement class management procedures;

9. set up the system; and

10. maintain and update the virtual classroom.

Identified stages of virtual learning include the following. (a) Instructors' stages of virtual learning are: excited; apprehensive; questioning; determined; over stimulated; questioning revisited; and exhausted. (b) Students' stages are: confused; shocked; timid; frustrated; and "Eureka" (McGonigle & Eggers, 1998). Major paradigm shifts take time to bring about changes.

DESIGN

Careful thought went into the design of this study. Elements of design have included the content of the videos, the design of the technology, and the design of the experiences of the participants.

Content Design of the Videos

The content of the videos was determined from state and national standards in elementary mathematics. Twenty-three real children from grades kindergarten through fifth grade were interviewed about their concepts/ problems in mathematics in the areas of: patterning and algebraic thinking, space and geometry, measurement, early number concepts, number and operations, rational numbers, and probability and statistics. Two hundred and sixty eight videos were then created of approximately five minutes each. The videos were organized so that they could be accessed and examined by child, grade level, and/or content. Each child chose his or her own pseudo name that is used in the videos.

Technology Design of the Videos

Interviews with the 23 children were videotaped using a digital video camera. The videos are streamed to allow for a variety of accesses. The videos are stored on a RealServer G2 (see http://www.real.com/products/), which allows for high-speed access. The videos should be used with Realplayer G2 which can be downloaded from: http://www.real.com/products/. An up-to-date browser such as either Netscape, Version 4.7 or higher, or Internet Explorer, Version 5.0 or higher.

Regretfully the only microphone that was used for video tapings was the microphone from the video camera. Sometimes people have trouble with the audio portion of the videotapes. They have been given the following directions:

There are several different places where audio can be adjusted, including:

* On the task bar there may be a megaphone or other audio icon. Participants can place the mouse on that icon and see if a volume control appears. If so adjust it.

* In the control panel of the computer there should be either an audio or a multimedia option. If so, participants can adjust the volume there.

* The monitor, earphones, or audio output may have capabilities of adjustment, as well.

* The audio or multimedia software itself often has the ability to adjust the audio volume.

* The audio can be heard but sometimes caution needs to be used to be sure that optimal volume is available.

Design of Prospective Teachers' Experiences

The prospective teachers were asked to watch the videos in a variety of settings, involving various children, grade levels, and content. They initially viewed videotapes along content strands to see how the content evolved from grade level to grade level. Then they responded to more pedagogically oriented questions to help bring out clearer connections between theory and practice. The following are sample questions that prospective teachers have responded to:

Using the videotapes of interviews illustrating children's mathematical understandings, discuss the following (when responding to the following questions to provide a case, provide the child' grade level, name and the task, and then give specific incident within the task):

1. Identify three cases, if possible, of children having relational understanding in the videotapes and justify.

2. Examine Dan (3rd Grade, Blocks) and Art (4th Grade, Addition). What are your observations about these two cases? Discuss.

3. Describe three cases, if possible, where the interviewer uses scaffolding to help the student solve the problem at hand.

4. Do you feel that these students have come from a "rule based" curriculum or a "model based/problem-solving" curriculum? Discuss with specific incidences.

5. Identify and describe three cases above, if available, where the students seemed to be using dynamic imagery and describe nonverbal cues that suggest such.

6. Describe the potential evolution of a content strand, e.g., division concepts, from K through 5th grade. Identify and describe levels of understanding.

7. Select a grade level and describe the range of understandings within that grade.

As the prospective teachers were analyzing the videotapes, they were also working with manipulatives and technology, themselves. After these experiences the prospective teachers worked directly with real children.

SAMPLE RESPONSES OF PROSPECTIVE TEACHERS

While working through the videos and answering questions, the prospective teachers needed to think through the relationship between theory and practice. The following are sample responses of the prospective teachers concerning the videos.

Relational Understanding

Relational understanding involves the students making connections between related ideas and/or concepts. Sample participant responses included:

* In answering this question, I used the definition from page II of the Flake text (Flake, in progress) for relational understanding, which is "making connections or relationships from one situation to another." Based on this definition, Lisa (second grade, addition) was able to take the addition problem 23 + 18 from paper and convert it into blocks. She was able to represent 23 as two rows of ten and 3 individual blocks. She was also able to represent 18 blocks as one row of ten blocks and 8 individuals. She added the two groups and represented 31 as three rows of ten and one individual block. This shows her ability to connect the relationship of blocks or actual things to the values of the numbers.

* Relational understanding refers to "webs of ideas." Whereby one idea is associated with another, or other ideas, thereby enriching understanding (van de Walle, 1998). I believe one example of relational understanding is Matthew (fifth grade, division) who seemed to readily be able to solve the problem using rules (although from listening to him I am not sure what the rules are), but equally he was able to associate the numbers on the page to models (the blocks). Further, in the latter minutes of the tape he seemed to be able to overcome the limitations of the blocks by constructing a decimal system.

* Justin (kindergarten, division) did not have the need to use the blocks when "sharing fairly." He did most of his problem solving in his head. I feel he must have a clear understanding since he was able to make connections of the sharing fairly concept without having to use a manipulative to "verify" his answer.

* Rose (kindergarten, division & number sense) demonstrated a good relational understanding with division and number sense. With regard to number sense, she gave a "good estimate" for the number of blocks; she was able to answer correctly "the number left when 1 was taken away" and "another taken away" and finally "add another back." Rose also demonstrated good relational understanding with division when asked to share fairly; "one person leaves, how many cookies would each person receive?" The fact that she was able to base her answers on the understanding that if one person left, there would be more cookies for the remaining folks reflects comprehension of relational understanding.

* Lisa (second grade, feely bag) impressed me very much with the connections she made between the shapes in the bag and her drawings on paper. She was correct every time and was very confident in her answers. She found all the shapes in the bag as well. She knew what the shapes were and then she found relationships in order to draw them on her paper.

* Relational understanding is displayed by Fog in his transformations video (fourth grade, transformations/space sense). The interviewer presented the student with a shape that was transformed in six different ways. The student was asked to describe what was done to the shape to make it look different. Fog observed that the shape was sometimes flipped (after the term was provided by the interviewer) and sometimes turned. He also observed that the size changes. Through his responses, it was apparent that Fog understood that the shape remained the same, though it was transformed in some way. He was displaying his understanding of what has happened to the shape to make it appear different.

* Nathan (fourth grade, addition) first did addition on paper and then with blocks with three-digit-numbers. He did very well with blocks. He easily counted the blocks, while he was adding and put them together. He counted tens, ten by ten, to 100 and said this is one hundred and put it near other hundreds. I think he had relational understanding.

Through these experiences the prospective teachers have been able to make the connections between the theory of relational understanding and the reality of the children's understandings. By giving specific examples they address their concepts of the theory.

Specific Cases

A couple of specific cases of children were identified for the prospective teachers to study. In these cases things did not go exactly as expected and illustrated pitfalls that a number of children make. The following are samples of responses to these specific cases.

* Dan (third grader, blocks) worked with blocks to solve mathematical problems. The interviewer initially place blocks out on the table. He was asked to estimate how many were there, and he guessed 20. In actuality there were 26 blocks present, so his estimation was fairly reasonable. He was asked what the 2 in the 26 meant, and he said it was 20. He then thought that the 6 meant a 60, but he was then led to see that it was 6. Next he put the blocks into groups of 3 then counted them, then put the blocks in groups of 10 and recounted them. When given the subtraction problem 26-18, he couldn't solve it on paper; however, he knew that he couldn't take 8 from 6. He solved the problem using blocks and came up with the correct answer of 8. He believed the blocks more than he did his own answer (on paper), displaying proof that he could learn well using manipulatives and by using visualization. He didn't display understanding of concepts on paper, as well as the borrowing rule in subtraction.

* Art (fourth grade, addition) was given a paper problem with 234 + 389 =?. The interviewer asked Art to solve the problem (after stating he had seen problems like it). Art used his fingers to count and answered it (solved the problem quickly, using his fingers). He was then asked to demonstrate the problem with blocks. He didn't have any difficulty showing 234 and 389 with blocks. He used 2 flats (100/each flat), 3 longs (10/ each long), and 4 singles and 3 flats, 8 longs, and 9 singles. He actually counted some of the longs to make sure there were 10. He counted 4 + 9 (pulled out 10 singles and exchanged for I long); then stated 1 + 3 = 4 and 4 + 8 12 (used fingers); exchanged 1 long and 10 singles (combining the paperwork problem-carrying numbers-with blocks).

* Art continued trying to "carry" with the blocks-but he tried to carry a long into the one hundred's place rather than a flat or 10 longs-illustrating some confusion about the place value system. The interviewer suggested he exchange for different size blocks by asking, "what could you trade for a big flat?" (Leading question) Art knew 10 would trade so he then exchanged for a big flat giving 6 flats.

While it is nice to have prospective teachers see examples of students doing well, it is perhaps even more instructive for these prospective teachers to see cases where the students make common errors. These two, cases particularly illustrate some confusion on the part of these students.

Scaffolding

Scaffolding is the notion of a more experienced person helping students, by asking questions, probing, doing things that the students might not figure out completely on their own. Ultimately the notion is that the scaffolding is gradually removed and the students function independently. The following are sample participant responses to uses of scaffolding:

* In answering this question, I based my understanding of scaffolding on the Flake text from Chapter 2, which states that scaffolding can be accomplished by, "other students, a mentor, a computer, manipulative materials, or other models." These are tools that can aid in the students' sense making. The teacher's role can be to ask focus questions, when necessary, to help the students redirect their attention." An example of scaffolding was with Brittany (first grade, transformations). Brittany was asked how to make the cat on the left side of the paper look like the cat on the right side of the paper. When she was unable to perform this task, the interviewer picked up the paper and showed Brittany how the picture could be turned to match the other paper and showed Brittany how the picture could be turned to match the other picture. When Brittany was presented with the second problem, she was able to say, "Maybe I could turn it."

* As it relates to education, the word "scaffolding" generally encompasses the notion that students sometimes need "support" or assistance when attempting to solve a problem or achieve a goal that may be too difficult without the help of a skilled partner (often the teacher.) As the student begins to understand the concept, the skilled partner gradually withdraws assistance so that the student can begin to solve the problem unaided. In the videos, the interviewer often uses the technique of scaffolding to help the student solve problems. Scaffolding is used in Rosie's measurement video (first grade, estimating length/measurement). The interviewer first asked Rosie to estimate the width of a poster board without actually using the measuring apparatus (difficult to see, looked like cubes linked together). After she completed that task, the interviewer asked that she actually measure the width. After Rosie was done, the interviewer suggested that she measure again, and this time, overlap the measuring device "li ke footstep, one over the other..." When Rosie still had difficulty measuring, the interviewer physically assisted Rosie by placing her finger where the measuring tool should go next. Again scaffolding was used because some elements within the task may have been too difficult for Rosie to complete by herself.

* The first case of scaffolding I observed was with Andy (second grade, division) and the area of division. Andy was given the paper problem of 23 divided by 4 but did not know how to proceed working the problem. She began by moving the blocks randomly (distributing more blocks in some groups and less in other groups). The interviewer mentioned that it appeared some groups did not have as many (or were "missing some"). Andy counted the blocks and "discovered" the 2 of the groups had more blocks that the 2 groups. She then proceeded to redistribute some of the blocks. She counted and recounted to make sure she had the same number in each group. The interviewer then "assisted" Andy with transferring the concept of the blocks to the paper problem. I couldn't hear the response to determine if she was correct with paper problem.

* Mark (fifth grade, missing values) illustrated that he already knew the basic concept of missing values when he said, "I do this all the time." The first problem he was asked to solve he did the quickness and ease, but the successive problems increased gradually in difficulty, while still testing the same skill. The interviewer seemed to assess at what level does he seem to have the most difficulty and that's when she intervened, she was working at his Zone of Proximal Development.

From using the videos the participants were more clearing able to understand the role of scaffolding at a very concrete level. They were able to identify ways that the interviewer helped the children bridge to the concepts.

"Rule-based" Curriculum or "Model-based/problem-solving" Curriculum

A more traditional view of mathematics instruction is to provide the students a rule followed by a number of practice problems, a "rule-based curriculum." However, a more current view of mathematics learning is to engage students in problem-solving activities and give the students models that will allow them to figure out the problem themselves, a "model-based! problem-solving curriculum." Sample participant responses to whether they thought the children had come from a "rule-based" curriculum or "model-based/problem-solving" curriculum.

* A rule-based curriculum is a curriculum based upon giving students the rule followed by practice problems, reasoning that the more practice, the more proficient. However, a problem-solving curriculum encourages students to use models or develop their own, to see patterns and relationships. The video featuring Louisa has sufficient clues to infer the students come from a rule-based curriculum. First, she got confused early in the process. Second, I see no dynamic imagery used. Third, she did not seem comfortable with the manipulatives. As discussed elsewhere, Louisa did not have any familiarity with the base 10 blocks.

* I can honestly say that I feel, for the majority, that these students have been learning in a rule-based curriculum. They are dependent on working out mathematical solutions by using rules and formulas rather than by thinking of alternative routes to solving. I myself think I am a victim of a rule-based curriculum, and at times find it difficult to think of alternative ways to solve problems. Math has always been a source of stress in my educational career, and I think that had I been taught differently, with a more model-based/problem solving strategy. I truly hope to incorporate a more model-based/problem solving strategy in my classroom, using blocks, tangrams, and candy (to make it fun).. .the sky is the limit.

* I feel the majority of these kids, especially the younger ones, come from a model-based/problem solving curriculum. For the most part, they seem quite able to use the manipulatives without too much direction from the interviewer. There weren't too many questions about "how do you want me to show you?" or "how can I do that with the blocks?" I believe a child might have a difficult time demonstrating the manipulatives for a mathematical problem if they were only accustomed to "doing it on paper."

* My overwhelming response to this question is that the majority of these students have come from a "rule-based" curriculum with some exposure to "model-based" ideas. Though some students such as Lisa and Art performed problems using manipulatives with relative ease, most of the other students either could not understand the relationship between the blocks and their problems on paper or they were very uncomfortable using the blocks. Even Lisa and Art, however, worked their problems on the paper and tried to count on fingers and in the air rather than using the blocks to answer. My ideas are based on the following specific examples:

--Felicia (Kindergarten, division) did not grasp the idea that these blocks were useful in helping her figure out the problem. She started playing with the blocks as if they were people and had no idea of their use in dividing. This led me to believe she had liffle or no exposure to "model-based problem solving.

--Lisa (second grade, addition) was able to show what she did with the blocks when asked by the interviewer, but she did not initially use the blocks to solve the problems. She relied on what she already knew about solving addition problems, the rules, to come up with a solution to 23 + 18.

--Art (fourth grade, addition), when trying to solve the problem 623 - 234, tried to use his fingers and looked into space rather than getting the blocks to try to solve his problem first. Just like Lisa, he was able to show how the blocks relate to the numbers, which lead me to believe he has been exposed to some "model-based" ideas, but he still attempted to solve his problem by rules first.

--Louisa (third grade, division), when given a division problem, rather than using the blocks to "share fairly," tried to use multiplication tables to find what would be closest to the number she is dividing. This is a very "rule-based" idea and when Louisa finally used the blocks, she easily found the answer to the problem.

As these examples indicated, these students tried to solve these problems by rules first and then, either because they were encouraged or asked to by the interviewer, they used the manipulatives in the "model-based" style to solve the problems, often getting results much easier and faster. It is as a result of these observations that I feel these learners are learning under a "rule-based" curriculum.

* In my opinion, these students have come from a "rule-based" curriculum. Because I can say most of the students can do addition-subtraction and the other processes on paper correctly. But they do not know what that means in real life. For example, they can do multiplication on paper easily but they could not do it with blocks, and they could not understand multiplication has the same process with addition.

* I feel that these children have come from a "rule-based" curriculum. The rule-based curriculum is based on giving students the rule followed by practice problems. The model-based curriculum is based on using models that students apply for problem solving and to figure out patterns and relationships. The children are not familiar with the use of models. The smaller children don't seem to have a very good connection between what they are doing on paper and a mathematics model. Examples: Brittany (first grade, place value)-When she was asked what the 1 in 16 meant, she said, "One cookie." Brandon (first grade, blocks) counted out 19 blocks and wrote the number 19 on a sheet of paper. When asked what the one meant, he said it meant one block. The older children can work problems on paper, but they really don't seem to know why. They are working the "rules" and coming up with an answer. Examples: Art (fourth grade, addition) was able to solve the problem easily on paper, but was very unsure with the blocks. He k new there were number places, but I think he had trouble visualizing them and actually physically manipulating them. Fog (fourth grade, multiplication) was able to solve the problem quickly by using the rules and even talked the problem through. When he was asked to solve the problem using blocks, however, he had a hard time. I don't think he had had much experience with manipulatives. He didn't seem to understand at first that "four times," meant that the same number would be added together four times.

Although the participants saw different behaviors and justified their responses using different data and came up in varying cases with different conclusions, they did see the importance of how the children were taught. By observing these children's behaviors it was much clearer to the participants that they should help children learn to reason their way through mathematical ideas rather than simply giving the children a rule and asking them to follow the rule.

Dynamic Imagery and Nonverbal Cues

Underlying many areas of mathematics learning is the concept of dynamic imagery, where students can manipulate their images by taking images of objects apart, turning them around stretching or shrinking them, and so forth. The following were participant responses to the children using dynamic imagery and nonverbal cues:

* My answers to this question are based on my understanding of dynamic imagery from the Flake text, which stated that dynamic imagery is "the ability to mentally manipulate images in one's mind." The best example of dynamic imagery in the videos were in the transformations section where learners were literally asked to tell how they would manipulate or "transform" one image to make it look like another. One example of the use of dynamic imagery is Sam (fifth grade, transformations). Sam used his hands to give clues to the processes taking place in his head when explaining how the images on the page flip or turn to become the next image.

* Fog (fourth grade, algebraic & missing values) concentrated hard when given the 3 missing values problem. With 43 + N = 67, Fog looked around, eyes were moving as though he was picturing something: he got the correct answer and the interviewer asked him how? He replied with "I counted." When given the next problem, 40 x N = 80, Fog whistled, looked all around, again trying to picture the answer (or at least trying to picture how he should work the problem). Again, he got the correct answer: 20. When asked how, he replied that he knew (4) 10's = 40, so (4) 20's would = 80. He also worked both problems on paper, but I was unable to see his work. The final problem 4 x N - 3 = 97, seemed to concern Fog at first. He made the comment "Holy Cow!" After spending a few moments looking around, eyes and lips moving, hitting his pencil on the table he seemed to visualize and then verbalize the problem. He got the correct answer "25" by saying he knew (4) x 25 100 and if he subtracted "3" from 100 he would have 97. You could almost "feel" Fog working/thinking the problem in his mind!

* Within the realm of mathematics, dynamic imagery refers to a student's ability to transform, reverse, or change perspective when solving a problem. Some nonverbal cues that suggest a student is using such imagery include a student's hand motion and/or a student looking up while thinking about a problem. One example of dynamic imagery is found in Fog's volume video (fourth grade/volume/measurement). The interviewer provided the student with a clear container filled with cubes and asked the student to estimate how many cubes were in the container. Fog looked at the container and then briefly looked up as though he were imaging the number of cubes. After he asked for a piece of paper, Fog began working out an answer. Again, as he wrote, he looked around using dynamic imagery to visualize the answer. Through the use of dynamic imagery, it is possible for a student to envision what needs to occur m order to solve a specific problem.

* Louisa (third grade, feely bag) used imagery while drawing the objects on paper. She felt the shapes in the bag and drew them very small on the paper. I would say it is absolutely related to using imagery in this process.

* I used Sarah's video to represent a video for dynamic imagery. Sarah was working with transformations. Dynamic imagery is exemplified in her instruction as she was working to solve how her shapes must be manipulated. She did not have a physical block to work with but she used hand gestures to develop an image of what must be done to get the shapes she was working with to move it into new positions.

* I selected Matthew (fifth grade, drawing a house on a hill). In this process every student tried to draw a house on a hill. Some of the students paid attention to the slope of the hill while drawing the house, but turning the paper, by turning their heads or by turning their bodies with the hill. Some of the students did not pay any attention. But I can say all of the students were using dynamic imagery. Why I selected Matthew for this question is that we can easily understand and see his using dynamic imagery from his picture. He paid the most attention for the hill, and he thought it was better to draw a house like this on this hill.

By actually examining videos of the children, the prospective teachers can begin to develop a concrete understanding of student's possible thinking. This, in turn, can help them learn to develop experiences that will help students develop and use such a dynamic imagery.

Evolution of a Content Strand from K through Fifth Grade

By being able to look at children's performances at each grade level the participants were more concretely able to see the evolution of ideas developed from grade to grade and varying from child to child. In this particular case they examined a content area (division) that traditionally is not dealt with in Kindergarten through second or third grade but the interviews illustrated that children could deal with such concepts, and that there were transition steps that could help the children evolve into the standard views. For example, with division, the younger children did understand "sharing fairly" as a means to dividing in a case such as, "There are nine cookies and four children, how many cookies does each child get if they share fairly?" From that concept a transition move can be made to, "How many cookies would it take to go around once?" This in turn shifts to repeated subtraction, which then leads to the more traditional view of division of multiple repeated subtractions. These ideas do not develop qui ckly with children. It takes a lot of experiences, appropriate transitions, and maturation on the part of the children.

Range of Understandings within that Grade

Again it was constructive for the participants to look carefully at the ability ranges of the children within one grade. Many classroom teachers say that their classes often have about a five-grade range in abilities within a single grade level. When the children were selected for this project there were deliberate efforts to get a wide ability range. As the participants examined the videos they were able to see that ability range.

PROSPECTIVE TEACHERS' EVALUATION OF EXPERIENCE

The participants found the videotape experiences very useful. When they moved to the actual experiences with the children they felt much better prepared than earlier groups because of what they had gained from the video experiences.

At the same time there were some concerns. The videotape viewing took a considerable amount of time. However, it is likely that it would have taken much longer for them to view the same behaviors in the schools with actual children. Also there were some technical problems.

Some of the participants complained about the audio portion, but the author was not sure that the participants understood the variety of ways that the audio could be adjusted. Other students seemed to function just fine with the audio. This would suggest that some students needed to explore more ways to adjust their audio on their computers. Posted on the web site now are multiple ways of adjusting the audio so that future classes can make better use of the audio. Also in future tapings each child will have his/her own microphone to help improve the audio.

Most of the students were using telephone connections rather than high-speed lines, such as ADSL, DSL, or cable lines. As a matter of fact, when these participants took part in this study, ADSL and DSL were not available in this area, now they are available in limited parts of the area. Video streaming is a major help in this direction. The author believes that the high-speed lines will become far more dominant in time.

DISCUSSION

While there were some technical problems, the author believes that this method holds much potential for the future. The videotaping took time, but it helped to have the interviews carefully conceptualized so the variety of experiences did appear as they did with these children. For prospective teachers, the awareness of a variety of behaviors and abilities is very important for helping them understand the challenge that is ahead of them.

The technical piece will greatly improve as these capabilities become more and more integrated into our society. The author anticipates that within five years, television, telephone, and the Internet will be quite integrated. Greater access to online video will exist and allow for prospective teachers to easily sort for examples of a variety of levels of understanding in the comfort of their own home or alternative sites. Further, for those at a great distance from campus, the reality base of online videos can greatly enhance other forms of communication in terms of understanding children's levels of understanding.

SUMMARY AND CONCLUSION

This study investigated the use of online videos of children for helping prospective teachers understand how children think and the children's mathematical understandings. While this is an early stage of investigation, the author believes that it presents enormous potential for teacher education or other forms of education. While it is not quite at the form of a full-blown simulation where the participants "instruct" the students and make changes based upon their previous behaviors, it does provide them with experiences that help bridge between the theory and practice and allows them a transition between academic class instruction and the real world of schools and children. Research and development in this area should continue to discover how to effectively use the technology that we have available and that will be continuing to develop.

Acknowledgements

The author gratefully acknowledges the children, their teachers, and the prospective teachers who have willingly participated in this study and given permission to use their information. Florida State University is also enthusiastically acknowledged for providing the technical capabilities and support for this project.

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Author: | Flake, Janice L. |
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Publication: | Journal of Computers in Mathematics and Science Teaching |

Geographic Code: | 1USA |

Date: | Mar 22, 2002 |

Words: | 7002 |

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