A Non-standard Mathematics Program for K-12 Teachers

Patricia Baggett Andrzej Ehrenfeucht

Presented at 29th Conference of the International Group for the Psychology of Mathematics Education, July 10-15, 2005,

A university program in mathematic for practicing and future K-12 teachers is described. The program was started in 1995 and currently consists of six one-semester courses. All courses are run in a laboratory format and are based on a "modern" view of mathematics. They use calculator and computer technology, and strongly lean toward direct applications of mathematics which are meaningful for school-age students. There are no textbooks for the courses, but practicing and future teachers are provided with lesson plans for all material covered which they can use in their classrooms. Participants consistently evaluate the program as relevant and interesting, but a formal evaluation of its effectiveness has not yet been done.

INTRODUCTION

Studies in the psychology and pedagogy of mathematics seem to concentrate on the status quo. The psychological underpinnings of the concept of number deal exclusively with whole numbers (e.g., Butterworth, 1999; Dehaene, 1997; Geary, 1994), and studies in pedagogy put a huge stress on common fractions (Kieren, 1988; Kilpatrick, Swafford, & Findell, 2001; Ma, 1999). Both of these topics dominate current early- and middle-grade curricula. There is a dearth of studies that anticipate curricular changes or that concentrate on concepts that will possibly dominate school mathematics programs in the future.

Theoretical mathematics has experienced very fast evolutionary changes during the last two hundred years, and applied mathematics has recently gone through a computer revolution. During the same period, school mathematics, especially in early and middle grades, which was based on the commercial mathematics of the eighteenth century (e.g., Daboll, 1800; Ostrander, 1823; Pike, 1809), has changed very little. Many reforms (e.g. NCTM, 2000) have significantly changed the way mathematics is taught in schools, but they have not changed its content.

The main reason for this "stability", which can also be termed stagnation, is the fact that elementary and middle school teachers get their mathematical knowledge mainly during their own school years. And this knowledge is rather reinforced than modified by any "teacher-oriented" courses they take in college.

At

At present we offer six courses covering topics that can be used from kindergarten through high school. Each course has a central focus, and each can be taken both at the graduate level (primarily, but not exclusively, by practicing teachers) and at the undergraduate level (primarily by students from the

The foci of these courses are: Arithmetic with geometry, and Geometry with arithmetic, which are designated mainly for elementary teachers, and which fulfill the basic math requirement for future K-6 teachers; Algebra with geometry, and Use of technology, which are directed toward middle school teachers; and Mathematics and science, and Algebra with geometry II, directed toward high school teachers.

Below we try briefly to answer three questions: What do we try to teach? Why? And how?

WHAT TO TEACH AND LEARN

The current curricula in mathematics for early and middle grades developed from eighteenth century vocational mathematics (see references cited in the Introduction) which was based on several number systems (whole numbers, common fractions, and decimals), practical geometry, and commercial arithmetic. Today's high school mathematics is enriched by the content of early algebra, which deals with linear and quadratic equations and irrational numbers (surds), and a selection of more "modern" and "advanced" topics such as analytic geometry, probability, discrete math, and calculus. Euclidean geometry, which played a central role in high schools during the nineteenth century and the first half of the twentieth century, has mostly disappeared from school curricula.

The current program in school mathematics is highly fragmented, and riddled with contradictions (many properties of different number systems are mutually inconsistent), and it doesn't mesh well with calculator and computer technology. Skills in elaborate written computations and their applications to everyday problems have mostly become obsolete. At the same time, more valuable skills, such as mental computation, flexible use of the language of algebra, and the use of calculator and computer technology, together with skills in sketching and drawing, and using graphing tools, are not learned in any systematic way. Our program doesn't attempt to change school curricula, but to provide current and future math teachers with better, more useful modern knowledge of mathematics. Without making any claims that our approach is "better" than other possible and existing approaches, we describe below the basic mathematical content of our six courses for teachers.

1. All courses are based on one concept of number, namely the concept of *real numbers*. Whole numbers, integers, rational numbers, and so on are treated as special and important subsets of the system of real numbers. Of course this doesn't mean that we try to tell middle school students about Cauchy sequences or Dedekind cuts. But it means that on all levels we talk about numbers in a way that is *consistent* with the theory of real numbers. So even a first grade teacher knows that 0 is not the smallest number. Zero is the smallest whole number, but there are other numbers that are less than 0, which are going to be discussed in other grades. Such consistency is a *prerequisite* for using technology in early grades. On several occasions we have observed that children who are introduced to calculators without an appropriate introduction to the system of real numbers have formed persistent misconceptions. A typical one was treating the minus sign as an error message. "The calculator tells you that you cannot subtract a bigger number from a smaller one."

2. Algebra is not treated as a separate topic. It provides a language for the arithmetic of real numbers. It describes relationships among physical variables, and it serves as a programming language for scientific and graphing calculators. So algebra is an integral part of mathematics from the very beginning, long before specific algebraic techniques are introduced.

3. Calculator technology is used at all levels, and it forms an essential part of the development of arithmetic and algebraic skills. In early grades four-operation calculators are used, together with mental computations. In middle grades, when writing becomes the main mode of communication, two-line-display scientific calculators, which have algebraic notation, are used. High school material often requires the use of graphing calculators and specialized computer programs.

4. Other domains of mathematics are treated more as applications and not as stand-alone topics. Among them, geometry is the most important. It provides the majority of problems that deal with design, measurements, and creative problem solving. Technically speaking, the main focus is on three-dimensional Euclidean metric spaces. So geometry is built around the concepts of length and distance. We have tried to incorporate topics from probability theory and statistics, because they are strongly represented in current school curricula and teachers ask for them. But the results have been mixed at best.

5. The presentation of topics is guided by applications and not by "internal cohesion" of mathematics. A topic is divided into units. One unit is typically introduced through a task to be done. Its mathematical aspects do not even have to be stated at first. The mathematics enters when the task is analyzed and it happens that it requires using mathematical techniques. Problems presented in this manner are often very rich in content. But because of the entanglement of several techniques and concepts, their solution often lacks the simplicity and clarity of artificial problems that are created to illustrate just one idea.

WHY TO TEACH AND LEARN MATHEMATICS?

There are so many reasons given for teaching and learning mathematics that it is hard to list them. We mention here only a few examples. "The country needs a math-savvy workforce." "To get a job in modern society you need to know mathematics." "Citizens need mathematics to participate in the democratic process." "Mathematics teaches logical thinking." "Mathematics teaches how to solve all kinds of problems." "Mathematics has intrinsic beauty."

But when we talk with school children, we see different patterns. They like math when it relates to their everyday interests. In early grades they like all problems related to food. For example, weighing a banana and a banana peel, and computing what fraction of a banana is eaten is a fascinating problem for fourth graders^{1} (especially if they can use the bananas for a snack). "Make believe" stories are also favored. Drawing a spider web^{2} and measuring and computing the length of the "thread" is a very popular problem. In middle grades the focus changes. Making artifacts, objects, patterns, or puzzles (e.g., constructiong three congruent pyramids that form a cube^{3}) which are taken home and shown to others creates great interest. In high school, interests vary. But students still prefer mathematics that is directly applied to their current interests. Lessons combining experiments in physics, which yield data that can be mathematically analyzed, were well received by high school students (Baggett & Ehrenfeucht, 2004). Dynamic simulations of mathematical processes that can be programmed on graphing calculators are strongly favored. (See also Freudenthal, 1987.)

When we talk with school teachers who are taking a math course as a part of a professional development program, and with future teachers who are taking preparatory math courses, the picture is even simpler. The main question they ask about the math they are required to learn is, "Am I going to use it in my classroom?" They find material that they can directly use worth learning and interesting. Other material, even if it is interesting for some other reason, has low priority. (See also Garet, Porter, Desimone, Birman, & Yoon, 2001.)

Using materials that are directly related to students' interests changes the classroom dynamics, both in schools and in courses offered for teachers at the college level. Learning becomes "goal oriented". Sometimes, but not always, these changes are clearly visible. Here are a few specific examples take from the early elementary grades.

First graders were making Puffy Stars^{4}. They were to take their stars home later. The teacher was helped by two student teachers. The activity lasted two hours without a break. According to one student teacher, "During the second hour it was not fun, but they (the children) were determined to finish the task."

A day after a lesson called Ants' Roads^{5}, a second grade teacher gave her children a practice session in drawing lines with a ruler and measuring their lengths. The children spent the whole period just drawing and measuring. They wanted to become "good at it".

But children's desire to practice and improve their skills can also be misplaced. During a fifth grade lesson about tossing two dice, children understood expected value to be "the value you try to get", and in writing about the lesson one child said, "I'm still not good at it. But I'll practice at home until I can get seven (the expected value) all the time."

Young children like to practice skills that have meaning to them. But they are missing one thing. They almost never ask, "Why?" Even middle school and many high school students treat mathematics as factual knowledge that can be mastered, and not as a web of mutually dependent logical conclusions. In this respect, teachers and students at the college level are very different. Teachers are not interested in practicing skills, but they always want to know why something "works", how we know that "it is true", and so on. This curiosity is strong as long as they are genuinely interested in the topic at hand. It is persistent even if the answers to their questions are unsatisfactory. This often happens because some proofs are too difficult to present in classrooms, and also many algebraic and trigonometric derivations are purely "rule based", and they don't provide much insight.

HOW TO TEACH

Here we discuss our six university classes for practicing and future teachers, and not what teachers do in their classrooms with the university materials. Also we ignore many social aspects of pedagogy, and we concentrate on the relationships among classroom practice, the content of learning, and teachers and their students' short-range goals.

In the courses for teachers we mainly stress the applied aspects of mathematics, and we want to provide teachers with materials that they can use directly in their own classrooms. So all courses are run in a laboratory format, in which students, working in groups, actually do all the tasks and become familiar with the smallest details and possible pitfalls of a lesson. There is no required text book; instead students get very detailed handouts for each unit.

A typical unit is presented as follows. It contains a task which often has a large "hands-on" component, the relevant mathematics, and a description of the technological tools and other supplies that are needed. Students start with creating their own solutions, and there is an atmosphere of respect for their questions, ideas, and contributions. But if students are not successful, the instructor provides them with a "model" solution and explains how it was found. Students often struggle with, but also frequently become intellectually engaged with, the important mathematical concepts embedded in the lesson. (See Hennigsen & Stein, 1997.) The handouts also include lesson plans with varying numbers of details, including one or more "model" solutions. A typical lesson presented in the university class requires from one to three periods of time in a school classroom. The lesson plans enable teachers to use the material immediately in their classrooms. This approach happens to be very successful. Most lessons are tried with children within one or two weeks after they are presented to teachers. Teachers adapt them to the levels of their particular students.

Future teachers (students in the College of Education) act as teachers' apprentices. They are required to spend a prescribed number of hours (ten in each semester) in their mentors' classroom. They observe, help, and even teach under the guidance of their mentors. At the end of a one-semester course they have a portfolio of up to fifty lesson plans. We know that many alumnae and alumni who are now practicing teachers still use the lesson plans that they originally studied in these courses.

Most homework consists of writing. Both practicing and future teachers write classroom reports that focus on the content of the lessons and how they were presented in the university class or in teachers' classrooms. Often children's artifacts, their recalls of the lessons, and other data are included. Students' reports are corrected for content and style, extensively commented on, and returned to the students. No numerical or letter grades are assigned to students' work during the semester.

Final grades, required by the university, are assigned at the end of the semester. For practicing teachers the grades are rather a formality. Teachers who take these courses are competent professionals who want to improve their knowledge. Because they are tutoring future teachers, they are also the instructor's assistants, and not just her students.

The situation for future teachers is different. If they pass these courses, it is a certification that they have sufficient knowledge of mathematics to become teachers. So they are graded rather harshly, and this brings a considerable level of tension to some of them.

The laboratory format of teaching that is described above allows us to cover material in minute detail, but the total amount that is covered is considerably smaller than in classes taught in a lecture format. So decisions concerning what to include and what to omit are crucial. There is one important factor that guides these decisions. We omit topics that are a review of material that teachers and students already know, and we concentrate on those that are new for them. So, for example, we don't cover solutions of quadratic equations, which play a very big role in high school mathematics; instead we show teachers the fixed-point method for solving equations, which they have never seen before.

All these courses are multi-level, in the sense that the background knowledge of students varies considerably. We often have in the same class high school math specialists and first grade teachers, and undergraduate and graduate students from different colleges. This creates some obvious problems, but it also has some advantages. When students work in groups, multi-level teams are usually preferable. Also participants get a better overview of school mathematics as a whole. The applied approach to mathematics also helps. Many tasks can be done at many levels. Problems concerning volumes such as "Ball in a Box"^{6} can be solved with rice in the second grade, and with calculus in the twelfth grade.

CONCLUSIONS

The mathematics program for teachers described above is successful in the following sense. Participants uniformly evaluate the materials as relevant and interesting, and the course as valuable and useful. Elementary, middle, and high school students are highly motivated by this kind of mathematics, (see Wigfield, Eccles, & Rodriguez, 1998), and when tested for knowledge acquired during specific lessons, they show good understanding of the advanced material that is presented, weeks and even months later. (Compare Boaler, 1998.) But this program is not directly related to any specific curriculum. Therefore teachers who use the materials still follow one or more of the curricula that are used in American public schools. So the overall and long-term effects of this approach cannot be measured, at least at present.

Also there is one problem that arises at the college level. Running classes in a laboratory format, preparing handouts for each session, and correcting students' writings, take at least twice as much time and effort on the part of an instructor, as running a class in the more traditional lecture format. So even college instructors who like this idea in principle are reluctant to put it into practice, especially before receiving tenure.

In the presentation we intend to include examples of specific lesson plans, show samples of the work of school students from different grades together with their recalls from lessons they were taught, and discuss student evaluations of the courses.

References

Baggett, P. & Ehrenfeucht, A. (2004; 1995). *Breaking Away from the Math Book: Creative Projects for Grades K-6*, Lanham, MD: Scarecrow Press.

Baggett, P. & Ehrenfeucht, A. (1999-2005). Breaking Away from the Math Book: Creative Projects for K-8. http://math.nmsu.edu/breakingaway

Baggett, P. & Ehrenfeucht, A. (2004; 1998) Breaking Away from the Math Book II: More Creative Projects for Grades K-8, Lanham, MD: Scarecrow Press.

Baggett, P. & Ehrenfeucht, A. (2001) Breaking Away from the Algebra and Geometry Book: Original and traditional lessons for grades K-8, Lanham, MD: Scarecrow Press.

Baggett, P. & Ehrenfeucht, A. (2004) *Breaking Away from the Math and Science Book: Physics and other projects for grades three through twelve*. Lanham, MD: Scarecrow Press. (A review of this book by James M. Boyd is in *Mathematics Teacher*, vol. 98, no. 3, October 2004, pp. 206-7.)

Boaler, Jo (1998). Open and closed mathematics: Student experiences and understandings. *Journal for Research in Mathematics Education*, 29 (1), 41-62.

Butterworth, Brian (1999). *What Counts: How Every Brain Is Hardwired for Math*. New York: Free Press.

Daboll, Nathan (1800). *Daboll's Schoolmaster's Assistant*. New London, CT: Samuel Green.

Dehaene, Stanislas (1997). *The Number Sense: How the Mind Creates Mathematics*. Oxford: Oxford University Press.

Freudenthal, H. (1987). Mathematics starting and staying in reality. In I. Wirszup & R. Street (Eds.), *Proceedings of the USCMP conference on mathematics education on development in school mathematics education around the world*. Reston, VA: National Council of Teachers of Mathematics.

Garet, M., Porter, A., Desimone, L., Birman, B., & Yoon, K. (2001). What makes professional development effective? Results from a national sample of teachers. *American Educational Research Journal *Winter, 38 (4), 915-945.

Geary, David C. (1994). *Children's Mathematical Development: Research and Practical Applications*. Washington, DC: American Psychological Association.

Henningsen, M. & Stein, M. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. *Journal for Research in Mathematics Education* 28 (5), 524-549.

Kieren, T. (1988). Personal knowledge of rational numbers : Its intuitive and formal development. *Number concepts and operations in the middle grades.* Reston, VA: National Council of Teachers of Mathematics (NCTM).

Kilpatrick, J., Swafford, J. & Findell, B. (2001). *Adding it up: Helping children learn mathematics*. Washington, DC: National Academy Press.

Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahway, NJ: Erlbaum.

NCTM (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM.

Ostrander, T. (1823) The elements of Numbers or Easy Instructor; being a new and comprehensive system of Practical Arithmetic. Canandaigua: J.D. Bemis & Co.

Pike, Nicolas (1809). A New and Complete System of Arithmetic, composed for the use of the Citizens of the United States. Edited by Nathaniel Lord. Seventh edition. Boston: Tommas & Andrews.

Wigfield, A. J.S. Eccles, & D. Rodriguez (1998). The development of children's motivation in school context. *Review of Research in Education* 23 (73-118). Washington, DC: AERA.

Footnotes.

^{1}The lesson Bananas! is at http://math.nmsu.edu/breakingaway

^{2 }The lesson Spider web is chapter 17, pp. 63-68, in Baggett & Ehrenfeucht, 1995; 2004.

^{3} The lesson Three congruent pyramids that form a cube is chapter 56, pp. 249-251, in Baggett & Ehrenfeucht, 1998; 2004.

^{4}The unit Puffy Stars is Lesson 37, pp. 107-108, in Baggett & Ehrenfeucht, 2001.

^{5}The lesson Ants' Roads is chapter 16, pp. 57-62, in Baggett & Ehrenfeucht, 1995; 2004.

^{6} Ball in a box can be found at http://math.nmsu.edu/breakingaway