by John Kellermeier. Published in: (1996). Feminist Teacher, 10(1), 8-11.
I teach statistics in a math department and also teach an introductory women's studies course. A while ago, while I was teaching my women's studies course, I put students into small groups to discuss our readings on sexual abuse and rape. While the students were talking in their small groups, I mentioned to my teaching assistant, Luanne Williams, some of the problems I had been having that day in my introductory statistics course getting students to talk. I told her that I attempted to make my classroom a riskable classroom, that is, a safe place for students to risk doing mathematics. She suggested a parallel between the way people who were sexually abused act (as we had been reading about) and the way my statistics students were acting. In other words, she was suggesting that I look at my statistics students as people who were math abused. While persons who are survivors of sexual abuse need a safe place to come to grips with what happened to them, and to learn to take the risks of forming relationships with people, survivors of math abuse also need a safe place to relearn how to deal with mathematics.
Speaking of children who are sexually abused, Sears (1991) describes the emotional damage done by the violation of trust. She says, "Over time they frequently develop a sense that they can do nothing 'right.' They withdraw into a world in which they must be able to do everything and never ask others for help. They can't trust or ask adults to be nurturing." For many people a similar thing happens in the math classroom. When only the "right" answer gotten by the "right" method is given any credence, when questions are labeled "stupid," when saying, "I don't understand" gets looks of annoyance and disgust, when failure to give the "right" answer in quick fashion earns chastisement and even physical punishment, students learn that they can do nothing "right." They learn that they cannot trust math teachers to be nurturing--of them or their mathematical abilities. Just as sexual abuse interferes with one's ability to develop sexually, math abuse interferes with one's ability to develop mathematically, that is to further one's experience with and learning of mathematics. Sears says "the process of recovery from sexual abuse is one of retrieving one's personal power and pride of self." In a similar way, overcoming math abuse involves a process of personal empowerment and pride within a math classroom or environment. While I do not mean to imply that math abuse is the same as sexual abuse, the analogy does offer insight into the effects of math abuse.
That math abuse exists is evident by the level of math anxiety in our students and in the public. Rodgers (1990) describes studies that show the level of "maths anxiety" among the population in Britain. In the U.S., Sheila Tobias (1980, 1987) has documented math anxiety among college-age and older students. The report "Everybody Counts" from the National Research Council (1989) states "mathematics is seen not as something that people actually use, but as a best forgotten (and often painful) requirement of school. For most members of the public, their lasting memories of school mathematics are unpleasant--since often the last mathematics course they took convinced them to take no more." Among our students, math anxiety seems to be most prevalent among general education students. They make such comments as "I was never good in math" or "I can't do well in math." Further discussions usually show that they were taught to believe these statements in earlier math classes.
I propose that college mathematics teachers recognize that many of our students, particularly those in general education mathematics classes, come to us with a history of math abuse. This suggests to me the need to create the riskable classroom. Students who were math abused need to relearn how to relate to mathematics, to relearn how to learn mathematics. They need to re-establish their own personal power in encountering mathematics. This requires the ability to take "mathematical risks." In doing and learning mathematics, mathematicians readily take "mathematical risks." That is, they use conjecture, guessing and trial and error. According to the mathematician Halmos (1968), "the mathematician at work makes vague guesses, visualizes broad generalizations, and jumps to unwarranted conclusions." Buerk (1986) describes how many students view mathematics as "a discipline of rules that was not creative and which was cold and distant" while she herself "value(s) the personal, creative and intuitive side of mathematics." In order to be successful in their encounters with mathematics, math abused students must learn to engage this creative and intuitive side of mathematics; and in order to do this they must be willing to take these "mathematical risks." A mathematics teacher can help this process by providing a nurturing, supportive environment where students are encouraged to express themselves and find their own personal power and knowledge. This is what I call the riskable classroom.
The riskable classroom is particularly important to women and people of color. Mathematics is still viewed as a white male domain. In fact, it is a white male domain. Roughly three of every four doctoral degrees in the mathematical sciences awarded to U.S. citizens go to white males, and only about one in a hundred go to women of color (National Research Council, 1989). People of color and all women are more likely to have been told, either directly or indirectly, that they "can't do math." Women of color are most at risk from this.
The following are some strategies I have used for creating the riskable classroom. These are strategies that have worked for me and my students. My focus is always empowering the students to feel good about their own abilities to learn and discover mathematics and to help students give birth to their own knowledge.
Regardless of class size, I learn all my students' first names and call them by that in class. This makes students feel more personally involved since they realize that they are not just another face sitting in my class. It gives them a sense that I care about them individually and their learning of mathematics. I also make a point of telling my students I am going to learn their names. It often takes me to the middle of a semester to fully memorize all the students' first names, but students are aware of the change in classroom dynamics when their instructor sets out to learn their names.
I also ask students to call me by my first name. I realize that as a white male I have access to power and privilege that may make it more easily possible for me to "give away" some of that power.
The point is to break down the usual view of math teachers (or mathematicians) as more powerful or important or smarter. This use of first names on both sides of the classroom introduces a more human aspect to the classroom and breaks down the authority structure.
I start out a new class by first briefly going over the syllabus and giving the information needed for grades, etc. Then I hand out and have students read the Stainless Steel Fence, (Buerk, 1986).
The Stainless Steel Fence
And on the eighth day, God created mathematics. He took stainless steel, and he rolled it out thin, and he made it into a fence forty cubits high, and infinite cubits long. And on this fence, in fair capitals, he did print rules, theorems, axioms and pointed reminders. "Invert and multiply." "The square on the hypotenuse is three decibels louder than one hand clapping." "Always do what's in the parentheses first." And when he was finished, he said, "On one side of this fence will reside those who are good at math. And on the other will remain those who are bad at math, and woe unto them, for they shall weep and gnash their teeth."
Math does make me think of a stainless steel wall--hard, cold, smooth, offering no handhold, all it does is glint back at me. Edge up to it, put your nose against it, it doesn't take your shape, it doesn't have any smell, all it does is make your nose cold. I like the shine of it--it does look smart, intelligent in an icy way. But I resent its cold impenetrability, its supercilious glare.
This reading makes an excellent spring board for discussing how students feel about doing and confronting mathematics. I then hand out and discuss the Math Student's Bill of Rights (adapted from the Math Anxiety Bill of Rights by Sandra Davis, in Donaday & Auslander, 1980).
Math Student's Bill of Rights
I usually combine this with my first attempt to learn students' names. I have each student in turn give me her or his first name and their response to the Stainless Steel Fence and the Math Student's Bill of Rights. Students always respond favorably to these handouts. They are often then willing to express their fear or dislike of math. From these initial comments students begin to learn that the classroom governed by the Bill of Rights will be a safe place for them to experience and learn mathematics. In fact, Ayers-Nachamkin (1992) describes a response to the Bill of Rights by a traditional math instructor that gives "a very clear measure of just how empowering this document is for the student and how very threatening it is for the patriarchal instructor."
I end the first class by emphasizing that I believe in the Bill of Rights and intend to run a class with them in mind. All of this is to set a tone on the first day of class.
I have students hand in a one page journal entry each week. I tell them they can write on anything they want. I give them a few questions to work from if they wish.
These journals help to keep communication open between myself and my students and gives students a low risk way to ask questions. As long as I makes comments on the journals before returning them, they also serve to show the students that I care how well they are learning. I have found that this is true even if I only make short comments such as "Good" or "Great Idea."
At the end of every class I have students fill out a sheet of paper that has a space for their name, the date and two questions.
Besides providing me with a way to take attendance, this helps to keep channels of communication open. It also gives students another low-risk way to ask questions. It keeps me apprised of the material that they know and that with which they are still having trouble. The language they use in telling me what they learned also helps me to see how well they understand. I usually answer the questions asked at the next class.
I lecture as little as possible and only as a last resort when everyone is lost on some topic. I expect students to read about new material before it is discussed in class. Then we can develop new material together. I do this by directing students to discover it. This is where a class that is willing to speak up is important.
Recently a student in a journal told me that she liked the way I was running class. I had been asking for volunteers to put problems on the board each day. As I came into class, I routinely found two to five problems already waiting to be reviewed. Then I would go over the problems and review each one asking students to comment, asking the authors to explain what they had done and so on. The student commented in her journal, "It's almost like we're teaching ourselves."
I give verbal credit for students' ideas. Usually in working through an example problem, I will have the students tell me the steps to do. Then later in the problem when someone says, "I don't understand. How did you get that step?," I can go back to the student who first told me what to do and ask them to explain why they told me to do this. In this way, students get credit for their thoughts and contributions and are continually explaining mathematics to each other.
I give students two extra credit points for putting a problem on the board. I emphasize that they will get their credit regardless of whether the problem is correct or incorrect. I suggest that, in fact, they should put up the problems for which they have only partial solutions. In this way, I emphasize the value of the process of working out a problem and the value of taking risks in problem solving.
I encourage students to form study groups and to work together on take-home quizzes. In class, I will often have students work in groups on problems. When I hold office hours in the college learning center, I will often have a group of students asking questions. I encourage students to ask each other their questions and to answer each other.
I give stamps or stickers for 10s (perfect scores) on take home quizzes and for A's (90 or more points) on tests. I find that students get very excited to get a sticker or stamp on their tests and quizzes. It gives them a sense of self esteem. Several students have even reported putting their papers with stickers on their refrigerator doors or even sending them home to their parents to put on their refrigerators. Often this takes students back to the feelings they had in early grades when learning mathematics was still exciting. And in the end, stickers and stamps are fun! Learning best takes place when people are having fun (Rodgers, 1990).
Red ink has had so many bad connotations that I never grade with red ink. I have a collection of pens in a variety of colors: pink, lavender, orange, green, gold, brown. I use these colors for grading.
Whenever I give an exam, I have students prepare one sheet of paper to bring to the test with them. This alleviates the need to memorize formulas. But more than this, it gives them a way to focus their work, to organize and realize what they know. Students will routinely put together masterpieces of "cheat sheets" complete with color coding. Then they find that they hardly need to use it. They discover that the process of creating the sheet gives them the knowledge and the confidence to work through a test.
After each in-class exam, I give students a chance to earn a rebate of up to 25% of the points lost on the exam. They must rewrite the entire solution of any problem for which they wish to receive rebate points. The reworked solutions and the original graded exam are handed in during the next class period. Again this emphasizes that we are always in a process of doing and redoing in order to learn mathematics. Risk taking is acceptable because we can rework our errors.
Belenky, et al. (1986) says, "Midwife-teachers encourage students to use their knowledge in everyday life." Relevant word problems, examples taken from the daily lives of my students, do just this. Students are more willing to talk about problems when they feel they have "a real face," as one student said. I have been working as well on using word problem content for curriculum inclusion (Kellermeier, 1992, 1993). This also gives students a sense of the relevance of statistics to their own lives. They feel that these word problems represent the important issues of the times, and consequently, they are more willing to engage the material.
I do not use examples prepared ahead of time. Instead, I ask for questions from the students and do problems that students ask to see. In doing this, I model problem solving by asking students to direct me. If they go off path (or try a different path), I let it go. Eventually, someone will ask a question about it. This allows me to model looking over one's reasoning to determine if there are any problems. Sometimes this will lead to ways of solving problems that I never occurred to me.
For example, in a recent semester students showed me that there was another way to graph data from contingency tables other than the way I was presenting. This was developed during a discussion of a graph that was in error.
These are some of the strategies I have used to create the riskable classroom. This concept of the riskable classroom is akin to the connected classroom and the midwife-teacher developed by Belenky, et al. (1986). They say,
Midwife-teachers "assist the students in giving birth to their own ideas, in making their own tacit knowledge explicit."
"The midwife-teacher's first concern is to preserve the student's fragile newborn thoughts, to see that they are born with their truth intact, that they do not turn into acceptable lies."
"Midwife-teachers focus not on their own knowledge (as the lecturer does) but on the student's knowledge. They contribute when needed, but it is always clear that the baby is not theirs but the student's."
In the mathematics classroom it is necessary for the midwife-teacher to address attention to creating an environment where "the student's fragile newborn thoughts" can even be expressed. Most of the strategies I have outlined above have evolved out of an attempt to bring connected teaching into my statistics classroom.
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