Undergraduate pre-service elementary teachers at LSU take 12 credit hours of mathematics in four courses, each lasting one semester. The first is college algebra. This is followed by a course in data, probability and statistics. Then come two courses specifically designed for students in the elementary education curriculum. They are described in the 2000-2001 LSU Course Catalogue as follows:
Number
Sense and Open-Ended Problem Solving. … Cardinality and integers; decimal representation of the number line;
exploratory data analysis; number sense; open-ended problem solving strategies;
written communication of mathematics.
Geometry,
Reasoning and Measurement. …
Synthetic and coordinate geometry in two and three dimensions; spatial
visualization and counting procedures; symmetries and tilings; history of
geometry; written communication of mathematics.
In a recent discussion with 10 pre-service teachers who were finishing their field experience at an inner-city Baton Rouge elementary school, the participants agreed that these two courses had been the most valuable component of their undergraduate preparation for teaching math. Most of the credit for shaping these courses is due to Lynne Tullos, who was an instructor in the LSU Department of Mathematics until 2001, when she accepted an executive position at LaSIP. At present, the courses are taught by instructors who were mentored by Tullos in a program supported by LaCEPT. In this essay, I will give some of the history of the courses, describe the structure of the classroom experience and present a brief analysis.
History. In the late 1980s and early 1990s, Dr. Richard Anderson (who was at that time already LSU Boyd Professor Emeritus) worked with the LSU Department of Mathematics to address the mathematical needs of teachers, helping to design and carry out several externally funded projects. The following passage, quoted from a proposal written during that period, shows the philosophy that guided these efforts:
Teaching practices must switch from a formal lecture and ‘memorize and show your work’ format toward much increased student-teacher dialogue about ideas, increased consideration of finding different (perhaps easier) but not necessarily formal ways of doing problems, and increased cooperative learning procedures in the classroom. (Anderson, 1990), page 8.
Tullos acted as site coordinator for two projects in which teachers attended summer workshops of several weeks duration. In this capacity, she observed classes conducted by Anderson, LSU Professor Ron Retherford and myself, and she spent many hours interacting with teachers. She reports that when she later designed and tested courses for elementary teachers, these experiences provided a rich source of guidance.
In 1993, Retherford and Tullos worked together to prepare a mathematics course for teachers that was presented in a series of television broadcasts for Louisiana Public Broadcasting. Around the same time, Anderson and Retherford outlined two undergraduate courses for intending elementary teachers, which were to be offered as part of the Holmes Program then starting in the LSU College of Education. Tullos began developing the Number Sense and Geometry courses and taught them for the first time in the 1993-94 academic year.
The following excerpt from the course description that Tullos prepared for the initial offerings will give a clear indication of the goals that shaped the courses. Similar language has occurred in the course descriptions for every section of these courses that has been offered since:
This course is specifically designed to provide the student with a working knowledge of teaching models of mathematics, the use of concrete teaching materials and the importance of “understanding” before practice. …Writing, communication, problem solving, reasoning, and making mathematical connections will be…vital… The course will emphasize open-ended problem solving, using manipulatives for concept development and understanding…[and]…will provide a wide variety of learning activities involving critical thinking skills and cooperative learning groups …. The spirit of the NCTM Curriculum and Evaluation Standards for grades K--8 will permeate the course … .
According to Tullos, the early years of testing and development were challenging and often frustrating. Many students were not receptive to the new curriculum, and some even appeared hostile to it. In retrospect, this does not seem surprising. Similar reactions have been observed frequently in many course reforms. The explanation seems to be that students develop rather specific conceptions of their roles and responsibilities. Reform classes aim to change student roles and make demands that are different from those to which many students have previously become accustomed in procedure-driven courses. Even at the present time, students still find it challenging to adapt to the new demands of these courses, but their preparation seems to be improving, and instructors are developing good techniques for helping students to adapt.
Lesson Structure. One of the most intriguing things about the Number Sense and Geometry courses is the novel structure of classroom lessons. Tullos describes the typical lesson as follows:
Each class period includes the collection and distribution of manipulatives and written work. The “lesson” for the day is usually introduced through a situation presentation where the class is asked to find a way to solve the problem. The “need” arises to know “why” a strategy will or will not work. “Using a formula” is not sufficient. “What does that formula represent? Why will it work? Will it always work? Can you think of a situation where it wouldn't work? What do you think would happen if such-and-such changed? Why? Why? Why?” are questions continually asked in my classroom. I ask questions of my students and force them to think and reason and observe patterns.
The classroom in which the courses are taught is furnished with about 8 round tables. The lesson begins with the students sitting at tables in groups of 3 or 4. Equipment such as geoboards, paper for folding, geometric models, etc., has been distributed. The instructor opens the day's activities by presenting a problem using an overhead projector. Often this is a problem that the students have worked on in a previous class or in their homework. The instructor leads a discussion that includes a lot of interaction with the class. Much of what the instructor says consists of interpretation, clarification, and commentary on remarks from students. Some students ask questions seeking clarification and elaboration. Others talk quietly among themselves.
This preliminary discussion may last five minutes or a little longer. After the problem has been defined and the majority of students have come to an understanding of the criteria that a solution will be expected to meet, it is “handed over” to the class. The instructor may remain at the overhead, soliciting solution proposals from students and recording the ideas offered by the class on the overhead screen. If students have worked on the problem previously, the recording of student work will proceed very quickly. If not, the instructor may wander about the room quietly, stopping from time to time to exchange commentary with the groups at different tables, looking at student work, or responding to student questions individually until most tables either have a solution or at least are well on the road to a solution.
As student work accumulates, there is constant monitoring and encouragement. Eventually, students begin presenting their work---often speaking from their seats. While they do so, the instructor asks for clear statements of assumptions, justifications for steps and demonstrations that complete solutions have been achieved. As closure, the instructor registers approval of one or more of the solutions that have been displayed, and verifies that the class understands and appreciates the reasons why the solutions are acceptable. The instructor, of course, is always in possession of a more sophisticated critical understanding of the problem than any of the students, and accordingly the class recognizes her as the final arbiter and judge of the quality of student work.
We can characterize the entire transaction as a “cycle of moderated problem-solving.” The entire cycle, from the time the problem is introduced to the point of closure typically takes about 20 minutes, though longer and shorter cycles occur. Cycles of moderated problem-solving form the most important structural unit in these courses. Each class consists of several cycles in succession, with other activities---such as quizzes---occasionally inserted in between.
Analysis. How does this structure compare with other, more traditionally taught courses? Like the courses we are describing, the typical college algebra class breaks up into discrete episodes, but in college algebra the episodes tend to be much shorter, ranging from just a couple of minutes to a maximum of 10 or 15. Some of the episodes in college algebra involve introducing vocabulary or demonstrating procedures. There are also episodes that involve posing a problem, working on it, and reaching a solution. The typical college algebra problem concerns a specialized, abstract mathematical situation, e.g., simplify an expression, solve an equation, graph something, etc. Solutions to problems of this kind seem to sort themselves into specific, well-defined types. The goal is often to produce a pattern of an acceptable kind, rather than an explanation. Solving an equation is a typical example: the solution consists of a sequence of expressions leading ultimately to one stating the value of the variable. Students are seldom expected to discuss the meaning of the sequence of symbols. In fact, many learn to produce the sequence without having a full understanding of its meaning, even when instructors have provided explanations. Problems in the Number Sense and Geometry courses, in contrast, are often motivated common experience and are posed in colloquial terms. A good deal of explanation is sometimes needed in order make a bridge between the problem situation and the mathematics. At the same time, there is much more variety in problem types, and solutions do fall into patterns that are easily anticipated.
Experienced college algebra instructors easily anticipate the form that solutions to problems will take, and it is often the case that there is a routine procedure, which if followed accurately, will generate a solution. Accordingly, the instructor spends a good deal of time demonstrating or modeling routines and leading the class through them. This contrasts with the Number Sense and Geometry classes, where there is seldom a routine to follow and where students are clearly expected to search on their own for solution methods and while doing so to sustain the momentum. In the algebra class, the question was often whether or not the solution method was executed properly. In Number Sense and Geometry, the instructor waits for students to propose solution methods, possibly making some suggestions, but seldom, if ever, declaring that a certain method or approach is the “right” way to proceed.
In an algebra class taught by an experienced instructor, there is a lot of interaction between instructor and class. It is, however, of a different kind from that which is seen in the courses we are considering. The college algebra instructor will check frequently to see that students are in command of the relevant vocabulary and are able to carry out the “subroutines” from which a more complex routine is built. Thus, the instructor's questioning seems intended to determine, “Is the class with me?” or, “Do the students have at hand the component skills from which this new, more complex procedure is built?” Much of the questioning seems intended to remind the class of these things (as if constant prompting were needed) or to activate specific skill that have been learned in the past, but might be “out of hand” at the moment. In courses we are considering, the questioning has a different purpose. It is not intended to verify that students are “on track,” or to place them back on track when they stumble, but rather to expose the deeper levels of student thinking to as wide an audience as possible and to enable everyone in the class to appreciate the best ideas produced in the classroom.
In comparing American mathematics teachers with their Japanese counterparts, TIMSS researchers Stigler and Hiebert remark that:
U.S. teachers appear to
feel responsible for shaping the task into pieces that are manageable for most
students, providing all the information needed to complete the task and
assigning plenty of practice.
Providing sufficient information means, in many cases, demonstrating how
to complete a task just like those assigned for practice. …[In contrast], … Japanese
teachers lead class discussions, asking questions about the solution methods
presented, pointing out important features of students' methods, and presenting
methods themselves. Because they
seem to believe that learning mathematics means constructing relationships
between facts, procedures and ideas, they try to create a visual record of
these different methods as the lesson proceeds.
(Stigler & Hiebert, 1999), page 92.
In many ways, Number Sense and Geometry seem much closer to the Japanese ideal than to the traditional American. However, there are ways in which these courses differ from Japanese lessons. For example, overhead projectors, though common in U.S. classrooms, are rare in Japan. I suspect that many of these differences are due to the transitional nature of these courses. As I mentioned before, these courses challenge students in ways that are very new to them.
Mentoring. If an individual teacher manages to teach in a new way, it is not necessarily a change in culture. A true cultural change must be transmissible, must replicate itself. The culture of Number Sense and Geometry is propagated through a mentoring program that is supported by LaCEPT funding. LSU Instructor Nell MacAnelly, who interned with Tullos in fall 1995, gives a vivid sense of the impact of the program in an essay she wrote in spring 2000:
I entered
the mentor experience thinking that there would be little, if any, change in
the way I taught. Mainly, I
planned to enhance the teaching skills and approaches that I already used. After all, in [multi-section] courses,
my students already consistently ranked well above the median scores. Was I surprised! … Through the experiences and
“hands-on” strategies, we had acquired much more depth of conceptual
understanding than I could have ever imagined. The traditional questions seem almost “trivial”
in their assessment. The change
was not something that anyone could have “instructed” me to
do. I had to “experience
it” for myself and truly “feel the need.” The primary impetus came in my
analyzing my own learning to see what engaged me the most. It amazed me that I had so much fun and
yet learned at a far deeper level than I had before.
Conclusions. The data presented here indicate that Number Sense and Geometry embody a novel classroom culture. In my view, it is a good culture, in the sense that experiencing this culture is likely to be socially advantageous. I think that people who are more in the habit of looking for reasons and explanations and who are able to communicate better about them will make better citizens. In this essay, however, it has not been my purpose to argue for the social value of the courses, but only to demonstrate exactly how they differ from more traditional courses. They are based on giving students entirely different roles and responsibilities from traditional courses, and the classrrom environment is finely structured and tuned to support these new roles and responsibilities. I feel confident that readers who have thought seriously about what it means to understand mathematics will agree that this is the kind of environment where understanding is achieved.
Anderson, R., (1990). Middle-School Mathematics Revitalization Project. Proposal submitted to NSF by the LSU Department of Mathematics.
Stigler, J. & Hiebert, J. (1999). The Teaching Gap. New York: The Free Press.