The Importance of Teachers' Beliefs
Setting, Subjects, and Methods of the Study
Appendix A: The Lesson Packet
Appendix B: The Writing Assignment
Appendix C: The Consent Form
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According to Paul Ernest (1989), "Mathematics teachers' beliefs have a powerful impact on the practice of teaching." Ernest was writing primarily about teacher's beliefs concerning the nature of mathematics, but the statement is equally true of teachers' beliefs about diversity, equity, and the pedagogy appropriate for diverse groups. Indeed, beliefs about the subject matter are inextricably intertwined with beliefs about teaching, learning, and students. Certain views of mathematics can lead to dangerously negative views of students. Gregg (1995) notes: "[W]here learning is equated with internalizing objective facts and procedures, it appears that no matter what one tries, some students lack the ability to learn mathematics." Thompson (1984), studying mid-career teachers, discovered that teacher beliefs about mathematics and pedagogy are affected by the extent to which the teacher reflects on the relationships between various beliefs and experiences.
As Irvine and York (****) discovered, teachers make different attributions to explain the success and failure of students from different racial and ethnic groups. These attributions affect the ways teachers interact with their students, which in turn influence the students' achievement in mathematics. Gutierrez (1996, p. 510) discovered that, in schools that were organized for advancement of all students, the teachers held constructive conceptions of students and flexible conceptions of the learning process. Teachers also shared responsibility for learning with students and held high expectations of students. Teachers in schools that were not organized for advancement were much less likely to hold these beliefs.
Secada (1991) identifies teacher beliefs about diversity and equity as an area in which more work is needed. On page 33 he writes "What is missing from their studies, and indeed from most similar accounts of teaching, is what teachers think, believe, and do as a function of their diverse student populations." This paper describes a study of pre-service teachers' views of equity and diversity. Understanding these beliefs and the ways they are formed can guide the faculty of teacher-preparation programs in their efforts to encourage the formation of attitudes that lead to more effective and equitable teaching.
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Curriculum and Instruction 101/219 is a course for juniors, seniors, and graduate students who intend to become mathematics teachers at the middle school and high school levels. The students observe mathematics classes in local public schools and discuss their observations in class. They also read, write, and learn to use various technological aids. They do much of their work in groups of three or four students, with the same members remaining in each group throughout the semester. In the Fall semester of 1996, there were two sections of the course. One section was taught by Professor Ramirez, who specializes in equity issues in mathematics education. The other section was taught by graduate teaching assistant Mr. Thomas, who is interested in curriculum development in mathematics.
Near the end of the semester, I led each class through a lesson, which I had written, using ultrasonic motion detectors and computers to introduce the concept of a derivative. The unit was designed for a first-semester calculus class. The students in C & I 101/219 had completed first-semester calculus and several courses beyond it, but had not previously used this particular unit. A few of the students had seen motion detectors used in the classes they were observing.
The lesson was written to require as little computer expertise as possible. When it was used in the calculus class, the students were required only to press the return key when they wanted to begin taking data. The C & I 101/219 students had laptop computers that they had been using all semester, so these were used for the lesson. This meant that the equipment could not be set up until the students arrived, which caused a little delay and exposed the students to more of the technical details than the calculus students had seen.
Using their computers and the motion sensors, the students worked through exercises described in a twelve-page packet (Appendix A). The packet represents about three hours' work for calculus students, two hours in class and an hour of homework. The C & I students were told that they were not expected to finish the packet. Except for two small groups that combined to make a group of five students, the students worked in the same groups they had been using all semester. After the students had spent about an hour working through the calculus lesson, the instructor and I led a discussion of the experience, concentrating on the mathematical ideas that the students had recognized in the lesson, their feelings about the lesson, and the ways they thought the technology could be used in various math courses. Equity and diversity were not mentioned in the discussion.
After the discussion, the students were given a writing assignment, to be submitted in five days or a week, depending on the section. A copy of the instructions for the assignment is given in Appendix B. The assignment was required as part of the final exam in the course.(Footnote 1) It was also used for this study of pre-service teachers' concepts of equity and diversity and the role of technology. To participate in the study, the students signed a consent form (Appendix C). Two students in Mr. Thomas' section did not return the consent form, and so were not included in the study. This left 19 subjects from Mr. Thomas' section and 21 from Prof. Ramirez' section.
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Aspects of Diversity
Neither the printed assignment nor the class discussion on the day of the study mentioned specific aspects of diversity. The subjects were not led toward racial issues versus gender issues, for example. The fifteen aspects of diversity listed below were identified in the subjects' writing.
Appendix D contains several graphs of the students' responses by gender and section. No statistically significant differences were found between the responses of male and female subjects.
Differences between sections are not apparent from the graphs, because the graphs show simply which aspects of diversity were mentioned, rather than which ones were emphasized. When the emphasis is taken into account, differences between the sections emerge. Prof. Ramirez' students generally stressed cultural aspects of diversity, while Mr. Thomas' students concentrated on diversity of learning styles. Indeed, many of Mr. Thomas' students mentioned cultural differences only in the context of stating that differences in learning styles are more important.
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Sources of Conceptions of Diversity
The differences between the sections are most easily explained by the differences between the instructors. Prof. Ramirez, as a Latina, represents a different culture or ethnic group for most of her students, who are white. Also, her research interests are in the area of equity issues, so that she is both knowledgeable and interested in the effects of cultural differences. It is likely that she stressed these issues more in class than did Mr. Thomas.
Mr. Thomas is interested in curriculum development and pedagogy in mathematics, but without any particular emphasis on cultural differences. He is thus more likely to emphasize learning styles, and other issues that relate directly to pedagogy. He is a white and American-born, as are most of his students, so he does not represent a different culture or ethnic group to them.
Several of the aspects of diversity mentioned by the subjects had been discussed in class or described in required readings for C & I 101/219. This could account for the similarity of many of the responses, particularly within a section.
The day I presented the unit in Prof. Ramirez' class, I was wearing a shirt advocating gay rights. During the class, a subject asked me to turn around so she could read both sides of the shirt, and we briefly discussed what it meant. Several other subjects were listening to this conversation. This was not planned as part of the experiment, but in retrospect it appears likely that it would influence the subjects, causing them to be more conscious of sexual orientation issues than they would otherwise have been. In a group of this size, it is quite probable (Footnote 2) that at least one subject is of a minority orientation, and even more likely that at least one subject has a friend or relative with a minority orientation. Difficulties in school are common for young people of minority sexual orientations, but none of the 40 subjects mentioned this aspect of diversity in their papers. Perhaps their idea of what qualifies as diversity for purposes of classroom discussion is limited by what they have read in their textbook or other required readings, or heard their professor address. Since sexual orientation has not received this stamp of approval as a recognized aspect of diversity in the classroom, they would be unlikely to mention it in their papers.
The concept of diversity was not always generalized from one area to another. For example, ethnic or cultural diversity was the most frequently mentioned in Prof. Ramirez' section, with 19 of the 21 subjects referring to it, and eight of those discussing it in some depth. In contrast, religious diversity was mentioned by only two subjects in that section, neither of whom discussed it beyond a brief reference. One might think that culture and religion are similar in the types of influence they could have on students, but they did not get the same level of attention from these subjects. This is probably due in part to the fact that religious diversity was not mentioned in this class, and that the subjects are not likely to have taken a class in which it was discussed. By contrast, cultural diversity had received attention in class and in the required readings.
Many subjects mentioned a personal experience or course that had increased their awareness of a particular dimension of diversity and the related equity issues. In some cases the subjects were themselves members of the disadvantaged or minority group. Subject R3 became aware of the potential for gender bias in word problems when her statistics class used many sports examples.
| For example, giving a word problem that talks about baseball batting averages to girls may not interest them and may not teach them anything of significance unless they have an interest in baseball. This happened to me when I was a high school student in statistics class. We looked at many different statistics of basketball, baseball, and hockey teams and I had no interest in learning any of it because it did not apply to anything significant in my life. All I remember is not liking the class and not liking statistics. So, from that time on, I still do not like statistics and do not have a desire to learn it further. |
Subject R7 described her school's method for coping with economic differences. She mentions that students went to teachers privately for assistance. Since she knows what transpired in those meetings, it seems likely that she was one of the economically disadvantaged students, or had such a student as a friend.
| Many students cannot ask their parents to buy them additional items. I like to see teachers loaning out calculators when I observe at the middle school. In my school, using graphing calculators in our math class was a big deal. My school district never wanted social class issues to be apparent in the classroom. The way the teacher dealt with the graphing calculators was for those students who could not afford it to come and see her privately, and she would make sure to loan them one. |
Subject R10 favored cooperative learning as a way to reach diverse populations because of personal experiences in school.
| I have clear memories of those times in class when a classmate asks a question that the teacher just doesn't understand, but myself and several other students understood the asker's confusion and were dying to jump in and answer it. In a traditionally run class, the students don't have much opportunity to explain things to each other, but with group work cooperative learning can occur and thrive. |
Subject R11 found mathematics difficult to learn, and needed extra help. This caused her to consider availability for such assistance to be a valuable quality of a teacher of diverse populations.
| Being available to the students outside of class is important for those students who feel they need extra help in order to grasp an idea. . . . I personally found I was able to do mathematics in high school because I was given the option of being able to come in and get extra help on the parts I didn't understand. |
In some cases, the subjects were influenced by situations they observed or studied in their courses, even though the subjects were not directly affected. Subject R21, who is white, gained awareness of discrimination against African-American males while observing a class for C & I 101/219.
| In general, a teacher of any subject material can make the learning process more equitable by considering the individual needs and strengths of each student in the class. This includes making a conscious attempt to eliminate any personal biases that a teacher might have in regard to the differences in students. For example, even though my cooperating teacher at [name of school] claims that she is not biased, she usually disciplines the African American males most harshly. Therefore, constant reflection on ones actions as a teacher is very important. |
Subject R14 made a similar observation in his own high school.
| In many current school systems minority students do not have the chance to take any challenging, "higher level" math classes (I personally saw this in my high school where the AP classes were composed entirely of white and Asian students, and the remedial classes were just the opposite). This can be attributed to the tracking system and student course selection. |
Subject R6 reports learning about diversity in a University course.
| The idea of making math applicable to a diverse population goes far beyond just racial or gender equity. I think that teachers need to take into consideration students with disabilities. Whether these disabilities be physical or cognitive, teachers need to recognize and adjust for them. I took Special Education 218 this semester and it really opened my eyes to what teaching a diverse population entails. |
Subject R11 mentioned the same course. Subjects R6 and R11 were two of the six students in their section who mentioned physical ability or disability as a dimension of diversity. It seems likely that there is a connection between taking the course and expressing an awareness of the issue.
The subjects did not all express the same concept of what aspects of diversity are important, nor were they equally explicit about what was meant by diversity. One subject mentioned eleven of the aspects of diversity, while two other subjects mentioned only two aspects each. Some subjects concentrated on racial, ethnic, and cultural aspects, while others concentrated on learning styles and physical or mental abilities and disabilities. Many of the subjects mentioned personal experiences or connections to the types of diversity they discussed, indicating that these experiences helped form their concepts of diversity. Aspects of diversity that had been discussed in class were mentioned more frequently than those that were not.
It appears that the subjects' conception of diversity was strongly influenced by their coursework and their personal experiences. The influence of coursework is encouraging, because it implies that universities can create greater awareness of issues of diversity and equity by incorporating relevant material into teacher preparation courses. It also gives us reason to take seriously our responsibility to include a wide variety of diversity and equity issues in the curriculum. Students cannot be expected to generalize. If we use only a few aspects of diversity repeatedly in our examples, neglecting others, our students will learn that these are the only aspects of diversity that matter. In particular, issues of immigration status, religious affiliation, sexual orientation, and urban/suburban/rural differences seem to have been underemphasized in these subjects' education.
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When asked how to make mathematics relevant to diverse populations, most of the subjects responded by addressing issues of quality. While the subjects generally related these specific quality issues to equity and diversity, it also appears that they see a strong link between equity and general quality of the lesson. Implicit in this is the idea that there is no equity in evenly distributing bad materials; equity is promoted by improving the quality of the lesson for everyone.(Footnote 3) The subjects appeared to have some understanding of this concept, but did not articulate it.
Two of the quality issues addressed by the subjects are examined in more detail in the following sections. These are the extent to which the lesson related mathematics to real life, and the advantages and disadvantages of using cooperative learning. Other quality issues discussed by the subjects are not presented in detail here due to considerations of length and time. These issues include the advantages of discovery learning and active, hands-on exploration, the use of easily understandable language with a minimum of jargon, the variety of questions, and the fact that this unit represents a change from the traditional modes of mathematics instruction. The subjects noted that with students of multiple intelligences and diverse learning styles, variety in presentation is very important.
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Equity and Relevance
The most commonly mentioned strategy for making mathematics applicable to a diverse population was relating the mathematics directly to the everyday lives of the students.(Footnote 4) A common response to the question of how the unit fails to make mathematics applicable to diverse populations was that the lesson does not contain any problems relating to the outside world. The subjects recommended that such problems be added, with an emphasis on the fact that different students have different life experiences to which the lesson should relate.
In a sense, the unit was perfectly equitable as it was; there were no references to anything outside of the classroom, so there was no possibility for any group's experiences to be more or less represented than any other group's. Still, the subjects were nearly unanimous in their insistence on making the lesson relate to the students' "real life" experiences. Subject R16 wrote: "The most important element of making mathematics applicable to a diverse population is to relate mathematics to everyday life." Subject R14 agreed, saying: "To make mathematics applicable to a diverse population teachers must also use classroom situations and examples that are directly related to the lives of people from varying backgrounds." Subject R13 supported this point with a reference to a reading assignment from earlier in the course.
| Further, in the article, "Mathematics for All," Volmink made an excellent point that mathematics is not removed from life, but rather it helps to explain it. Therefore, making mathematics applicable to a diverse group of students should involve demonstrating how math explains certain concepts in life. |
There was some disagreement about what constitutes real life. Some considered the motion sensor work to be part of real life; the real live student moves in the room and sees a custom-made graph of that motion appear on the screen. Relating the graph directly to the student's movement, and encouraging the student to experiment with different motions to understand their graphs, was considered by these subjects to ground the lesson firmly in reality. This was the dominant view in Mr. Thomas' section. In the words of subject T13:
| This lesson may also help teachers make math applicable to diverse students because . . . it avoids any of the unrealistic "real world" applications that often accompany math activities. . . . Also, this lesson was based on an actual, rather than a hypothetical, problem. The distances and times measured actually occurred in the classroom and involved the student. Therefore, students should be able to see the importance of this application. They are not dealing with an imaginary object falling off of a building, rather they are dealing with their own velocity as they walk around the classroom. |
Subject R11 agreed:
| This assignment was also connected to real-life and this led to a greater understanding of the graphs and what they represented. By moving and seeing the graph in the screen, it was easier to understand that the graphs were distance over time and velocity over time. It was also easier to know what the graphs should look like because you could test your hypothesis with the motion detector and then you could see your results. |
Others viewed the classroom setting as too artificial and contrived to count as real life. They wanted the lesson to relate to the world outside of the school, and viewed it as having failed in this regard. This was the majority opinion in Prof. Ramirez' section. Subject R9 said: "I was disappointed that she [the researcher] had not stressed the reality of the derivative in our everyday lives." Subject R14 agreed:
| This unit also does not relate directly to the lives of the students. Instead, it somewhat follows the traditional abstract form of mathematics that we already know has failed to meaningfully [sic] in teaching mathematics to a diverse student body. |
Subject R4 thought the teacher should explain the connection between the lesson and the real world.
| In order to make math applicable for a diverse population, a lesson also needs to ensure that all people can use the math. One way to accomplish this is by making sure the problems being solved in class are problems that people with diverse backgrounds and needs will all need to solve. In the real world, people do not need to know the appearance of a graph of distance vs. time, so the exercise is not necessary in itself. It is important however as a tool to teach the concept of slopes, velocity, and differentiation. The teacher ought to explain this to the students so that they see that although they would not need to solve the given problems in the outside world, they will need to know the concepts. |
Many subjects sought to connect the lesson to the real world by adding word problems. Subject R8 expressed this view: "One way we could encourage under-represented students could be to include real world examples that incorporate different cultural views." Subject R2 agreed:
| Also, another possible suggestion would be to inject some more practicality into the lesson. This would enable a wider variety of kids to really relate to the unit, give them the security of knowing what math is being used, why it works and where it shows up in the outside world. |
Subject R3 elaborated:
| How many times have teachers heard students say, "When am I ever going to use this in my life?" This is a question that teachers should not hear in their classrooms because it is their job to make mathematics applicable to every one of their students. . . . But, in order to make this more applicable to a diverse population, it would help to set up a few word problems that integrated the relationship of distance and velocity so that students can see how these types of graphs would be used in a real world situation. As discussed before, the word problems would need to include topics with which the students can relate and be familiar. Also, if those types of word problems are given, then students will be able to bring in their own cultural experiences and problem-solving skills into the solution process. |
Despite general agreement that the lesson should relate to real life, possibly through the use of word problems, there were few practical suggestions for exactly how that could be done. In some cases, the attempt to bring in "real life" led to suggestions for replacing the concrete and immediate with the abstract and hypothetical, as when subject R14 suggested:
| This unit could be improved by making some of the problems relate directly to student's lives. For instance, instead of the instructions saying get a graph to come up that looks like this, have a problem where a girl named Yolanda has to go to different stores along a street to get a graph that looks like this. |
To relate more directly to the life of a particular student, we should not ask the student to personally move and make a graph, but instead ask the student to imagine the movements of an unknown girl along an unknown street? It appears that this subject was firmly convinced that it is necessary to relate mathematics lessons to life outside the school, but not sure exactly how to accomplish that goal.
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Equity and Cooperative Learning
The fact that the unit required the subjects to work in groups was frequently mentioned as a strong point, making the lesson more applicable to diverse populations. The same subjects often also described group work as a problem or potential source of problems. Subject R1 was one of many expressing this view.
| I also believe by structuring the activity so that students were allowed to work in small groups the goal of making mathematics applicable to a diverse population was obtained. . . . By letting the students work in groups you have taken away some of the anxiety which is often a strong feeling which is evolved when doing math. The group setting lets students feel at ease because they now have someone to direct their questions towards. . . . if the groups are not determined wisely then some students in the group may get left out and not fully participate in the activity. This could occur if one of the students in the group is very dominating and takes the workload upon themselves. |
The subjects felt that group work could have several benefits. Subject R4 emphasized the value of talking to peers about mathematics. "To be able to use math, both inside and outside the classroom, students will need to be able to communicate their ideas." Subject R11 wrote that group work provided increased opportunities for students to use the equipment, and a safe environment for students to express their opinions.
| The group work involved in the lesson made it applicable for more students. If it was done in front of the whole class, less people could be involved in the actual process of making the graphs. Many people are more intimidated in asking questions in front of the large group compared to small groups. In groups, each student's mathematical needs could be met, oftentimes by other members of the group. |
Subject R2 thought that the group work could help compensate for differences in mathematical backgrounds.
| [The packet's] structure combined with the group nature of the class insures that students at different levels of math knowledge had a more equal opportunity to complete the packet correctly. That was certainly the case in our group, where each of us participated and brought a better recollection of the calculus and graphical knowledge at different stages of the activity. |
Subject R3 saw group work as an opportunity for effective peer tutoring.
| Furthermore, students working together in groups strengthens this unit's ability to reach diverse students. According to Preston Dinkins in his article, Multicultural Teaching Strategies for Simplifying Mathematical Concepts and Principles, cooperative learning groups give students the opportunity to formulate their understanding in different ways. . . . The teacher can only convey ideas and concepts in a limited way, but with a diverse student population in the classroom, students can learn so much from each other. In our experience with this unit, it helped to have the whole group's input on the answers to the questions. |
On the down side, subjects feared that group work could allow dominant students to take control, leaving less assertive students on the sidelines. Subject R12 was one of many expressing this concern.
| As for personality, some students can feel threatened by working in groups. This unit calls for interaction between groups members and promotes discussion, which can be intimidating for quiet students. The outgoing students in the group might dominate the discussion and control the answers that are put on the handout packet. |
The solutions offered for this problem generally took the form of adding more structure to the activity, as subject R5 suggests.
| Group activities when structured with specific tasks for each member and specific group goals also promotes a sense of cooperation and provides a means for students to get to know each other personally. . . . In addition the group method provided students access to high tech technology in small non-threatening environment. Group work seemed to ease math anxiety . . . By assigning each group member certain responsibilities or even allowing the groups themselves to delegate the responsibilities helps ensure a more equitable activity. |
Subject R12 also suggested more structure. "In order to keep one student from dominating the teacher can assign each person a role in the group and after each question has been answered then each student's role changes."
Subject R8 was one of several who indicated that the composition of the groups is important.
| For example, when the teacher groups the students she should take into account the differences that exist in the classroom. She should recognize and respect which students work well together and which students cannot work together but also allow for variety in the groups and not always use the same groups for group work. |
Several subjects agreed with subject R18, who thought it would be best to pair students who have a firm grasp of the concepts with those who do not.
| A suggestion that could be used to help this situation is probably having the students working in groups such that those who understand the various concepts will be grouped with those who have yet to understand. This way, the students who already have grasped the concept will be able to reinforce their understanding by working with the other students while these students will be getting extra help to aid them in achieving understanding. . . . Group work pairing students of different levels could be very effective. |
There was one dissenting vote; subject R10 thought that the groups should be homogeneous.
| The student who is likely to take over the equipment should be grouped with students who are outgoing and likely to stand up for themselves. The slower students can be grouped together, and also given a chance to come back outside of class if they don't finish. |
With minor variations, the subjects appeared to be in agreement on cooperative learning. They felt that it could be very beneficial, particularly with diverse populations, but that there are some potential problems. Their ideas for minimizing these problems centered around adding structure to the activity, giving thought to the best groupings of the particular students in the class. If the teacher plans carefully to avoid the pitfalls, cooperative learning can promote mathematical communication skills, give everyone an opportunity to participate, provide a non-threatening environment for discussion, reduce the effects of differences in mathematical backgrounds, and facilitate peer teaching.
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When this unit was used in a first semester calculus class, I went to the lab before class, set up the equipment, turned everything on, and started the software. All the students had to do was press one button, which I demonstrated for them in the first few minutes of class. No previous computer experience was required. All students were successful in using the equipment.
This study used laptop computers which the subjects brought with them to class; the subjects had been using the laptops all semester. This prevented me from connecting the ULIs to the computers and loading the software before class, but it also meant that I was working with people who could be expected to do this for themselves. In fact, they did more than I had expected. I dispensed disks and packets, with the intention of leading the entire class through the brief set-up procedure once the materials were distributed. By the time I was ready to start instructing them, most of the groups had finished connecting the equipment and turning it on. Rather than disrupt their work, I gave individual assistance to the groups that had not managed to start the program on their own.
This had the unfortunate effect of giving the subjects the impression that their level of computer competency was necessary for this unit to be used. Many complained about the lack of the introduction that they so clearly had not needed. Some subjects related this to equity and diversity, saying that underprivileged students would be less likely than their peers from higher socioeconomic groups to have the necessary computer experience to feel competent with this lesson. Subject T1 was one of a very few who suggested that this unit could play a part in helping students to gain experience and develop confidence with computers.
| One big drawback that this unit could have is that it uses computers. Students will have variable experience with computers, which could put some students at a disadvantage. Students that have had limited, or no, exposure to computers may not feel comfortable doing this lesson. However, the more I think about it,the more I realize that this is actually a strength of the lesson. It does not require much computer use except for "pressing the start key". As long as the teacher is aware of particular students who have limited experience, the teacher can give more guided instructions. In reality, this lesson becomes a way of allowing all students access to computers. |
Most subjects focused exclusively on the potential barrier presented by the perceived need for computer competency. They seemed not to consider the possibility that this lesson could present an opportunity for the teacher to introduce economically deprived students to technology with which other students are familiar, thus reducing the differences in the students' experience and confidence, and thereby promoting equity. A few subjects recognized this aspect of the issue. The subjects who are able to see the potential opportunity in what at first appears to be a problem are likely to be the most effective in using this type of unit.
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Students with Disabilities
Many subjects wrote that, since the unit is based on motion, it would present difficulties for students with physical disabilities. Some subjects suggested that disabled students watch their classmates perform the exercises, but noted that this was not very satisfactory. Other subjects described practical ways of including disabled students in the motion, such as changing the scale on the axes so that hand motion would suffice to make a graph for a student who cannot walk. The problem of how to modify the lesson for use by blind students was more difficult. Fewer subjects addressed this issue. Of those who did, most concluded that the lesson simply was not suitable for blind students.(Footnote 5)
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Gender Differences
Many subjects said that mathematics has traditionally been taught and viewed in ways that make female students less likely to succeed. Subject T7 gave a popular suggestion for dealing with this situation: "The lessons should contain equal amounts of male and female references." Excessive use of sports-related examples was criticized. Most subjects addressing this issue felt that the best approach to gender differences was to treat all students alike, regardless of gender. In the words of subject T5:
| The inequality of gender can be overcome because it allows females the same opportunities as males. The females will have the same equipment and the same tools to work with. This will also make the mathematics more interesting and approachable to the females. As long as the teacher does not demonstrate any gender bias, this lesson will effectively not produce and inequalities in gender. |
This is in marked contrast to the responses of many subjects on the topic of cultural diversity, which is discussed below.
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Diverse Learning Styles
The subjects in Mr. Thomas' section placed more emphasis on diverse learning styles than did the subjects in Prof. Ramirez' section. Subjects in both sections agreed on the best approach to dealing with this aspect of diversity: vary the mode of instruction. They generally praised the unit for placing the students in an active role, in contrast to the passive role of the student in traditional mathematics instruction. Other ways of varying the mode of instruction were also suggested.
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Incorporating Diverse Cultures
The subjects in Prof. Ramirez' class wrestled with the question of what is needed to reach a culturally diverse population. Is it necessary to make specific references to each student's culture, to make that student feel included? Is it better not to make any cultural references at all, so that no culture can be considered to have been preferred? The unit presented in this study was written without conscious consideration of these questions. It did not involve any personal names or other explicit cultural references, but this was not by design; I simply didn't have any reason to put such references in. The unit was not contorted to remove cultural references, nor were extraneous references inserted.
In a sense, the unit could be said to be bias-free; all cultures were equally absent. The unit was based on the common human experience of physical motion. Some students saw this as an asset for teachers trying to make mathematics applicable to a diverse population. Subject R7 wrote: "The type of data we took was not biased toward any particular group. Everyone is familiar with moving back and forth at different speeds." Subject T17 agreed, saying:
| I think this lesson allows for diverse students to learn, and learn effectively. The graphs are not culturally biased, as any student with a pre-calculus background can understand them. Motion is a very basic principle, so the students easily can relate to moving, and how the graphs of their motion would be drawn. The questions asked are straightforward, and they do not seem to be from one perspective of another. So I feel that this lesson does well not to exclude members of a minority group, and thus can be very helpful for a teacher seeking to appeal to a diverse population. |
Even the technology was viewed as being culture-free, as subject R2 wrote:
| First and foremost among the benefits, the concept of a motion detector is something that really doesn't favor one background or another. Actually, it is a pretty universal--albeit universally unfamiliar--accessory in the classroom. |
Subject R6 concluded that the absence of personal names in the lesson avoided cultural bias:
| This type of activity helps rid most cultural biases. The fact that no examples with names were given was good because that sometimes is looked upon as discriminatory. Some cultures might think that names which are traditionally related to their culture are not used enough. Others might think a certain name is funny because it is not traditionally related to their culture. |
In contrast, several subjects objected to the lack of names and other cultural references, indicating that the unit was inadequately sensitive to culturally diverse students because it did not contain these references. Subject R16 wrote: "I find it difficult to understand how this unit makes mathematics applicable to a diverse population. The lesson does not relate to the students' different backgrounds." Subject R15 agreed, saying:
| The issue of how this activity fails to be applicable to a diverse population is a difficult one. I say this because there is really no interaction with personal issues in this activity. . . . There is really no way to make [the unit] apply or not apply to a diverse population because there is no interaction with the outside world. |
To remedy this defect, subject R14 suggested:
| For instance, instead of the instructions saying get a graph to come up that looks like this, have a problem where a girl named Yolanda has to go to different stores along a street to get a graph that looks like this. |
Subject R13 also suggested an exercise:
| Compare two people driving to a Latino-American Festival. One driver lives 20 miles from the festival, and the other driver lives 45 miles from the festival. The first driver can only travel 30 mi/hr and has to stop every 5 miles for 1 minute each stop because of stop signs. The second driver can drive 55 mi/hr and only has to stop twice for a total of 3 minutes. If both drivers leave at the same time, assuming there is not much traffic and the weather is nice, who will get to the festival first? |
Both of these suggested exercises drag a cultural reference into the statement of the problem, without making it intrinsic to the problem or the solution; neither the name of the girl nor the destination of the drivers has any bearing on the mathematics of the situations. These are fairly standard, "canned" exercises, with token cultural references, but they do not really belong in this lesson. Mathematically, neither exercise fits very well into the unit. Neither one advances the mathematical objectives of the lesson, and neither one uses the motion sensor.(Footnote 6) They represent an improvement only if one accepts that it is necessary to work some reference to Latino/a culture into the unit.
Recognizing the superficial nature of this approach, subject R15 hesitantly suggested:
| However, there are ways to incorporate issues of equity for a diverse population into an activity like this in your class. One way which is actually quite simple and shallow, but still goes farther into a diverse population than this activity does now, is to use word problems in the handouts that the students get. In these word problems, you could use examples which incorporate a diverse population. For example, you could say something like 'This culture does a certain dance during this celebration. Show how this dance would be registered on the motion detector.' Then the teacher would describe the dance on the handout and the student would, in effect, have to simulate the dance in order to put it into the motion detector. This may not be the best example because in some ways it labels a culture. However, used tactfully and accurately, it may be helpful in introducing students to a different culture. |
Unless the dance were performed by moving along a straight line, the motion sensor wouldn't give a meaningful graph. It is not clear to me how this exercise would advance a student's understanding of the derivative. The subject's goal in suggesting this exercise seems not to be related to the mathematics of the unit; the goal is "introducing students to a different culture". The subject acknowledges that this introduction is "quite simple and shallow", but seems to feel that some effort in this direction is necessary. It is interesting that the subject would suggest adding this sort of exercise to promote equity, after stating earlier in the paper: "One thing that this unit incorporates that is positive for issues of equity for a diverse population is that it excludes no one."
Subject R18 struggled with the question of whether it is appropriate to insert gratuitous cultural references that have nothing to do with the mathematics of the lesson:
| In terms of making the activity more culturally integrated, one can change the questions of the unit around to somehow include various cultural issues (this would take a lot of creativity on the part of the teacher). One drawback with this method is the trivializing of the culture and ending up with the "heroes and holidays" approach. Maybe a little pre-lesson on the culture at the beginning of the class would help, but would probably still be ineffective for this unit. |
Later in the same paper, subject R18 indicated that a "band-aid" approach to diversity in this lesson was not needed, since the unit was not injured:
| It does not appear that this unit has too much of a problem of being applicable to a diverse population. As mentioned before, the unit appears to be straightforward with no real way of directly excluding anyone. |
Subject R7 tried to reconcile the idea of including students by making specific references to their cultures with the idea that some lessons can convey mathematical concepts to all students without any explicit cultural references:
| Although I find it important to address all groups in the classroom, there are those activities that are just for students, not for African American students or Latino students, just plain old students. When it comes down to it, I think the most important thing to look at is whether the students learned what you expected them to learn. A lesson plan can be packed with diversity, but if it doesn't achieve the goal you have set out for, then it doesn't do anyone any good. |
It appears that many of these subjects equate making mathematics applicable to a diverse population with making explicit references to specific cultures. Several subjects stated that the unit was not biased, and none described any indications of cultural bias in the unit, but many still felt these cultural references are necessary. Even if it means inserting references that bear little or no relation to the lesson, these subjects want to see Yolanda and the Latino-American Festival.
The subjects in Mr. Thomas' section did not object to the lack of cultural references. Some of them commented on the absence of such references as a positive feature of the unit, since no culture was favored. Most of them did not address this issue at all. Only one suggested that cultural references might be a good idea, and he was ambivalent on that point. This difference in the sections' opinions on cultural references reflects the difference in their conception of which aspects of diversity are relevant to teaching and learning. Many subjects in Mr. Thomas' section were quite emphatic on the point that learning styles are more important than ethnic of gender differences. They felt that if lessons were planned with sufficient variety and quality to reach students of all learning styles, then equity would be achieved. By contrast, Prof. Ramirez' students saw ethnic and cultural issues as a major component of diversity, and so believed that it is necessary to appeal to the students' cultural backgrounds.
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These subjects had different views of diversity; some stressed individual differences, such as learning styles, while others stressed group differences, such as gender or ethnic identity. Their concepts of diversity appeared to have been influenced by their instructors, their coursework, and their life experiences. This indicates that faculty of teacher-preparation programs can expect to have an impact on their students by addressing diversity issues in the curriculum.
The subjects generally agreed that quality is an important way of achieving equity. Variety in presentation, relevance and applicability of content, effective use of cooperative learning, and other attributes of high-quality instruction were seen as vital to achieving equity in the classroom. This indicates that the subjects could be very receptive to a program which includes some consideration of diversity issues in a variety of different courses. Since they see strong ties between equity and quality of instructional techniques and content, they are likely to be ready to integrate the study of equity into other aspects of the study of education.
The subjects were divided on the necessity of cultural references and other appeals to the group identities of students. They agreed that where these references are used, they should reflect all groups, rather than just the majority or dominant group. Some expressed concern that they would not have the knowledge or resources to include appropriate references from cultures other than their own. More study on how and when to incorporate such references could be very helpful.
Several themes emerged from this study to suggest that teacher-preparation programs have a very real opportunity to expand pre-service teachers' understanding of equity and diversity. There is much work left to be done, but there is strong reason to believe that pre-service teachers' conceptions and awareness of equity and diversity issues are already strong and have the potential to become even stronger.
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Ernest, P. (1989). The impact of beliefs on the teaching of mathematics. In Ernest, P. (Ed.), Mathematics Teaching: The State of the Art, 249-254. New York: Falmer.
Gregg, J. (1995). The tensions and contradictions of the school mathematics tradition. Journal for Research in Mathematics Education, 26, 442-466.
Gutierrez, R. (1996). Practices, beliefs and cultures of high school mathematics departments: Understanding their influence on student advancement. Journal of Curriculum Studies, 28, 495-529.
Irvine, J. J., & York, D. E. (****). Teacher perspectives: Why do African-American, Hispanic, and Vietnamese students fail? In **** (Ed.), Schooling in Urban America, 161-173. ****
Secada, W. G. (1989). Educational Equity Versus Equality of Education: An Alternative Conception. In Secada, W. G. (Ed.), Equity in Education, 68-88. New York: Falmer.
Secada, W. G. (1991). Diversity, equity, and cognitivist research. In Fennema, E., Carpenter, T. P., & Lamon, S. J. (Eds.), Integrating Research on Teaching and Learning Mathematics, 17-53. State University of New York Press.
Thompson, A. G. (1984). The relationship of teachers' conceptions of mathematics and mathematics teaching to instructional practice. Educational Studies in Mathematics, 15, 105-127.
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C & I 101/219
Computer-Based Laboratory Demonstration:
Introduction to the Derivative
This CBL unit was not written with equity issues in mind. It may have implications for equity that the writer has not considered. Think about this unit and your knowledge of mathematics classrooms, both in your field experience and in your role as a student. Consider what equity means, how it could be achieved, and the effect this or a similar unit could have. Write a total of 4 to 5 pages (double-spaced) answering the three questions given below. You may write either three short essays or one long one addressing all three questions together. Please be specific. Support your answers with personal experience and/or research you have read. Grades will be based more on content than on form; the important thing is to think about the questions and explain your ideas.
1. What does it mean to make mathematics applicable to a diverse population?
2. In what ways could this unit help a teacher accomplish that goal?
3. In what ways does this unit fail to be applicable to a diverse population? How do you think the unit could be improved?
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This paper is intended to serve two functions: it forms part of your final exam in C & I 101/219, and it can help Lisa with research into pre-service teachers' concepts of equity in mathematics education as it relates to technology use. You are required to write the paper for class, but you are not required to participate in Lisa's research. If your paper is used for the research, it will be used in a confidential way. Only Lisa and your instructor will read your work. Your name will not be used in Lisa's paper. Your words may be quoted in sentences like "One student wrote . . .", or "Over half of the female students reported . . .", but not "John Smith wrote . . . " .
Please check one of the boxes below and sign this page. Attach it to your paper when you turn it in. Lisa will get only those papers with the first box checked. Your grade will not be affected by your choice. Thank you.
[] I give my permission for Lisa Murphy to read the attached essay(s) for use in her work as described above.
[] I do not want anyone other than my instructor to read the attached essay(s).
Signed _____________________________ Date ___________________
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Footnote 1
The writing assignments says it is to be counted as two short writing assignments, but this was changed by an announcement to the class.
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Footnote 2
If one accepts the commonly quoted assertion that 10% of the population is gay, lesbian, or bisexual, the likelihood of 40 randomly selected people all being straight is a mere 1.5%. Even using the more conservative estimate that only 5% of the population are of minority sexual orientations, the probability that 40 subjects selected at random are all heterosexual is only 12.9%.
[Several semesters after writing this paper I met one of the subjects in another context, and discovered that this subject is homosexual. The subject indicated that issues related to sexual orientation, including what to tell classmates, professors, cooperating teachers, and potential employers, had been of concern and had directly affected the educational experience. This is not unexpected. It confirms that some subjects were omitting all mention of sexual orientation diversity despite personal knowledge that this type of diversity does affect students and teachers.]
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Footnote 3
Secada (1989, p. 74) discusses the link between quality and equity.
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Footnote 4
Seventeen of the 21 subjects in Prof. Ramirez' section gave some form of this response, which may have been elicited by the repeated use of the word "applicable" in the instructions for the writing assignment.
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Footnote 5
I don't have an answer to that one either. I think I could make this unit appropriate for students who use wheelchairs, but I am at a loss for what to do with blind students.
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Footnote 6
One might imagine using the motion sensor in the Yolanda problem, but since the sensor measures only along a straight line, is doesn't really apply to going in and out of stores. Perhaps if Yolanda simply stopped to look into the store windows, one might manage to bring the motion sensor into this problem.
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I wrote this paper for a course called Research in Mathematics Education: Historical and Contemporary Issues in Equity. Due to the constraints of the schedules this course and my subjects' course, I got the first half of the subjects' essays only a week before my paper was due. The other half of the essays were not available until three days before my paper was due. I did what I could in the available time, but I think I could do a lot more with that data. I have no plans to pursue this, because at present I have no reason to put more time into it. If someone is interested enough in this to want to see more, please let me know. Thank you.
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This page last revised February 3, 1999