Preservice teachers learning to teach proof through classroom implementation: Successes and challenges

Proof and reasoning are central to learning mathematics with understanding. Yet proof is seen as challenging to teach and to learn. In a capstone course for preservice teachers, we developed instructional modules that guided prospective secondary mathematics teachers (PSTs) through a cycle of learning about the logical aspects of proof, then planning and implementing lessons in secondary classrooms that integrate these aspects with traditional mathematics curriculum in the United States. In this paper we highlight our framework on mathematical knowledge for teaching proof and focus on some of the logical aspects of proof that are seen as particularly challenging (four proof themes). We analyze 60 lesson plans, video recordings of a subset of 13 enacted lessons, and the PSTs’ self- reported data to shed light on how the PSTs planned and enacted lessons that integrate these proof themes. The results provide insights into successes and challenges the PSTs encountered in this process and illustrate potential pathways for preparing PSTs to enact reasoning and proof in secondary classrooms. We also highlight the design principles for supporting the development of PSTs’ mathematical knowledge for teaching proof.     

Examining preservice teachers’ noticing of equity-based teaching practices to empower students engaging in productive struggle

Our study examined ways preservice teachers (PSTs) make connections between teaching practices and use of student resources that support productive struggle and promote equity. Our research questions are: (1) How do PSTs notice and describe the equity-based mathematics teaching practice of leveraging student resources to support student struggles? and (2) In what ways do PSTs make connections to and interpret the role student resources play in the resolution of students’ mathematical struggles? The qualitative study examined 39 PSTs in a mathematics content course for PSTs. Data come from a video analysis assignment where PSTs described their mathematical interpretations of the student struggle(s) and teacher’s use of student resources to support the struggle resolution. Findings show that PSTs noticed teacher moves that leveraged student’s mathematical thinking and linguistic funds of knowledge and based the productive level of the struggle on actions built upon peers, linguistic knowledge and prior mathematical knowledge.     

Elementary Preservice Teachers’ Transitional Conceptions of Partitive Division with Proper-Fraction Divisors

An understanding of partitive division is foundational for numerous other mathematics topics, including unit rate, slope, and probability. However, research has shown that learners tend to have a limited understanding of partitive division when the divisor is a proper fraction. To extend research on conceptions of partitive division in this study, we used Moschkovich’s (1999) transitional conceptions perspective to examine how conceptions of partitive division evolve. This article reports on preservice teachers’ (PSTs) conceptions of partitive division with proper-fraction divisors before and after they explored partitive division in a mathematics content course for elementary teachers (N = 17). Our analysis of pre- and post-interviews revealed an initial transitional conception and two levels of refinement of their conceptions. Furthermore, we identified two perturbations that PSTs experienced during refinement of their conceptions. By identifying ordered levels of refinement and associated perturbations, this exploratory study extends the transitional conceptions perspective. Furthermore, this study adds new insights into the conceptual complexities that the partitive model for division of fractions presents to PSTs (and to learners in general) and suggests new hypotheses about ways that conceptions of partitive division undergo refinement.     

Applying NAEP to Improve Mathematics Content and Methods Courses for Preservice Elementary and Middle School Teachers

This study explored how mathematics content and methods courses for preservice elementary and middle school teachers could be improved through the integration of a set of instructional materials based on the National Assessment of Educational Progress (NAEP). A set of eight instructional modules was developed and tested. The study involved 7 university instructors and 542 preservice teachers (PSTs) from three different universities. A quasi‐experimental nonequivalent groups design was used for this study in which the following data sources were collected and analyzed. Three versions of a Learning Mathematics for Teaching test were given to assess PSTs‘ mathematical content knowledge for teaching: (a) Elementary Number Concepts and Operations—Content Knowledge; (b) Elementary Geometry—Content Knowledge; and (c) Middle School Number Concepts and Operations—Content Knowledge. In addition, the Mathematics Teacher Efficacy Beliefs Instrument was given to assess PSTs’ teacher efficacy beliefs. Test results were analyzed using paired samples t‐tests. Findings suggest that use of instructional materials, based on NAEP, with PSTs results in increases in their mathematical content knowledge for teaching and in their teaching efficacy beliefs.     

Transferring specialized content knowledge to elementary classrooms: preservice teachers’ learning to teach the associative property

This study explores how preservice teachers (PSTs) transfer the intended specialized content knowledge (SCK) to elementary classrooms. Focusing on the case of the associative property of multiplication, we compared three PSTs’ SCK during enacted lessons in fourth grade classrooms with their own learning in professional development (PD) settings. Findings revealed the PSTs’ successes and challenges in unpacking an example task, especially in areas of making connections between concrete and abstract representations and asking deep questions that target quantitative interactions. Factors that may have supported or hindered PSTs’ SCK transfer include the complex nature of teacher knowledge, the PD effort and the outside factors such as the support from textbooks and cooperating teachers. Implications for teacher education and directions for future research are discussed.     

The Growing Awareness Inventory: Building Capacity for Culturally Responsive Science and Mathematics With a Structured Observation Protocol

This study represents a first iteration in the design process of the Growing Awareness Inventory (GAIn), a structured observation protocol for building the awareness of preservice teachers (PSTs) for resources in mathematics and science classrooms that can be used for culturally responsive pedagogy (CRP). The GAIn is designed to develop awareness of: how students use language in classrooms; relationships between teacher questioning patterns and student participation; messages conveyed by the classroom environment; and ways to incorporate students’ interests into lesson plans. The methodology took the form of a multiple case study design with fourteen mathematics PSTs as one case and five science PSTs as the other case. The participants' response to the GAIn and lesson plans served as data sources. Findings reveal that the GAIn scaffolded PSTs’ awareness of their students, their own attitudes, and several elements of CRP. However, there were key areas of CRP that were neither explored with the GAIn nor identified by the participants. Consistent with design‐based research, outcomes include a design framework for revision of the GAIn and a theory of action that situates it within a teacher education course that includes a field placement.     

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.     

A Problem-Solving Based Mathematics Course and Elementary Teachers' Beliefs

This study examined the impact a problem-solving based mathematics content course for preservice elementary education teachers (PSTs) had on challenging the beliefs they held with respect to mathematics and themselves as doers of mathematics. Nine PSTs were interviewed to gain insight into changes that occurred to their belief systems and what aspect(s) of the course were instrumental in producing those changes. Surveys to measure how strongly PSTs subscribed to five mathematical beliefs were administered to 137 PSTs who were enrolled in the course. Significant positive changes (p < .01) were observed for four of the five beliefs. When changes were studied by achievement level, students with final grades of A or B showed statistically significant changes (p < .005) in three of the five beliefs. Students who were interviewed consistently reported increased confidence in their mathematical abilities as a result of the course.     

Preservice teachers’ pictorial strategies for a multistep multiplicative fraction problem

Previous research has documented that preservice teachers (PSTs) struggle with understanding fraction concepts and operations, and misconceptions often stem from their understanding of the referent whole. This study expands research on PSTs’ understanding of wholes by investigating pictorial strategies that 85 PSTs constructed for a multistep fraction task in a multiplicative context. The results show that many PSTs were able to construct valid pictorial strategies, and the strategies were widely diverse with respect to how they made sense of an unknown referent whole of a fraction in multiple steps, how they represented the wholes in their drawings, in which order they did multiple steps, and which type of model they used (area or set). Based on their wide range of pictorial strategies, we discuss potential benefits of PSTs’ construction of their own representations for a word problem in developing problem solving skills.     

Computational estimation skill of preservice teachers: operation type and teacher view

Despite the importance of computational estimation skill for the improvement of number sense, little research exists on preservice teachers’ estimation skills and their view on estimation in the US context. This study examined the computational estimation skill of 58 preservice elementary teachers (PSTs) and its relationship to their views of the meaning of estimation and the importance of teaching it. Three sets of instruments were used: an estimation task, a computational task, and a belief survey. Results indicated that PSTs performed differently depending on the types of operations on the estimation test. It was also found that different types of problems elicited different strategies. Furthermore, the intervention of the study, along with five other factors were found to significantly correlate with estimation skills. The five factors include PSTs’ mathematical knowledge, their reported confidence about estimation skills, their self-reported knowledge about calculator use in instruction, their views of estimation in teaching mathematics, and their definition of estimation. A negative correlation was documented for the knowledge of calculator use in instruction, and positive correlations were present for other factors. Implications are discussed in accordance with these findings.     

Preservice mathematics teachers' conceptions and enactments of modeling standards

Mathematical modeling has been highlighted recently as Common Core State Standards for Mathematics (CCSSM) included Model with Mathematics as one of the Standards for Mathematical Practices (SMP) and a modeling strand in the high school standards. This common aspect of standards across most states in the United States intended by CCSSM authors and policy makers seems to mitigate the diverse notions of mathematical modeling. When we observed secondary mathematics preservice teachers (M‐PSTs) who learned about the SMP and used CCSSM modeling standards to plan and enact lessons, however, we noted differences in their interpretations and enactments of the standards, despite their attendance in the same course sections during a teacher preparation program. This result led us to investigate the ways the M‐PSTs understood modeling standards, which could provide insights into better preparing teachers to teach mathematical modeling. We present the contrasting ways in which M‐PSTs presented modeling related to their conceptions of mathematical modeling, choices of problems, and enactments over an academic year, connecting their practices to extant research. We consider this teaching and research experience as an opportunity to make significant changes in our instruction that may result in our students enhanced implementation of mathematical modeling.     

Preparation for Practice: Elementary Preservice Teachers Learning and Using Scientific Classroom Discourse Community Instructional Strategies

Despite historical national efforts to improve elementary science education, science instruction continues to be marginalized, varying by state. This study was designed to address the ongoing challenge of educating elementary preservice teachers (PSTs) to teach science. Elementary PSTs are one of the science education community's major links to schools and science education reform. However, they often lack a strong background in science, knowledge of effective science teaching strategies, and consequently have low confidence and self‐efficacy. This investigation explored the initial learning of elementary PSTs using an interdisciplinary model of a scientific classroom discourse community during a science methods course. Findings post‐methods course suggested that the PSTs gained confidence in how to teach inquiry‐based elementary science and recognized inquiry‐based science as an effective means for engaging student learning. Additionally, PSTs embraced the interdisciplinary model as one that benefits students' learning and effectively uses limited time in a school day.     

Supporting disciplinary and interdisciplinary knowledge development and design thinking in an informal,pre‐engineering program: A Workplace Simulation Project

In this paper, we examined students’ engagement in an implementation of a Workplace Simulation Project (WSP). The WSP was designed to actively engage students in learning disciplinary content by inviting engineers from industry to have a physical presence within the school building to collaborate with teachers and students to complete projects which simulate the tasks authentic to their work. We focus on the first year implementation of the program that partnered a high school in the rural Midwest with an engineering unit of a government organization. Using a multiple methods study design, we analyzed disciplinary and interdisciplinary pre and posts test along with students’ interviews to determine learning gains as well as students’ interpretations of creative and critical thinking as experienced in the project and their knowledge of the engineering design process. Effect sizes showed that students in the WSP group had notable gains over the control group participants. Additionally, students’ knowledge of core elements of the design process were identified in inductive analyses of the interviews. Findings from this study will provide usable knowledge about effective ways to support systems and design thinking and ways to support expert‐novice collaboration to ensure success.     

The generation and use of graphical examples in calculus classrooms: The case of the mean value theorem

We analyzed video data of five instructors teaching the Mean Value Theorem (MVT) in a first-semester calculus course as part of a broader project investigating how active learning strategies were being implemented and supported in calculus courses. We sought to identify the ways examples of functions that did or did not satisfy the conclusion of MVT were generated and used in instruction. Using thematic analysis, we identified four themes that serve as characterizations of examples, which then allowed for the analysis of trends and patterns. We propose that attention to the generation and use of examples serves as one lens for considering how students can be engaged in the mathematical activity of the classroom, with implications for learning. This work contributes to an evolving notion of what is entailed in students’ active learning of mathematics and the role of the instructor in facilitating active learning opportunities.     

Cultivating Culture: Preparing Future Teachers for Diversity Through Family Science Learning Events

Preservice teachers (PSTs) participated in Family Science Learning Events (FSLEs) at a university designated as a Hispanic Serving Institution. PSTs were required by the instructor to conceive and design culturally relevant science activities as well as implement these activities with K‐8 students and their families during three separate FSLEs each semester. After school venues included elementary and middle schools located in ethnically diverse school districts. Data collected from these future teachers included qualitative PST reflections, lesson plans, project board/activity evaluation by peers, and a quantitative survey instrument (modified SEBEST) to assess PSTs perceptions of teaching diverse learners. Results suggest that using FSLEs as an integral component of teacher preparation can be a powerful facilitator of learning for all involved, increasing excitement for learning, confidence in using culturally relevant activities and valuable experience in working with family members, particularly Hispanics. In addition, using culturally relevant science activities deepened content knowledge and gave PSTs the opportunity to use culturally responsive activities with Hispanic students and their families, increasing feelings of self‐efficacy in science teaching with diverse learners.     

Opportunity to learn problem solving in Dutch primary school mathematics textbooks

In the Netherlands, mathematics textbooks are a decisive influence on the enacted curriculum. About a decade ago, Dutch primary school mathematics textbooks provided hardly any opportunities to learn problem solving. In this study we investigated whether this provision has changed. In order to do so, we carried out a textbook analysis in which we established to what degree current textbooks provide non-routine problem-solving tasks for which students do not immediately have a particular solution strategy at their disposal. We also analyzed to what degree textbooks provide ‘gray-area’ tasks, which are not really non-routine problems, but are also not straightforwardly solvable. In addition, we inventoried other ways in which present textbooks facilitate the opportunity to learn problem solving. Finally, we researched how inclusive these textbooks are with respect to offering opportunities to learn problem solving for students with varying mathematical abilities. The results of our study show that the opportunities that the currently most widely used Dutch textbooks offer to learn problem solving are very limited, and these opportunities are mainly offered in materials meant for more able students. In this regard, Dutch mainstream textbooks have not changed compared to the situation a decade ago. A textbook that is the Dutch edition of a Singapore mathematics textbook stands out in offering the highest number of problem-solving tasks, and in offering these in the materials meant for all students. However, in the ways this textbook facilitates the opportunity to learn problem solving, sometimes a tension occurs concerning the creative character of genuine problem solving.     

Pre-service elementary teachers’ strategies for tiling and relating area units

It has been established that preservice elementary school teachers (PSTs) often employ procedural methods when solving measurement problems without conceptual understanding or flexibility, but a significant gap exists in the literature identifying why. Through the lens of discrete and continuous interpretations of area, this study extends the research base by describing strategies PSTs use to tile a two-dimensional space with varying size tiles and what these strategies imply about PSTs’ conceptions of area measurement. These strategies and implied conceptions enable further discussion on the multiple purposes of the area model as an illustrative measure for mathematics throughout the elementary school curriculum.     

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.     

Beyond statistical methods: teaching critical thinking to first-year university students

We discuss a major change in the way we teach our first-year statistics course. We have redesigned this course with emphasis on teaching critical thinking. We recognized that most of the students take the course for general knowledge and support of other majors, and very few are planning to major in statistics. We identified the essential aspects of a first-year statistics course, given this student mix, focusing on a simple question, ‘Given this is the last chance you have to teach statistics, what are the essential skills students need?’ We have moved from thinking about statistics skills needed for a statistician to skills needed to participate in today's society. We have changed the way we deliver the course with less emphasis on lectures and more on alternative resources including on-line tutorials, Excel, computer-based skills testing, web-based learning materials and smaller group activities such as study groups and example classes. Feedback from students shows that they are very receptive and enthusiastic.     

Identification and definition of lexically ambiguous words in statistics by tutors and students

Lexical ambiguity arises when a word from everyday English is used differently in a particular discipline, such as statistics. This paper reports on a project that begins by identifying tutors’ perceptions of words that are potentially lexically ambiguous to students, in two different ways. Students’ definitions of nine lexically ambiguous words are also collected at the beginning and end of a semester of introductory statistics study, in a complex design taking account of multiple tutors and multiple words in multiple contexts. Tutor perceptions and actual student difficulties at the beginning of a semester are compared. The lexical ambiguity associated with the word ‘significance’ is shown to be evident in students even after completing an introductory statistics course.