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Investigating Preschool Teachers' Pedagogical Content Knowledge of Number Comparison

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Title: Investigating Preschool Teachers' Pedagogical Content Knowledge of Number Comparison
Language: English
Authors: Xia Li, Colleen Maas, Colleen Oppenzato
Source: Early Education and Development. 2025 36(2):249-264.
Availability: Routledge. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Page Count: 16
Publication Date: 2025
Document Type: Journal Articles
Reports - Research
Education Level: Early Childhood Education
Preschool Education
Higher Education
Postsecondary Education
Descriptors: Preschool Teachers, Pedagogical Content Knowledge, Number Concepts, Mathematics Education, Mathematics Teachers, Preservice Teachers, Knowledge Level, Student Needs, Teacher Effectiveness, Ability Identification, Mathematics Achievement, Numeracy, Metropolitan Areas, Teacher Education Programs, Early Childhood Education, Undergraduate Students, Graduate Students
DOI: 10.1080/10409289.2024.2389362
ISSN: 1040-9289
1556-6935
Abstract: Research Findings: The aim of this study was to investigate U.S. preschool teachers' math pedagogical content knowledge in number comparison. Seventy-four in- and pre-service teachers completed a set of scenario-based questions from a researcher-designed questionnaire that is composed of learning scenarios. Through a mixed-method approach, the study revealed the following findings: a considerable portion of participants (89%) successfully identified the concept of number comparison embedded in a learning scenario, showcasing their specialized content knowledge. However, when it came to knowledge of content and students, only 24 participants (32%) were able to pinpoint both a child's mathematical strength and area of need. In terms of knowledge of content and teaching, more than half of the participants (51%) recommended the utilization of hands-on materials and/or visual aids as part of their remediation techniques, while 49% did not reference these educational tools. Notably, no significant differences were observed between the pedagogical content knowledge of pre-and in-service teachers concerning number comparison. Practice or Policy: The findings indicate that teacher education and professional development should focus on helping teachers understand children's learning trajectories on number comparison and include instructions on using concept specific developmentally appropriate strategies, and indicate that teaching experience cannot replace professional training.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1457421
Database: ERIC
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  Value: <anid>AN0182209680;h4j01feb.25;2025Jan16.01:28;v2.2.500</anid> <title id="AN0182209680-1">Investigating Preschool Teachers' Pedagogical Content Knowledge of Number Comparison </title> <p>Research Findings: The aim of this study was to investigate U.S. preschool teachers' math pedagogical content knowledge in number comparison. Seventy-four in- and pre-service teachers completed a set of scenario-based questions from a researcher-designed questionnaire that is composed of learning scenarios. Through a mixed-method approach, the study revealed the following findings: a considerable portion of participants (89%) successfully identified the concept of number comparison embedded in a learning scenario, showcasing their specialized content knowledge. However, when it came to knowledge of content and students, only 24 participants (32%) were able to pinpoint both a child's mathematical strength and area of need. In terms of knowledge of content and teaching, more than half of the participants (51%) recommended the utilization of hands-on materials and/or visual aids as part of their remediation techniques, while 49% did not reference these educational tools. Notably, no significant differences were observed between the pedagogical content knowledge of pre-and in-service teachers concerning number comparison. Practice or Policy: The findings indicate that teacher education and professional development should focus on helping teachers understand children's learning trajectories on number comparison and include instructions on using concept specific developmentally appropriate strategies, and indicate that teaching experience cannot replace professional training.</p> <p>Number comparison is a fundamental aspect of early numeracy, encompassing both non-symbolic and symbolic comparison, with each playing a crucial role in mathematical development (Jordan et al., [<reflink idref="bib12" id="ref1">12</reflink>]; Van de Walle et al., [<reflink idref="bib36" id="ref2">36</reflink>]). Both non-symbolic and symbolic comparison skills were found to be longitudinal predictors for math achievement, with symbolic comparison being a stronger predictor than non-symbolic comparison (Vanbinst & De Smedt, [<reflink idref="bib33" id="ref3">33</reflink>]; Xenidou-Dervou et al., [<reflink idref="bib37" id="ref4">37</reflink>]). Specifically, symbolic number comparison was found to be predictively related to single-digit arithmetic competence, which cannot be explained by children's intellectual ability, general mathematics knowledge, or working memory, indicating an important and unique role of symbolic number comparison in children's arithmetic development (Vanbinst et al., [<reflink idref="bib34" id="ref5">34</reflink>]). Moreover, the consistent longitudinal association between symbolic number comparison and arithmetic development (e.g., Dowker, [<reflink idref="bib7" id="ref6">7</reflink>]; Siegler & Lortie-Forgues, [<reflink idref="bib30" id="ref7">30</reflink>]) suggests that screening on this skill might be helpful to identify at risk children and that fostering number comparison might be a remedial intervention focus (Vanbinst et al., [<reflink idref="bib35" id="ref8">35</reflink>]).</p> <p>Despite its significance, preschool teaching practices often fail to adequately address number comparison. Research indicates a discrepancy between teaching approaches in the U.S. and countries like China, with American preschoolers receiving less emphasis on number relations (Li et al., [<reflink idref="bib19" id="ref9">19</reflink>]). Moreover, unlike other concepts such as enumeration or pattern/shape, number comparison was not observed occurring spontaneously during children's play (Seo & Ginsburg, [<reflink idref="bib27" id="ref10">27</reflink>]). In order to counter the lack of adequate exposure to number comparison, preschool teachers should actively introduce the concept and be equipped with adequate teaching knowledge including identifying the emergence of this concept from children's everyday activities, analyzing children's mathematical thinking, and designing activities to support its development. The purpose of the present study was to investigate preschool teachers' math pedagogical content knowledge (PCK) of number comparison, encompassing those three aspects. This study makes the first such study that examines preschool teachers' PCK in the specific numeracy area of number comparison; its results can reveal preschool teachers' specific knowledge on number comparison including teachers' strength and needs, which will be informative for effective teacher training or professional development.</p> <hd id="AN0182209680-2">Theoretical Framework</hd> <p>Shulman ([<reflink idref="bib28" id="ref11">28</reflink>]) proposed three ways to categorize what teachers know: content knowledge, pedagogical content knowledge (PCK), and curricular knowledge (Shulman, [<reflink idref="bib28" id="ref12">28</reflink>]). PCK was identified as the central piece of teachers' knowledge that "goes beyond knowledge of subject matter per se to the dimension of subject matter knowledge <emph>for teaching</emph>" (Shulman, [<reflink idref="bib28" id="ref13">28</reflink>], p. 9).</p> <p>Ball et al. ([<reflink idref="bib1" id="ref14">1</reflink>]) further developed Shulman's PCK concept (Shulman, [<reflink idref="bib28" id="ref15">28</reflink>], [<reflink idref="bib29" id="ref16">29</reflink>]) and identified two discernible subdomains. One subdomain is knowledge of content and students (KCS), "knowledge that combines knowing about students and knowing about mathematics ... knowledge of common student concepts and misconceptions about particular math content" (Ball et al., [<reflink idref="bib1" id="ref17">1</reflink>], p. 401). Researchers have also described this subdomain as <emph>interpreting</emph> children's math actions or conversations during classroom activities. Teachers with strong KCS understand the degree and depth of children's mathematical thinking (e.g., Lee, [<reflink idref="bib17" id="ref18">17</reflink>]).</p> <p>The other subdomain of PCK is knowledge of content and teaching (KCT), "combines knowing about teaching and knowing about mathematics ... Teachers evaluate the instructional advantages and disadvantages of representations used to teach a specific idea and identify what different methods and procedures afford instructionally" (Ball et al., [<reflink idref="bib1" id="ref19">1</reflink>], p. 401). Similarly, other researchers have described this subdomain as <emph>planning</emph> appropriate activities within the child's zone of proximal development to <emph>enhance</emph> children's math thinking (e.g., Dunekacke et al., [<reflink idref="bib9" id="ref20">9</reflink>]; Lee, [<reflink idref="bib17" id="ref21">17</reflink>]). Moreover, Ball et al. ([<reflink idref="bib1" id="ref22">1</reflink>]) expanded on Shulman's ([<reflink idref="bib28" id="ref23">28</reflink>]) content knowledge to introduce specialized content knowledge (SCK), which is a type of math knowledge that "involves an uncanny kind of unpacking of mathematics that is not needed – or even desirable – in settings other than teaching" (p. 400).</p> <p>Those three subdomains of knowledge – SCK, KCS, KCT, constitute an important framework for the present study.</p> <p>In the field of early childhood education, existing instruments primarily focus on</p> <p>assessing certain aspects of preschool teachers' math PCK. For example, Platas ([<reflink idref="bib24" id="ref24">24</reflink>]) instrument evaluates preschool teachers' knowledge of children's development on number and operations, McCray and Chen ([<reflink idref="bib20" id="ref25">20</reflink>]) measure teachers' sensitivity to math concepts in children's free play and skills to support those concepts, and Gasteiger et al. ([<reflink idref="bib11" id="ref26">11</reflink>]) measures knowledge of children's mathematical thinking and ways to support their mathematical thinking. Li ([<reflink idref="bib18" id="ref27">18</reflink>]) however proposed that math PCK for preschool teachers should and can consist of three aspects – SCK, KCS, and KCT. Indeed, if searching for the least common multiple of math PCK components from the existing instruments, we can find those three components: identifying math concepts [from a learning scenario such as children's play] (Dunekacke et al., [<reflink idref="bib8" id="ref28">8</reflink>]; McCray & Chen, [<reflink idref="bib20" id="ref29">20</reflink>]), analyzing children's mathematical thinking/development (Gasteiger et al., [<reflink idref="bib11" id="ref30">11</reflink>]; Platas, [<reflink idref="bib24" id="ref31">24</reflink>]), and supporting children's math learning (Dunekacke et al., [<reflink idref="bib8" id="ref32">8</reflink>]; Gasteiger et al., [<reflink idref="bib11" id="ref33">11</reflink>]; McCray & Chen, [<reflink idref="bib20" id="ref34">20</reflink>]), which reflects SCK, KCS, and KCT (Ball et al., [<reflink idref="bib1" id="ref35">1</reflink>]), respectively. Therefore, supported by both theories (Ball et al., [<reflink idref="bib1" id="ref36">1</reflink>]; Shulman, [<reflink idref="bib28" id="ref37">28</reflink>]) and empirical research in the early childhood field, we adopted those three components as the construct of math PCK in the present study and operationally defined it as consisting of <emph>identifying [math concepts]</emph>, <emph>analyzing [children's mathematical thinking]</emph>, and <emph>supporting [children's math learning]</emph> (see Table 1 for definition and examples). We are mindful that this definition is different from Ball et al. ([<reflink idref="bib1" id="ref38">1</reflink>]) whose PCK includes knowledge of content and students, knowledge of content and teaching, and knowledge of content and curriculum.</p> <p>Table 1. Definition and example for the math PCK components.</p> <p> <ephtml> <table><thead><tr><td /><td>Ball et al. (<xref ref-type="bibr" rid="bibr1">2008</xref>) Definition</td><td>Our Definition</td><td>Our Example</td></tr></thead><tbody><tr><td>Identifying (math concepts) –Specialized Content Knowledge (SCK)</td><td>"The mathematical knowledge and skill unique to teaching ... involves an uncanny kind of unpacking of mathematics that is not needed – or even desirable – in settings other than teaching" (p. 400)</td><td>Identifying <italic>the most relevant</italic> math concepts and their unpacked sub concepts in a learning context, in contrast to perceiving <italic>any relevant</italic> math concepts</td><td>Counting objects involve four sub skills: correct number sequence, one to-one correspondence, keeping track of the items counted, and labeling the set size</td></tr><tr><td>Analyzing (children's mathematical thinking) – Knowledge of Content and Students (KCS)</td><td>"Knowledge that combines knowing about students and knowing about mathematics ... knowledge of common student concepts and misconceptions about particular mathematical content" (p. 401)</td><td>Identifying children's common errors or misconceptions with particular concepts</td><td>Identifying a child's counting errors such as cardinality error or one-to one correspondence error</td></tr><tr><td>Supporting (children's math learning) –Knowledge of Content and Teaching (KCT)</td><td>"Combines knowing about teaching and knowing about mathematics. Many of the mathematical tasks of teaching require a mathematical knowledge of the design of instruction ... They choose which examples to start with and which examples to use to take students deeper into the content" (p.401)</td><td>Proposing or choosing tasks that either advance a child's learning on a particular concept or target a child's specific errors, as well as involving the use of motivating contexts and concrete/visual materials</td><td>For a cardinality error, teaching should focus on counting small sets and emphasizing the last number word ("three, so there are three cars") rather than practicing the correct counting sequence, and teaching should involve the use of a motivating context and concrete objects, for example, counting how many cookies Cookie Monster eats</td></tr></tbody></table> </ephtml> </p> <hd id="AN0182209680-3">Literature Review on Number Comparison</hd> <p>In this section, we review what preschool teachers are expected to know <emph>about</emph> number comparison, children's development on number comparison, and developmentally appropriate teaching methods and procedures that target number comparison.</p> <hd id="AN0182209680-4">Specialized Content Knowledge: About Number Comparison</hd> <p>Specialized content knowledge refers to the knowledge <emph>about</emph> number comparison that preschool teachers uniquely need. Pedagogy textbooks and teaching guidelines (Brownell, [<reflink idref="bib3" id="ref39">3</reflink>]; Council of Chief State School Officers, [<reflink idref="bib6" id="ref40">6</reflink>]; Van de Walle et al., [<reflink idref="bib36" id="ref41">36</reflink>]) recommend two pieces of basic knowledge for preschool teachers. One is that number comparison entails three core relations of more than, less than, and equal to (e.g., Council of Chief State School Officers, [<reflink idref="bib6" id="ref42">6</reflink>]; Van de Walle et al., [<reflink idref="bib36" id="ref43">36</reflink>]). Although this knowledge might be implicit or vague for non-teachers, preschool teachers need to comprehend it explicitly to effectively teach number comparison. Effective educators, for instance, emphasize the importance of children recognizing not just the larger set, but also identifying the smaller one and understanding the concept of equivalence through both non-symbolic (e.g., dot cards) and symbolic mediums (e.g., numeral comparison). Secondly, preschool teachers should comprehend that number comparison encompasses the assessment of physical or visual sets (non-symbolic comparison) as well as the evaluation of written numerals (symbolic comparison), with the former preceding the latter (Brownell, [<reflink idref="bib3" id="ref44">3</reflink>]; Council of Chief State School Officers, [<reflink idref="bib6" id="ref45">6</reflink>]).</p> <p>As stated previously, number comparison is not taught adequately in preschools. Why so? From the aspect of <emph>identifying</emph> (<emph>specialized content knowledge)</emph>, we ask: is it because teachers cannot identify the number comparison concepts in children's play or other daily scenarios? Existing literature does not offer much insight. McCray and Chen's ([<reflink idref="bib20" id="ref46">20</reflink>]) instrument did not include the concept of number comparison, nor did Gasteiger et al. ([<reflink idref="bib11" id="ref47">11</reflink>]). In the present study, we employed a structured learning scenario centered on number comparison and evaluated preschool teachers' specialized content knowledge by their ability to identify the embedded concepts.</p> <hd id="AN0182209680-5">Knowledge of Content and Students: Children's Development on Number Comparison</hd> <p>Preschool teachers are expected to have some knowledge on children's development of number comparison. Three pieces of knowledge can be identified from pedagogy textbooks and teaching guidelines: 1) the <emph>more than</emph> relation is relatively easier than <emph>less than</emph> for young children (Van de Walle et al., [<reflink idref="bib36" id="ref48">36</reflink>]), 2) children's number comparisons progress from smaller numbers (zero-to-five) to larger numbers (five-to-10; Council of Chief State School Officers, [<reflink idref="bib6" id="ref49">6</reflink>]; Sarama & Clements, [<reflink idref="bib26" id="ref50">26</reflink>]); and 3) children progress from comparing objects (non-symbolic comparison) to comparing written numerals (symbolic comparison; Sarama & Clements, [<reflink idref="bib26" id="ref51">26</reflink>]). Many children have difficulty comprehending the less than relation because in everyday life situations involving more items are more relevant to children's interest (e.g., most children want more snacks, more snacks, etc.). Additionally, children face challenges in generalizing number relations with larger numbers, as such relations are based on cardinality. Children typically become comfortable with larger numbers, such as 6–10, later than with smaller numbers.</p> <p>Preschool teachers are expected to know what is described above in order to effectively support children. For example, a lack of understanding regarding the natural progression from non-symbolic to symbolic comparison might lead a preschool teacher to bypass concrete object comparisons and directly introduce numeral comparisons, which could be counterproductive. Despite its importance, we do not know much about preschool teachers' knowledge of children's development on number comparison. Number comparison is not included in any measurement instruments (e.g., Gasteiger et al., [<reflink idref="bib11" id="ref52">11</reflink>]; McCray & Chen, [<reflink idref="bib20" id="ref53">20</reflink>]; Platas, [<reflink idref="bib24" id="ref54">24</reflink>]). For example, Platas's ([<reflink idref="bib24" id="ref55">24</reflink>]) instrument was designated to measure preschool teachers' knowledge on children's number development that includes verbal counting sequence, counting/numerosity, ordinal number words, addition/subtraction, division of sets, and written number symbols and words, but not number comparison. The present study entails not only a designated scenario on number comparison but also designated questions on preschool teachers' knowledge on children's development.</p> <hd id="AN0182209680-6">Knowledge of Content and Teaching: Developmentally Appropriate Teaching Method on Number Comp...</hd> <p>Similar to other number concepts, teaching number comparison should incorporate visual or concrete items because preschool children are preoperational thinkers (Piaget, [<reflink idref="bib23" id="ref56">23</reflink>]). Particularly with number comparison, the pedagogy should employ a matching and/or a counting method (Council of Chief State School Officers, [<reflink idref="bib6" id="ref57">6</reflink>]; Sarama & Clements, [<reflink idref="bib26" id="ref58">26</reflink>]). The former involves matching two sets based on one-to-one correspondence, and the latter involves counting each set to compare. The combined method is to match and then count. Moreover, teachers should ensure that the numbers being compared fall within an appropriate range of numbers (e.g., zero-to-five for three-year-olds). It is also pertinent to progress from far comparisons (e.g., one vs. many, or one versus five) to close comparisons (four versus five) and from comparing similar items to comparing dissimilar items (Krasa et al., [<reflink idref="bib14" id="ref59">14</reflink>]).</p> <p>Presently, there is a considerable research gap concerning preschool teachers' awareness and implementation of developmentally appropriate teaching strategies for number comparison. In the present study, we assessed this knowledge by analyzing how preschool teachers' proposed teaching methods are aligned with the developmentally appropriate guidelines depicted in this section.</p> <hd id="AN0182209680-7">The Present Study</hd> <p>Most previous studies of preschool teachers' math PCK have focused on general math content and general teaching knowledge, such as sensitivity to overall math in children's play (e.g., McCray & Chen, [<reflink idref="bib20" id="ref60">20</reflink>]) or on one aspect of pedagogical content knowledge, such as knowledge of children's mathematical thinking (e.g., Platas, [<reflink idref="bib24" id="ref61">24</reflink>]). Some studies (Korkmaz & Şahin, [<reflink idref="bib13" id="ref62">13</reflink>]; Sahin & Korkmaz, [<reflink idref="bib25" id="ref63">25</reflink>]) targeted specific content areas like geometric shapes or quantitative concepts, yet they often evaluated only one aspect, such as knowledge of students' mistakes. Li ([<reflink idref="bib18" id="ref64">18</reflink>]) was one of the few studies that has focused on a specific content area, namely, counting and cardinality, and investigated preschool teachers' math PCK from a full set of aspects: knowledge on children's mathematical thinking (e.g., misconceptions), knowledge of teaching, and also specialized content knowledge of particular concepts. This study builds upon the research trajectory initiated by Li ([<reflink idref="bib18" id="ref65">18</reflink>]), providing further insights into another vital content area: number comparison. Specifically, we address the following three research questions:</p> <hd id="AN0182209680-8">Identifying (SCK):</hd> <p>To what extent can preschool teachers identify the number comparison concepts that are embedded in a math game?</p> <hd id="AN0182209680-9">Analyzing (KCS):</hd> <p>To what extent can preschool teachers analyze a child's performance on number comparison in the math game including the child's strengths and areas of need?</p> <hd id="AN0182209680-10">Supporting (KCT):</hd> <p>To what extent can preschool teachers propose developmentally appropriate tasks that target number comparison?</p> <p>Addressing these three research questions can reveal preschool teachers' specific math PCK knowledge regarding number comparison, thus offering guidance for teacher training and preparation programs.</p> <hd id="AN0182209680-11">Method</hd> <p>The present study is part of a larger study that involved the use of a researcher-designed questionnaire (Li, [<reflink idref="bib18" id="ref66">18</reflink>]) that was designed to measure preschool teachers' PCK in the three content areas of early numeracy. Specifically, Section one addresses counting and cardinality, Section two addresses number comparison, and Section three addresses number composition and addition. Each section depicts a learning scenario, following which are two sets of questions: one consisting of short answer questions and the other of multi-choice questions. This paper only reports on responses to Section two, which presents a preschool classroom scenario of a number comparison game.</p> <hd id="AN0182209680-12">Procedure</hd> <p>The questionnaire was distributed to participants in education classes or at their school sites. Participants completed the questionnaire either during the researcher's visit or at another time and returned it to the mailbox that was outside the researcher's office within three weeks. After completing Section one, the participants were then instructed – by an instruction page in the questionnaire – to proceed to Section two number comparison if they were preschool teachers or planning to become one, and to Section three if they were teachers of 1st or 2nd grade. Altogether, it took about 30 minutes for the participants to complete the whole questionnaire.</p> <hd id="AN0182209680-13">Participants</hd> <p>One hundred participants completed the study, and 74 of them completed Section two. Participants were recruited from graduate and undergraduate teacher preparation programs and also from preschools in a major metropolitan city.</p> <p>Of the 74 participants, 50% (<emph>N</emph> = 37) were pre-service teachers and 50% (<emph>N</emph> = 37) were in-service preschool teachers. The 37 pre-service teachers were either undergraduate or graduate students enrolled in an early childhood teacher preparation program with no formal teaching experience. These pre-service teachers either had not taken any math pedagogy courses or were currently enrolled in a math pedagogy course. For the latter, participants completed the questionnaire early in the semester before any specific math topics had been taught. Among the 37 in-service preschool teachers, 24 were head teachers and 13 were teaching assistants; some of them were enrolled in a master's program in early childhood education. The participant pool represents a diverse ethnic/racial population with 15% Asian or Pacific Islanders, 37% Black, 14% Hispanic, 24% White, and 10% other. Participants also varied across ages: 24% 21–30 years old, 37% 31–40, 14% 41–50, and 25% 51 +.</p> <hd id="AN0182209680-14">Instrument</hd> <p>Section two starts with a scenario depicting a guessing game played by a teacher and a four-year-old named Lui. Depicted below is the scenario:<bold>Ms. G:</bold> hides four Lego people under a box and asks:"Would you please guess how many people are under the box?"<bold>Lui:</bold>I don't know.<bold>Ms.G:</bold>Take a guess.<bold>Lui:</bold>Seven.<bold>Ms.G:</bold>Less than seven.<bold>Lui:</bold>Eight.<bold>Ms. G:</bold>Less than eight.<bold>Lui:</bold>10.<bold>Ms. G:</bold>It is less than 10.Lui pauses<bold>Ms. G:</bold>Can you think of a smaller number, a small number?<bold>Lui:</bold>Two or three?<bold>Ms. G:</bold>Yeah, those are small numbers. Do you think it might be two or three?<bold>Lui:</bold>Two.<bold>Ms. G:</bold>It is more than two.<bold>Lui:</bold>Three!<bold>Ms.G:</bold>It is more than three.<bold>Lui:</bold>Four?<bold>Ms.G:</bold>It is four!<emph>...</emph></p> <p>After reading the full scenario depicted above, participants answered four short-response questions designed to measure their PCK knowledge. Question one targets participants' SCK, as measured by their ability to <emph>identify</emph> the key concept of number comparison embedded in the guessing game – giving a larger or smaller number based on the verbal clues of "less than" or "more than." Questions two and three target participants' KCS as measured by their abilities to identify Lui's strengths and areas to be improved; specifically, Lui understood the concept of "more than" and was able to name small numbers but had difficulty with "less than." Lastly, question four targets participants' KCT, as measured by their ability to describe a developmentally appropriate activity to help Lui overcome his difficulties with number comparison.</p> <hd id="AN0182209680-15">Coding</hd> <p>For the first three short-answer questions, a deductive approach to coding was followed, in which key concepts were identified before data analysis; the key concepts were identified based on analysis of both the scenario and a subset of the data. See Table 2 for a list of deductive codes and examples of participant responses. For the fourth question, responses were coded according to a rubric. There were three coders (the three authors). Inter-rater reliability for all four questions among the three coders was established for 38% of the dataset at κ =.65, which <emph>is substantial</emph> (.61–0.80;.81–1.0, <emph>perfect</emph>; Cohen, [<reflink idref="bib5" id="ref67">5</reflink>]). Disagreements were discussed and the remaining data were divided and coded separately by the coders.</p> <p>Table 2. Coding categories for questions one, two, and three.</p> <p> <ephtml> <table><thead><tr><td /><td>Code</td><td>Criteria</td><td>Sample Participant Response</td></tr></thead><tbody><tr><td>Q1: What math skills are required of a child to play this game?</td><td>Primary skill</td><td>What the child must know to play the game; for this scenario, it is the skill to compare numbers by more than/less than</td><td>"greater than/less than" "Comparing smaller/bigger numbers"</td></tr><tr><td>Secondary skill</td><td>A less essential math concept that nevertheless helps facilitate game play; for this scenario, it is to know the order of numbers</td><td>"Number order" "ordinal relation"</td></tr><tr><td>Incorrect response</td><td>Not correctly identifying either the primary or secondary math skills</td><td>"Counting backwards" "guessing numbers"</td></tr><tr><td>Q2: Please evaluate Lui's mathematical performance</td><td>Strengths</td><td>1) Identifying "more than" with smaller numbers 2) Knowing what a small number is</td><td>"He understands more than, but ... " "Lui struggles ... but shows an understanding of small numbers"</td></tr><tr><td>Weakness</td><td>Identify the difficulty with less than, especially with large numbers</td><td>"He does not understand less than, he understands ... " "He did not know the term less than, but he knew ... "</td></tr><tr><td>Incorrect</td><td>Name neither the child's mathematical strength nor area to be improved</td><td>"He is guessing, performing trial and error" "Good. He needs prompting and scaffolding from a teacher"</td></tr><tr><td>Q3: What is Lui's difficulty?</td><td>Specific</td><td>Describe Lui's difficulty in a specific manner</td><td>"Naming numbers 'less than' another number" "Comparing numbers greater than 4 or 5"</td></tr><tr><td /><td>Acceptable</td><td>Does not specify the difficulty with <italic>less than</italic> or the specific number range</td><td>"Lui has difficulty with understanding the concept of more/less" "He did not know the relation between numbers which one biggest or small"</td></tr><tr><td /><td>Incorrect</td><td>All other responses</td><td>"Lui could not count backwards – would count forward even when Ms. G. said less than"</td></tr></tbody></table> </ephtml> </p> <hd id="AN0182209680-16">Coding for Question One</hd> <p>Participants' written responses to question one were coded to determine whether participants accurately identified the primary and/or secondary math skill involved in the game or if they incorrectly identified a math skill not addressed in the game. The primary skill is the concept that children must know to play the game, and the secondary skill is a less essential math concept that nevertheless helps facilitate game play. In the present scenario, comparing numbers is the primary skill because the game explicitly involves naming a number larger or smaller than another number. The secondary skill is ordering numbers. All other skills unrelated to comparing or ordering numbers were coded as incorrect answers. For example, one participant wrote that the game involved "comparing numbers and ordinal relations." This response was coded as correctly identifying both the primary and secondary skills in the game. Another participant wrote that the game involved "guessing, number identification, and hidden objects." This response was coded as not correctly identifying either the primary or secondary math skills and providing an incorrect answer.</p> <hd id="AN0182209680-17">Coding for Questions Two and Three</hd> <p>Responses to question two were coded to determine whether participants accurately identified the child's math strengths and areas needing improvement. Lui in the scenario demonstrated two discernible strengths – identifies more than with smaller numbers, and knows what a small number refers to, and one discernible weakness – does not know "less than" [with larger numbers]. To give an example, one participant wrote: "He struggles with the concepts of more and less, but shows an understanding of smaller numbers." This response was coded as correctly naming both the child's strength and area to be improved. Another participant wrote: "Lui had trouble with the game and did not do that well," which was coded as incorrect because the participant neither named the child's mathematical strength nor area to be improved.</p> <p>Question three asked participants "What is Lui's difficulty?" Lui's main difficulty is his inability to name numbers less than another number. Responses that indicate this difficulty in a specific manner such as "naming numbers 'less than' another number" or "comparing numbers greater than 4 or 5" were coded as specific responses. Responses that only indicate comparing numbers but does not specify the difficulty with <emph>less than</emph> or the specific number range were coded as acceptable responses. The other responses such as "verbal counting," "numeral recognition" was coded as incorrect responses.</p> <hd id="AN0182209680-18">Coding for Question Four</hd> <p>Finally, responses to question four were scored using a rubric (see Table 3) that was developed by the authors to determine the degree to which participants described a developmentally appropriate task which targets the child's needs, includes details, involves the use of hands-on manipulatives or visuals, and is likely to help improve the child's understanding of number comparison. As an example, one participant wrote:</p> <p>She should teach the concept of more and less at the same time. She should line up say five children in a row and count them, then ask one to step down, helping him to have a visual representation of the account going smaller.</p> <p>Table 3. Rubrics and participants' responses to question four: supporting the learning of number comparison (KCT).</p> <p> <ephtml> <table><thead><tr><td>Category</td><td>Criteria</td><td>All Teachers (<italic>N</italic> = 74)</td><td>Pre-service Teachers (<italic>N</italic> = 37)</td><td>In-service Teachers (<italic>N</italic> = 37)</td></tr></thead><tbody><tr><td>Targets area of need</td><td>The response targets the child's area of need.</td><td>44 (59%)</td><td>24 (65%)</td><td>20 (54%)</td></tr><tr><td>The response somewhat targets the child's area of need.</td><td>17 (23%)</td><td>7 (19%)</td><td>10 (27%)</td></tr><tr><td>The response does not target the child's areas of need.</td><td>12 (16%)</td><td>6 (16%)</td><td>6 (16%)</td></tr><tr><td>Level of detail</td><td>The response is very detailed.</td><td>5 (7%)</td><td>1 (3%)</td><td>4 (11%)</td></tr><tr><td>The response is mostly detailed.</td><td>23 (31%)</td><td>6 (16%)</td><td>17 (46%)</td></tr><tr><td>The response is somewhat detailed.</td><td>21 (28%)</td><td>16 (43%)</td><td>5 (14%)</td></tr><tr><td /><td>The response is not detailed.</td><td>14(38%)</td><td>3 (9%)</td><td>11 (30%)</td></tr><tr><td>Hands-on or visual</td><td>The suggestion includes the use of hands-on manipulatives, visual representations, etc.</td><td>38 (51%)</td><td>15 (41%)</td><td>23 (62%)</td></tr><tr><td>The suggestion does not include the use of hands-on manipulatives, visual representations, etc.</td><td>36 (49%)</td><td>22 (60%)</td><td>14 (38%)</td></tr><tr><td>Benefit for learning</td><td>The suggestion is very likely to help improve the child's ability to compare numbers.</td><td>2 (3%)</td><td>1 (3%)</td><td>1 (3%)</td></tr><tr><td>The suggestion is likely to help improve the child's ability to compare numbers.</td><td>10 (14%)</td><td>7 (19%)</td><td>3 (8%)</td></tr><tr><td>The suggestion is somewhat likely to help improve a child's ability to compare numbers.</td><td>38 (51%)</td><td>14 (44%)</td><td>24 (65%)</td></tr><tr><td>The suggestion is not likely to help improve the child's ability to compare numbers.</td><td>24 (32%)</td><td>15 (41%)</td><td>9 (24%)</td></tr></tbody></table> </ephtml> </p> <p>That was coded as <emph>targeting</emph> the child's primary areas of need, a detailed response, includes hands-on manipulatives or visual representations, and is likely to help improve the child's ability to compare numbers. Another participant wrote: "She should explain the terms more carefully." This response was coded as somewhat targeting the child's primary areas of need, not detailed, not including hands-on manipulatives or visual representations, and unlikely to help improve the child's ability to compare numbers.</p> <hd id="AN0182209680-19">Results</hd> <p>A series of chi-square tests of independence were performed to examine the relation between role (pre-service or in-service preschool teacher) and teachers' abilities to identify the math concepts and evaluate the child's performance in the scenario. The analysis revealed no significant differences between pre-service and in-service teachers; thus, data from the two groups were combined and analyzed together to answer the three research questions. A more detailed account of these results is described further below.</p> <hd id="AN0182209680-20">Identifying (SCK): To What Extent Can Preschool Teachers Identify the Number Comparison Conce...</hd> <p>Question one asks participants to identify the math skills required to play the guessing game depicted in the classroom scenario. Table 4 lists the number and percentage of participants who correctly named the primary and/or secondary skill, along with those who provided an incorrect response. Forty three participants (58%) identified either the primary or secondary skill, whereas twenty-six participants (35%) named both the primary and secondary skills, demonstrating a more complete understanding of the math skills required in the game compared to naming either the primary or secondary skill alone. Only a few participants (7%) identified neither the primary nor the secondary skill. Due to low expected cell counts, a Fisher's exact test was performed instead of a chi-square test of independence. The test was used to examine the association between accurate and inaccurate responses. Results indicated that significantly more participants were able to accurately identify the primary math skill in the scenario than were unable to name either the primary or secondary skill, <emph>p</emph> <.001. This suggests that, to a significantly large extent, preschool teachers can identify the number comparison concept in a game involving comparing numbers.</p> <p>Table 4. Participants' responses to question one: identifying number comparison (SCK).</p> <p> <ephtml> <table><thead><tr><td>Responses</td><td>All Teachers (<italic>N</italic> = 74)</td><td>Pre-service Teachers (<italic>N</italic> = 37)</td><td>In-service Teachers (<italic>N</italic> = 37)</td></tr></thead><tbody><tr><td>Identifies the primary skill but not secondary skill</td><td>40 (54%)</td><td>17 (46%)</td><td>23 (62%)</td></tr><tr><td>Identifies the secondary skill but not primary skill</td><td>3 (4%)</td><td>2 (5%)</td><td>1 (3%)</td></tr><tr><td>Identifies both the primary and secondary skills</td><td>26 (35%)</td><td>15 (42%)</td><td>11 (30%)</td></tr><tr><td>Does not identify either the primary or secondary skills</td><td>5 (7%)</td><td>3 (8%)</td><td>2 (5%)</td></tr><tr><td>Incorrect response</td><td>47 (64%)</td><td>19 (51%)</td><td>28 (76%)</td></tr><tr><td>Incorrect response and DOES NOT identify either primary or secondary skill.</td><td>5 (7%)</td><td>3 (8%)</td><td>2 (5%)</td></tr><tr><td>Incorrect response and DOES identify either primary or secondary skill</td><td>42 (57%)</td><td>16 (43%)</td><td>26 (70%)</td></tr></tbody></table> </ephtml> </p> <p>We further examined participants' incorrect responses. The three most frequent erroneous responses that participants listed were verbal counting (<emph>N</emph> = 15, 20%), followed by cardinality (<emph>N</emph> = 13, 18%) and numbers (<emph>N</emph> = 10, 14%). While not irrelevant, those responses were vague and did not indicate the specific math skills required to play the game.</p> <p>We ran two conditional probabilities to determine the likelihood of a participant providing an incorrect answer, given their ability to identify the number comparison concepts involved in the game. The goal was to ascertain whether participants who are better at identifying the number comparison concepts are more or less likely to make errors compared with participants with weaker identifying abilities. Of the 47 participants whose responses contained an error, 36 did not identify the primary and secondary skills in the game, while 11 did identify the primary and secondary skills. This represented a 77% probability of participants providing an erroneous response when they could not name the primary and secondary math skills in the game. Conversely, participants had a 23% probability of providing an erroneous response when they could successfully name both the primary and secondary math skills. These results suggest that participants with stronger identifying ability are less likely to include errors in their responses, demonstrating a robust knowledge of number comparison concepts in early math.</p> <hd id="AN0182209680-21">Analyzing (KCS): To What Extent Can Preschool Teachers Analyze a Child's Performance on Numbe...</hd> <p>Both question two and question three targeted analyzing a child's mathematical thinking regarding number comparison but from slightly different angles. Question two was used to elicit participants' spontaneous evaluation of the child's performance during the game, while question three explicitly asked participants to diagnose the child's difficulties.</p> <hd id="AN0182209680-22">Question Two</hd> <p>In question two, 29 participants (39%) accurately identified at least one of the child's strengths, and 42 participants (57%) accurately identified the child's areas of need. Only 24 participants (32%) named both a math strength and need, demonstrating a more complete understanding of the child's math performance during the game than naming either strength or need alone. In contrast, twenty-seven participants (37%) did not correctly name any of the child's strengths or areas of need. See Table 5 for a complete list of participants' response categories and corresponding counts and percentages. Results of a proportion difference test did not reveal a statistically significant difference between the percentage of participants who identified both the child's strength and need and that of those participants who identified neither a strength nor need, <emph>z</emph> = 0.52, <emph>p</emph> =.60.</p> <p>Table 5. Participants' responses to question two: analyzing mathematical thinking (KCS).</p> <p> <ephtml> <table><thead><tr><td>Responses</td><td>All Teachers (<italic>N</italic> = 74)</td><td>Pre-service Teachers (<italic>N</italic> = 34)</td><td>In-service Teachers (<italic>N</italic> = 34)</td></tr></thead><tbody><tr><td>Mentions Lui's strengths (at least 1)</td><td>29 (39%)</td><td>16 (43%)</td><td>13 (35%)</td></tr><tr><td>Mentions Lui's difficulties</td><td>42 (57%)</td><td>22 (59%)</td><td>20 (54%)</td></tr><tr><td>Mentions Lui's strengths but not Lui's difficulties</td><td>5 (7%)</td><td>2 (5%)</td><td>3 (8%)</td></tr><tr><td>Mentions Lui's difficulties but not Lui's strengths</td><td>18 (24%)</td><td>8 (22%)</td><td>10 (27%)</td></tr><tr><td>Mentions both Lui's strengths and difficulties</td><td>24 (32%)</td><td>14 (38%)</td><td>10 (27%)</td></tr><tr><td>Mentions neither Lui's strengths nor difficulties</td><td>27 (36%)</td><td>13 (50%)</td><td>14 (38%)</td></tr><tr><td>Incorrect response</td><td>35 (47%)</td><td>20 (54%)</td><td>15 (41%)</td></tr><tr><td>Incorrect response and <italic>does not</italic> identify either a strength or difficulty</td><td>23 (31%)</td><td>11 (30%)</td><td>12 (32%)</td></tr><tr><td>Incorrect response and <italic>does</italic> identify either a strength or difficulty</td><td>12 (16%)</td><td>9 (24%)</td><td>3 (8%)</td></tr></tbody></table> </ephtml> </p> <p>Thirty-five participants gave incorrect responses, for example, "He did not understand the game," "She's guessing, performing trial and error." Of the 35 participants whose responses were incorrect, 33 participants did not identify the child's strength and area of need, whereas two participants did identify both. This represented a 94% probability of demonstrating an incorrect response when participants could not accurately assess the child's math performance in the game. Conversely, participants had a 6% probability of providing an erroneous response when they could successfully evaluate the child's math performance in the game.</p> <hd id="AN0182209680-23">Question Three</hd> <p>On question three, 26 participants (35%) identified the child's difficulty in a specific manner, 29 (39%) identified the child's difficulty but not adequately specific, and 18 (24%) did not correctly name the difficulty at all.</p> <p>Twenty-eight participants (38%) included an incorrect response to question three, for example, "[the child] has no numerical concept of numbers," "his difficulty is counting backwards." Of the 28 participants, 23 participants did not correctly name the child's difficulty, while five accurately described the child's difficulty. This represented an 82% probability of demonstrating a misconception when participants could not accurately explain how the child struggled in the game. In contrast, participants had an 18% probability of providing an erroneous response when they could successfully explain how the child struggled in the game.</p> <p>Considering questions two and three together, either on a spontaneous level (Question 2) or explicit level (Question 3), only a small proportion of participants (32% [Question 2], 1% [Question 3]) could completely and accurately assess the child's math performance The majority of participants demonstrated only partial knowledge by identifying either the child's strengths or areas of need. Another proportion of participants (36% [Question 2]; 24% [Question 3]) could not give any relevant and specific diagnosis.</p> <hd id="AN0182209680-24">Supporting (KCT): To What Extent Can Preschool Teachers Propose Developmentally Appropriate T...</hd> <p>Question four asked participants to describe how the teacher in the scenario should help remediate the child's difficulties. Table 3 lists the rubric categories of participants' responses and the corresponding data for each category.</p> <p>The first category addresses whether a participant's proposed remediation technique targets the child's primary area of math difficulty. Most responses (<emph>N</emph> = 44, 60%) targeted the child's primary areas of need. The second category addresses the level of detail that the participant provided to describe the proposed remediation technique. Responses were nearly evenly split among three of the rubric levels: mostly detailed (<emph>N</emph> = 23, 31%), somewhat detailed (<emph>N</emph> = 21, 28%), and not detailed (<emph>N</emph> = 25, 34%). Only five participants (7%) provided a very detailed response.</p> <p>The third category addresses whether or not the participant's proposed remediation technique includes the use of hands-on manipulatives and/or visual representations of math processes, which are considered to be a developmentally appropriate strategy in the early childhood mathematics classroom when used appropriately (Osana & Pitsolantis, [<reflink idref="bib22" id="ref68">22</reflink>]). Responses were almost evenly split between the two levels of this rubric category. Thirty-eight participants (51%) included hands-on materials and/or visuals in their suggested remediation technique. In comparison, 36 participants (49%) did not mention the use of hands-on materials and/or visuals, which is surprisingly high given that hands-on learning is strongly advocated in early childhood education (e.g., New York State Early Childhood Advisory Council and the New York State Council on Children and Families, 2012), and the participants must have been exposed to this tenet in various contexts.</p> <p>The fourth category addresses whether or not the proposed remediation technique benefits the child's number comparison development. The highest percentage of responses (<emph>N</emph> = 38, 51%) were <emph>somewhat likely</emph> to help improve a child's ability to compare numbers. The second-highest percentage of responses (<emph>N</emph> = 24, 32%) were <emph>not likely</emph> to help improve a child's number comparison knowledge. In contrast, only two responses (3%) were scored at the highest level–<emph>very likely</emph> to help improve knowledge of number comparison.</p> <hd id="AN0182209680-25">Discussion</hd> <p>The present study examined preschool teachers' PCK of number comparison, an essential aspect of children's early math development. Although most other studies have explored teachers' PCK more broadly across several math domains within a single instrument (e.g., Gasteiger et al., [<reflink idref="bib11" id="ref69">11</reflink>]; Lee, [<reflink idref="bib17" id="ref70">17</reflink>]; McCray & Chen, [<reflink idref="bib20" id="ref71">20</reflink>]; Torbeyns et al., [<reflink idref="bib32" id="ref72">32</reflink>]), our study focused specifically on number comparison. The goal was to identify preschool teachers' strengths and areas of improvement in the three subdomains of PCK regarding number comparison to inform pre-service teacher education and in-service professional development.</p> <hd id="AN0182209680-26">Areas of Strengths in Preschool Teachers' PCK of Number Comparison</hd> <p>Several areas of strength emerged from the evaluation of preschool teachers' responses to the instrument. First, teachers exhibited a robust ability to identify number comparison elements from the game, indicating a stronger Specialized Content Knowledge compared to Knowledge of Content and Students or Knowledge of Content and Teaching. Specifically, most teachers (89%) accurately identified that the game involved number comparison, and only fewer teachers (7%) could not do so. In contrast, only about one third (32%) of preschool teachers identified the child's strength and areas of improvement, with an even smaller percentage (3%) proposing activities highly likely to aid the child. This aligns with Lee's finding in an observational study ([<reflink idref="bib16" id="ref73">16</reflink>]) and a survey study ([<reflink idref="bib17" id="ref74">17</reflink>]) that preschool teachers are more successful at noticing instances of math than interpreting children's math thinking. This finding, however, is different from Li ([<reflink idref="bib18" id="ref75">18</reflink>]) that in counting and cardinality, preschool teachers are relatively weak in identifying the count-out concept, but better in proposing activities that help with children's cardinality error or one to one correspondence error. In addition, Lee ([<reflink idref="bib17" id="ref76">17</reflink>]) found that teachers noticed examples of classification, number sense, and measurement embedded in a scenario more frequently than examples of geometry and operations, lending support to the suggestion that specialized content knowledge is not a static trait but varies across math contents. Taken altogether, this indicates a variation in teachers' ability across different mathematical concepts. It implies that although teachers showcase strength in identifying number comparison elements, it does not necessarily translate to other mathematical contents or other PCK subdomains within the same area.</p> <p>Another area of strength concerns an aspect of supporting KCT. Specifically, about 60% preschool teachers successfully suggested a remediation activity that targeted the child's primary areas of need on number comparison. This demonstrates teachers' abilities to base their instructional decisions on identifying and analyzing the math situation.</p> <hd id="AN0182209680-27">Areas of Improvement</hd> <p>Conversely, the data also revealed several areas needing improvement, especially on the KCS and KCT questions. Specifically, on question two (KCS), a substantial number of teachers (36%; Table 5) were unable to name neither the child's strengths nor math difficulties, which aligns with previous findings that pre-service preschool teachers were only somewhat successful at diagnosing children's errors related to quantity (Sahin and Korkmaz [<reflink idref="bib25" id="ref77">25</reflink>]) and geometry (Korkmaz & Şahin, [<reflink idref="bib13" id="ref78">13</reflink>]). Li ([<reflink idref="bib18" id="ref79">18</reflink>]) also found that when the child's error (e.g., cardinality error) is not salient, teachers have difficulties detecting it. Therefore, evaluating children's performance on number comparison tasks and identifying their difficulties appears to be an area of improvement for preschool teachers.</p> <p>On question four (KCT), many preschool teachers appeared to have difficulty in proposing an appropriate activity for remediating the child's difficulties even though most knew what to target. Specifically, in the present study, many teachers (38%; Table 3) provided vague responses that lacked specific details. Vague answers included, for example, "He needs more exposure to numbers," "Use visuals to help Lui understand the concept of more and less," and "Practice and explain more than and less than concepts." It was unclear from these responses how the teacher would explain concepts such as <emph>more than</emph> or <emph>less than</emph> to the child. This finding is consistent with the previous finding that preschool teachers produced insufficient solutions for addressing children's math mistakes (Korkmaz & Şahin, [<reflink idref="bib13" id="ref80">13</reflink>]; Sahin & Korkmaz, [<reflink idref="bib25" id="ref81">25</reflink>]) or planned activities unlikely to promote an understanding of early math concepts (Dunekacke et al., [<reflink idref="bib8" id="ref82">8</reflink>]; Figueiredo et al., [<reflink idref="bib10" id="ref83">10</reflink>]). In addition, about half of the responses (49%; Table 3) did not reference the use of tangible manipulatives or visuals. Previous studies have also uncovered a preference for "direct instruction" methods that favor explicitly demonstrating the skill to preschoolers rather than teaching through discovery, games, storytelling, or the use of hands-on materials (Figueiredo et al., [<reflink idref="bib10" id="ref84">10</reflink>]; Korkmaz & Şahin, [<reflink idref="bib13" id="ref85">13</reflink>]).</p> <p>Lastly, although teachers demonstrated strength in identifying the presence of number comparison concepts (SCK), few identified the secondary skill of ordering numbers that was evidenced in the game. Therefore, preschool teachers may notice the more salient math concepts in a classroom scenario (e.g., more than and less than), but may not be as successful at detecting the less obvious but no less important math concepts that are inherent in the game (e.g., ordinal relations of numbers).</p> <hd id="AN0182209680-28">Comparing Pre-Service and In-Service Teachers' Performance</hd> <p>Interestingly, no statistically significant differences were found between pre-service and in-service teachers across any of the three subdomains of PCK. But note that although the aggregate PCK scores were not statistically different in our study, in-service teachers did outperform pre-service teachers on individual items (e.g., level of details, hands-on or visual), which indicates that in-service teachers gained some useful teaching knowledge due to their teaching experience, even if their overall PCK is not significantly higher than that of pre-service teachers. Related to but different from our findings, years of experience have previously been associated with greater levels of PCK among pre-service preschool teachers (Sahin & Korkmaz, [<reflink idref="bib25" id="ref86">25</reflink>]; Torbeyns et al., [<reflink idref="bib31" id="ref87">31</reflink>]) and in-service preschool teachers (Lee, [<reflink idref="bib17" id="ref88">17</reflink>]). These divergent findings might be attributed to the nature of the in-service participants in this present study (see details in limitation). Nevertheless, the findings indicate that although teaching practice (indicating the difference between pre- and in-service teachers) may increase teachers' ability to plan appropriate math learning opportunities, it may not be adequately enough to systematically improve all the subdomains of PCK, necessitating targeted professional training.</p> <hd id="AN0182209680-29">Limitations and Future Directions</hd> <p>A few limitations apply to the present study. One limitation is the small size, which may limit generalizability of the results as well as statistical power of the study (e.g., pre- vs. in-service contrast). In addition, the data were obtained from a single university and regional area and thus might not represent the knowledge of all preschool teachers in the United States. Moreover, the in-service sample includes a high percentage of assistant teachers or teachers of 3-year-olds or younger, and therefore may underestimate in-service preschool teachers' overall PCK knowledge. Furthermore, information about the participants' years of teaching experience is not provided. A larger, more representative sample may provide a more complete picture of teachers' ability to notice, interpret, and enhance children's understanding of number comparison in the preschool classroom and the interplay between those three subdomains of PCK. In addition, to gain a deeper understanding of preschool teachers' PCK regarding particular concepts, interviews could be incorporated into the assessment.</p> <p>Another limitation is the exclusion of additional variables, including affective-motivational characteristics such as math beliefs, math anxiety, and self-efficacy (Blömeke et al., [<reflink idref="bib2" id="ref89">2</reflink>]; Dunekacke et al., [<reflink idref="bib9" id="ref90">9</reflink>]; Oppermann et al., [<reflink idref="bib21" id="ref91">21</reflink>]) and math content knowledge (Capraro et al., [<reflink idref="bib4" id="ref92">4</reflink>]; Dunekacke et al., [<reflink idref="bib9" id="ref93">9</reflink>]; Oppermann et al., [<reflink idref="bib21" id="ref94">21</reflink>]) that are potentially related to math PCK.</p> <hd id="AN0182209680-30">Conclusion and Implications</hd> <p>Despite the limitations, this study serves as the first attempt to examine preschool teachers' PCK in the specific numeracy area of number comparison, which uncovers informative findings and generates important implications for teaching number comparison to young children.</p> <p>First, teachers in the study were better at identifying and describing the number comparison concepts in the scenario than analyzing a child's math performance or proposing developmentally appropriate tasks. Therefore, teacher education and professional development should focus on helping teachers understand children's learning trajectories on number comparison and notice a child's strengths as well as difficulties when comparing numbers. In addition, teacher education and professional development should include instructions on using concept specific developmentally appropriate strategies. Many preschool teachers in our study did not include the use of manipulatives or visual representations in their proposed remediation strategies. Therefore, preschool teachers need support to plan instructions that include manipulatives and need knowledge about which manipulatives are most likely to improve learning, as research has found that not all manipulatives are equally effective for promoting early math knowledge across all situations (Laski et al., [<reflink idref="bib15" id="ref95">15</reflink>]).</p> <p>Second, teaching experience cannot replace professional training. This study shows that although in-service teachers performed better than pre-service teachers in a few categories related to planning effective activities, the overall differences were insignificant. This finding suggests that teaching experience cannot replace professional training, which is consistent with Platas ([<reflink idref="bib24" id="ref96">24</reflink>]) finding that eight years of experience equate with the knowledge gained from completing a single math development course. Moreover, based on the present finding, in-service teachers might be more skilled than pre-service teachers at enhancing children's learning but both groups are equally inadequate at analyzing children's performance.</p> <p>Third, the present study also shows that increasing teachers' overall understanding of specialized content knowledge of number comparison will also decrease incorrect knowledge and misunderstandings on the corresponding SCK. Therefore, teacher education and preparation should capitalize on teachers' strengths in SCK and help them translate and expand this knowledge to the other subdomains of PCK, including evaluating children's performance (KCS) or designing appropriate tasks (KCT). Although SCK may not automatically transfer to KCS or KCT (Lee, [<reflink idref="bib16" id="ref97">16</reflink>], [<reflink idref="bib17" id="ref98">17</reflink>]), a strong SCK is instrumental because teachers first need to know what to focus on and then analyze a child's performance on the particular concept and design corresponding tasks.</p> <p>In summary, additional research exploring teachers' knowledge in specific math content areas will help teacher education and professional development build on areas of knowledge and support gaps in understanding. Consequently, teachers will be strongly equipped to provide rich opportunities for math learning, which will translate into learning gains for young children.</p> <hd id="AN0182209680-31">Disclosure Statement</hd> <p>No potential conflict of interest was reported by the author(s).</p> <ref id="AN0182209680-32"> <title> References </title> <blist> <bibl id="bib1" idref="ref14" type="bt">1</bibl> <bibtext> Ball, D., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59 (5), 389 – 407. https://doi.org/10.1177/0022487108324554</bibtext> </blist> <blist> <bibl id="bib2" idref="ref89" type="bt">2</bibl> <bibtext> Blömeke, S., Dunekacke, S., & Jenßen, L. (2017). Cognitive, educational and psychological determinants of prospective preschool teachers' beliefs. European Early Childhood Education Research Journal, 25 (6), 885 – 903. https://doi.org/10.1080/1350293X.2017.1380885</bibtext> </blist> <blist> <bibl id="bib3" idref="ref39" type="bt">3</bibl> <bibtext> Brownell, J. O. (2014). Big ideas of early mathematics: What teachers of young children need to know. Pearson.</bibtext> </blist> <blist> <bibl id="bib4" idref="ref92" type="bt">4</bibl> <bibtext> Capraro, R. M., Capraro, M. M., Parker, D., Kulm, G., & Raulerson, T. (2005). The mathematics content knowledge role in developing preservice teachers' pedagogical content knowledge. Journal of Research in Childhood Education, 20 (2), 102 – 118. https://doi.org/10.1080/02568540509594555</bibtext> </blist> <blist> <bibl id="bib5" idref="ref67" type="bt">5</bibl> <bibtext> Cohen, J. (1960). A coefficient of agreement for nominal scales. Educational and Psychological Measurement, 20 (1), 37 – 46. https://doi.org/10.1177/001316446002000104</bibtext> </blist> <blist> <bibl id="bib6" idref="ref40" type="bt">6</bibl> <bibtext> Council of Chief State School Officers. (2010). Common core state standards for mathematics. National Governors Association Center for Best Practices.</bibtext> </blist> <blist> <bibl id="bib7" idref="ref6" type="bt">7</bibl> <bibtext> Dowker, A. (2005). Early identification and intervention for students with mathematics difficulties. Journal of Learning Disabilities, 38 (4), 324 – 332. https://doi.org/10.1177/00222194050380040801</bibtext> </blist> <blist> <bibl id="bib8" idref="ref28" type="bt">8</bibl> <bibtext> Dunekacke, S., Jenßen, L., & Blömeke, S. (2015). Effects of mathematics content knowledge on pre-school teachers' performance: A video-based assessment of perception and planning abilities in informal learning situations. International Journal of Science and Mathematics Education, 13 (2), 267 – 286. https://doi.org/10.1007/s10763-014-9596-z</bibtext> </blist> <blist> <bibl id="bib9" idref="ref20" type="bt">9</bibl> <bibtext> Dunekacke, S., Jenßen, L., Eilerts, K., & Blömeke, S. (2016). Epistemological beliefs of prospective preschool teachers and their relation to knowledge, perception, and planning abilities in the field of mathematics: A process model. ZDM - Mathematics Education, 48 (1–2), 125 – 137. https://doi.org/10.1007/s11858-015-0711-6</bibtext> </blist> <blist> <bibtext> Figueiredo, M. P., Gomes, H., & Rodrigues, C. (2018). Mathematical pedagogical content knowledge in early childhood education: Tales from the 'great unknown' in teacher education in Portugal. European Early Childhood Education Research Journal, 26 (4), 535 – 546. https://doi.org/10.1080/1350293X.2018.1487164</bibtext> </blist> <blist> <bibtext> Gasteiger, H., Bruns, J., Benz, C., Brunner, E., & Sprenger, P. (2020). Mathematical pedagogical content knowledge of early childhood teachers: A standardized situation-related measurement approach. ZDM - Mathematics Education, 52 (2), 193 – 205. https://doi.org/10.1007/s11858-019-01103-2</bibtext> </blist> <blist> <bibtext> Jordan, N. C., Kaplan, D., Locuniak, M. N., & Ramineni, C. (2007). Predicting first-grade math achievement from developmental number sense trajectories. Learning Disabilities Research and Practice, 22 (1), 36 – 46. https://doi.org/10.1111/j.1540-5826.2007.00229.x</bibtext> </blist> <blist> <bibtext> Korkmaz, H. I., & Şahin, Ö. (2020). Preservice preschool teachers' pedagogical content knowledge on geometric shapes in terms of children's mistakes. Journal of Research in Childhood Education, 34 (3), 385 – 405. https://doi.org/10.1080/02568543.2019.1701150</bibtext> </blist> <blist> <bibtext> Krasa, N., Tzanetopoulos, K., & Maas, C. (2022). How children learn math: The science of math learning in research and practice. Routledge.</bibtext> </blist> <blist> <bibtext> Laski, E. V., Jor'dan, J. R., Daoust, C., & Murray, A. K. (2015). What makes mathematics manipulatives effective? Lessons from cognitive science and Montessori education. SAGE Open, 5 (2), 215824401558958. https://doi.org/10.1177/2158244015589588</bibtext> </blist> <blist> <bibtext> Lee, J. E. (2014). A study of pre-kindergarten teachers' knowledge about children's mathematical thinking. Australasian Journal of Early Childhood, 39 (4), 29 – 36. https://doi.org/10.1177/183693911403900405</bibtext> </blist> <blist> <bibtext> Lee, J. E. (2017). Preschool teachers' pedagogical content knowledge in mathematics. International Journal of Early Childhood, 49 (2), 229 – 243. https://doi.org/10.1007/s13158-017-0189-1</bibtext> </blist> <blist> <bibtext> Li, X. (2020). Investigating U.S. preschool teachers' math teaching knowledge in counting and numbers. Early Education & Development, 32 (4), 589 – 607. https://doi.org/10.1080/10409289.2020.1785226</bibtext> </blist> <blist> <bibtext> Li, X., Chi, L. P., DeBey, M., & Baroody, A. J. (2015). A study of early childhood math teaching in U.S. and China. Early Education & Development, 0, 1 – 29. https://doi.org/10.1080/10409289.2015.994464</bibtext> </blist> <blist> <bibtext> McCray, J. S., & Chen, J. Q. (2012). Pedagogical content knowledge for preschool mathematics: Construct validity of a new teacher interview. Journal of Research in Childhood Education, 26 (3), 291 – 307. https://doi.org/10.1080/02568543.2012.685123</bibtext> </blist> <blist> <bibtext> Oppermann, E., Anders, Y., & Hachfeld, A. (2016). The influence of preschool teachers' content knowledge and mathematical ability beliefs on their sensitivity to mathematics in children's play. Teaching & Teacher Education, 58, 174 – 184. https://doi.org/10.1016/j.tate.2016.05.004</bibtext> </blist> <blist> <bibtext> Osana, H. P., & Pitsolantis, N. (2019). Supporting meaningful use of manipulatives in kindergarten: The role of dual representation in early mathematics. In K. M. Robinson, H. P. Osana, & D. Kotsopoulos (Eds.), Mathematical learning and cognition in early childhood (pp. 91 – 113). Springer International Publishing. https://doi.org/10.1007/978-3-030-12895-1_7</bibtext> </blist> <blist> <bibtext> Piaget, J. (1965). The child's conception of number. Norton.</bibtext> </blist> <blist> <bibtext> Platas, L. M. (2015). The Mathematical Development Beliefs Survey: Validity and reliability of a measure of preschool teachers' beliefs about the learning and teaching of early mathematics. Journal of early childhood research, 13 (3), 295 – 310.</bibtext> </blist> <blist> <bibtext> Sahin, Ö., & Korkmaz, H. I. (2019). Pre-service preschool teachers' pedagogical content knowledge on quantity concepts in terms of children's mistakes. Educational Research Quarterly, 43 (2), 55 – 94.</bibtext> </blist> <blist> <bibtext> Sarama, J., & Clements, D. (2009). Early childhood mathematics education research: Learning trajectories for young children. Routledge.</bibtext> </blist> <blist> <bibtext> Seo, K. H., & Ginsburg, H. P. (2004). What is developmentally appropriate in early childhood mathematics education? Lessons from new research. In D. H. Clements, J. Sarama, & A. M. DiBiase (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics education (pp. 91 – 104). Erlbaum.</bibtext> </blist> <blist> <bibtext> Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15 (2), 4 – 14. https://doi.org/10.2307/1175860</bibtext> </blist> <blist> <bibtext> Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57 (1), 1 – 22. https://doi.org/10.17763/haer.57.1.j463w79r56455411</bibtext> </blist> <blist> <bibtext> Siegler, R. S., & Lortie-Forgues, H. (2014). An integrative theory of numerical development. Child Development Perspectives, 8 (3), 144 – 150. https://doi.org/10.1111/cdep.12077</bibtext> </blist> <blist> <bibtext> Torbeyns, J., Verbruggen, S., & Depaepe, F. (2020). Pedagogical content knowledge in preservice preschool teachers and its association with opportunities to learn during teacher training. ZDM - Mathematics Education, 52 (2), 269 – 280. https://doi.org/10.1007/s11858-019-01088-y</bibtext> </blist> <blist> <bibtext> Torbeyns, J., Verbruggen, S., & Depaepe, F. (2020). Pedagogical content knowledge in preservice preschool teachers and its association with opportunities to learn during teacher training. ZDM Mathematics Education, 52, 269 – 280. https://doi.org/10.1007/s11858-019-01088-y</bibtext> </blist> <blist> <bibtext> Vanbinst, K., & De Smedt, B. (2016). Individual differences in children's mathematics achievement: The roles of symbolic numerical magnitude processing and domain-general cognitive functions. Progress in Brain Research, 227, 105 – 130. https://doi.org/10.1016/bs.pbr.2016.04.001</bibtext> </blist> <blist> <bibtext> Vanbinst, K., Ghesquière, P., & Bert De Smedt, B. (2015). Does numerical processing uniquely predict first graders' future development of single-digit arithmetic? Learning & Individual Differences, 37, 153 – 160. https://doi.org/10.1016/j.lindif.2014.12.004</bibtext> </blist> <blist> <bibtext> Vanbinst, K., Ghesquièrea, P., & De Smedt, B. (2019). Is the long-term association between symbolic numerical magnitude processing and arithmetic bi-directional? Journal of Numerical Cognition, 5 (3), 358 – 370. https://doi.org/10.5964/jnc.v5i3.202</bibtext> </blist> <blist> <bibtext> Van de Walle, J. A., Lovin, L. H., Karp, K. H., & Bay Williams, J. M. (2018). Teaching student-centered mathematics: Developmentally appropriate instruction for grades pre- K-2 (3rd ed., Vol. I). Pearson Education, Inc.</bibtext> </blist> <blist> <bibtext> Xenidou-Dervou, I., Molenaar, D., Ansari, D., van der Schoot, M., & van Lieshout, E. C. D. M. (2017). Nonsymbolic and symbolic magnitude comparison skills as longitudinal predictors of mathematical achievement. Learning & Instruction, 50, 1 – 13. https://doi.org/10.1016/j.learninstruc.2016.11.001</bibtext> </blist> </ref> <aug> <p>By Xia Li; Colleen Maas and Colleen Oppenzato</p> <p>Reported by Author; Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib12" firstref="ref1"></nolink> <nolink nlid="nl2" bibid="bib36" firstref="ref2"></nolink> <nolink nlid="nl3" bibid="bib33" firstref="ref3"></nolink> <nolink nlid="nl4" bibid="bib37" firstref="ref4"></nolink> <nolink nlid="nl5" bibid="bib34" firstref="ref5"></nolink> <nolink nlid="nl6" bibid="bib30" firstref="ref7"></nolink> <nolink nlid="nl7" bibid="bib35" firstref="ref8"></nolink> <nolink nlid="nl8" bibid="bib19" firstref="ref9"></nolink> <nolink nlid="nl9" bibid="bib27" firstref="ref10"></nolink> <nolink nlid="nl10" bibid="bib28" firstref="ref11"></nolink> <nolink nlid="nl11" bibid="bib29" firstref="ref16"></nolink> <nolink nlid="nl12" bibid="bib17" firstref="ref18"></nolink> <nolink nlid="nl13" bibid="bib24" firstref="ref24"></nolink> <nolink nlid="nl14" bibid="bib20" firstref="ref25"></nolink> <nolink nlid="nl15" bibid="bib11" firstref="ref26"></nolink> <nolink nlid="nl16" bibid="bib18" firstref="ref27"></nolink> <nolink nlid="nl17" bibid="bib26" firstref="ref50"></nolink> <nolink nlid="nl18" bibid="bib23" firstref="ref56"></nolink> <nolink nlid="nl19" bibid="bib14" firstref="ref59"></nolink> <nolink nlid="nl20" bibid="bib13" firstref="ref62"></nolink> <nolink nlid="nl21" bibid="bib25" firstref="ref63"></nolink> <nolink nlid="nl22" bibid="bib22" firstref="ref68"></nolink> <nolink nlid="nl23" bibid="bib32" firstref="ref72"></nolink> <nolink nlid="nl24" bibid="bib16" firstref="ref73"></nolink> <nolink nlid="nl25" bibid="bib10" firstref="ref83"></nolink> <nolink nlid="nl26" bibid="bib31" firstref="ref87"></nolink> <nolink nlid="nl27" bibid="bib21" firstref="ref91"></nolink> <nolink nlid="nl28" bibid="bib15" firstref="ref95"></nolink>
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  Data: Investigating Preschool Teachers' Pedagogical Content Knowledge of Number Comparison
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  Data: <searchLink fieldCode="AR" term="%22Xia+Li%22">Xia Li</searchLink><br /><searchLink fieldCode="AR" term="%22Colleen+Maas%22">Colleen Maas</searchLink><br /><searchLink fieldCode="AR" term="%22Colleen+Oppenzato%22">Colleen Oppenzato</searchLink>
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  Data: <searchLink fieldCode="SO" term="%22Early+Education+and+Development%22"><i>Early Education and Development</i></searchLink>. 2025 36(2):249-264.
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  Data: Routledge. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
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  Data: 16
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  Data: 2025
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  Data: Journal Articles<br />Reports - Research
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  Label: Education Level
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  Data: <searchLink fieldCode="EL" term="%22Early+Childhood+Education%22">Early Childhood Education</searchLink><br /><searchLink fieldCode="EL" term="%22Preschool+Education%22">Preschool Education</searchLink><br /><searchLink fieldCode="EL" term="%22Higher+Education%22">Higher Education</searchLink><br /><searchLink fieldCode="EL" term="%22Postsecondary+Education%22">Postsecondary Education</searchLink>
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  Data: <searchLink fieldCode="DE" term="%22Preschool+Teachers%22">Preschool Teachers</searchLink><br /><searchLink fieldCode="DE" term="%22Pedagogical+Content+Knowledge%22">Pedagogical Content Knowledge</searchLink><br /><searchLink fieldCode="DE" term="%22Number+Concepts%22">Number Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Education%22">Mathematics Education</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Teachers%22">Mathematics Teachers</searchLink><br /><searchLink fieldCode="DE" term="%22Preservice+Teachers%22">Preservice Teachers</searchLink><br /><searchLink fieldCode="DE" term="%22Knowledge+Level%22">Knowledge Level</searchLink><br /><searchLink fieldCode="DE" term="%22Student+Needs%22">Student Needs</searchLink><br /><searchLink fieldCode="DE" term="%22Teacher+Effectiveness%22">Teacher Effectiveness</searchLink><br /><searchLink fieldCode="DE" term="%22Ability+Identification%22">Ability Identification</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Achievement%22">Mathematics Achievement</searchLink><br /><searchLink fieldCode="DE" term="%22Numeracy%22">Numeracy</searchLink><br /><searchLink fieldCode="DE" term="%22Metropolitan+Areas%22">Metropolitan Areas</searchLink><br /><searchLink fieldCode="DE" term="%22Teacher+Education+Programs%22">Teacher Education Programs</searchLink><br /><searchLink fieldCode="DE" term="%22Early+Childhood+Education%22">Early Childhood Education</searchLink><br /><searchLink fieldCode="DE" term="%22Undergraduate+Students%22">Undergraduate Students</searchLink><br /><searchLink fieldCode="DE" term="%22Graduate+Students%22">Graduate Students</searchLink>
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  Data: 10.1080/10409289.2024.2389362
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  Data: 1040-9289<br />1556-6935
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  Data: Research Findings: The aim of this study was to investigate U.S. preschool teachers' math pedagogical content knowledge in number comparison. Seventy-four in- and pre-service teachers completed a set of scenario-based questions from a researcher-designed questionnaire that is composed of learning scenarios. Through a mixed-method approach, the study revealed the following findings: a considerable portion of participants (89%) successfully identified the concept of number comparison embedded in a learning scenario, showcasing their specialized content knowledge. However, when it came to knowledge of content and students, only 24 participants (32%) were able to pinpoint both a child's mathematical strength and area of need. In terms of knowledge of content and teaching, more than half of the participants (51%) recommended the utilization of hands-on materials and/or visual aids as part of their remediation techniques, while 49% did not reference these educational tools. Notably, no significant differences were observed between the pedagogical content knowledge of pre-and in-service teachers concerning number comparison. Practice or Policy: The findings indicate that teacher education and professional development should focus on helping teachers understand children's learning trajectories on number comparison and include instructions on using concept specific developmentally appropriate strategies, and indicate that teaching experience cannot replace professional training.
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  Data: 2025
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        Value: 10.1080/10409289.2024.2389362
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      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 16
        StartPage: 249
    Subjects:
      – SubjectFull: Preschool Teachers
        Type: general
      – SubjectFull: Pedagogical Content Knowledge
        Type: general
      – SubjectFull: Number Concepts
        Type: general
      – SubjectFull: Mathematics Education
        Type: general
      – SubjectFull: Mathematics Teachers
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      – SubjectFull: Preservice Teachers
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      – SubjectFull: Mathematics Achievement
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      – SubjectFull: Numeracy
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      – SubjectFull: Metropolitan Areas
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      – SubjectFull: Teacher Education Programs
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      – TitleFull: Investigating Preschool Teachers' Pedagogical Content Knowledge of Number Comparison
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