Kay Toliver Math Program
This article was first published in the Journal of Negro Education in 1993. In this first person account, Kay Toliver lays out the fundamental principles that underlie her teaching methods.
Kay Toliver, East Harlem Tech, Public School Number 72, New York City
I have been teaching for over 25 years. I have taught mathematics and communications arts at East Harlem Tech, the junior high school component of Public School 72, a “school of choice” in New York City’s school district four. Prior to entering the junior high program, I taught elementary grades at P.S. 72. I believe I am a good teacher because I have had many successes over the years. Yet, when others ask me why I am successful, I often find it difficult to put my reasons into words. How does one explain something that has become second nature over the years? If I had to encapsulate my methods, they would be stated simply as “teaching and learning through listening, speaking, touching and writing.” Still, I probably could not explain what I mean by “good teacher” any more than I could explain what it is like growing up in East Harlem; or explain why, after all these years, I am not (as some put it) “burnt out.” I can state with conviction, however, that my methods stem from my deep love of teaching, part of which springs from my larger love for learning. After 25 years, I still see myself as a student striving for greater knowledge and a better understanding of both my particular role in life and life in general. Perhaps having this frame of mind gives me my longevity, but can it be taught? I can only hope that in offering some possibilities from my methods, I will inspire others to create their own.
Caring is the foundation of good teaching. In my classes, caring can and does take many forms. The first form, perhaps, is in the giving of my time. Many students are under the impression that teachers are not willing to invest extra time because they have seen that when the school day ends, too many of their teachers have other things to do and are not interested in spending this time with their students. Unfortunately, where this is the case, these teachers are missing an important opportunity. I generally stay after school or come in early, depending on my students’ preference, to help students with their homework or work with them individually if they have questions about the concepts covered in class. I see this as an opportunity for me to discover where students are lacking as well as to ensure they are not falling behind the pace of the class. The other side of this is that if I am giving of my time, I expect my students to invest some of their own. Recently, I have also begun to come to school early to work with students who want extra help but are unable to stay after school.
Caring also includes being a willing listener. My students know that they may come to me with problems, questions, or realizations, whether school-related or not. They view me as someone they can talk to, and I serve as counselor or preacher as needed. In turn, I share my own experiences with my students. For instance, when I came back from a summer of study, I told them of the difficulties I faced going back to school and concentrating on my studies. Additionally, my students all have my home number, and they are free to call me anytime. After all, their lives do not go “on hold” after the school day – neither does my status as a teacher. Because of this openness, my students are not hesitant to open up to me.
Often, it is only through a deep caring on the part of the teacher that a great student appears. In that regard I am reminded of a former student of mine named Xavier. When I think of Xavier, I see a talent who might easily have been missed. Anyone who is familiar with New York City’s homeless shelters knows that they are terrible, and Xavier was living in a shelter on 42nd Street when he first came to my class. (I believe his previous home had burned down, leaving his family with no other recourse than to seek public aid.) He was very introverted due to his difficult personal experiences, and I suspect that the other students secretly ridiculed him because he lived in a shelter. He would not talk to anyone.
Xavier began coming to school very early in the morning to get out of the shelter. One morning I found him crouched in the corner of the hall, and I told him that if he was going to come in early he could help me prepare for class. Thus, he became my “assistant” for that year. I would help him get breakfast, and soon he started to open up and began speaking to me. The other students, seeing how much I cared for Xavier, began to gravitate toward him and include him in their conversations. By the end of the school year, he was much less introverted, and it turned out he was both an excellent student and a wonderful artist. Consequently, I managed to get him into an art program for high school.
Students like Xavier are easily missed. Their brilliance is almost buried under the problems they are having in their environments and at home. They often appear to be unwilling to learn or hopeless to teach, and many slip “through the cracks” of the educational system, never reaching or even seeing their potential. Only a caring teacher can bring out their true intelligence and vitality.
Caring is the basis on which a class can be built. If students perceive that their teachers are there to help them and willing to assist them in getting through difficulties, it is then impossible not to teach them. If they perceive that their teachers are bored with their jobs, their subject matter, or their students, then it is no surprise if teaching becomes a chore.
High expectations are vital to success, especially for students who are regularly told they are "disadvantaged" and who, by the time they reach my junior high school class, may have the impression that mathematics in particular is a subject beyond their abilities. That is why I tell each and every one of my students, from the first day of class, that anyone can learn math. I put them on notice that if they approach the subject with an attitude of "I can’t do it" or "I don’t like it," then their prophecies will probably come true. I believe every student can succeed in mathematics, even if they have never before been successful. "Forget what has happened in the past," I tell them at the beginning of the school year, "This is a new day and we will work from here." I do not believe in failure. Mathematics may be hard to learn–it takes dedication and hard work–but I let my students know two things from the beginning: (1) I am with them to teach, and (2) I expect to be met halfway.
Some children enter my seventh-grade mathematics classes unprepared. Factors such as unstable households, crime, and drug-infested communities as well as poverty help to create students with uneven educational backgrounds. These situations can make it very easy for children to do poorly in school. However, I believe I would be doing a disservice to my students if I let them use their problems as excuses for failure or for not reaching their potential. On the other side of the coin, our school system does not always have money for the supplies, equipment, and other resources that would make teaching a little easier, but I never use this as an excuse for poor teaching nor do I allow my students to use it as an excuse for poor studenting. Despite these "disadvantages," I seek to provide them with the best schooling possible. I do not pity my students’ school or home situations because doing so simply will not achieve my objective, which is to make them more successful and able students. This may seem coldhearted, and perhaps it is, but I believe it is far more coldhearted to allow these children to "fall back" on their troubles and miss the opportunity to succeed.
This does not mean I am oblivious to their difficulties; I myself grew up in the South Bronx and East Harlem. I also realize that often my students lack money for paper and pencils and other essentials, but these are not reasons for failure. I can and do get them many of the supplies they need and I help where I can with their other problems because I believe the greater the problems surrounding the students, the more important it is that they triumph in their education and thus receive the key to a better life.
In a home, community, and school environment that is often chaotic, students find many reasons to miss class. However, I have found that the students who come to class always succeed to some degree. Therefore, it is my job to get them "hooked," to entrance them with schooling to the point that they anxiously look forward to coming to and participating in class.
Consequently, my math classes might seem a little odd to a newcomer; certainly, my new students find them so. My classes do not follow the mold of traditional math classes. The first oddity is that I do not use the textbook to teach the class. That, in my opinion, is my job. Though I consider textbooks valuable for homework practice or extra drill problems, I have developed a general order of lessons and I use the textbook only occasionally to support me in this. Secondly, my lesson topics are not as clearly delineated as is usual. I view the whole body of mathematics study as a unified subject in my class, and therefore I present the various subdivisions as interrelated facets of the same principles. For example, topics such as measurement, volume, square roots, fractions, and decimals may be covered in the study of geometry by applying each of these subjects to geometric shapes and formulas. At other times, students’ pursuit of a greater understanding of mathematics may take the class into a discussion of history, a tour of literature, or an exercise in language skills. Also, I make every effort to show my students the wholeness of mathematics and how it relates to every aspect of life in "the real world." I only present my students with problems that require a sensible, well-thought-out answer. I also point out that many problems do not have only one correct answer; indeed, it is often quite the reverse. Thus, they become problem solvers and learn the real value of using mathematics as a tool.
Many times students are still asleep when the first class starts at 8:00 AM in the morning, and I have to wake them up and get their interest. This may entail getting them to laugh, for if they are laughing at me, at least I know they are paying attention. For example, for my lesson on interior angles, I wear a "Zorra" (the female version of "Zorro") outfit made by my students. When I dramatically slash a big "Z" on the board, my students are immediately captivated and soon fascinated by the angles we begin to find in that letter. My use of costumes is not designed to turn math class into entertainment, but rather to add a little spice to the lesson and get my students’ attention–even if they are only looking to find out whether or not I am crazy!
However, mathematics is not only fun and games, as my students also come to know. The fun of it must be followed by hard work, but it does not have to be drudgery. My math classes are full of conversation, with increasing amounts of idea sharing and question posing as students begin to feel that the class is theirs as well as mine and that mathematics is a subject of real value to their lives.
Discipline as such is a foreign concept in my classroom. I have a reputation for accepting no bad behavior, which, I believe, must contribute significantly to my success. My students learn very quickly that the only thing that makes me angry is when they fail to do their best. I believe students who desire to learn will not present problems unless that desire is disappointed, so I strive never to disappoint them in that regard. As a result, my students never present discipline problems for me (though occasionally some do fall asleep in class because they have been "hanging out" in the streets the previous night). Even students who are considered "troublemakers" in other classes are wonderful students when they get to my math class.
The best form of discipline in my view is to keep students very busy, so I come to class ready to teach. As soon as they enter the classroom they are presented with a "Do-Now" problem that I write on the chalkboard. These Do-Nows are warm-up exercises to get them into a "mathematics frame of mind," and eliminate any time for nonsense. They include problems such as "see how many different shapes you can make on a geo-board" or "represent the number 43 using seven 7’s and any operations." One Do-Now asked students to take several multiples of 3 (3, 6, 9, 12, etc.), multiply them by 37, and describe any pattern they noticed. Student Maria Sanchez* wrote the following response upon solving the problem: "I discovered that for every multiple of 37, the answer is 3 digits all the same numbers and the more you multiply the higher the numbers." Another student, Lucy Reyes, a seventh-grader and Special Education student, submitted the following proof, replete with a six-step diagram as illustration, regarding the area of an isosceles triangle:
I made an isosceles triangle then a (sic) made an altitude line then I cut it in half and I had two triangles then I put it together and made a square and the area is A= 1/2 b x h.
This last example is especially impressive because it shows that the student discovered for herself how the formula was derived. Most students merely memorize this formula, but in this student’s case, the Do-Now heightened her understanding of the formula’s origin and made it much more meaningful to her.
*Pseudonyms are used throughout this article to protect students’ identities.
Assessment takes many forms in my classes; yet, because I do not expect my students to learn by rote, I do not assess by rote. I do not adhere to the rigid practice of numerically graded, paper-and-pencil assessment tests, nor do I prejudge my students’ performance based on cumulative report card grades or other teachers’ comments. I have found that such tests rarely provide accurate measures of my students’ progress, understanding, or true knowledge in mathematics; while report cards or other reports are often based on the children’s previous test scores or ability (or lack of ability) to memorize. Consequently, test scores are of secondary importance to my goal of developing in my students a basic appreciation for mathematics. Many public school systems are beginning to move away from reliance on standardized tests, or at least they are beginning to question the value of such tests. I think this is very much a step in the right direction.
I believe that constant contact and communication are the best tools for monitoring and assessing student performance. They are also my tools for helping students improve their performance. Therefore, I try to create a classroom atmosphere in which students feel comfortable about answering problems, whether they are right or wrong. According to my philosophy, there is no point in a student finding a correct answer if he or she has no idea why the answer is correct or does not know how to prove it is right. In daily classroom work, my students know that they must justify each answer with a proof. Conversely, the process by which students err is equally as important as that by which they succeed. I am therefore very interested in how students arrive at wrong answers (an interest that often confuses my new students) because I believe that uncovering the reasons for incorrect answers is the key to helping students find the correct answers.
Hence, my tests are never of the "fill-in-the-blank" variety; rather, they are performance based. The result, however is that students’ test papers are often as diverse as their personalities. For example, some students respond to test questions with step-by-step directions explaining the processes by which they arrived at an answer; others write their answers in narrative form or compose a story about the problem. They are also required to illustrate their answers with diagrams where applicable. This flexibility allows students to show their creativity along with their understanding. Though students are graded on the number of correct answers they provide, their ability to explain the processes through which they arrived at those answers is far more heavily weighted. Class grades are based on students’ performance test scores and on the quality of the classroom and homework assignments they submit.
No student can expect to sit quietly through a semester in my class and receive a good grade at the end of it. However, any student who participates and produces excellent work will receive an excellent grade. My goal is to bring about a change of attitude toward mathematics. I want my students to learn that mathematics is not an impossible subject; that it is not something that belongs only in school or in a classroom. I know my students are succeeding when they begin to see mathematics in the world around them. I know they are making progress when they begin to write sensible explanations to accompany their answers. If I can change their attitudes and make mathematics a subject they truly understand and for which they gain an affinity, I know that good test scores will follow.
Success in the middle grades often sets the precedent for children’s success as they move into high school. Yet, many parents of middle school-age children begin to take less interest in their children’s education during this critical period, perhaps because they believe these children are at last big enough and old enough to take care of themselves. This could not be further from the truth. Children entering middle school must walk a difficult line between childhood and young adulthood, and parental support at this juncture is extremely important. Many of the parents of my students have told me that they cannot help their children with my homework assignments because middle school mathematics is a subject they do not understand. Others are afraid of mathematics and their math anxieties are carried over to their children. However, I want parents to understand what I do in my classes; more importantly, I want them to gain an appreciation for mathematics themselves. Their help on homework projects is always welcome, but even if they are not able to help their children with homework, a positive attitude on the part of parents can make a world of difference to how children approach the subject. A parent who is supportive and who actively contributes to a child’s education usually increases the self-confidence of the student immensely.
To assist parents in this goal, I instituted "family math" sessions during parents’ meetings. During these sessions I present "nonthreatening" activities to involve children and their parents in joint mathematics learning. I also encourage parents to call me at home to talk about any problems or difficulties their children are having in my class. Even when the reasons for the calls are very minor problems, I always appreciate hearing from parents because, to me, education is a mutual effort. It is always most effective when parents and teachers work together. Unfortunately, many parents are accustomed to hearing from teachers only when their child has done something bad. While I do not hesitate to call a student’s parent if the child has been regularly absent or if some point of behavior should be corrected, I often call parents just to share good news about their children’s performance. I am generally able to achieve a happy medium in my communications with parents by sharing with them "both sides of the coin" regarding their children.
Perhaps the way in which my approach most diverges from the norm is in its emphasis on blending history, culture, literature, writing, and other subjects with the study of mathematics. As I have stressed before, mathematics is not an isolated subject. If students are ever to view it as a tool for their futures, it will be because they understand how mathematics can be used in their lives and how it relates to their daily activities. Thus, I require my students to record, in individual journals, their thoughts about each day’s lesson. In these journals they are to write about any difficulties or successes they may be experiencing in a classroom activity or note anything else they wish. For instance, sometimes, when the class is in the middle of figuring out a problem and it appears that several students are at the height of frustration over it, I tell the class to stop, pull out their journals, and write down how or what they feel about the problem. I have noted a wide range of responses, and have discovered that many of my students have difficulty expressing themselves in writing. What better way, in my view, to give them practice in expressing themselves on paper than to have them write about their troubles and thoughts in mathematics class?
Some will write, "I’m so frustrated and I wish she’d go on," or "I can’t stand this problem–I don’t like what she’s doing." Others might make comments about my zany outfits (i.e., "Why does she dress up like [Zorra]?") or about my teaching methods. Others, such as Anna Colon, wrote the following entry in her math class journal:
I learned that a mathematical sentence is a statement that states a fact, is judged to be true or false, contains an idea. This is called logic. Logic is thinking, reasoning, and using common sense. Ex.: She fell from her bike is …an open sentence. Another ex.: A non-mathematical sentence is 6 + 2. The purpose of using logic is in every day life because we are always thinking.
Indeed, the point of this journalizing is to get students to write! In addition to providing writing practice, however, the journals also provide me with wonderful assessments of how well students understand my lessons and how well my lessons are reaching them. By collecting and reviewing the journals once a week, I can find out very clearly how well my students are doing. Their comments often show me those areas about which students are confused or do not fully understand. They reveal gaps in instruction that I might have otherwise missed, thereby allowing me to adjust my teaching methods accordingly.
Literature often provides me with an excellent means of introducing mathematical topics and motivating students to discuss them. Students frequently read intriguing poems and short stories in my class as part of the mathematics lesson. For example, a "Dr. Seuss" story on the creation of a mysterious substance called "oobleck" provides the entry for a lesson on complex fractions. After reading a short excerpt from the story with my students, I introduce the idea of a complex substance; from there, I introduce the concept of complex fractions as it relates to the story line.
I frequently require students to give oral presentations about mathematical problems because I have found that children can often "go through the motions" of solving a problem without having any real understanding of why the operations they apply work. If a student can coherently and confidently discuss a mathematical concept or process, then I am more certain of their understanding of the solution they give for a math problem. I also assign research projects on mathematics and mathematicians so that my students can gain background information on the lessons at hand. In this way students come to learn that mathematics has existed for many thousands of years, and that it has developed in various ways through diverse cultures. Awareness of the history of mathematics and of the great minds who have made outstanding contributions to the field lends the subject a fullness and an overall importance it may not otherwise have.
Not surprisingly, my students often ask me just what class they are in; no other teacher has made them read and write in mathematics class, some protest. In response I merely tell them that the world is changing, and they simply will have to adjust. However, I have found that when students begin to succeed in my mathematics class, they also start doing better in their other classes. Often as a result of my class, students’ attitudes about school, their levels of competence in all areas of study, and their confidence in themselves as students begin to change for the better.
I teach in a community where the students do not hear a lot of positive comments about the area in which they live. Our school building is usually referred to by visitors as a "relic." To counter these negative assessments, I try to show my students that there is a wealth of mathematics in our neighborhood. I developed a lesson called the "Math Trail" to give students an appreciation for the community as well as an opportunity to see mathematics at work.
To create a Math Trail, the class must first do some research on the history of East Harlem. Then, they are instructed to plot a course, starting from the school plaque in the lobby of the building, that leads the class through the community and back to school, with stops along the way to visit several interesting sites and create math problems about various real-life situations. For example, on one such walk, students created a math problem centered around the number of times a bus stopped at a particular bus stop in one hour. Another concerned the ratio of "gypsy cabs" (freelance taxis) to Yellow Cabs in the neighborhood. Other problems examined the geometric shapes and the angles on the buildings we passed. One set of student-generated Math Trail problems instructs students to visit the school yard and asks questions about the mathematics to be found at that site:
1) There are 14 steps on the right hand side of the ramp. If I counted the steps every time I went up and I counted 126 steps, how many times did I go up the stairs?
2) How many steps does it take to walk from the bottom ramp to the fence across the yard?
3) How many feet of fencing is needed to cover the perimeter of the yard?
4) If the ramp takes up two-fourths of the school yard, what fractional part of the school yard is left?
To culminate the Math Trail exercise, the class prepares a book, complete with photographs and illustrations, of their Math Trail problems. Students must proofread their work rigorously to ensure that their problems are clearly written as well as mathematically interesting. Thus, the importance of grammar and spelling in expressing mathematical situations is highlighted, and students’ skills in all three areas are simultaneously enhanced.
I have discovered that some students who are not the best test takers have a very good understanding of how to apply the mathematical concepts involved in these Math Trail exercises. I recall one girl who pointed out two triangles on a rooftop and explained to me that they must be similar. When I asked her what one would have to know to prove this, she explained that the sides would have to be proportional and angles congruent. She then went on to explain that the angles would remain the same regardless of the size of the triangle. Imagine my surprise to hear this from a girl who had failed three-quarters of the tests I had given in class! I knew then that she had really learned a great deal of mathematics and learned it well enough to apply it in a practical fashion to the world around her. With other students, I have noticed the reverse. They do well on tests but have difficulty applying what they have studied. Math Trail exercises force these students to make a connection between the ideas they memorize for the tests and the real world. However, as one group of students wrote in the introduction to their book of Math Trail problems, the purpose of this experience is "to prove that the classroom is NOT the only place to learn math."
In coming semesters, I hope to bring parents along on the Math Trail as guides and participants. I believe it will be another effective way in which parents can gain a greater appreciation for mathematics with their children.
A few years ago I initiated math fairs at P.S. 72. These events are similar to science fairs but involve students in creating and displaying projects relating to mathematics. The first such fair was an opportunity for my eighth-grade students to create original projects that would showcase the knowledge they had acquired in eight years of math, with an emphasis on the topics they were learning in the eighth grade. Participants had to be able to explain thoroughly the mathematical theories and concepts behind their projects, which were placed on display at the school so that students from the lower grades could examine the older students’ research. Several students created mathematics games such as "Dunking for Prime Numbers," "Fishing for Palindromes," and "Black Jack Geometry." Some did research projects. One girl researched the Fibonacci sequence and made a display of examples of this sequence’s occurrence in nature in certain leaves, pine cones, and so forth. The next year, at the request of my principal, I expanded the fair to include all grade levels, even down to the pre-kindergarten students, who made a display on sorting and colors. The event has developed into a week-long celebration of mathematics, through which students of all ages learn by doing and creating. It provides a wonderful way to introduce younger students to the excitement of mathematics.
For the past five years I have worked part-time as a staff developer for my school and district, giving workshops and sharing with other teachers information on the ways I have been successful. I am also pleased to be working with Jaime Escalante and the Foundation for Advancements in Science and Education (FASE) to create a multimedia computer product that will allow Escalante, me, and one other teacher to disseminate some of our successful methods to teachers across the United States. The product will be designed to relate various pre-algebra topics covered in the middle grades to applications in the real world; for example, by helping teachers develop projects like the Math Trail for use in their own classes. It will also include video footage drawn from Escalante’s public television network educational video series, "Futures," which showcases professionals who use mathematics on the job.
In my own future, I hope to become more involved in training teachers. Though I cannot imagine fully removing myself from the classroom, I hope I will one day have the opportunity to pursue projects like these full-time. In addition to teacher training, it is my sincere hope that I will have the opportunity to recruit more minorities into teaching mathematics. Specifically, I would like to be the link that brings inspired young people of color into the field of mathematics teaching; yet I have found that minority students often do not want to be teachers for one reason or another–most would rather go into business or pursue other careers. Still, teaching contributes to every career, and good teachers are the backbone of any field. While I do not believe ethnic minority students must be taught by persons of like ethnic background, I do believe it helps for students to see someone like themselves in the role of teacher, someone about whom students can think, "If she (or he) did it, maybe I can, too." Ultimately, however, it is the spirit teachers bring to class, not their skin color, that determines their success. Beneath all methods and rationales, a teacher must have the desire to teach. His or her ultimate goal must be that of helping students achieve understanding, and there are as many ways of doing this as there are "good" teachers.
I have wanted to be a teacher since I was in second grade and have never regretted my choice. I feel very privileged to be able to open up new worlds to the bright minds of tomorrow. My rewards are many and they increase with every student who succeeds. My hope is that teachers will find a few really useful ideas in what I do and will be able to incorporate these into their classes. If my experience and success can help them in this way, I will feel I have made a valuable contribution.
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