Mathematical Modeling for Preservice Teachers: A Problem from Anesthesiology

The study reported in this article deals with the observed actions of prospective Swedish mathematics teachers as they were working with a modeling situation. These prospective teachers were preparing to teach in Grades 4 to9 or in the gymnasium (Grades 10 to 12) and were students in a course in mathematical modeling. The larger study of which this study was a part focused on these students' understanding of modeling and how they related mathematical models to the real world. This article also attempts to illustrate how mathematics is intertwined with many other subjects, in this case medicine. This revised version was published online in July 2006 with corrections to the Cover Date.     

Using Performance Assessment to Engage Preservice Teachers in Mathematical Discourse

This article describes one mathematics educator's inclusion of an oral performance assessment into mathematics methods courses for preservice teachers at both the elementary and middle school levels. The nature of the assessment is described, issues related to implementation are discussed, and reflections about the assessment from the instructor and the students are shared.     

Cultivating Computational Thinking Practices and Mathematical Habits of Mind in Lattice Land

There is a great deal of overlap between the set of practices collected under the term “computational thinking” and the mathematical habits of mind that are the focus of much mathematics instruction. Despite this overlap, the links between these two desirable educational outcomes are rarely made explicit, either in classrooms or in the literature. This paper presents Lattice Land, a computational learning environment and accompanying curriculum designed to support the development of mathematical habits of mind and promote computational thinking practices in high-school mathematics classrooms. Lattice Land is a mathematical microworld where learners explore geometrical concepts by manipulating polygons drawn with discrete points on a plane. Using data from an implementation in a low-income, urban public high school, we show how the design of Lattice Land provides an opportunity for learners to use computational thinking practices and develop mathematical habits of mind, including tinkering, experimentation, pattern recognition, and formalizing hypothesis in conventional mathematical notation. We present Lattice Land as a restructuration of geometry, showing how this new and novel representational approach facilitates learners in developing computational thinking and mathematical habits of mind. The paper concludes with a discussion of the interplay between computational thinking and mathematical habits of mind, and how the thoughtful design of computational learning environments can support meaningful learning at the intersection of these disciplines.     

Writing Papers in Math Class: A Tool for Encouraging Mathematical Exploration by Preservice Elementary Teachers

This paper describes the author's attempt to design assignments that engage preservice elementary teachers in original mathematical thinking. In particular, the choice of integer operations as the focus of a structured writing assignment that takes students two weeks to complete is explained and justified. Exemplary student work is quoted.     

Preservice Teachers of High School Mathematics: Success,Failure, and Persistence in the Face of Mathematical Challenges

The goal of this study was to identify variables related to success and resilience in an undergraduate, high school mathematics teacher education program. Over a five‐year period, we tracked the academic performance and achievement motivation goals of multiple cohorts of students. Students who successfully completed their degrees had higher grade point average (GPAs) upon entering the program, earned higher grades in their first college mathematics course, and failed fewer courses than students who left the program or university. Learning and performance motivational goals did not predict success in the program. Performance goals decreased over time. Nearly half the successful students repeated one or more mathematics courses. Ten students completed their degrees, obtained a teaching license, and are teaching despite the need for multiple repetitions of the same mathematics courses. These persistent students did not differ from their peers in motivational goals. Our results suggest that although students with higher GPAs and initial mathematics grades were more likely to complete the program, students who experienced challenges in mathematics courses were able to succeed. We discuss the implications of these results for recruiting, advising, and retention of students in mathematics education programs.     

Toward Improving the Mathematics Preparation of Elementary Preservice Teachers

Research suggests the importance of mathematics knowledge for teaching (MKT) for enabling elementary school teachers to effectively teach mathematics. MKT involves both mathematical content knowledge (M‐CK) and mathematical pedagogical content knowledge (M‐PCK). However, there is no consensus on how best to prepare elementary preservice teachers (PSTs) to achieve M‐CK and M‐PCK. This study builds on research related to MKT by investigating influences of mathematics content courses designed specifically for elementary PSTs (IMPACT courses—Impact of Mathematics Pedagogy and Content on Teaching) on their attitudes (i.e., confidence and motivation) toward M‐CK and M‐PCK. Results suggest that the PSTs who participated in these IMPACT courses not only acquired high levels of confidence and motivation toward M‐CK, but also showed significant and greater gains in attitudes toward M‐PCK, after taking the required mathematics methods course, than their counterparts. Further, the findings suggest that these IMPACT courses provided a mathematical foundation that allowed the PSTs to engage in mathematics teaching methods better than those PSTs who did not have such a foundation. These results suggest potential course experiences that may enhance M‐CK and M‐PCK for elementary PSTs.     

Motivation and Attitude of Preservice Elementary Teachers toward Mathematics

This study investigated preservice elementary teachers' achievement goal orientations for learning mathematics and the relationship of those goals to their attitudes toward mathematics. Self‐report instruments were administered to assess three achievement goals—mastery, performance‐approach, and performance‐avoid, and three constructs of attitude—confidence in learning mathematics, usefulness of mathematics, and mathematics as a male domain. The preservice teachers were higher in mastery goals than in performance goals, and performance‐avoid goals were higher than performance‐approach goals. Mastery goals correlated positively to all three constructs of attitude. Since mathematics classes are traditionally performance‐oriented, these results suggest a mismatch between personal and classroom goals that could result in negative attitudes and the adoption of maladaptive performance‐avoid goals. These findings suggest that mathematics content courses for preservice elementary teachers should be taught in a classroom climate that supports and encourages mastery goals.     

Exploring Probability and Statistics with Preservice and Inservice Teachers

NCTM's mathematics curriculum and evaluation standards (1989) have provided educators with the challenge of revamping high school mathematics curricula as well as pedagogies by which content is taught. This article presents a lesson designed for preservice and inservice teachers that permits participants to: (a) strengthen their conceptual understanding, and (b) experience learning in a cooperative environment that encourages communication. The lesson engages participants in the collection and representation of probabilistic data using dice with 4, 6, 8, 10, 12, and 20 faces. Opportunities are provided for participants to discover patterns and construct mathematical knowledge concerning theoretical probability. Teacher educators can facilitate reform of mathematics education by developing and delivering such lessons.     

Learning the Learning Cycle: The Differential Effect on Elementary Preservice Teachers

This study examined the difficulties and factors that led to understanding the learning cycle teaching strategy. Participants included 83 preservice elementary teachers (PTs) enrolled in multiple sections of a science methods course taught by the same professor, one of the authors of this study. Analysis revealed that there were four categories of PTs, ranging from Enthusiastic to Fearful students, distinguished by their science content background and attitudes to science. High achieving students, successful in science courses, felt confused by the learning cycle that was so different from their previous science learning experiences and formed mindsets against learning it. Average students who expressed disinterest or even fear of science embraced it as their first successful science learning experience. Multiple exposures to the learning cycle were necessary to overcome these mindsets. Most PTs in all categories increased in their understanding of the learning cycle and perceived it as an effective method for allowing students to construct their understanding of science.     

A Tutorial Program: Collaboration Between Preservice and Inservice Teachers

Many elementary school teachers face severe time constraints working individually with students who need extra help or attention in mathematics. Sometimes, children who need this attention do not receive it. At the same time, critics of teacher education programs contend that preservice teachers do not receive enough hands-on experience teaching children. The Mathematics Tutorial Program is one attempt to address both of these issues. In this program, elementary grade children are identified by their classroom teachers as needing extra help in mathematics. They are paired with preservice elementary school teachers. Each preservice teacher tutors one or more elementary children for two 30-minute sessions each week for 6 to 24 weeks. They use manipulatives and a hands-on approach. The tutors reported professional gains from their contacts with classroom teachers, mathematics educators, and elementary children. Classroom teachers involved with the program commented on the children's improved selfesteem and confidence in mathematics. The children eagerly looked forward to the time with their tutors.     

How Preservice Teachers Interpret and Respond to Student Geometric Errors

Recognizing and responding to students' thinking is essential in teaching mathematics, especially when students provide incorrect solutions. This study examined, through a teaching scenario task, elementary preservice teachers' interpretations of and responses to a student's work on a task involving reflective symmetry. Findings revealed that a majority of preservice teachers identified the student's errors from conceptual aspects of reflection rather than from procedural aspects. However, when they responded to the student's errors, preservice teachers tried to cope with them by invoking procedural knowledge. This study also revealed the three types of responses and two different forms of address by preservice teachers to student errors; these categories might provide insight into the difficulties arising in communication between students and teachers.     

Level of Inquiry as Motivator in an Inquiry Methods Course for Preservice Elementary Teachers

Of great importance for achieving science education reform may be teachers' interest in science and enjoyment of science. This study explores the motivational qualities (rated for interest, fun, and learning value) of different levels of inquiry of hands‐on class activities. The participants, 53 preservice teachers in two sections of a science methods course, rated the activities at the end of each class. At the end of the course, these activities were categorized by level of inquiry (levels 0–3), with 30% rated as level 0 (no inquiry), 40% as level 1, 22% as level 2, and 8% as level 3, according to how much choice was given for posing questions and designing investigations. Ratings of each hands‐on activity indicated that participants perceived activities of higher levels of inquiry to be more fun and more interesting. They also perceived that they had learned more. These findings suggest that course instructors should determine level of inquiry when planning course activities, and degree of participant input into course activities may be important in the development of interest in science. A focus on hands‐on learning especially at higher levels of inquiry may serve both to capture the interest of teachers and to model how they can make science more authentic and engaging for children.     

Relationship Between Science and Mathematics Teaching Efficacy of Preservice Elementary Teachers

The purpose of this study was to investigate the change in teacher efficacy beliefs about mathematics and science teaching during participation in methods courses and student teaching, as well as the relationship between mathematics and science teaching efficacy. Data revealed that, as science and mathematics teacher education in a methods course progressed, science and mathematics teaching efficacy significantly increased. This effect appeared to decrease slightly by the end of student teaching. Analysis of data indicated a significant difference in both the personal mathematics and personal science teaching efficacy scores, as well as mathematics outcome expectancy. Additionally, preservice teachers' personal mathematics and science teaching efficacies were directly related, as were their mathematics and science teaching outcome expectancies.     

Traditional vs. Retrospective Pretests for Measuring Science Teaching Efficacy Beliefs in Preservice Teachers

The difference in gain scores produced by traditional pretests and those produced by retrospective pretests when compared to posttest scores on the Science Teaching Efficacy Belief Instrument for preservice teachers was investigated in this study. Results indicated that gain scores using the traditional pretest produced significant improvement in one factor, but not in the other, which is congruent with most other results for this instrument found in the literature. However, gain scores produced by using retrospective pretests produced a greater magnitude of significance for the first factor, and also produced significant gains for the second factor. Follow‐up interviews with participants provided evidence for greater internal validity for the retrospective pretest. Findings support the notion that retrospective pretests may produce gain scores with greater validity and greater statistical power.