Reconceptualizing Mathematics:
Courseware for Elementary and Middle Grade Teachers

San Diego State University

Purpose and Goals

This project was a research-based curriculum development project, focused on the development of courseware and instructional materials for mathematics courses for prospective elementary and middle school teachers, and for the professional development of practicing teachers. Our goal was to provide instructional materials for mathematics courses for prospective and practicing elementary and middle school that will help these teachers to reconceptualize the mathematics they often think they already know. We believe that teachers should know mathematics at a much deeper level than most people do. They need to know the mathematics they teach in a way that allows them to hold conversations about mathematical ideas and mathematical thinking with their students. To do so, teachers must be able to understand their students’ reasoning and to build on this reasoning in a manner that allows mathematical understanding to develop. To attain this knowledge of mathematics, the coursework must address beliefs known to be obstacles to students' understanding of mathematics as knowledge arising from the natural attempts of humans to understand their intellectual and physical worlds. It must also address misconceptions and invalid knowledge known to be obstacles to students' understanding of mathematical concepts that are foundational to the elementary mathematics curriculum.

Background and Considerations

One obstacle to preparing teachers to understand mathematics at a depth that will allow them to teach it to others is their narrow view about what mathematics is. All too often they believe mathematics is a set of rules to be learned, rules that they do not and even cannot understand. We also know that many prospective and practicing teachers carry with them some of the same misconceptions and invalid knowledge that we know to be obstacles to students’ understanding of mathematics. We try to address these beliefs and misconceptions throughout the materials.

We have attempted to do more than simply reorganize the usual content of mathematics courses for elementary and middle grade teachers. We have selected and developed topics and problems that exemplify mathematical reasoning and problem solving, make mathematical connections clear, clarify and overcome common misconceptions, provide opportunities to communicate mathematically, and promote greater confidence in one's ability to deal with mathematics (all thrusts of NCTM's Principles and Standards for School Mathematics, 2000). In designing these materials, we intend that teachers will come to make sense of mathematics.

Four major considerations guided our work.

Consideration 1. We want to help teachers develop a strong mathematics background.

Although the module headings listed later may sound familiar, the content within them looks quite different from most traditional mathematics courses for elementary teachers. We have quite deliberately tried to challenge students to reason, to make sense of the mathematics they are learning, and to reflect on what they know. We do this in part by asking students to provide reasons for what they do, to formulate good questions, to come to understand the thinking of their peers, in short, to communicate about mathematics.

A teacher’s knowledge should go beyond the mathematics that is presented in elementary and middle grade textbooks. Truly knowledgeable teachers can deal with students' insightful moments or their misconceptions, can recognize and seize opportunities for fruitful digressions, can reject textbook organization when it does not match their own understanding, and can choose instructional representations that reflect a deep understanding of the topic and that invoke basic concepts and principles. The choosing of appropriate representations is seen by many as a hallmark of an effective teacher, one that is highly dependent on a deep understanding of content. Creating, recognizing, and using appropriate instructional representations provide ways by which teachers can communicate their own knowledge to students, and require that the teachers have a deep understanding not only of the mathematics, but also of the manner in which they themselves understand that mathematics.

A teacher’s knowledge of mathematics is intricately interwoven with her or his beliefs about the nature of mathematics itself. As Hersh (1986) has explained: "One's conception of what mathematics is affects one's conception of how it should be presented" (p. 13). We attempt in these materials to challenge teachers’ conceptions of the nature of mathematics.

Consideration 2. Instructors who use these materials need to understand our goals, and need assistance in carrying out the objectives of this program.

Mathematics department instructors of mathematics courses for prospective and practicing elementary and middle grade teachers often have no experience with teaching other than lecturing, but are interested in materials that will assist them in making their courses more meaningful to teachers. These course materials are designed to provide instructors of preservice and inservice teachers with the opportunity to model the types of instructional delivery expected of teachers in grades K-8: engaging students in rich mathematical discourse in both large and small groups, enhancing instruction through a variety of tools, creating supportive learning environments, helping students become responsible for their own learning, and building on intuitive and prior knowledge during instruction. Instructor notes are specifically designed to clarify, along the way, why we have chosen to organize the materials the way we have. The student materials contain problems or sets of learning activities that lend themselves to a variety of instructional settings. Many of the activities and questions deal with problems and situations that are best undertaken in paired or group settings. Some are designed to act as catalysts for in-depth discussions by the entire class. Others will be useful in promoting individual and group projects. And, when appropriate, some materials provide instructors with the background material for a lecture. Each CD ROM contains a few videoclips intended to assist instructors. The clips portray instructors teaching the materials, prospective and practicing teachers working in groups with the materials, elementary students learning mathematics, and interviews with students who use the materials. Many of the instructors portrayed are graduate students who have only begun their college teaching careers, but who have found that these materials help them understand their students’ thinking and successfully implement instructional styles other than lecture.

Consideration 3. Technology must play an appropriate role in courses that prepare teachers.

We view technology as a tool, to be used to clarify mathematical concepts and to perform time-consuming computations. Calculators should be used when appropriate. We have limited our incorporation of computer software programs into our materials so that instructors would not be dependent on software at their particular sites. Except for the Chance and Data module, where it is necessary to go beyond our materials to obtain needed software, we use only programs that project personnel have developed, or which are freeware. However, when other appropriate software is available at a site, we strongly encourage its use.

Consideration 4. Research about learning mathematics must affect our decisions on what to include.

We, the developers of this program, are also researchers. Our knowledge of what research tells us about learning, about teaching, about the curriculum, and about the effect of classroom culture on student beliefs has influenced both the student and the instructor materials--implicitly in the sense of how we organize and select lessons, activities, and exercises, and explicitly in the sense that we many times refer, particularly in the instructional materials, to relevant research.

The expected audience for these materials

We developed these materials with both preservice and inservice teachers in mind, and we have extensively piloted the materials with both groups.

Preservice elementary teachers commonly have mathematics backgrounds and confidence levels that cover an enormous range; some should be math majors, others are still struggling with fractions and decimal points, some welcome any challenging problem, others feel that they should be expected only to imitate worked-out examples. As with most required mathematics classes, the instructors’ toughest jobs is to convince the "weaker" students that they can indeed succeed and to inspire the indifferent students to enjoy mathematics and appreciate mathematical thinking.

Practicing elementary and middle school teachers who have discovered that their mathematics understanding is too weak to teach in the manner they want are seeking courses that cater to their needs as teachers. All too often the only courses available to them are courses such as college algebra, which seldom leads them to a better understanding of the mathematics they are teaching. We have found that they welcome courses based on these materials.

Sometimes students are expecting a methods-of-teaching course. Although there are messages about teaching methods built into the modules, for example, some work in small groups, hands-on with manipulatives, explanations as a part of mathematics, and an occasional exercise or note on children’s work or misconceptions--our primary concern in developing these materials is the future teachers’ understanding of mathematics, not of how to teach mathematics.

Our expectations for students

"Sense-making" is the dominant theme throughout the materials, and explanations and justifications are expected from all students. Students should not expect to be shown "the right way," as though imitation is the only way one can learn mathematics and as though there is only one way in which certain tasks can be accomplished. Some of the exercises or discussion questions are rich enough that different solution methods are possible and even likely; a variety of approaches or thinking sometimes comes up in group work, and such occurrences should be noted to the class in a positive fashion. Attention to "how I was thinking about the question" is something that many students do not automatically do. Using these materials, most students will need to change their beliefs about what mathematics is and how they themselves learn it.

Our theory of learning

We believe that mathematical ideas need to be connected to be learned. The more instructors can help their students make these connections, the better their students will understand. Unfortunately, some students continue to try to memorize disconnected bits of information and procedures. We also believe that students learn by doing, and then reflecting on what they have done. Writing out explanations forces this reflection and clarifies the ideas in the mind of the writer. The result should be that the student makes sense of the mathematics.

Every theory of learning assumes that the student is engaged. Discussions usually demand more involvement than the taking of notes during lectures does. Questions calling on manipulative materials seem to accomplish the same thing. "Engagement in what?" is our ultimate concern. Exercises in sense-making, not just exercises in memory retrieval, should be routine. We believe that work that focuses on conceptual development and explanations is more important for prospective and practicing teachers than only knowledge of vocabulary and other conventions.

Technology. For our modules on Uncertainly and on Data, we used software developed by Cliff Konold, with his permission. The software was developed only for Macintosh computers, and so we subcontracted with a software company to develop PC versions of the two software programs. The programmer quit the company before completing the programs, and the cost was excessive--much more than had been estimated. Cliff Konold now has what work was done, and is arranging for another company to continue this effort. If he succeeds, we will be able to use in with our courses. We also use place value software developed by Janet Bowers, one of our senior faculty. We list other appropriate software, but do not require it. We also intend to incorporate new Palm Pilot programs now being written (for an IERI grant) on understanding and using fractions.

Description of the six modules developed

1. Number and Number Sense Module

This module focuses on conventional mathematics but from slightly unconventional ways. There is a strong focus on number sense: coming to understand numbers and to be able to operate on them in non-standard ways. Teachers with whom we have used this module report that they found the mental computation and estimation, the examination of student work, and the different approach to fractions extremely helpful.

Prospective teachers too often feel that work with whole numbers, decimal numbers, and fractions is review for them. They will find, however, that there is little review here--many of the ideas will be quite new and challenging to them. The underlying theme of this module is to provide teachers, and students who are preparing to teach, opportunities to reconceptualize their notions about numbers and operations. The "Notes for Teaching" sections address some of the issues that are subjects of the current debate: the use of calculators in schools and the role of traditional algorithms. Prospective teachers have strong opinions about these issues, but their opinions are often uninformed and have an emotional component. It is imperative that these issues be discussed, and that prospective teachers begin to reason through them.

2. Shapes and Measurement Module

Beginning this module with three dimensional work was a deliberate choice. Because a rigorous, proof-oriented course is not intended and because the students are assumed to have some familiarity with geometry, the gradual build-up from points, lines, and planes is not essential; one can deal with the relatively unfamiliar 3D shapes early, building in any review of terms necessary. The main reason for choosing to start with something unfamiliar is that the tone for the effort to be expended on a course is often set during the first few class sessions. Our experience suggests that when the start of a course treats "familiar" material first, some students fall into low-effort patterns. Three-dimensional shapes are "new" to most of the students, so more effort is immediately called for than would be the case with a from-the-ground-up development.

After the introduction to polyhedra, properties of polygons are reviewed in terms of faces of polyhedra. Symmetry, tessellations and size changes in a plane and in space follow the introductory sections. Congruent polyhedra are visited informally, then polyhedra are considered in terms of hierarchical taxonomies. A rather long section on measurement concepts builds on ideas presented in previous sections. Finally, we present a study of geometric transformations. We have followed a "Cook’s tour" approach to geometry, as opposed to a "less is more" treatment. A "less is more" course is certainly possible, using some of the sections here as starting points or reference materials. We hope that we have combated the "broad but shallow" criticism of a typical Cook’s tour by attention to some central ideas and skills for the elementary teacher.

3. The Uncertainty and Its Quantifications Module

We begin this module by introducing students to some "famous" probability problems that are not actually solved until the end of the unit. They are intended to introduce discussion on chance. We then attempt to clarify for students the issue of what a probabilistic situation is. Often when we as instructors discuss the probability of a situation, we are not careful about the language we use, and as a result students do not realize that a probability statement is made only about situations in which a process is repeated a large number of times. We introduce traditional ways of assigning probabilities to situations, then simulate probabilistic situations (using the software ProbSim or a random number table). The next section deals with independence and is the most difficult section for students. These ideas developed here will take some time, with many examples and a good deal of class discussion. The section on conditional probability is a favorite of students because they begin to realize the worthwhileness of knowing some probability. With this background, students are able to tackle the problems presented at the beginning of the module. Further sections deal with expected value, and with the fundamental counting principle, combinations, and permutations. (Note: ProbSim is available only for Macintosh computers.)

4. The Collecting, Representing, and Interpreting Data Module

As was true of the module on uncertainty, there are multiple goals for these sections on statistics. The first and primary goal is to provide teachers with the background they need to teach the many statistical ideas now found in elementary and middle school mathematics curricula. Second, the units are intended to help teachers as educated citizens come to understand statistics to the degree needed to interpret common, everyday statistical statements, that is, to be statistically literate. A third goal is to provide some background that will help them understand statistical ideas they need as professional educators. For example, topics such as percentiles, the normal curve, and z-scores are included not because they are found in some of the new curriculum projects, but rather because teachers need to be able to interpret some of the statistical ideas used in measurement and testing.

The first section focuses on variability in sampling, and is the most problematic, conceptually, for students. The second and third sections focus on organizing and interpreting data with one variable, and then two variables. Students have access to DataScope to carry out statistical procedures and provide representations of data. There is, however, the expectation that they will carry out some statistical procedures for representing data before they use statistics software to undertake more complex representations. Throughout the module, conceptual understanding is valued, and the exercises are challenging and demand understanding rather than procedural skills. (Note: DataScope is available only for Macintosh computers.)

5. The Quantitative Reasoning Module

Quantitative reasoning is reasoning about objects and their measurements and the relationships among these quantities. When students are presented with a quantitative situation, they too often try to reason only with the numbers, trying one operation and then another, before they have a full understanding of the situation. The approach we advocate in these materials involves asking students to carefully identify the quantities involved in a problem situation and to map out their relationships, often in the form of a diagram. Only then do they attach numerical values to the quantities, and, at that time, the operations needed to solve the problem are usually clear. Based on extensive piloting research, we have found that students often find this approach at first to be difficult. In their minds, it is "not like the mathematics they know." But most come to understand and value the power of this approach if the instructor supports students' development of quantitative reasoning. This involves continually asking students to identify the quantities and their relationships within a given situation. Another aspect of this module that is different from most coursework for prospective teachers is the explicit distinction made between additive and multiplicative reasoning and the need to identify which is the appropriate reasoning within a given situation. For example, a student who compares two slopes by finding the differences between the height and length rather than the ratio of the height and length is reasoning additively rather than multiplicatively. Multiplicative reasoning supports the development of proportional reasoning.

The goal of this module is to help students develop habits of reasoning quantitatively. The ideas in this module are quite new to students and students are at first resistant to them. But many of our students have claimed that after this module, they were able to solve problems they were never able to solve before.

6. The Describing Change Module

This module investigates ideas of algebra primarily through a study of graphing. Graphing and algebraic symbols are used to represent quantitative relationships, thus tying back to the module on quantitative reasoning. A deep understanding of slope is constructed through the describing of the relationships between time, distance, and rate, in some instances using motion detectors. The ideas are all used to then develop and interpret qualitative graphs of situations, and creating explanations involving quantitative relationships.

Suggested Pathways

One Semester Mathematics Content Course
for Preservice Elementary Teachers

Note to the instructor:
Below we have listed a suggested path for a one semester mathematics content course for preservice elementary teachers. The two strands we have chosen to focus on are Number and Quantitative Reasoning. Both modules maintain a focus on sense-making in mathematics. We have not dealt with every mathematical topic an elementary teacher may encounter; instead, we believe that having students grapple with two strands substantively will serve them better as teachers than providing a potpourri of activities from several strands. The materials were written to facilitate 1) students' understanding of the content in the materials; and 2) the development of habits of mind to make sense of the mathematics that they learn now and in the future. As you read through the materials, you will find that there is some overlap in content between the Number and Number Sense module and the Quantitative Reasoning module. We have noted the sections of overlap in the side notes of the instructor's version of the modules. We have included only one section of overlap (Section 1.1 of both modules) in the path below.

Number and Number Sense module

Section 1 Expressing Values of Quantities (only include section 1.2-scientific notation if time permits)
Section 2 Numeration Systems
Section 3 Students' Understanding of Operations and Methods for Computing
Section 4 Some Conventional Ways of Computing
Section 5 Using Numbers in Sensible Ways
Section 6 Meanings for Fractions
Section 7 Computing with Fractions

Quantitative Reasoning module

Section 1 What is a Quantity? (You many want to quickly review this section; remind your students that you discussed the meanings of quantity and value in section one of the Number and Number Sense module)
Section 2 Quantitative Analysis
Section 5 Additive Combinations and Comparisons
Section 6 Multiplicative Comparisons
Section 7 Ratio as a Measure

Two Quarters of Mathematics Content Courses
for Preservice Elementary Teachers

Note to the instructor:
Below we have listed a suggested path for a two quarter mathematics content course sequence for preservice elementary teachers. The three strands we have chosen to focus on are Number, Quantitative Reasoning, and Geometry. All three modules maintain a focus on sense-making in mathematics. We have not dealt with every mathematical topic an elementary teacher may encounter; instead, we believe that having students grapple with three strands substantively will serve them better as teachers than providing a potpourri of activities from several strands. The materials were written to facilitate 1) students' understanding of the content in the materials; and 2) the development of habits of mind to make sense of the mathematics that they learn now and in the future. As you read through the materials, you will find that there is some overlap in content between the Number and Number Sense module and the Quantitative Reasoning module. We have noted the sections of overlap in the side notes of the instructor's version of the modules. We have included only one section of overlap (Section 1.1 of both modules) in the path below.

First Quarter

Number and Number Sense module

Section 1 Expressing Values of Quantities (only include section 1.2-scientific notation if time permits)
Section 2 Numeration Systems
Section 3 Students' Understanding of Operations and Methods for Computing
Section 4 Some Conventional Ways of Computing
Section 5 Using Numbers in Sensible Ways
Section 6 Meanings for Fractions
Section 7 Computing with Fractions

Second Quarter

Quantitative Reasoning module

Section 1 What is a Quantity? (You many want to quickly review this section; remind your students that you discussed the meanings of quantity and value in section one of the Number and Number Sense module)
Section 2 Quantitative Analysis
Section 5.1 Additive Combinations and Comparisons-Quantitative Analysis
Section 6 Multiplicative Comparisons
Section 7 Ratio as a Measure

Shapes module

Section 5.1 Size Changes in Planar Figures (relate back to multiplicative reasoning/ratio as a measure)
Section 9.1 Key Ideas of Measurement
Section 9.2 Length and Angle Size
Section 9.3 Area
Section 9.5 Counting Units Fast (This is a particularly long section. Discuss formulas for volume as time permits)

Two Semesters of Mathematics Content Courses
for Preservice Elementary Teachers

Note to the instructor:
Below we have listed a suggested path for a two semester mathematics content course sequence for preservice elementary teachers. The four strands we have chosen to focus on are Number, Quantitative Reasoning, Geometry, and Probability and Statistics. All four modules maintain a focus on sense-making in mathematics. The materials were written to facilitate 1) students' understanding of the content in the materials; and 2) the development of habits of mind to make sense of the mathematics that they learn now and in the future. As you read through the materials, you will find that there is some overlap in content between the Number and Number Sense module and the Quantitative Reasoning module. We have noted the sections of overlap in the side notes of the instructor's version of the modules. We have included only one section of overlap (Section 1.1 of both modules) in the path below.

First Semester

Number and Number Sense module

Section 1 Expressing Values of Quantities (only include section 1.2-scientific notation if time permits)
Section 2 Numeration Systems
Section 3 Students' Understanding of Operations and Methods for Computing
Section 4 Some Conventional Ways of Computing
Section 5 Using Numbers in Sensible Ways
Section 6 Meanings for Fractions
Section 7 Computing with Fractions

Quantitative Reasoning module

Section 1 What is a Quantity? (You many want to quickly review this section; remind your students that you discussed the meanings of quantity and value in section one of the Number and Number Sense module)
Section 2 Quantitative Analysis
Section 5.1 Additive Combinations and Comparisons-Quantitative Analysis
Section 6 Multiplicative Comparisons
Section 7 Ratio as a Measure

Second Semester

Shapes module

Section 5.1 Size Changes in Planar Figures (relate back to multiplicative reasoning/ratio as a measure)
Section 9.1 Key Ideas of Measurement
Section 9.2 Length and Angle Size
Section 9.3 Area
Section 9.5 Counting Units Fast (This is a particularly long section. Discuss formulas for volume as time permits)
Section 9.6 The Pythagorean Theorem (at time permits)

Chance and Data module

Section 1.2 Methods of Assigning Probabilities
Section 1.3 Simulating Probabilities
Section 1.4 Determining More Complicated Probabilities
Section 2.3 Finding a Good Sample
Section 2.5 Measuring Group Characteristics
Section 3.1 Graphical Representations of (Univariate) Data (Do only the sections on bar graphs, stem-and-leaf plots, and histograms
Section 3.2 Examining the "Spreadoutness" of Data
Section 3.3 Means and Standard Deviations

We also use these materials for semester-long professional development courses for practicing teachers. Workshops with teachers could focus on brief selections of the units, such as the sections on mental computation and estimation or on students' understanding of operations and methods of computing from Numbers and Number Sense, from the sections on measurement in Shapes and Measurement, or on the first few sections of Quantitative Reasoning. The Change module is a particularly good one to use with middle grade teachers.

Development Team

Professors Judith T. Sowder (Project Director), Larry Sowder, Alba Thompson (now deceased), Patrick Thompson, Janet Bowers, Joanne Lobato, and Randolph Philipp.
Graduate students Jamal Bernhard, Lisa Clement, Melissa Mellissinos, Susan Nickerson, and Daniel Siebert.

For more information, contact Judith Sowder by e-mail (jsowder@sciences.sdsu.edu), phone (619-594-1587), or at the following address:
Judith Sowder
Center for Research in Mathematics and Science Education
6475 Alvarado Road, Suite 206
San Diego, CA 92120

References

Hersh, R. (1986). Some proposals for revising the philosophy of mathematics. In T. Tymoczko (Ed.), New directions in the philosophy of mathematics (pp. 9-28). Boston: Birkhäuser.

National Council of Teachers of Mathematics. (2000). Principles and Standards of School Mathematics. Reston, VA: Author.

Note: These materials were developed with funding from the National Science Foundation, ESI-9354104. The developers are solely responsible for the contents of this courseware. The materials do not necessarily reflect the position, policy, or endorsement of the National Science Foundation.

Last modified on January 8, 2001.