Investigating and Measuring Preservice Elementary Mathematics Teachers' Decision about Lesson Planning after Experiencing Technologically-Enhanced Methods Instruction.
Research on teaching and learning mathematics indicates that offering an appropriate constructivist-based teaching model of mathematics instruction to preservice teachers enhances their teaching abilities and confidence in their future teaching. The purpose of this study was to investigate and measure decisions about teaching strategies made by preservice elementary teachers (N=28) in a methods course, which emphasized constructivism and used problem-solving multimedia. The experience consisted of six hours of multimedia-enhanced instruction over two weeks. This study focuses on three mathematics education topics taught to the students: problem-solving, technology, and ratios/proportions. The article describes the development and use of the teaching strategy test.Historically, several instructional theories and approaches have been used to help teachers and students learn mathematics. One current theory that has been successful in understanding student learning and success in mathematics is constructivism. If the student is actively involved in the activity, the student is more likely to learn the mathematics content of the activity (Carpenter & Moser, 1982; Cobb, Yackel, & Wood, 1991; Lesh, 1981; Schoenfeld, 1985, 1989, 1992). However, this learning process requires teachers to pose meaningful and worthwhile tasks (National Council of Teachers of Mathematics (NCTM), 1991) to facilitate learning. In the original Curriculum and Evaluation Standards for School Mathematics (1989), the NCTM asserted that knowing mathematics is doing mathematics and what students learn depends to a great degree on how they learn it. Implementing this approach to the learning process requires changing teachers' beliefs about the nature of mathematics teaching and learning.
When multimedia is used in a constructivist fashion by a teacher, it can be used to facilitate students' construction of learning. Facilitating K-12 teachers' abilities to implement the NCTM Standards with or without multimedia requires giving them powerful experiences in mathematical thinking and conceptual understanding (Hyde, 1989; NCTM, 1991). Moreover, most preservice elementary teachers enter the classroom with very little knowledge about effective and appropriate uses and integration of technologies in teaching mathematics.
With the increasing availability of computer-related technologies, research about preservice teachers experiences with investigations of technology-based influences on teaching and learning are needed. In particular, instructional technologies such as computer-supported learning provide a means by which students can construct their own knowledge of the mathematical concept under consideration (Papert, 1980; Schoaff, 1993). Furthermore, as computer-supported learning has contributed to educators' efforts to provide meaningful mathematics learning experiences, students have become involved in authentic real-situation problems and situations. Even though the effect of such technology-based materials has been a focus of studies in mathematics education (Kaput, 1992; Papert, 1980), such research has rarely included implementing the NCTM Standards.
THEORETICAL FRAMEWORK
Constructivism is a theory that explains how learners make sense of their environments and experiences to create their own knowledge (Fosnot, 1992; Yackel, Cobb, Wood, Wheatley, & Merkel, 1990). All knowledge is constructed (Noddings, 1990). The instruments of that construction include cognitive structures that are either innate (Chomsky, 1968; 1971) or are themselves products of developmental construction (Piaget, 1970). While constructing knowledge, learners are pondering the world from their own perceptions and experiences, which imply dependence on the learner's unique previous knowledge (Duffy & Jonassen, 1992).
As a result, constructivism calls for the creation of teaching-learning contexts and environments in which learners actively engage in the construction of their own individual knowledge through meaningful interactions with teacher, peers, and the physical environment (Bruner, 1986; Roth, 1993; von Glasersfeld, 1990). Although constructivist views have gained creditability among mathematics educators, there still exist mathematics classrooms where learning is viewed as a process where students passively receive information and formulas by repeated practice and memorization. However, constructivism maintains that learning mathematics does not take place by passive absorption (Cobb, 1991; Kamii, 1989; NCTM, 1991; Resnick, 1987; Schifter & Fosnot, 1993).
On the contrary, the teacher must purposely structure the intellectual and social climate of the classroom so students can discuss, reflect on, and make sense of their tasks in ways meaningful to them (Kamii, 1989; Yackel et.al, 1990). The role of the teacher is critical in implementing such classroom activity. But, most practicing teachers were expected to learn mathematics through traditional methods, such as transmission of knowledge by lecture. It may be difficult for many teachers to understand learning by way of constructivism.
To further complicate the issue, most constructivists agree that experiences with concepts and relations in school mathematics are typically different from experiences with those concepts in the real world. If the learners in school are to make sense of the worlds around them, then the school-world and the real-world need to intersect. To this end, constructivists emphasize, "situating" cognitive experiences in authentic activities (Duffy & Jonassen, 1992). There are several projects providing real-world episodes that may help practicing teachers better facilitate constructivist environments for example, the Problem-Centered Mathematics Project (Cobb, Wood, & Yackel, 1990; Cobb, Yackel, & Wood, 1991), and the Educational Leaders in Mathematics Project (Simon & Schifter, 1991). Using such problem-based materials from the real world, teachers should be able to establish a mathematical environment in which they themselves are able to "learn the mathematical knowledge of their students and how to harmonize their teaching methods with the nature of that mathematical knowledge" (Steffe & Wiegel, 1992, p. 17). That is, materials such as these should provide the practicing teachers with the tools needed to modify their teaching methods.
Another effective strategy for changing the practice of teachers is to provide them with exceptional collegiate experiences as part of their teacher preparation program well before they become practicing teachers. In typical programs, most preservice teachers have not had the opportunity to participate in their own formal study of mathematics through constructivism (Feldt, 1993). This limits both their knowledge of mathematics content (Simon & Blume, 1994: Ball & Wilson, 1990) and their abilities to design creative and appropriate ways to teach children (Showalter, 1994).
Teachers tend to teach as they were taught (Schifter, 1997; Scholz, 1995; Russell, 1997). Schifter (1997) claims that teaching in line with constructivism "requires a qualitatively different and significantly richer understanding of mathematics than most teachers currently possess" (p.2). Ball & Wilson (1990) compared preservice elementary teachers' knowledge and beliefs about mathematics to mathematics major's knowledge and beliefs about mathematics. They found neither group to be prepared "to teach mathematics for understanding nor to teach mathematics in a way that differs the traditional pedagogy of telling and drilling algorithms into students" (p. 10). Clearly, many preservice elementary teachers graduate without having well-developed mathematical concept knowledge and are not prepared for teaching based on a constructivist view of the learner
The use of innovative problem-solving multimedia technology to learn both mathematics and methods for teaching mathematics may hold potential for improving these situations. With appropriate guidance about meaningful use of technology, preservice elementary teachers may reconstruct their comprehension of how children learn as well as develop a working knowledge of mathematics concepts.
According to Brook and Kopp (1989), "If teacher education is to meet its responsibility to prepare teachers for the information age, then teacher educators have a professional responsibility to provide leadership in developing the full potential of existing and emergent technologies in teacher training" (p. 2). Beichner (1993) suggested that educators must show preservice teachers how to take advantage of the capabilities of educational technology, while also making them aware of its limitations. Teacher preparation programs must be designed so that student teachers understand, develop, and practice the skills they need to acquire a meaningful understanding and use of integrated technology throughout the entire program (Barone, Berliner, Blanchard, Casanova, & McGowan, 1996; Willis & Mehlinger, 1996).
Willis and Mehlinger (1996) indicated that most preservice teachers know very little about effective uses of technology in education. The most critical factor for capitalizing on the instructional potential of computer-related technologies is to focus upon the role these students will play once they become practicing teachers (Clack, 1983; Fulton, 1989). As teachers, they must understand that technology is only a partial answer to moving toward student-centered education. Teachers of the 21st century are faced with the need to enter the profession with the ability to effectively use and integrate technologies as they provide a learning environment, which fosters the construction of mathematics knowledge.
Problem Solving Multimedia.
The Cognition and Technology Group at Vanderbilt (1991, 1992, 1993) developed the Jasper Series, The Adventures of Jasper Woodbury, to help facilitate a real-world learning environment for middle school students. The Jasper Series is a valuable and potential multimedia-based program designed to "motivate students and help them learn to think and reason about complex problem" (Cognition and Technology Group at Vanderbilt, (1992), p.29 1). When teachers use the story to guide construction of mathematics knowledge, the complex problems from video-based, narrative adventures naturally encourage students to attempt to identify and solve problems based on information embedded in the narrative. Each 15-20 minute adventure in the Jasper Series provides multiple opportunities for students to experience problem solving, reasoning, communication, and making connections to other areas such as science, social science, literature, and history (NCTM, 1989; Cognition and Technology Group at Vanderbilt, 1992).
Due to the powerful impact of teachers' beliefs on their teaching (Thompson, 1985), research must address preservice elementary teachers' understandings and beliefs about constructivism as a theory to explain how children learn mathematics. The abilities of new teachers to plan learning experiences consistent with constructivist theory must be studied. Also, research is needed to further describe how to use and integrate computer-related technology into elementary mathematics teacher education programs and to provide worthy examples of the uses and integration of technology into constructivist-based mathematics classrooms. Finally, a method to measure lesson-planning ability in line with constructivism is needed. The purposes of this study were to investigate the mathematical lesson planning of preservice elementary teachers and to evaluate lesson plans written after completing a mathematics methods course emphasizing constructivism and the use of problem solving multimedia.
METHOD
Sample
The participants were 28 preservice teachers enrolled in an elementary education mathematics methods course at a large mid-western university. At the beginning of the course, students were given a demographic questionnaire (Appendix A) designed to provide information on both academic background and computer experience. These results are shown in Table 1. All students completed the course.
Procedures
The instructor used the NCTM curriculum standards (1989) and professional standards (1991) in planning and teaching the course. All participants studied the NCTM standards, applied them in assignments, and made extensive use of a variety of mathematics manipulatives. The focus of this study is on the two-week portion of the course typically dedicated to teaching about problem-solving, technology, ratio, and proportion.
The students encountered the technology-enhanced unit on ratios and proportions after seven weeks of instruction. During those two weeks of the course, all participants learned about concept knowledge as well as pedagogical knowledge for problem solving, technology, ratio, and proportion. The instructor not only encouraged preservice teachers to think about how the videodisc experience improved their understanding of constructivism in learning/teaching mathematics in meaningful ways, but also encouraged them to model and reflect that comprehension of constructivism in their conceptions of their future classrooms. At the conclusion of study of these topics, all students completed the teaching strategy test.
The procedures for two weeks are described as follows: During the course, students experienced several technologically-rich lessons designed to provide an environment in which students could construct both mathematical and pedagogical knowledge. Rescue at Boone's Meadow is one of the Adventures of Jasper Woodbury (Learning Technology Center, 1992). The Jasper Series provides anchored instruction by posing complex real-world problems from video-based, narrative adventures. At the end of each story, the major character is faced with a challenge that the students must solve before they are allowed to see how the character solved the challenge. In cooperative groups, students solve the problems using information embedded in the narrative (Cognition and Technology Group at Vanderbilt, 1990).
The story in Rescue at Boone's Meadow begins with Jasper's friend, Larry, teaching another friend, Emily, to fly an ultralight airplane. Later, Jasper and these friends discuss Jasper's upcoming hike to Boone's Meadow over a meal in a restaurant. On the trip, Jasper finds a badly wounded eagle that needs emergency care. Jasper radios his dilemma to Hilda and Emily. Students are challenged to develop solutions for Emily to help Jasper rescue the eagle in a timely fashion. The mathematical skills underlined in Rescue at Boone's Meadow are whole number operations, fractions, decimals, ratios/proportions measurement, distance/length, time, and problem solving. An example of one group's work is shown in Figure 1.
Next, students watched the video-tape, Similarity! made by the California Institute of Technology (Project MATHEMATICS!, 1990). This computer-animated videotape shows students the mathematical concept of similarity in ways that cannot be done with chalkboard or with a textbook. Part of the animation effectively showed how ratios and proportions could exist in the real world of biology.
Then, students watched a clip from Raiders of the Lost Ark (Marshall & Spielberg, 1981). Bransford, Goin, Hasseibring, Kinzer, Sherwood, & Williams (1988) and Van Haneghan, Barron, Young, Williams, Vie, & Bransford (1992), used Raiders of the Lost Ark as a context for discussing a trip to South America where students encountered some obstacles. In this study, students were asked how to determine the width of the pit that Indiana Jones needed to jump. The pit is only a few inches wide on the screen. Students used the proportional relationship of a known object to the unknown object. For example, if Indiana Jones is 6 feet tall, then the pit is 12 feet wide.
Then, an activity from the NCTM Addenda series (NCTM, 1994) was used. In the film, Honey, I Shrunk the Kids (Cox, Yuzna, Landau, & Johnston, 1989), the concepts of ratio and proportion were used to answer the question "Do you agree with the statement that the kids have been shrunk to 1/4 inch?" After students discussed in their groups, they presented their group solutions to the problem. One of their solutions is shown in Figure 2.
The previous activities focused on enhancing mathematics knowledge through multimedia. The two-week experience culminated with viewing and discussing a videotape focused on enhancing methods knowledge with multimedia. Double-Column Addition (Kamii, 1987) shows real classroom interaction between a 2nd grade elementary teacher and her students as they constructed knowledge about double-column addition. The essence of this style of teaching is based on Piaget's theory to foster the children's own natural thinking and to encourage them to exchange their points of view (Kamii, 1987). For instance, Kamii (1987) doesn't teach a universal procedure, such as adding from right to left, for double-column addition but encourages students to invent many different ways based on their personal experiences.
Materials
One of the purposes of the study was to evaluate preservice elementary teachers' lesson plans for teaching mathematics according to constructivism. The teaching strategy test (See Appendix B) was developed to measure participants' abilities to plan teaching strategies which would provide a constructivist-based learning environment for ratios and proportions. The scoring rubric for this test instrument was based on the theory of constructivism and the NCTM Standards (NCTM, 1989, 1991).
Four categories (Cobb, Wood, & Yackel, 1990; NCTM, 1990) were developed to score the lesson plans written in response to the teaching strategy test:
Category 1: Using appropriate authentic content of mathematical experience of students emphasizing context
Category 2: Communicating mathematically with students by justifying and supporting their own views with multiple perspectives
Category 3: Approaching student-centered instruction
Category 4: Appropriate use of manipulative, diagrams, computer-related technologies, or alternative representations
The sub-categories for each of the main categories are described in Table 2 and each category is scaled for scoring in Table 3.
The scales for the scoring grew from the Assessment of Constructivism in Mathematics Instruction (ACMI) by Simon and Schifter (1991), adapted from Hall et al. (1975).
The original scale included four levels and was designed for practicing teachers, "to determine whether teachers' decision making was based on a constructivist view" (Simon & Schifter, 1991, p. 324). For this study, two changes to the scales were necessary to accommodate future teachers. First, the highest level of the original ACMI scoring scale read: "assists or collaborates with colleagues to implement instruction..." (Simon & Schifter, 1991, p. 325). Future teachers do not have this experience, and as such, this level was deleted in favor of a different descriptor. Simon and Schifter's (1991) definition of constructivism provided the following criterion: "Maximizes opportunities for students to construct mathematical metacognition" (p. 325). Because this is their definition of constructivism, it served as the measure of the subjects' demonstration of the highest level of constructivist-like teaching and was used as the highest level (level III).
Second, one portion of the original second highest level classification included, "Has modified teaching style to include regular activities to foster construction by students ..." (p. 325). Because the participants in this study were future teachers, it seemed likely that they would not have a teaching style to modify. So, this criterion was deleted from our level II classification leaving three other criteria for level II. In all other instances, the levels (0 and 1) were used verbatim from the existing scoring rubric. The resulting scale fell across four categories. The score within each category ranged from 0 points through 3 points. A possible high score for the teaching strategy test was 12 points (3 points x 4 problems).
Three people, a mathematician, and the two researchers (a mathematics educator and a research assistant), independently scored the teaching strategy test. Then, after all independent scores were reported, the three raters discussed any variance in scoring of each participant's lesson plan until agreement was reached. One of the scorers was consistently more strict than the other two and one of the scorers was consistently more lenient. The scorers opted not to simply find the arithmetic mean. Rather, they discussed each students' lesson plan score for which there was a discrepancy. Extreme scores were adjusted after agreement about interpretation of the rubric was reached for each lesson plan.
RESULTS
The Teaching Strategy Test
The scores on the teaching strategy test ranged from 0 to 8 out of 12 total possible points. The cumulative mean for the group was 5.25 (Table 4).
Thus, the average future teacher functioned only slightly above level I. That is, in general, students focused their lesson plans on student learning rather than teaching behavior to shape instruction from a constructivist perspective. However, analyzing the data within categories allowed a more particular investigation of the preservice teachers' lesson plans.
Category I focused on measuring the participants' willingness to use appropriate content and mathematical experiences. This category was used to measure whether or not the preservice teachers' lesson plans emphasized meaningful, authentic, contextualized, and situational tasks. In addition, category 1 was used to determine the extent to which the participants' used real-world situations with relevant content and which would with support multiple solution strategies. The scores on category 1 ranged from 0 to 3 out of 3 total possible points and the mean was 1.71. These students functioned, on average, at or near level II. They demonstrated a working knowledge of constructivism, writing plans to include relevant contexts and providing opportunities for open-ended, child-centered learning.
Category 2 focused on measuring the participants' willingness to encourage their future students to communicate mathematically with the teacher as well as their classmates. This category was used to measure whether or not the lesson plans elicited reflective mathematical thinking and communication such as justifying and supporting views with multiple perspectives emphasized in the process of reaching conclusion. The scores on category 2 ranged from 0 to 2 out of 3 total possible points and the mean was .71. These students functioned, on average, at or near level I. These future teachers made shallow attempts to modify instruction that seemed to reflect a rather vague understanding of constructivism. They did not typically describe planned activities to draw children toward reflective communication of their learning.
Category 3 focused on measuring the participants' willingness to approach student-centered instruction to teaching ratios/proportions. This category was designed to measure the extent to which lesson plans encouraged children to take responsibility for their own learning, to explore alternative problems, and invent many different ways of re-solving a situation. Moreover, the ability of the preservice teachers to foster in their future students the ability to reflect on their own thinking was a component for this category. The scores on category 3 ranged from 0 to 3 out of 3 total possible points and the mean was 1.50. These students functioned, on average, directly between level I and level II. They tended to view learning as being explained by constructivism, attempted to modify instruction toward student centered, but seemed unsure about how to let go of teacher control.
Category 4 focused on measuring the participants' willingness to appropriately use instructional materials such as manipulatives, diagrams, computer-related technologies, or alternative representations. The scores on category 4 ranged from 0 to 3 out of 3 total possible points and the mean was 1.32. These students functioned, on average, slightly higher than level I. They mentioned using materials such as technology and manipulatives but that use seemed to be forced or indecisive and tended to be neither specific nor reflective of a mature understanding of how children learn.
Examples of the Highest Scores on the Teaching Strategy Test
Samples of students' responses (Figure 3) to the teaching strategy test are presented in this section. These composite values also provide scores against which future researchers, who choose to use this test, may compare their results.
DISCUSSION
The purpose of this study was to investigate and measure the extent to which preservice elementary teachers' lesson plans demonstrated constructivism after these students completed a methods course, which emphasized constructivism and included the use of problem-solving multimedia. During the experiment, the instructor explicitly modeled constructivist-based instruction and used problem-solving multimedia. However, the variance in the students' levels, as measured by the Teaching Strategy Test, suggests transfer on a limited basis.
The four categories identified in the measurement instrument appeared important in identifying weaknesses and strengths in students' lesson planning. Some clustering did occur. That is, several students tended to be at or near the same general level across the four categories. But, in no cases were students always at the same level across categories. As is often the case with open-ended response data, the results appear messy. Students don't fall nicely and neatly into one level or another. The four categories appeared to be needed to appropriately investigate and measure the consistency of the students' lesson plans with constructivist ideology.
Some of the preservice elementary teachers in the study were able to articulate a constructivist approach to decisions about teaching ratios/proportion. Constructivism research with practicing teachers suggests that constructivist-based instruction through an inservice program in mathematics education can facilitate the development of teaching strategies consistent with recent reform movements (Simon & Schifter, 1991). But in a preservice methods course, the extent to which preservice teachers can make plans consistent with constructivism was highly variable. However, this instrument helped to identify those aspects of a professional methodology developed by preservice teachers' in their teaching of mathematics.
In this study, the issue of appropriate tasks, which included context and real-world problem solving, tended to be the category with the highest average. It is interesting that this category more strongly emerged given that students experienced a specifically designed, problem-based multimedia experiment. The category with the second highest average was the category measuring students' lesson plans for planned use of appropriate instructional aids, including technology. Students' scores indicated that they were, in general, knowledgeable about the importance of using materials, but were not able to clearly articulate a strategy. They seemed to be somewhat aware of the consistency of using these materials with practicing teaching that is in line with constructivism. However, much more research is needed in this area before solid claims can be identified. In so doing, future researchers may choose to isolate the technology treatment from the real-world situations and further refine the teaching strategy test.
This study described a potentially powerful test for measuring constructivist-based instruction. This study also offers a potential teaching model of constructivist-based instruction, which includes the use of multimedia to help preservice elementary teachers understand technologically enhance mathematics teaching. As Handler and Pigott (1994), Hess (1992), and Wetzel (1993) note, the use and integration of technologies across the entire teacher education curriculum enables future researchers to use the constructivist-based model including the use of multimedia as a foundation to explore other topics besides ratios and proportions.
Educators are facing fundamental issues related to teaching meaningful mathematics to students to learn mathematics in their rapidly changing technological information society. Constructivism has been regarded as a valuable and powerful theory in implementing the reform in mathematics education that learners actively engage in the process of making sense of their environment and experiences so that they can create their own knowledge. In order to investigate these ideas, this research has demonstrated an initial test for measuring the constructivist nature of preservice teachers' lesson planning. However, constructivism itself takes a long time to learn as well as to show an effect on the teaching and learning of mathematics. Professional development of teachers' mathematics methodology with constructivistbased instruction and technology is an area of research that deserves continued attention.
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Demographic Information About Participants attribute Percentage of students (n = 28) Gender 89% female Major 71% will earn teaching license for K-6 endorsement Year in College 4% juniors High School 4% 29% Mathematics 1 course 2 courses Background Computer 82% 86% Software data bases spread sheets Experience 100% 100% word processing desk top publishing Self- 3.21 0.448 Reported mean standard deviation GPA attribute Gender 11% male Major 25% will earn teaching license for age 0 - 8 years endorsement Year in College 93% seniors High School 54% 11% Mathematics 3 courses 4 courses Background Computer 89% 75% Software programming educational software Experience 100% 100% graphics tools electronic Self- networks Reported GPA attribute Gender Major 4% will earn both licenses Year in College 4% 5th year High School 4% Mathematics 5 courses Background Computer 93% Software interactive multimedia Experience Self- Reported GPA note: Some groups do not add to 100% due to rounding.
Categories of Scoring for the Teaching Strategy Test
1. Using appropriate content and mathematical experiences emphasizing context
- authentic, contextualized (situational) tasks
- meaningful to students
- set real world situation problems with relevant content
- support multiple approaches
2. Encouraging mathematical communication
- encourage students to justify and support their own views with multiple perspectives
- encourage students to communicate mathematically among themselves
- use cooperative learning appropriately
- emphasize process of reaching conclusion
3. Approaching student-centered instruction
- encourage students to assume responsibility for own learning
- encourage students to explore alternative problems and invent many different ways with individual and diverse starting points
- connect mathematics to other areas
- encourage students to reflect on their thinking
4. Appropriate use of manipulatives, diagrams, computer-related technologies, or alternative representations
Level Expectations Level 0 - Does not have/use a constructivist epistemology. Level I - Attempts to modify instructon based on a general view that instruction should involve students in active construction; struggles with how to integrate this view with teaching style and curriculum Level II - Focuses on student learning rather than teaching behaviors to shape instruction from a constructivist perspective. Level III - Maximize opportunities for students to construct mathematical metacognition
One group's solution to the problem
The Scores for Level Categories of the Teaching Strategy Test N Category 1 Category 2 Category 3 Category 4 Total M (SD) M (SD) M (SD) M (SD) M (SD) 28 1.71 (.71) .71 (.66) 1.50 (.51) 1.32 (.55) 5.25 (1.88)
Investigate rescue plan options and list what you found and describe step by step for your solution. (Show all of your calculation for each rescue plans.)
Flying Fields
Our Information:
60 miles from Hilda's to Doc's by vehicle, 60 minutes; by flight 120 minutes
18 miles from Hilda's to Boone's Meadow on foot, about 5 hours; by flight 36 minutes
Total weight: Max payload = 220 (pilot, fuel and cargo) 180 (pilot 120, gas 30, eagle 15, box 10, gas can 5)
35 hp engine burns regular gas (Flying Fields)
5 gal extra gas tank weighs 5 pounds when full (Flying Fields)
engine weighs 30 lbs and holds 5 gal (Flying Fields)
Fuel and cargo box weigh 10 pounds (Flying Fields)
We can fly 1 mile every 2 minutes (Flying Fields)
It requires 100 yards to get off the ground and land (Flying Fields)
Eagle is panting and has lost blood (Boone's Meadow)
Eagle weighs 15 pounds (Doc's)
Boone's meadow is 1000 feet (Doc's)
Boone's Meadow is 65 miles from Doc (Doc's)
Eagle is 5 hour walk from Hilda (Boone's Meadow)
Larry landed near Hilda in the past (Flying Fields)
Hilda sells gas (Hilda)
Emily weighs 120 pounds (Restaurant)
Our Plan:
Fly to Boone's meadow from Flying fields (carry 5 gallons extra gas) get bird (refuel & leave gas can behind with Jasper), fly to Hilda's, land, get more fuel, then fly to Doc's.
If we can find a place to land and Larry meets us along the highway, we will land and give him the eagle because he can drive twice as fast.
A group's efforts to resolve "Honey I shrunk the kids" question Size in the pictures: Real-life sizes (if the child were really shrunken): Hole in Cheerio piece 17 mm 4 mm Child's head 3 mm x mm Child's Height 24 mm ?
Comparing the size in the picture and the real-life size, we found that:
17:3 = 4:x
x = 12/17 = 0.7
We considered the fact that the height of an adult can be calculated by multiplying the head size by 8. So we multiplied the kids's real life head by 8.
0.7 mm x 8 (= 5.6 mm = 0.22 inches).
On the other hand, 1/4 inch = 6.35 mm = 0.25 inches
Therefore, we agree with the statement that the kids had indeed been shrunk to 1/4 inch.
Three Samples of Students response
Example 1: Highest score (8)
Answer: I would use a strategy similar to the Barbie strategy used in class. I would bring in a doll house and show the students many of the things in the doll house examples. I would show the chairs and table, the sofa, the Christmas tree (I really own a doll house with all of these things!). I would assign groups of 2 to a few items or a room full of items depending on the group size. I would let the student figure out what the sizes of the furniture would be in a real world setting.
Level 0 Level 1 Level 2 Level 3 Category 1: Using appropriate X content and mathematical experiences Category 2: Encouraging X mathematical communication Category 3: Approaching X student-centered instruction Category 4: Appropriate use of X manipulatives, diagrams, computer-related technologies, or alternative representations Total points = 8
Mean score (5)
Answer: The best way to teach is to give the students a situation where they have to solve a problem. Like when we viewed the video tape to get the bird home. Let the students plan a trip give them some variable they have to use and let them plan the rest.
Level 0 Level 1 Level 2 Level 3 Category 1: Using X appropriate content and mathematical experiences Category 2: Encouraging X mathematical communication Category 3: Approaching X student-centered instruction Category 4: Appropriate use X of manipulatives, diagrams, computer-related technologies, or alternative representations Total points = 5
Lowest score (2)
Answer: Use the base 10 blocks to develop strategies and solve problems involving numbers. Work with fraction, hands on, using things to explain fractions. Use example students keep with problems for whole number operations, use counting beans, sticks etc. to help the process.
Level 0 Level 1 Level 2 Level 3 Category 1: Using appropriate X content and mathematical experiences Category 2: Encouraging X mathematical communication Category 3: Approaching X student-centered instruction Category 4: Appropriate use X of manipulatives, diagrams, computer-related technologies, or alternative representations Total points = 2
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| Author: | KIM, MIN KYEONG; SHARP, JANET |
|---|---|
| Publication: | Journal of Computers in Mathematics and Science Teaching |
| Date: | Dec 22, 2000 |
| Words: | 6705 |
| Next Article: | Glimpsing the Future of Mathematics Education: One Undergraduate's Story. |
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