The development of instructional guidelines for elementary mathematical writing

Mathematical writing recently has been defined as writing to reason and communicate mathematically. But mathematics instructional resources lack guidance for teachers as to how to implement such writing. The purpose of this paper is to describe how methods of design-based research were used to develop an instructional resource when one does not currently exist. Thirty-four participants—including teachers, mathematics coaches, mathematics curriculum developers, literacy coaches, a mathematician, and academics in elementary mathematics education, mathematics education, writing education, and science education—participated in a multi-step process to recommend, revise, and confirm instructional guidelines for elementary mathematical writing. The development process began with thirty-two recommendations from science writing and language arts writing. Through multiple cycles of feedback, five instructional guidelines and related considerations and techniques for implementation of elementary mathematical writing emerged.     

Preservice Elementary Teachers' Use of Mathematics in a Project‐Based Science Approach

The use of a project‐based science (PBS) approach to teaching encourages students to integrate mathematics and science in meaningful ways as they create projects. As a beginning study of how students use mathematics in such an approach, an analysis of 23 projects developed by preservice elementary teachers enrolled in an elementary science course was conducted. Findings showed that students made a number of different types of mathematical errors and underutilized data representation and summary forms. Implications included the importance of developing methods for supporting the use of mathematical tools in utilizing a project‐based approach and considering ways that such tools mediate scientific thinking.     

‘I Finally Get It!’: developing mathematical understanding during teacher education

A deep conceptual understanding of elementary mathematics as appropriate for teaching is increasingly thought to be an important aspect of elementary teacher capacity. This study explores preservice teachers’ initial mathematical understandings and how these understandings developed during a mathematics methods course for upper elementary teachers. The methods course was supplemented by a newly designed optional course in mathematics for teaching. Teacher candidates choosing the optional course were initially weaker in terms of mathematical understanding than their peers, yet showed stronger mathematical development after engaging in the extra hours the optional course provided.     

Elementary teachers’ mathematical beliefs and mathematics anxiety: How do they shape instructional practices?

This quantitative study investigated the relationships among practicing elementary teachers’ (N = 153) beliefs about mathematics and its teaching and learning, mathematics anxiety, and instructional practices in mathematics. When viewed singly, the findings reveal the teachers with higher levels of mathematics anxiety tend to use less standards‐based instruction and those with beliefs oriented toward a problem‐solving view of mathematics reported more standards‐based teaching. A combined analysis shows that after controlling for mathematical beliefs, teaching longevity, and educational degree attainment, there is no relationship between teachers’ mathematics anxiety and instructional practices. These findings suggest a spurious relationship between anxiety and practices, with beliefs having the strongest relationship with practices. Several suggestions for positively influencing the mathematical beliefs and affect in general of elementary teachers while learning mathematics are offered.     

Mathematics Anxiety and Preservice Elementary Teachers' Confidence to Teach Mathematics and Science

Sixty‐five preservice elementary teachers' math anxiety levels and confidence levels to teach elementary mathematics and science were measured. The confidence scores of subjects in different math anxiety groups were compared and the relationships between their math anxiety levels and confidence levels to teach mathematics and science were investigated. The results suggest that low math anxious preservice teachers are more confident to teach elementary mathematics and science than are their peers having higher levels of math anxiety. Negative correlations were found between preservice teachers' math anxiety and their confidence scores to teach elementary mathematics (r = ?.638) and between preservice teachers' math anxiety and their confidence scores to teach elementary science (r = ‐.417). Also, personal math and science teaching self‐efficacy scores of participants were found to be correlated at .01 level (r =.549).     

Construction of Teacher Knowledge in Context: Preparing Elementary Teachers to Teach Mathematics and Science

The purpose of this study was to further the understanding of how preservice teachers construct teacher knowledge and pedagogical content knowledge of elementary mathematics and science in a school‐based setting and the extent of knowledge construction. Evidence of knowledge construction (its acquisition, its dimensions, and the social context) was collected through the use of a qualitative methodology. The methods course was content‐specific with instruction in elementary mathematics and science. Learning experiences were based on national standards with a constructivist instructional approach and immediate access to field experiences. Analysis and synthesis of data revealed an extensive acquisition of teacher knowledge and pedagogical content knowledge. Learning venues were discovered to be the conduits of learning in a situated learning context. As in this study, content‐specific, school‐based experiences may afford preservice teachers greater opportunities to focus on content and instructional strategies at deeper levels; to address anxieties typically associated with the teaching of elementary mathematics and science; and to become more confident and competent teachers. Gains in positive attitudes and confidence in teaching mathematics and science were identified as direct results of this experience.     

Using a Functional Model to Develop a Mathematical Formula

The unifying theme of models was incorporated into a required Science Capstone course for pre‐service elementary teachers based on national standards in science and mathematics. A model of a teeter‐totter was selectedfor use as an example of a functional model for gathering data as well as a visual model of a mathematical equation for developing the mathematical relationship for a Class 1 lever, M1D1=M1D1. In this study, 20 student groups (n=72) collected data using the model in an inquiry‐based activity. All groups developed the qualitative relationship, 13 groups developed a correct mathematical formula, 6 groups developed one‐half of the relationship (X = mass × distance), and 1 group attempted to develop a procedural relationship. The pre‐service elementary teachers used a variety of model types in the activity including visual/pictorial, functional/physical and mathematical‐both graphs and formulas. The use of the teeter‐totter model as a visual and functional model of a mathematical formula was a factor in developing the mathematical relationship.     

Teachers’ responses to student mathematical thinking in Chinese elementary mathematics classrooms

Recent research on teachers’ use of student mathematical thinking (SMT) and recommendations for effective mathematics instruction claim that how teachers respond to SMT has great impact on student mathematical learning in the classroom. This study examined some Chinese mathematics teachers’ responses to student in-the-moment mathematical thinking that emerged during whole class discussion. The findings of this study revealed that the majority of Chinese elementary mathematics teachers in the data involved the whole group of students to make sense of in-the-moment SMT. They either invited students to digest SMT involved in the instance or provided an extension of the instance to further develop student mathematical understanding.     

Teaching Science as Science Is Practiced: Opportunities and Limits for Enhancing Preservice Elementary Teachers' Self‐Efficacy for Science and Science Teaching

Science teaching in elementary schools, or the lack thereof, continues to be an area of concern and criticism. Preservice elementary teachers' lack of confidence in teaching science is a major part of this problem. In this mixed‐methods study, we report the impacts of an inquiry‐based science course on preservice elementary teachers' self‐efficacy for science and science teaching, understanding of science, and willingness to teach it in their future careers. Our findings suggest that for some students, the inquiry‐based science course positively influenced their self‐efficacy for science and science teaching. Gains were made in a majority of students' conceptual understanding of science, understanding of the science process and scientific research, and confidence with science and science teaching. The subjects did not experience the course uniformly, however. Rather, there appeared to be two distinct groups, one on a trajectory of improving their outlook on science teaching and one worsening. The results presented here therefore provoke some interesting questions regarding preservice elementary teachers' preparation for science teaching.     

Preservice teachers’ use of mathematics and science learning processes

Mathematics and science have similar learning processes (SLPs) and it has been proposed that courses focused on these and other similarities promote transfer across disciplines. However, it is not known how the use of these processes in lessons taught to children change throughout a preservice teacher education course or which are most likely to transfer within and between disciplines. Three hundred and ninety lesson plans written by 113 preservice teachers (PSTs) from 10 sections of an elementary mathematics/science methods course were analyzed. PSTs taught an eight‐lesson sequence to children: five science lessons followed by three mathematics lessons. The findings suggested that: (a) PSTs needed to only teach three mathematics lessons, after five science lessons, to reach the same number of SLPs used in the five science lessons; (b) some SLPs are highly correlated processes (HCPs) and are more likely to transfer within and between science and mathematics lessons; and (c) PSTs needed to teach no mathematics lessons, after four science lessons, to reach the same number of HCPs used in the four science lessons. Implications include centering courses on multiple and varied representations of learning processes within problem‐solving, and HCPs may be essential similarities of problem‐solving which promote transfer.     

Challenging Preservice Teachers' Mathematical Understanding: The Case of Division by Zero

Preservice elementary school teachers' fragmented understanding of mathematics is widely documented in the research literature. Their understanding of division by 0 is no exception. This article reports on two teacher education tasks and experiences designed to challenge and extend preservice teachers' understanding of division by 0. These tasks asked preservice teachers to investigate division by 0 in the context of responding to students' erroneous mathematical ideas and were respectively structured so that the question was investigated through discussion with peers and through independent investigation. Results revealed that preservice teachers gained new mathematical (what the answer is and why it is so) and pedagogical (how they might explain it to students) insights through both experiences. However, the quality of these insights were related to the participants' disposition to justify their thinking and (or) to investigate mathematics they did not understand. The study's results highlight the value of using teacher learning tasks that situate mathematical inquiry in teaching practice but also highlight the challenge for teacher educators to design experiences that help preservice teachers see the importance of, and develop the tools and inclination for, mathematical inquiry that is needed for teaching mathematics with understanding.     

A false belief about fractions – What is its source?

This paper presents the results of interviews with 174 participants solving a problem of elementary mathematics, connected with the part–whole approach to fractions. The motive for the investigation was a specific kind of difficulty observed during a case study conducted to verify the elementary school student's understanding of the concept of fractions. The authors decided to examine the problem in a broader population of mathematics learners at different levels of education: from elementary school to university students and graduates of science majors. Approximately 65% of respondents reported the wrong answer immediately after reading the fraction problem taken from the fourth grade of elementary school. Detailed analysis of the respondents’ performance showed that the source of many wrong answers was a false belief about fractions: The only way to get 1/n of a given whole is to divide this whole into n equal parts, not yet described in educational literature.     

The Influences of Three Interventions on Prospective Elementary Teachers' Beliefs About the Knowledge Base Needed for Teaching Mathematics

To meet the challenge to reform mathematics education, effective opportunities to learn are needed to promote prospective elementary school teachers' development of the knowledge base that supports teaching for mathematical proficiency. This article describes three professional development interventions and their influence on prospective teachers' beliefs about mathematics, how children learn mathematics, and mathematics teaching. The three interventions consisted of problem‐solving journals, structured interviews, and peer teaching that were integrated in a PreK‐6 mathematics methods course. Results of precourse and postcourse survey data are included that measured 24 prospective teachers' beliefs about the knowledge base needed to teach elementary school mathematics. Data indicated that using these interventions and other course experiences facilitated change in the prospective teachers' beliefs, with a shift toward reform‐oriented mathematics education perspectives.     

Problems Identifying Independent and Dependent Variables

This paper discusses one step from the scientific method—that of identifying independent and dependent variables—from both scientific and mathematical perspectives. It begins by analyzing an episode from a middle school mathematics classroom that illustrates the need for students and teachers alike to develop a robust understanding of independent and dependent variables. It then outlines four rationales (two from science and two from mathematics) for identifying independent and dependent variables. Finally, it reports the results of a textbook analysis that used these rationales to examine the extent to which typical mathematics textbook problems support or supplant a sensible view of independent and dependent variables. The findings indicate that often, mathematics textbook problems misrepresent the sense‐making aspect of identifying independent and dependent variables, possibly setting students up to develop misconceptions about this step from the scientific method.     

Writing as a Tool to Demonstrate Mathematical Understanding

In this study, I examine how using a writers' workshop model in mathematics creates a space for students to write about their mathematical thinking and problem solving and how their writing impacts instruction. This case study of one classroom with one teacher spanned 6 weeks and included 18 implementations of an adapted version of the Writers' Workshop (WW) in a fourth‐grade mathematics class. On a biweekly basis, the data were reviewed and changes made to the model. The analysis of the students' writing revealed (a) their understandings and misunderstandings of the mathematical content, (b) their readiness for more challenging tasks, and (c) their connections to prior knowledge. Students used writing to demonstrate their understanding of mathematics and show their mathematical processes. In some cases, examining only the numerical work failed to illuminate the students' understanding, their writing provided deeper insight. Students recognized writing as a tool for learning; this was evident in interview responses.     

Promoting Preservice Elementary Teachers' Awareness of Learning and Teaching Mathematics Conceptually Through KTEM

In this article, we describe preservice elementary teachers' reactions to Liping Ma's (1999) book, Knowing and Teaching Elementary Mathematics (KTEM), from five universities. Ma's discussion of solely teaching elementary mathematics procedurally and its consequences awakens the preservice teachers' memories of learning elementary mathematics. Ma's analysis of and solution to the problem ignites strong emotions in the preservice elementary teachers and promotes a desire to teach elementary mathematics conceptually. Through the analysis of writing assignments, we summarize how reading and reflecting on KTEM gives preservice teachers an opportunity to examine their beliefs about teaching and learning elementary mathematics conceptually.     

Developing teacher content understanding by integrating pH and logarithms concepts

Logarithms are notorious for being a difficult concept to understand and teach. Research suggests that learners can be supported in understanding logarithms by making connections between mathematics and science concepts such as pH. This study investigated how an integrated chemistry and mathematics lesson impacted 29 teachers’ understanding of the logarithmic relationship and pH. Pre- and post-test data indicated 23 teachers improved their understanding of logarithms and 28 improved their understanding of pH, suggesting that teacher educators in both science and mathematics context can use this approach to foster better understanding with their teachers and ultimately school students. Our analysis also identified professional development components and teacher characteristics associated with gains in understanding of pH and logarithms, which mathematics and science teacher educators can use to strategically adapt and implement the lesson within other teacher education settings.     

Enhancing student engagement to positively impact mathematics anxiety,confidence and achievement for interdisciplinary science subjects

Contemporary science educators must equip their students with the knowledge and practical know-how to connect multiple disciplines like mathematics, computing and the natural sciences to gain a richer and deeper understanding of a scientific problem. However, many biology and earth science students are prejudiced against mathematics due to negative emotions like high mathematical anxiety and low mathematical confidence. Here, we present a theoretical framework that investigates linkages between student engagement, mathematical anxiety, mathematical confidence, student achievement and subject mastery. We implement this framework in a large, first-year interdisciplinary science subject and monitor its impact over several years from 2010 to 2015. The implementation of the framework coincided with an easing of anxiety and enhanced confidence, as well as higher student satisfaction, retention and achievement. The framework offers interdisciplinary science educators greater flexibility and confidence in their approach to designing and delivering subjects that rely on mathematical concepts and practices.     

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.     

The Tabular Mode: Not Just Another Way to Represent a Function

Understanding mathematical functions as systematic processes involving the covariation of related variables is foundational in learning mathematics. In this article, findings are reported from two investigations examining students' thinking processes with functions. The first study focused on seven middle school students' explorations with a dynamic physical model. Students were videotaped during the 20‐ to 45‐minute sessions occurring two or three times per week over a period of 2 months, and students' written work was collected. The second investigation included 19 preservice elementary and middle school teachers enrolled in a course focusing on a combination of mathematical content and pedagogy. Participants' written problem‐solving work and reflective writing were collected, and participants were individually interviewed in 50‐minute videotaped sessions. Results from both investigations indicated that students often relied on a table, or some variation of a table, as a cognitive link advancing the development of their reasoning about underlying function relationships.