Ana Maria Klein
Finding each other in a hall of mirrors:
Negotiating goals and values in language
I want to begin by bouncing these words around a bit, attempting to distill the essence of their message while massaging them into my reality.
"Finding each other in a hall of mirrors."
How does this relate to me and my work?
I can easily weave these words into my actual work and my classroom experience with children. I can also like to relate it to other professionals, who like me, continuously wonder how it is that children learn and figure things out. My recent research experience with children has proven, once again, that when allowed to interact in a dynamic and nurturing classroom, they too find each other in a hall of mirrors. They engage in buoyant activity where, along with their peers they perform artful and playful chants and rhythmic sing-song. As they become more in tune with what they are doing--or, more concentrated, they drift off into a world of make-believe. In this trance-like stage they seem to know where to go within that hall of mirrors. As they solve problems, they start to envision a passageway that leads to the exit of that hall. If you have ever experienced this trance-like, heightened sense of knowing and acute perception, or watched a child in this total abstraction, you'll know what I mean.
A friend referred to it as her niece's "e-key-stage", meaning that while the little girl started playing, she would become so engrossed in what she was doing that she would begin humming in the musical key of E (Mi). In this stage, when the youngster reaches such a heightened sense of awareness, nothing else seems to matter. She is so totally absorbed coloring, piling things up, cutting and pasting. Fantasy meets invention, and hand in hand, they find they way into a reality interpreted by the child as a small theory, an invented word , a possible pattern to discern a solution to a problem-- a glorious "I did it."
I found, at my research site, that children are capable of finding solutions to the most intricate mathematics problems. They are also capable to sustain a high level of concentration when performing tasks that interest them. In so doing, they use gibberish or invented language forms that become place-holders for ideas or words. Children also seek you out to share with them. They want you to be their interlocutor so they can hear themselves out-loud. They will seek other presences to share wish or invent them.
This need for somebody to "tell it to" is a natural form of learning, often thwarted in classical classrooms where children aren't supposed to find their way in a hall of mirrors, let alone, negotiate values and language.
"Negotiating goals and values in language"
Magdalene Lampert (1990), who recently presented a paper at AERA (American Educational Research Association), a conference held in Montreal this spring. She addresses this problem and the need for reform in mathematics instruction in today's schools. There is not enough negotiation between students, teachers and between peers. Youngsters need somebody to help them interpret what they are learning. They need listeners and interlocutors. In her pre-service teacher program at Michigan State University, she insists on showing teachers how to cultivate non-judgmental discourse in the classroom. Educational mathematics researchers general (Resnick, 1989; Shoenfeld, 1989; Schifter & Fosnot, 1993;Walkerkine 1989) insist that mathematics is learned well when there are no wrong or right answers, but a rich, descriptive explanations. Students explain what it is they haven't yet written down on paper or think is a possible answer.
I'm going to share a short excerpt of a classroom episode where you will see this happening. As you watch, think about the role of invented language, the way the adult presence participates as interlocutor and the way the problem's solution unfolds. The title of this section is also an interesting back-drop as we explore values across generations in the excerpt. What is the teacher's role, and how does the student approach her?
Bernie beckons me to come over to see what he has done. In his mathematical problem, he has color coded a series of lines so that each line represents something. In his verbal explanation to me, he says that he "minused" several times until the number appeared. While he was "minusing" he realized what the number pattern was and could guess what the next number would be.
As you have probably noticed, Bernie was proud of what he was doing and I just happened to be there in close proximity. As he holds up his notebook for me to see his work, he says, "I minused this..." the first reaction of any classroom teacher would probably be to correct him telling him to substitute "minus" for "subtract", right? But that's not important in this particular classroom. What's important here is that Bernie is confident about what he is doing and is almost ready to produce a theory that may or may not work, and in explaining it to me or somebody else, he has a chance to understand his theory better. In actual fact, once he'd talked to a few people, he became even more engaged improving what he'd originally done because he'd realized an even better system worked.
I'd like you to think about what learning math was like for you as a youngster, and perhaps to inkshed after this morning's session about your feelings of adequacy, your coping strategies with demanding mathematical problems and with your problem-solving potential. Were you able to have this discourse with your teacher? Were you allowed to talk to your peers in class? I'd also like you to think about whether you've explained something and in doing so have figured out a better way.
Lampert, M. (1990).When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27, 29-63.
Resnick, L.B., & Ford, W.W. (1981). The psychology of mathematics instruction. Hillsdale, NJ: Lawrence Erlbaum Associates, Publishers.
Schoenfeld, A.H. (1989). Problem solving in context(s). In R.I. Charles & E.A. Silver (Eds.)The Teaching and assessing of mathematical education.
Schifter, D, & Fosnot, C. T. (1993). Reconstructing mathematics education stories of teachers meeting the challenge of reform. NY: Teachers College Press.
Walkerdine, V. (1989). Counting girls out. London: Virago Press.