Teacher of Mathematics Issues for Today and Tomorrow : Proceedings of a Conference (1987) / Chapter Skim
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THE VISIONS AND THE CHALLENGES
THE VISIONS AND THE CHALLENGES
Pages 17-64
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Cooney University of Georgia As I understand the central issue before us, we are concerned with the question of how the National Council of Teachers of Mathematics' (NCTM's) Curriculum and Evaluation Standards for School Mathematics*
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have argued that observational studies that detail carefully what happened in classrooms during the modern mathematics movement are virtually nonexistent, thereby making it difficult to determine just what the nature or effectiveness of the reform movement really was. These points provide some historical perspective for our current situation and some sense of the problems associated with previous reform movements.
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on the achievement of middle school Japanese students underscores further the need for reflection on our middle school mathematics programs. Miwa presented the following results: Table 1 Achievement Items for Japanese Fifth- and Sixth-Grade Students Fifth Grade Find the value of X which satisfies each X x 4 - 2 - 6 (85.8% correct responses)
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The Cambridge Conference on School Mathematics (1963) focused on curricular reform by addressing the sequencing of topics, but dismissed the practicality of classroom teaching and related teacher education problems as issues that could be resolved when the time came to do so.
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A humanistic orientation emphasizes the teacher as a decision-maker who determines what mathematics students are capable of learning and what strategies are appropriate, given the mathematical maturity of the students. Can you imagine any greater task for teacher education than educating teachers to make such decisions?
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44.) In the 5-8 standards, it is stated that: Teachers need to provide a caring environment in which students can feel free to explore mathematical ideas, to ask questions, to discuss their ideas, and to make mistakes.
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The implication for teacher education in educating teachers to teach as suggested in the Standards is, in my mind, the ultimate challenge for mathematics teacher educators. To accomplish this, two significant obstacles must be addressed: teachers' conceptions and students' conceptions of mathematics and the teaching of mathematics.
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Similarly, preservice teachers have certain conceptions about how they ought to teach mathematics. Secondary preservice teachers provided the following diary entries in reflecting on their own teaching experiences: Activities must be done in a class in which the teacher has total control because we all know that, when unruly children are given a chance to move around and talk in the classroom, they will go crazy and no learning will take place.
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A common perspective among preservice teachers seems to be that teaching mathematics is primarily a matter of providing a broadcast. The most prevalent word used by our secondary mathematics methods students to describe their teaching is "present." "If only I can find the right way of presenting mathematics," says the intern, "life in the classroom will be o.k." When the presentation goes awry, as it typically does for the neophyte, it is easy for the preservice teacher to reach the conclusion that students lack sufficient internal motivation to receive the broadcast; their antennas are not up.
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The Influence of Students' Conceptions A second factor to be considered with respect to realizing the Standards is the students' conceptions about what constitutes appropriate teaching of school mathematics. In a case study I conducted with a beginning mathematics teacher (Cooney, 1985)
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So, argues Fletcher, is it the case with teachers of school mathematics. They, too, must have a general knowledge of mathematics.
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It is difficult to imagine that any other issue could speak more directly to mathematics teacher education than that of how the philosophy set forth in the Standards can be engendered in teachers. The Standards call into question how much we "lie" to students when we teach mathematics, particularly in the face of the finding from the Second International Mathematics Study that 50 percent of the eighth-graders surveyed believed that learning mathematics is basically a matter of memorizing rules.
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What is in question is the practice in mathematics teacher education of dichotomizing the study of advanced mathematics on the one hand and of general pedagogy on the other so as to limit the teachers' mathematical experiences to only those involving the study of wholesale new mathematical ideas. Teachers need experiences constructing the same mathematics that they will be teaching.
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It provides the basis for teaching mathematics in a way consistent with Goffree's fundamental use of realistic textual material. Yes, teacher education should consist of substantial study of higher mathematics and of general pedagogical techniques that ^ emphasize such things as maintaining an appropriate learning environment, preventing or dealing with discipline problems, and being aware of what students are doing, that is, Kounin's (1970)
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Such a conception of teacher education requires considerable introspection, determination, and a good dose of patience and fortitude on the part of us all. The stakes are high, but the rewards are many.
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S "Preservice Secondary Mathematics Teachers' Knowledge about Teaching Mathematics and Decision-making During Teacher Training." Doctoral dissertation, University of Georgia, 1982.
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Miwa, T "Algebra Teaching in Japanese School Mathematics." Paper presented at the conference on "The Teaching and Learning of Algebra," University of Georgia, Athens, Georgia, March 1987.
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E "A Study of Four Preservice Secondary Mathematics Teachers' Constructs of Mathematics and Mathematics Teaching." Doctoral dissertation, University of Georgia, 1982.
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G "Teachers' Conception of Mathematics and Mathematics Teaching: Three Case Studies." Doctoral dissertation, University of Georgia.
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This is an indication of good mathematics teaching because of the nature of mathematical knowledge and the procedures used in the discipline to verify whether knowledge is true. In order to teach mathematics, a teacher needs to be confident that what he or she is teaching and what the students are learning make mathematical sense.
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It might result in a correct answer, but it will not result in learning how to compare fractions. What does it take for mathematics teachers to wean themselves and their students away from answer books and algorithms, and replace them with public mathematical conjecturing and arguing among students about plausibility?
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There are a number of things, such as knowing: • how to get a large group of students in a small space interested and engaged in doing intellectually challenging work; • how to manage a rather complex set of interactions, both between teacher and students as a whole class and among students as they work together in problem-solving groups; • how students think about mathematical phenomena and knowing how to respond to that thinking in ways that are both supportive and challenging; • how to listen to students and how to organize the classroom so that students can express their thinking and listen to one another with respect; • where the mathematics teaching and learning processes are headed, not in the linear sense of one topic following another, but in the global sense of a network of big ideas and the relationships among those ideas and between ideas, facts, and procedures; • a variety of ways in which to represent big ideas to students, drawing on concrete, pictorial, verbal, and contextual as well as abstract modalities; • how to assess student understanding and being able to represent that assessment in terms that students, parents, and administrators can understand and accept. More probably could be added to this list, but the above are difficult requirements already.
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In that project, I have been observing and interviewing a group of secondary school mathematics teachers in a wide variety of high school settings who have chosen to experiment with technology designed at the Educational Development Center at Newton, Massachusetts (a member of the Educational Technology Center consortium) to support the process of "guided discovery" in classrooms.
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Are the calculators and computers that are available being used in ways that engage students in mathematical activity? Are connections being made among different concrete representations of a mathematical idea and between the manipulation of concrete representations and symbolic strategies for performing mathematical operations?
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At that time, we will be able to develop a more conventional set of indicators. But, at the moment, an appropriate indicator of whether or not a mathematics teacher is moving in the direction of supporting the kind of mathematics learning that all of the reform documents embrace might be whether the following are among the problems he or she is facing: • managing the social structure of the classroom, including large group, teacher-led lessons, small group activities, and individual work on problems, in a way that supports inquiry and coherence; • keeping track of what content is taught and determining how to move through the appropriate mathematical terrain at a compatible pace; • assessing student learning in a way that takes into account both individual understanding and issues of equity, and negotiating the terms of assessment with students and other concerned parties.
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We have a public divided over public education and it may be giving public schools one last chance. Now, a new wrinkle on tuition tax deductions and credits has been added through state, rather than federal, action.
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There are a lot of pieces to the Thatcher plan and one piece is this: if the parents of any of the children in any public school decide that they want to vote on whether or not they like the way the school is being run, the government has a procedure for it, very much like a collective bargaining action. If the majority of the parents say they do not like the way that a school is being run, that school can be removed from the authority of the local education agency, and the parents will be able to elect their own board of education for that school alone.
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But where are we really? Let us look for a few minutes at some of the results of the National Assessment of Educational Progress.
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If an employer is looking for someone reliable, it will impress him or her to be told that the applicant has held three jobs and has never missed a day's work, even coming in when sick. If the employer wants someone who can handle money, the student needs to be alert to point to experience such as working in a pharmacy and being the treasurer of the Boy Scouts troop.
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From page 47... ...
The French used to say to me, "There are absolutely no problems in our schools; no child leaves illiterate." Then, under Mitterand a few years ago, an attempt was made to remove subsidies from parochial schools. About one million people came to a rally about the number of illiterates graduating from public schools.
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No, they are not, because the way that children enter school is through an arbitrary date. If a child's birthday is before a certain date, he or she goes to school this year.
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One of the questions we have to ask is: Can we build a learning environment that does not create unfair competition at an age when children usually cannot handle it? It was better in the old days when groups of children were admitted to school twice a year.
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Basically, what the results of the National Assessment of Educational Progress tell us is not that these children are stupid, and it does not tell us that their teachers are doing a poor job of organizing their material, or that their textbooks are no good. These results show that the number of people who can sit still and absorb words, either from the written page or from someone else's mouth, and respond by turning them into complex and meaningful pictures and patterns which help to organize experience, is very small.
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A final example. Most children enter school in September, but the final grade is not given until the following June.
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So I will share with you the story of that great educational philosopher, Father Guido Sarducci. He does a comic routine where he stands in front of an audience and says, "I have opened a college.
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One, they would forget about birds a few weeks after that, a la Sarducci, and, two, they would probably end up hating birds. So I thought of a different experience I had had -- through the Boy Scouts bird study merit badge.
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From page 54... ...
What might a school look like if we were to make one where children were not learning mainly from lectures or textbooks, if they were not all starting on the same day, if they did not have to sit still and be quiet, if they were not exposed to public humiliation, especially at the earlier grades, if there were privacy, if their time were not planned a year in advance- -a school incorporating all of the items I have mentioned, as well as others? It probably would look far more like a Boy Scout troop.
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This cannot be done as an add-on; it can only be done by restructuring schools and the teaching profession. It cannot be done by making teachers work "harder" or through some additional programs.
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Privatization will not make education any better. Most private schools are like most public schools, only private schools are able to choose and reject their students.
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-57Reference Goodlad, John I A Place Called School.
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and elementary school mathematics (fractions, especially equivalence as taught in the fifth grade)
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Another exercise stipulates that the candidate bring a lesson plan to the assessment center, where she or he discusses the plan with an examiner, teaches the lesson to a group of six students who have been coached for the exercise, then reflects on the lesson, and finally discusses what happened with the examiner. Additional exercises include such tasks as critiquing a videotaped portion of a real mathematics class; discussing instructional uses of a standard mathematics manipulative, a computer program, and a collection of everyday objects with potential for mathematics applications; describing alternative classroom routines; analyzing and critiquing a textbook; and others relevant to the teaching of elementary mathematics.
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From page 61... ...
Educational institutions in general will have to find ways in which to enable increased numbers of minority candidates to meet higher standards in the teaching profession. In summary, the board's assessment and certification processes will be worthwhile only to the extent that they encourage individual teachers, the teaching force as a whole, teacher educators, and their institutions to achieve higher professional standards.
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