Keywords

1 Introduction

This chapter deals with one of the essential online variables of research on teaching (i.e., pre-existing teacher characteristics) promoted by Donald M. Medley in his seminal work on the evolution of research on teaching (Medley, 1987). Medley developed a framework of variables that research in teaching from a presage-process–product perspective must be concerned with to effectively contribute to the understanding and improvement of teaching. This framework provided the theoretical basis for framing this book on “evolution of research on teaching mathematics” (Manizade, Buchholtz, & Beswick, Chap. 1, this volume). As Manizade et al. explained,

Medley’s framework is still valuable as it gives an orientation to all possible variables that become apparent qua the chain of effects from teacher behavior to student achievements. Moreover, the abiding challenges associated with the conceptualization, instrumentation, operationalization, and research design that Medley described are still complex, despite recent advances in technology and research methodology in the digital era. (p. 5)

However, for this book, Manizade et al. updated the framework to take into consideration cultural and epistemological contexts and digital contexts and to situate it within research on teaching mathematics (see Fig. “Updated framework of research on teaching mathematics”, Manizade et al., this volume).

Medley’s (1987) framework includes six types of essential “online variables”, that is, “ones which lie along a direct line of influence of the teacher on pupil learning” (p. 105). Medley labelled and sequenced these variables from Type F to Type A. This chapter deals with the Type F variable that is at the beginning of this direct line. Figures two and three in the introductory chapter illustrate these variables as presented by Medley for research on teaching and adapted by Manizade et al. for research on teaching mathematics (Manizade, Buchholtz, & Beswick, Chap. 1, this volume).

According to Medley (1987):

Pre-existing teacher characteristics include abilities, knowledge, and attitudes that a candidate for admission to a teacher preparation program possesses on entry; they make up a candidate’s aptitude for teaching. Part of it consists of the characteristics a teacher needs in order to acquire those competencies that training and experience can provide; part of it consists of those competencies that a teacher must possess on entry. (p. 105)

In relating it to mathematics teaching, Manizade et al. (this volume, p. 6) defined the Type F variable as “a mathematics teacher’s beliefs and aptitude for teaching, characteristics needed to acquire professional competencies during training.” This definition was adapted in this chapter to explore research of pre-existing mathematics teacher characteristics [PMTC] that prospective teachers possess on entry into a teacher education program or mathematics teacher education [MTE] as a necessary stage in understanding the mathematics teacher and mathematics teaching.

In addition, Medley’s four factors regarding methodological issues that research on teaching must deal with were adapted in this chapter to discuss the evolution of research on PMTC. These factors, discussed later, are conceptualization, instrumentation, design, and analysis. Medley explained that “evolution of research on teaching depends on advances made in how each has been dealt with” (p. 106).

In general, the chapter provides an overview of research that addressed PMTC of prospective teachers of mathematics [PTs] through a systematic review and synthesis of relevant published empirical studies for the period 2000 to 2020. It begins with an overview of the scope of the literature review, followed by an overview of the types and nature of PMTC covered in the studies reviewed, then a discussion of the evolution of the research on PMTC and suggestions regarding future evolution of research on PMTC.

2 Scope of Literature Review to Determine Studies of PMTC

Given the large body of literature on PTs, it was decided to focus only on high profile peer-reviewed international journals (Williams & Leatham, 2017) that likely included studies on PTs’ PMTC. They included: Educational Studies in Mathematics (ESM), Journal for Research in Mathematics Education (JRME), Journal of Mathematics Teacher Education (JMTE), Journal of Mathematical Behavior (JMB), International Journal of Science and Mathematics Education (IJSME), Mathematical Thinking and Learning (MTL), and ZDM—Mathematics Education. The author and a trained research assistant conducted a search of these journals for the period 2000–2020 using various combinations of keywords that included: prospective teachers; future teachers; teacher candidates; preservice teachers; student teachers; characteristics; competencies; abilities; knowledge; attitudes; beliefs; identity; and recruitment. Based on our review of titles and abstracts, we prepared a list of articles with potentially relevant studies. We examined these articles to determine whether participants were at the beginning of their MTE. This process produced very few studies. We then decided to focus on studies that addressed PTs’ characteristics at the beginning of a course or prior to an intervention during a course or in situations that reflected the nature of their background knowledge or ability (e.g., interpreting students’ work or evaluating tasks), which seemed to be a more promising approach to obtain studies of PMTC. The assumption was that these studies would suggest characteristics the PTs held prior to entering MTE if these characteristics were directly related to their school experiences with mathematics (e.g., mathematics curriculum content and pedagogy).

We obtained a large list of these studies by examining the methodology section of articles in our list. We then examined these studies to determine if the findings provided information that was clearly related to PMTC to identify those studies to exclude. Many intervention studies highlighted the changes resulting from the intervention but not the initial characteristics of the PTs and were removed from the list. Studies at the beginning of a course that investigated characteristics that were related to prior mathematics or mathematics education courses in the program were also removed from the list. In keeping with the theme of this book, all studies not situated in a Western context were also later removed. This process resulted in a list of 51 studies from the above-noted journals, to which were added a few studies from other journals based on citations of relevant studies in articles on this list. These studies were situated mainly in the USA, with some from different regions internationally. To highlight this, in reporting the studies, the countries are noted for those that were not situated in the USA.

For each article on the final list, the author and research assistant identified and recorded the PTs’ characteristics explicitly investigated based on the aim of the study. On examining the characteristics, we determined that they generally involved PTs’ mathematics knowledge, pedagogical knowledge, or beliefs, which became initial categories used to group the characteristics. The content of these categories consisted of, for each study, the type of characteristics, the aim of the study related to the characteristics and key findings regarding the nature of the characteristics. Further examination of the content of each of the three categories and cross-checking of findings for agreement between the researcher and research assistant, resulted in sub-groups of characteristics consisting of different types of mathematics concepts and skills, different types of pedagogical knowledge and ability, and different types of beliefs or conceptions, respectively. This process also validated that all the characteristics were appropriately accounted for and could be represented by three broad categories: pre-existing mathematical knowledge and skills, pre-existing mathematics-related pedagogical knowledge and ability, and pre-existing mathematics-related beliefs. These final categories, described in the next section, provided a landscape of PMTC related to the Type F variable that were researched in the period 2000–2020. We also documented examples of research tools, design and analysis that formed the basis of discussion of the evolution of research on the PMTC.

3 Landscape of PMTC Researched in 2000–2020

The studies reviewed provided a landscape of several PMTC researched in 2000–2020 regarding what PTs knew or were able to do on entering MTE. These PMTC, grouped in three categories, are presented in this section in terms of the aims of the studies, with examples of key findings of the studies related to the PTs’ aptitude for teaching mathematics. The goal is to provide an overview of PMTC for the three categories: pre-existing mathematical knowledge and skills, pre-existing mathematics-related pedagogical knowledge and ability, and pre-existing mathematics-related beliefs.

3.1 Pre-existing Mathematical Knowledge and Skills

This category consists of studies that investigated PTs’ mathematical knowledge and skills connected to school mathematics in the period 2000–2020. Collectively, these studies included primary, elementary, middle, and secondary school PTs and their knowledge of different content areas (i.e., fractions, whole number operations, geometry, algebra) and skills (i.e., problem posing). They addressed one category of PMTC that is central to teaching mathematics and important for PTs to have on entering teacher education. The following overview of these studies is organized by each content area and skill to highlight the extent to which they were addressed in terms of the aims of the studies and nature of the PTs’ PMTC and in reversed chronological order to indicate distribution in the period beginning with most recent studies.

3.1.1 Fractions

These studies on fractions focused mostly on elementary school PTs and addressed their knowledge of fractions in a variety of ways. During the second 10 years of the period: Lee and Lee (2020) investigated elementary school PTs’ exploration of model breaking points in fractions that included the area model of fraction addition. Most of the PTs represented fraction addition well with simple fractions but had difficulty representing fraction addition with improper fractions or fractions with unlike and relatively large denominators and tended to use algorithm-based thinking. The area models drawn by several of the PTs revealed various misconceptions. Lovin et al. (2018) investigated elementary and middle school PTs’ understanding of fractions as they were starting their first required mathematics course and found that they relied on procedural knowledge. Most of them had constructed the lower-level fraction schemes and operations but less than half had constructed the more sophisticated ones. Baeka et al. (2017) investigated elementary and middle school PTs’ pictorial strategies for a multistep fraction task in a multiplicative context. They found that many of the PTs were able to construct valid pictorial strategies that were widely diverse regarding 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 the type of model they used (area or set). Whitacre and Nickerson (2016) investigated elementary school PTs’ fraction knowledge at the beginning and end of their first mathematics content course. In the beginning, the PTs used predominantly standard strategies with weak performance and flexibility in comparing fractions. Lin et al. (2013) explored an intervention for enhancing elementary school PTs’ fraction knowledge and found that, prior to the intervention, the PTs held procedural understanding of basic fractional ideas and basic fractional operations, including equivalent fractions and addition, subtraction, multiplication, and division of fractions. Finally, Osana and Royea (2011) explored an intervention centered on problem solving to support Canadian elementary school PTs’ learning of fractions. The PTs were initially challenged to generate word problems for number sentences involving fractions, construct meaningful solutions to fraction problems, and represent those solutions symbolically.

Regarding the first 10 years of the period: Newton (2008) studied elementary PTs enrolled in a course on elementary school mathematics to obtain a comprehensive understanding of their fraction knowledge. Findings at the beginning of the course indicated that they had limited and fragmented knowledge of fractions. For example, they misapplied fraction algorithms, attended to superficial conditions when choosing a solution method, and demonstrated little flexibility in solving problems. Although they remembered many procedures, such as cross-multiplying and finding a common denominator, they were using them in inappropriate ways. Their most common error was to keep the denominator the same when it was not appropriate to do so. Tirosh (2000) investigated fraction division and found that in a class of Israeli elementary PTs, most of them knew how to divide fractions but could not explain why the procedure worked.

3.1.2 Whole Number Operations

This group of studies addressed elementary school PTs’ knowledge of addition, subtraction, multiplication, and division of whole numbers. Norton (2019) examined Australian primary school PTs’ mathematics knowledge at the beginning and end of their education course. Findings indicated that the PTs had low levels of knowledge of whole numbers at the beginning of the course. The most challenging whole-number computation for them was division by a double-digit divisor. Kaasila et al. (2010) investigated Finnish elementary PTs’ conceptual understanding, adaptive reasoning, and procedural fluency based on a non-standard division problem and concluded that division seemed not to be fully understood. Less than half of the PTs were able to produce complete or mainly correct solutions. The main reasons for their issues in understanding the task consisted of staying on the integer level, inability to handle the remainder, difficulties in understanding the relationships between different operations, and insufficient reasoning strategies. Thanheiser (2010) examined PTs’ responses to standard addition and subtraction place-value tasks and found that, at the beginning of their MTE, the PTs were often able to perform but not explain algorithms. For example, they had incorrect views of regrouped digits that included: interpreting all regrouped digits consistently as having the same value (all as 1 or all as 10); treating the value of the digits as dependent on the context (addition or subtraction); interpreting the digits consistently within but not across contexts (i.e. all as 10 in addition but all as 1 in subtraction); and interpreting the digits inconsistently depending on the task (i.e. the same digit was interpreted in multiple ways).

Thanheiser (2009) also reported on the PTs’ knowledge of multidigit whole numbers in the context of standard algorithms for addition and subtraction prior to their first mathematics course in their MTE. Most of the PTs did not have a deep understanding of numbers and struggled relating the values of the digits in a number to one another. They did not provide mathematical explanations of the algorithms. They referred to the digit in the tens place as ones rather than in terms of the reference unit tens or the appropriate groups of ones. While some drew on a conception that enabled them to explain the algorithm in at least one way, few exhibited an understanding of numbers that enabled them to explain the algorithm flexibly, including why the digits in any column can be treated as ones and why we can treat any pair of adjacent digits as if they were ones and tens.

3.1.3 Geometry

These two studies addressed different aspects of elementary and middle school PTs’ knowledge of geometry concepts. Miller (2018) analyzed PTs’ definitions of types of quadrilateral based on a survey of elementary school PTs who, since high school, had not yet studied geometry in their MTE. Findings included that the majority of the PTs’ definitions contained necessary attributes, but not sufficient or minimal attributes. The PTs were most comfortable with squares, followed by parallelograms, then rectangles, trapezoids, rhombi, and finally kites. They did not include hierarchical relationships as a means of defining one shape in terms of another and often created definitions that were aligned with emergent concept images of the shape types with only typical examples. Yanik (2011) investigated middle school PTs’ knowledge of rigid geometric translations and found that the PTs had difficulties recognizing, describing, executing, and representing geometric translations. They viewed geometric translations mainly as physical motions based on their previous experiences, that is, as rotational motion, translational motion, and mapping. They interpreted the vector that defines translations as a force, a line of symmetry, a direction indicator, and a displacement. Many of them knew that a vector has a magnitude and a direction but did not conclude that vectors define translations.

3.1.4 Algebraic Concepts and Thinking

This group of studies addressed elementary and middle school PTs’ knowledge of algebraic concepts and their ability to think algebraically. Hohensee (2017) examined the insights and challenges elementary school PTs experienced when exploring early algebraic reasoning. Findings indicated that they were challenged conceptually to identify the relationships contained in algebraic expressions, to distinguish between unknowns and variables, to bracket their knowledge of formal algebra, and to represent subtraction from unknowns or variables. You and Quinn (2010) investigated elementary and middle school PTs’ knowledge of linear functions and found that they were stronger on procedural than conceptual knowledge of linear functions. They were weak in representation flexibility, for example, ability to transfer flexibly: (i) between visual and algebraic representations to recognize relevant properties of algebraic and visual representations and to make connections among them when treating functions as an entity; (ii) from functions to a word problem situation; and (iii) from word problem situations to various forms of functions. Richardson et al. (2009) studied how pattern-finding tasks promoted elementary school PTs’ learning of how to generalize and justify algebraic rules from an emergent perspective to support their teaching of early algebra concepts. They found that most of the PTs, in their only mathematics methods course, initially focused on numerical data in tables and had difficulty providing a valid justification for their generalizations. Nearly all of the PTs generalized explicit rules using symbolic notation but had trouble with justifications early in the experiment. Pomerantsev and Korosteleva (2003) investigated the typical mistakes elementary and middle school PTs made as they progressed through their courses. They found that the PTs had difficulties recognizing structures of algebraic expressions at the introductory level of the courses.

3.1.5 Problem Posing

This group of studies addressed elementary and lower secondary school PTs’ problem posing knowledge or ability. Crespo and Sinclair (2008) investigated elementary school PTs’ problem-posing practices prior to planned interventions. They found that a majority of the problems the PTs posed consisted of assignment problems as opposed to the more complex relational or conditional problems for one task and factual problems (involving the recall of names and properties, the identification of properties, the application of measurement formulae, or the counting of shapes) for another task. The purpose was mainly to elicit information. Problem structure included clarity (problems not confusing, misleading, or under- and over-stated) and simplicity (numbers or shapes common and uncomplicated and right answers). Rizvi (2004) investigated Australian lower secondary school PTs’ ability to pose word problems for mathematical expressions involving division before an instructional intervention. She found that none of the PTs was able to pose word problems for the expressions where the divisors were fractions. They posed only sharing type word problems for the expressions where the divisor was a whole number. While many were aware of the repeated subtraction, no participants posed any word problem based on the repeated subtraction model for any division expression. Crespo (2003) investigated elementary school PTs’ beginning approaches to posing problems and found that they consisted of: making problems easy to solve (e.g., the narrow mathematical scope of the original version of the problem and the work of students); posing familiar problems (e.g., quick-translation story problems or computational exercises); and posing problems blindly (i.e., unawareness of the mathematical potential and scope of problem).

3.1.6 Summary

The overview of studies in this section on pre-existing mathematical knowledge and skills offers insights of the nature of the PTs’ content knowledge at the point of entry into a teacher education program. The studies investigated the PTs’ knowledge of different content areas (i.e., fractions, whole number operations, geometry, algebra) and their problem-posing skills. There was more attention on elementary than secondary PTs and on fractions than the other areas. Those studies dealing with fractions focused on meaning of fractions, arithmetic operations with fractions, strategies for solving fraction tasks and models of representing fractions. They indicated that the PTs’ fraction knowledge contained many misconceptions and was generally limited, fragmented, weak, low level, and procedural. Studies dealing with whole numbers focused on the arithmetic operations (addition, subtraction, multiplication, division). They indicated that the PTs did not have deep understanding of these procedures. Studies dealing with geometry focused on two-dimensional shapes and rigid motions. They indicated that the PTs’ had superficial knowledge or difficulties in dealing with these concepts. Studies dealing with algebraic concepts addressed algebraic expressions, linear functions, and algebraic rules. They indicated that the PTs had weak knowledge of the concepts, were challenged conceptually, and had difficulties with the concepts. Problem posing received the least attention with these studies focusing on posing word problems. The studies indicated that the PTs’ problem-posing ability was limited to posing problems of low level of cognitive demand. Overall, the studies highlighted that the PTs’ pre-existing knowledge of mathematical content was plagued with difficulties and low conceptual understanding of specific mathematics concepts that are central to school mathematics curricula and their future teaching.

3.2 Pre-existing Mathematics-Related Pedagogical Knowledge and Ability

This second category consists of studies that investigated PTs’ mathematics-related pedagogical knowledge and ability in the period 2000–2020. These studies collectively included early childhood and primary, elementary, middle, and secondary school PTs and their pedagogical ability (e.g., to notice, observe, analyze, and/or interpret teaching situations). They addressed another category of PMTC that is important for a teacher to function effectively in mathematics teaching situations and that PTs should have on entering MTE. The following overview of these studies is organized based on their foci on the PTs’ knowledge or ability involving (i) observing and analyzing teaching, (ii) noticing and interpreting students’ work or thinking, and (iii) evaluating tasks, to highlight the extent to which each was addressed in terms of the aims of the studies and the nature of the PTs’ PMTC. The studies are presented in reversed chronological order to indicate distribution in the period beginning with the most recent studies.

3.2.1 Observing and Analyzing Teaching

This group of studies addressed elementary, middle, and secondary school PTs’ ability to observe and/or analyze teaching by engaging the PTs in exploring videos of mathematics lessons. Star and Strickland (2008) investigated the impact of video viewing as a means to improve secondary school PTs’ ability to be observers of classroom practice. Their findings of the pre-assessment indicated that the PTs generally did not enter teaching methods courses with well-developed observation skills. They were astute observers of classroom management regarding what the teacher did to maintain control in the classroom and what students did that might influence the teacher’s ability to maintain control. They were also reasonably attentive to the actions of the teacher to support the lesson objectives, such as her use of notes, her presentation of the material, how she structured the group work, and her assignment of homework. However, their ability to notice other aspects of the classroom was not as strong. They did not attend to features of the classroom environment and/or did not feel that such features needed their attention. They were weak in observing the mathematical content, for example, questions about the representation of the mathematics, the examples used, and the problems posed. They did not notice subtleties in the ways that the teacher helped students think about content. In general, the PTs were very attentive to issues of classroom management but mostly unaware of static features of the classroom environment and the subtleties of classroom communication and mathematical content. Stockero (2008) investigated the use of a video-case curriculum in a middle school mathematics methods course for PTs to develop a reflective stance to enable them to analyze classroom interactions. Findings early in the course indicated that the PTs’ level of reflection or observation was at the two lowest levels of reflection; that is, describing and explaining levels. Their reflection focused on describing and explaining what they observed in the videos and did not demonstrate the higher levels of reflection consisting of theorizing, confronting, and restructuring. They also tended to analyze classroom events based on affective measures instead of pedagogical and mathematical reasons for instructional decisions.

Morris (2006) provided the only study that focused on this observing/analyzing PTs’ ability when the PTs entered their MTE compared to others that considered it prior to an intervention in a course later in their MTE. She investigated the “learning-from-practice skills” that elementary and middle school PTs possessed by requiring them to analyze a videorecorded mathematics lesson regarding the effects on student learning, to support their analysis with evidence, and to use their analysis to revise the lesson. She found that many of the PTs could carry out a cause-effect type of analysis of the relationships between specific instructional strategies and student learning and could use this analysis to make productive revisions to the instruction. But their ability to collect evidence that supported their analysis was less developed. Their analysis of the effects of instruction on the students’ learning was dependent on the video-task conditions. For example, when the task instructions indicated that the lesson was not successful, the PTs attended to both teacher and students and could make some elementary claims about how teaching and learning might be connected, but specific types of deficiencies in their evidence-gathering were apparent including the ability to collect evidence that supported conjectures about the effects of instruction. When the condition allowed the PTs to decide whether the lesson was successful and which instructional activities worked well or not, most of them focused primarily on the teacher, implying that students learn what the teacher explains. For example, they saw a teacher giving explanations and children giving correct responses, concluded that the children understood the teacher’s explanations, and made minimal revisions to the lesson. In general, the PTs’ support of hypotheses about student learning involved: no references to students’ responses, referring to students’ responses that were marginally related to the claims, attributing a wide range of understandings to students based on little or no objective evidence, and failing to refer to students’ responses that provided the most access to students’ thinking.

3.2.2 Noticing and Interpreting Students’ Work and Thinking

This group of studies collectively addressed early childhood and primary, elementary, and secondary school PTs’ knowledge of, and ability to notice and interpret, students’ mathematical work and thinking. Regarding the second 10 years of the period: Shin (2020) examined secondary school PTs’ noticing of students’ reasoning about mean and variability. Findings indicated that the PTs had difficulties noticing students’ reasoning about variability. None of the PTs explicitly interpreted the students’ limited understanding of variability when comparing data sets with unequal sample sizes. Some showed no evidence of differentiating between students’ different levels of reasoning. Superfine et al. (2019) investigated different facilitation moves to support the elementary school PTs in noticing children’s mathematical thinking and found that they generally did not discuss their noticing at a high-level and there were few instances where they provided evidence for their noticing. Sánchez-Matamoros et al. (2019) examined the relationships between how secondary school PTs in Spain attended to the mathematical elements in students’ solutions and interpreted students’ understandings for the derivative of a function at a given point. Their findings indicated that the PTs had different levels of pre-existing ability consisting of those who provided general comments about students’ learning, who found it difficult to recognize characteristics of the students’ understanding, and who had difficulties in using mathematical elements in students’ solutions to recognize differences among students’ understanding. Callejo and Zapatera (2017) investigated Spanish primary school PTs’ noticing, describing, and interpreting of students’ mathematical thinking in their solution to a pattern generalization task. They found that the PTs were able to name various mathematical elements to describe the students’ answers but did not always use them to interpret the understanding of pattern generalization of each student. Some PTs could not recognize the understanding of the students.

In addition, Simpson and Haltiwanger (2017) investigated how secondary school PTs made sense of students’ mathematical thinking of an algebra and function mathematics problem, when professional noticing was not a formal part of their MTE. They found that the PTs exhibited a lack of rigorous evidence when interpreting what the students may or may not have understood. The PTs discussed only what the students understood in terms of the written work. They did not consider misconceptions or errors in the students’ mathematical thinking. Sánchez-Matamoros et al. (2015) examined the ability of secondary school PTs in Spain to notice students’ understanding of the derivative concept in the beginning and end of a “training module”. At the beginning, the PTs’ noticing was limited to describing students’ answers in the graphical and analytical modes of representation but without identifying the relevant mathematical elements and interpreting the students’ understanding by making general comments related to “the good or bad understanding of the student.” Lastly, Son (2013) examined the secondary and elementary school PTs’ interpretations of and responses to a student’s error(s) involving finding a missing length in similar rectangles through a teaching scenario task. Findings indicated that although the student’s errors came from conceptual aspects of similarity, a majority of the PTs identified the errors as stemming from procedural aspects of similarity and consequently drew on procedural knowledge as a way to guide the students.

Regarding the first 10 years of the period: Harkness and Thomas (2008) investigated early childhood PTs’ mathematical understanding of a student’s invented multiplication algorithm and found that a majority of the PTs relied on procedural and memorized explanations rather than using mathematical properties to describe the validity of the algorithm. Generally, their responses demonstrated a procedural or memorized understanding of the invented algorithm. Crespo’s (2000) study on how elementary school PTs in Canada interpreted their students’ work indicated that their interpretations were initially from a limited focus on the correctness of the students’ solutions and not meaning.

3.2.3 Evaluating Tasks

This group of studies addressed elementary and middle school PTs’ knowledge of, and ability to evaluate, features of mathematical tasks to support students’ learning. Magiera et al. (2013) explored middle school PTs’ ability to recognize opportunities to engage students in algebraic thinking. They found that the PTs demonstrated limited ability to recognize the full potential of algebra-based tasks to elicit algebraic thinking in students, recognizing only some features in the analyzed tasks. Stephens (2006) examined elementary school PTs’ awareness of equivalence and relational thinking to assess their initial preparedness to engage students in these aspects of early algebraic reasoning. She found that the PTs collectively demonstrated an awareness of relational thinking in identifying opportunities offered by the tasks to engage students in this thinking. But in proposing difficulties students might have with selected tasks, few of them demonstrated an understanding that many students have misconceptions about the meaning of the equal sign. Osana et al. (2006) examined the nature of elementary school PTs’ evaluations of elementary mathematics problems using a model designed to discriminate among tasks according to their cognitive complexity. Results demonstrated that, overall, the PTs had more difficulty accurately classifying problems considered to represent high levels of cognitive complexity compared to less complex problems. They were influenced by the surface characteristics of task length and tended to label short problems as less cognitively demanding and long problems as more so.

3.2.4 Summary

The overview of studies in this section on pre-existing mathematics-related pedagogical knowledge and ability offers insights of the nature of the PTs’ knowledge and ability related to teaching and learning mathematics at the point of entering a teacher education program. The studies addressed the PTs’ knowledge of how to observe and analyze teaching, notice and interpret students’ work or thinking, and evaluate tasks. Those dealing with observing and analyzing classroom behaviours of teachers and students indicated mostly weaknesses in the PTs’ ability to observe and make appropriate conclusions or suggestions regarding instruction. For example, they were strong in observing classroom management but weak in noticing other aspects of the classroom and demonstrated lowest levels of analysis of classroom interactions. Those studies dealing with noticing and interpreting students’ mathematical thinking and work indicated that the PTs demonstrated several areas of difficulties in noticing or recognizing students’ reasoning, providing rigorous evidence to support their noticing, and discussing noticing at a high level. Studies dealing with evaluating or interpreting features of tasks to support students’ learning indicated that the PTs had limited ability to recognize potential of tasks or difficulties students could experience with a task or classifying a problem of high level of cognitive complexity. Overall, the studies highlighted that there were much more weaknesses than strengths in the PTs’ pre-existing ability to notice, analyze, and interpret mathematics classroom behaviors of teachers and students and students’ mathematical work.

3.3 Pre-existing Mathematics-Related Beliefs

This third and final category consists of studies that investigated PTs’ mathematics-related beliefs in the period 2000–2020. The studies collectively included primary, elementary, middle, and secondary school PTs and their content and pedagogical beliefs (conceptions and perceptions). They addressed another category of PMTC that are important to how teachers conceptualize and enact their teaching of mathematics and PTs would have on entering MTE. The following overview of these studies is organized based on their foci on the PTs’ beliefs about: (i) nature of mathematics, (ii) teaching and learning mathematics, (iii) use of technology, (iv) mathematical processes, and (v) mathematics concept, to highlight the extent to which they were addressed in terms of the aims of the studies and the nature of the PTs’ PMTC. The studies are presented in reversed chronological order to indicate distribution in the period beginning with the most recent studies.

3.3.1 Nature of Mathematics

This group of studies addressed primary, elementary, middle, and secondary school PTs’ beliefs of the nature of mathematics. Weldeana and Abraham (2014) investigated an intervention to change beliefs of middle school PTs. They found that before the intervention a majority of the PTs did not hold progressive beliefs related to the nature of mathematics. For example, they believed that for every problem of mathematics, there is one unique approach leading to its solution. Shilling-Traina and Stylianides (2013) investigated changes in the beliefs about mathematics held by elementary school PTs in a mathematics course and found that their initial beliefs largely reflected instrumentalist and Platonist views. Conner et al. (2011) investigated secondary school mathematics PTs’ beliefs about mathematics. Findings indicated that their initial views of mathematics were primarily Platonist and instrumentalist. Some of the most prevalent descriptors of mathematics across participants were mathematics is logical and less subjective than other disciplines, and mathematics is unambiguous in the sense that, while multiple solution paths are possible, each problem has a single, correct answer. Finally, Bolden et al. (2010) investigated primary school PTs in the UK early in their education course at the beginning of the program and found that the PTs held narrow, absolutist views of mathematics as a subject. Most conceived mathematics as a subject of a set body of knowledge that offered little or no room for freedom of expression, imagination, and independence. Most also believed that mathematics was not a creative subject, and it was difficult to encourage creativity in mathematics.

3.3.2 Teaching and Learning Mathematics

This group of studies addressed primary, elementary, middle, and secondary school PTs’ beliefs or conceptions about teaching and learning mathematics, including qualities of teachers and learners, mathematical behaviour and creativity, and doing and understanding mathematics. Regarding the second 10 years of the period: Stohlmann et al., (2014/2015) investigated changing elementary school PTs’ beliefs about mathematical knowledge. At the beginning of the course, the majority of the PTs showed little or no evidence of the belief that conceptual understanding of mathematics is more powerful or generative than remembering mathematical procedures and they appeared to be focused on understanding mathematics in terms of procedural fluency. The majority of them also showed weak or no evidence of beliefs that: (i) One’s knowledge of how to apply mathematical procedures does not necessarily go with the understanding of the underlying concepts. (ii) Understanding mathematical concepts is more powerful and more generative than remembering mathematical procedures. (iii) If students learn mathematical concepts before they learn procedures, they are more likely to understand the procedures when they learn them. If they learn procedures first, they are less likely ever to learn the concepts. For (i), very few PTs showed evidence or strong evidence of the belief that if a child knows procedures, they may not understand the underlying concepts. Weldeana and Abraham (2014) investigated an intervention to change beliefs of middle school PTs. They found that before the intervention a majority of the PTs did not hold progressive beliefs related to the way mathematics is learned, taught, and practiced. The PTs began with a strong belief that mathematics can be learned and understood through memorization of facts and formulae. They held many traditional beliefs related to knowing in mathematics (e.g., step-by-step procedure; getting the right answers quickly; retrieving information quickly; and figuring out formulae and equations to solve problems immediately).

In addition, Conner et al. (2011) investigated secondary school PTs’ beliefs about mathematics teaching. They found that the PTs’ initial beliefs of characteristics of effective mathematics teachers included: having positive affective characteristics (a good teacher is nice, patient, friendly) and mathematical knowledge, attending to student needs, and facilitating student participation. They also initially held the belief that students should participate in class, asking questions and working together, but sometimes they described a teacher centered view of student participation that included direct instruction as the primary method for teaching new content. Bolden et al. (2010) investigated conceptions of creativity of primary school PTs in the UK early in their education course at the beginning of their MTE. Findings indicated that the meaning of creativity in primary school mathematics was not well understood by the PTs based on their conception of it. Their conceptions were narrow, predominantly associated with the use of resources and technology, and tied to the idea of “teaching creatively” rather than “teaching for creativity”. They viewed creativity in terms of the types of resources used and the way in which they were used to teach mathematical topics and the way in which real-life examples were used to explore mathematical concepts.

For the first 10 years of the period: Ambrose (2004) investigated an intervention to build elementary school PTs’ beliefs. Findings indicated that initially the PTs held beliefs that teaching involves explaining things to children, which most of them continued to hold following the intervention. They initially equated doing mathematics with using memorized procedures. Their actions indicated a teaching-as-telling belief along with a belief about mathematics learning as the acquisition of standard symbolic procedures. Szydlik et al. (2003) explored elementary school PTs’ beliefs about the nature of mathematical behavior both at the beginning and end of the education course. At the beginning, the majority of the PTs believed that, as learners, they could not “figure out” mathematics for themselves. They could not imagine being asked to do a problem significantly different from those in the textbook or having a teacher who did not first show them how to do similar problems. They believed that they must memorize formulas, procedures, or template problems in order to work on new problems.

3.3.3 Use of Technology

These two studies addressed middle and secondary school PTs’ beliefs about the use of technology. Wachira et al. (2008) assessed middle school PTs’ beliefs about the appropriate use of technology in mathematics teaching and learning prior to taking the methods course. They found the PTs’ beliefs to be limited to the use of technology as computational tools and for checking the accuracy of these computations. The PTs’ conceptions indicated a lack of understanding of technology as powerful tools to help students gain knowledge, skills, deeper understanding and appreciation of mathematics. They did not provide specific ways on how technology could be used to promote learning. None indicated that technology could be used to explore patterns, discover more about mathematics concepts or investigate mathematical relationships, which suggested that they lacked understanding of how technology could be used appropriately to develop concepts. Leatham (2007) investigated secondary school PTs’ beliefs about teaching mathematics with technology later in their program after taking an education course on technology and found that their beliefs about the nature of technology in the classroom were about the availability of technology, the purposeful use of technology, and the importance of teacher knowledge of technology.

3.3.4 Mathematical Processes

These two studies addressed elementary school PTs’ beliefs (conceptions, views) of two mathematical processes: problem solving and representing mathematical concepts or situations. Son and Lee (2020) examined elementary PTs’ problem-solving conceptions and performances. They found that a majority of the PTs held conceptions of problem solving as a means to a solution, that is, they expressed a skill-based or means-to-an-end view by focusing on solutions or procedural steps. Their conceptions of problem solving were related to their performances. Dreher et al. (2016) investigated views about using multiple representations held by British and German elementary school PTs at the beginning of their first year of their MTE. They found that the PTs showed little awareness for the role of representations for mathematical understanding. They viewed the role of multiple representations for understanding mathematics as less important than other non-discipline reasons for using multiple representations. They mostly were not able to recognize the learning potential of tasks focusing on conversions of representations, in comparison with tasks including rather unhelpful pictorial representations, to which they tended to assign a higher learning potential.

3.3.5 Algebra

One study addressed PTs’ conceptions of algebra. Stephens (2008) examined conceptions of algebra held by elementary school PTs enrolled in their only course addressing the teaching of mathematics. Findings suggested that their conceptions of algebra as subject matter were narrow. Most of them equated algebra with the manipulation of symbols. Very few identified other forms of reasoning, in particular, relational thinking, with algebra. Several made comments implying that student strategies that demonstrated traditional symbol manipulation might be valued more than those that demonstrated relational thinking, suggesting that what was viewed as algebra is what will be valued in the classroom. Tasks were often judged to be algebra or non-algebra problems by the presence or absence of a variable or letter, and students were often judged to have used or not used algebra based on how closely their work matched the symbol-manipulation model.

3.3.6 Summary

The overview of studies in this section on pre-existing mathematics-related beliefs offers insights of the nature of the PTs’ pre-existing beliefs or conceptions related to mathematics and mathematics pedagogy at the point of entering a teacher education program. The studies addressed the PTs’ beliefs about the nature of mathematics, teaching and learning mathematics, use of technology, mathematical processes, and mathematics concepts. Those dealing with beliefs about the nature of mathematics indicated that the PTs held beliefs of a Platonist or absolutist perspective of mathematics. Those dealing with beliefs about teaching and learning mathematics indicated that the PTs’ beliefs mostly reflected a traditional or ‘teacher-centered’ perspective of teaching and learning mathematics. Those dealing with beliefs about mathematics concepts and processes indicated that the PTs held narrow conceptions of algebra, problem solving, and multiple representations. Those dealing with technology indicated that the PTs lacked understanding of use of technology to support students’ learning and to develop concepts and held beliefs that could significantly limit their use of technology in teaching mathematics. Overall, the studies highlighted that the PTs’ pre-existing beliefs were mostly inappropriate to support contemporary perspectives of reform-based mathematics education.

3.4 Summary of Landscape of PMTC

Table 1 provides a summary of the PMTC that were addressed by studies investigating PTs’ characteristics in the period 2000–2020. The table includes the three categories of PMTC researched and the types of PMTC researched for each category. These PMTC could directly impact PTs’ learning during initial teacher education and their teaching as future teachers. They are further discussed in the sections that follow concerning how research has evolved in PMTC.

Table 1 Landscape of PMTC of PTs Researcher in 2000–2020
Full size table

4 Evolution of Research on PMTC

The preceding section outlined studies relevant to Medley’s (1987) Type F variable that provided a landscape of the types of PMTC they addressed. These studies formed the basis to consider the evolution of research on PMTC in the period 2000–2020, based on what was done (i.e., the scope of research) and how it was done (i.e., methodological factors) in establishing and advancing research in this area of mathematics education. The scope of research involves the extent to which PMTC were studied. The methodological factors involve those that Medley proposed are necessary to consider the evolution of research on teaching, adapted to address research on PTs’ PMTC. These factors are conceptualization (e.g., of good teaching), instrumentation (e.g., valid tools), design, and analysis. This section is organized in terms of the scope of research on PMTC and methodological factors.

4.1 Scope of Research on PMTC

The scope of research suggests an evolution of research on PMTC in terms of the extent of the types of PMTC researched, the extent to which PMTC were addressed by the studies, and the extent to which the studies were framed at the point of entry into MTE.

4.1.1 Types of PMTC Researched

The studies suggested a shift from researching teacher candidates’ characteristics, such as level of school mathematics they completed, mathematics courses they completed, their overall grade point average (GPA), and their mathematics GPA, which were not considered in any of the studies for 2000–2020. They also suggested growth in research on PMTC in terms of different types of PTs’ characteristics that were investigated. They addressed specific aspects of nine types of PMTC associated with three categories of characteristics (Table 1) the PTs held on entering MTE. These characteristics included PTs’ knowledge, skills, and beliefs that were connected to what they would have learned, directly or indirectly, as students in school mathematics classrooms. For example, prior to entering teacher education, PTs would have developed knowledge of mathematical:

  • Content—directly, based on what was taught.

  • Processes—directly, based on what they engaged in.

  • Learning—indirectly, based on how they were engaged and their personal orientation.

  • Teaching—indirectly, based on how they were taught and assessed.

  • Technological tools—directly or indirectly, based on how they were used in their learning.

  • Beliefs—indirectly, based on what was taught and how it was taught.

While the studies touched on all of these areas of PTs’ learning, aspects of them were not explored enough or at all to add depth to the body of research on PMTC. For example, little or no consideration was given to secondary school mathematics concepts, mathematical problem solving and thinking skills, technological tools, and assessment of learning. The studies also did not address contextual variables that impacted the characteristics, in particular, cultural context and technological context (discussed later). Thus, while the studies offered insights of some important PMTC held by PTs on entering MTE, in advancing the field of research on the mathematics teacher and teaching, they were limited in types and number of PMTC covered in a recent 20-year period.

4.1.2 Attention to PMTC by Studies

The extent to which the studies attended to the PMTC researched, based on number of studies, suggested relative levels of evolution of each of the three categories and types of PMTC (Table 1) for the period 2000–2020. For example, the pre-existing mathematical content and skills category received the most attention, suggesting higher interest in content-related characteristics, in particular, knowledge of fractions that represented almost half of the studies in the category, with most of them occurring in the second half of the period. The pre-existing mathematics-related pedagogical knowledge and ability and the pre-existing mathematics-related beliefs categories received the same level of attention, but when combined were higher by about 20% more studies than the mathematical content and skills category. This suggested that overall, research focused on PMTC regarding pedagogical ability and beliefs had grown more than for content knowledge.

In particular, in the pedagogical ability category, there was significant attention in the studies on noticing and interpreting students’ work and thinking, which formed about two-thirds of the studies in this category, were the largest group of studies for the three categories, and were mostly occurring in the second half of the period. For the beliefs category, beliefs about teaching and learning received the most attention, but was third behind the interpreting students’ work and the fractions PMTC. Most of the studies for these three types of PMTC were also in the second half of the period, compared to the other types of PMTC that collectively had more studies in the first half of the period. All of the studies on problem posing, use of technology, and the ability to observe and analyze teaching, and most of those on ability to evaluate tasks and knowledge of whole number operations were in the first half of the period. Hence, there was a shift in focus from the first to the second half of the period that suggested a shift in research on PMTC that may be considered a partial growth regarding some PMTC researched and a limitation regarding lack of continuation of attention to those that are of ongoing importance to support effective teaching of mathematics.

4.1.3 Studies at Point of Entry

The extent to which the studies were framed at the PTs’ point of entry into MTE provided another perspective of the level of growth on research on PMTC for the period 2000–2020. While there is a large body of mathematics education research on PTs in this period, it is lacking in addressing PMTC of teacher candidates at the point of entering MTE. Only about 8% of the studies addressing PMTC in 2000–2020 focused on PTs at the beginning of their programs, which was not defined. Hence, as previously discussed, the majority of the studies were framed at the beginning of mathematics education courses, or prior to research-based instructional interventions in a mathematics education course or based on activities in mathematics for teachers or mathematics education courses that depended solely on the use of prior knowledge related to school experience. This framing suggested that, in 2000 -2020, there was no growth of research of PTs’ PMTC at the point of entry into MTE or there was an evolution in terms of a shift to, or consideration of, more practical approaches of obtaining access to PTs and their PMTC (e.g., participants in mathematics education courses at different points in a program).

Given the importance of understanding teacher candidates’ PMTC, the little attention of research on them could partly be related to challenges associated with the point of entry, which could be messy regarding accessing information on teacher candidates for one discipline and dealing with complexities associated with different admission requirements, different academic backgrounds of candidates, and different programs. For example, in a Western cultural context, mathematics teacher candidates could enter a teacher education program directly from high school with or without a college entrance test/exam, or after receiving an undergraduate or graduate degree in mathematics or some other related degree, or while jointly working on a mathematics education degree and another related degree. They could have completed only middle or high school mathematics, or a mathematics degree, or a mathematics-related degree (e.g., physics, engineering), or some mathematics or mathematics-related courses prior to beginning a teacher education or mathematics education program. These various groups of PTs would need to be considered separately, in addition to the various groups according to different school levels for which PTs are preparing (e.g., elementary, middle, or high school) to obtain a reliable and meaningful picture of PMTC.

A related issue is what is the point of entry for candidates in mathematics education–the beginning of a general teacher education program that includes mathematics education, the beginning of a specialized mathematics education program, the beginning of mathematics education courses, and mathematics courses for teachers? All of these possibilities could lead to different versions of the nature of PMTC held by PTs. Although the studies on PMTC in 2000–2020 addressed school levels, there was an underlying assumption that the PTs in a course formed a homogeneous group in terms of their academic backgrounds, which might or might not have been the case. However, the intent of most of these studies was not explicitly to address PMTC at the point of entry into an MTE, but to obtain baseline information to evaluate their instructional approaches, which could be another way of considering how research on PMTC has evolved.

Another possible challenge for researching teacher candidates’ PMTC involves the use of institutions’ recruitment criteria or admission requirements as a basis of their point-of-entry PMTC, since what institutions want and what they get may not fully align. For example, as Artzt and Curcio (2008) explained, regarding recruiting high school students for secondary school mathematics education programs in the USA:

There were several obstacles we faced in recruiting talented mathematics secondary students; it requires finding students from the intersection of three sets: those who love mathematics, those who want to become teachers, and those who are interested in attending Queens College …. Overcoming the barriers requires a multitude of recruitment strategies. (p. 249)

Schmidt et al. (2012) also raised concerns about selectivity for institutions in the USA. They noted:

There is great variation in what secondary mathematics individuals have had before entering teacher preparation. (p. 265).

Variation was especially large for the college entrance mathematics score, … revealing a very large selectivity factor across institutions. (p. 270)

From a policy perspective, selectivity relates to differences in mathematics knowledge among future teachers before they began their teacher preparation—the issue of who enters teaching. This is manifested by large differences among institutions. The policy issue related to selectivity includes recruiting more mathematically able students into primary teacher preparation no matter which institution they might attend. (p. 275)

Selectivity could also be an issue within an institution where admission could be based on a combination of grades, interviews, portfolio, etc. which adds another layer of complexity in using admission requirements to determine PMTC. Research on PMTC in 2000–2020 did not address recruitment criteria or admission requirements of institutions, which could also be considered as representative of a shift in interest of the type of PMTC that seemed to be more relevant in this period.

In general, then, the lack of research on PTs’ PMTC in 2000–2020, based on the journals reviewed, could be related to challenges in addressing variables associated with a point of entry into an education program or a shift in interest from considering PMTC at point of entry to indirectly addressing PMTC for PTs based on their participation in mathematics education courses regardless of where they are situated in a teacher education program.

4.2 Methodological Factors

Each of the four methodological factors adapted from Medley (1987) is discussed in this section regarding the evolution of research on the PTs’ PMTC. Conceptualization is interpreted in terms of relationship to ‘good teaching’, teacher education, and technology and culture. Instrumentation is interpreted as procedures or tools used in collecting the data. Design is interpreted as what was used or done to support the data collection process and analysis is interpreted as the means used to extract information from the data. These interpretations are appropriate to address the information provided in the studies.

4.2.1 Conceptualization of PMTC in Relation to ‘Good Teaching’

In adapting Medley’s (1987) framework as a guide for research on teaching, it is important for studies to provide a conceptualization of good teaching. For the period 2000–2020, while good or effective teaching of mathematics was not explicitly conceptualized in the studies on PMTC, it was implied based on the theoretical bases framing the PMTC being investigated for the three categories of mathematical knowledge and skills, pedagogical knowledge, and beliefs. These theoretical perspectives were related to reform-based perspectives of mathematics education that promoted significant shifts in school mathematics curriculum, teaching, and learning and were associated with effective teaching of mathematics. Collectively, the studies directly or indirectly considered PMTC concerning specific elements of these perspectives that include: (a) standards and principles for mathematics education (NCTM, ); (b) mathematical proficiency (Kilpatrick et al., 2001); (c) mathematical thinking (Mason et al., 2010; Schoenfeld, 1992); (d) mathematics knowledge for teaching (Ball et al., 2008); (e) teaching practices (NCTM, 2014); and (f) beliefs about the nature of mathematics (Ernest, 1989) and teaching and learning mathematics (Beswick, 2012).

The studies, then, indicated an evolution in conceptualizing PMTC to reflect contemporary perspectives of teaching and learning mathematics, with implications about the nature of the PMTC required for a PT to become “the teacher who has a set of personal characteristics closest to those of the ideal teacher” (Medley, 1987, p. 106). This implication seemed to underlie most of the studies considering the nature of the PTs’ PMTC mainly from a deficit perspective. The result was an evolution of research in the period to highlight what was wrong with the PMTC of the PTs on entering MTE.

Regardless of whether addressing PTs at the primary, elementary, middle, or secondary school level, the studies showed an ongoing focus on issues and limitations in their PMTC that indicated what they did not know or were not able to do at the beginning of their MTE. For example, collectively, the PTs: did not have deep conceptual or relational understanding of arithmetic and algebraic concepts; could not pose complex relational or conditional problems; were not able to think mathematically beyond a low level; were not able to observe details of the classroom environment, mathematical content of a lesson, and subtleties of classroom communication and mathematics content beyond a surface level; were not able to collect, beyond surface level, appropriate evidence to support analysis of instruction and learning; could not base their analysis of instructional decisions on pedagogical and mathematical reasons instead of affective reasons; could not reflect on a level to theorize or restructure; were not able to analyze or interpret students’ solutions with depth, notice students reasoning based on meaning of student work, identify appropriate evidence for their noticing, identify conceptual aspect of errors, and classify tasks of high levels of cognitive complexity for students; and did not hold views of inquiry-based, constructivist-oriented perspectives of mathematics, teaching and learning, technology as tool to support deeper understanding and appreciation of mathematics, nature of genuine problems and problem solving in terms of their openness, conceptual use of multiple representation to support deep learning, and algebra as reasoning and relational thinking.

The deficit perspective of the PTs’ PMTC also suggested limited evolution of school mathematics teaching based on reform recommendations since the different types of PMTC involved were directly related to what the PTs should have learned or experienced directly (e.g., mathematics content) and indirectly (e.g., mathematics pedagogy and beliefs) in school prior to entering teacher education. This probably limited evolution of teaching was also suggested by the studies based on what they noted or implied about what the PTs knew or were able to do, which reflected learning from traditional, pre-reform-oriented teaching of mathematics. For example, the PTs’ PMTC included: procedural or instrumental understanding of the some key school mathematics concepts; ability to pose problems of low level of cognitive complexity; ability to conduct instrumental analysis of relationship between instruction and student learning; ability to observe, describe and explain generic instructional issues, classroom management, instructional tasks, and affective factors of classroom interactions; ability to notice, interpret and describe students’ work or thinking on a procedural level; ability to identify tasks of low cognitive demand and attend to surface characteristics of tasks; beliefs of mathematics as absolute, teaching and learning as teacher-centered (e.g., teaching as telling, learning as memorizing), technology as computational tool; use of representations on an instrumental level; problem solving as a means to a solution involving procedural steps; and algebra as manipulation of symbols and in terms of surface features (e.g., variable or letter).

In general, whether the PTs’ PMTC were viewed from a perspective of what they knew or did not know, were able or not able to do, the studies indicated continued issues with their PMTC when the PMTC were conceptualized in relation to reform expectations for effective teaching of mathematics in the period 2000–2020. This outcome suggested that the impact of the reform movement in school mathematics had not materialized in this period and many PTs were entering teacher education with PMTC that did not align with appropriate knowledge, skills, and beliefs in relation to effective teaching. But this conclusion might not be representative of the actual situation given the limitations of the studies regarding small sample sizes and little information on those PTs with PMTC that reflected reform-based teaching at the point of entry into MTE.

4.2.2 Conceptualization of PMTC in Relation to Teacher Education

Based on the definitions of Type F variable (Medley, 1987; Manizade et al., this volume), research should also conceptualize PMTC in relation to “the characteristics a teacher candidate needs in order to acquire those competencies that formal education and experience can provide” (Medley, p. 105). Hence, as Medley indicated, Types F and E variables should be combined in research on teaching, where Type E involves mathematics teacher competencies, knowledge, and skills to function effectively in mathematics teacher education (Manizade et al.). In addition, Medley explained:

Type FE research is the proper research to provide support for selective admission to teacher preparation, and may be called research in teacher selection. What characteristics of entering students identify teachers who will acquire the competencies they need as a result of training? (p. 111)

His perspective suggested that PMTC, in addition to being conceptualized in relation to good or effective teaching, should be conceptualized in relation to good or effective teacher education, for example, the types and nature of the PMTC that are consistent with the role of MTE or are most needed to support PTs’ learning in MTE.

The studies on PMTC in 2000–2020 did not address this relationship or lacked clarity about it. While they suggested that the PTs’ PMTC were not adequate in relation to effective teaching, there was less clarity regarding whether or not the PMTC were adequate to support their learning in MTE. But, based on the design of many of the studies, there were underlying assumptions of the relationship between the PTs’ PMTC and possible roles of MTE to help PTs develop the knowledge and competencies for effective teaching of mathematics. For example, the intervention studies with focus on fixing deficiencies in the PTs’ PMTC implied PMTC were conceptualized in relation to a remedial role of the teacher education programs. In general, the studies did not suggest that PMTC were conceptualized in relation to a constructivist role of MTE in which the PMTC were viewed as resources to build on and not deficiencies to fix. For both of these roles, the quality of the PMTC on entering MTE might not be important beyond some minimum standard required to complete high school and/or to enter an education program. Overall, then, while conceptualization of PMTC in relation to teacher education was unclear or limited for research in 2000–2020, there was a shift in the underlying implication of the studies that the role of mathematics teacher education was important to researching and understanding the PMTC PTs needed on entering MTE for them to succeed in it.

4.2.3 Conceptualization of PMTC in Relation to Technology and Culture

Context is important to understanding the mathematics teacher and teaching and should be considered in the conceptualization of teachers’ PMTC in research on teaching. Medley (1987) identified four categories of context-related variables, also adapted by Manizade et al., (this volume) for mathematics education research, that should be considered, but he associated these categories with practicing teachers and did not directly connect any with the Type F variable of PMTC. Thus, context-related factors that could have impacted the PMTC prior to being engaged in teacher education were not highlighted in Medley’s model. However, in a digital age and a twenty-first century society, the evolution of research on PMTC in 2000–2020 should reflect the availability of technology and the cultural context in and outside of classrooms. This view means that, for research in this period, the conceptualization of PTs’ PMTC should also be related to technology and culture. This was not, however, reflected in the studies, directly or indirectly, based on theories to support the importance of culture and technology to mathematics teaching. For example, equity and technology are two of the six principles recommended by NCTM (2000) as fundamental to high-quality mathematics education.

Regarding Technology. NCTM (2000) promoted technology as being essential in teaching and learning mathematics. In addition, NCTM (2011) explained:

Technological tools include those that are both content specific and content neutral. In mathematics education, content-specific technologies include computer algebra systems; dynamic geometry environments; interactive applets; handheld computation, data collection, and analysis devices; and computer-based applications. These technologies support students in exploring and identifying mathematical concepts and relationships. Content-neutral technologies include communication and collaboration tools and Web-based digital media, and these technologies increase students’ access to information, ideas, and interactions that can support and enhance sense making, which is central to the process of taking ownership of knowledge. (NCTM, 2011)

Despite this range of tools and importance of technology, there was no study in the last ten years and only two studies in the early 2000s that considered technology in relation to PMTC. There was, therefore, a lack of information regarding the influence of technology on PTs’ PMTC based on technology in general or the different types of technology the PTs would have encountered in or out of the classroom. The two studies on technology focused on PTs’ beliefs about it in a general sense (without consideration of specific types), but neither considered the relationship between the use of technology in learning and the PMTC.

Of the two studies, only one (Wachira et al., 2008) focused explicitly on pre-existing beliefs at the beginning of a semester based on students’ responses to two prompts: (a) to indicate their experiences with instructional technology use in mathematics, and (b) to provide compelling arguments for the use of technology in mathematics learning. Both studies highlighted the limitations of the PTs’ beliefs, which suggested that their exposure to technology was not enriching to their PMTC. But these studies occurred in the early 2000s when access to technology was not as available in schools as later in the period. They may not, therefore, be representative of the significant changes and access to technology in western cultural contexts and the impact on teacher candidates’ learning and thinking on entering MTE. Overall, there was a lack of growth in conceptualizing PMTC in relation to technology.

Regarding Culture. The actions of teaching and learning exist in cultures that vary greatly from society to society, from school to school, and even from classroom to classroom in the same school. Thus, culture could be a problematic variable regarding its meaning in researching teaching and a basis of conceptualizing PMTC. It is considered here in relation to the classroom. The culture of a mathematics classroom determines and is determined by the type of learning that takes place, affects the types of experiences students engage in, and could interact with students’ personal cultural (e.g., home or societal culture) experiences in positive or negative ways. Thus, in the period 2000–2020, there has been the promotion of culturally responsive teaching in general (Gay, 2010; Taylor & Sobel, 2011) and specific to mathematics education (Greer et al., 2009; Presmeg, 2007) and of equity as a principle in school mathematics education (NCTM, 2000) to meaningfully address diverse student population based on various cultural heritage and social backgrounds of students in the western cultural context.

Culture then should be of importance in considering the evolution of research on PMTC in relation to its impact on PTs’ experience and learning of mathematics that are connected to the nature of their PMTC on entering MTE. These PMTC include the mathematical identity PTs developed based on the classroom culture and the personal cultural-based resources they bring to MTE with implications for the type of teacher they will become. None of the studies attended to PMTC in relation to culture. This lack of conceptualization of PMTC in relation to culture suggested a significant deficiency in the evolution of research on beginning PTs. There seemed to be a lack of a humanistic perspective in framing the studies that made culture irrelevant in considering the PMTC. There was also a lack of focus on affective factors that are directly associated with culture. Regardless of whether the PTs were from culturally homogenous classrooms with homogenous cultural backgrounds, culture still mattered regarding their identity as a teacher and the nature of their PMTC. Thus, overall, there was a lack of growth in conceptualizing PMTC in relation to culture.

4.2.4 Instrumentation in Researching PMTC

Instrumentation is the second factor Medley (1987) indicated as important in considering the evolution of research on teaching, interpreted here as procedures or tools used in collecting the data. In earlier studies, Medley, for example, noted that instrumentation focused on surveys consisting of closed response questionnaires or brief written response items about teaching. For the period 2000–2010, while the studies on PMTC continued to use surveys, they also used a variety of tools for data collection. This evolution of instrumentation mirrored the evolution in the conceptualization of PMTC in relation to contemporary perspectives of effective mathematics teaching. Four categories of instruments, discussed in turn in the following paragraphs, were used in the studies to determine the PTs’ PMTC at the beginning of a program or course or prior to an intervention to support their learning.

Questionnaires. Some studies used only questionnaires, for example: motivation questionnaire (Newton, 2009); beliefs questionnaire (Dreher et al., 2016; Weldeana & Abraham, 2014); questionnaire to analyze student work (Simpson & Haltiwanger, 2017); diagnostic questionnaire (Tirosh, 2000); content knowledge questionnaire (Lee & Lee, 2020); questionnaire on concept maps and definitions (Miller, 2018); and questionnaire with true/false, multiple choice, and short answer questions (Star & Strickland, 2008). Other studies used open-ended questionnaires with semi-structured interviews, for example: regarding conceptions and beliefs of mathematics, problem solving or creativity (Bolden et al., 2009; Conner et al., 2011; Son & Lee, 2020; Szydlik et al., 2003) and mathematics backgrounds and teaching interests (Stephens, 2008).

Interviews. Some studies used only semi-structured interviews based on participants solving mathematical tasks (Stephens, 2006; Thanheiser, 2009, 2010; Yanik, 2011). In addition to interviews being combined with questionnaires, other studies combined interviews with written responses to analyse students’ written work (Magiera et al., 2013; Shin, 2020) and to prompts on beliefs (Shilling-Traina & Stylianides, 2013).

Written Responses. Some studies used only written responses, including: journals on mathematics problem posing (Crespo, 2003); mathematical autobiographies (Harkness et al., 2007; Wachira et al., 2008); responses to interpreting students’ solutions to mathematical tasks (Callejo & Zapatera, 2017; Sánchez-Matamoros et al., 2015, 2019); responses to incorrect students’ solutions to the same problem solved by PTs (Son, 2013); responses on reflecting on a student’s invented algorithm (Harkness & Thomas, 2008); responses to written standard place-value-operation tasks (addition and subtraction) (Thanheiser, 2010); and responses to the analysis of video recorded mathematics lessons (Morris, 2006; Star & Strickland, 2008) and analysis of a mathematics video curriculum (Stockero, 2008).

Mathematics Tests and Tasks. Some studies used content knowledge tests on rational numbers and computations (Lovin et al., 2018); fractions (Lin et al., 2013; Osana & Royea, 2011); whole number operations (Kaasila et al., 2010; Norton, 2019); linear functions (You & Quinn, 2010); and algebraic language (Pomerantsev & Korosteleva, 2003). One study combined a number sense test with interviews (Whitacre & Nickerson, 2016). A few studies used students’ work on mathematical tasks involving solving algebra tasks (Hohensee, 2017); posing mathematics problems (Crespo & Sinclair, 2008); solving pattern-finding tasks (Richardson et al., 2009; and sorting mathematics problems (Osana et al., 2006).

The different ways of collecting data outlined above were used throughout the period. There were about the same number of studies that used questionnaires, interviews, written responses, and tests and tasks, alone or in different combinations. Overall, the growth in instrumentation consisted of very little use of only interviews and an increased use of combinations of questionnaires and interviews, open written responses, and tests or tasks which have the potential to produce more valid data regarding the types of PMTC that were studied.

4.2.5 Design and Analysis in Researching PMTC

Design and analysis are the last two factors Medley (1987) indicated were important in considering the evolution of research on teaching. However, based on Medley’s perspective, they were problematic to address for the studies on PTMC in 2000–2020, most of which were not specifically designed to address PTs’ PMTC on entering MTE, but had broader goals. Thus, the design was considered in terms of what the studies used to support the data collection process at the beginning of a course or prior to an intervention and analysis as the means used to obtain information from the data.

Design. There was an evolution in the design of research in the period in terms of the use of school students’ mathematical work and videos of mathematics teaching to engage the PTs in situations to apply their PMTC. Students’ work included: actual solutions to various mathematical tasks (Callejo & Zapatera, 2017; Magiera et al., 2013; Sánchez-Matamoros et al., 2015, 2019; Simpson & Haltiwanger, 2017); hypothetical written solutions (Shin, 2020); incorrect solutions to the same problem solved by PTs (Son, 2013), and a student’s invented algorithm (Harkness & Thomas, 2008). Videos included videotaped mathematics lessons (Morris, 2006; Star & Strickland, 2008) and a video-case curriculum (Stockero, 2008).

There was also evolution in terms of significant variations in the design of instrumentation (e.g., questionnaires, written responses, tests and tasks) to match the different types of PMTC and in terms of the combination of interviews with other instruments to obtain reliable data. One area of limitation involved studies not being designed solely for researching PMTC at the beginning of an education program, which could have resulted in aspects of the PMTC not being identical to the PTs’ PMTC on entering the program. A few studies were designed at the beginning of courses, while most were designed as intervention studies with a pre-post-intervention design. The intent of the intervention studies was more about promoting the intervention as a way of impacting change and less about the nature of the PMTC. Thus, they tended to provide little information on the pre-intervention characteristics, with the emphasis being on the post-intervention. The design also tended to use convenient samples of PTs enrolled in specific courses and small sample sizes regardless of the nature of the instrumentation. Thus, the studies did not necessarily provide a representative picture of PMTC within an institution or a region, even though they offered useful insights about the PMTC.

Analysis. The analysis approaches used in the studies depended on the instrumentation and thus showed an evolution of approaches consisting of both quantitative and qualitative strategies. These approaches varied within and across categories of instruments depending on the design of the instrument. For example, some questionnaires used the Likert scale (e.g., Dreher et al., 2016; Newton, 2009; Szydlik et al., 2003; Weldeana & Abraham, 2014) while others used open-ended items with a rubric or scale for scoring or categories to compile and rank frequencies (e.g., Conner et al., 2011; Lee & Lee, 2020; Miller, 2018; Son & Lee, 2020). Interviews by themselves were semi-structured and based on PTs solving mathematics tasks (Stephens, 2006; Thanheiser, 2009, 2010; Yanik, 2011); when combined with questionnaires they were semi-structured and based on following up on questionnaire items or ideas (Ambrose, 2004; Conner et al., 2011; Son & Lee, 2020; Stephens, 2008; Szydlik et al., 2003). Interviews as well as open written response tasks (Crespo, 2003, Harkness et al., 2007; Sánchez-Matamoros et al., 2015; 2019; Son, 2013; Thanheiser, 2010; deCallejo & Zapatera, 2017; Harkness & Thomas, 2008; Morris, 2006; Star & Strickland, 2008, Stockero, 2008) were generally analyzed through coding to produce themes or categories. Tests, which dealt with mathematics content, used scoring schemes that indicated the level of correctness or error in participants’ responses (e.g., Lin et al., 2013; Lovin et al., 2018; Norton, 2019; Osana & Royea, 2011; You & Quinn, 2010). While Medley (1987) suggested more use of technology in analysis, this was not reflected in the studies because of the shift to more qualitative approaches or quantitative approaches with small sample sizes that did not necessarily require complex statistical analysis.

4.3 Summary of Evolution of PMTC Research

Overall, consideration of the scope and the methodological factors of the studies on PMTC for the period 2000–2020 indicated that there were both growth and limitations or gaps in research on PTs’ PMTC at the point of entry in MTE. There was evolution in the scope of research in terms of the extent of the types of PMTC researched and the extent to which PMTC were addressed by the studies. For example, while the mathematical knowledge and skills category of PMTC received the most attention in the studies, suggesting ongoing interest in content-related characteristics, a significant shift was the pedagogical skills category regarding studies on noticing and interpreting students’ work and thinking that was the largest group of studies for the three categories of PMTC (Table 1).

There was also evolution of aspects of methodology based on Medley’s (1987) four factors of the conceptualization, instrumentation, design, and analysis. For example, conceptualization of PMTC evolved to reflect contemporary perspectives of teaching and learning mathematics. There was a shift in instrumentation from a focus on surveys with large samples in early studies to a variety of tools, used alone or in different combinations. There was growth in design regarding the use of school children’s work and of videos on teaching as bases of obtaining pedagogical-related data. The analysis also shifted from mainly statistical approaches to including qualitative approaches particularly for interviews and written responses. Limitations and gaps in the evolution of the research on PMTC are addressed in the following section in relation to what needs to be considered in future research.

5 Future Evolution of Research on PMTC

Future evolution of research on PMTC refers to what should be considered to move the field forward in this area. Medley (1987) indicated that the methodological factors of conceptualization, instrumentation, design, and analysis should also be used in considering the future evolution of research on teaching, which includes research on PMTC. Consistent with the preceding section, the scope of research is also relevant. The preceding discussion of evolution on research on PMTC in the period 2000–2020 indicated that there were limitations or lack of attention regarding scope and methodological factors, which suggest areas that need attention to support future evolution of research on PMTC. The following summary highlights the key areas that future research should consider.

Regarding scope, much more attention is needed on research of PMTC for PTs at the point of entry into a teacher education program as opposed to other points during the program, which could be affected by confounding factors associated with their experience in the program in general. There also needs to be more scope and depth of the types of PMTC researched within and beyond the categories of mathematical content and skills, pedagogical knowledge, and beliefs. There are other aspects of PTs’ mathematical abilities, knowledge, and attitudes, as well as aptitude for teaching that are important to understand PTs and their PMTC at point of entry MTE. One area the studies in 2000–2020 were particularly lacking in addressing, that needs future consideration, was affective factors such as PTs’ attitudes and what they value. For example, do they value collaboration, know how to connect and form collaborative groups, have the skills needed to create an environment of working with others? As Blanton (2002) also asked, do they value discourse as an active process in which students use the collective knowledge of a group to build understanding (i.e., dialogic discourse)? What is their level of competence to reflect and to be curious? In addition, as Strutchens et al. (2017) also suggested, there is a need to consider the various identities that PTs have prior to their participation in preservice education.

Another area not attended to, but that is of significance in the context of the current digital age and twenty-first century society and in need of future attention, is the impact of technology and culture on the PTs’ PMTC at the point of entry into MTE. Both are important to the nature of PTs’ PMTC and culture, in particular, to their developing mathematical identity. Finally, regarding beliefs, the scope was limited to types of beliefs but future attention could also be given to PTs’ ability to reflect on them.

Regarding conceptualization, more attention is needed to conceptualize PMTC in relation to teacher education, for example, regarding the types and nature of PTs’ PMTC on entering teacher education that are consistent with the role of the teacher education program or are most needed to support the PTs’ learning in the program. In addition, PMTC should be conceptualized in relation to technology and culture regarding specific characteristics of the latter that influence the nature of the PMTC.

Regarding instrumentation, design, and analysis, the future evolution will depend on the scope and characterization of future studies on PMTC. Some considerations are: designing studies with the sole aim of exploring PMTC at point of entry to MTE, which may also require different or “better instruments” (Medley, 1987) and analysis; designing studies that are more humanistic in focusing on what PTs’ know and can do based on their PMTC, which could be more practical when using convenient samples and qualitative instruments; and the use of more rigorous mixed methods research design with more rigorous statistical analysis and use of technology.

To conclude, overall, the studies suggested that there have been significant changes in research on teaching with a focus on the Type F variable regarding PMTC of candidates MTE. But ongoing work is necessary for this area of research given its importance to understanding the selection of teacher candidates, the mathematics teacher, teaching of mathematics, and teacher education. Since PMTC at the point of entry are important starting points of PTs’ formal education to become a teacher, then more attention is needed to understand these PMTC and how to work with them in mathematics teacher education.